Mathematics Anxiety What Is Known, and What is Still Missing [1 ed.] 9780367190330, 9780367190392, 9780429199981

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MATHEMATICS ANXIETY

Feelings of apprehension and fear brought on by mathematical performance can affect correct mathematical application and can inf luence the achievement and future paths of individuals affected by it. In recent years, math anxiety has become a subject of increasing interest in both educational and clinical settings. This ground-breaking collection presents theoretical, educational and psychophysiological perspectives on the widespread phenomenon of mathematics anxiety. Featuring contributions from leading international researchers, Mathematics Anxiety challenges preconceptions and clarifies several crucial areas of research, such as the distinction between math anxiety from other forms of anxiety (i.e., general or test anxiety); the ways in which math anxiety has been assessed (e.g. throughout self-report questionnaires or psychophysiological measures); the need to clarify the direction of the relationship between math anxiety and mathematics achievement (which causes which). Offering a re-evaluation of the negative connotations usually associated with math anxiety and prompting avenues for future research, this book will be invaluable to academics and students in the psychological and educational sciences, as well as teachers working with students who are struggling with math anxiety. Irene C. Mammarella is Associate Professor at the University of Padua, Italy. Her research interests include the role of working memory and emotional aspects in specific learning disorders, and other neurodevelopmental disorders. She is the cofounder of a clinical university centre for neurodevelopmental disorders (LabDA srl) and coauthored the book Nonverbal learning disabilities (Guilford Press, 2016). Sara Caviola is a Lecturer in Developmental Psychology, at the School of Psychology, University of Leeds, UK. She won a Marie Skłodowska-Curie fellowship and spent two years at the Centre for Neuroscience in Education, University of Cambridge. Her main interests include analyses of cognitive and emotional underpinnings of mathematical cognition, in both children and adult populations. Ann Dowker is University Research Lecturer at the Department of Experimental Psychology, Oxford University, UK. She has edited and coedited several books, and is the author of Individual Differences in Arithmetic: Implications for Psychology, Neuroscience and Education (Psychology Press, 2005; second edition to be published in 2019). She is the lead researcher on the Catch Up Numeracy intervention project.

MATHEMATICS ANXIETY What is Known and What is Still to be Understood

Edited by Irene C. Mammarella, Sara Caviola and Ann Dowker

First published 2019 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 52 Vanderbilt Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2019 selection and editorial matter, Irene C. Mammarella, Sara Caviola and Ann Dowker; individual chapters, the contributors The right of Irene C. Mammarella, Sara Caviola and Ann Dowker to be identified as the authors of the editorial material, and of the authors for their individual chapters, has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book has been requested ISBN: 978-0-367-19033-0 (hbk) ISBN: 978-0-367-19039-2 (pbk) ISBN: 978-0-429-19998-1 (ebk) Typeset in Bembo by Apex CoVantage, LLC

CONTENTS

Preface 1 Models of math anxiety Mark H. Ashcraft

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2 Different ways to measure math anxiety Krzysztof Cipora, Christina Artemenko and Hans-Christoph Nuerk

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3 Psychophysiological correlates of mathematics anxiety Chiara Avancini and Dénes Szűcs

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4 Mathematics anxiety and performance Ann Dowker

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5 Acquisition, development and maintenance of maths anxiety in young children Dominic Petronzi, Paul Staples, David Sheffield and Thomas Hunt

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6 Mathematics anxiety and working memory: what is the relationship? Maria Chiara Passolunghi, Marija Živković and Sandra Pellizzoni

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7 The different involvement of working memory in math and test anxiety Ee Lynn Ng and Kerry Lee

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Contents

8 Math anxiety in children with and without mathematical difficulties: the role of gender and genetic factors Sara Caviola, Irene C. Mammarella and Yulia Kovas

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9 Probing the nature of deficits in math anxiety: drawing connections between attention and numerical cognition Orly Rubinsten, Hili Eidlin Levy and Lital Daches Cohen

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10 Gender stereotypes, anxiety, and math outcomes in adults and children Carlo Tomasetto

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11 The role of parents’ and teachers’ math anxiety in children’s math learning and attitudes Julianne B. Herts, Sian L. Beilock and Susan C. Levine

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Concluding remarks Irene C. Mammarella, Sara Caviola and Ann Dowker Index

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PREFACE

During a cold December afternoon, we were discussing a research project related to mathematical learning and related emotional difficulties, drinking cups of tea and coffee, when we realized the absence of a handbook entirely dedicated to this topic. Almost simultaneously we looked at each other (Irene and Sara) and had the same insight: why not try to edit a book on this topic? Once we realized that we had decided to follow through on the idea, we found ourselves lost (deep) in conversation, trying to list, organize and select all the fundamental topics we thought were worth including in the book. Thus, we wrote to Ann, who possesses tremendous expertise in this field, and asked her to join us. Happily, she immediately embraced the project. The final structure of this book followed on from a symposium that we organized for the first international conference on Mathematical Cognition and Learning Society held in Oxford (Sara and Ann) in April 2018, and includes chapters from leading researchers in psychology, neuroscience and education from all over the world. There is a shared understanding that learning mathematics involves a complex interplay of cognitive, motivational and emotional processes (Carey, Hill, Devine, & Szücs, 2016; Dowker, Sarkar, & Looi, 2016; Hill et al., 2016; Mammarella, Hill, Devine, Caviola, & Szűcs, 2015). Indeed, mathematical difficulties may be associated with not only specific mathematical learning disorders but also domain-general cognitive weaknesses (e.g. phonological memory, working memory, executive functions) and negative emotions (Maloney & Beilock, 2012; Vukovic, Kieffer, Bailey, & Harari, 2013). Interest in the interference of these negative affective factors, usually defined in the literature by the term ‘mathematics anxiety’, has grown from the students’ mathematics outcomes observations: these feelings of apprehension and fear aroused during a mathematical performance can hamper or even impede its correct execution (Ashcraft & Kirk, 2001; Dowker et al., 2016).

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For the domain of mathematics as a whole, long-standing quantitative research concerning the relationship between students’ math anxiety and their general mathematical achievement has been carried out, and the literature has mostly revealed a substantial negative relationship between the two (Ashcraft, Krause, & Hopko, 2007; Ashcraft & Moore, 2009; Carey et al., 2017; Hembree, 1990; Ma, 1999). In his meta-analysis, Hembree (1990) pointed out that cognitive-behavioural interventions developed for the treatment of test anxiety or general anxiety were effective in reducing or eliminating the effects of math anxiety on mathematics performance. Interventions merely focused on changes in classroom curricula, relaxation therapy or group counselling were less effective. Later studies have investigated the effects of more transitory disruptions involving ‘choking’ in response to threat, and have focused on relieving the cognitive symptoms of anxiety, and particularly their impact on working memory resources, with some promising results (Ramirez & Beilock, 2011; Supekar, Iuculano, Chen, & Menon, 2015). Since the overall aim of the book is to gain a greater understanding of math anxiety and, consequently, of ways to prevent or ameliorate the phenomenon, it is important to obtain converging evidence from as many fields as possible. Developmental psychologists, educationists, neuroscientists, educational and clinical psychologists, teachers and policymakers often tend to proceed independently, sometimes neglecting relevant findings from the research and practice outside their own disciplines. Just as with mathematical cognition in general, math anxiety research intersects with a wide array of sub-fields, such as cognitive and educational psychology, neuroscience and developmental psychology. In this book, we broadened the perspective, bringing together converging international researchers working on different areas with the aim to shed light on a) the theoretical background of math anxiety, b) the development of this phenomenon in both typical and atypical populations, c) the main cognitive processes involved and d) the importance and role of different social contexts. The result is a collection of eleven essays and constitutes a comprehensive survey of state-of-the-art studies on important facets of math anxiety. The book begins with two chapters about the theoretical backgrounds and the psychophysiological consequences of math anxiety. The first chapter, by Mark H. Ashcraft, provides a comprehensive summary of four major approaches to understanding the phenomenon and highlights the strengths and weaknesses of each of them. The chapter can be viewed as an overview and introduction to the in-depth chapters that comprise the entire volume. It is followed by Cipora, Artemenko and Nuerk’s chapter, which presents an exhaustive review of math anxiety measurement techniques, and by the chapter of Avancini and Szűcs, who discuss how math anxiety induces physiological reactions within individuals, and how psychophysiological measures may offer new ways to assess this phenomenon without relying on self-report questionnaires. The subsequent group of chapters discuss the development features of math anxiety. In particular, Ann Dowker points out the relationships and differences

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between math anxiety and other related constructs (e.g. general- and test-anxiety, and attitudes to mathematics), and discusses reasons for the well-established negative relationship between math anxiety and maths performance. Dominic Petronzi and colleagues, after discussing issues regarding math anxiety measurement, propose a review of the literature surrounding the onset of math anxiety, focusing on several factors that may inf luence the development of such feelings of apprehension: e.g. negative evaluation from peers and teachers, pessimistic attitudes, low self-efficacy and reduced motivation. Another important aspect involves the cognitive processes, especially working memory processes often associated with math anxiety. Two chapters specifically address this topic, providing a detailed summary of the current research. Passolunghi and colleagues discuss the relationship between math anxiety and mathematical performance by reviewing studies of both children and adult populations. The relationship between math anxiety and mathematics achievement is further analysed by Ng and Lee. In their chapter, they focused on not only math anxiety but also test anxiety, and discuss the overlap between these two constructs. A different insight is offered by the subsequent chapter, where Rubinsten and colleagues, after defining the meaning of attentional bias, report the most recent studies aimed at investigating whether math anxiety is characterized by an attentional bias toward math-related stimuli. Caviola, Mammarella and Kovas give an overview of the literature in not only typical but also atypical populations, and in particular on children with mathematics difficulties or developmental dyscalculia. Their chapter tries to provide a fresh framework of the individual differences (considering both genetic and environmental factors) involved in young children’s low mathematics performance. In the following chapter, Tommasetto presents the most recent research on how gender differences in math anxiety are moderated by gender stereotypes (and self-concept beliefs). Finally, the last chapter, by Herts, Beilock and Levine, provides a detailed description about the social determinants of children’s and adolescents’ math anxiety by examining their parents’ and teachers’ math achievements and attitudes. Parents’ and teachers’ math anxiety is conceptualized as a moderator, determining the strength and direction of the relationships between children’s math anxiety and math education outcomes. Thus, the purpose of this book is to stimulate theoretical ref lection on the ways in which math anxiety can inf luence the achievement and consequently the future paths of individuals. Findings with regard to gender differences, cognitive networks, types of assessment and psychophysiological correlates may help generate a better definition of math anxiety and clarify what is known about it. In this respect, a novel contribution of this book is to bring together different research fields into one single volume. In doing so, we also hope to challenge preconceptions about math anxiety and offer a re-evaluation of the negative connotations usually associated with the term. September 2018

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References Ashcraft, M. H., & Kirk, E. P. (2001). The relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology. General, 130 (2), 224–237. https://doi.org/10.1037/0096-3445.130.2.224 Ashcraft, M. H., Krause, J. A., & Hopko, D. R. (2007). Is math anxiety a mathematical learning disability? In D. B. Berch & M. M. M. Mazzocco (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 329–348). Baltimore, MD: Paul H Brookes Publishing. Ashcraft, M. H., & Moore, A. M. (2009). Mathematics Anxiety and the Affective Drop in Performance. Journal of Psychoeducational Assessment, 27(3), 197–205. https://doi. org/10.1177/0734282908330580 Carey, E., Devine, A., Hill, F., Szűcs, D., Ng, P., & Chan, M. (2017). Differentiating anxiety forms and their role in academic performance from primary to secondary school. PLoS One, 12(3), e0174418. https://doi.org/10.1371/journal.pone.0174418 Carey, E., Hill, F., Devine, A., & Szücs, D. (2016). The chicken or the egg? The direction of the relationship between mathematics anxiety and mathematics performance. Frontiers in Psychology, 6( JAN), 1–6. https://doi.org/10.3389/fpsyg.2015.01987 Dowker, A., Sarkar, A., & Looi, C. Y. (2016). Mathematics Anxiety: What Have We Learned in 60 Years? Frontiers in Psychology, 7, 508. https://doi.org/10.3389/fpsyg.2016. 00508 Hembree, R. (1990 ). The Nature, Effects, and Relief of Mathematics Anxiety. Journal for Research in Mathematics Education, 21(1), 33–46. https://doi.org/10.2307/749455 Hill, F., Mammarella, I. C., Devine, A., Caviola, S., Passolunghi, M. C., & Szűcs, D. (2016). Maths anxiety in primary and secondary school students: Gender differences, developmental changes and anxiety specificity. Learning and Individual Differences, 48, 45–53. https://doi.org/10.1016/j.lindif.2016.02.006 Ma, X. (1999). A Meta -Analysis of the Relationship between Anxiety toward Mathematics and Achievement in Mathematics. Journal for Research in Mathematics Education, 30 (5), 520–540. https://doi.org/10.2307/749772 Maloney, E. A., & Beilock, S. L. (2012). Math anxiety: Who has it, why it develops, and how to guard against it. Trends in Cognitive Sciences, 16(8), 404–406. https://doi.org/ 10.1016/j.tics.2012.06.008 Mammarella, I. C., Hill, F., Devine, A., Caviola, S., & Szűcs, D. (2015). Math anxiety and developmental dyscalculia: A study on working memory processes. Journal of Clinical and Experimental Neuropsychology, 37(8), 878–887. https://doi.org/10.1080/13 803395.2015.1066759 Ramirez, G., & Beilock, S. L. (2011). Writing about testing worries boosts exam performance in the classroom. Science (New York, N.Y.), 331(6014), 211–213. https://doi. org/10.1126/science.1199427 Supekar, K., Iuculano, T., Chen, L., & Menon, V. (2015). Remediation of Childhood Math Anxiety and Associated Neural Circuits through Cognitive Tutoring. The Journal of Neuroscience : The Official Journal of the Society for Neuroscience, 35(36), 12574– 12583. https://doi.org/10.1523/JNEUROSCI.0786-15.2015 Vukovic, R. K., Kieffer, M. J., Bailey, S. P., & Harari, R. R. (2013). Mathematics anxiety in young children: Concurrent and longitudinal associations with mathematical performance. Contemporary Educational Psychology, 38(1), 1–10. https://doi.org/10.1016/j. cedpsych.2012.09.001

1 MODELS OF MATH ANXIETY Mark H. Ashcraft

Mathematics anxiety: “a feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations.” (Richardson & Suinn, 1972, p. 551)

There is good reason to begin this volume with an introductory chapter on models of math anxiety – discussing the several models that have guided investigations of math anxiety almost necessarily involves a general review of the research on math anxiety, or at least the highlights of that research. As such, this chapter can serve as an introduction to the more specific, in-depth chapters that follow. As this introduction will show, the models that have been proposed for understanding math anxiety ref lect researchers’ varying viewpoints about expected consequences of math anxiety as well as factors suspected of inf luencing it, researchers’ own theoretical orientations, and, to a degree, developments in the field that have made new kinds of research possible. Thus, the models show how our thinking about math anxiety has evolved and how different research orientations have enriched our understanding of math anxiety. In the title of their recent review, Dowker, Sarkar, and Looi (2016) asked rhetorically “What have we learned in 60 years?” in our research on math anxiety. The 60 years in question date from the first modern research paper on the topic by Dreger and Aiken (1957) in which those authors tentatively advanced the term “number anxiety” as a label for the emotional reaction to numbers and mathematics. To be sure, there were precursors to this 1957 article; for example, Browne’s (1906) report on performance on the four arithmetic operations made passing reference to emotional reactions to math, and Gough (1954) contributed anecdotal evidence about students’ “mathemaphobia” (along with advice

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for other teachers). But the precursors were largely anecdotal or clinical; indeed, several were psychoanalytic writers who suggested that “failure in arithmetic may be related to maternal overprotection” ( Dreger & Aiken, 1957, p. 344). But the Dreger and Aiken paper was the first clear example of an empirical research approach. In their study, Dreger and Aiken added three math-focused questions to the Taylor Scale of Manifest Anxiety (and dropped three questions from the Taylor scale that had low validity), and also collected scores on an intelligence test, and final grades in a university math course; a subsample (n = 40) of the 704 participants were also given an arithmetic test while Galvanic Skin Response (GSR) def lections were recorded. All measures were inter-correlated, and the three math-focused questions were factor analyzed. The results showed that “number anxiety” appeared to be a separate construct from more general anxiety, that it was unrelated to general intelligence, and that it correlated negatively with math grades. All of these results have been replicated many times since this original report.

Math anxiety as a personality construct The Dreger and Aiken (1957) paper began a tradition of research on math anxiety that treated math anxiety as a personality construct, that is, as a factor or dimension of the individual that needed to be explored in relation to other personality characteristics, factors, traits, or differences. An early effort in the research, not surprisingly, involved assessment. What test or survey was to be used to measure math anxiety? Although several different tests were devised, the Mathematics Anxiety Rating Scale (MARS), by Richardson and Suinn (1972), became the most widespread assessment tool for determining an individual’s level of math anxiety; it, along with its various revisions and versions for younger individuals, was the underlying test used in over half of the studies that appeared in Hembree’s (1990) and Ma’s (1999) inf luential meta-analyses on math anxiety. The original MARS was a 98-item test, with the 98 items describing scenes or situations that might invoke math anxiety (having to reconcile a checkbook, checking a restaurant bill that you think has overcharged you, getting ready to take a math quiz). The test asked for a self-report of how anxious each situation would make the respondent feel, on a Likert scale of 1 (not anxious) to 5 (very anxious). Later versions of the test, including the sMARS (s for shortened), by Alexander and Martray (1989), a 25 item test extracted from the original MARS, and Hopko, Mahadevan, Bare, and Hunt’s (2003) Abbreviated Math Anxiety Scale (AMAS), a 9-item test, are also in wide use. All have good to excellent reliability and show substantial inter-correlations, ranging from .50 to .85 ( Dew, Galassi, & Galassi, 1983; Hopko et al., 2003; see Chapter 2 for a full discussion). Beyond the basic work on assessing math anxiety, considerable effort was devoted across the ensuing two decades to determine whether math anxiety was in fact a separate construct from general anxiety or the more specific construct of test anxiety. The well-known meta-analysis by Hembree (1990) devoted appreciable effort to this question and argued that math anxiety is indeed a separate

Models of math anxiety

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construct, although one that overlaps with test and general anxiety to a degree (for further detail, see Chapter 7 in this volume). Repeatedly, the correlations between math anxiety and test anxiety were reasonably high, but not as high as those between alternate tests of math anxiety – for instance, various math anxiety tests tend to inter-correlate in the range of .50 to .85, whereas the overall math-to-test anxiety correlation (Hembree, 1990) is .52 (see also Dew, Galassi, & Galassi, 1984). The correlation between math anxiety and general anxiety is usually smaller; in Hembree’s (1990) meta-analysis, the value was .35. Of more interest, research summarized in Hembree’s (1990) and Ma’s (1999) meta-analyses covered the relationships investigated since the advent of the MARS (and its successors) and a variety of personality and achievement factors. This work revealed an extensive list of worrisome correlations with math anxiety (see Table 1.1 for a list of factors and correlations). On the educational side, math anxiety correlates negatively with math achievement at both the precollege and college levels, and also negatively with high school and college math grades. Math anxiety also correlates negatively with the extent of math taken in

TABLE 1.1 Selected correlations with math anxiety (MARS) summarized in Hembree’s

(1990) and Ma’s (1999) Correlation between MARS and: Measures of anxiety General anxiety Trait anxiety State anxiety Test anxiety

r .35 .38 .42 .52

Math attitudes Usefulness of math Enjoyment of math (pre-college) Enjoyment of math (college) Math Self-confidence (pre-college) Math Self-confidence (college) Motivation

−.37 −.75 −.47 −.82 −.65 −.64

Avoidance Extent of high school math Intent to enroll (college)

−.31 −.32

Performance measures IQ Verbal aptitude / achievement Math achievement (pre-college) Math achievement (college) High school math grades College math grades (Adapted from Hembree, 1990; Ma, 1999)

−.17 −.06 −.27 −.31 −.30 −.27

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high school (elective coursework), and individuals’ intent to enroll in elective math courses in college; these correlations are routinely interpreted as indicators of avoidance. Note, however, that math anxiety has a fairly low correlation with overall intelligence (−.17), and is uncorrelated with IQ when only verbal aptitude or achievement is considered (−.06). The correlations between math anxiety and attitudes concerning math are more strongly negative, and are also considered as supportive evidence for an overall pattern of avoidance. Math anxiety correlates negatively with enjoyment of math, self-confidence in math, motivation to learn math, and views about the usefulness of math. Hembree’s (1990) paper considered the theoretical models for test anxiety as a guide for initial models of math anxiety and focused on two models in particular, the interference model and the deficits model. According to the interference model, test anxiety was thought to disrupt recall of prior learning. The model also claimed that interference included an individual’s worry during test taking, which would divert attention away from the test itself. The deficits approach, in contrast, claimed that an individual’s lower scores on a test were due to poor study habits and deficient test-taking skills. The individual in a test-taking situation, accordingly, would remember previous poor test performance, and this would cause test anxiety in the present moment. Because his earlier work on test anxiety had supported the interference account, Hembree proposed that math anxiety too might be better approached from the standpoint of the interference model. Interestingly, very little of the research leading up to the time of Hembree’s meta-analysis appeared to advance theoretical proposals or models of math anxiety. Instead, the research focused on two general topics. First, researchers explored other personality characteristics and factors with which math anxiety was associated (for a related perspective on the lack of such theoretical work, see McLeod, 1989). This is the work just discussed, such as studies of the associations between math anxiety and factors like self-confidence in math, enjoyment of math, selfefficacy, and so forth. The other focus during this period was research relating math anxiety to educational outcomes, that is, math achievement. A variety of studies examined the negative association between math anxiety and grades, and between math anxiety and math achievement, with the overall correlations (in Hembree, 1990) found to be −.30 (pre-college) and −.27 (college) for grades, and −.27 (pre-college) and −.31 (college) for math achievement. Relationships of similar magnitude continue to be obtained, although the relationship is now believed to be more nuanced than a simple overall negative relationship (see, e.g., Ramirez, Chang, Maloney, Levine, & Beilock, 2016, and Chapter 4, this volume).

Math anxiety as a cognitive construct Early on, researchers and theorists acknowledged that math anxiety, along with other forms of anxiety, involved a cognitive component (e.g., Dew et al., 1983).

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Early writings routinely noted that test anxiety involved both an affective component, emotionality, and a cognitive component, conscious worry. The theoretical model that brought this thinking into the realm of cognitive psychology was the important processing efficiency theory by Eysenck (1992; Eysenck & Calvo, 1992). According to this theory, worry is an internal process that occupies consciousness during an anxiety reaction. Critically, this preoccupation was predicted to consume the resources of the limited Working Memory system. Thus, Eysenck predicted quite specifically that an anxious individual should show disruption on a cognitive task to the extent that the task relies on working memory resources. Interestingly, just before Hembree’s (1990) important meta-analysis on math anxiety, and likewise just before Eysenck’s (1992) theory, Ashcraft and Faust (1988) presented a conference report on an initial study concerning the cognitive consequences of math anxiety. Their study examined the underlying cognitive processes of doing mental arithmetic by individuals varying in their math anxiety; as noted when the study was subsequently published (Ashcraft & Faust, 1994), it appeared to be the first to pose the question whether math anxiety actually inf luenced the mental processing involved in doing arithmetic. In this exploratory work, Ashcraft and Faust presented simple addition and multiplication problems (e.g., 4 + 3 = 7, 8 × 4 = 38), two-digit addition problems with and without a carry (e.g., 24 + 17 = 43), and a set of complex problems containing all four arithmetic operations (e.g., 18 + 16 = 34, 47 – 18 = 19, 12 × 14 = 168, 156 ÷ 12 = 13), all for true/false judgments. Participants were given the MARS assessment and were divided into four math anxiety groups. For the most part, the simple addition and multiplication problems revealed no math anxiety effects, these problems showing only the standard effects found in regular tests of addition and multiplication, namely, that latencies and errors increased as the problems grew larger. But the two-column addition problems revealed two particularly interesting math anxiety effects. First, the higher anxiety groups were considerably slower to these problems than the lowest anxiety group. Second, the higher anxiety groups seemed particularly slowed down by the presence of a carry problem; that is, when a problem involved a carry, only the low anxiety group demonstrated efficient performance, whereas performance in groups 2, 3, and 4 was particularly disrupted. In a second set of studies, Faust, Ashcraft, and Fleck (1996) replicated and extended the exploratory studies, and found several additional effects of math anxiety on cognitive performance. Three results deserve particular mention. First, in the initial experiment, we again studied performance of simple addition problems in the true/false task. But we expanded the range of values (termed “split”) for the incorrect answers; here, incorrect answers could be wrong by 1, 5, 9, or 23 (e.g., 7 + 5 = 35). Unlike the typical result, which shows performance improving as the split grows larger, we found the higher anxiety groups actually made more errors (and had more extreme scores) when the split grew larger. This suggested to us a deficiency in “number sense,” in that we expected

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an unreasonable answer like 35 for the problem 7 + 5 to be rejected immediately based on a plausibility judgment (see Suarez-Pellicioni, Nunez-Pena, & Colome, 2013, for a replication using event-related potentials methodology). Second, we again found that the carry operation in two-column addition led to slower and more error prone performance for the higher levels of anxiety. And finally, to rule out a simple math competence as a confound in explanations of the math anxiety effects, we tested all of the experimental stimuli in an untimed, paperand-pencil format in a separate study, thus removing the time pressure of the laboratory methods. In this study, there was no relationship between performance and measured level of math anxiety at all, either in correlations or in analysis of variance with the four-level math anxiety groups. Mainstream research in mathematical cognition had firmly established the important role of working memory in procedural aspects of performance, that is, carrying in two-column addition (e.g., Ashcraft & Stazyk, 1981; Widaman, Geary, Cormier, & Little, 1989). Thus, we were confronted with evidence that this working memory-intensive aspect of performance was especially troublesome for those higher in math anxiety. Accordingly, we extended Eysenck’s (1992) processing efficiency theory to the realm of math anxiety and conducted a direct test of the hypothesis that math anxiety causes a disruption in working memory processing while doing math, visible only when the math task relies on the resources of working memory. To perform this test, Ashcraft and Kirk (2001) placed participants in a dual-task setting, asking them to perform addition problems of increasing difficulty: basic facts (single digit operands), medium size problems (a two-digit plus a one-digit operand), and large problems (two two-digit operands). Half of the problems required a carry operation. In a letteronly control condition, either two or six random letters were shown prior to the addition problem, and then had to be recalled after the addition problem was removed from the screen. Because we showed the answer to the addition problem for the participant to read aloud, no actual math processing was required in this condition. But in the dual-task condition, the letters had to be held in working memory while solving the addition problem, and then recalled from memory after giving the answer to the problem. Our prediction was that the larger addition problems, especially those involving the carry operation, would be the most taxing of working memory resources, and thus would interfere the most when six letters had to be held in working memory for recall. This is the classic dual-task interference effect, of course. But our prediction went further. The prediction we derived from Eysenck (1992) was that the dual-task interference effect should be especially pronounced for the high math anxious group, which would essentially be performing in a triple-task situation – difficult carry problems being solved under a heavy working memory load while their working memory was being drained by their anxiety-induced worries. In short, this is exactly what the results demonstrated. Higher working memory loads led to more errors, especially when the addition problem involved carrying, but this of course was only found in the dual-task condition; letter

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recall was nearly perfect in the letter-only control condition. But the critical prediction involving math anxiety was confirmed; the high math anxious group scored almost 40% errors on the working memory task when working memory load was high and the problem involved a carry, compared to 18% for the low anxious group. And these two groups had identical error rates in the six-letter load condition when the problem did not involve carrying. Thus, the critical role of working memory in the procedure of carrying was confirmed, and the crippling of working memory on the part of highly math anxious participants was also demonstrated (see Chapters 6 and 7 for additional information on working memory; for related work showing working memory’s involvement in algebra problem solving, see Trezise & Reeve, 2014). Ramirez, Shaw, and Maloney (2018) have termed this model of math anxiety the disruption account; math anxiety disrupts math performance because it reduces the working memory resources necessary for successful performance. Of course, working memory is the attentional and executive control system that manipulates and maintains the limited amount of information immediately relevant for a task; it keeps important information available while inhibiting attention to irrelevant information (see, e.g., Baddeley, 1986; Engle, 2002; Miyake & Shah, 1999). In this disruption account, it is the worry and ruminations about one’s math anxiety that cause the disruption in working memory, either that working memory’s resources are in some sense consumed by those worries and ruminations, or perhaps equivalently that one fails to inhibit attention to those worries and ruminations. Recall that in Faust et al. (1996), the final study gave participants an untimed, pencil-and-paper test on the math problems and found no differences in performance (accuracy) at the different levels of math anxiety. This is consonant with the implication of the disruption account, that the math anxiety effect is one of a transient disruption of performance due to interference in working memory, rather than a global disruption due to overall lower competence at math. As Ashcraft and colleagues have been careful to note, however, this assumption about competence cannot be made at higher levels of math difficulty, given the wealth of evidence that people higher in math anxiety avoid taking higherlevel (elective) math coursework. There surely are math competence differences between levels of math anxiety when truly high-level math is involved; however, to date, only relatively lower-level math problems have been studied in experimental work, where competence, in terms of educational attainment, seems to be equivalent across the levels of math anxiety (though see Trezise & Reeve, 2014). But when higher levels of math are concerned, given the clear evidence that individuals with higher levels of math anxiety take fewer math courses (e.g., Hembree, 1990), we would expect a lower level of competence as a function of math anxiety. This has been described as an education-based reduced competency account of math anxiety by Ramirez et al. (2018). In an offshoot of this work, several studies have examined what particular aspect of working memory might be disrupted by math anxiety. A likely

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candidate, based on reported studies, is the inhibition function, that is, the aspect of the central executive system within working memory that inhibits one’s attention to task-irrelevant stimuli. In this account, information contained in the math problem along with one’s own memory concerning math are relevant to the math task, while worries and ruminations are clearly irrelevant to the problem solving task. Thus, attention to those worries should be inhibited in order for math processing to proceed in an uncompromised fashion. Accordingly, several investigators have asked whether high math anxious individuals have difficulty in inhibiting their attentional processes to threat-related stimuli when their math anxiety is aroused. In one study, Hopko, Ashcraft, Gute, Ruggiero, and Lewis (1998) had participants read passages of italicized text into which irrelevant distractor words had been inserted in regular font (strings of x’s were used in the control condition); participants were instructed to read the italicized text aloud. Notably, in many of the tested paragraphs, the topic was generally about math, and mathrelated words were often used as distractor words (e.g., a paragraph about balancing a checkbook, distractor words “negative exponent” and “add ten formula”). Although reading times increased for all groups when distractor words were present, the medium and high math anxiety groups took considerably longer to read the texts when either unrelated or related distractor words were present, compared to the low math anxious group. And on the follow-up test on content accuracy, the medium and high math anxious groups had considerably higher error rates when distractor words had been present, again compared to the low anxious group. In other words, during processing, the higher math anxious participants were far less able to inhibit their attention to the irrelevant distractor words, and then, despite their slower reading, still answered fewer of the questions correctly on the final test for content. Related work has shown comparable evidence of attentional bias toward math-related stimuli (i.e., emotionally negative for a math anxious individual) using a Stroop task (Suarez-Pellicioni, Nunez-Pena, & Colome, 2015) and the dot probe task (Rubensten, Eidlin, Wohl, & Akibli, 2015). Again, high math anxious participants preferentially process highly math anxious stimuli, or the location where such stimuli were presented, as if unable to inhibit attention to those anxiety-related stimuli. (For similar evidence concerning weaknesses in inhibitory control on the part of high math anxious secondary school students, see Passolunghi, Caviola, De Agostini, Perin, & Mammarella, 2016, and Chapter 8 in this volume.) On the other hand, a more recent line of research questions whether there might be a more central competency deficiency in those with high math anxiety. This possibility, also referred to as a reduced competency account (Ramirez et al., 2018), suggests that high math anxious individuals simply do not have the math skills of their lower math anxious peers, that their lower competency leads to poorer learning and performance, thus yielding math anxiety. Interestingly, this version of reduced competency does not appeal to education-based reasons, but

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suggests that individuals are simply lower in math skills, particularly in basic numerical and spatial skills. Evidence for this view comes especially from a series of studies by Maloney and colleagues ( Ferguson, Maloney, Fugelsang, & Risko, 2015; Maloney, Ansari, & Fugelsang, 2011; Maloney, Risko, Ansari, & Fugelsang, 2010). In one, these researchers had high and low math anxiety groups of participants perform a straightforward enumeration task, counting the number of squares displayed on the screen. The groups were equivalent in performance in the subitizing range (1–3 or 4 squares), but the high math anxious group was significantly slower in the counting range (more than 4) of the task. In another, participants completed the number comparison task, either deciding which of two presented digits was the larger, or, in a second study, comparing the single presented digit to an unchanging standard (5). In both cases, high math anxious participants showed a steeper numerical distance effect than low anxious participants, that is, they had more difficulty in choosing the larger of two digits when the digits were closer in value. In the third series, participants were tested on a spatial orientation task. As in the other studies, high math anxious participants were far less accurate than their low math anxious peers. The simplicity of these tasks, their presumed reliance on rather primitive number representations (as opposed to formally taught arithmetic) or elementary procedures (counting), led Maloney and colleagues to suggest that math anxiety may well be due to basic skill and numeracy deficiencies, deficiencies that then become more visible later in schooling when math content becomes more demanding. Note that none of the effects investigated in these cognitive approaches to math anxiety challenge or dispute the findings discussed in the personality approach to math anxiety. The cognitive theorists have not rejected claims about math anxiety’s associations with self-confidence, motivation, and test anxiety and have not argued that the personality approach should be overthrown in order to promote a cognitive approach. Instead, in most respects the cognitive approach has sought to both advance plausible explanations that stem from the personality approach and derive testable hypotheses from that approach. For example, the demonstrated avoidance characteristics of math anxiety (e.g., Liew, Lench, Kao, Yeh, & Kwok, 2014), including the avoidance of elective coursework, should lead to deficits in knowledge, as proposed by the reduced competency account of math anxiety. Noting that worry and ruminations routinely accompany an anxiety reaction led to specific predictions about cognitive processing requiring working memory resources. And as we discuss two final models of math anxiety, the same will be true there. These two approaches to understanding math anxiety do not overthrow what has come before in terms of offering alternative explanations of the phenomenon. Instead, they f lesh out or provide further insights into possible reasons for the onset of math anxiety in the first place, potential risk factors in other words, leading to additional ideas to be explored for possible interventions.

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Math anxiety as a sociocultural construct It seems true, almost by definition, that it will be difficult to determine the causal factors that lead to math anxiety. As we all know, a truly experimental study, with random assignment, is needed to determine causality, which is impossible with a variable like math anxiety, and the correlational designs that are largely available to us to study math anxiety are simply not suitable for studying causal relationships. Researchers have devised clever manipulations, however, that border on cause-effect results, or at least results that strike many as far more persuasive than correlational studies by themselves. Some such manipulations have already been discussed, for example the dual-task study by Ashcraft and Kirk (2001), testing a specific prediction about math anxiety (reduced working memory resources) in a specific math setting (two-column addition with carrying). But now we turn to studies of possible inf luences on the individual’s math anxiety due to social and cultural factors. It has long been suspected, or even asserted based on anecdotal evidence, that teachers play a role in the development of their students’ math anxiety. Factors such as pedagogical practice, teacher affect, and the teacher’s own math anxiety have all been considered as possible negative inf luences on students’ attitudes and math anxiety. As is commonly known, Hembree’s (1990) meta-analysis reported that the highest mean math anxiety score, by college major, was found for students majoring in elementary education. It is thus plausible that elementary school teachers, or teachers more generally, affect their students’ attitudes toward math by their own behaviors and attitudes; and if the teachers are highly math anxious, then the inf luence being transmitted would likely be negative. In what is now a classic demonstration of this, Beilock, Gunderson, Ramirez, and Levine (2010) assessed first- and second-grade teachers’ math anxiety, and then examined their students’ math achievement and stereotype endorsement about math (e.g., “boys are good at math, and girls are good at reading”). Although there were no differences in children’s attitudes at the beginning of the school year, by the end of the year girls exhibited much more gender-stereotyped beliefs and lower math achievement when their teachers were higher in math anxiety; boys, however, were unaffected, perhaps not a surprise since children tend to imitate same-gender models (all teachers were female). Thus, teacher math anxiety was directly related to students’ ability beliefs and math achievement. Particularly because gender stereotypes concerning women and math are so prevalent, this result regarding stereotyped beliefs is a particularly worrisome result with respect to women’s life-long pursuit and success in math-related careers. In related work, Maloney, Ramirez, Gunderson, Levine, and Beilock (2015) studied a different possible source of children’s math anxiety: their parents. These researchers reasoned that parents might spend time at home helping their children with their math homework. For parents with low math anxiety, this could easily be beneficial. But for parents with high math anxiety, one might easily

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expect that the parents themselves could be transmitting anxious, negative attitudes about math to their children, to the extent that the parents displayed frustration, lack of understanding, and anxiety themselves during the help sessions. In a large field study spanning a school year, parents’ math anxiety was assessed, and parents kept careful logs of the frequency and duration of their homework help sessions with their children. The results confirmed the research hypothesis; higher math anxious parents who helped frequently with math homework had children who, at the end of the year, were higher in math anxiety themselves, compared to children whose high math anxious parents seldom helped with homework, or to children who were helped by non-anxious parents (for a review of parent effects, see Batchelor, Gilmore, & Inglis, 2017; Chapter 11 in this volume discusses both parent and teacher effects on math anxiety). Finally, it seems clear that peers – and society at large – play a role in shaping children’s and adults’ attitudes about math, and likely their math anxiety as well. As many authors have noted, poor attitudes about math abound in Western societies. Ashcraft (2002) noted that math is frequently viewed as inherently difficult (as Barbie dolls used to say, “Math class is hard”), aptitude is considered as more important than effort (e.g., Geary, 1994, chap. 7), and mastering math is often viewed as either unimportant or even optional by many (see the correlations in Table 1.1 concerning motivation and intent to enroll, for example). Unfortunately, there appears to be no clear evidence concerning peer inf luence on math anxiety. At the global level, however, there is now evidence that economically developed and more gender equal countries have lower overall levels of math anxiety than less developed, less equal countries (Stoet, Bailey, Moore, & Geary, 2016) – but even so, those more favored countries show a larger national gender difference in math anxiety compared to less developed countries.

Math anxiety and the role of gender The role of gender in math anxiety is perplexing. The research routinely finds that females score higher on standard assessments of math anxiety than do males; see, for example, Betz (1978), Wigfield and Meece (1988), and Hopko et al. (2003), and the full meta-analysis account in Hembree, 1990. In the classic figure from Hembree (1990), showing mean math anxiety levels from grades 6 through college, the curve representing female participants shows that females score approximately 20 points higher than males at every grade level. But despite the negative connection between math anxiety and math performance, there are generally no overall differences between the genders in actual mathematical performance (e.g., Spelke, 2005); or, when a gender difference is found, it is usually rather small (e.g., Kovas et al., 2007). Interestingly, however, females tend to rate themselves as having lower math performance than males rate themselves (e.g., Dowker, Bennett, & Smith, 2012; Goetz, Bieg, Ludtke, Pekrun, & Hall, 2013; Hill et al., 2016).

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It may be that the tendency for females to show higher levels of math anxiety is related to the general tendency for females to show generally higher levels of anxiety, and for females to have a higher prevalence of clinical anxiety disorders (e.g., Beidel & Alfano, 2011; Feingold, 1994; McLean, Asnaani, Litz,  & Hofmann, 2011). If so, then this would conceivably be due to the overlap between math anxiety and more general anxiety reactions. Another possibility, as articulated in Ashcraft (2002), is that women may feel more comfortable reporting anxiety than men do. This could explain the puzzling reason why the gender difference in math anxiety is not accompanied by a gender difference in math performance. Yet another alternative, explored more fully in a later section (see also Chapter 10 in this volume), relates to the notion of stereotype threat, the phenomenon that one’s performance will dwindle if a negative stereotype about one’s group, say gender, is brought to mind at the time of a test. It has been shown ( Beilock, Rydell, & McConnell, 2007) that women do more poorly on a math-based test if a negative stereotype about women and math is aroused prior to testing (“we are trying to understand why men do better than women on this task”). Interestingly, in this study it was only on problems requiring the resources of working memory that the effects of math anxiety were apparent; thus the cognitive interpretation of the effect was parallel to that of the Ashcraft and Kirk (2001) study, that anxiety caused a drain on working memory resources. A final possibility, of course, is that women have internalized to some degree the social and cultural attitudes involving the stereotypical belief that math is a male domain. When inevitable challenges arose during education, they may have interpreted their difficulties as evidence of a lack of ability, and developed a degree of math anxiety as a result. According to Dweck’s (1999) social-cognitive theory of intelligence, this would demonstrate an entity theory of intelligence on the part of such females, a belief that their abilities in math are largely fixed and unable to change. When confronted with negative feedback, say, poor math test performance, such individuals often withdraw their effort from the task, that is, they adopt an avoidance pattern of behavior. Of course, as noted earlier, avoidance is a prime characteristic of math anxious individuals. And as demonstrated by Burkley, Parker, Stermer, and Burkley (2010), female students who endorse this entity or fixed abilities belief system were less likely to enjoy math, to pursue a college major requiring math, or to consider career paths involving math. These characteristics, of course are prominent in the Table 1.1 correlations involving math anxiety. They may be especially prominent, or exacerbated, among females. (The opposite set of beliefs in Dweck is the incremental theory, the belief that intelligence and abilities are malleable, and can be improved with practice.)

Math anxiety as a neuro-biological construct Very little research has been conducted on the possible biological, that is, genetic, bases of math anxiety, although one study that has been reported is rather

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compelling. Wang et al. (2014) tested 216 monozygotic and 298 same-sex dizygotic twins, from an ongoing longitudinal project; at the time of testing the twins’ average age was 12.25 years. The tests that were administered included a math anxiety test, a test of general anxiety, a math problem solving test, and a reading comprehension test. Overall, the results showed that approximately 40% of the variation in math anxiety was accounted for by genetic factors, with the remaining 60% accounted for by individually specific environmental factors. Math anxiety was inf luenced by the genetic and environmental risk factors associated with general anxiety. There was also evidence for an independent genetic component related to math problem solving. In their conclusion, the authors note that their results suggest a rather worrisome possibility, the possibility of a dynamic, downward spiraling process. In particular, they suggest that the genetic inf luence related to poor math problem solving, along with the genetic and individual-specific environmental inf luences for high general anxiety, serve as risk factors in the development of math anxiety. When these risk factors combine, there may be further impairment in math performance, with attendant negative consequences (low grades, emotionality, decreases in math self-concept). Such reactions can in turn exacerbate the individual’s math anxiety (e.g., avoidance, negative attitudes), setting up another cycle of declining performance and increasing anxiety. Three other well-known studies also deserve mention here, not because they explored possible genetic determinants of math anxiety, but because they examined brain mechanisms and regions underlying the anxiety reaction. And, for those who might have doubted that math anxiety was truly a documentable phenomenon, they provided brain-based evidence of its existence and operation. In the first study, Young, Wu, and Menon (2012) tested children ages 7–9 in an fMRI environment on addition and subtraction problems; problems were shown with answers, and children made true/false judgments. Simple problems for both operations had an operand of 1 (e.g., 5 + 1 = 6; 7 – 1 = 5). Beyond obtaining conventional results on behavioral measures (errors, latencies), Young et al. also found rather definitive outcomes in terms of neural activations. Math problem-solving was associated with greater activation in the right amygdala in the high math anxious group, compared to the low anxious group. The right amygdala also showed greater effective connectivity with brain areas associated with social and general anxiety. On the other hand, the results showed lower activation for the high anxious group, compared to low anxious participants, in many cortical areas associated with typical math processing, including the intraparietal sulcus (IPS), superior parietal lobule, and right dorsolateral prefrontal cortex. Likewise, the right amygdala in high math anxious participants showed less effective connectivity to the typical “math” regions, namely, the posterior parietal cortex, including the IPS and angular gyrus. Thus, Young et al. determined that math anxiety is associated with hyperactivity in the amygdala, a region of the brain typically associated with processing fear and negative emotions. Furthermore, their evidence revealed that high math anxious children also

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showed reduced brain activations in regions that have repeatedly been found to be associated with mathematical and numerical reasoning in children and adults (see also Klados, Simos, Micheloyannis, Margulies, & Bamidis, 2015). In the second two studies, Lyons and Beilock (2012a, 2012b) tested low and high math anxious adults in an fMRI setting. In both studies, participants were given a cue several seconds before receiving a block of either math or word trials, with the cue alerting them to which kind of trial would follow. Math stimuli had to be judged as true or false; easy problems had operands less than 5, and hard problems had operands of 5 or greater. The word trials showed either 4 or 7 letter strings; participants responded true if the letters spelled an English word when reversed (e.g., yrestym would yield mytsery, and participants should respond “false”). In one study ( Lyons & Beilock, 2012b), the focus was on the different regions and networks that become activated during the cue period, that is when the participant is anticipating either having to do math or having to do the word task. The compelling result was that high math anxious participants showed increased bilateral dorso-posterior insula and mid-cingulate cortex activity during the anticipation period, that is, during the cue period prior to actually processing the math problem. The critical point here is that these regions are associated with visceral threat detection, and even with the experience of pain itself. Thus, anticipating an upcoming math problem registers in the brain as an experience of visceral pain, in regions of the brain associated with threat to the physical body (see also Pletzer, Kronbichler, Nuerk, & Kerschbaum, 2015). In a paired study, Lyons and Beilock (2012a) noted a corollary result; while the high math anxiety group averaged nearly twice as many errors on difficult problems as the low math anxious group, some of the high math anxious participants seemed relatively unimpaired in their math performance. Lyons and Beilock wondered if the key to this result might be found during the anticipation phase, after participants had been given the cue telling them which kind of trial would follow. Perhaps these participants were engaging in some form of preparation, some kind of cognitive restructuring that enabled them to then process the math problem unhindered by the interference of anxiety. Indeed, when the fMRI results were examined in terms of behavioral differences, those high math anxious participants who showed better math performance (a smaller discrepancy between their math and word performance) showed increased neural activity in frontoparietal regions during the anticipation period, that is, during the cue phase when they were preparing for the upcoming trial. Frontoparietal regions would be just the regions one would expect to be more active during math processing, suggesting that indeed these participants were ramping up their cognitive resources in preparation for the upcoming math problem. Lyons and Beilock suggested, accordingly, that it is perhaps not necessarily one’s level of math anxiety but, instead, one’s ability to engage cognitive control processes before problem solving has even begun that predicts math performance. Such control might include reappraising one’s approach to problem solving, increasing one’s motivation, or even marshaling deliberate control

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to inhibit attention to distracting worries (for additional evidence concerning anticipatory anxiety, see Klados, Pandria, Micheloyannis, Margulies, & Bamidis, 2017).

Math anxiety in the future – the interpretation account Ramirez et al. (2018) have recently proposed an overarching account of math anxiety that departs in several significant ways from current thinking, in that it emphasizes the individual’s interpretation of previous math experiences and outcomes, rather than the individual’s avoidance, reduced competency, or worries per se, as determiners of math anxiety. The approach stems from existing appraisal theory (e.g., Lazarus, 1991), in its view that emotional outcomes and attitudes are based on one’s interpretations of events, internal states, physiological cues, and the like. Consider, for example, studies showing that earlier achievement has a stronger effect on subsequent math anxiety than earlier math anxiety has on subsequent math achievement (Meece, Wigfield, & Eccles, 1990). The reasoning here is that students may believe that their math outcomes, that is, their math performance on standardized tests, are indicators of their abilities. As such, students form expectations about their future success – or lack of success – based on those performance measures. Thus, those who attribute their poor performance to lower ability may be at greater risk for developing math anxiety than are those who attribute their poor performance to lower effort, or those who acknowledge that mistakes are routine when learning math. Note the similarities here to Dweck’s (1999) distinction between entity versus incremental theories of intelligence, described earlier; someone with an entity theory does poorly, attributes that to fixed (and low) abilities, and then fails to engage in the practice and effort that would improve performance, thus falling further behind and likely developing worse attitudes. While the Ramirez et al. approach is a novel approach to explaining math anxiety, it is nonetheless far too new to evaluate how well it explains all of the extant evidence that has now accumulated about math anxiety. It does, however, make some clear claims about designing interventions for math anxiety. They all generally have to do with reinterpretation or reappraisal of the experiences and outcomes that lead an individual to adopt math anxious attitudes, or that perpetuate the anxiety and attitudes. And in general, although there are only a handful of studies that have tested such manipulations, they do seem to show promise. For example, Jamieson, Peters, Greenwood, and Altose (2016) tested community college students who were enrolled in remedial mathematics. Two exams were given to the classes, one prior to the manipulation to serve as a pre-test (which showed group equivalence). The reappraisal intervention was given prior to Exam 2. In it, students were told about the adaptive functions of stress; they were told that increased arousal was not harmful, that stress responses evolved to help address acute demands, and that increased arousal aids performance. The control group was simply told that the best way to improve outcomes during

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stressful test situations was to ignore the stress, and to ignore negative thoughts associated with stress during the test ( Jamieson et al., 2016, p. 581). Students in the reappraisal group performed significantly better on Exam 2 than did students in the control group, and they also reported lower math evaluation anxiety during Exam 2 compared to controls. Thus reappraisal of the stress experienced by high math anxious students during the second math test led to improved performance. (For similar effects observed after a short period of expressive writing, see Park, Ramirez, & Beilock, 2014.) Whatever the ultimate fate of the Ramirez et al. (2018) interpretation account of math anxiety, its different approach, along with the other models of math anxiety has made it clear that math anxiety is a multifaceted phenomenon, which has profited from the multiple approaches researchers have taken in attempts to understand it. In a very real sense, the several models that have been summarized here are all mutually compatible – no single model or construct contradicts another model in major ways. Instead, the models and different approaches have been largely supportive, and have enriched the study of math anxiety. As an example, determining that the anxiety response consumes essential working memory resources, thus compromising the cognitive effort that can be devoted to math problem solving, led directly to brain imaging searches for evidence both of the emotional, anxiety-based processing and the standard cognitive-based math processing. And the personality-based research revealed clear evidence of avoidance and lower math achievement, important warnings for researchers in terms of the potential confounding factors between math anxiety and math competence at higher levels of math difficulty. This chapter was intended to be an overview, a brief introduction to the various topics that are relevant to a full understanding of math anxiety. Having been introduced to those topics, the reader is now invited to full-length, in-depth treatments of these topics, from the leading researchers in the field.

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Batchelor, S., Gilmore, C., & Inglis, M. (2017). Parents’ and children’s mathematics anxiety. In U. Xolocotzin Eligio (Ed.), Understanding emotions in mathematical thinking and learning (pp. 315–336). San Diego: Academic Press. Beidel, D. C., & Alfano, C. A. (2011). Child anxiety disorders: A guide to research and treatment. New York: Routledge. Beilock, S. L., Gunderson, E. A., Ramirez, G., & Levine, S. C. (2010). Female teachers’ math anxiety affects girls’ math achievement. Proceedings of the National Academic of Sciences, 107, 1860–1863. Beilock, S. L., Rydell, R. J., & McConnell, A. R. (2007). Stereotype threat and working memory: Mechanisms, alleviation, and spillover. Journal of Experimental Psychology: General, 136, 256–276. Betz, N. E. (1978). Prevalence, distribution, and correlates of math anxiety in college students. Journal of Counseling Psychology, 25, 441–448. Browne, C. E. (1906). The psychology of the simple arithmetical processes: A study of certain habits of attention and association. American Journal of Psychology, 17, 1–37. Burkley, M., Parker, J., Stermer, S. P., & Burkley, E. (2010). Trait beliefs that make women vulnerable to math disengagement. Personality and Individual Differences, 48, 234–238. Dew, K. M. H., Galassi, J. P., & Galassi, M. D. (1983). Mathematics anxiety: Some basic issues. Journal of Counseling Psychology, 30, 443–446. Dew, K. M. H., Galassi, J. P., & Galassi, M. D. (1984). Math anxiety: Relation with situational test anxiety, performance, physiological arousal, and math avoidance behavior. Journal of Counseling Psychology, 31, 580–583. Dowker, A., Bennett, K., & Smith, L. (2012). Attitudes to mathematics in primary school children. Child Development Research, 124939. doi:10.1155/2012.124939 Dowker, A., Sarkar, A., & Looi, C. Y. (2016). Mathematics anxiety: What have we learned in 60 years? Frontiers in Psychology, 7, #508, 1–16. doi:10.3389/fpsyg.2016.00508 Dreger, R. M., & Aiken, L. R., Jr. (1957). The identification of number anxiety in a college population. Journal of Educational Psychology, 48, 344–351. Dweck, C. S. (1999). Self-theories: Their role in motivation, personality, and development. Philadelphia: Psychology Press. Engle, R. W. (2002). Working memory capacity as executive attention. Current Directions in Psychological Science, 11, 19–23. Eysenck, M. W. (1992). Anxiety: The cognitive perspective. Hove, UK: Erlbaum. Eysenck, M. W., & Calvo, M. G. (1992). Anxiety and performance: The processing efficiency theory. Cognition and Emotion, 6, 409–434. Faust, M. W., Ashcraft, M. H., & Fleck, D. E. (1996). Mathematics anxiety effects in simple and complex addition. Mathematical Cognition, 2, 25–62. Feingold, A. (1994). Gender differences in personality: A meta-analysis. Psychological Bulletin, 116, 429–456. Ferguson, A. M., Maloney, E. A., Fugelsang, J., & Risko, E. V. (2015). On the relation between math and spatial ability: The case of math anxiety. Learning and Individual Differences, 39, 1–12. Geary, D. C. (1994). Children’s mathematical development: Research and practical applications. Washington, DC: American Psychological Association. Goetz, T., Bieg, M., Ludtke, O., Pekrun, R., & Hall, N. C. (2013). Do girls really experience more anxiety in mathematics? Psychological Science, 24, 2079–2087. Gough, M. F. (1954). Mathemaphobia: Causes and treatments. Clearing House, 28, 290–294. Hembree, R. (1990). The nature, effects, and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21, 33–46.

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Hill, F., Mammarella, I. C., Devine, A., Caviola, S., Passolunghi, M. C., & Szucs, D. (2016). Maths anxiety in primary and secondary school students: Gender differences, developmental changes and anxiety specificity. Learning and Individual Differences, 48, 45–53. Hopko, D. R., Ashcraft, M. H., Gute, J., Ruggiero, K. J., & Lewis, C. (1998). Mathematics anxiety and working memory: Support for the existence of a deficient inhibition mechanism. Journal of Anxiety Disorders, 12, 343–355. Hopko, D. R., Mahadevan, R., Bare, R. L., & Hunt, M. A. (2003). The Abbreviated Math Anxiety Scale (AMAS): Construction, validity, and reliability. Assessment, 10, 178–182. Jamieson, J. P., Peters, B. J., Greenwood, E. J., & Altose, A. J. (2016). Reappraising stress arousal improves performance and reduces evaluation anxiety in classroom exam situations. Social Psychological and Personality Science, 7, 579–587. Klados, M. A., Pandria, N., Micheloyannis, S., Margulies, D., & Bamidis, P. D. (2017). Math anxiety: Brain cortical network changes in anticipation of doing mathematics. International Journal of Psychophysiology, 122, 24–31. Klados, M. A., Simos, P., Micheloyannis, S., Margulies, D., & Bamidis, P. D. (2015). ERP measures of math anxiety: How math anxiety affects working memory and mental calculation tasks? Frontiers in Behavioral Neuroscience, 9, 282. doi:1010389/fnbeh.2015.00282 Kovas, Y., Haworth, C. M. A., Dale, P. S., Plomin, R., Weinberg, R. A., Thomson, J. M., & Fischer, K. W. (2007). The genetic and environmental origins of learning abilities and disabilities in the early school years. Monographs of the Society for Research in Child Development, 72(3), vii, 1–160. Lazarus, R. S. (1991). Progress on a cognitive-motivational-relational theory of emotion. American Psychologist, 46, 819–834. Liew, J., Lench, H. C., Kao, G., Yeh, Y., & Kwok, O. (2014). Avoidance temperament and social-evaluative threat in college students’ math performance: A mediation model of math and test anxiety. Anxiety, Stress, & Coping, 27, 650–661. Lyons, E. M., & Beilock, S. L. (2012a). Mathematics anxiety: Separating the math from the anxiety. Cerebral Cortex, 22, 2102–2110. Lyons, E. M., & Beilock, S. L. (2012b). When math hurts: Math anxiety predicts pain network activation in anticipation of doing math. PLoS One, 7(10), e48076. https:// doi.org/10.1371/journal.pone.0048076 Ma, X. (1999). A meta-analysis of the relationship between anxiety toward mathematics and achievement in mathematics. Journal for Research in Mathematics Education, 30, 520–541. Maloney, E. A., Ansari, D., & Fugelsang, J. A. (2011). The effect of mathematics anxiety on the processing of numerical magnitude. The Quarterly Journal of Experimental Psychology, 64, 10–16. Maloney, E. A., Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2015). Intergenerational effects of parents’ math anxiety on children’s math achievement and anxiety. Psychological Science, 26, 1480–1488. Maloney, E. A., Risko, E. F., Ansari, D., & Fugelsang, J. (2010). Mathematics anxiety affects counting but not subitizing during visual enumeration. Cognition, 114, 293–297. McLean, C. P., Asnaani, J. A., Litz, B. T., & Hofmann, S. G. (2011). Gender differences in anxiety disorders: Prevalence, course of illness, comorbidity and burden of illness. Journal of Psychiatric Research, 45, 1027–1035. McLeod, D. B. (1989). The role of affect in mathematical problem solving. In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving (pp. 20–36). New York: Springer-Verlag.

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Meece, J. L., Wigfield, A., & Eccles, J. S. (1990). Predictors of math anxiety and its inf luence on young adolescents’ course enrollment intentions and performance in mathematics. Journal of Educational Psychology, 82, 60–70. Miyake, A., & Shah, P. (1999). Models of working memory: Mechanisms of active maintenance and executive control. New York: Cambridge University Press. Park, D., Ramirez, G., & Beilock, S. L. (2014). The role of expressive writing in math anxiety. Journal of Experimental Psychology: Applied, 20, 103–111. Passolunghi, M. C., Caviola, S., De Agostini, R., Perin, C., & Mammarella, I. C. (2016). Mathematics anxiety, working memory, and mathematics performance in secondaryschool children. Frontiers in Psychology, 4, 42. doi:10.339/fpsyg.2016.00042 Pletzer, B., Kronbichler, M., Nuerk, H. C., & Kerschbaum, H. H. (2015). Mathematics anxiety reduces default mode network deactivation in response to numerical tasks. Frontiers in Human Neuroscience, 9, 202. doi:10.3389/fnhum.2015.00202 Ramirez, G., Chang, H., Maloney, E. A., Levine, S. C., & Beilock, S. L. (2016). On the relationship between math anxiety and math achievement in early elementary school: The role of problem solving strategies. Journal of Experimental Child Psychology, 141, 83–100. Ramirez, G., Shaw, S., & Maloney, E. A. (2018). Math anxiety: Past research, promising interventions, and a new interpretation framework. Educational Psychologist, 53, 145–164. Richardson, F. C., & Suinn, R. M. (1972). The Mathematics Anxiety Rating Scale. Journal of Counseling Psychology, 19, 551–554. Rubinsten, O., Eidlin, H., Wohl, H., & Akibli, O. (2015). Attentional bias in math anxiety. Frontiers in Psychology, 6, 1539. doi:10.3389/fpsyg.2015.01539 Spelke, E. (2005). Sex differences in intrinsic aptitude for mathematics and science? A critical review. American Psychologist, 60, 950–958. Stoet, G., Bailey, D. W., Moore, A. M., & Geary, D. C. (2016). Countries with higher levels of gender equality show larger national sex differences in mathematics anxiety and relatively lower parental mathematics valuation for girls. PLoS One. doi:10.1371/ journal.pone.0153857 Suarez-Pellicioni, M., Nunez-Pena, M. I., & Colome, A. (2013). Mathematical anxiety effects on simple arithmetic processing efficiency: An event-related potential study. Biological Psychology, 94, 517–526. Suarez-Pellicioni, M., Nunez-Pena, M. I., & Colome, A. (2015). Attentional bias in high math-anxious individuals: Evidence from an emotional Stroop task. Frontiers in Psychology, 6, 1577. doi:10.3389/fpsyg.2015.01577 Trezise, K., & Reeve, R. A. (2014). Working memory, worry, and algebraic ability. Journal of Experimental Child Psychology, 121, 120–136. Wang, Z., Hart, S. A., Kovas, Y., Lukovski, S., Sodden, B., Thompson, L. A., . . . Petrill, S. A. (2014). Who is afraid of math? Two sources of genetic variance for mathematical anxiety. Journal of Child Psychology and Psychiatry, 55, 1056–1064. Widaman, K. F., Geary, D. C., Cormier, P., & Little, T. D. (1989). A componential model for mental addition. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15, 898–919. Wigfield, A., & Meece, J. L. (1988). Math anxiety in elementary and secondary school students. Journal of Educational Psychology, 80, 210–216. Young, C. B., Wu, S. S., & Menon, V. (2012). The neurodevelopmental basis of math anxiety. Psychological Science, 23, 492–501.

2 DIFFERENT WAYS TO MEASURE MATH ANXIETY Krzysztof Cipora, Christina Artemenko and Hans-Christoph Nuerk

General remarks Most of the objects we encounter on an everyday basis can be seen or touched. Therefore, we do not need specific measurement instruments to prove a specific object’s existence and distinctiveness from other objects. Unfortunately, this is not the case for psychological constructs. They cannot be directly assessed via sensory experiences; they have to be indirectly assessed by specific measurement instruments. When attempting to measure psychological constructs one needs not only to show that the measurement is reliable, but also that the measurement of the construct is valid (e.g., by showing that different measures of the same construct are related, and that measures of different constructs are not too strongly related). Math anxiety (MA) is such a psychological construct. Here, we provide an overview of the different ways to measure MA. Before we go into the details, we want to emphasize some basics about MA: MA can be reduced neither to poor math skill (i.e., one declares to be anxious about math solely because of having poor skills in this domain) nor to anxiety in general (i.e., one declares to be anxious about math solely because they are anxious about virtually everything). However, individual MA assessment is necessary for MA research, diagnosis, intervention planning and evaluation. In our overview of MA measurement instruments, we will focus mostly on selfdescriptive measures, as they are the most popular methods. Additionally, we discuss other approaches for measuring MA including behavioral and (neuro) physiological measures. We will also point to some new avenues in MA measurement that should be considered in future investigations.

Structure of math anxiety Despite more than fifty years of intense MA research, the question about its structure is far from being resolved. Discussing the structure of MA thoroughly

Different ways to measure math anxiety 21

goes largely beyond the scope of this chapter (for more details on this matter, see Chapter 1). Nevertheless, we need to introduce the debate on that issue, because the postulated MA structure determines, to a certain extent, the way in which measurement instruments are built. As one can see in Table 2.1, subscales of the MA measurement instruments differ considerably. Usually it is claimed that MA comprises two factors: anxiety related to using math in everyday (learning) situations and anxiety related to being evaluated in math. This structure was proposed in early research, starting in the 1970s (see Suinn & Edwards, 1982). Similarly, two factors (termed numerical anxiety and math test anxiety) account for the results of the Mathematics Anxiety Rating Scale (MARS) – the first instrument aimed specifically at MA measurement (see below). A two-factor solution (learning math anxiety and math evaluation anxiety) was also found in its several revisions. These two factors are usually highly correlated (about .70; Hopko, 2003), and are also ref lected in the Abbreviated Math Anxiety Scale (AMAS; Hopko, Mahadevan, Bare, & Hunt, 2003; see below). Nevertheless, there are some valid alternatives to the two-factorial solution. Several studies showed that MA comprises three components. For instance, three factors best explain the variance in the shortened MARS (sMARS) scale (Alexander & Martray, 1989). These factors are (1) math test anxiety, (2) numerical task anxiety, and (3) math course anxiety. Three factors also best account for the pattern of scores in the Mathematics Anxiety Scale – UK (MAS-UK) developed for the British population (Hunt, Clark-Carter, & Sheffield, 2011). These factors were termed (1) maths evaluation anxiety, (2) everyday/social maths anxiety, and (3) maths observation anxiety. Furthermore, abstraction anxiety (anxiety related to abstract mathematical content) is considered to be another MA component (Ma & Xu, 2004). Some other researchers postulate that MA considers not only negative but also positive affect toward math (e.g., Bai, 2011; Kazelskis, 1998). Some recent studies suggest a more complex structure of MA. Based on a confirmatory factor analysis of MARS30-Brief scale, Pletzer, Wood, Scherndl, Kerschbaum, and Nuerk (2016) concluded that a simple distinction between numerical anxiety and testing anxiety does not satisfactorily fit the data. The best fit was obtained by a model consisting of six factors: (1) evaluation anxiety 1 proper  – taking math exam; (2) evaluation anxiety 2 – thinking of upcoming exam; (3) learning math anxiety; (4) everyday numerical anxiety; (5) performance anxiety; and (6) social responsibility anxiety. Another recent study (Yánez-Marquina & VillardónGallego, 2017) showed that MA is a hierarchical construct. First, two general factors, everyday life math anxiety and academic math anxiety, were differentiated. The latter was further subdivided into math learning anxiety and math test anxiety. This new model is theoretically interesting as it includes MA related to both academic and everyday life situations, however, it requires further investigation. The models discussed above tried to capture MA related to different contexts and situations. Nevertheless, they treated anxiety as being perceived as unitary in each of these contexts and situations. Another approach differentiates two MA dimensions even within a context: an affective and a cognitive one. The affective dimension refers to feelings of nervousness, dread etc., while the cognitive

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dimension refers to the worry component of anxiety (Ho et al., 2000). It is worth noting that this MA framework corresponds well to the understanding of anxiety and its inf luence on cognition in general (see Calvo & Eysenck, 1992). There is much less evidence on MA structure in children of early school age. For a long time it was assumed that MA develops only around sixth grade when negative experiences with math accumulate and the math content becomes more difficult. However, more recent studies show that MA is present much earlier and can be accurately measured even at the onset of school age (see Harari, Vukovic, & Bailey, 2013; Krinzinger, Kaufmann, & Willmes, 2009). Nevertheless some studies suggest a componential MA structure in these early ages such as (a) math learning and (b) math evaluation anxiety (Carey, Hill, Devine, & Szűcs, 2017) or (a) negative emotions, (b) numerical confidence, and (c) worry (Harari et al., 2013). In sum, the factorial structure of math anxiety remains relatively unclear, and different structural proposals currently exist in the literature. However, most researchers agree that it is not a unidimensional construct. The variety of models of MA structure requires researchers to carefully select measurement instruments to meet their particular needs. On the other hand, one must keep in mind that the selection of a particular instrument automatically restricts theoretical claims and possibilities for differential diagnosis and tailored intervention.

Paper-and-pencil self-descriptive math anxiety measurement instruments Measuring math anxiety in adults and adolescents MA questionnaires have been developed since the 1960s. Ashcraft and Moore (2009) briefly summarize the history of MA measurement (see also Eden, Heine, & Jacobs, 2013; Yánez-Marquina & Villardón-Gallego, 2017 for overviews of different instruments). The first instrument used for measuring MA was a modification of the Taylor Manifest Anxiety Scale proposed by Dreger and Aiken (1957), who introduced three specific MA items to that scale. MARS ( Richardson & Suinn, 1972) was the first tool specifically designed to measure math anxiety. It comprises 98 items and is characterized by satisfactory psychometric properties. It covers both everyday life situations and academic settings. However, the large number of items makes its administration rather time consuming. Numerous modifications of the MARS have aimed at shortening the scale without loss of psychometric quality. For instance, MARS-R (MARSRevised; Plake & Parker, 1982) comprises only 24 items. It is characterized by very good psychometric properties and strongly correlates with the original MARS (.97). It comprises a composite score and two subscale scores for learning math anxiety and math evaluation anxiety. Subsequent confirmatory factor analysis of MARS-R items by Hopko (2003) suggested further shortening it to 12 items without loss in psychometric properties. The sMARS, another very popular tool (Alexander & Martray, 1989), is a 25-item scale which highly correlates (.97)

Different ways to measure math anxiety 23

with the MARS. It is also characterized by excellent psychometric properties. This scale is the most popular among MA researchers. Numerous instruments originating either directly or indirectly from the MARS got very popular and for a long time were utilized in the vast majority of MA studies. They are all characterized by very good psychometric properties and are well established in the field. Nevertheless, they are all copyrighted and cannot be used free of charge either for research purposes or for diagnosis. As an alternative, numerous freely available instruments were developed. Their psychometric properties are also very good, and they are increasingly gaining popularity (see Table 2.1 for an overview of such instruments). Interestingly, as Ashcraft (2002) notes, participants’ answers to the single question about their math anxiety on a 10-point scale correlates from .49 up to .85 with the sMARS scores. Very good validity and reliability of a single-item scale (termed the SingleItem Math Anxiety Scale – SIMA) was also shown in a subsequent systematic study. Participants are required to give an answer to the following question: “On a scale from 1 to 10, how math anxious are you?” ( Núñez-Peña, Guilera, & SuárezPellicioni, 2014). It is worth noting that a similar single question approach was chosen to assess math anxiety in the PISA (Programme for International Student Assessment) studies (see Lee, 2009; OECD, 2013). However, while the correlations with some overall math anxiety questionnaires is high, the SIMA only allows for a general assessment of MA level, but no differential diagnosis, in which context and situations are affected (e.g., whether MA equally strongly affects everyday situations and being tested in math). Therefore, such extremely short instruments might be used as a general measure for an unstructured covariate math anxiety, but they are not suitable for differential diagnostics and adequate, tailored intervention planning. In general, most MA instruments are characterized by very good psychometric properties. The selection of a MA questionnaire for a particular research purpose should also be guided by the research questions, especially whether one wishes to tap only a very general MA level or expects, rather, to glean relations to specific MA components and have a targeted sample.

The Abbreviated Math Anxiety Scale The AMAS is one of the most commonly used math anxiety measures today (Hopko, Mahadevan, et al., 2003). Due to its popularity and potential for being utilized in numerous linguistic and cultural contexts, various age groups, and administration modes, we describe it in more detail in following paragraphs. The AMAS comprises nine items contributing to the math learning and math testing scales and calculates an overall sum score. The AMAS originates from a reanalysis of the MARS-R by Hopko (2003). The widespread use of the AMAS is possibly due to its very short form (administration takes no more than 5 minutes), and to its very good psychometric properties: high reliability as measured by internal consistency and test-retest method, construct validity as measured

Abbreviated Math Anxiety Scale

Math Anxiety Scale – UK

Single-Item Math Anxiety Scale

Children’s Anxiety in Math Scale

Child Math Anxiety Questionnaire The Fennema-Sherman Mathematics Attitudes Scales – Short Form

AMAS

MAS-UK

SIMA

CAMS

CMAQ

FSMAS-SF

Full name

Abbreviation

51

8

16

1

23

9

No. of items

Total score General math anxiety Math performance anxiety Math error anxiety Total score

• Total score • Attitude towards success in mathematics • Mathematics as a male domain • Parent’s attitudes • Teacher’s attitudes • Mathematics-related affect • Usefulness of mathematics

• • • • •

Measures aimed for adolescents and children

• Total score • Maths evaluation anxiety • Everyday / social maths anxiety • Maths observation anxiety NA

Measures aimed for adults and adolescents • Total score • Learning subscale • Testing subscale

Subscales

TABLE 2.1 Examples of freely available MA measurement instruments

Two age groups: 12-year-olds, 15-year-olds

7-year-olds

Grades 1–5 (no age reported)

Adults and adolescents

Adults and adolescents, in some studies from 8-year-olds Adults

Target group

Ramirez, Gunderson, Levine, and Beilock (2013) Mulhern and Rae (1998)

Jameson (2013)

Núñez-Peña et al. (2014)

Hunt et al. (2011)

Hopko et al. (2003)

Reference

Math Anxiety Questionnaire1 Mathematics Anxiety Scale – Revised Scale for Assessing Math Anxiety in Secondary education

MAQ

Scale for Early Mathematics Anxiety

Mathematics Anxiety Scale for Young Children

SEMA

—

SAMAS

MAS-R

Modified Abbreviated Math Anxiety Scale

mAMAS

12

20

25

14

11

9

• • • • • • • • • • • • • • • • • • •

Total score Math evaluation subscale Math learning subscale Negative affective reactions Worry Negative affect towards math Positive affect towards math Total score Everyday life’s math anxiety Academic math anxiety Math learning anxiety Math test anxiety Total score Numerical processing anxiety Situational and performance anxiety Total score Negative reactions Numerical confidence Worry 6–7-year-olds

8-year-olds

Validated in one age group: 12–16-yearolds

Validated in one age group: 8–13-year-olds Grades 5–11 (no age reported) Grades 7–8

Wu, Barth, Amin, Malcarne, and Menon (2012) Harari et al. (2013)

Yánez-Marquina et al. (2017)

Bai (2011)

Wigfield and Meece (1988)

Carey et al. (2017 )

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by exploratory and confirmatory factor analyses, and convergent and discriminant validity (Hopko, Mahadevan, et al., 2003). Numerous subsequent studies have confirmed these results (see Cipora, Willmes, Szwarc, & Nuerk, 2017 for a summary). Crucially, good psychometric properties of the AMAS were shown in various cultural and linguistic contexts: (1) American (e.g., Ferguson, Maloney, Fugelsang, & Risko, 2015, Hopko, Mahadevan, et al., 2003), (2) Iranian ( Vahedi & Farrokhi, 2011), (3) Spanish ( Brown & Sifuentes, 2016), (4) Italian (Caviola, Primi, Chiesi, & Mammarella, 2017; Primi, Busdraghi, Tomasetto, Morsanyi, & Chiesi, 2014), (5) German ( Dietrich, Huber, Moeller, & Klein, 2015; Schillinger, Vogel, Diedrich, & Grabner, 2018, and also yet unpublished studies conducted in our lab), and (6) Polish (Cipora, Szczygiel, Willmes, & Nuerk, 2015; Cipora et al., 2017). The AMAS has also been translated into other languages such as French, Chinese, and Russian. Another advantage of the AMAS is that it can be administered to a variety of age groups. It seems to work well with adolescents at the high school level. There are also some studies showing its suitability for MA measurement even with younger individuals. For instance, it was successfully administered to 11-yearolds ( Devine, Fawcett, Szűcs, & Dowker, 2012). In a modified form (mAMAS, Carey, Hill, et al., 2017), it was even used to measure the MA of 8–13-year-olds. Its short form and ease of administration makes the AMAS a perfect instrument for online measurements (see Reips, 2002 for guidelines for online testing). The AMAS has been administered online in several studies (see e.g., Ferguson et al., 2015; Jones, Childers, & Jiang, 2012). A systematic study by Cipora and colleagues (2017) has shown equivalence of the online administration mode to the traditional pencil-and-paper version. The only caveat was that mean scores were lower in the case of online administration, however, effect sizes related to these differences were small (d s ≤ .16). This finding is partly at odds with early observations that MA scores are higher when administered via a computer (Ashcraft & Faust, 1994). This discrepancy may be related to the relative inaccessibility of and high-level of technical knowledge required to operate computers when the study was conducted compared to now.

Questionnaires for math anxiety in children Most of the instruments, aimed at MA measurement in adults, can also be used in adolescents. However, measures have also been specifically designed for children and adolescents. The first of them is the MARS-A (Suinn & Edwards, 1982) aimed at MA measurement in adolescents. A 26-item MARS-E (Suinn, Taylor, & Edwards, 1988) was developed for younger children, specifically 5th through 6th graders. When it comes to younger children, it was argued that one cannot differentiate MA from other anxiety types and that MA does not correlate with math performance at this stage of development. However, subsequent studies have shown

Different ways to measure math anxiety 27

that MA can be reliably measured even at the beginning of elementary school (see Krinzinger et al., 2009 for an overview of the debate). Furthermore, it was also shown that MA affects performance even in the early years of education (Harari et al., 2013; Ramirez et al., 2013). Nevertheless, one needs to heed some caveats, when MA of very young children (e.g., 1st graders) is measured. First, not all of them might have the necessary self-awareness needed for self-rating questionnaires. Second, reading skill and linguistic comprehension is limited so that not all questions might be well understood, especially when given in written format. Third, children of that age are not always motivated to complete assessments. This is a problem for performance tests, but inadequate responses can also be a problem in personality and other self-descriptive questionnaires. Nevertheless, MA measurement seems to be possible and needed also in early grades, but such results have to be interpreted with care, especially when it comes to individual diagnosis. Luckily, several psychometrically evaluated instruments allowing such measurements are available (cf. Table 2.1). When constructing these instruments, several means were taken to make instruments suitable for the administration to children, such as (a) minimizing number of items, (b) shortening the response scale, and replacing numbers with emotional expressions (e.g., of cartoon characters) depicting increasing worry (see e.g., Krinzinger et al., 2009), and (c) decreasing dependency on reading skills by reading the items aloud to the children (see e.g., Harari et al., 2013).

Conclusions To conclude, in recent years one could observe two opposing trends in MA measurement. First, there was a tendency to develop and use instruments requiring less time to administer, such as, for instance, the AMAS ( Hopko, Mahadevan, et al., 2003) or SIMA (Núñez-Peña et al., 2014). A short administration time and very good psychometric properties are undisputable advantages of these instruments. On the other hand, for obvious reasons such instruments consider very few (if any) MA subcomponents. Therefore, the risk of using them and skipping more multi-factorial instruments is that developments in understanding MA will be biased toward a simplified picture of the construct, which is both theoretically and practically problematic for diagnosis and interventions. The other quite recent trend is the development of instruments allowing the measurement of more fine-grained MA components (e.g., Pletzer et al., 2016) or the postulation of a hierarchical structure of MA ( Yánez-Marquina & Villardón-Gallego, 2017). It seems that both trends might lead to valuable insights into the nature and consequences of MA. The first one seems to be valuable in the context of incorporating MA as a potential measure of interest in a broad range of numerical cognition research. It might facilitate tracing cognitive mechanisms of MA as well as lead to better understanding of elementary number processing itself (see e.g., Georges, Hoffmann, & Schiltz, 2016 for evidence that MA inf luences elementary number processing). Developing such tools will also simplify screening

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students for elevated MA levels. On the other hand, the development of more fine-grained instruments will be particularly helpful for individual diagnosis and intervention planning, especially for individuals who were initially diagnosed with elevated MA level using a more general instrument. Undoubtedly, self-descriptive tools are the most popular means of MA measurement. Recent research has also shown that MA can be effectively measured even at the onset of schooling. Nevertheless, there are also other types of measures, which do not rely on self-report but also have been successfully utilized for MA measurement, and they address some disadvantages of self-descriptive measures.

Behavioral, physiological, and neuroscientific methods Self-reports in personality psychology, like the abovementioned math anxiety questionnaires, have their problems – as shortly outlined below. • •

•

They require functioning self-inspection and can therefore be problematic with children and patient groups. They require good reading skills and thorough linguistic understanding and can therefore be problematic with children and adults who are functionally illiterate. They can be manipulated consciously like in selection situations, but also unconsciously (social desirability effects).

Often self-report measures correlate poorly with other measures, like ratings by another person or physiological and neuroscientific variables. Such poor correlations can be due to error variance in either variable, but most likely, they are due to f laws in both variables. In a more theoretical framing, one could say that there is no perfect method for assessing the underlying construct of MA. Therefore, it is worthwhile to view different methods of assessment as complimentary approaches to uncover as much as possible about MA. For this reason, we wish to give an overview of other behavioral, physiological, and neuroscientific correlates of MA that can potentially serve as MA measures in the future. There were several attempts to measure MA by using behavioral, physiological, and neuroscientific methods. Due to the large variety of these methods, however, there is a lack of systematics in the investigation of MA by these methods, none of them are as well established as the self-report-based questionnaires for MA, and usually they are not considered as a substitute for, but rather a complement to, the use of questionnaires. Regarding behavioral measures, the most prominent are reaction time (RT) and accuracy (ACC) assessments during math tasks or when handling mathrelated material. Interestingly, the relation of speed and accuracy is dependent on the level of math anxiety (Ashcraft & Faust, 1994): individuals with low MA show a fast RT and high ACC, individuals with medium MA show a slow RT (with relatively high ACC), and individuals with high MA show a low ACC

Different ways to measure math anxiety 29

(with relatively fast RT). Future studies should explore whether specific strategies in terms of speed and accuracy trade-off (e.g., much above average speed and much below average accuracy) can be a valid measure of an individual’s MA. Another possible way to measure MA with RT methods are interference effects. For instance, it was shown that individuals with high MA reveal stronger interference effects when they are to inhibit task irrelevant numerical information (Hopko, McNeil, Gleason, & Rabalais, 2002). As MA is separate from math skill, typical math performance tasks cannot be used as a MA measure. However, as it has been shown, the inf luence of MA on math performance depends on certain conditions, for instance, it increases with time pressure (see Ashcraft & Moore, 2009). After establishing that individuals can master some easy problems when tested in a relaxed and untimed condition (Faust, Ashcraft, & Fleck, 1996), introducing time pressure leads to a drop in performance for those with medium or high MA (see Ashcraft & Moore, 2009). For this reason, future studies can explore whether discrepancies between performance in stressful and non-stressful conditions can be considered as a valid measure of the individual MA level. Other interesting approaches might be taken from the field of stress research, since the reaction of math-anxious individuals to math-related situations also includes a stress reaction. Therefore, measures of cortisol secretion (e.g., MattarellaMicke, Mateo, Kozak, Foster, & Beilock, 2011; Pletzer, Wood, Moeller, Nuerk, & Kerschbaum, 2010; Sarkar, Dowker, & Cohen Kadosh, 2014) and postural control (e.g., Doumas, Morsanyi, & Young, 2018) are of use in MA research. Regarding physiological measures, methods from the context of emotion processing have been applied in MA research. In particular, heart rate (measured by electrocardiography) and blood pressure as well as skin conductance (electrodermal activity) and temperature might be associated with increased arousal during a math task caused by math anxiety (e.g., Dew, Galassi, & Galassi, 1984; Hopko, McNeil, et al., 2003; Osborne, 2006; Salvia et al., 2013). However, up-to-date evidence for psychophysiological markers of math anxiety is rather thin and requires additional research (see Chapter 3 for a more extended discussion on this topic). Neuroscientific approaches have included several neuroimaging, neurophysiological, and brain stimulation methods used to investigate the neural correlates of MA (see Artemenko, Daroczy, & Nuerk, 2015 for an overview). By using (functional) magnetic resonance imaging, altered activation patterns before and during math tasks (e.g., Lyons & Beilock, 2012a, 2012b; Pizzie & Kraemer, 2017), altered functional connectivity ( Young, Wu, & Menon, 2012), altered default mode network deactivation ( Pletzer, Kronbichler, Nuerk, & Kerschbaum, 2015), and altered structural correlates ( Hartwright et al., 2018) were detected related to MA. Furthermore, studies using electroencephalography detected differences in event-related potentials (e.g., Klados, Simos, Micheloyannis, Margulies, & Bamidis, 2015; Núñez-Peña & Suárez-Pellicioni, 2014; Suárez-Pellicioni, Núñez-Peña, & Colomé, 2013a, 2013b, 2014) and brain oscillations ( Klados, Pandria, Micheloyannis, Margulies, & Bamidis, 2017) between

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individuals with high and low MA. For instance, the neural signature of standard numerical effects, such as the distance effect and the size effect, were shown to differ due to MA ( Núñez-Peña & Suárez-Pellicioni, 2014). Finally, one single study (Sarkar et al., 2014) aimed at intervening the math performance deficit of individuals with high MA using non-invasive brain stimulation. Thereby, they applied transcranial direct current stimulation over a brain region responsible for emotion regulation and were able to reduce the cortisol level, indicating the stress level, and to improve the math performance of math-anxious individuals (Sarkar et al., 2014 see also Lyons & Beilock, 2012a). Further ongoing studies (e.g., registered report of Soltanlou et al., 2018) investigate the potential of brain stimulation for intervention for MA. Taken together, behavioral, psychophysiological, and neurophysiological methods can be successfully used to detect differences in neural processing between individuals with high and low MA and to provide interventional approaches on a physiological level. However, due to the diversity of the methods, analyses, and results in these studies and the exclusively conducted analyses on a group level, individual diagnosis of MA is (still) out of scope for these measures (Artemenko et al., 2015). Therefore, we recommend more systematic research on the diagnostic usefulness of the physiological markers of MA in the future. It is possible that such measures can provide complimentary evidence to self-descriptive reports.

Challenges and unanswered questions Who is math anxious? One might ask how many people are math anxious. The issue of MA prevalence is vital for curricula development, policy making, and intervention planning. However, there is no consensus on the issue so far. As always in psychological research, we are looking at a continuously distributed variable without a visible gap between two dichotomous groups (as with the binary entailed by many medical diseases). An inspection of results of large scale studies shows that the distributions of MA scores represent a normal rather than a bimodal distribution (e.g., Cipora et al., 2015, Fig. 1; 2017, Fig. 1). For this reason, MA should be considered as a continuous, not a binary trait. One pole of this continuum would represent extreme anxiety towards math. While MA level gets lower along this continuum, it still exists. Sometimes it is even claimed that the majority of the population might encounter some negative feelings related to math (see Chang & Beilock, 2016). As a side note, we can add that it is not agreed on what represents the positive pole – being neutral toward math, or rather positive affective responses toward math and solving math problems (e.g., see Sacks, 1985, especially the chapter “The Twins”). As for all non-dichotomous variables, MA prevalence depends on the criteria adopted – these criteria differ between studies and organizations, and therefore prevalence estimates can also differ greatly.

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MA is not considered in existing medical classifications such as DSM-5 or ICD-10. There are no strict, agreed upon diagnostic criteria for MA. Some researchers provide estimates of MA prevalence based on results of the PISA study (e.g., Morsanyi et al., 2016). For instance, estimates of prevalence based on this data are around 30% for the proportion of individuals who (strongly) agree with the statement that they “get very nervous doing mathematics problems” (see OECD, 2013, p. 99). A bit lower prevalence estimates appeared in early studies: Richardson and Suinn (1972) have suggested that in about 11% of the university student population’s MA is high enough that they would need counseling. Further estimates seem to be similar, however, one must keep in mind that usually all these estimates were at least to some extent based on statistical criteria, such as top x% (e.g., 10% like the ICD often uses) of a population or above n SD s from population mean (see Dowker, Sarkar, & Looi, 2016 for a recent summary). When investigating the relation of MA to math performance, the problem of defining high vs. low MA groups was permanently present in basic research. Ashcraft and Ridley (2005) explain that groups are usually composed based on a median split or selection of extreme groups (e.g., 20%/25% scoring lowest vs. 20%/25% scoring highest) from a pool of initially screened participants. The latter method, assuming sufficient variance in the sample and a reliable MA measurement, is especially adequate for basic research. However, the absolute values of the cut-off criteria depend solely on characteristics of the sample being tested in the particular experiment. For this reason, the results of different experiments can be hardly comparable. In extreme cases, a particular score might be assigned to either the low or high MA group depending on the sample being screened. Furthermore, despite some usefulness in basic research, such a relative approach does not allow for estimating the prevalence of MA. This is because classifying a given percentage of participants as math anxious is an arbitrary decision of the researcher. Additionally, it does not allow for the interpretation of individual scores for diagnostic purposes. Based on the above discussion, one can see that prevalence and individual diagnostic cut-offs of MA are still far from being resolved. As argued by Ashcraft and Ridley (2005), MA meets the criteria for a genuine phobia, so in the long run it might be possible to adapt the diagnostic criteria for a specific phobia for MA diagnosis. Based on such somehow objectivized criteria (e.g., the degree to which MA affects well-being in academic settings, or how much it inf luences one’s life or career choice), one could estimate MA prevalence and allow for individual diagnosis. However, as MA seems to be a continuous trait, individual diagnosis could also be based on normative data ref lecting MA levels in the population. Having established valid and reliable MA measurement instruments, the next step for their practical usefulness is constructing norms. Creating norms for widely used MA instruments would be possible, for instance, by collapsing existing datasets of individual scores, and building norms based on these numerous samples. Using these norms (adjusted to a given linguistic and cultural context) as a reference can turn out to be useful in basic research; researchers can ensure that

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samples are comparable between studies in regard to MA. It will be also helpful for diagnostics and intervention planning. Such efforts have already been made for the AMAS (see e.g., Caviola et al., 2017; Cipora et al., 2017), SEMA ( Wu et al., 2012), and the German version of MAQ ( Krinzinger et al., 2007).

Are we always math anxious? One of the most important distinctions in anxiety research is the differentiation between trait and state anxiety, first introduced by Spielberger (see Endler & Kocovski, 2001 for a discussion). While trait anxiety is considered as a stable disposition to respond with anxiety to a wide range of situations, state anxiety refers to feelings being related to particular circumstances. New developments in the field of MA suggest that this differentiation seems to be valid also for MA. Several studies show discrepancies between state and trait MA ( Bieg, Goetz, Wolter, & Hall, 2015; Roos et al., 2015). This means that MA differs not only between individuals but also between situations. For example, we have recently observed that state anxiety changes depending on the difficulty of arithmetic problems individuals are asked to solve. In sum, the accumulating body of research shows that it is worthwhile to consider both the trait and state aspects related to MA in basic research and for individual diagnosis. At best, a psychometrically evaluated, freely available measurement instrument considering the state aspects of MA needs to be introduced, or at least adapted from state anxiety scales, and evaluated in different cultural and linguistic contexts.

A broader perspective on math anxiety measurement and its practical usefulness As discussed above, MA in principle can be measured reliably and validly in various populations and cultures, and by different kinds of measures, which proves the existence of MA as a construct distinct from poor math skill and general trait anxiety. The link between MA and math performance has been shown in numerous countries and cultures (see Foley et al., 2017 for a review). However, a growing body of evidence indicates that not only MA level but also specific configurations of other traits play a role in the relation between MA and math performance. These observations set new challenges for MA measurement because an elevated MA level might have different consequences in different individuals. In a study conducted in Poland with a group of adult university students of different faculties, Cipora and colleagues (2015) found a correlation between MA and self-assessed math skill, school math grades, and affinity for math and sciences. MA further correlated with persistence in attempting to solve math problems and being likely to use non-allowed help when struggling with math problems. MA also correlated with trait and state anxiety. The size of these correlations was similar to the correlations reported in other published studies. However, when the sample was split depending on field of study into two groups

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studying math-related (e.g., math, physics, economics, engineering) or mathunrelated subjects (e.g., literature, pedagogy), it turned out that the pattern of correlations differed considerably between groups. Specifically, MA correlated with self-assessed math skill, math grades, persistence in solving math problems, and an affinity for math only for individuals who studied math-unrelated subjects, while such a relation was not present for students of math-related subjects. This was not due to f loor or ceiling effects in any of the variables. Importantly, in the math-related group MA correlated with state and trait anxiety. Similar results were found in two follow-up studies conducted in our lab. When interpreting these results, one must keep in mind that field of study selection can be inf luenced not only by skills but also interests, attitudes, and potentially MA. For instance, one can observe that individuals characterized by relatively high MA, who nonetheless chose to study math-related subjects, do not perform worse than those characterized with lower MA. For this reason, such a difference in correlation patterns requires a more thorough investigation. In recent years, several studies have suggested that the correlation between MA and math performance can be moderated by several constructs. We will brief ly describe some of them: 1

2

3

Self-math overlap is the extent to which one incorporates math into one’s self-structure (Necka, Sokolowski, & Lyons, 2015). Across participants who declared high self-math overlap, the correlation between MA and math performance was smaller than in those who characterized themselves with low overlap. Math motivation ( Wang et al., 2015). A uniform linear relationship between MA and math performance was only observed in individuals characterized by low intrinsic math motivation. On the other hand, for those characterized by high intrinsic motivation there was an inverted U-relation between MA and math performance: math performance was highest in moderately anxious individuals. Math self-concept ( Justicia-Galiano, Martín-Puga, Linares, & Pelegrina, 2017). The relationship between MA and math performance in children was mediated by working memory capacity and math self-concept.

Another line of evidence showing that the relationship between MA and math performance is not unitary across individuals comes from studies utilizing a method called Latent Profile Analysis (LPA). It allows detection of the configurations of multiple traits (see Hickendorff, Edelsbrunner, McMullen, Schneider, & Trezise, 2018 for examples of use in differential psychology). LPA allows to look at individual differences not only in terms of traits but also in terms of specific configurations. Below we list some of these findings: 1

Carey, Devine, Hill, and Szűcs (2017) differentiated four different latent profiles depending on MA, test anxiety, and general anxiety. The link between MA and math performance was more pronounced in individuals

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characterized by the so-called academic anxiety profile (elevated MA and test anxiety), than in individuals characterized by the so-called high anxiety profile (elevated all anxiety types). Hart et al. (2015) differentiated five latent classes based on configurations of MA, and different measures of math performance. None of the groups were found to be above average both in math performance and MA. Mammarella, Donolato, Caviola, and Giofrè (2018) found different anxiety profiles in grades 3–6. However, contrary to the findings from Carey et al. (2017), the profiles were not characterized by dissociations between levels of different anxiety types. Results from the study by Trezise and Reeve (2018) indicated stability in currently perceived anxiety depending on task difficulty as one of the distinctive features of profiles they differentiated. Interestingly, only two out of five profiles (very high and very low MA) were consistent across types of tasks. Three other profiles revealed f luctuations in MA depending on task difficulty, way of presenting a problem, and presence of time pressure. Wang, Shakeshaft, Schofield, and Malanchini (2018) showed that individuals of the same MA level but differing in certain aspects of math motivation can vary considerably in their math performance.

The studies discussed above provide evidence that adequate MA measurement requires looking at more than MA levels. To obtain sufficient prognostic validity (i.e., being able to predict math performance based on MA level) one needs to consider MA in the context of other individual characteristics. A full list of these characteristics still needs to be established. It must be noted that all of these developments are relatively recent and some work still needs to be done to translate them into practice and to implement them into commonly used measurement instruments. Potentially, a battery of tests measuring different anxiety types and using a standardized scale could be developed. Using such a unified battery would allow for calculation of critical differences such that one could check whether standard scores differ significantly between tests (see Cipora et al., 2017 for an example). Such a differential profile could potentially serve as a proxy for individual anxiety profiles and could be used for diagnostic purposes, as well as for tailoring interventions. These results also suggest that when designing interventions, one should also target potential mediators of the MA–math performance relationship. However, the feasibility and validity of such approaches needs further investigation. Nevertheless, we consider the development of a diagnosis of MA that would also consider other relevant constructs as one of the most urgent challenges of MA measurement. This entirely new avenue of research needs to take developmental aspects into consideration because in early school years MA and general math anxiety are less differentiated (or at least can be assessed as such) and the added value of using latent profiles seems to be limited in this age group (Mammarella et al., 2018). Furthermore, recent results suggest that considering

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different anxiety profiles can also inform studies on the state-trait distinction of MA ( Trezise & Reeve, 2018).

Concluding remarks This chapter provides an overview of MA measurement techniques, spanning traditional self-descriptive questionnaires to their computerized implementations, as well as behavioral and (neuro)physiological measures. The literature review shows that all of these methods can be used to reliably and validly measure MA in most populations. In other words, it suggests that a reliable and valid indicator of general MA is relatively simple (e.g., by directly asking a single question on whether one is afraid of math). However, such an indicator falls short of capturing the heterogeneity of the MA construct and is not useful for differential individual diagnosis or tailored adaptive intervention planning. For instance, if a child has only math test anxiety, but no math anxiety in everyday math situations, it would probably be a waste of resources to tackle everyday math situations. What still needs to be done, both to improve the understanding of MA and to implement these developments in practice, is to make the research instruments, especially freely available ones, suited to individual diagnostic purposes by establishing norms. Furthermore, tracing the temporal dynamics of MA and considering situational factors can be insightful for both basic research and individual diagnostics. Finally, one of the biggest imminent challenges is an adequate MA measurement in a context where other constructs might inf luence its relationship with math skill.

Acknowledgments KC was supported by a DFG grant [NU 265/3–1] to HCN. KC, CA, and HCN are members of the LEAD Graduate School & Research Network [GSC1028], which is funded within the framework of the Excellence Initiative of the German federal and state governments. We thank Zoë Kirste for language proofreading.

Note 1 Note that another questionnaire with the same name was developed by Thomas and Dowker (2000).

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Trezise, K., & Reeve, R. A. (2018, February). Patterns of anxiety in algebraic problem solving: A three-step latent variable analysis. Learning and Individual Differences, 66, 78–91. https://doi.org/10.1016/j.lindif.2018.02.007 Vahedi, S., & Farrokhi, F. (2011). A confirmatory factor analysis of the structure of abbreviated math anxiety scale. Iranian Journal of Psychiatry, 6(2), 47–53. Wang, Z., Lukowski, S. L., Hart, S. A., Lyons, I. M., Thompson, L. A., Kovas, Y., . . . Petrill, S. A. (2015). Is math anxiety always bad for math learning? The role of math motivation. Psychological Science, 26(12), 1863–1876. https://doi.org/10.1177/095679 7615602471 Wang, Z., Shakeshaft, N., Schofield, K., & Malanchini, M. (2018). Anxiety is not enough to drive me away: A latent profile analysis on math anxiety and math motivation. PLoS One, 13(2), 1–16. https://doi.org/10.1371/journal.pone.0192072 Wigfield, A., & Meece, J. L. (1988). Math anxiety in elementary and secondary school students. Journal of Educational Psychology, 80 (2), 210–216. https://doi.org/10.1037/ 0022-0663.80.2.210 Wu, S. S., Barth, M., Amin, H., Malcarne, V., & Menon, V. (2012, June). Math anxiety in second and third graders and its relation to mathematics achievement. Frontiers in Psychology, 3, 1–11. https://doi.org/10.3389/fpsyg.2012.00162 Yánez-Marquina, L., & Villardón-Gallego, L. (2017). Math anxiety, a hierarchical construct: Development and validation of the Scale for Assessing Math Anxiety in Secondary education. Ansiedad y Estrés, 23, 59–65. https://doi.org/10.1016/j. anyes.2017.10.001 Young, C. B., Wu, S. S., & Menon, V. (2012). The neurodevelopmental basis of math anxiety. Psychological Science, 23(5), 492–501. https://doi.org/10.1177/0956797611429134

3 PSYCHOPHYSIOLOGICAL CORRELATES OF MATHEMATICS ANXIETY Chiara Avancini and Dénes Szűcs

Introduction Anxiety is an emotional state that occurs when an individual faces a threat to self-preservation ( Johnson-Laird & Oatley, 1989). In the specific case of maths anxiety (MA), it refers to a debilitating negative emotional reaction to mathematical tasks, which may occur in children and adults (Ashcraft, 2002) MA is typically measured by self-report questionnaires in which individuals rate their agreement or disagreement on a series of sentences referring to everyday situations involving maths. While these questionnaires have satisfactory psychometric properties (Alexander & Martray, 1989; Carey, Hill, Devine, & Szűcs, 2017; Hopko, Mahadevan, Bare, & Hunt, 2003a; Plake & Parker, 1982; Richardson & Suinn, 1972), they are subject to general limitations regarding explicit self-report measures of anxiety. First, questionnaires assume that individuals are honest in their answers. This might be problematic especially in studies investigating gender differences. For example, the stereotype that boys are better at maths than girls (Spencer, Steele, & Quinn, 1999) might result in girls being more willing to report difficulties in maths-related tasks than boys. Unfortunately, the tools available do not provide any scale enabling questionnaire invalidation where biased answers are detected. Second, self-report measures require a certain level of metacognition that might differ between ages, genders and abilities. Third, scores are not easily comparable between culturally and linguistically different samples. One way to overcome these limitations would be to use implicit measures of MA which do not rely on self-report but rather automatized reactions. A characteristic of anxiety is to cause the body to activate cognitive processes and induce physiological arousal that will allow the prompt detection of danger and a rapid behavioural response that is evolutionary advantageous ( Eysenck, Derakshan, Santos, & Calvo, 2007; Marks & Nesse, 1994). Therefore, understanding the physiological response in individuals who experience anxiety when faced

Psychophysiological correlates of mathematics anxiety 43

with maths would provide an insight into those implicit measures that may help overcome the limitations of self-reports. The aim of the present chapter is to review studies that have investigated implicit measures of MA. In particular, we will review the literature that has used autonomic responses, electroencephalography and functional magnetic resonance to assess automatic reactions of maths-anxious individuals when exposed to maths.

Autonomic measures and cortisol levels Autonomic measures and cortisol Symptoms of physiological arousal caused by anxious states are thought to ref lect activity of the autonomic nervous system (ANS) ( Thayer, Friedman, & Borkovec, 1996). The ANS can be divided into the sympathetic and parasympathetic branches that, from the brain stem and through the spinal cord, innervate the cardiovascular and visceral systems ( Figure 3.1). Such coordinated activity prepares the body to respond to threat in emergencies or stressful situations ( Jänig & McLachlan, 1992). Anxiety is characterized by sympathetic activation and parasympathetic deactivation which cause a series of physiological reactions including increase in heart rate (HR), increase in blood pressure, decrease in finger temperature, increase in respiratory rate and oxygen consumption, and increase in the variation of the electrical properties of the skin in response to sweat secretion (skin conductance) ( Benedek & Kaernbach, 2010; Kreibig, 2010). A consequence of the activation of the sympathetic nervous system during a stressful situation is the activation of the hypothalamus-pituitary-adrenal (HPA) axis which results in an increase of cortisol concentration in the blood and in the saliva (Gaab, Rohleder, Nater, & Ehlert, 2005; Sapolsky, Romero, & Munck, 2000).

Studies that used autonomic measures and cortisol as implicit measures of MA The literature available on HR is limited to a handful of studies which overall point to the suggestion that HR (calculated as beats per minute) is mostly uncorrelated to self-report measures and other implicit measures of MA. The first study that used HR to investigate MA was Dew, Galassi, and Galassi (1984). The study focused on whether in test-like situations MA interfered with performance. They assessed the relationship between MA and performance in relation to physiological arousal and avoidance behaviour. Self-report data were collected by asking participants to complete the following questionnaires: the Mathematics Anxiety Rating Scale (MARS, Richardson & Suinn, 1972), the Mathematics Anxiety Scale (MAS; Fennema & Sherman, 1976), and the Anxiety Toward Mathematics Scale (ATMS; Sandman, 1979). During the testing, the participants solved 20 arithmetic computation problems and 15 words problems with no time limits. Subsequently, a third set of problems was administered under test-like

Stressor

Cortex

Limbic system

HPA axis

Brain stem

Cranial nerves Periphereal nerves

Sympathetic activation

Parasympathetic deactivation

Stress response

Stress response

Cortisol concentration

Cardiovascular Respiratory Electrodermal

Autonomic response to stress. In the presence of a stressor, information is first processed by cortical areas and then transmitted to subcortical structures. Through the activation of the hypothalamic-pituary-adrenal (HPA) axis, neurohormones induce the production of cortisol by the adrenal glands. The stress reaction is also transmitted through the brain stem and then cranial and peripheral nerves which innervate internal organs. The coordination of the sympathetic and parasympathetic divisions of the autonomic nervous system modulate cardiovascular, respiratory and electrodermal stress responses.

FIGURE 3.1

Psychophysiological correlates of mathematics anxiety 45

conditions: the need for speed and accuracy was emphasized, the task was timed, and it was communicated that the aim was to assess maths ability. The problems in the third set were most likely arithmetic problems, although no clear description of the task is given in the paper. HR was uncorrelated to most self-report measures with the exception of a negative correlation with the ATMS. Such correlation is isolated and not supported by further research. Two other studies have found little sensitivity of HR to MA levels. In a study (Hopko et al., 2003b) in which anxiety was induced by means of carbon dioxide inhalation (CO2), HR increased as a result of the experimental manipulation but there was no difference in physiological response between high maths-anxious individuals (HMAs) and low maths-anxious individuals (LMAs). The study suggests that the manipulation of anxiety by CO2 administration does not inf luence HR in HMAs more than non-clinical participants. It is however also possible that HMAs do not specifically respond to inhalation of CO2 , at least no more than any other population. Finally, Hopko, Hunt, & Armento (2005) found that differences in MA selfratings to the Abbreviated Maths Anxiety Scale (AMAS, Hopko et al., 2003a) did not correspond to differences in HR modulation when participants were asked to perform attentional tasks. While these attentional tasks mostly included tasks unrelated to performing maths (such as the Stroop task1), they did include the PASAT-C ( Lejuez, Kahler, & Brown, 2003) which requires one to perform mental additions. Again, HR after the completion of the PASAT-C was not modulated by pre-experimental levels of MA. Similarly to HR results, little evidence shows that skin conductance is sensitive to MA levels. In the experiment of Dew et al. (1984) described above, data on skin conductance was obtained for two indices: skin conductance fluctuations and skin conductance levels. The first reflected the rate of spontaneous voltage fluctuations per minute which occur as a result of the activity of the sudorimotor nerve. Increased nerve activity increases the number of sweat glands opening. Because the activity of the sudorimotor nerve is thought to ref lect sympathetic arousal, f luctuations of skin conductance may ref lect anxious responding ( Bach, Friston, & Dolan, 2010). The second index, was obtained by comparing the value of post-test skin conductance to a pre-test baseline value. Higher levels of skin conductance post-test would ref lect stronger anxious responding during the testing. Only skin conductance levels correlated with scores to the MARS and the ATMS, suggesting that participants with high MA have higher skin conductance during testing than during a baseline period. While these results suggest the sensitivity of skin conductance levels to maths anxious responding, no other study supported these findings. For example, a group of participants that had anxiety induced by the inhalation of CO2 showed increased skin conductance levels compared to the control group. However, no differences were found between HMAs and LMAs (Hopko et al., 2003b). Furthermore, pre-experimental MA levels assessed with the AMAS did not modulate the skin conductance response during attentional tasks (Hopko et al., 2005). A third physiological measure that has been traditionally used to assess anxious arousal and that researchers have tried to study as a potential implicit measure

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of MA, is the concentration of salivary cortisol. The comparison between the concentration in samples collected before and after a stressful event provide a measure of anxious arousal. Salivary cortisol has been found to be associated with maths performance as a function of MA and working memory (WM) capacity (Mattarella-Micke, Mateo, Kozak, Foster, & Beilock, 2011). WM is a brain system that allows one to temporary store and manipulate information used to carry out complex cognitive tasks and is typically thought to have limited capacity ( Baddeley, 1992; Cowan, 2010). Hence, if anxiety has a disruptive effect on WM, those individuals who interpret their physiological response as mathsrelated distress would perform worse in tasks that rely on WM. In the study of Mattarella-Micke et al. (2011), participants judged the correctness of arithmetic problems in the form “x ≡ (y mod z)”. Problems had two levels of difficulty: high demand and low demand. The first type included a carry and could not be solved with simple heuristics, while the latter included no carry and could be solved with simple heuristics. Then, WM capacity was assessed with the automated Reading Span task (RSPAN; Conway, Kane, & Al, 2005) which requires one to read a series of sentences followed by a letter and to judge whether the sentences make sense or not. After two to five sentences, the participants were asked to recall as many letters at the end of the sentences as possible. Scores were calculated on the total numbers of letters recalled. Finally, MA was assessed with the shortened MARS (sMARS; Alexander & Martray, 1989). Cortisol samples were taken prior the testing and after the arithmetic task. With increasing levels of cortisol, maths performance decreased when high-WM participants were also HMAs, while performance increased when they were also LMAs. On the other hand, maths performance did not vary as a function of cortisol and MA in low-WM participants. This suggests that the effect of physiological arousal on maths performance depends on the interacting effects of MA and WM capacity and on the difficulty of the task. Such relationships emerged when the operations presented were high demand and not when the operations were low demand. The study provides an indication of the potential of cortisol as a measure of anxiety in maths-related context when WM and task difficulty are also taken into consideration. The association between salivary cortisol levels and maths performance as a function of MA was also reported by Pletzer, Wood, Moeller, Nuerk, and Kerschbaum (2010). They tested the association between cortisol concentration levels and MA during statistics examinations. In particular, participants were categorized into cortisol profiles according to the variation of cortisol levels pre and post examination compared to a baseline. Only participants whose cortisol levels increased pre-examination and decreased post-examination showed a negative correlation between MA levels (assessed with the MARS30-brief; Suinn & Winston, 2003) and maths performance. The authors argued that examinationinduced cortisol responses facilitate the association between MA and maths performance. Both the results of Pletzer et al. (2010) and Mattarella-Micke et al. (2011) suggest that the interaction between MA as assessed by self-report measures, salivary cortisol and performance depends on the interaction between these factors rather than their individual contribution.

Psychophysiological correlates of mathematics anxiety 47

Cortisol has also been found to be sensitive to MA levels when the performance of maths is associated with emotional processing (Sarkar, Dowker, & Kadosh, 2014). Sarkar et al. (2014) looked at reductions of cortisol levels in a study in which the dorsolateral prefrontal cortex (dlPFC), area associated with emotional processing, was stimulated in order to inhibit negative and facilitate positive emotions by means of transcranial direct stimulation. A dissociation between HMAs and LMAs was found: in HMAs, reductions in cortisol levels (calculated by subtracting prestimulation from poststimulation levels and then dividing by prestimulation levels) occurred only during real stimulation, while for the LMAs greater reductions in cortisol levels occurred only during sham stimulation. To date, findings on HR and skin conductance point toward a lack of sensitivity of these measures to MA levels as assessed by self-report measures. On the other hand, literature examining cortisol responses seem more promising. In particular, cortisol seems to play a role in the relationship between MA and maths performance ( Table 3.1). However, the mechanisms through which

TABLE 3.1 Indices and results of studies investigating autonomic reactions to MA

Measure Heart Rate

Study

Dew et al. (1984) Hopko et al. (2003b) Hopko et al. (2005) Skin Dew et al. Conductance (1984)

Salivary Cortisol

Index

Results

Beats per min Beats per min

Inverse correlation with ATMS scores in test-like situations Not sensitive to the anxiety group

Beats per min

No effects

Skin conductance Positive correlation with MARS levels: poststimulus and ATMS scores during test-like – prestimulus situations Hopko et al. Not specified Not sensitive to the anxiety group (2003b) Hopko et al. Mean skin No effects (2005) conductance levels Mattarella- Post-task cortisol In HMAs and high-working Micke et concentration memory individuals, negative al. (2011) correlation between cortisol levels and performance Correlation between MA and Pletzer et al. Cortisol changes: performance depends on (2010) Pre-examination individual cortisol profile levels – baseline; Post-examination levels – baseline. Sarkar et al. Cortisol changes: In HMAs cortisol levels reduced (2014) (Poststimulation – after stimulation of the dlPFC prestimulation)/ In LMAs cortisol levels reduced after sham stimulation of the dlPFC prestimulation

Note: ATMS = Anxiety Toward Mathematics Scale; MARS =Mathematics Anxiety Rating Scale; HMA= high mathematics anxiety; dlPFC = dorsolateral prefrontal cortex

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cortisol, MA and performance interact are still unclear, as the literature is still scarce and mostly correlational.

Electrophysiology Electroencephalography Electroencephalography (EEG) is a technique that records variations in the electrical activity (electrical potentials) produced by groups of neurons firing together. When the morphology and alignment of a certain number of cortical neurons allows for the summation of their activity, electrical potentials can be recorded with scalp electrodes ( Bressler, Ding, Bressler, & Ding, 2006). Because such activity is very fast, scalp EEG records electrical changes in the scale of milliseconds, allowing for great time resolution. On the other hand, a single electrode records the summed activity of millions of neurons, resulting in low spatial resolution. This issue can be partially overcome by using a large number of recording electrodes that may allow to roughly approximate from what area of the cortex the signal is produced. The electrical oscillations produced by the neurons in the cerebral cortex are continuous and variations in electrical potentials time-locked to the presentation of a stimulus are called event-related brain potentials (ERPs). ERPs are obtained by averaging the electrical changes produced by consistent number stimuli that are hypothesized to elicit the cognitive functions of interest. Because variations that are time-locked to an event will happen at a specific time after the presentation of the stimulus and will have the same polarity (either positive or negative) after each stimulus presentation, averaging such signals will produce an ERP wave. ERP waves are time-locked to a stimulus, and they are characterized by positive or negative polarity, an amplitude range (usually between 0.5 and 20 μV) and peaking at a specific latency after stimulus presentation. On the other hand, activity that is not specifically elicited by the stimuli will not be time-locked to them and their average will minimize their signal, making it negligible. Different components have been attributed to different cognitive processes. In the next section of the chapter we will review studies that used ERPs as measures of MA. When a new ERP component is mentioned, the cognitive function to which it has been attributed will also be explained.

Studies that used event-related brain potentials as implicit measures of MA The P2 component is a positive def lection peaking at around 200ms after stimulus onset. Several studies have found it to differ between levels of MA. For example, is was modulated by MA in a study investigating the precision of magnitude representation ( Núñez-Peña & Suárez-Pellicioni, 2014). In a number comparison task ( Núñez-Peña & Suárez-Pellicioni, 2014), the P2 amplitude was larger in

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HMAs than in LMAs when comparing trials in which the numerical difference between the stimuli was small to trials in which the difference was large. Similarly, it was larger in HMAs than in LMAs when comparing trials with small numbers to trials with large numbers. According to well-established research in maths cognition, the processing of arithmetic problems with large solutions is harder than the processing of problems with small solutions (Groen & Parkman, 1972) because large numbers are represented more vaguely and therefore are more difficult to discriminate ( Restle, 1970). Furthermore, the comparison between numbers that are numerically close to each other is harder than the comparison of number further apart because the representations of close numbers overlap (Moyer & Landauer, 1967). Both these effects have been attributed to the ability of accessing magnitude representations. The P2 had previously been identified as a measure of effort to distinguish two overlapping number representations as it had been shown that its amplitude is greater when the difference between two numbers is smaller (Szűcs & Soltész, 2008; Temple & Posner, 1998). Hence, the increased P2 amplitude in the HMAs group may ref lect less precise magnitude representation which may result in recruitment of more cognitive resources in order to perform the task. Larger prefrontal P2 amplitude in HMAs compared to LMAs has also been recorded in a multi-digit verification task (Suárez-Pellicioni, Núñez-Peña, & Colomé, 2015). The prefrontal P2 amplitude has been found to be greater after the presentation of emotionally negative stimuli, and it has been suggested to be a correlate of the employment of attentional resources toward negative stimuli (Carretié, Hinojosa, Martín-Loeches, Mercado, & Tapia, 2004; Carretié, Mercado, Tapia, & Hinojosa, 2001). The authors argued that enhanced P2 amplitude in HMAs may ref lect the mobilization of attentional resources toward difficult arithmetic operations which are attributed negative emotional value. While increased P2 amplitudes have suggested that HMAs allocate more attentional resources when performing tasks involving maths, contrasting results were reported by Klados, Simos, Micheloyannis, Margulies, and Bamidis (2015) in a study using a verification task. While the authors did not specifically refer to the P2 components, the frontocentral positive amplitude around 180–220ms poststimulus was bigger for LMAs than HMAs. Because a similar effect of MA was found in a control attentional task, the authors suggested that there was more cortical activation in LMAs than in HMAs and this effect seemed to be linked to general WM skills. As noted, while other studies have suggested that the MA effect on the P2 may show that HMAs allocate more attentional resources toward maths stimuli ( Núñez-Peña & Suárez-Pellicioni, 2014; Suárez-Pellicioni et al., 2015), Klados et al. (2015) suggest instead that LMAs are spontaneously allocating more resources than HMAs when performing maths. It is still necessary to clarify whether these conf licting results are simply inconsistent or whether they ref lect different processes, such as resources availability versus abnormal resources allocation.

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Another component that may be sensitive to MA levels is the P3b, which is a positive def lection peaking around 300ms after stimulus presentation with temporal/parietal topography (Polich, 2007; Verleger, Heide, Butt, & Kömpf, 1994). It is considered to be sensitive to attentional resources allocation and memory processing ( Polich, 2007). For example, in demanding tasks that require more attentional resources or with high memory load, the P3b has smaller amplitude and later latency than in undemanding tasks ( Bailey, Mlynarczyk, & West, 2016; Kok, 2001). In an arithmetic verification task, HMAs had longer P3b latencies during the processing of large-split solutions at parietal sites. Furthermore, a positive correlation was found between latency and the sMARS scores for largesplit solutions. Because P3b latency, as well as longer RTs, is thought to ref lect stimulus evaluation time ( Polich, 2007) the study suggests that higher MA is ref lected in a higher need of engaging attentional resources in order to perform arithmetic (Suárez-Pellicioni, Núñez-Peña, & Colomé, 2013a). Similarly to the P2, a positive def lection between 380ms and 420ms was found to be more positive in HMAs than in LMAs ( Klados et al., 2015), which is in contrast with SuárezPellicioni et al.’s (2013a) findings. However, the frontocentral topography of such positive def lection compared to the parietal topography of the P3b suggests that the two components may represent different processes. Other components have been studied as possible correlates of MA. However, such components have been investigated by single studies and therefore only provide anecdotal evidence. Macarena Suárez-Pellicioni, Núñez-Peña, and Colomé (2013b) looked at the error-related negativity (ERN) and the correctresponse negativity (CRN) to assess whether HMAs showed abnormal error monitoring during the performance of an arithmetic task. The ERN and CRN were selected as components of interest as they have been found to ref lect error monitoring. The ERN is a frontocentral negative component peaking around 50–100ms after an erroneous response has been given ( Falkenstein, Hohnsbein, Hoormann, & Blanke, 1991; Gehring, Goss, Coles, Meyer, & Donchin, 1993; Yeung, Botvinick, & Cohen, 2004). It has been observed that subjects diagnosed with anxiety disorders show bigger ERN ( Luu, Collins, & Tucker, 2000), suggesting that increased subjective sensitivity to errors produce enhanced ERNs ( Vidal, Hasbroucq, Grapperon, & Bonnet, 2000). The CRN is a similar negativity to the ERN but elicited after correct responses (Cordes, Gelman, Gallistel, & Whalen, 2001). Studies on other anxiety disorders such as obsessive-compulsive disorder have reported that abnormal error monitoring in anxious participants was not ref lected in behavioural measures, but rather in electrophysiological data ( Endrass et al., 2010; Gehring, Himle, & Nisenson, 2000; Hajcak, McDonald, & Simons, 2004). Hence, MA effect on the ERN and the CRN would ref lect abnormal error monitoring regardless of any compensation that my result in no behavioural effect. In a numerical Stroop task and a classical Stroop task in which the physical characteristics of the stimuli could match or mismatch the semantic information of the stimuli (e.g. deciding which one between 4 and 2 is the bigger

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number in the numerical Stroop task; naming the colour of the word “red” when written in green ink in a classical Stroop task), HMAs showed enhanced ERNs in the numerical compared to the classical Stroop task. Furthermore, to assess the difference in processing errors and corrected answers, a difference wave was obtained by subtracting the CRN from the ERN. The MA groups differed in the numerical Stroop task only. Finally, the higher the self-reported levels of MA with the sMARS, the more negative the ERN and the bigger the difference wave amplitude. Hence, the ERN and CRN may suggest that HMAs show abnormal error monitoring that is specific to numerical stimuli. The P1, the N450 and the conf lict sustained potential (Conf lict-SP) were investigated to look at whether MA yields abnormal cognitive control when conf licting information is presented as a result of poor attention inhibition (Suárez-Pellicioni, Núñez-Peña, & Colomé, 2014). Indeed, these components were selected as they are sensitive to conf lict processing. The P1 is a positive wave peaking around 100ms after stimulus onset. It is thought to be a correlate of the processing of low-level stimuli features (Zhu, Zhang, Wu, Luo, & Luo, 2010). The N450 is a frontocentral negativity peaking around 450ms after stimulus onset. It is considered to represent the detection of conf licting stimulus information (Szűcs & Soltész, 2012). For example, it has been found to be elicited in the incongruent condition in a numerical Stroop task (Szűcs, Soltész, & White, 2009). The Conf lict-SP is a positive wave following the N450 and it has been found to be modulated by conf lict level ( Lansbergen, van Hell, & Kenemans, 2007 ). In a numerical Stroop task, these components of interest were analysed by grouping the trials according to whether the previous trial was congruent or not. The N450 and the Conf lict-SP, but not the P1, differed between HMAs and LMAs (groups selected using the sMARS), showing that the interference of a previous incongruent trial impacts HMAs more than LMAs participants. Several ERP components have been investigated in relation to MA. However, at the time of writing this book chapter, the literature is still too limited to allow for any satisfactory conclusion on electrophysiological correlates of MA. Overall, the positivity peaking around 200 ms after stimulus onset seem to be the components that most consistently showed a difference between HMAs and LMAs ( Klados et al., 2015; Núñez-Peña & Suárez-Pellicioni, 2014; NúñezPeña & Suárez-Pellicioni, 2015; Suárez-Pellicioni et al., 2013a). This component is thought to ref lect attentional resources allocations and memory processing, possibly inf luenced by emotional value of the stimulus (Carretié et al., 2004, 2001). This might suggest that MA bears an impact on attentional and memory resources. Furthermore, the positivities around 200 and 400ms found in Klados et al. (2015) have been associated to the relation between MA and memory as well. It is however still to be clarified whether such positivities are generated by the same cerebral structures and induced by the same cognitive functions as those found in other studies.

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Finally, it should be noted that comparisons between groups have been done on the basis of groups that had been previously selected with self-report measures. Therefore, the presence or absence of electrophysiological differences between HMAs and LMAs might be related to covert characteristics of self-report questionnaires. The limited literature, the variety of components investigated and the dependency of electrophysiological results to self-report questionnaires calls for replication of the available findings and further research on the topic.

fMRI studies fMRI Magnetic resonance imaging (MRI) exploits different tissues’ ability to absorb electromagnetic energy to obtain images of those tissues. MRI scanners use changing magnetic gradients and oscillating magnetic fields to produce energy that is absorbed by hydrogen nuclei found in the water molecules present in the human tissues. Put in very simple terms, the magnetic field produced by the MRI scanner causes the protons of the hydrogen nuclei to align parallel to the direction of the magnetic field, when the protons are in a state of low energy. Then, an excitation pulse is applied, which causes certain protons to invert their orientation, absorbing energy. When the excitation pulse stops, the protons return to their initial state of low energy, parallel to the magnetic field vector. During this transition, protons release the absorbed energy, which is detected by the scanner. The amount of energy released depends on the numbers of hydrogen nuclei. Because different human tissues have different quantities of hydrogen nuclei, the energy released by different tissues allow for discrimination between them. Moreover, in the MRI scanner the strength of the magnetic field is spatially controlled, and hence different locations in space contribute differently to the signal. Therefore, the final signal detected by the MRI scanner not only can discriminate between different tissues but also carries spatial information, allowing us to obtain a precise image of the tissues scanned. The smallest unit in the reconstructed 3D image is called a voxel.2 Psychologists and cognitive neuroscientists are, however, interested not only in obtaining a structural image of the brain but also in localizing the neural origins of cognitive functions. Functional MRI (fMRI) attempts to identify what brain area is activated when a certain cognitive function is employed. The main concept behind fMRI is that the activation of brain areas require supplies of oxygen provided by the haemoglobin in the blood f low. The consumption of oxygen during neural activity causes the outgoing blood f low to deoxygenate. Deoxygenated blood has greater magnetic susceptibility than oxygenated blood ( Pauling & Coryell, 1936). Hence, the contrast between oxygenated and deoxygenated blood can be detected by the scanner and it is called blood-oxygenationlevel dependent (BOLD) contrast. The BOLD contrast can therefore be used as a measure of brain activation (Ogawa, Lee, Kay, & Tank, 1990).

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It is important to stress that fMRI is an indirect measure of brain activation. Indeed, it does not detect neuronal activity, but rather metabolic activity. Changes in blood flow are slower than electrical impulses. Therefore, while fMRI is an imaging technique with great spatial resolution, its temporal resolution is low.

Studies using fMRI to investigate MA fMRI studies have mainly focused on investigating whether HMAs and LMAs differed in the activation of brain areas associated with emotional processing and the anxious anticipation of maths. In anticipating the execution of a maths task, HMAs seem to activate brain areas linked to cognitive control, emotional processing and pain processing. When HMA participants (selected with the sMARS) anticipated an arithmetic verification task, better performance was associated with more activation of bilateral inferior frontal junction and the bilateral inferior parietal lobe ( Lyons & Beilock, 2012a). These areas are thought to be involved in cognitive control ( Brass, Derrfuss, Forstmann, & von Cramon, 2005) and regulation of emotions ( Bishop, 2007). Therefore, it is suggested that for HMAs better performance is associated with increased recruitment of areas linked to cognitive control and emotion regulation when anticipating a maths task. Moreover, the activation of the pain network such as the bilateral dorsoposterior insula and the mid-cingulate cortex before task execution was found to be positively correlated to sMARS scores in HMAs. The authors argued that anticipatory anxiety of maths simulates pain in HMA individuals ( Lyons & Beilock, 2012b). Activity of the default-mode network (DMN) has also been found to differ between levels of MA ( Pletzer, Kronbichler, Nuerk, & Kerschbaum, 2015). Decrease in deactivation of the areas of the DMN has been linked to emotional processing and it has been found to be proportional to the increase of cognitive control (Greicius, Krasnow, Reiss, & Menon, 2003). In the context of MA, differences due to MA levels (assessed using the MARS30-brief; Suinn & Winston, 2003) emerged in the precuneus during a number comparison task and in the anterior cingulate gyrus during a number bisection task. In the number comparison task, trials could either be unit-decade compatible where the smaller numbers also had the smaller unit digit (e.g. 23 vs. 68) or unit-decade incompatible where the smaller number had the bigger unit digits (e.g. 28 vs. 63). LMAs activated the left inferior frontal gyrus and insula, the left dorsolateral prefrontal cortex, and the supplementary motor area more during incompatible items than compatible items, while HMAs did not. The study suggests that MA is associated with less deactivation of the DMN when inhibitory control is required, such as with incompatible items in arithmetic tasks, highlighting poorer cognitive control in HMAs. The activity of areas supervising emotion regulation have been found to be modulated by MA in children as well. Young, Wu, and Menon (2012) tested

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whether children with high MA showed hyperactive amygdala activity during maths problem-solving and stronger connectivity between the amygdala and prefrontal cortex. These regions and structures were chosen because they have been found to be activated during the view of negative stimuli (Sabatinelli et al., 2011) and therefore enabled investigation into whether numeric stimuli were perceived as negative by HMAs in developmental age. Children aged 7–9 were administered the Scale for Early Mathematics Anxiety ( Wu, Amin, Barth, & Malcarne, 2012). Participants took part to two fMRI testing sessions in which they had to perform an arithmetic verification task. Activation in the right amygdala and anterior hippocampus was stronger for HMAs than LMAs. On the contrary, HMAs showed less activation in the intraparietal sulcus, right dlPFC, and the bilateral caudate and putamen nuclei of the basal ganglia. Moreover, greater deactivation of the ventromedial prefrontal cortex was seen in HMAs. Finally, the authors used connectivity analysis aimed at identifying whether HMAs and LMAs differed in how the amygdala interacts with other brain regions. Compared to LMAs, HMAs showed greater connectivity of the right amygdala to regions associated with social and general anxiety such as the left amygdala, the ventromedial prefrontal cortex and the anterior thalamic nucleus. On the other hand, they showed less connectivity between the right amygdala and the posterior parietal cortex. The study suggests that differences in brain activation associated with different levels of MA are apparent as early as in 7-to-9-year-old children. Specifically, limbic areas were more activated in HMAs. Parietal, prefrontal and basal ganglia were less activated in HMAs. Ventromedial prefrontal areas were instead more deactivated in HMAs. Finally, MA also modulated the connectivity of the right amygdala to adjacent areas of the brain. Similarly to EEG literature, studies investigating brain correlates of MA through fMRI are still scarce. Overall, the focus has been drawn toward brain areas linked to emotional processing and cognitive control. However, findings should be interpreted with care considering some methodological aspects of the studies reviewed. One issue concerns the study by Lyons and Beilock (2012a), in which better performance was associated with more activation of areas involved in cognitive control when anticipating a maths task. In the study, performance was defined as in terms of “maths deficit”, namely the difference in performance between an arithmetic verification task and a word task: the smaller the difference in performance between the two tasks, the smaller the “maths deficit” as a result of better cognitive control. First, such an approach is problematic because the outcome is dependent on not only an impairment in maths but also linguistic task ability. Further, the score can also be interpreted as a “verbal advantage” score and in that case, for example, it could be argued that the identified brain areas ref lect happiness about executing verbal rather than mathematical tasks. A second point of concern is that in Lyons and Beilock (2012a, 2012b), the analysis is based on double dipping, a practice in which the data are first analysed to select a subset of significant data and then those data are reanalysed to obtain

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results. This practice constitutes a circular process in which the assumption of independency of the data is violated and therefore the results distorted ( Kriegeskorte, Simmons, Bellgowan, & Baker, 2009). In Lyons and Beilock (2012a), the authors identify brain areas which showed activity correlated with maths deficit scores in HMAs only. After this they tested these brain areas for significant MA effects. Provided that the areas were pre-selected by a significant association defined by the HMAs group, this clearly constitutes invalid double dipping. A similar procedure was used in Lyons and Beilock (2012b). It is important to note that the selected regions are consistently called regions of interest which usually implies that regions are defined before the analysis is run. However, in this case it is clear that regions of interest were defined during analysis and they cannot be considered independently defined. Overall, the most reliable fMRI findings concern an increased and possibly dysfunctional emotional processing as shown by hyperactive limbic areas ( Young et al., 2012) and impaired cognitive control as shown by a reduced deactivation of the DMN in MA participants ( Pletzer et al., 2015). Importantly, these studies will need replication before any strong conclusion can be drawn on the implication of these areas in MA cognitive functioning. A final issue on fMRI studies is the interpretation of the imaging data in terms of differences in specific cognitive processes between HMAs and LMAs, which constitutes a case of inverse inference. Inverse inference is a backward reasoning in which the engagement of a cognitive function is inferred by the activation of a specific brain area. However, this type of inference can only be drawn with relative confidence if the region is activated selectively for one specific cognitive function ( Poldrack, 2006). Given that several cognitive processes are engaged during arithmetic tasks (Avancini, Soltész, & Szűcs, 2015), specific claims about differences in cognitive functioning between MA groups should be specifically tested in the future.

Conclusions The studies reviewed suggest a potential for implicit measures as a tool to investigate MA. Regarding autonomic measures, salivary cortisol has so far provided the most evidence of differences between HMA and LMA individuals. On the other hand, HR and skin conductance have not consistently differed between the two groups. However, the literature is still very scarce and not yet conclusive. A further complicating factor is that there is no standard procedure to analyse physiological data and several indexes can be extrapolated from the same component ( Kreibig, 2010). Hence, the lack of significant results on HR and skin conductance may either represent a true lack of sensitivity to MA or be a consequence of methodological choices such as the selection of the most reliable and valid index. In terms of electrophysiological measures, several components have been identified as possible markers of MA, most of which had previously been associated with attentional processes and cognitive control. The components that have

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been reported most extensively are the P2 and the P3, which are measures of allocation of attentional resources. Other components that have been suggested to be modulated by MA constitute only isolated evidence as a result of the scarcity of available literature. Limited are also the studies that have investigated brain correlates of MA. The literature so far has identified brain areas responsible for emotional processing and cognitive control. Results should, however, be interpreted with care given issues related to methodological choices such as double-dipping practices. For both fMRI and ERP results, the reader should refrain from drawing conclusions on the cognitive processes differing between MA groups on the basis of brain area activation or modulation of specific ERP components. Nevertheless, these studies offer interesting insight into questions to be directly tested in future studies. While available literature uncovers how physiological measures may offer insight into implicit processes of MA functioning, evidence for physiological correlates of MA are still scarce and fragmented. An important task of future studies is to identify measures that are the most sensitive to MA variations, reliable and ecologically valid. To do so, replication of the current findings is imperative, as well as further investigation of potential physiological indexes. Furthermore, establishing implicit measures of MA would allow for investigating those aspects of MA that are not under conscious control. A further limitation of the studies reviewed in this chapter is that comparison between MA levels has been carried out between HMA and LMA groups that had been a-priori selected based on scores from MA questionnaires. As mentioned in the introduction, self-report measures may be biased by metacognitive skills, individuals’ response honesty and cultural and linguistic factors. Therefore, the comparison of physiological measures between groups still relies on the construct validity of the self-report questionnaires. This raises the question of whether the physiological effects observed after group selection are a true representation of implicit mechanisms directly linked to MA or a result of the inf luence of covert factors. Relying on questionnaires is inevitable at such early stages in the field, but future research should aim at validating physiological markers as independent measures of MA. Finally, the ultimate goal of research on MA should be developing tools that the general population can benefit from in order to overcome obstacles resulting from MA, be these obstacles performance difficulties or potential deterrents to choosing careers requiring the mastering of maths skills. With a new understanding the physiology of MA, neurofeedback and biofeedback techniques may be explored as accessible and low-cost treatment options for MA.

Notes 1 The Stroop task (Stroop, 1935) was designed to assess the ability to inhibit a well-learned response. In Hopko et al. (2005), a card version was used (Golden, 1994) in which participants saw names of colours written in black ink, semantically meaningless symbols

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written in black ink and names of colours written in conflicting colours (red written in blue ink). Participants had to read the word or the ink colour as quick and accurately as possible. 2 A voxel is the analogue of a pixel in three-dimensional space.

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4 MATHEMATICS ANXIETY AND PERFORMANCE Ann Dowker

Attitudes to mathematics vary a lot between individuals. Some people are extremely enthusiastic about mathematical activities; but a significant number of people have very negative attitudes toward mathematics, often including considerable anxiety (Ashcraft, 2002; Dowker, Looi, & Sarkar, 2016; Hembree, 1990; Maloney & Beilock, 2012).

What is mathematics anxiety? Mathematics anxiety has been defined as ‘a feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in .  .  . ordinary life and academic situations’ ( Richardson & Suinn, 1972). Although ‘mathematics anxiety’ is the usual term, it most commonly refers to anxiety about numbers and arithmetic. Possibly the early term ‘number anxiety’ (Dreger & Aiken, 1957) is a more accurate one. Most studies do not focus on anxiety about non-numerical aspects of mathematics. Mathematics anxiety is often treated as a single entity, but in fact it consists of more than one component; and this needs to be taken into account, as different components appear to have different relationships to performance. In particular, Wigfield and Meece (1988) found two separate dimensions of mathematics anxiety in elementary and secondary school pupils. There were the cognitive and affective dimensions, similar to those that had been previously identified in the area of test anxiety by Liebert and Morris (1967). The cognitive dimension was labelled as ‘worry’, and is similar to what is sometimes termed ‘performance anxiety’, involving concerns about how one is performing and the fear of failure. The affective dimension was labelled as ‘emotionality’ and refers to emotions of fear, nervousness and tension with their associated physiological reactions, which occur in the presence of numerical stimuli, whether or not there is a threat of failure or evaluation.

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Relationships between mathematics anxiety and mathematics performance Most studies have indicated that there is a negative relationship between mathematics anxiety and mathematics performance. Higher levels of mathematical anxiety have been linked to lower performance in school and college tests ( Hembree, 1990; Ma & Kishor, 1997). Although most studies have looked at both performance and anxiety as single entities, there is some evidence that anxiety and performance can show specific correlations with regard to particular aspects of mathematics. For example, Evans (2000) found that, among a group of mature students, anxiety about mathematics tests and courses could be distinguished from anxiety about using numbers in everyday contexts. He also found that the former was more related to school mathematics performance, and the latter to practical mathematics performance. There are several possible reasons why mathematics anxiety might affect performance. One reason may be that the negative emotions associated with mathematics make people avoid mathematics-related activities as much as possible (Ashcraft, 2002; Hembree, 1990). As a result, they get less practice in mathematics than those with more positive attitudes toward mathematics, and thus have less opportunity to learn and improve. Indeed, adolescents and adults who score high on mathematics anxiety measures are less likely than their less anxious peers to take optional mathematics courses in high school and college (Ashcraft, Kirk, & Hopko, 1998; Hembree, 1990) or to plan careers in science, even after controlling for scores on quantitative tests (Chipman, Krantz, & Silver, 1992). Moreover, extremely avoidant or panicky reactions to mathematics may not only lead to reduced mathematics learning by discouraging practice but also have an immediately adverse effect on performance. For example, Ashcraft and Faust (1994) found that people with high mathematics anxiety solved mathematics problems more quickly but less accurately than less anxious people. Far from making people more careful and vigilant, mathematics anxiety seems to result in people rushing to finish the task as fast as possible, even at the cost of accuracy. This is likely to be because of both a desire to get an unpleasant task out of the way quickly and feeling under pressure to produce an answer as quickly as possible. Mathematics anxiety seems often to involve concerns about not only accuracy but also speed and a feeling that one is in a sort of race. For example, some studies have indicated that mathematical anxiety is stronger for timed than untimed mathematics tests. Another possible reason for the negative effects of mathematics anxiety on performance is that higher levels of mathematical anxiety lead to greater mental rumination and preoccupying thoughts, and that this consumes working memory resources that could otherwise have been applied to mathematical learning and mathematical problem-solving (Ashcraft & Kirk, 2001; Ashcraft & Ridley, 2005; Eysenck & Calvo, 1992). Ashcraft and Kirk (2001) found that people with high maths anxiety demonstrated smaller working memory spans

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than people with less maths anxiety, especially in tasks that required any sort of calculation. In particular, people with high maths anxiety were much slower and made many more errors than others in tasks where they had to do mental addition at the same time as keeping numbers in memory. Ashcraft and Kirk (2001) concluded that maths anxiety does indeed disrupt working memory in mathematical tasks.

The other way round: Could performance deficits lead to anxiety? But it could also be that the performance deficits come first and cause the anxiety. Mathematics anxiety is at least partly related to fear of failure, so that repeated experiences of failure in mathematics – whether involving low scores in formal assessments, or personal experience of confusion and bewilderment in mathematical activities – may lead to anxiety. Certainly, people tend to experience more enjoyment and less anxiety with activities at which they are successful. Some studies suggest that people with mathematics anxiety have more difficulty than less mathematically anxious people with even elementary numerical abilities, suggesting that their numerical difficulties might be there from the beginning and lead to later anxiety. For example, Maloney, Ansari, and Fugelson (2011) gave groups of people with high and low mathematics anxiety two tasks involving comparisons of written numbers. The people with high mathematics anxiety were less precise in their number comparisons than those with lower mathematics anxiety. Núñez-Peña and Suárez-Pellicioni (2014) also found that people with high mathematics anxiety showed a larger distance effect (longer reaction times to comparisons of numbers that are close together in magnitude) as well as a larger size effect (longer reaction times to comparisons involving larger numbers) than those with low mathematics anxiety. The authors of both studies suggested that those with high mathematics anxiety may have a primary deficit in basic numerical abilities that interferes with the development of higher mathematical skills and the resulting mathematical difficulties lead to mathematics anxiety. However, one cannot rule out the effect of experience on even low-level numerical abilities. Studies have shown that at least in young children, practice with numerical activities can inf luence number comparison skills. For example, Ramani and Siegler (2008) found that giving young children experience with numerical board games improved their number comparison performance. It may be that experience inf luences such abilities even at older ages, and that mathematics anxiety may cause number comparison weaknesses by causing people to avoid practicing number-related activities. Moreover, there is evidence that working memory contributes significantly to both symbolic and non-symbolic number comparison (Simmons, Wills, & Adams, 2012) as do some other related executive function abilities (Merkley,

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Thompson, & Scerif, 2016). It may be that mathematics anxiety disrupts number comparison, just like more complex arithmetic, by interfering with working memory. A study by Lee and Cho (2018) provides some evidence for this hypothesis. They found that mathematics anxiety affected both magnitude comparison and complex arithmetic, but not simple arithmetic problems that were likely to be solved by retrieval of overlearned facts and were not dependent on working memory.

A bidirectional relationship between anxiety and performance As can be seen, the difficulty of determining whether mathematics anxiety is the cause or result of poor performance is a pervasive issue in research on this topic. In fact, the relation between mathematics anxiety and performance appears be bidirectional. In addition to mathematics anxiety having a negative impact on mathematics performance, poor performance in mathematics is also likely to increase anxiety (Carey, Hill, Devine, & Szucs, 2016; Maloney & Beilock, 2012; Maloney et al., 2011). Longitudinal studies provide evidence for both types of relation. For example, Ma and Xu (2004) found evidence of stronger correlations between performance and mathematical anxiety than between anxiety and performance across time. On the other hand, Cargnelutti, Tomasetto, and Passolunghi (2017), in a longitudinal study of progression from second to third grade, found a stronger predictive relation of mathematics anxiety to performance than the other way round. Thus, there is evidence that mathematical anxiety both inf luences and is inf luenced by performance. Jansen et al. (2013) argue that this leads to a vicious cycle where higher anxiety can lead to suboptimal strategy use or avoidance behaviour, which might then negatively impact performance and increase anxiety. Even as early as the first two years of school, Gunderson, Park, Maloney, & Levine (2018) have found that reciprocal relationships are already operating between mathematics anxiety and mathematical performance, and creating vicious and virtuous circles. The strongest evidence for a bidirectional relationship between mathematics anxiety and performance comes from intervention studies, which indicate that interventions that improve either of these can also improve the other. Some studies suggest that treating mathematics anxiety can improve performance (Hembree, 1990; Kamann & Wong, 1993), even if mathematics itself is not trained. There are also studies that indicate that interventions that improve children’s mathematical performance can reduce anxiety (Supekar, Iuculano, Chen, & Menon, 2015). Maloney and Beilock (2012) proposed that mathematics anxiety is likely to be due both to pre-existing difficulties in mathematical cognition and to social factors, e.g. exposure to teachers who themselves suffer from mathematics anxiety. Additionally, they propose that those with initial mathematical difficulties are also likely to be more vulnerable to negative social inf luences, and that this may create a vicious cycle (see Chapter 10).

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Not only anxiety: other types of attitudes toward mathematics Most studies have focused on the more negative attitudes toward mathematics, especially anxiety. However, positive attitudes such as enjoyment of mathematics and self-confidence in mathematics are also important topics to study, and cannot be reduced to the simple absence of anxiety. Enjoyment of mathematics is an important construct in itself because it may inf luence the extent to which children are willing to practice mathematical activities, which may enable them to improve in the subject. Van der Beek, Van der Ven, Kroesbergen, and Leseman (2017) found that mathematics achievement correlated positively with mathematics enjoyment and negatively with mathematics anxiety in a Dutch adolescent sample. Pinxten, Marsh, De Fraine, Vane den Noortgate, and Van Danne (2014) carried out a longitudinal study of Belgian upper primary school children, and mathematics enjoyment was a mild positive predictor of later mathematics achievement and later perceived effort in mathematics. Self-rating of one’s own mathematical ability is also generally found to be positively related to performance (Abed & Alkhateeb, 2000; Pinxten et al., 2014). Indeed, some studies suggest that in primary school children, mathematical performance may correlate more with self-rating than with anxiety ( Krinzinger, Kaufmann, & Willmes, 2009; Wood et al., 2012) and that at all ages self-rating of one’s own mathematical ability may have a significant mediating role for the relations between mathematical performance and mathematics anxiety. For example, Dowker, Bennett, and Smith (2012) gave the Mathematics Attitude and Anxiety Questionnaire and the British Abilities Scales Basic Number Skills test to 44 Year 3 (7-to-8-year-old) children and 45 Year 5 (9- to 10-year-old children from English primary schools. The Mathematics Attitude and Anxiety Questionnaire includes pictorial scales of Anxiety; Unhappiness at Failure/Poor Performance. There were few year group differences in attitudes. Performance correlated strongly with liking for mathematics and self-rating of mathematical performance, and liking and self-rating also correlated with one another. Mathematics anxiety correlated strongly with unhappiness at poor performance, but neither measure correlated with actual performance. Similar results were obtained using the same questionnaire with German ( Krinzinger et al., 2007; Krinzinger et al., 2009), Brazilian ( Wood et al., 2012) and Finnish children (Sorvo et al., 2017). Thus, at this age, mathematics performance seems to be much more closely related to self-rating than to mathematics performance anxiety. As often, the ‘chicken and egg’ question arises: self-rating may correlate with performance because it ref lects a somewhat accurate assessment of the child’s mathematical attainment, and/or because higher self-rating leads to greater motivation to improve. Relationships between mathematical attainment, mathematics performance anxiety and unhappiness at poor performance may only come later, and possibly as a result of the initial relationship between mathematical attainment and self-rating, performance and anxiety and other attitudes ( Pinxten et al., 2014; Van der Beek et al., 2017), perhaps in combination

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with the early-developing relationship between mathematical performance and the affective component of mathematics anxiety. However, one study of children at the end of their first year at school ( Dowker, Cheriton, Horton & Mark, in press) did find a significant negative relationship between mathematics performance and unhappiness at poor performance, even at this young age. How closely related are self-rating and anxiety? In young children, the relationship does not seem to be strong, at least as regards the ‘worry’ dimension (performance anxiety). Dowker et al. (2012) and Krinzinger et al. (2009) found that self-rating was much more closely related to liking mathematics than to anxiety. In older children and adults, a negative relationship is usually found between mathematics self-concept and mathematics anxiety (Goetz, Cronjaeger, Frenzel, Ludtke, & Hall, 2010; Hembree, 1990; Hoffman, 2010; Pajares & Miller, 1997). However, as most of these studies are not longitudinal, there are problems determining the direction of causation. Does anxiety lead to lower self-rating in mathematics, or does low self-rating cause anxiety? Ahmed, Minnaert, Kuyper, and Van Der Werf (2012) carried out a longitudinal study of 495 seventh-grade pupils, who completed questionnaires about both anxiety and self-concept three times during a school year. Structural equation modelling suggested that each characteristic inf luenced the other over time, but that the effect of self-concept on subsequent anxiety was significantly greater than the effect of anxiety on subsequent self-concept. As with the relationship between anxiety and performance, it appears that the relationship between mathematics anxiety and mathematics self-concept is reciprocal: each inf luences the other. Another attitude that may be related to both mathematics anxiety and mathematical performance is self-efficacy. Self-efficacy is a concept closely related to selfrating but has additional elements. Self-efficacy has been defined as having positive beliefs in one’s ‘capabilities to organize and execute the courses of action required to produce given attainments’ (Bandura, 1994). Thus, it implies not rating one’s performance in mathematics highly, but regarding oneself as able to take steps successfully to learn new mathematical topics and improve mathematical performance in the future. Most studies suggest a positive relationship between self-efficacy in mathematics and mathematical performance (Klassen, 2002; Pajares, 1996; Passolunghi, 2011; Skaalvik, Federici, & Klassen, 2015; Tariq & Durrani, 2012), though some have found only a relatively modest relationship (Norwich, 1987).

Developmental changes in mathematics anxiety and performance The majority of studies of attitudes to mathematics and mathematics anxiety have been carried out in secondary school children and adults. Studies that have included younger children typically suggest that primary school children’s attitudes to mathematics tend to be positive, but that they deteriorate with age during childhood and adolescence ( Krinzinger et al., 2009; Ma & Kishor, 1997; Wigfield & Meece, 1988). There is some variability in results. In some studies, even

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first-graders have been shown to report and demonstrate mathematics anxiety ( Jameson, 2014; Ma & Kishor, 1997; Ramirez, Gunderson, Levine, & Beilock, 2013; Vukovic, Kieffer, Bailey, & Harari, 2013). Petronzi (2016) found significant mathematics anxiety in children as young as five. While most studies show increased anxiety with age, Sorvo et al. (2017) found that Finnish children in Year 2 showed greater anxiety in Year 2 than children in later elementary grades. If levels of mathematics anxiety change, and usually increase, with age, does the relationship between mathematics anxiety and performance also change with age? There is indeed some evidence that the relationship between mathematics anxiety and performance increases with age, and that performance in younger children may be more related to other aspects of attitudes toward mathematics, as described in the previous section. However, the evidence is conf licting, and it may depend in part on what aspects of anxiety are being studied. A meta-analysis by Ma and Kishor (1997) indicated that the relationship between attitudes and performance increases with age. Some studies suggest that among young children, performance is not significantly related to anxiety (Cain-Caston, 1993; Krinzinger et al., 2009; Dowker et al., 2012; Wood et al. 2012), but is more related to self-rating. However, there are also studies that do show a significant relationship between anxiety and performance even in young children (Dossey, Mullis, Lindquist, & Chambers, 1988; Newstead, 1998; Wu, Barth, Amin, Malcarne, & Menon, 2012; Ramirez et al., 2013; Vukovic et al., 2013). Most of the studies that show a relationship between anxiety and performance in young children have been carried out in the United States. This may reflect some cultural or educational factors that differentiate the United States from other countries, but it is not clear what these might be. A more likely explanation is perhaps that different mathematics anxiety tests have been predominantly used in different places. Studies that use measures based on Richardson and Suinn’s (1972) Mathematics Rating Scale (MARS) or MARS-Elementary (Suinn, Taylor,  & Edwards, 1988) have typically shown such a relationship in elementary school children (Wu et al., 2012; Vukovic et al., 2013), which may be because that such measures emphasize affective dimension of mathematics anxiety. Those that have used the Mathematics Attitude and Anxiety Questionnaire (MAQ) developed by Thomas and Dowker (2000) have tended not to show such a relationship in younger children (Krinzinger et al., 2007, 2009; Dowker et al., 2012; Wood et al., 2012), which could reflect the fact that this measure places more emphasis on the cognitive (‘worry’/performance anxiety) aspect of mathematics. The few studies that have included both dimensions of mathematics anxiety have suggested that performance in young children is related to the affective but not to the cognitive dimension (Sorvo et al., 2017). Changes in the relationships between mathematics anxiety and performance may also reflect changes in working memory, and in the ways in which it is used in arithmetic. Vukovic et al. (2013) carried out a longitudinal study of 113 children, who were followed up from second to third grade. Mathematics anxiety showed a negative relationship with mathematical performance from second to third grade,

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but only for children with higher levels of working memory. This may seem surprising in view of the studies that suggest that mathematics anxiety interferes with performance by overloading working memory: one might have expected that this would have the most deleterious effects on children, who have lower working memory to start with. However, it may reflect differences in strategy use. Perhaps young elementary school children with high levels of working memory are more likely to be already using arithmetical processes that depend strongly on working memory, such as mental calculation, while those with lower working memory may rely more on concrete counting strategies. The children who use more mental calculation may be the ones whose progress is most affected by mathematics anxiety. Similar factors might explain age differences in the relationship between anxiety and performance: older children are using more mental calculation and fewer concrete strategies than younger children, and may therefore be more impeded in their performance by mathematics anxiety. Interestingly, a study of adult undergraduate students ( Johnson & Gronlund, 2009) found the reverse relationship to that found by Vukovic et al. (2013): individuals with lower working memory scores showed a higher relationship between mathematics anxiety and mathematics performance than did individuals with higher working memory scores. This may be because presumably all participants by that age were using mental calculation strategies, and those with better working memory may have been doing so more efficiently, and thus have been less vulnerable to disruption by anxiety. There is also, of course, the possibility that the increase with age in relationships between mathematics anxiety and performance ref lects the reverse direction of causation. As children gain increasing experience of mathematical success and failure, mathematics anxiety may increase in those children whose poor performance results in repeated failure experiences, but not to the same extent as in children who experience greater success in mathematics.

Cultural and national differences in mathematics anxiety and performance International comparisons consistently reveal international differences in mathematics performance: in particular, East Asian pupils usually perform better in mathematics than European and American pupils (Mullis, Martin, & Loveless, 2016; Mullis, Martin, Foy, & Hooper, 2016). Is this ref lected in mathematics anxiety and other attitudes? Mathematics anxiety could be related to a country’s level of achievement in more ways than one. Children in higher-achieving countries could have positive attitudes to mathematics because they are doing well (and/or may do well because they have positive attitudes). On the other hand, they could be more anxious about mathematics, and rate themselves lower, because high-achieving countries often attach high importance to mathematics and to academic achievement in general, and this may increase academic pressures on children and lead

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to greater fear of failure. Moreover, such children might rate themselves lower in mathematics because they are comparing themselves with their own peers, who are likely to be high-achieving. In fact, studies comparing mathematics anxiety in different countries (Askew, Hodgen, Hossain, & Bretscher, 2010; Dulaney, Herts, Borgonovi, & Beilock, 2017; Lee, 2009) have found inconsistent relationships between a country’s overall mathematics achievement level and the average level of mathematics anxiety among children in that country. Children in high-achieving East Asian countries, such as Korea and Japan, tend to demonstrate high mathematics anxiety; while those in high-achieving Western European countries, such as Finland, the Netherlands, Liechtenstein and Switzerland, tend to demonstrate low mathematics anxiety. It is of course difficult to tell to what extent responses might be inf luenced by cultural differences in the perceived social desirability of expressing confidence versus anxiety with regard to academic subjects. Despite the inconsistency in the relationships between national mathematics achievement and national mathematics anxiety levels, a negative relationship between anxiety and performance has consistently been found within any given country ( Dulaney et al., 2017; Lee, 2009; Ma & Kishor, 1997). There is no evidence that the national level of mathematics achievement, or indeed any other national or cultural factor, has a significant inf luence on the relationship between anxiety and performance.

Gender, attitudes and performance Most studies today indicate that, where girls and boys have similar educational opportunities in mathematics, they perform similarly in mathematics (Spelke, 2005). However, it is generally found that females have more negative attitudes toward mathematics, rate themselves lower in mathematics, and are more anxious about mathematics (Callan, 2015; Devine, Fawcett, Szucs, & Dowker, 2012; Hembree, 1990; Kyttälä & Björn; Wigfield & Meece, 1988). This may be partly due to females having greater general anxiety than males. Most studies indicate that females score higher on general anxiety measures and on the closely related personality trait of neuroticism than males (e.g., Costa, Terracciano, & McCrae, 2001; Chapman, Duberstein, Sörensen, & Lyness, 2007) and show higher prevalence of clinical anxiety disorders (McLean, Asnaani, Litz, & Hofmann, 2011). There may be gender differences in the relationship between attitudes and performance. There are, however, some conflicting results as to exactly how these might differ. Devine et al. (2012) gave 433 British 11-to-15-year-olds mental mathematics tests and questionnaires measuring mathematics anxiety and test anxiety. Girls and boys did not differ in mathematics performance, but girls showed higher mathematics anxiety and higher test anxiety than girls. Both boys and girls showed a negative correlation between mathematics anxiety and mathematics performance. However, once the researchers controlled for test anxiety, this

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relationship disappeared for boys. Only girls continued to show an independent relationship between mathematics anxiety and mathematics performance, after controlling for test anxiety. However, not all studies have given similar results. Hembree (1990) and Ma and Xu (2004) obtained results suggesting that math anxiety is more negatively related to achievement in males than in females. There are several studies that have found no gender differences in the relationship between mathematics anxiety and performance (Ma, 1999). Most of these studies did not, however, control for test anxiety. Also, most studies have not looked at whether there could be gender differences between the relationships between mathematics anxiety and performance on different aspects of mathematics. Miller and Bichsel (2004) did look at this issue in adults and found that mathematics anxiety was more related to basic mathematics scores in males but to applied mathematics scores in females.

Is mathematics anxiety always bad? Could there be an ‘optimum’ level of mathematics anxiety? There is no doubt that very high mathematics anxiety has a negative effect on achievement, as found in many studies as discussed above. It is still possible that mild mathematics anxiety – involving concern about performance rather than true fear – might be associated with higher achievement than complete indifference. If this does turn out to be the case, it is likely to be the case for what Wigfield and Meece (1988) termed the ‘worry’ than the ‘emotionality’ component of mathematics anxiety: performance anxiety, rather than strong adverse emotional reactions to any situation involving numbers. Some studies do give some evidence for such a relationship. Evans (2000) found that an inverted U best described the relationship between mathematics anxiety and mathematics test performance in a group of mature British students in higher education, suggesting that there may be an ‘optimum’ level of anxiety, rather than that all anxiety is deleterious. Wang et al. (2015) obtained similar results in both adolescent and adult samples, but only for those who also had high intrinsic motivation for mathematics. People who had low intrinsic motivation for mathematics showed a negative association between mathematics anxiety and mathematics performance at all levels of mathematics performance. Thus, it appears that mild levels of mathematics anxiety may have a beneficial effect in people who have a strong intrinsic motivation to learn and to do well in mathematics, but not in those who are chief ly concerned with external consequences and other people’s reactions.

Anxiety and performance can dissociate This chapter has emphasized the mostly negative relationships between anxiety and performance. However, it must be remembered that the correlations are by no

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means perfect; some people who are doing well at mathematics are highly anxious about mathematics, and some low attainers in mathematics are not anxious about it. Although severe mathematical difficulties are significantly associated with mathematics anxiety (Rubinsten & Tannock, 2010), dyscalculia and mathematics anxiety often dissociate. Devine, Hill, Carey, and Szucs (2018) carried out a large-scale study of school pupils and found that mathematics anxiety was twice as common in dyscalculics as in those without severe mathematical difficulties, but that many dyscalculics did not exhibit mathematics anxiety, and 77% of highly mathematicsanxious pupils were achieving at a typical or high level in mathematics.

Conclusions There is a significant bidirectional relationship between mathematics anxiety and low attainment in mathematics. However, the relationship is more complex than sometimes represented, and not all people with mathematics anxiety are low-attaining in mathematics, or vice versa. It is important for future research to investigate the circumstances under which mathematics anxiety and performance do and do not interact to create vicious cycles, as this may help us to understand how to intervene to prevent them from developing. It is also desirable to look more closely at which components of mathematics are most strongly affected by mathematics anxiety. Additionally, there needs to be more research on how the relationships between anxiety and performance interact with other attitudes (e.g. motivation and self-rating) and with cognitive abilities such as working memory. Finally, it is important to understand why attitudes toward mathematics appear to change for the worse with age and, in particular, why mathematics anxiety appears to increase with age.

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Askew, M., Hodgen, J., Hossain, S., & Bretscher, N. (2010). Values and variables: Mathematics education in high-performing countries. London: Nuffield Foundation. Bandura, A. (1994). Self-efficacy. In V. S. Ramachaudran (Ed.), Encyclopedia of human behavior (Vol. 4, pp. 71–81). New York: Academic Press Cain-Caston, M. (1993). Parent and student attitudes toward mathematics as they relate to third grade mathematics achievement. Journal of Instructional Psychology, 20, 96–101. Callan, S. (2015). The fear factor: Maths anxiety in girls and women. London: Maths Action. Carey, E., Hill, F., Devine, A., & Szucs, D. (2016). The chicken or the egg? The direction of the relationship between mathematics anxiety and mathematics performance. Frontiers in Psychology, 6, 1987. Cargnelutti, E., Tomasetto, C., & Passolunghi, M. C. (2017). How is anxiety related to math performance in young students? A longitudinal study of grade 2 to grade 3 children. Cognition and Emotion, 31, 755–764. Chapman, B. P., Duberstein, P. R., Sörensen, S., & Lyness, J. M. (2007). Gender differences in Five Factor Model personality traits in an elderly cohort. Personality and Individual Differences, 43, 1594–1603. Chipman, S., Krantz, D., & Silver, R. (1992). Mathematics anxiety and science careers among able college students. Psychological Science, 3, 292–295. Costa, P. T. Jr., Terracciano, A., & McCrae, R. R. (2001). Gender differences in personality traits across cultures: Robust and surprising findings. Journal of Personality and Social Psychology, 81, 322–331. Devine, A., Fawcett, K., Szucs, D., & Dowker, A. (2012). Gender differences in mathematics anxiety and the relation to mathematics performance while controlling for test anxiety. Behavioral and Brain Functions, 8, 33. Devine, A., Hill, F., Carey, E., & Szucs, D. (2018). Cognitive and emotional math problems largely dissociate: Prevalence of developmental dyscalculia and mathematics anxiety. Journal of Educational Psychology, 110, 431–444. Dossey, J. A., Mullis, I. V. S., Lindquist, M. M., & Chambers, D. L. (1988). The mathematics report card: Are we measuring up? Trends and achievement based on the 1986 national assessment. Princeton: Educational Testing Service. Dowker, A., Bennett, K., & Smith, L. (2012). Attitudes to mathematics in primary school children. Child Development Research, Article ID 124939. Dowker, A., Cheriton, O., Horton, R., & Mark, W. (in press). Relationships between attitudes and performance in young children’s mathematics. Dowker, A., Looi, C. Y., & Sarkar, A. (2016). Mathematics anxiety: What have we learned in 60 years? Frontiers in Psychology, 7, 508. Dreger, R. M., & Aiken, L. R., Jr. (1957). The identification of number anxiety in a college population. Journal of Educational Psychology, 48(6), 344–351. http://dx.doi. org/10.1037/h0045894 Dulaney, A., Herts, J. B., Borgonovi, F., & Beilock, S. (2017). The math anxiety performance link: A global phenomenon. Current Directions in Psychological Science, 26, 52–58. Evans, J. (2000). Adults’ mathematical thinking and emotions: A study of numerate practices. London: Routledge. Eysenck, M. W., & Calvo, M. G. (1992). Anxiety and performance: The processing efficiency theory. Cognition and Emotion, 6, 409–434. Goetz, T., Cronjaeger, H., Frenzel, A. C., Ludtke, O., & Hall, L. C. (2010). Academic self-concept and emotion relations: Domain specificity and age effects. Contemporary Educational Psychology, 35, 44–58.

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Gunderson, E. A., Park, D., Maloney, E. A., Beilock, S. L., & Levine, S. C. (2018). Reciprocal relations among motivational frameworks, math anxiety, and math achievement in early elementary school. Journal of Cognition and Development, 19, 21–46. Hembree, R. (1990). The nature, effects and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21, 33–46. Hoffman, B. (2010). “I think I can, but I’m afraid to try”: The role of self-efficacy beliefs and mathematics anxiety in mathematics problem solving efficiency. Learning and Individual Differences, 20, 276–283. Jameson, M. M. (2014). Contextual factors related to math anxiety in second-grade children. Journal of Experimental Education, 82, 518–536. Jansen, B. R. J., Louwerse, J., Straatemeier, M., Van der Ven, S. H. G., Klinkenberg, S., & Van der Maas, H. L. J. (2013). The inf luence of experiencing success in math on math anxiety, perceived math competence, and math performance. Learning and Individual Differences, 24, 190–197. Johnson, D. R., & Gronlund, S. D. (2009). Individuals lower in working memory capacity are particularly vulnerable to anxiety’s disruptive effect on performance. Anxiety, Stress and Coping, 22, 201–213. Kamann, M. P., & Wong, B. Y. (1993). Inducing adaptive coping self-statements in children with learning disabilities through self-instruction training. Journal of Learning Disabilities, 26, 630–638. Klassen, R. (2002). A question of calibration: A review of the self-efficacy beliefs of students with learning disabilities. Learning Disability Quarterly, 15, 88–102. Krinzinger, H., Kaufmann, L., Dowker, A., Thomas, G., Graf, M., Nuerk, H. C., & Willmes, K. (2007). German version of the math anxiety questionnaire (FRA) for 6- to 9-year-old children. [German]. Zeitschrift fur Kinder und Jugendpsychiatrie und Psychotherapie, 35, 341–351. Krinzinger, H., Kaufmann, L., & Willmes, K. (2009). Math anxiety and math ability in early primary school years. Journal of Psychoeducational Assessment, 27, 206–225. Lee, J. (2009). Universals and specifics of math self-concept, math self-efficacy, and math anxiety across 41 PISA 2003 participating countries. Learning and Individual Differences, 19, 355–365. Lee, K., & Cho, S. (2018). Magnitude processing and complex calculation is negatively impacted by mathematics anxiety while retrieval-based simple calculation is not. International Journal of Psychology, 53, 321–329. Liebert, R. M., & Morris, L. W. (1967). Cognitive and emotional components of test anxiety: A distinction and some initial data. Psychological Reports, 20, 975–978. https:// doi.org/10.2466/pr0.1967.20.3.975 Ma, X. (1999). A meta-analysis of the relationship between anxiety toward mathematics and achievement in mathematics. Journal for Research in Mathematics Education, 30, 520–541. Ma, X., & Kishor, N. (1997). Assessing the relationship between attitude toward mathematics and achievement in mathematics: A meta-analysis. Journal for Research in Mathematics Education, 28, 26–47. Ma, X., & Xu, J. (2004). Determining the causal order between attitude toward mathematics and achievement in mathematics. American Journal of Education, 100, 256–281. Maloney, E. A., Ansari, D., & Fugelsang, J. A. (2011). The effect of mathematics anxiety on the processing of numerical magnitude. Quarterly Journal of Experimental Psychology, 64, 10–16. Maloney, E. A., & Beilock, S. (2012). Math anxiety: Who has it, why it develops, and how to guard against it. Trends in Cognitive Sciences, 16, 404–406.

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McLean, C. P., Asnaani, A., Litz, B. T., & Hofmann, S. G. (2011). Gender differences in anxiety disorders: Prevalence, course of illness, comorbidity and burden of illness. Journal of Psychiatric Research, 45, 1027–1035. Merkley, R., Thompson, J., & Scerif, G. (2016). Of huge mice and tiny elephants: Exploring the relationship between inhibitory processes and preschool math skills. Frontiers in Psychology, 6, 1903. Miller, H., & Bichsel, J. (2004). Anxiety, working memory, gender, and math performance. Personality and Individual Differences, 37, 591–606. Mullis, I. V. S., Martin, M. O., Foy, P., & Hooper, M. (2016). TIMSS 2015 international results in mathematics. Boston, USA: International TIMSS and PIRLS Study Centre. Mullis, I. V. S., Martin, M. O., & Loveless, T. (2016). 20 years of TIMSS: International trends in mathematics and science achievement, curriculum, and instruction. Boston, USA: International TIMSS and PIRLS Study Centre. Newstead, K. (1998). Aspects of children’s mathematics anxiety. Educational Studies in Mathematics, 36, 53–71. Norwich, B. (1987). Self-efficacy and mathematics achievement: A study of their relation. Journal of Educational Psychology, 79, 384–387. Núñez-Peña, M. I., & Suárez-Pellicioni, M. (2014). Less precise representation of numerical magnitude in high math-anxious individuals: An ERP study of the size and distance effects. Biological Psychology, 103, 176–183. https://doi.org/10.1016/j. biopsycho.2014.09.004 Pajares, F. (1996). Self-efficacy beliefs and mathematical problem-solving of gifted students. Contemporary Educational Psychology, 21, 325–344. Pajares, F., & Miller, M. D. (1997). Mathematics self-efficacy and mathematical problem solving: Implications of using different forms of assessment. Journal of Experimental Education, 65, 213–228. Passolunghi, M. C. (2011). Cognitive and emotional factors in children with mathematical learning disabilities. International Journal of Disability, Development and Education, 58, 61–73. Petronzi, D. (2016). The development of the numeracy apprehension scale for children aged 4–7 years: Qualitative exploration of associated factors and quantitative testing (Ph.D. Thesis). University of Derby, Derby, UK. Pinxten, M., Marsh, H., De Fraine, B., Vane den Noortgate, W., & Van Danne, J. (2014). Enjoying mathematics or feeling competent in mathematics? Reciprocal effects on mathematics achievement and perceived math effort expenditure. British Journal of Educational Psychology, 84, 152–174. Ramani, G. B., & Siegler, R. S. (2008). Promoting broad and stable improvements in lowincome children’s numerical knowledge through playing number board games. Child Development, 79, 375–394. https://doi.org/10.1111/j.1467-8624.2007.01131.x Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2013). Math anxiety, working memory and math achievement in early elementary school. Journal of Cognition and Development, 14, 187–202. Richardson, F. G., & Suinn, R. M. (1972). The Mathematics Anxiety Rating Scale: Psychometric data. Journal of Counseling Psychology, 19, 551–554. Rubinsten, O., & Tannock, R. (2010). Mathematics anxiety in children with developmental dyscalculia. Behavioral and Brain Functions, 6, 46. Simmons, F. R., Wills, S., & Adams, A. M. (2012). Different components of working memory have different relationships with different mathematical skills. Journal of Experimental Child Psychology, 111, 139–155.

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Skaalvik, E. M., Federici, R. A., & Klassen, R. M. (2015). Mathematics achievement and self-efficacy: Relations with motivation for mathematics. International Journal of Educational Research, 72, 129–136. Sorvo, R., Koponen, T., Viholainen, H., Aro, T., Raikkönen, E., Peura, P., . . . Aro, M. (2017). Math anxiety and its relationship with basic arithmetic skills among primary school children. British Journal of Educational Psychology, 87, 309–327. Spelke, E. S. (2005). Sex differences in intrinsic aptitude for mathematics and science? A critical review. American Psychologist, 60, 950–958. http://dx.doi.org/10.1037/0003066X.60.9.950 Suinn, R. M., Taylor, S., & Edwards, R. W. (1988). Suinn Mathematics Anxiety Rating Scale for Elementary School Students (MARS-E): Psychometric and normative data. Educational and Psychological Measurement, 48, 979–986. Supekar, K., Iuculano, T., Chen, L., & Menon, V. (2015). Remediation of childhood math anxiety and associated neural circuits thought cognitive tutoring. The Journal of Neuroscience, 35, 12574–12583. Tariq, V. N., & Durrani, N. (2012). Factors inf luencing undergraduates’ self-evaluation of numerical competence. International Journal of Mathematical Education in Science and Technology, 43, 337–356. Thomas, G., & Dowker, A. (2000, September 15). Mathematics anxiety and related factors in young children. Paper presented at British Psychological Society Developmental Section Conference, Bristol. Van der Beek, J., Van der Ven, S., Kroesbergen, E., & Leseman, P. (2017). Self-concept mediates the relation between achievement and emotions in mathematics. British Journal of Educational Psychology, 87, 478–495. Vukovic, R. K., Kieffer, M. J., Bailey, S. P., & Harari, R. R. (2013). Mathematics anxiety in young children: Concurrent and longitudinal associations with mathematical performance. Contemporary Educational Psychology, 38, 1–10. Wang, Z., Lukowski, S. L., Hart, S. A., Lyons, I. M., Thompson, L. A., Kovas, Y., . . . Petrill, S. A. (2015). Is mathematical anxiety always bad for math learning? The role of math motivation. Psychological Science, 26, 1863–1876. Wigfield, A., & Meece, J. L. (1988). Math anxiety in elementary and secondary school students. Journal of Educational Psychology, 80, 210–216. Wood, G., Pinheiro-Chagas, P., Júlio-Costa, A., Micheli, L. R., Krinzinger, H., Kaufmann, L., & Haase, V. G. (2012). Math anxiety questionnaire: Similar latent structure in Brazilian and German school children. Child Development Research, Article ID 610192. Wu, S. S., Barth, M., Amin, H., Malcarne, V., & Menon, V. (2012). Math anxiety in second and third graders and its relation to mathematical achievement. Frontiers in Developmental Psychology, 3, 162.

5 ACQUISITION, DEVELOPMENT AND MAINTENANCE OF MATHS ANXIETY IN YOUNG CHILDREN Dominic Petronzi, Paul Staples, David Sheffield and Thomas Hunt

Setting the scene Now, we all know that a good understanding of maths concepts is an important part of our education and that a certain ability in the subject can define a person’s career opportunities. As we move into a more technologically dependent world, there is high international competition, and mathematically brilliant minds are perhaps more important than ever. However, according to the official Programme for International Student Assessment (PISA) rankings (2015), the United Kingdom’s performance in maths has dropped a position and is currently ranked 27th. In contrast, Singapore is ranked in first position and the United Kingdom has attempted to replicate how maths is taught in East Asian countries. So, what’s going on? If you ref lect upon your own classroom experiences of maths, you might perhaps recall thinking that the curriculum demands were excessive. Now place yourself in the position of a primary-school-aged child in 2018. This child now has to be able to understand Pythagoras’s theorem, creating more pressure for the child and, in particular, their teacher. This pressure can create a perfect breeding ground for anxiety. With the curriculum facing consistent criticism for increasing the difficulty level in primary schools, it’s perhaps easier to understand why some children who encounter small problems in the subject can start on a path toward maths anxiety. Researchers are increasingly drawing attention to the classroom experiences in early education. As early as 1986, Skemp believed that maths anxiety developed at the age of 5–6 years in response to the classroom environment. He drew particular attention to the use of rote memorisation. This style of teaching maths prevents deeper learning of concepts and their wider application. It has also been suggested by Rossnan (2006) that maths anxiety can develop at any age and that fear is often linked to a child’s first experiences of maths.

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If a child’s first experiences of something are not enjoyable, they are unlikely to look forward to it or aspire to make the most of it. In fact, it is highly likely that they will avoid it and eventually give up! We can all think of something that we have not started off enjoying and never gave our full effort. This can be the same with maths, and there may be other reasons for this too. Other factors that have been theorised and evidenced by researchers include the following: •

Previous negative classroom experiences associated with teachers and rigid rules of maths. (Hadfield & McNeil, 1994)

•

Personality, in particular shyness and low self-esteem. (Hadfield & McNeil, 1994)

•

Intellectual factors including attitude, self-doubt, confidence and viewing maths as unnecessary. (Hadfield & McNeil, 1994)

•

Strained relationships between students and teachers and being placed under pressure. (Mata, Monteiro, & Peixoto, 2012)

•

Children having an awareness of an ability deficit. ( Erdogan, Kesici, & Yuksel-Sahin, 2011)

•

Acceptance of failure. (Ashby, 2009)

To explore important factors in maths anxiety, research has focused on the numeracy1 experiences of children aged 4–7 years. This is discussed in more detail later in the chapter, but it is important to highlight some of the key findings. The data, which are based on discussion groups with children, point towards other reasons as to why they may feel worried when working with numbers. These include the following: • • • •

Children becoming aware of an intelligence hierarchy. Fear of failure. Dependence on friends and competing with peers. Numeracy being used as a punishment.

These factors suggest that the roots of maths anxiety begin in the classroom. However, the findings from research with younger children ( Petronzi, Staples, Sheffield, Hunt, & Fitton-Wilde, 2017) cannot be directly compared with those of previous research. This is because maths anxiety research, until very recently, had not really considered the early years of education and had not used qualitative

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methods to gain direct insight with regard to feelings and experiences. This was an omission, as interviews and focus groups with children have recently shown that much can be found through discussions, and we’ll explore this later in the chapter.

A closer look at influential factors Maths anxiety is an emotional response that is often felt to greater extremes by those whose ability is already poor ( Witt, 2012) and can lead to the avoidance of working with numbers. It is the perception of being incapable of learning and applying concepts that leads to f light responses. In addition to withdrawal from maths, an individual may also experience feelings of guilt and particularly shame when faced with failure. Such emotions can reinforce the desire to avoid maths, so a vicious cycle ensues. The avoidance of maths work has been described as the ‘no-attempt’ error (Chinn, 2012). This means that rather than putting effort into solving maths tasks, some simply do not try, or perhaps cannot. The no-attempt error relates to the internalisation of persistent negative feedback as a consequence of failure. Essentially, individuals blame themselves for their lack of success/ability, which relates to feelings of inadequacy (Mutodi & Ngirande, 2014). Such feelings intensify when confronted with and failing to complete complex maths problems (Ashcraft, 2002). Indeed, failure is a key maths anxiety factor that links with many others. It is important to recognise that a person does not simply have or not have maths anxiety. At this point, try to think of anxiety as more of a ladder that ascends to higher anxiety (Figure 5.1). Each negative experience a person encounters in maths can be imagined as a step on the ladder, and naturally, a high frequency of negative experiences in the subject can lead to increased anxiety. Harari, Vukovic, and Bailey (2013) believed that maths anxiety would emerge following a significant period of time in which a person has internalised failure and had largely negative experiences. However, we cannot say how many experiences lead to a higher level of anxiety. Bearing the previous point in mind, and to complicate matters further, consider each person’s ladder has a different height. Thus, some can develop anxiety very quickly, whereas others may be more resilient and for them, it may take more negative experiences before they become highly anxious if they become anxious at all. In this sense, maths anxiety becomes a relative matter that is also defined by individual differences. This also explains observed differences in children’s early maths experiences and performance. In the last 10 years, articles and reports have referred to a growing maths competency problem in the United Kingdom. This, coupled with research in the area suggests that a negative correlation exists between maths anxiety and performance in children and adults (Ashcraft & Kirk, 2001; Maloney & Beilock, 2012). Government statistics reveal that almost half of the English population only have primary school maths skills, with the chairman of BT describing poor numeracy as “the

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Higher Anxiety

Lower Anxiety FIGURE 5.1

Maths anxiety can be visualised as a ladder

hidden problem that blights the economy and ruins individuals’ chances in life.” (Burns, 2012). It has also been claimed that this is very much a British based issue. Further, primary care providers (teachers and parents) can inf luence maths anxiety. The significant role that parents play has been well documented by research (e.g. Erden & Akgul, 2010; Gunderson, Ramirez, Levine, & Beilock, 2012) and suggests that it is not only the classroom environment that contributes toward early maths anxiety.

Maths anxiety and emotional responses To immediately emphasise the observed link between maths anxiety and achievement, Ashcraft and Moore (2009) stated that no other relationship is as troublesome. We often witness athletes underperforming due to sports performance anxiety, and this is the same as in maths. However, athletes are likely to receive support from sports psychologists, whereas students and others living with maths anxiety often do not understand their own thought processes (cognitions). Like athletes in their area of specialism, high levels of anxiety have been shown to impact performance in maths (Maloney & Beilock, 2012) yet we know that athletes have an underlying ability, even when they underperform. This can

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be the same case for individuals with maths anxiety. Essentially, their anxiety can mask their true ability. This has been evidenced in research such as that of Sheffield and Hunt (2007), who reduced high levels of anxiety in students, coupled with a performance increase, through a simple intervention. For young children who are immediately baseline assessed in reception, if they already have some anxieties, this may be mistaken for lower ability. We now have a greater understanding of the thought processes involved in maths anxiety. Emotional responses can also range in severity from small frustration to overwhelming disruption, such as panic, paralysis and mental disruption (e.g. Tobias & Weissbrod, 1980). This would be the point at which a person begins to cry and possibly verbalise their feelings. These may be aimed at themselves, such as, “I’m absolutely useless and everyone else can understand this except me”. This is not a response exclusive to children; indeed, many adults can be brought to tears by maths. This highlights the emotional element. In research, individuals with maths anxiety have demonstrated poor performance when solving maths problems, even though they perform as well as others in most thinking and reasoning tasks (Maloney & Beilock, 2012). Lundberg and Sterner (2006) consider that arithmetic performance is inf luenced by a number of motivational and emotional factors such as helplessness, depression, anxiety and self-esteem. Based on such findings, the emotional element of maths anxiety seems particularly inf luential and, again, points toward a clear disadvantage for the maths anxious.

Anxiety, achievement and brain mechanisms In 2009, Luo, Wang and Luo conducted research with middle school students in the United States and found that a negative correlation existed between maths anxiety and maths performance. The overall results demonstrated that those who had higher levels of maths anxiety performed poorly on maths tasks and that negative emotional elements were also apparent; the anxiety levels left learners in a cognitively passive state. Additionally, Lyons and Beilock (2011) have shown that when those with high levels of maths anxiety simply anticipate a maths task, they show greater activity in a particular brain network – the frontoparietal network – that is involved in the regulation of emotion. So, if this is applied to a school setting, when an already maths anxious child thinks about the upcoming maths lesson, they can start to experience a negative emotional response. In research with children aged 7–9 years, Young, Wu, and Menon (2012) also found a link between maths anxiety and the amygdala – part of the frontoparietal network that regulates emotions. Specifically, within the highly maths anxious, the link between the amygdala with cortical regions that process negative emotions was particularly evident in relation to lower problem-solving accuracy. For those with low maths anxiety, brain regions involved with emotions showed greater deactivation, meaning that they experienced less of an emotional response to working with numbers and solving problems. They could therefore

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perform at a much higher level than those with higher anxiety. However, when those with high maths anxiety are taught strategies to regulate negative emotions, they can perform at almost the same level as the low maths anxious. This demonstrates that an emotional element is inf luential in maths anxiety but can be alleviated to reveal genuine ability.

Factors associated with children’s maths anxiety Until very recently, research has overlooked the early years of education when maths anxiety may begin to develop. This might seem like quite a large oversight, but research has been able to suggest and point to what can inf luence early attitudes toward maths. We will now look at some of these important factors, and later in the chapter, some direct feedback from children aged 4–7 years will be considered. A main point to consider is that children in early education can sometimes encounter negative evaluation from peers and teachers, particularly if the children consistently underperform in maths ( Beck, 1989; Ashcraft & Krause, 2007). Evaluation anxiety is thought to be another individual factor associated with maths anxiety. Donaldson, Gooler, and Scriven (2002) consider this to be inherent in human beings and encapsulated by emotions such as embarrassment and ridicule. Potential negative consequences of evaluation can lead to responses such as avoidance, and negative evaluation is considered to be a particular issue in early childhood ( Beck, 1989). In relation to maths anxiety in children, Ashcraft and Krause (2007) suggest this is learnt in the classroom and consider negative evaluation by peers, and in some cases teachers, to cause embarrassment when publicly performing maths poorly (Hadley & Dorward, 2011). This increases their chances of developing a negative self-image (or attitude), which can impact performance ( Nicolaidou & Philippou, 2003; Dowker, Bennet, & Smith, 2012). These negative thoughts will drain essential working memory capacity to the point that there is nothing left to devote to task completion. Haase et al. (2012) have found that when children with maths difficulties have self-assessed their ability and feelings, their performance often negatively ref lects this. In contrast, children with a more positive outlook on their ability show higher performance. What’s more, research has found that children with optimistic attitudes toward maths in primary school show higher performance later into their secondary education ( Yates, 2002). The opposite was found for children with pessimistic maths attitudes in primary education. This indicates that early educational attitudes can remain consistent over many years and that a good start is important. These ideas relate to ‘self-efficacy’, a theory that corresponds to social cognitive theory (Bandura, 1986). This is based on the idea that individuals will more likely engage in activities if they believe they have the capability to complete them. Selfefficacy can therefore be an outcome-dependent factor in a wide range of activities, including maths. When applying this theory to maths anxiety, some children have been found to have higher self-efficacy than their peers (Meece, Wigfield, &

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Eccles, 1990). The effect of self-efficacy on maths was shown by Pajares and Graham (1999) who found that US children in sixth grade with higher self-efficacy (UK age 10–11 years) were more persistent, interested and performed better at the end of the school year in comparison to children with lower self-efficacy. Thus, there is some evidence of self-efficacy as a key determinant of maths achievement. This also has an association with maths anxiety (Linder & Smart, 2010), such that those with higher self-efficacy are more likely to persist with maths, particularly when it becomes difficult. As described earlier, increasing difficulty in maths can be the point at which a person can start to experience and show more signs of anxiety – and this can happen in early education. Indeed, in research with children aged 9–11 years, Hunt, Bhardwa, and Sheffield (2017) found systolic blood pressure increases when mental arithmetic problems became more difficult; such reactivity was also shown to be related to self-reported maths anxiety. Ashby (2009) found that children failed to understand the wider practicalities of maths and this can be linked to decreased motivation ( Tella, 2007), negative attitudes, avoidance and lower performance. This is important, as motivation, task completion and seeking help are necessary for success but are badly affected by anxiety. In a questionnaire study, Anthony (2000) collected data from 92 undergraduate students and 26 lecturers to gauge agreement or disagreement with statements based on factors inf luencing success in maths. Poor performance was associated with students being uncertain of the skills required for maths courses or being unable to apply these appropriately. This revealed that the difficulties faced by adults relate to those identified in children and further suggests that anxiety may develop in early education and persist into adulthood. This highlights the importance of understanding the maths experiences of children. The consequences of low motivation in the early years can also persist into later education, as shown by Zakaria and Nordin (2008). By measuring students’ maths anxiety in relation to achievement and motivation, they observed that low achievers often have high maths anxiety and reduced motivation, as well as lower performance. So, to put this all together, maths anxiety in the later years can relate to low motivation in earlier educational years and may be the outcome of an apparent lack of understanding of the purpose of maths in the wider world. However, we cannot be certain of the causality of factors; we do not know whether maths anxiety is the cause or the effect. This again demonstrates the complexity of maths anxiety. Research has typically shown that higher levels of maths anxiety are linked to decreased performance. Yet in more recent research showing the inf luence of motivation, Wang et al. (2015) found that moderate levels of maths anxiety actually facilitate maths performance, but only when the learner has high motivation to learn, highlighting the role of individual differences. The avoidance of maths is another factor that can inf luence anxious responses and performance. Chinn (2012) found that if children perceive a maths task to be complex, this can induce anxiety and fear which is linked to task avoidance and what he called ‘no attempts’. This is exactly what it sounds like: children simply

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do not answer a question because they see this as a better option than providing an entirely wrong answer. Specifically, subtraction, multiplication and division place greater demand on working memory capacity, and children’s avoidance of these more complex tasks suggests that some may not have a positive belief in their ability (low self-efficacy) and may fear failure. Indeed, Chinn considered that many maths procedures are unforgiving of inaccurate memories. In light of this, it is important to remember that young children’s memories are going through a development process and are restricted in how much information they can store in the short term (Croker, 2012). Some children may be better than others at storing and accessing the required information when doing maths and is another point to consider. Ramirez, Chang, Maloney, Levine, and Beilock (2016) have explored the association between maths anxiety and achievement, with an emphasis on the problem-solving strategies that children employ. Young children use basic strategies, such as finger counting, but following repeated use, develop “problemanswer associations” ( Ramirez et al., 2016, p. 84) such as understanding that 3 + 3 = 6. Building on this, children begin to use strategies such as retrieval, decomposition and reconstruction, which are suggested to be more intensive working memory strategies ( Ramirez et al., 2016). In terms of the relationship strategies have with working memory and anxiety, Ashcraft (2002) has previously shown that simple arithmetic does not require significant working memory processing, but more advanced problems, including carrying and borrowing operations, place greater demand on working memory and increase anxiety. Higher levels of anxiety are associated with complex maths, such as algebraic equations, due to the processes involved in order to achieve the correct solution.

Measuring maths anxiety in children: self-assessment scales for older children Despite the focus on adults, some research has attempted to measure the degree of maths anxiety experienced by older children. Suinn, Taylor, and Edwards (1988) developed the Maths Anxiety Rating Scale for Elementary School Students (MARS-E). This consists of 26 questions (items) and uses a five-point Likert system. This means that there are five options that children can respond to for each item. The MARS-E was developed with 1,119 fourth (age 9–10, UK Year 5), fifth (age 10–11, UK Year 6) and sixth graders (age 11–12, UK Year 7) from six schools in the United States. Children were asked to indicate the level of anxiety they experienced in certain situations, for example, when reading a maths chapter. The total score, indicating either higher or lower anxiety, was correlated with SATs scores. This suggests that children’s self-assessment of anxiety scores were reflected in their maths performance. Specifically, it was found that children had anxieties about maths tests and their performance in general. However, the age range that this scale focused on still overlooked the earlier educational years when anxiety

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may have already begun to develop. Moreover, maths test and general performance anxiety can potentially be the consequences of earlier negative experiences. To demonstrate this point, maths experts have discussed observing performance anxiety in much younger children when working in front of others (Petronzi et al., 2017). This implies that certain aspects of maths anxiety develop in the early years and can persist over time. Chiu and Henry (1990) also developed a maths anxiety scale for children that consisted of 22 items: the Maths Anxiety Scale for Children (MASC). However, this again overlooks the earlier years of education. The MASC was developed with 562 children from fourth grade (age 9–10, UK Year 5) to eighth grade (age 13–14, UK Year 9) in the United States. As with the MARS-E, children rated their level of nervousness in response to questions that considered, for example, their feelings when taking a maths test and when asked to make sense of maths questions. Children’s self-assessment scores were compared against their maths results from the school term. The results demonstrated that children who scored highly on the MASC (higher maths anxiety) had lower achievement in maths and higher test anxiety. This clearly has implications in a school setting and shows how maths anxious children can be disadvantaged, particularly in test scenarios. Specifically, children were found to have concern with maths evaluation, learning maths and solving problems, as well as feelings of anxiety towards their teachers. While the findings were useful, the assessment scale overlooks several years of potential development of a negative attitude toward maths. In recent years, there have been some notable attempts to focus measurement on children’s maths experiences, including Dowker, Bennett, and Smith’s (2012) measure of attitudes toward maths for primary school children (aged 7 years and older) and the Modified Abbreviated Math Anxiety Scale (Carey, Hill, Devine, & Szucs, 2017) for children aged 8–13 years.

Measuring maths anxiety in younger children Researchers in the United Kingdom have developed a reliable self-assessment scale to be used specifically with children aged 4–7 years ( Petronzi, Staples, Sheffield, Hunt, & Fitton-Wilde, 2018). The purpose of the Children’s Mathematics Anxiety Scale-UK (CMAS-UK, see the Appendix for the full scale) is to support the assessment of children who may be showing signs of a negative attitude toward maths at an early age. This was the outcome of a project to develop a comprehensive understanding of how children experience dayto-day maths and what inf luences their experiences. Early identification can enable educators or primary care providers to intervene and support children in working toward a more positive attitude and success. The CMAS-UK was developed with 470 UK children and was shown to correlate anxiety levels with performance scores. More precisely, children who obtained a higher anxiety score on the scale typically demonstrated lower performance on a numeracy task.

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A core premise of the CMAS-UK relates to assumptions that maths anxiety, a construct evidenced as relating to the experiences of older children and adults, is applicable to the experiences of younger children. It is therefore suggested that the term ‘numerical apprehension’ better describes the anxieties of children in early education. This is more representative, as at this educational stage, numeracy, rather than maths, is taught in the classroom and research data suggest that maths anxiety is a subsequent and further developed issue, due to an association with complex mathematical procedures – difficult multiplications, algebra and fractions (Ashcraft, 2002).

Maths anxiety in younger children: feedback from 4 to 7-year-olds We now turn attention to the feedback provided by UK children aged 4–7 years (Reception, Year 1 and Year 2) who discussed their numeracy experiences and attitudes as part of a research project. This was an attempt to address our limited understanding of the origins of maths anxiety. Specifically, this research explored the factors that can shape children’s attitudes toward numeracy as well as primary care providers’ observations of children’s attitudes and responses to working with numbers. The research sample included children aged 4–7 years in reception (N = 11), year one (N = 18) and year two (N = 12) at three primary schools in the United Kingdom, and parents (N = 7), teachers (N = 9) and maths experts (N = 2). Insight was obtained from the use of focus groups. Four main themes emerged as being part of children’s numeracy experiences and attitudes: (1) emotional responses, (2) coping strategies, (3) teachers/teaching and (4) influence/perception of others (Figure 5.2). For each factor, there was a clear difference in the comments made by children with a positive numeracy attitude and those with a negative attitude. It is noteworthy that children who showed a positive numeracy attitude did not make any statements in relation to punishment, avoidance, failure or fear. The following sections present findings from children and maths experts.

Theme 1: Emotional responses to numeracy Young children’s accounts of their numeracy experiences suggested a number of inf luential and interacting factors. In particular, emotions became a salient point and a difference became clear between those with negative and positive attitudes.

Success Achievement in numeracy is an ambition and motivational force for a number of children, and success promotes positive emotional responses. A sense of attainment and other positive emotions can be reinforced by reward systems that promote rivalry and clearly identify the most numerically competent

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Emotional Responses

Positive

Success

Numeracy at Home

Negative

Perception of High Ability

Avoidance

Fear

Perception Difficulty / of Low Confusion Ability

Positive

Negative

Failure Coping Strategies

Friends / Copying Others

Removal of Strategies

Parents

Teachers / Teaching

Negative

Positive (Guidance, Motivation & Rewards)

Punishment & Provokes Apprehension Helps with Work Influence / Perception of Others

Friends

Comparison / Competition & Awareness of Hierarchy FIGURE 5.2

Parents

Attribution of Blame

Receiving Help or Not & Work with Maths

Four main themes of children’s numeracy experiences

children. In year one (UK age 5–6 years), children seem to have an awareness that enjoyment of numeracy results in doing more of it, making their own connections between attitude, quantity of work and performance. This suggests that those who enjoy the subject are higher achievers, in part, due

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to additional practice. Examples are provided here and for subsequent themes that are discussed: Happy, because if you like doing numeracy, you’ll have to do lots of it. I get a bit confident when I answer some right, then I do the same with the others.

High ability Children independently recognise that positive emotions link to a sense of high ability, confidence and successful completion of numeracy work. A perception of high ability seemingly enables children to recognise limitations in their knowledge without suffering any negative emotional consequences and performance effects. This is supported by children expressing belief in their ability when faced with a high amount of maths work, evidencing greater resilience.

Worry/fear Some children feel worried when presented with numerical tasks, typically believing that they lack the required level of ability – even at an early age. These worrisome thoughts place the child at risk of developing negative attitudes toward numeracy, and in relation to the hierarchy of needs (Maslow, 1943), will restrict a child in establishing safety in education and self-actualisation. Moreover, fear and perceived inability has an association with a fixed mindset (Dweck, 2006). For these children, attention diverts from work to focussing on worries and can inhibit working memory (Ashcraft & Krause, 2007) which results in a greater performance decline. Physiological anxiety is also associated with numeracy apprehension, such as “feeling your heart beating so fast”, and avoidance tactics present struggling children with the opportunity to conceal difficulties. In this scenario, a lack of motivation may develop through consistent low performance and a lack of enjoyment (see the following examples): Sometimes you’ve got like twenty answers and you think, how am I going to do this? Sometimes I’m a little bit bored and then I want to get out.

Failure Failure in numeracy is a prospect that produces fear and critical self-evaluation in children. Constant apprehension about failing links to a decline in task completion. Thus, if a child consistently fails to complete work, their attitude toward numeracy may ref lect this. Moreover, genuine ability may be masked by a focus on failure, with some visualising the negativity failure will bring, rather than the positivity of success, for example: A bit ashamed if I got it wrong. I always get it wrong.

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Low ability Some children genuinely have no confidence in their numeracy ability and although they attempt work, they may also have pessimistic outcome expectations. In this scenario, children seem to be aware of their specific weaknesses, and negative emotions intensify when these aspects are called upon in lesson, for example, with multiplication. Furthermore, some children aspire to complete numeracy work competently, but often feel incapable of doing so. This shows that despite some children developing a low sense of self, their motivation and attitude can remain intact and there are opportunities for intervention, for example: I’d work it out, but probably get the wrong answer.

Difficulty/confusion Children expressed significant emotional negativity when discussing experiences of attempting hard work. The difficulty of numeracy work is either met as a positive challenge by those who are secure with numeracy or as an obstacle by others. Anger and frustration was expressed toward difficult numeracy. In younger children, confusion was based on number recognition and representation and further contributed to negative attitudes toward numeracy. Pressure was expressed as a consequence of confusion, and children were conscious that when they are unable to do the work, they fall behind others – also highlighting the competitive nature of the subject. A sense of being under pressure arose from a perception of the teacher or fellow pupils observing their struggle, like in these examples: When I’m writing the numbers I sometimes get them back to front and I don’t really notice. Like if it’s two, you say five because they look the same.

Theme 2: Coping strategies Children uncomfortable with numeracy revealed methods they would employ to cope with numeracy lessons. Significant emphasis was placed on friends and how their help eased the pressure of the situation, whilst other children discussed codeveloped tactics to achieve the correct answer without being ‘found out’. This may explain how some children’s discomfort with numeracy and low performance can remain unnoticed in the early school years. Additionally, when coping strategies were removed, apprehension became dominant. Parents acted as another coping strategy for some children and were utilised whenever numeracy became difficult; for others, though, this strategy was either not in place or was employed to a lesser extent.

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Friends Friends are a clear vessel of help in the classroom and typically have a positive impact on children’s emotional responses. However, for some, having to accept help from peers caused negativity as this seemingly confirmed their inability to understand and complete the work. Others were able to recognise that their lower ability was not ideal and could accept help in an attempt to learn from this, while others were simply happy to ‘survive’ the lesson. Demonstrating this survival approach, some children will identify those that are confident with numeracy and exploit this ability for their own needs – typically by copying. Indeed, a great deal of effort is placed on numeracy lesson ‘survival’ rather than attempting to complete the work itself, as expressed in the following example: You just wait until they say the answer or you can just copy their work.

Removal of strategies When presented with the situation of numeracy strategies being disallowed (typically for testing scenarios) increased worry can arise in children who are most at risk of numeracy apprehension. Without their coping strategies, they may feel unable to attempt work and leave questions unanswered. Some children believe that when help is forbidden, their work will be wrong, and the teacher will be frustrated with them. Others have expressed concern about isolation from their friends when help is removed, as expressed in the following: Sometimes the teacher says you’re not allowed to be helping people so you just carry on with your own work and it’s kind of difficult and you can’t really concentrate properly.

Parents Highlighting the importance of numeracy support outside school, children secure in their ability typically receive help from parents which facilitates their learning. Children with parents in a position to help with numeracy will be spared from feeling incapable and apprehensive, particularly when the difficulty level increases, for example: If it was really hard, I would ask my dad to help me.

Theme 3: Teachers and teaching For some children secure with numeracy, teachers were regarded as the instructor who guided them through work, and they stated that listening to instructions results in work being correct, demonstrating their consideration of behaviour as a

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determinant of success. These children also take comfort knowing that the teacher is observing them and feel valued and supported. In contrast, observation causes apprehension for those uncomfortable with numeracy. Some children have also expressed happiness when explaining answers to the teacher as this provides an opportunity to demonstrate their knowledge. However, some children experience fear and apprehension in response to all aspects of their teacher’s presence.

Theme 4: Influence/perceptions of others It has been established that children’s numeracy experiences are inf luenced by others and the child’s perception of those persons being either a facilitator of work or a threat. A strong emphasis was placed on friends as a coping strategy in numeracy, with the highly apprehensive relying heavily on their input to ease pressure and remain unidentified. However, other factors surfaced relating to friends, particularly competing with them, and attributing blame to them for failure. Though parents were also previously discussed as a coping strategy, more specific details were expressed by children and clearly inf luenced their approach and attitude to numeracy.

Friends: Comparison/competition and awareness of a hierarchy Children who are secure in their numeracy ability may view the subject as a competition and regard fellow students as rivals. However, the findings suggest that numeracy is a subject that all children acknowledge as highly contested, yet those of lower ability are often unable to maintain the pace and workload of higher performing children. Indeed, by necessity, their focus is simply surviving each lesson. The competitiveness of numeracy results in negativity when children finish work after others and some children internalise the cause of their failure. The speed by which numeracy can be completed is a key factor, and for some, if answers are different to those of others, they typically assume that their work is wrong which exemplifies pessimistic attitudes (Yates, 2002), as in the example below: They might have got it right and we might have got it wrong. Also revealing a sense of competition, children in years one and two (UK age 5–7 years) have indicated an awareness of an intelligence hierarchy established through numeracy. As an example, a child conceded that a group of children received more challenging work as they were ‘brainier’ than others. This demonstrates how children in early education are already comparing themselves against the ability of others, using numeracy as the subject to define intelligence, for example: I sometimes beat them, I just think it’s a race. Rockets are more brainier than stars, so stars get different work from rockets.

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Parental support The assurance that parents can help children with numeracy at home can prevent panic and fear. Although secure with numeracy, children typically require assistance from parents when difficulty increases. This allows them to overcome issues immediately and without experiencing negativity. This is in contrast to children who do not receive much help at home. It is worth considering that as techniques and terminology have altered significantly in numeracy, the methods of some parents are now out-dated and meaningless to children, potentially confusing them further. The research showed that a child’s comfort with completing numeracy at home resulted from their parents working with maths. This demonstrates the value of parents providing an intellectually stimulating environment in the household (Mazzocco, 2007).

Motivation and educational psychology theories If we apply numeracy apprehension to motivational and educational psychology theories, such as Maslow’s hierarchy of needs (1943) (Figure 5.3), early difficulties and negative self-evaluation impede children in achieving esteem and selfactualisation. Indeed, many children will struggle to surpass the ‘safety’ phase of their hierarchy of needs as their anxiety denotes a fear of working with numbers and is related to similar factors, such as failure. In the same way, children cannot achieve educational stability and security when they are experiencing worrisome thoughts in a classroom setting. Although dated, this theory continues to inform new ideas around children’s needs and education. More recently, Dweck’s (2006) Mindset Theory relates to children viewing their intelligence as stationary and effort as ineffective. Applying this theory, Hierarchy of Needs

Selfactualisation Esteem Love/belonging Safety Physiological FIGURE 5.3

Maslow’s hierarchy of needs

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children who experience worrisome thoughts will avoid challenges as their failure highlights a lack of intelligence. In contrast, a growth mindset applies to children who regard intelligence as changeable and embrace numeracy challenges as a platform for improvement. We have seen evidence of this in children’s feedback as they consider effort as entirely worthwhile.

Insight from maths experts We have explored children’s feedback, but let’s now consider insight from maths experts obtained in the same study ( Petronzi et al., 2017). Working as university academics, the experts that took part had also taught and observed maths attitudes and behaviours in primary, secondary and further education. Their insight was obtained in interview settings which allowed for a detailed conversation about factors deemed inf luential in how children experience numeracy in early education.

Main findings Pressure and numeracy in public Maths experts discussed occasions where they had witnessed children being asked to solve numeracy problems in front of others, placing them in a negative pressure situation. Expanding on this, males were identified as being generally more confident when completing numeracy in front of others and being less affected by mistakes in comparison to females. This is in line with Devine, Fawcett, Szucs, & Dowker (2012), who found males to have more persistence and resilience following mathematical mistakes. Females were discussed as showing behavioural signs of anxiety in classrooms when attempting to avoid solving a numeracy problem in front of others (Mutodi & Ngirande, 2014), for example: Males are generally more confident and also more confident at making mistakes. Specifically relating to apprehension, a maths expert discussed the consequences for children who became overwhelmed with the pressure of numeracy, claiming that sudden reactions such as crying have been observed and are specific to the subject of numeracy. Crying, literally crying. I could say almost in no other subject, only in maths. A maths expert expressed their own feelings of anxiety when teaching the subject, demonstrating an inherent anxiety associated with numeracy and when solving numerical or mathematic problems in front of others ( Beilock, Gunderson, Ramirez, & Levine, 2010). The inherent nature of numeracy and negative public experiences may place some children at risk of developing aversive

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emotional responses to working with numbers, and will impact on their learning, ability and relationship with the subject (Ashcraft & Krause, 2007).

Difficulty of numeracy Maths experts discussed the perceived difficulty of numeracy as contributing to the formation of how learners experience the subject, including why some seek to avoid work. Polarities in feelings toward numeracy are often apparent and reinforced with self-critical or positive evaluations, as we have seen from children’s insight, such as: If people actually begin to perceive it as being a hard subject, that will increase their anxieties. The right or wrong nature of numeracy was also identified as a further issue of the subject that links to difficulty and apprehensive reactions (Chinn, 2012). Once anxiety develops, maths experts conceded that this can take some time to alleviate. Ref lecting upon their experience in primary schools, the method of teaching numeracy procedurally was also discussed as contributing to a perception of difficulty and ultimately, apprehension. Additionally, a proportion of the difficulty of numeracy may be attributed to the language used in the subject, particularly when children encounter worded problems (Ginsburg, 1977). Maths experts further discussed concepts, such as learning and working with the teen numbers, subtraction and division as being particularly difficult for children and causing a degree of apprehension (Chinn, 2012). For example: Moving into teen numbers is a problem for young children and subtraction as an operation is also tricky for children. There’s a similar issue with division. Demonstrating that negative numeracy attitudes are being learnt in the early years when some children encounter difficulty, maths experts revealed that undergraduate students’ years bring with them established negative attitudes toward the subject.

Influence of teachers (negative) Some teaching methods have been implicated in causing boredom and producing a lack of motivation, resulting in counterproductive behaviours such as talking with friends. Maths experts spoke of children encountering difficulties with incorrect numeracy terminology used by teachers who may not be confident with the subject. Thompson and Rubenstein (2000) also consider that teachers often forget that mathematical language is foreign to many students and identify issues with the vocabulary used in maths to convey the ‘surface structures’ that

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help students form ideas as they progress to the ‘deeper structures’ of maths concepts. Furthermore, maths experts believed that some student teachers underestimate the importance of being able to teach basic numeracy/maths and concept understanding in a primary school classroom ( Uusimaki & Kidman, 2004).

Low sense of ability Maths experts revealed some of the behaviours and attitudes of children who felt they were of low numeracy ability, which typically include disengagement from the subject and refusing to do the work. This demonstrates a pessimistic explanatory style and a decrease in persistence and assertiveness ( Yates, 2002). For example: There was a sense that they felt that they couldn’t do it and therefore, weren’t prepared to engage with it. In these cases, children may have adopted the belief that if they do not engage with numeracy, they have not failed to complete the work, as they have chosen not to do it.

High sense of ability Maths experts discussed the belief of ability as being a motivator to challenge the subject further ( Pajares & Graham, 1999) and also claimed that the performance gap would increase between children who were secure with numeracy and those who were not. Maths is quite an intuitive area; if you have ability, you’re quite keen to move on and challenge yourself. Maths experts considered age seven to be the point that children become aware of their ability and attitudes towards numeracy, potentially determining their trajectory in the subject for the remainder of their education. However, this contradicts Mcleod (1993) who considered ages 9–11 years to be the critical stage for the development of maths attitudes and emotional reactions. Additionally, it was discussed that those with a high sense of ability may not necessarily understand a concept immediately but have an intrinsic motivation to learn and find a solution.

Influence of parents (negative) Maths experts discussed the role of parents in inf luencing how children experience numeracy and identified that many parents in the United Kingdom have faced difficulty in their own numeracy/maths experiences. They considered the possibility of negative attitudes being transferred to children (Gunderson et al., 2012). Developing the idea of transference of attitudes, maths experts revealed

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experiences of parents stating that due to their own inability in numeracy, they knew their child would also have difficulties with the subject (Gunderson et al., 2012). If the child is aware that they are not expected to perform well, it was thought that they will respond according to this self-fulfilling prophecy. A lot of parents come into school and say; I wasn’t good at maths so they’re not going to be good at maths. The child listens to this. Offering an alternative perspective, maths experts brief ly considered the pressure placed on children to perform as having a detrimental effect. This may lead to self-consciousness about numeracy performance related to not meeting parental expectations ( Yuksel-Sahin, 2008).

Comparison/competition Maths experts discussed children as comparing themselves against others in numeracy lessons and how this would regulate how they perceived their own performance. Discussions revealed that when comparing ability, the children who have difficulty with numeracy begin to develop a negative self-perception (Erdogan et al., 2011) as they realise that other children are more secure with learning and doing numeracy. It also emerged that children who are secure with their numeracy ability may negatively perceive children who have difficulties, with one maths expert considering them to be patronising. When children who struggle with numeracy are aware of how they are negatively perceived by the more secure children, this is likely to further affect how they perceive themselves and their ability.

Fear/anxiety Failure is a main component of the fear associated with numeracy. In some cases, children are unsure of what method to use and are fearful to write it down, with their uncertainty being further compounded by time restrictions ( Wong & Evans, 2007). In other instances, maths experts had observed some children sitting back in lessons and not attempting the work to hide their failings. In this case, the children have adopted the belief that numeracy work is beyond their ability and choosing not to work is more self-preserving than to be stigmatised as failing (Chinn, 2012). A fear of failure in numeracy was also attributed to the right or wrong nature of the subject (Chinn, 2012). If experience has taught children to be fearful of wrong answers, they are unlikely to attempt their numeracy work and likely to disengage from the subject. This disengagement can be evident through the remainder of their education.

Avoidance Maths experts discussed in detail the numeracy avoidance behaviours that children have displayed. This is supported by previous research, in which a significant

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negative correlation was shown between being nervous about maths and the intention to take on more maths (Meece et al., 1990). An initial point was that those who enjoy the subject are eager to make the teacher aware of this ( Petronzi et al., 2017), whereas those who are not secure with their numeracy ability will attempt to hide their work. Maths experts stated that children with incorrect work will often cry and they also discussed extreme behaviours – revealed by parents – that children would display at home to avoid doing numeracy work, such as tantrums, although these behaviours were not evident in the classroom. However, they revealed that some children would ask to go to the toilet when given numeracy work and others would simply begin to display negative behaviours and become disruptive as a form of avoidance. Avoidance will, of course, only serve to increase the gap in concept knowledge and performance between them and other children.

Chapter summary Until recently, the extant literature on maths anxiety has paid relatively little attention to the early years of numeracy education, which has limited knowledge surrounding the onset of maths anxiety and its possible associations with factors in the early educational years. Several factors may inf luence the development of such apprehension, such as negative evaluation from peers and teachers, pessimistic attitudes, low confidence, low self-efficacy, a resignation to failure, worry relating to performance, reduced motivation and avoidance of perceived complex tasks. Norwood (1994) and Hadfield and McNeil (1994) considered that maths anxiety is a product of multiple factors, rather than one inf luential cause; we support this idea. The research findings presented in this chapter suggest that numeracy apprehension may develop as a precursor to maths anxiety. The findings indicate that children in early education can begin to develop worrisome thoughts about working with numbers and this can occur much earlier than once thought. The findings of Petronzi et al. (2017 ) relate, however, to only a single area within the United Kingdom, and thus more widespread research would be beneficial. Nonetheless, it is necessary for future researchers to consider the multitude of potential inf luences on the development of numeracy apprehension and evaluate intervention techniques that are known to be efficacious with older children to minimise the development of numeracy apprehension at its roots. Moreover, novel approaches to promoting and maintaining positive attitudes toward numeracy should be explored and measured in future research in the area.

Note 1 Within this research, numeracy is referred to as the ability to reason with numbers using basic mathematical skills (e.g. addition, subtraction, multiplication) and applying these to everyday life situations. Maths is referred to as the ability to use numbers and specialised operations in complex subfields (e.g. algebra, equations).

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Petronzi, D., Staples, P., Sheffield, D., Hunt, T., & Fitton-Wilde, S. (2018). Further development of the children’s mathematics anxiety scale UK (CMAS-UK) for ages 4-7 years. Educational Studies in Mathematics, 1–19. doi:10.1007/s10649-018-9860-1 Ramirez, G., Chang, H., Maloney, E. A., Levine, S. C., & Beilock, S. L. (2016). On the relationship between math anxiety and math achievement in early elementary school: The role of problem solving strategies. Journal of Experimental Child Psychology, 141, 83–100. Rossnan, S. (2006). Overcoming math anxiety. Mathitudes, 1, 1–4. Sheffield, D., & Hunt, T. E. (2007). How does anxiety inf luence maths performance and what can we do about it? MSOR Connections, 6, 19–23. Skemp, R. R. (1986). The psychology of learning mathematics. Harmondsworth: Penguin. Suinn, R. M., Taylor, S., & Edwards, R. W. (1988). Suinn Mathematics Anxiety Rating Scale for Elementary School Students (MARS-E): Psychometric and normative data. Educational and Psychological Measurement, 48, 979–986. Tella, A. (2007). The impact of motivation on students’ academic achievement and learning outcomes in mathematics among secondary school students in Nigeria. Eurasia Journal of Mathematics, Sciences and Technology Education, 3, 149–156. Thompson, D. R., & Rubenstein, R. N. (2000). Learning mathematics vocabulary: Potential pitfalls and instructional strategies. Mathematics Teacher, 93, 568–574. Tobias, S., & Weissbrod, C. (1980). Anxiety and mathematics: An update. Harvard Educational Review, 50, 63–70. Uusimaki, L., & Kidman, G. (2004, July 4–11). Challenging math-anxiety: An intervention model. Reviewed paper presented at the International Conference of Mathematics Teacher Education (ICME), Copenhagen, Denmark. Wang, Z., Lukowski, S. L., Hart, S. A., Lyons, I. M., Thompson, L. A., Kovas, Y., . . . Petrill, S. A. (2015). Is mathematical anxiety always bad for math learning: The role of math motivation. Psychological Science, 26, 1863–1876. Witt, M. (2012). The impact of mathematics anxiety on primary school children’s working memory. Europe’s Journal of Psychology, 8, 263–274. Wong, M., & Evans, D. (2007). Improving basic multiplication fact recall for primary school students. Mathematics Education Research Journal, 19, 89–106. Yates, S. M. (2002). The inf luence of optimism and pessimism on student achievement in mathematics. Mathematics Education Research Journal, 14, 4–15. Young, C., Wu, S., & Menon, V. (2012). The neurodevelopmental basis of math anxiety. Association for Psychological Science, 1, 1–10. Yuksel-Sahin, F. (2008). Mathematics anxiety among 4th and 5th grade Turkish elementary school students. International Journal of Mathematics Education, 3, 179–192. Zakaria, E., & Nordin, N. M. (2008). The effects of mathematics on matriculation students as related to motivation and achievement. Eurasia Journal of Mathematics, Science and Technology Education, 4, 27–30.

APPENDIX

1. When my friends finish their number work before me, I feel . . . 2. If I am the last to finish numeracy work on my table, I feel . . . 3. If I make a mistake in numeracy, I feel . . . 4. When I can’t do my numeracy work, I feel . . . 5. When I have to explain a numeracy problem to my teacher, I feel . . . 6. If I think I can’t do my numeracy work, I feel . . . 7. When I see lots of numbers, I feel . . . 8. When I have to explain a numeracy problem to my friends, I feel . . . 9. If I have to finish all my numeracy work in lesson, I feel . . . 10. Listening to the teacher in my numeracy class make me feel . . .

NAME_________________________ Total Score_________________________

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Children’s Mathematics Anxiety Scale-UK The items in the questionnaire refer to day-to-day numeracy situations which may cause apprehension for children aged 4–7 years. For each item, children can place a circle around the face which describes how they feel in relation to the situation. 11. If I answer questions and get them wrong, I feel . . . 12. If I have to tell the teacher that I don’t understand my numeracy work, I feel . . . 13. If other children know that I find numeracy hard, I feel . . . 14. When I watch or listen to my teacher explain a numeracy problem, I feel . . . 15. If I don’t finish my numeracy work in class, I feel . . . 16. If other children finish their numeracy very quickly, I feel . . . 17. When I explain how I got my answer to my teacher, I feel . . . 18. When my teacher wants me to do numeracy at home, I feel . . . 19. Walking into the numeracy class makes me feel . . .

6 MATHEMATICS ANXIETY AND WORKING MEMORY What is the relationship? Maria Chiara Passolunghi, Marija Živković and Sandra Pellizzoni

Mathematics is all around us. Mathematical competence is one of the most important abilities that a person needs to master in life both for professional choices – especially for careers in science, technology and engineering ( Dougherty, 2003; Rivera-Batiz, 1992) – and for properly mastering everyday activities, (e.g., cooking, banking, shopping; Mazzocco, 2008) as well as personal wellbeing (e.g., Reyna & Brainerd, 2007; Reyna, Nelson, Han, & Dieckmann, 2009). Further, mathematical abilities not only affect the individual’s life but are also crucial for the entire society given their effects on economic opportunity and on meaningful participation in the community (Moses & Cobb, 2001; Peterson, Woessmann, Hanushek, & Lastra-Anadòn, 2011). In fact, there is a correlation between the development of society and mathematics: the more complex a society is, the more it requires advanced mathematical abilities. Recent progress in research on mathematical performance has given the professionals that work with mathematical abilities the opportunity to evaluate different areas that underline mathematical achievement, including both cognitive and emotional aspects. The role of these two factors is extensively studied in literature even though there is still lack of consensus concerning the beginning of the inf luence of math anxiety on mathematical achievement. Moreover, the reciprocal inf luence of emotional and cognitive aspects of mathematical task performance is far from clear. Scientific studies of the topic are scarce, provide limited information and are not conclusive, especially with regard to young children. It is crucial to investigate this topic, from both a developmental and a clinical point of view, in order to understand the developmental trajectories of these factors and to gain important information that may facilitate work with children with mathematical difficulties. Therefore, the chapter will provide an update and critical revision of the state of the art concerning the reciprocal effects between working memory (WM) and math anxiety (MA).

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To provide a clear overview of the theme, the chapter will be organized in different sections specifically addressing the different factors (mathematical abilities, WM and MA) that we are going to consider and then the specific relations between these aspects. In particular: 1 2

3

The first part will provide a description of WM processes; The second part will provide a short description of how different mathematical abilities are inf luenced by cognitive factors, with particular focus on WM; The third part will focus on the affective factors that inf luence mathematical learning. It also examines the reciprocal interplay between MA and WM, evaluating this relationship from a developmental perspective.

I. Working memory WM is a limited-capacity system that holds information for a brief period while simultaneously manipulating it ( Baddeley & Hitch, 1974). Baddeley and Hitch (1974) proposed an explanation for WM as a three-way system composed of the central executive (assumed to be an attention-controlling system), the visuo-spatial sketchpad, which manipulates visual images, and the phonological loop, which stores and rehearses speech-based information. Baddeley (2000) added a fourth element: the episodic buffer, a limited-capacity system that integrates and provides temporary storage of information from the two subsystems and long-term memory (see Figure 6.1). Studies of this element are very limited, especially within developmental psychology. For this reason, our analysis will not focus on this component.

FIGURE 6.1

Working memory as a four-way system

( Baddeley, 2000)

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The distinctions between the central executive system and specific memory storage systems (i.e., the phonological loop and visuo-spatial sketchpad) in some ways parallel the distinction between working memory and short-term memory. Working memory is a term that refers to a processing resource of limited capacity that is involved in the preservation of information while simultaneously processing the same or other information ( Baddeley & Logie, 1999). Another model of WM, named the continua model (Cornoldi & Vecchi, 2000, 2003), proposes that WM consists of two dimensions: the horizontal continuum and the vertical continuum. The horizontal continuum is related to verbal, visual and spatial material and receives sensory information from the outside world, but is simultaneously connected with the representations stored in the long-term memory. The vertical continuum involves the processes that require active elaboration between information from various sources and is defined by the level of processing information needed (“active” or “passive”). The conical structure represents several subsystems at the “passive” dimension of the horizontal continuum, completely independent of each other. Along the vertical continuum, it can be observed that the processes become progressively more independent of the form in which the information is presented. It will be much easier to understand this model through examples: a forward span task as an example of a “passive” task and a listening span task as an example of an “active” task. In the forward span task the participant is asked to remember increasingly longer series of words, figures or syllables (usually from 2 to 7 for children), and to reproduce them from memory in the same order of presentation. These tasks can be considered as “passive” because the material must be reproduced in the same format in which it was presented, and does not need to be manipulated or transformed to any significant degree. The listening span task is requires the participant to judge the truth or falsity of a series of sentences immediately after hearing them. The participant has to reproduce the last word of the sentence, and of each previous sentence in order, as far back as they can remember. The task is composed of an increasing number of sentences. The listening span task considered “active” because it requires both storage and a high level of control and processing of information. Some experimental data show that the level of difficulty of both types of task can be manipulated and that “passive” tasks can be made significantly more difficult than active ones (see Figure 6.2, Cornoldi & Vecchi, 2000, 2003; Passolunghi & Siegel, 2001, 2004).

II. Mathematical abilities Research has extensively studied the different components that characterize mathematical ability. For example, Geary (2004) listed the following components: word problems, calculation procedures, number fact retrieval and visuospatial abilities. Karagiannakis and colleagues (Karagiannakis, Baccaglini-Frank, & Papadatos, 2014) proposed a different way to approach mathematical abilities (and disabilities) proposing core numbers, memory, mathematical reasoning and

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FIGURE 6.2

Continua model

(Cornoldi & Vecchi, 2003)

visual spatial abilities as the foundations of mathematical development. Geary’s approach (2000, 2004) mainly focuses on domain-specific abilities as the basis of the different mathematical domains, while Karagiannakis’s model puts more emphasis on domain-general cognitive abilities ( Karagiannakis et al., 2014) at the base of mathematical learning. Though the debate over whether domaingeneral vs. domain-specific abilities are the main foundations of mathematical development is still ongoing, the literature agrees that mathematical learning requires a variety of different abilities in relation to the different topics. In this chapter we describe some specific aspects of mathematical learning that have been extensively studied in the literature. These aspects are strongly related to WM and recent studies suggest that they may be affected by MA: 1) the approximate number system, 2) calculation skills and 3) problem-solving.

The approximate number system Numbers represent a basic way of describing a world and are abstractions that can be applied in different real or imaginary situations (four pieces of cake, four peers, four windows, four trees, four times, four minutes, four unicorns) (see, e.g., Spelke, 2017). Despite the ubiquity and simplicity of natural number concepts, their origins are debated and their cognitive foundations are still unclear. A wealth of empirical studies have suggested that both human and many nonhuman animal species share an intuitive “number sense”, that comprises a variety

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of competencies, such as the ability to subitize and count to distinguish number patterns, to discriminate quantities and to switch between different numerical formats ( Jordan, Kaplan, Locuniak, & Ramineni, 2007; Libertus, Feigenson, & Halberda, 2011). One central component of this number sense is the approximate number system (ANS) (Feigenson, Dehaene, & Spelke, 2004). This begins as a primitive, non-verbal and noisy cognitive system that allows the representation, estimation and discrimination of numerical quantities in an imprecise and intuitive way, without using counting or numerical symbols (Gilmore et al., 2013). More specifically, ANS can be described as an innate, preverbal system, which is independent of language, and remains active during the entire lifespan ( Dehaene, 1997; Feigenson, Libertus, & Halberda, 2013). Several studies have investigated this topic through research on specific characteristics of the ANS. First of all, ANS-acuity involves the degree of accuracy of the internal quantity representation, and there are significant interindividual differences in ANS precision ( Park & Starns, 2015). Many studies have found correlations between this aspect of ANS and mathematical abilities (Geary, Hoard, Nugent, & Rouder, 2015; Libertus, Feigenson, & Halberda, 2011, 2013; Soto-Calvo, Simmons, Willis, & Adams, 2015; Van Marle, Chu, Li, & Geary, 2014). Other characteristics of ANS are the distance and size effect. The distance effect means that it is easier to discriminate numbers that are further apart in the numerical distance (2 vs. 8 is easier than 3 vs. 5); and the size effect means that it is easier to discriminate smaller numbers compared to larger ones with the same distance between them (3 vs. 5 is easier than 33 vs. 35). However, not all studies have shown a link between the ANS and mathematical achievement in children (Sasanguie, De Smedt, Defever, & Reynvoet, 2012a; Sasanguie, Van den Bussche, & Reynvoet, 2012b; Soltész, Szúcs, & Szúcs, 2010), and the evidence for this relationship in adults is mixed (Castronovo & Göbel, 2012; Feigenson et al., 2013; Price, Palmer, Battista, & Ansari, 2012). The direction of causation between ANS and mathematical ability is still unclear. In particular, some studies suggest that ANS is a precursor of later mathematical abilities (Gilmore et al., 2010; Lyons & Beilock, 2011; Park & Brannon, 2013). According to this view, more precise approximate number representations might make some individuals better at mathematics. Conversely, other research has found that stronger mathematical abilities (maybe due to differences in the quality and/or quantity of the mathematics education that individuals receive) sharpen ANS representations (Mussolin, Nys, Content, & Leybaert, 2014; Pica, Lemer, Izard, & Dehaene, 2004). A third possibility is that the relationship between the ANS and mathematical abilities found in some studies could be mediated by domain-general cognitive abilities (Fuchs & McNeil, 2013; Hyde, Khanum, & Spelke, 2014), such as inhibition (Gilmore et al., 2013) or working memory skills. The association between ANS and mathematics could also be moderated by participant age (Inglis, Attridge, Batchelor, & Gilmore, 2011; Rousselle & Noël, 2008) or by the specific mathematics measures used in the studies (e.g., De Smedt, Noël, Gilmore, & Ansari, 2013).

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Recent studies have investigated the relation between ANS and MA. Some studies indicate lower performance in symbolic magnitude processing in adults with high MA (Maloney, Risko, Ansari, & Fugelsang, 2010; Maloney, Ansari, & Fugelsang, 2011; Núñez-Peña & Suárez-Pellicioni, 2014) while no significant association with MA was found in the few studies that adopted non-symbolic tasks to assess magnitude processing in children (Gómez-Velázquez, Berumena, & González-Garrido, 2015; Hart, Logan, Thompson Kovas, McLoughlin, & Petrill, 2016). A recent study Cargenlutti, Tomasetto, and Passolunghi (2017b), which will be described in the next section, proposed an interesting model of the relationships between WM, MA, ANS and mathematical achievement.

Calculation skills Arithmetic calculation is an important academic skill that children learn when they start formal education. Gelman and Gallistel (1978) proposed five implicit counting principles at the base of calculation skills. The principles include oneto-one correspondence (only one word is assigned for each counted object: “one”, “two”, “three”, . . .), the stable order principle (the order of the words must be invariant across counted sets), the cardinality principle (the value of the final word represents the quantity of the items in the counted set), the abstraction principle (any kind of objects can be collected together and counted) and the order irrelevance principle (items in a set can be tagged in any sequence). Basic addition skills are a fundamental prerequisite for the development of multiplications skills and increasingly complex arithmetic abilities. A substantial body of research has focused on identifying the cognitive processes that underlie arithmetic calculation and has demonstrated the important role of working memory. For instance, to perform a mental calculation (e.g., 57 + 8), it is necessary to temporarily retain the phonological representations of the numbers. The next step may be to employ one or more procedures (e.g., counting) to combine the numbers and produce an answer. Alternatively, employing carrying or regrouping strategies will maintain recently processed information while performing other mental operations. For example 57 + 10 – 2, or 55 + 8 + 2. Finally, we would need to add the products held in working memory, resulting in the correct solution. This example clearly shows how the cognitive processes involved in performing arithmetic calculations are embedded within the working memory system. Even the simplest mathematical calculations require the temporary storage of problem information, the retrieval of relevant procedures and the processing of operations to convert the information into a numerical output. Studies have also shown that the different working memory components (e.g., visuo-spatial sketchpad, phonological loop and central executive) play specialized and unique roles in the arithmetic calculation (Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007). Higher working memory capacity is associated with higher accuracy in solving complex arithmetic calculation in adults as well as in children. In

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particular, children with higher working memory abilities tend to use more sophisticated strategies such as decomposition in preference to less sophisticated strategies such as finger counting (Geary, Hoard, Byrd-Craven, & Catherine DeSoto, 2004; Caviola, Mammarella, Pastore, & LeFevre, 2018). Studies indicate that the central executive plays a greater role in mental calculation than the phonological loop (e.g., De Rammelaere, Stuyven, & Vandierendonck, 2001). The phonological loop does play a major role when calculation involves storing temporary information, while carrying operations put a major demand on the central executive processes (Furst & Hitch, 2000). Only a limited number of studies have examined the role of the visuo-spatial component of the WM model ( Passolunghi & Mammarella, 2010, 2012; Mammarella, Caviola, Giofré, & Szűcs, 2018; Szűcs, 2016; Szűcs, Devine, Soltesz, Nobes, & Gabriel, 2013). These studies have shown that visuo-spatial WM is related to performance in written calculation. In particular, it is important during the initial stages of arithmetic calculation for encoding arithmetic problems presented visually. In relation to the possible interplay between calculation, MA and WM, some authors (Ashcraft, 2002; Ashcraft & Kirk, 2001) have proposed that, due to the effect of anxiety on the central executive component of WM, MA will be associated with lower performance, especially in more complex calculations. On the other hand, other accounts propose that MA only affects calculation performance in individuals with high working memory (Beilock, Holt, Kulp, & Carr, 2004). In the next section, the relationships between MA, WM and mathematical performance will be discussed in more detail with a specific focus on studies with young students.

Mathematical problem-solving Mathematical problem-solving is a core aspect of mathematics acquisition (Pimta, Tayruakham, & Nuangchalerm, 2009), crucial for analyzing and interpreting the world in mathematical terms and applying mathematics to everyday life to use their mathematical knowledge in solving daily problems. The word problems have been described as instruments that develop the students’ abilities and talents in solving mathematical problems (De Corte, Vershaffel, & De Win, 1989). It is for this reason that word problems are typically introduced in the earliest stages of mathematical instruction (Cummins, 1991). Word problems are defined as verbal descriptions of problem situations, which raise one or more questions, that can be answered by the application of mathematical operations to the numerical data available in the problem statement (Verschaffel, Greer, & De Corte, 2000). An example of such a task is the following: “John bought 4 pizzas with 8 slices each. He and his friend Bruce ate 12 slices of pizza. How many slices were left?” Geary (1995) states that children make more errors when solving word problems than solving comparable number problems. The reason for such inability is the fact that solving such word problems requires children both to perform mathematical computations and possess the linguistic knowledge necessary to understand the problems (Cummins, Kintsch,

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Reusser, & Weimer, 1988). Word problem-solving, in fact, depends on several factors: linguistic comprehension, understanding the situation, selecting relevant information while ignoring irrelevant information, working out the right formal procedure to be applied and the appropriate arithmetical operations to be carried out to reach the solution and, finally, computational ability (Fuchs et al., 2010; Mayer, 1992; Mayer, Larkin, & Kadane, 1984). From a domain-general cognitive point of view, there are also different factors involved in mathematical problemsolving, for example, intelligence (Giofré, Borella, & Mammarella, 2017), WM, specifically dual tasks (e.g., Cornoldi & Giofrè, 2014; Cornoldi, Drusi, Tencati, Giofrè, & Mirandola, 2012) and visuo-spatial WM (e.g., Bizzaro, Giofrè, Girelli, & Cornoldi, 2018), as well as processing speed (Demetriou et al., 2014; Passolunghi, 1999; Salthouse, 1996; Salthouse & Meinz, 1995). Several results have suggested that complex working memory skills are impaired in children with mathematical learning disabilities. In particular the findings suggest a working memory deficit in children with mathematical difficulties, mainly in the central executive component of Baddeley’s model, and, specifically, in the inhibitory and updating processes (Passolunghi, Cornoldi, & De Liberto, 1999; Passolunghi & Siegel, 2001, 2004; Passolunghi & Pazzaglia, 2005). Besides cognitive characteristics, an inverse correlation between MA and arithmetical problem abilities was found in a study by Passolunghi, Cargnelutti, & Pellizzoni (under review). Strikingly, this study found the correlation to be present even in tasks without a time limit. Despite the cognitive and emotional profiles characteristic of students with mathematical difficulties, studies of mathematical problem-solving have highlighted a complex interplay of variables contributing to the development of this skill and, in particular, to students’ abilities to make flexible choices in solving mathematical problems ( Verschaffel & De Corte, 1997; Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009). Notably, the characteristics of the tasks itself (e.g., amount of unnecessary information in the text, the verbal complexity of the problem, familiarity with the situation described), the specific abilities of the student (especially in relation to domain-general cognitive abilities), and contextual characteristics (e.g., educational methods, socioeconomic status) seem to be crucial to mathematical problem-solving performance. This approach to the topic of mathematical problem-solving implies that it is important to study not only the ways in which tasks are solved but also student and contextual characteristics (Acevedo Nistal, Van Dooren, Clarebout, Elen, & Verschaffel, 2009; Verschaffel, De Corte, de Jong, & Elen, 2011).

III. The relation between math anxiety and working memory Anxiety and math anxiety Anxiety, defined as a dispositional dysfunctional response to a situation perceived as threatening ( Lewis, 1970), has a variety of repercussions on achievement, as well as on mental health and well-being. In the school context, studies show that

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clinical levels of anxiety are present in 10% of children and are recorded even in kindergarten ( Egger & Angold, 2006). Anxiety has been observed especially often in students with learning difficulties, typically described as more anxious than their peers (Fisher, Allen, & Kose, 1996). The American Psychiatric Association (2013) defines anxiety as “an emotion characterized by feelings of tension, worried thoughts and physical changes like increased blood pressure”. Recent studies indicate relationships between anxiety and complex cognitive performance (Arnsten, 2009; Diamond, Campbell, Park, Halonen, & Zoladz, 2007). The studies showed that emotional factors have an impact on mathematical performance (Ashcraft, Kirk, & Hopko, 1998; Ashcraft, 2002; Ho et al., 2000) and that a form of anxiety specifically relates to mathematics (Ashcraft, 2002; Hembree, 1990; Maloney & Beilock, 2012). Richardson and Suinn (1972) defined MA as “a feeling of tension, and anxiety that interferes with manipulation of numbers and the solving of mathematical problems in ordinary life and academic situations”. Ashcraft (2002) suggested that individuals with high MA avoid situations in which they have to perform mathematical calculations. Unfortunately, avoidance of mathematical results in less exposure and practice in mathematics, resulting in lower achievement, which, in turn, makes the students more anxious. In college and university, students with higher mathematics anxiety take fewer mathematics courses and have more negative attitudes toward mathematics. MA can range from mild to severe, from seemingly minor frustration to overwhelming emotional (and physiological) disruption. There are some practices in the traditional mathematics classroom that cause great anxiety in many students, for example, authoritarian practices by teachers and school leaders, public exposure (e.g., being asked to perform a calculation in front of the class or to solve a problem at the blackboard) and time limits. As shown in the sections above, MA and WM are among the two factors that most inf luence mathematics achievement. In the next section, the reader will be introduced to the analysis of the literature that investigates the reciprocal effects of MA and WM on mathematics performance. The research will first describe the studies that evaluate the effects of these two factors on adults and will then discuss the issues from a developmental perspective.

Reciprocal effects of MA and WM on mathematical performance For decades the effects of MA and WM (considered respectively as emotional and cognitive factors) on mathematical performance have been studied separately. More recently, researchers have tried to evaluate the reciprocal effects of WM and MA on mathematical abilities, and they agree that WM resources, usually devoted to solving mathematical tasks resolution, are decreased by the presence of MA and that this detrimentally impacts performance (Ashcraft & Kirk, 2001; Beilock & Carr, 2005; Calvo & Eysenck, 1992; Miller & Bichsel, 2004; Young, Wu, & Menon, 2012). Ashcraft and Faust (1994) observed that a high level of MA did not affect the execution of easy mathematical tasks such as single-digit addition problems

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(e.g., 2 + 3) but that it negatively impacted the execution of more complex and less automatic tasks, such as those with operations requiring carrying e.g., 46 + 25). To explain this experimental evidence, Ashcraft (Ashcraft, 2002; Ashcraft & Kirk, 2001) proposed a significant role for the central executive: if this component of WM does not work properly because of anxiety, it prevents an adequate processing of the information relevant to the task along with concurrent inhibition of irrelevant information (Ashcraft et al., 1998). As to the question of who is most susceptible to WM disruption as result of high MA, studies show contrasting results. On the one hand, some authors posit that the better an individual’s WM, the better they can manage mathematical performance in the face of anxiety-driven thoughts (Ashcraft & Kirk, 2001). An opposite account, on the other hand, indicates that individuals with higher WM are more susceptible than others to performance failures caused by anxiety ( Beilock & Carr, 2005). With regard to the first hypothesis, the authors proposed that a higher level of WM leads to reduced susceptibility to the detrimental effects of anxiety, given that the greater WM capacity permits simultaneous attention to both the tasks and to the anxiety-driven thoughts. Miller and Bichsel (2004) found such effects especially with regard to spatial WM in both mathematical calculation skills and problem-solving. An alternative hypothesis suggests that high WM individuals are more affected by MA. High WM individuals, in fact, seem to devote fewer resources to the task execution when anxiety-related thoughts intrude on mathematical tasks ( Beilock & Carr, 2005; Mattarella-Micke, Mateo, Kozak, Foster, & Beilock, 2011). The authors observed such an effect especially when they asked participants to solve tasks in a stressful situation, and they referred to it as “choking under pressure” ( Beilock & Carr, 2005). The literature clearly shows the detrimental effect of MA on both WM and mathematical performance. Which individuals are most susceptible to WM disruption in high anxiety situations, is, however still a matter for debate. It should be noted that the relationships between WM and MA and the possible detrimental effect of MA on WM and mathematical performance have been less studied in young children. In the next section, we give a critical summary of developmental research on links between MA, WM and mathematical achievement.

MA, WM and children’s mathematical performance The initial attempts to investigate links between MA, WM and mathematical performance have been described in articles by Ramirez, Gunderson, Levine, and Beilock (2013) and by Vukovic and colleagues (2013). Ramirez et al. (2013) gave first- and second-grade students tests of WM problem-solving, and a newly adapted MA test. The results showed a negative correlation between MA and problem-solving, which was found only among children who scored relatively high on WM, but not among those who scored relatively low in WM. The second study ( Vukovic et al., 2013) extended the research to include different

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cognitive functions during the second and third grade. Once again, a challenge for their research was to develop a measure for MA, because existing measures were designed for children at or above the fourth grade. They developed an ad hoc scale composed of 12 items that adapted material from the MARS-E (Suinn, Taylor, & Edwards, 1988) and MAQ ( Wigfield & Meece, 1988). The reported reliability of the MA self-report was a Cronbach’s alpha around .70. General control variables were also tested (e.g., reading abilities and early numeracy and visuo-spatial WM). The results showed that MA was negatively correlated with applied mathematical problems. Similarly to Ramirez et al. (2013), who found MA to predict mathematical performance in students with high WM during the same school year, Vukovic and colleague (2013) highlighted that children with a high MA score in second grade demonstrated worse performance in mathematical performance in third grade. This effect was observed only in students with high WM abilities. Moreover MA does not inf luence all components of mathematical performance in the same way. Calculation skills and the ability to apply mathematical knowledge, but not geometric reasoning, were affected. The authors speculated that MA may specifically inf luence performance on tasks that involve understanding and manipulating numbers, rather than other aspects of mathematics. In another recent study of second and third graders (Cargnelutti, Tomasetto, & Passolunghi, 2017a), both cognitive and emotional factors were studied. In particular, the study examined the interplay between domain-general factors (e.g., WM), domain-specific cognitive factors (e.g., the approximate number system) and affective aspects, and their inf luences on mathematical ability. This was the first attempt to explore the interplay between general anxiety and MA, with regard both to WM and the non-symbolic approximate number system (ANS) in relation to mathematical performance. The research confirmed the significant negative relation between general anxiety and mathematical performance in young school children. Also, the negative relation between general anxiety and mathematical performance was observed to operate through the disruption of WM and ANS. Beyond providing further evidence for a direct negative relation between general anxiety and mathematical performance, the authors also investigated whether anxiety might also indirectly disrupt mathematical proficiency by negatively affecting the domain-general and the domain-specific cognitive correlates of mathematics. In fact, while the negative inf luence of anxiety on the execution of tasks involving WM is well established (at least in older children and adults; see Eysenck, Derakshan, Santos, & Calvo, 2007 ), its impact on more automatic and basic quantitative skills, such as those related to non-symbolic ANS, has so far received little study. Cargnelutti et al.’s (2017b) findings indicated that Backward digit recall and the Approximate addition skills, but not Estimation ability, were significant mediators of the relation between general anxiety and mathematical performance (see figure 6.3).

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FIGURE 6.3 Multiple mediation model. Unstandardized coefficient of the mediation model between general anxiety (Anxiety-Obs) and mathematical achievement (Math) through backward digit memory, approximate addition, and estimation (standard errors presented in parentheses). In squared parentheses, the total effect (with standard error) of anxiety observed on mathematical achievement.

Dashed lines indicate non-significant paths. † p < .10, *p ≤ .05, **p ≤ .01, ***p ≤ .001. Adapted by Cargnelutti et al. (2017b).

To sum up, MA, WM and mathematical achievement are already correlated in the first years of primary school. In very young students, there seems to be a detrimental effect of MA on WM and mathematical achievement, but this is only significant for children who start out with a high level of WM (Beilock & Carr, 2005; Mattarella-Micke et al., 2011). If these effects are found in the first three years of primary school, others are found during the last two years of primary school. Passolunghi, Cargnelutti, and Pellizzoni (under review) investigated the relation between cognitive and emotional factors in a group of fourth graders. This study assessed their problem-solving proficiency, their MA (MARS-R, Saccani & Cornoldi, 2005) and their WM levels. Again, cognitive and emotional factors significantly predicted problem-solving performance in a similar way to younger children (Cargnelutti et al., 2017b; Ramirez et al., 2013; Wu, Barth, Amin, Malcarne, & Menon, 2012), but another effect of MA was indicated. The results showed that children with high MA were more impaired in both problem-solving and WM tasks, and this occurred in an untimed problem-solving task. This appears to rule out the hypothesis that MA only exerts a negative role by increasing pressure during timed mathematical tasks (e.g., Faust, Ashcraft, & Fleck, 1996). Regarding the relation between WM and MA, the results of this study showed a strong negative effect of MA on both arithmetical problem-solving and WM. Children with high MA were not only more impaired in problem-solving but also in WM than those with low MA.

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Going from primary to secondary school students, Passolunghi, Caviola, De Agostini, Perin, and Mammarella (2016) investigated the calculation proficiency and the cognitive profiles of secondary school students with high versus low MA. The results showed that children with high MA performed less well both in verbal short-term and working memory tasks. The effect was specific: high MA predicted poor achievement in mathematics but not in other school subjects (e.g., reading, decoding, reading comprehension and writing accuracy). Implications of this study are that high MA and low WM capacity can be seen as risk factors for poor mathematical achievement. Middle school students with high MA are at greater risk than those with a lower level of MA in performing poorly in mathematical achievement measures, and this detrimental effect is specific for mathematical abilities. If studies of typically developing children provide interesting insights into the relation between MA and WM, so do studies of atypical development. Mammarella, Hill, Devine, Caviola, and Szucs (2015) tested verbal and visuo-spatial short-term memory and WM in children with developmental dyscalculia and high math anxiety, comparing them with typically developing children. There were differences between children with developmental dyscalculia and those with high MA with regard to verbal and visuo-spatial short-term memory and WM. The results showed that children with developmental dyscalculia showed specific impairments in the visuo-spatial WM task. On the other hand, the MA group was particularly impaired in the verbal WM task. These data seem to show how different emotional and neurocognitive aspects can be linked to specific strengths and weaknesses related to mathematical performance.

Specific treatment for math anxiety As seems evident from the previous sections, the role of anxiety in cognitive functions and achievement is crucial, though the literature on how to promote treatment for this condition is still scarce. The few contributions on the topic show that applying the expressive writing technique could help people control anxious feelings. This clinical approach encourages people to write freely about their thoughts and feelings concerning anxiety ( Pennebaker & Beall, 1986) with the effect of increasing availability of WM for performing tasks ( Klein & Boals, 2001; Yogo & Fujihara, 2008). In particular, writing about traumatic things (a mild manipulation of negative moods) appears to have an effect on cognitive WM, compared to positive or neutral writing. Thus, WM is not a static factor but can be manipulated (and increased) as a function of psychosocial manipulation, ref lecting variations in intrusive thoughts related to off-task issues. For adults, expressive writing exercises help specifically to alleviate the intrusive thoughts that result from anxiety related to particular situations ( Ramirez & Beilock, 2011). In a recent experiment, on the day of the final exam, Ramirez and Beilock (2011), asked half of the students to write openly for 10 minutes about their feelings toward the upcoming exam (the expressive writing group),

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while the other half of the participants were asked to write about a generic topic, different from their forthcoming exam, for 10 minutes (the control condition). The results indicated that the expressive writing exercise was indeed successful, as the students in the expressive writing group had higher overall scores than those who did not write about the upcoming test. While the expressive writing technique was first applied to address test anxiety, research indicates that it is also effective in increasing mathematical performance in mathematics-anxious students ( Park, Ramirez, & Beilock, 2014). In particular, Park and colleagues (2014) investigated the role of expressive writing in reducing the negative impact of MA on mathematical performance. This study reported significant benefits for participants with high MA, but not for participants with low MA. Authors underlined that participants with high MA who used more anxiety-related words in writing, demonstrated better mathematical performance on the challenging problems. This was not the case with low MA participants. Thus, some studies have suggested that some clinical practices are effective in adults. To the best of our knowledge, the first study that attempted to decrease MA in children was designed by Supekar, Iuculano, Chen, and Menon (2015). The research showed for the first time that an intensive eight-week intervention involving one-to-one mathematical tutoring reduces MA, and also remediates aberrant connectivity in emotion-related circuits associated with MA in primary school children. Before tutoring, children with high MA (compared with low MA children) showed aberrant functional response in the amygdala (Phelps & LeDoux, 2005). The tutoring program was adapted from MathWise (Fuchs et al., 2008; Fuchs & McNeil, 2013; Powell, Fuchs, Fuchs, Cirino, & Fletcher, 2009) and involved 15–20 hour-long sessions, over a period of between eight and nine weeks. The first four lessons focused on making students familiar with mathematical manipulatives. The fifth and sixth lessons trained them in using “mini strategies” for addition. From the seventh until the twenty-second lesson, students practiced these strategies with more difficult problems. After the training, differences in the connectivity of the amygdala disappeared, indicating remediation of impaired functional brain circuits in the children with high MA. In a recent study of fourth-grade students, Passolunghi and Pellizzoni (2018) compared two training programs involving eight weekly sessions. One group (anxiety training) was trained using a specific approach that attempted to reframe cognitive beliefs about math anxiety (based on Rational Emotive Behavior Therapy, proposed by Ellis (2006)). The second group (mathematical training) received training specifically focused on learning to apply mathematical metacognitive strategies to mathematical exercises (Poli, Molin, Lucangeli, & Cornoldi, 2006). The third group (the control group), carried out reading activities. The data showed that both anxiety and mathematical training decreased MA (measured using AMAS, Hopko, Mahadevan, Bare, & Hunt, 2003) but only the mathematical training had an effect on mathematical achievement at the end of the school year. These data seem to show that an effective MA training has to tap mathematical materials and procedures as well as anxiety as such.

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Lastly, apart from the specific cognitive and emotional characteristics of each student, recent literature shows that other factors can affect mathematical abilities: beliefs and attitudes of teachers and parents. Ramirez and colleague (2013) indicated that one of the most important ways in which we can prevent the development of MA in children is to work with teachers and parents to change their negative way of interacting with children about mathematics, to change their attitudes about mathematics and to help them conquer their own MA. In a recent study (Maloney, Ramirez, Gunderson, Levine, & Beilock, 2015), showed that parents’ MA and their children’s mathematical learning has a negative correlation only if the higher anxious parents help the students with mathematical homework. Previous research ( Hembree, 1990) demonstrated that adults with high MA express various negative attitudes toward mathematics (e.g., they state that mathematics is not useful, they have low selfefficacy and low self-motivation). With these kinds of attitudes, they can have a negative transgenerational impact on their children, demotivating them with regard to mathematics.

Conclusions and future directions The main aim of this chapter was to describe the interplay of WM and MA in young students in order to describe and define the different aspects of the issue, considering future research perspectives and the possibility of intervention programs. In concluding our review, we want to pinpoint three aspects that characterize the interconnection between MA and WM. First, it is possible to observe a strong relation between WM and MA even in young students, showing the early roots of the relation between the two components. Some authors have suggested that this connection only applies to high WM students (Ramirez et al., 2013; Vukovic et al., 2013). These researchers’ finding could be explained in different ways: 1) negative emotions disrupt the resources that high WM children rely on to retrieve basic facts from long-term memory and to inhibit competing answers (Geary, Hoard, Byrd-Craven, & Catherine DeSoto, 2004), 2) MA may make high WM children more susceptible to interference, resulting in a slower and less efficient retrieval process or 3) among young children, those with high WM are more aware of their emotions than those with low WM. The different perspectives from which this topic has been evaluated (e.g., analyzing the relation in children with high WM vs. children with high math anxiety) have produced different results that need to be more specifically evaluated. Indeed, other findings showed that high MA and low WM capacity can be seen as risk factors for poor mathematical achievement (Cargnelutti et al., 2017b; Passolunghi et al., 2016; Mammarella et al. 2015). For this reason, this topic requires further research. Second, a fundamental aspect to consider when analyzing the interplay between WM and MA is the involvement of the tools used during the evaluations. In particular, the reliability of the tests has to be considered carefully, given young students’ limited experience with self-evaluation, the developmental

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trajectory of children’s explicit awareness of their own thoughts and emotions and the different dimensions (e.g., performance anxiety vs. emotional response to numbers) assessed by the self-report measures (Dowker, Sarkar, & Looi, 2016). Finally, we strongly believe in the need to invest more in early interventions to prevent or ameliorate MA and to increase awareness of the condition among parents and teachers. Trying to shed light on how MA and WM interact and affect mathematical achievement in young students has great potential importance both from a scientific and an educational point of view. Identifying the risk factors for MA may enable us to develop ways of identifying and developing early interventions for at risk pupils and thus to enhance achievement and, in turn, enjoyment of mathematics.

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7 THE DIFFERENT INVOLVEMENT OF WORKING MEMORY IN MATH AND TEST ANXIETY Ee Lynn Ng and Kerry Lee

Research has shown that math anxiety (MA) and test anxiety (TA) are highly prevalent among school-aged children (Ma, 1999; Segool, Carlson, Goforth, Von Der Embse, & Barterian, 2013) and have disruptive effects on their academic development. For example, Hembree’s (1988) meta-analysis of 562 studies revealed a significant negative correlation between TA and mathematics performance from Grade 4 to post-secondary levels (mean r = −0.22). Where MA is concerned, two meta-analyses reported correlations of −0.27 (Ma, 1999) and −0.34 (Hembree, 1990) between MA and math performance from Grades 4 to 12. Together, these findings indicate that TA and MA are significant problems not only for adolescents and adults (McDonald, 2001), but also for younger children attending formal education. As most of the existing work on academic anxiety has been conducted with adolescents or undergraduates ( Hill et al., 2016), much less is known about the interplay between academic anxieties and task performance in children attending primary or elementary school. Nevertheless, in recent years, there has been increased interest in advancing our knowledge base in this area ( Korhonen, Nyroos, Jonsson, & Eklöf, 2017; Mavilidi, Hoogerheide, & Paas, 2014). In this chapter, we will conduct a selective review of the small but steadily growing literature on the involvement of working memory (WM) in young children. Through this review, we will demonstrate that there is still a gap in our understanding about the extent to which younger children are similarly affected by anxiety and whether their experience of anxiety affects mathematical learning and performance. Given what we know about the developmental trajectory of WM and its importance for mathematical performance in early childhood, a question of concern is the extent to which any age-related increase in test/math anxiety (which leads to an increased taxation of WM resources) is compensated by growth in WM capacity. We explore these issues from a developmental

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perspective by drawing on previous literature as well as our own research on WM, test anxiety, and mathematical achievement.

Test anxiety and math anxiety Anxiety refers to a basic human emotion that appears when individuals appraise a stressful event as being a threat and perceive themselves to be incapable of coping with the stressful event in a satisfactory way (Zeidner, 1998). In other words, anxious persons feel they cannot meet the demands of the stressful event. Within this broad framework, MA and TA refer to situation-specific types of anxiety commonly found in the academic context. Although some researchers have conceptualized MA as a form of TA (e.g., Richardson & Woolfolk, 1980), recent studies have shown that measures of MA correlate more highly with each other than with TA (e.g., Ashcraft & Ridley, 2005; Hembree, 1990; Kazelskis et al., 2000). These findings are consistent with the prevalent view that MA and TA are distinguishable by the stimulus properties of situations that are considered personally threatening ( Dowker, Sarkar, & Looi, 2016; Zeidner, 1998), but not the experience of anxiety per se. TA refers to a set of phenomenological, physiological, and behavioral responses that accompany concern about possible negative consequences or failure on an exam or similar evaluative situation (Sieber, O’Neil, & Tobias, 1977). It is usually elicited when students believe that their intellectual capabilities are taxed or exceeded by demands stemming from an assessment or test situation (Zeidner, 2007). In contrast, MA refers to feelings of tension, apprehension, or fear about manipulating numbers and solving mathematical problems in a variety of ordinary life and academic situations (Ashcraft, 2002; Richardson & Suinn, 1972). It is important to note that whereas MA includes a reaction to the content as well as to the performance evaluation, TA focuses only on the evaluative nature of a test (which may involve a variety of domains, including math, science, or language). Notwithstanding the uniqueness of MA and TA as specialized forms of anxiety, both share the common features of perceived possibility of failure and resultant disapproval by significant others, who are evaluating the person’s performance in relation to a standard of achievement (Zeidner, 1998). Furthermore, cognitive and physiological elements of anxiety are evident in both MA and TA (Liebert & Morris, 1967; Ma, 1999; Wigfield & Meece, 1988). The cognitive elements, collectively known as worry, comprise negative expectations, self-preoccupation, and concerns about the possible consequences of failing. The physiological elements refer to the affective concomitants of anxiety, which include an individual’s perception of the physiological elements of the anxiety experience, such as indications of autonomic arousal and unpleasant feeling states such as nervousness and tension. Following Spielberger’s (1972) trait-state theory of anxiety, researchers have distinguished between trait and state components of TA (Spielberger, 1972; Whitaker Sena, Lowe, & Lee, 2007) and MA ( Beilock, Rydell, & McConnell,

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2007; Ganley & Vasilyeva, 2011; Ma, 1999). The trait component ref lects a general tendency to feel anxious about test situations (i.e., trait TA) or mathematics (i.e., trait MA). High trait-anxious individuals have a stronger tendency to perceive test or math situations as personally threatening and to respond to such threats with state anxiety. The state component of both TA and MA refers to the tendency to react to a test or math situation, respectively.

Explanatory frameworks Cognitive theories of TA and MA attribute the anxiety-performance link to various cognitive factors, such as attentional control and WM. Among the variety of cognitive theories that have been proposed since the early 1970s, arguably the most prominent of these is the attentional control theory (ACT; Eysenck, Derakshan, Santos, & Calvo, 2007), and its predecessor, the processing efficiency theory (PET; Eysenck & Calvo, 1992). Before delving into the specifics of the theoretical frameworks, we first describe the structure and functions of WM.

Working memory WM refers to a limited capacity system responsible for the temporary storage and manipulation of information (Alloway & Gathercole, 2006). It functions as a mental workspace that can be f lexibly used to support the performance of a variety of cognitive activities that require moment-to-moment monitoring, processing, and maintenance of task-relevant information ( Baddeley & Logie, 1999). These include simple everyday tasks (e.g., mental calculation), as well as more complex tasks such as reading, problem-solving, and mathematics (Alloway & Gathercole, 2006). In this chapter, we adopt the model and terminology first used by Baddeley and Hitch (1974). The latest version of Baddeley and Hitch’s WM model comprises the phonological loop, visuospatial sketchpad, central executive, and an episodic buffer ( Baddeley, 2000). The phonological loop and the visuospatial sketchpad are responsible for the short-term storage of verbal and visuospatial information, respectively. The episodic buffer facilitates the integration of information from the storage systems and long-term memory into a unitary episodic representation. The central executive is a domain-general processing resource that controls and regulates the WM system ( Baddeley & Logie, 1999). Subsequent research suggests that the central executive also performs other functions. Miyake et al. (2000) found empirical support for three interrelated executive functions, namely, shifting, inhibition, and updating. Shifting refers to the ability to change between mental sets or tasks in multiple-task situations. Inhibition is the ability to inhibit dominant, automatic, or prepotent responses when necessary. Updating involves monitoring and coding of incoming information and ensuring that only relevant information is held in WM. Together, these executive functions form the basis of abilities such as problem-solving and

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f lexible thinking, thus allowing individuals to engage in goal-directed thought and action, and to respond f lexibly to the environment (Cragg & Gilmore, 2014).

ACT and PET A central assumption of both ACT and PET is that task performance can be assessed in terms of performance effectiveness and processing efficiency. Effectiveness refers to the quality of task performance and is typically operationalized as response accuracy. Efficiency is defined as the relationship between accuracy and the resources used to accomplish the task; it can be measured by the ratio of accuracy to response time on correct trials ( Hoffman, 2012). Both theories propose WM as the mechanism underpinning the effects of anxiety on cognitive performance. According to PET, anxiety leads to an increase in worry, which consumes WM capacity and leaves a smaller functional capacity for the task at hand. ACT expands on this explanation by specifying that anxiety increases the allocation of attentional resources to worry, thus reducing attentional focus on the current task. ACT further specifies that worry motivates anxious individuals to compensate for the restricted availability of WM by increasing their effort (e.g., allocate additional processing resources) and using auxiliary resources or strategies (e.g., rote learning or articulatory rehearsal). Therefore, anxiety typically impairs efficiency to a greater extent than it does effectiveness. Nevertheless, anxiety is expected to impair effectiveness as well if situational circumstances, such as limited time or the presence of a concurrent distracter task, prevent the employment of compensatory processing resources.

Anxiety, working memory, and task performance Although PET and ACT were developed to account for the effects of general anxiety on cognitive performance, they have been applied to TA (Eysenck et al., 2007; Mowbray, 2012) and MA (Ashcraft & Kirk, 2001). In this section, we provide a selective review of this literature to highlight what we currently know about the relations between TA and MA with task performance in children aged 6–12. Studies that have examined the general construct of trait anxiety are also included in this review. We start with a more in-depth description of the studies that we have conducted, followed by brief summaries of other related works.

Test anxiety We conducted a series of experiments to investigate the role of WM capacity in the relationship between TA and mental arithmetic task performance in 11-year-olds ( Ng & Lee, 2010, 2015). Both studies employed the same research design: trait TA, WM load, and situational stress were the independent variables. High and low trait TA children performed a mental arithmetic task under varying WM load conditions, and each child performed the task under high and

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low situational stress conditions. Situational stress was manipulated and was used to compare participants’ task performance under high and low state TA. This design enabled us to tease apart the joint and independent effects of trait and state TA on task performance. In both experiments, the role of WM capacity was addressed from an experimental perspective by manipulating the WM load associated with the arithmetic task. Specifically, participants were instructed to maintain a sequence of letters in memory while solving a mental arithmetic problem (see Figure 7.1 for an example of the task used in Ng & Lee, 2015). In each trial, participants were first presented with a memory load set, which consisted of six As in the low WM load condition (AAAAAA), a contiguous alphabet sequence in the medium WM load condition (e.g., ABCDEF), and a contiguous sequence presented in randomized order in the high WM load condition (e.g., DAEBCF). Participants were instructed to repeat the memory load set aloud while solving an arithmetic problem. They responded to the arithmetic problem by providing their answers for the units first, then the tens by clicking on boxes numbered 1–9 on the computer screen using a mouse (e.g., the correct responses for 53 – 37 are “6” and “1”). This eliminates the need to maintain intermediate results in memory, thus removing a potential source of interference during the maintenance of the memory load set. After solving the arithmetic problem, participants recalled the

Sequence of events for the experimental task in Ng and Lee (2015). RT = response time.

FIGURE 7.1

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memory load set by typing their responses on the keyboard. Measures of task accuracy and accuracy/response time served as indicators of performance effectiveness and processing efficiency, respectively. Across both studies, high trait TA children showed significantly poorer task performance compared to their low trait TA counterparts, especially when the WM load of the task was high. In other words, the relation between trait TA and task performance is moderated by the task’s WM demands. There were also some indications that the moderator effect of WM may occur only under high situational stress (e.g., Ng & Lee, 2010) but this finding was not replicated in our subsequent study ( Ng & Lee, 2015). A notable finding from these studies is the different patterns of findings between trait TA and performance effectiveness versus processing efficiency. In Ng and Lee (2010), the moderator effect of WM load was observed only on processing efficiency. A different set of findings was observed in Ng and Lee (2015): while the adverse effects of trait TA on efficiency was independent of WM load, trait TA deficits in effectiveness occur at higher levels of WM load. Studies that focused on general anxiety have also found significant associations between trait anxiety, WM, and academic achievement. In a study involving 50 children aged 11–12 years, Owens, Stevenson, Norgate, and Hadwin (2008) showed that verbal WM significantly mediated the association between trait anxiety and academic performance (effectiveness) comprising measures of math, quantitative reasoning, and non-verbal reasoning. Extending these findings, Owens, Stevenson, Hadwin, and Norgate (2012) demonstrated that the negative association between trait anxiety and academic performance in a sample of 12- to 13-year-olds may be linked to increased test-specific worry that impinges on WM resources. More recently, Korhonen et al. (2017) investigated the role of WM from both an experimental (i.e., tasks with high vs. low WM demands) and a differential perspective (i.e., individual differences in WM capacity). A sample of 624 children aged 9–10 years completed a test battery comprising a TA scale, WM tasks, and the Swedish national examination in mathematics. The latter comprised tasks with high WM demands (e.g., mathematical problem-solving) and low WM demands (e.g., basic arithmetic). Using structural equation modeling techniques, Korhonen et al. (2017) found that WM capacity significantly mediated the relation between TA and mathematics performance. Korhonen et al. (2017) also found a moderating effect of WM capacity on TA and mathematics performance, which varied as a function of the task’s WM demands. For high-demand problem-solving tasks, TA negatively affected task performance irrespective of children’s WM capacity. For low-demand basic arithmetic tasks, only children with lower WM capacity were affected by the negative effects of TA. Korhonen et al. (2017) attributed this finding to the different types of problem-solving strategies used by children with high versus low WM capacity. Whereas high WM capacity children tend to use more efficient strategies, low WM capacity children tend to use more WM-demanding

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strategies (e.g., counting-based strategy). Thus, the experience of TA co-opts a portion of low capacity children’s WM resources, which leaves them with insufficient resources to execute their problem-solving strategies. Taken together, the studies reviewed above provide empirical evidence in support of ACT and PET, which assumes WM as the mechanism underpinning the effects of TA on cognitive performance. While some studies examined the role of WM from an experimental perspective (e.g., Ng & Lee, 2010, 2015) or a differential perspective (e.g., Owens et al., 2012; Owens et al., 2008), Korhonen et al.’s (2017) findings suggest that both perspectives should be considered simultaneously to better understand the interplay between TA and WM on academic achievement.

Math anxiety Relatively few studies have examined the role of WM as a mediator between MA and math performance in children. A recent exception is a study by JusticiaGaliano, Martín-Puga, Linares, and Pelegrina (2017), which involved primary school children between the ages of 8 and 12 years (mean age = 9.44 years; N = 167). In addition to WM capacity, the authors also examined the possible mediating role of math self-concept in the relationship between MA and math performance. Children completed a set of questionnaires to assess MA and trait anxiety, math self-concept, and WM capacity, as well as measures of math f luency and math problem-solving. Teachers rated children’s math achievement. Hierarchical regression analyses revealed that MA accounted for an additional 3% to 5% variance in math f luency, math problem-solving, and math teacher assessment, after controlling for trait anxiety. Moreover, results of mediation analyses showed that both WM capacity and math self-concept mediated the relationship between MA and all math performance measures. While it is generally accepted that MA detrimentally affects mathematical performance by disrupting WM resources otherwise needed for task performance ( Vukovic, Kieffer, Bailey, & Harari, 2013), an issue that remains unclear is who is most susceptible to WM disruption because of MA. Two competing hypotheses have been proposed. The first account posits that individuals with higher WM capacity are less susceptible to performance deficits because they have more cognitive resources to manage anxiety-related thoughts and solve the mathematical task at hand (e.g., Ashcraft & Kirk, 2001). The second account postulates that individuals with higher WM capacity are more susceptible to performance deficits because they rely more heavily on WM resources to solve mathematical problems. Under high-pressure, anxiety-inducing conditions, their performance is specifically impaired because anxiety consumes the WM resources usually devoted to skill execution. While evidence supporting the first account has been obtained mainly from adult studies (e.g., Ashcraft & Kirk, 2001; Miller & Bichsel, 2004), empirical evidence in support of the second account has been reported in many child-based studies. These findings were obtained from studies using cross-sectional (e.g.,

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Ramirez, Gunderson, Levine, & Beilock, 2013) and longitudinal approaches (e.g., Ching, 2017; Vukovic et al., 2013). Based on a sample of 162 children aged 5–8 years (mean age = 7 years), Ramirez et al. (2013) found a pronounced negative relation between MA and math achievement for children with relatively higher WM capacity; this relation was not evident among children with relatively lower WM capacity. Further analyses revealed that the WM by MA interaction effect in high WM children was significant only when children’s performance on difficult math items (presumed to require more WM resources) was considered. In a one-year longitudinal study, Ching (2017) followed 246 Chinese children (aged 7–8 years) to examine the associations between MA at second grade and mathematical performance at third grade. Mathematical performance was assessed using two types of tasks: calculation and story problem-solving. Each of these tasks comprised easy and difficult items. Multiple regression analyses revealed a pronounced negative association between MA and performance on mathematical problems that demand more WM resources (i.e., difficult calculation and story problem-solving items). Critically, children with higher WM capacity were more vulnerable to the deleterious effects of MA.

A developmental perspective Early studies provide some indications about age-related differences in TA and MA, as well as the associations between TA and MA with task performance at different ages. Based on data from studies that examined TA differences from Grades 2 through 12, Hembree (1988) showed that TA increased in the early elementary school grades, stabilized near Grade 5, remained constant through high school, and declined a little in college. In addition, there was a significant negative correlation (mean r = -0.22) between TA and mathematics performance from Grade 4 to post-secondary levels; TA did not correlate with mathematics performance at Grade 2 (Hembree, 1988). Where MA is concerned, Hembree’s (1990) meta-analysis indicated that MA levels increased through junior high school, peaked near Grades 9–10, and leveled off in high school, and college. The negative relation between MA and math achievement is well established in older age groups, that is, from Grade 5 until college (e.g., Hembree, 1990; Ma, 1999). Despite these earlier findings, the onset of the relations between TA and MA with task performance remains unclear. Most of the existing studies of TA and MA have been cross-sectional in nature and often involved children in Grade 5 and above. Thus, little is known about how MA affects younger children who are beginning to learn math in a formal academic setting ( Wu, Barth, Amin, Malcarne, & Menon, 2012). While there is now a growing interest in the relation between MA and math performance in younger children (e.g., Harari, Vukovic, & Bailey, 2013; Ramirez et al., 2013) and how this relation develops over time (e.g., Hill et al., 2016), the findings are less consistent compared to those from studies of the older age groups. Some researchers have found significant links between MA and mathematics proficiency in the first stages of schooling ( Ramirez et al., 2013; Wu et al., 2012).

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For example, Wu et al. (2012) found a significant negative correlation between MA and mathematics achievement in second- and third-grade children (aged 7–8 years), even after controlling for trait anxiety. Harari et al. (2013) also found significant associations between MA and mathematics achievement in Grade 1 children (aged 6 years). The authors developed and used a self-report MA instrument that assessed multiple constructs encompassing negative reactions (to math), numerical confidence, and worry. This study revealed that negative reactions and numerical confidence appear to be the most salient dimensions of MA for young children. Moreover, each of these dimensions was associated with specific math domains: negative reactions were related to foundational mathematical concepts (e.g., counting skills and understanding of concepts such as “first” and “last”), whereas numerical confidence was related to computation skill. Hill et al. (2016) mapped developmental changes relating to MA and its link with mathematical performance in both primary (mean age = 9 years) and secondary school students (mean age = 12 years). There was a reliable negative correlation between MA and secondary students’ arithmetic performance, even after partialling out the association between MA and general anxiety. In contrast, although MA was present in primary school students, there was no relationship between MA and arithmetic performance in this age group. Hill et al. proposed that the negative MA/math performance link surfaces later in the educational timeline, likely because of the greater cognitive demands associated with the secondary school math curriculum and an increasing exposure to “high-stakes” testing. Cargnelutti, Tomasetto, and Passolunghi (2017) investigated the early association between MA and math performance in Grades 2 and 3, by accounting for general anxiety and inspecting the prevalent directionality of the anxietyperformance link. Results revealed that the anxiety-performance link was significant in Grade 3, with a prevalent direction from MA to performance (rather than the reverse). Longitudinal analyses also showed an indirect effect of MA in Grade 2 on subsequent math performance in Grade 3 (via MA in Grade 3). To our knowledge, only one published study failed to find an early relation between MA and math performance. Krinzinger, Kaufmann, and Willmes (2009) tested a sample of 140 children (mean age = 7 years) at the end of first grade, with repeated assessments every six months until the children were in third grade. At each assessment point, children completed measures of calculation ability and MA. Across all assessment points, there were no significant associations between MA and calculation ability. Krinzinger et al. (2009) proposed that the MA-calculation ability link possibly emerges only in individuals with extremely high MA concomitant with very pronounced math difficulties.

Issues for further research Our review has shown that there is a small but steadily growing body of literature on the involvement of WM in TA and MA in young children. Most of the

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research on TA has taken a cross-sectional approach. In terms of the role of WM, different patterns of findings have emerged, depending on whether WM is considered from an experimental or differential perspective, or both. In studies that focused on the experimental perspective (e.g., Ng & Lee, 2010; 2015), the adverse effect of TA on task performance was more pronounced when the task’s WM load was high. When individual differences in WM capacity is considered in conjunction with the task’s WM load (e.g., Korhonen et al., 2017), the pattern of relations becomes more complex. In particular, the moderating effect of WM capacity on TA and mathematics performance was observed only on tasks with low WM load. Turning to the research on MA, it was apparent that there is a mix of crosssectional and longitudinal studies involving children as young as age five. Regarding the role of WM, we found reasonably strong empirical evidence in support of the hypothesis that individuals with higher WM capacity are more susceptible to performance deficits because they rely more heavily on WM resources to solve mathematical problems. Critically, this account stands in contrast to Korhonen et al.’s (2017) results indicating a more pronounced effect of TA on task performance in low WM capacity children. As described earlier, Korhonen et al. explained that low WM capacity children tend to employ more WMdemanding strategies, which becomes a challenge to execute when a portion of their WM resources is co-opted by TA. Our review also indicated that research interest in MA and its relation to mathematics achievement in the early years of formal schooling is gaining traction. Although only a handful of studies thus far has examined this issue from a developmental perspective, several studies have reported significant negative associations between MA and mathematics achievement from ages 6–8 years (e.g., Cargnelutti et al., 2017; Harari et al., 2013; Wu et al., 2012). However, others have reported dissimilar findings. Hill et al. (2016) showed that although MA was present in 9-year-olds, it was not significantly associated with math achievement at this age. Instead, the negative correlation between MA and math was observed only in older children aged 12 years. To summarize, while it is clear that WM is implicated in the relationship between TA and MA surrounding academic achievement, there remain several gaps in our understanding of the role of WM in this context. We discuss some of these outstanding issues and provide suggestions for future research.

Longitudinal change in TA and MA One of the key takeaways of our literature review is the dearth of studies examining age-related changes in TA and MA. Although extant findings (e.g., Hembree, 1988; Ramirez et al., 2013) suggest that children as young as 6- and 7-year-olds experience MA and TA, little is known about the extent to which the severity of these anxieties increase or decrease over time. Several studies have shown that academic anxieties continue to be a significant problem through adulthood, with prevalence estimates ranging from 17% for MA (Ashcraft & Moore, 2009) and

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up to 40% for TA (Gregor, 2005). Thus, it is important to examine the extent to which anxious adults first develop TA or MA at an early age (e.g., during the early years of formal schooling). In addition, given the limited and inconsistent results about the pattern of associations between TA and MA with academic achievement over time, longitudinal studies may prove useful in disambiguating these inconsistencies, as well as facilitating our understanding of how and when academic anxieties affect academic achievement across development. In examining the association between academic anxieties and academic performance over time, it is also important to consider age-related changes in the WM-academic performance link. Given the central role of WM in explaining the relation between academic anxiety and performance, development of WM can be expected to complicate the relations between these constructs. In a recent 4-year cohort-sequential study involving four cohorts (kindergarten, 2nd, 4th, and 6th grades; N = 673), Lee and Bull (2016) examined the relations between WM and math performance from ages 6–15 years. Findings showed that WM capacity increases with age, but rate of growth in WM was invariant across individuals. In addition, WM capacity was a stronger predictor of general mathematical achievement than prior achievement for younger children (aged 6–8 years). In contrast, prior achievement plays a more central role for older children (aged 9–15 years) as they encounter more complex math problems. According to Lee and Bull (2016), these findings suggest that while WM is important for solving math problems throughout, its relative importance for learning how to solve these problems varies across grades. Lee and Bull’s (2016) findings raise a few issues worth considering within the broader aim of understanding the involvement of WM in TA and MA. If we assume that WM capacity increases over time but the rate of growth does not vary across individuals, it seem reasonable to expect that from a longitudinal perspective, children with higher WM capacity at kindergarten are more likely to show better math achievement. This hypothesis is based on the assumption that higher WM capacity translates into better preparedness for learning (Lee & Aunio, 2017). However, when MA and TA constructs are brought into the mix, it is unclear whether susceptibility to the adverse effects of MA and TA will change as a child’s WM capacity increases. Relatedly, does the degree of susceptibility to MA and TA vary depending on whether WM growth is truly invariant? Returning to the idea that high WM capacity translates into better preparedness for learning, children with higher WM capacity may also develop more positive attitudes about learning math (e.g., greater enjoyment in learning and higher levels of self-confidence), which likely puts them in a positive developmental trajectory for math achievement, and a lowered susceptibility to academic anxiety. In contrast, children with lower WM capacity may encounter significantly more challenges in learning math, which increases the likelihood of developing negative attitudes about learning and an increased susceptibility to academic anxiety. Admittedly, these hypothesized relationships do not fully capture the complex inter-relations amongst these constructs. Many other individual and

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contextual factors may inf luence children’s susceptibility to academic anxiety, such as math self-concept (e.g., Jameson, 2014) and parental attitudes toward math (e.g., Dowker et al., 2016).

Overlap between TA and MA As described earlier, MA and TA differ in terms of the stimuli that triggers state anxiety reactions in anxious individuals. In addition, several studies have shown that MA and TA are distinct constructs despite being moderately intercorrelated (Ashcraft & Ridley, 2005). Thus, an interesting question comes to mind: in a mathematical test situation, how would a child with both MA and TA respond to the test? Would this child exhibit significantly greater deterioration in test performance compared to another child who suffers only from MA or TA? Suárez-Pellicioni, Núñez-Peña, and Colomé’s (2016) review of the research on brain structures related to MA revealed that MA involves the same brain structures (e.g., amygdala, insula, and the prefrontal areas) previously reported for other anxiety types (e.g., PTSD, social phobia, and specific phobia). This evidence lends support to previous claims suggesting a common neurobiological pathway both for anxiety disorders and for anxiety-prone individuals (Stein, Simmons, Feinstein, & Paulus, 2007). Thus, it is highly likely that TA also involves the same brain structures as those involved in MA. However, this is an empirical question requiring further study. To the extent that there is an overlap between TA and MA from a neurobiological standpoint, follow-up investigations may explore the extent to which dual-anxiety children (i.e., those with high levels of MA and TA) show different patterns of behavioral responses during a math test compared to their single-anxiety counterparts. For instance, it may be that dual-anxiety children show significantly poorer performance effectiveness and processing efficiency due to their overtaxed WM capacity.

Conclusions Academic anxieties, such as MA and TA, are present in children attending the earliest stages of formal schooling. Given the central role of WM in explaining the associations between academic anxieties and task performance, there is now a small but steadily growing literature base focusing on the involvement of WM in TA and MA in young children. While extant studies have contributed some interesting insights, more research is needed to acquire a better understanding of the interrelations among these variables, particularly from a developmental perspective.

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Owens, M., Stevenson, J., Norgate, R., & Hadwin, J. A. (2008). Processing efficiency theory in children: Working memory as a mediator between trait anxiety and academic performance. Anxiety, Stress and Coping, 21(4), 417–430. doi:10.1080/10615800701847823 Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2013). Math anxiety, working memory, and math achievement in early elementary school. Journal of Cognition and Development, 14(2), 187–202. doi:10.1080/15248372.2012.664593 Richardson, F. C., & Suinn, R. M. (1972). The Mathematics Anxiety Rating Scale: Psychometric data. Journal of Counseling Psychology, 19(6), 551–554. doi:10.1037/h0033456 Richardson, F. C., & Woolfolk, R. L. (1980). Mathematics anxiety. In I. G. Sarason (Ed.), Test anxiety: Theory, research, and applications (pp. 271–288). Hillsdale, NJ: Erlbaum. Segool, N., Carlson, J. S., Goforth, A. N., Von Der Embse, N., & Barterian, J. (2013). Heightened test anxiety among young children: Elementary school students’ anxious responses to high-stakes testing. Psychology in the Schools, 50 (5), 489–499. doi:10.1002/ pits.21689 Sieber, J. E., O’Neil, H. E., Jr., & Tobias, S. (1977). Anxiety, learning, and instruction. Hillsdale, NJ: Erlbaum. Spielberger, C. D. (1972). Anxiety: Current trends in theory and research. New York: Academic Press. Stein, M. B., Simmons, A. N., Feinstein, J. S., & Paulus, M. P. (2007). Increased amygdala and insula activation during emotion processing in anxiety-prone subjects. American Journal of Psychiatry, 164 (2), 318–327. doi:10.1176/ajp.2007.164.2.318 Suárez-Pellicioni, M., Núñez-Peña, M. I., & Colomé, À. (2016). Math anxiety: A review of its cognitive consequences, psychophysiological correlates, and brain bases. Cognitive, Affective and Behavioral Neuroscience, 16(1), 3–22. doi:10.3758/s13415-015-0370-7 Vukovic, R. K., Kieffer, M. J., Bailey, S. P., & Harari, R. R. (2013). Mathematics anxiety in young children: Concurrent and longitudinal associations with mathematical performance. Contemporary Educational Psychology, 38(1), 1–10. doi:10.1016/j. cedpsych.2012.09.001 Whitaker Sena, J. D., Lowe, P. A., & Lee, S. W. (2007). Significant predictors of test anxiety among students with and without learning disabilities. Journal of Learning Disabilities, 40 (4), 360–376. Wigfield, A., & Meece, J. L. (1988). Math anxiety in elementary and secondary school students. Journal of Educational Psychology, 80 (2), 210–216. doi:10.1037/0022–0663.80.2.210 Wu, S. S., Barth, M., Amin, H., Malcarne, V., & Menon, V. (2012). Math anxiety in second and third graders and its relation to mathematics achievement. Frontiers in Psychology, 3, 162. doi:10.3389/fpsyg.2012.00162 Zeidner, M. (1998). Test anxiety: The state of the art. New York: Plenum Press. Zeidner, M. (2007). Test anxiety in educational contexts: Concepts, findings, and future directions. In P. A. Schutz, & R. Pekrun (Eds.), Emotion in education (pp. 165–184). Boston, MA: Elsevier Academic Press.

8 MATH ANXIETY IN CHILDREN WITH AND WITHOUT MATHEMATICAL DIFFICULTIES The role of gender and genetic factors Sara Caviola, Irene C. Mammarella and Yulia Kovas

Mathematics anxiety is generally defined as feeling tense, fearful and apprehensive about mathematics ( Richardson & Suinn, 1972). It is a multi-dimensional construct, characterized by different types of reactions: emotional (i.e. negative feelings); cognitive (e.g. intrusive concerns and thoughts); physiological (e.g. increased arousal, stress and agitation); and behavioural (e.g. avoiding contexts that require the use of mathematical skills, disengagement and off-task behaviours). From a different angle, math anxiety can reverse the effects of positive factors, such as interest in mathematics and self-efficacy (Moore, Rudi, & Ashcraft, 2014). Individuals with high levels of math anxiety tend to take fewer mathematics courses; get lower grades in those they do attend; and avoid, where possible, additional math classes (Ashcraft, 2002). In addition, highly math-anxious students are also more likely to avoid mathematically oriented college majors and career paths that require quantitative skills (Ashcraft, Krause, & Hopko, 2007; Ashcraft & Moore, 2009). Math anxiety seems to have serious long-term consequences, adversely inf luencing an individual’s choice of career, type of occupation and professional growth in adulthood (Ashcraft & Ridley, 2005; Beasley, Long, & Natalia, 2001; Hembree, 1990; Ho et al., 2000). Beyond consequences for an individual’s personal life, math anxiety also affects society. For example, in the United States, math anxiety may contribute to the shortage of graduates who want to work in the area of science, technology, engineering and mathematics – to meet the demands of a technology-dependent society – despite increased emphasis on improving mathematical education ( Beilock & Maloney, 2015). Because of its consequences in limiting people’s mastery of mathematics, math anxiety has become a subject of increasing interest in educational, rather than only clinical, settings. Many factors are involved in the links between math anxiety and mathematics. For example, these links depend on the nature of mathematics, such as increasing complexity of its contents during the school years. In

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the following sections, we summarize previous studies of math anxiety, focusing on gender differences, distinctions between mathematics difficulties related to math anxiety vs. those related to specific mathematics impairments, and the role of genetic factors.

Gender differences in math anxiety and mathematics performance Math anxiety has recently been investigated in the best-known international comparison of student achievement in mathematics, the Programme for International Student Assessment (PISA, OECD, 2013), which assessed the competencies of 15-year-old students from 65 different countries. The PISA investigation (OECD, 2013) asked students several questions about math anxiety, such as whether they get very tense or nervous when they have to do mathematics homework or mathematics problems. A considerable proportion of students reported feelings of helplessness and emotional stress when dealing with mathematics. Across OECD countries, around 33% reported that they get very tense when they have to do mathematics homework, 31% reported that they get very nervous doing mathematics problems and 61% reported that they worry about getting poor grades in mathematics. The proportion of students who reported high math anxiety has slightly increased from 2003 (29%) to 2012 (32%). Across most countries, differences in levels of mathematics anxiety related to gender are wide, with girls on average reporting higher math anxiety than boys. For example, Italian girls on average reported being less confident in their ability to learn mathematics and more anxious about mathematics than boys (e.g., 48.5% of the girls vs. 37.8% of the boys reported high levels of math anxiety; OECD, 2013). A similar trend could be observed for the UK adolescents (although the overall level of math anxiety was lower than in the Italian sample): 24.4% of the girls vs. 15% of the boys reported high levels of math anxiety. Across most countries, students who reported higher math anxiety had poorer performance in mathematics than students who reported lower levels of math anxiety (Foley et al., 2017). Several conclusions about math anxiety can be drawn from the existing literature. Similar findings to those observed in the PISA survey (OECD, 2013) have been reported in other studies. First, the phenomenon of math anxiety has been observed in in both Western and Eastern countries (Ching, 2017; Ho et al., 2000; Lee, 2009). Second, math anxiety is robustly negatively correlated with mathematics performance, in both boys and girls (roughly r = -.30; Hembree, 1990; Ma, 1999). Third, the negative relation between math anxiety and mathematics performance seems to be stronger in girls than in boys ( Else-Quest, Hyde, & Linn, 2010; Devine, Fawcett, Szűcs, & Dowker, 2012; Dowker, Sarkar, & Looi, 2016; Hill et al., 2016). Indeed, females, on average, have higher levels of math anxiety than males (Hembree, 1990; Karimi & Venkatesan, 2009), and this gender difference seems to increase in older students ( Hill et al. 2016).

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However, gender differences in mathematics performance and math anxiety seem to follow different trends. Recent studies revealed that gender-gap in maths performances is disappearing. For example, the study of Karimi and Venkatesan (2009) did not show gender differences in mathematics performance. Similarly, Devine and colleagues (2012), found no differences in maths performance between girls and boys in 482 students attending secondary school in the United Kingdom. However, the same study reported significant differences between math anxiety in girls and boys. Similar findings, that is, no gender differences in mathematics but in math anxiety, have been observed in a sample of 981 primary and middle schools students in Italy (Hill et al. 2016). Although several cross-sectional studies found no gender differences in math anxiety (e.g., Ahmed, 2018; Birgin, Baloğlu, Çatlıoğlu, & Gürbüz, 2010; Chiu & Henry, 1990; Newstead, 1998), a recent meta-analysis found that girls reported significantly higher math anxiety than boys despite negligible gender differences in mathematics performance ( Else-Quest et al. 2010). It is worth noting though that not all of these studies took into account an objective measure of mathematics performance (i.e., Ahmed, 2018). Different hypotheses have been proposed. A possible explanation for similar performance in mathematics between boys and girls, despite greater math anxiety in girls, is that girls may have greater mathematics potential than boys ( Devine et al. 2012). Alternatively, greater math anxiety in girls may relate to their lower maths self-perceptions and confidence (Cvencek, Meltzoff, & Kapur, 2014; Fredricks & Eccles, 2002; Marsh & Yeung, 1998; Pajares, 2005); the fact that boys are less likely to openly state their negative feelings (Ashcraft & Ridley, 2005); that boys’ answers are more affected by a recall bias as well as social desirability biases ( Dowker et al., 2016); or the presence of gender stereotypes about maths (Appel, Kronberger, & Aronson, 2011; Flore & Wicherts, 2015). More research is needed in order to clarify why female students frequently report higher math anxiety than do male students.

Mathematics anxiety vs. mathematical difficulties A large proportion of students have cognitive and/or emotional difficulties in dealing with mathematics (Hopko, McNeil, Gleason, & Rabalais, 2002). There are two major reasons for why a student fails in mathematics: the presence of a specific learning disorder in mathematics (i.e., developmental dyscalculia), or the presence of emotional issues that affect mathematics performance, such as mathematics anxiety. An unresolved question is whether math anxiety causes poor mathematics performance, or whether poor mathematics performance prompts math anxiety (Carey, Hill, Devine, & Szücs, 2016). According to some studies, mathematics anxiety is believed to cause negative thoughts and ruminations, often about the consequences of failure in maths tasks (Ashcraft & Kirk, 2001). Thus, cognitive resources, such as working memory, that are needed for success in mathematics, are impaired in individuals with high math anxiety (Ashcraft &

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Kirk, 2001; Ashcraft & Faust, 1994). In contrast, other authors argue that math anxiety is actually the outcome of poor mathematics performance. According to this account, a student’s reduced competency in maths leads to weaker learning and performance, which then contribute to the development of math anxiety (Maloney, 2016). Math anxiety is thus an outcome of poor mathematics skills. Another proposed framework within this account, suggests that students who have reduced maths abilities avoid taking maths classes and learning mathematics in general. This avoidance can contribute to successive failures and lead to the development of math anxiety. Consistent with this claim, students with high math anxiety take fewer maths courses (Ashcraft & Kirk, 2001; Hembree, 1988; LeFevre, Kulak, & Heymans, 1992). Thus, students with high math anxiety or with mathematics difficulties due to the presence of a specific learning disorder can be confused; or, from another viewpoint, it can be expected that math anxiety will be frequently co-morbid with developmental dyscalculia. This is actually one important question that needs to be further addressed in future research: do mathematical difficulties and math anxiety share a common root in terms of impairments or can they be identified as two separate and dissociable problems? Developmental dyscalculia is a specific learning disorder of mathematical ability in individuals showing average language and general cognitive skills ( Devine, Soltész, Nobes, Goswami, & Szűcs, 2013; Shalev & Gross-Tsur, 2001). Not much research to date has investigated the prevalence of co-occurrence between math anxiety and dyscalculia. Recently, Devine, Hill, Carey, and Szűcs (2018) investigated the co-morbidity of these conditions in a sample of 1757 primary and secondary school children in the United Kingdom. The results demonstrated that developmental dyscalculia and math anxiety largely dissociate, calling into question the idea that low mathematics performance is the primary cause of math anxiety. In particular, Devine and colleagues (2018) showed that, using a threshold of high math anxiety at or above the 90th percentile, 10% of students with typical mathematics performance had high math anxiety and 22% of students in the developmental dyscalculia had high math anxiety. The study also reported an equal distribution of boys and girls with developmental dyscalculia, in agreement with previous findings ( Koumoula et al., 2004; Lewis, Hitch, & Walker, 1994). In contrast, Devine and co-authors (2018) showed that more females than males fell in the sub-group with comorbid developmental dyscalculia and math anxiety, which, again, is in line with findings reporting higher math anxiety in girls compared to boys (reviewed in Devine et al., 2012, and Hill et al., 2016; for opposite results, see: Kucian et al., 2018). In another study, the prevalence of developmental dyscalculia was the same in boys and girls; however, gender differences emerged when the cut-off criteria was varied using a discrepancy threshold between reading and mathematics performances ( Devine et al., 2013). In other words, there were more girls with higher reading than mathematics performances, but when performances on maths were below 1 or 1.5 standard deviation (with average or above average reading skills), no differences between boys and girls emerged.

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Some studies have also examined the cognitive consequences of math anxiety by comparing high vs. low math anxiety levels in different mathematics achievement groups (e.g., Lai, Zhu, Chen, & Li, 2015; Passolunghi, 2011; Wu, Willcutt, Escovar, & Menon, 2014). However, these studies did not report the prevalence of co-morbid developmental dyscalculia and math anxiety or correlation between math anxiety and mathematics performance – as they focused on disentangling specific cognitive profiles of children with math anxiety vs. mathematics difficulties or developmental dyscalculia. In particular, several studies focused on working memory profiles of children with math anxiety and developmental dyscalculia (e.g., Mammarella, Hill, Devine, Caviola, & Szűcs, 2015; Passolunghi, Caviola, De Agostini, Perin, & Mammarella, 2016) or their inhibitory processes (Mammarella, Caviola, Giofrè, & Borella, 2018). For example, Mammarella et al. (2015) found that children with high math anxiety and developmental dyscalculia were impaired in working memory tasks in a different way: children with high math anxiety scored lower on a verbal working memory task, while children with developmental dyscalculia scored lower on a visuospatial working memory task. The impairment in verbal working memory in the group of children with high math anxiety is consistent with the hypothesis of Ashcraft and colleagues (Ashcraft & Kirk, 2001; Ashcraft & Faust, 1994; Hopko, Ashcraft, Gute, Ruggiero, & Lewis, 1998), who suggested that intrusive thoughts can interfere with the ability to perform mathematics tasks by usurping working memory resources of individuals with high math anxiety. Passolunghi et al. (2016) also found that children with high math anxiety made more intrusion errors than controls in a verbal working memory task, meaning that they were not able to properly inhibit irrelevant information while performing a verbal working memory task. Similarly, consistent with Mammarella et al. (2015), previous studies also indicated that children with developmental dyscalculia were more impaired in visuospatial than in verbal working memory tasks (e.g., Passolunghi & Mammarella, 2010; Szűcs, Devine, Soltesz, Nobes, & Gabriel, 2013; van der Sluis, van der Leij, & de Jong, 2005). As for inhibitory processes, Mammarella and colleagues (2018), compared 8–10-year-old children with math anxiety, with or without developmental dyscalculia. The results showed that children with math anxiety without developmental dyscalculia had more difficulty in resisting proactive interference (i.e., ability to ignore information that was no longer relevant for the execution of a particular task). This result is consistent with the attentional control theory ( Eysenck, Derakshan, Santos, & Calvo, 2007), which states that anxiety impairs processing efficiency because it reduces attentional control. Overall, these results indicate that math anxiety and developmental dyscalculia may largely stem from different factors. This, in turn, means that successful intervention methods should discriminate between emotional and cognitive factors (e.g., Caviola, Gerotto, & Mammarella, 2016; Ramirez & Beilock, 2011) because while they both lead to poor mathematics performance, they do so for different reasons.

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Genetic factors underlying mathematics anxiety The aetiology of math anxiety can be examined not only at the phenotypic level, such as examining causal links between math anxiety and mathematics performance, but also at the level of genetic and environmental factors. In recent years, a number of studies explored the genetic and environmental aetiologies of mathematical abilities and difficulties and, even more recently, math anxiety. Genetic studies can be broadly split into quantitative genetic methodologies (e.g., twin and adoption studies) and molecular genetic research. Quantitative genetic research allows for the exploration of the origins of individual differences in specific traits as well as of the covariation between multiple traits; molecular genetic studies aim to identify genes related to specific traits (e.g., Kovas, Malykh, & Gaysina, 2016). Most of the quantitative genetics studies used the twin method to estimate genetic and environmental inf luences on individual differences in observed variables (Plomin, DeFries, & McClearn, 2008). The twin method compares phenotypic (observed) similarity between monozygotic twins (who are 100% identical in their DNA) and dizygotic twins (who are 50% genetically similar for those parts of DNA that differ in humans). Heritability is defined as the proportion of individual differences in an observed trait explained by genetic variation, which can be due to additive or non-additive effects of genes. Additive genetic factors are the sum of the effects of all alleles at all loci contributing to the variation in a trait or to the covariation between traits. Shared environmental effects contribute to the similarity of twins growing up in the same family; whereas non-shared environmental effects contribute to differences in family members and include measurement error ( Kovas et al. 2016). The role of genetics factors in mathematical ability has been explored in a number of studies, both using the twin method (Kovas et al., 2013; Petrill, Kovas, Hart, Thompson, & Plomin, 2009; Rimfeld, Ayorech, Dale, Kovas, & Plomin, 2016) and molecular association (Davis et al., 2014; Docherty, Kovas, & Plomin, 2011). Kovas, Petrill, and Plomin (2007) revealed that for different mathematics domains the heritability estimates were moderate and similar across different domains (i.e., ranging from .30 to .45 for mathematical application, understanding of numbers, computation, etc.). Kovas et al. (2013) analysed literacy, numeracy and f luid intelligence ( g) in children at ages 7, 9, and 12. They found that about 65% of the differences in literacy and numeracy among children in the early school years could be explained by genetic differences and that the heritability of g was significantly lower than that of literacy and numeracy in primary school. According to the authors, heritability of literacy and numeracy could be viewed as an index of educational equality – the more equal/standardized environments are, the more differences in educational outcomes are explained by genetic differences. In fact, in younger students the effects of family environments on g were not mitigated by education. In contrast, in older students, the shared environmental inf luence on g declined and was no longer significantly different from that on literacy and numeracy.

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Regarding learning disabilities, studies revealed that monozygotic and dizygotic twins of individuals with developmental dyscalculia were twelve and eight times more likely to have the same disorder, respectively (Alarcón, DeFries, Light, & Pennington, 1997) and more than 50% of siblings of individuals with developmental dyscalculia also had the same disorder (Shalev et al., 2001). Plomin and Kovas (2005) in a review of the literature, suggested that most genes associated with specific learning disabilities, such as developmental dyslexia and dyscalculia, are generalists in their effects within and between learning disabilities and abilities. For example, genes that affect specific learning disabilities are largely the same genes responsible for normal variation in learning abilities. In addition, genes that affect a specific learning ability and disability are also likely to affect other learning abilities and disabilities. Moreover, research has suggested that mathematics ability is thought to be inf luenced by many genes that have small effects across the entire spectrum of ability ( Docherty et al., 2010). Very few studies to date have investigated the role of genetics factors on math anxiety. Wang et al. (2014) carried out one of the first studies, applying the twin method, which investigated the genetic contributions to math anxiety and to its association with general anxiety and maths performance, in a sample of 514 12-year-old children (216 monozygotic, and 298 dizygotic twins). The results revealed that genetic factors accounted for around 40% of the variation in math anxiety, with the remaining variance accounted for by child-specific (nonshared) environmental factors. Multivariate behavioural genetic models explored the aetiology of the overlap between general anxiety, maths problem solving, and math anxiety. The results showed that 9% of the total variance in math anxiety was associated with genetic inf luences in common with general anxiety, and 4% of the total variance was associated with non-shared environmental inf luences in common with general anxiety. An additional 12% of the total variance in math anxiety was associated with genetic inf luences related to maths problem-solving. No gender differences were found in the aetiology of anxiety. The authors concluded that although the origins of math anxiety partially overlap with general anxiety and maths performance, the causes of individual differences in these three traits are mostly independent ( Wang, et al., 2014). In a further study, Malanchini et al. (2017) analysed different forms of anxiety: general, spatial and math anxiety, with the aim of disentangling the origins of their association. Data on general, math and spatial anxiety were collected in a large sample of more than 1400 twin pairs, assessed online. The results revealed that all forms of anxiety were moderately heritable (from 30% to 41%). The genetic and environmental architecture of different forms of anxiety was similar for males and females, suggesting that the same factors are implicated in the aetiology of individual differences in anxiety in boys and girls. Finally, the results indicated a large degree of specificity in the aetiology of different forms of anxiety, suggesting the importance of studying and potentially treating anxiety forms separately. Overall, these findings suggest the importance of both genetic and environmental (e.g. socio-cultural and demographic) factors in people’s susceptibility to

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math anxiety. The literature investigating the social and other environmental aspects of math anxiety’s aetiology focused on individuals’ experiences inside the classroom and home experience related to maths, such as teachers’ mind-sets and their maths-related teaching practices ( Bursal & Paznokas, 2006; Gresham, 2008; Swars, Daane, & Giesen, 2006), parental involvement in maths education, parental expectations and parents’ math anxiety experiences ( Maloney, Ramirez, Gunderson, Levine, & Beilock, 2015). These studies are described to more extent and in further detail in Chapter 11. Future quantitative genetic research is needed to explore the genetic and environmental aetiology of covariation between math anxiety and maths performance. In addition, molecular genetic research is needed to identify specific genetic markers involved in the aetiology of math anxiety and its links to maths performance.

Overall conclusions and future directions The studies reviewed in this chapter suggest the complexity of the phenomenon of math anxiety. Although a moderate negative correlation exists between math anxiety and mathematics performance, the presence of the causal association and direction of its effects are still not completely clear. Knowing the direction of the math anxiety-maths performance link can have implications for education and for psychology research (Carey et al., 2016). Research has begun to clarify the differences in cognitive profiles and other characteristics between students who struggle with mathematics because of math anxiety and students who struggle with maths because of a specific learning disorder, such as developmental dyscalculia ( Lai et al., 2015; Mammarella et al., 2015, 2018; Passolunghi et al., 2016; Passolunghi, 2011). Additionally, recent research suggests that developmental dyscalculia and math anxiety largely dissociate ( Devine et al., 2018). As for gender differences in math anxiety, previous findings have suggested that gender differences may be partly related to gender stereotypes about maths (Appel et al., 2011; Flore & Wicherts, 2015), home environment (Maloney, et al. 2015) and attitudes and beliefs of teachers ( Bursal & Paznokas, 2006; Gresham, 2008; Swars et al., 2006). From an interactionist point of view, anxiety is the result of not only individual predispositions toward evaluating an event as threating but also situational demands, interpersonal interactions and expectations (e.g., Löwe et al., 2008). Interestingly, genetic studies did not find any gender differences in the aetiology of math anxiety between males and females. In terms of the aetiology of individual differences in math anxiety, genetically informative research suggests a high degree of specificity: largely non-overlapping genetic factors contribute to different forms of anxiety (general anxiety, spatial anxiety and math anxiety) and to maths performance (Malanchini et al., 2017; Wang et al., 2014). Further studies should not only consider other inf luential factors of the direction of math anxiety-maths performance (such as environmental variables and multiple source reports) but also examine in more depth the cognitive profiles of children with math anxiety or developmental dyscalculia.

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The complexity of the relationship between mathematics difficulties and math anxiety leaves several questions to be answered by future studies. For example, the factors that moderate the development, growth and impact of math anxiety should be further explored. The literature has shown that moderate levels of anxiety help focus attention and enhance working memory, whereas extremely high or low levels of anxiety are associated with insufficient cognitive resources allocated to the tasks (Arnsten, 2009; Diamond, Barnett, Thomas, & Munro, 2007). Wang et al. (2015) revealed that the facilitating and debilitating effects of math anxiety on maths performance vary not only across different levels of math anxiety, but also as a function of how motivated children are to perform well. Interestingly, Wang and colleagues (2015) observed the classical inverted-U relation between math anxiety and mathematics performance only in students showing a high intrinsic motivation, while students with low motivation revealed a negative linear relation between math anxiety and mathematics performance. In another study, Mammarella and colleagues (Mammarella, Donolato, Caviola, & Giofrè, 2018) highlighted the importance of considering self-concept and resilience in relation to different levels of anxiety risk profiles. Hence, further studies should investigate in more depth individual characteristics that could support and promote academic success and well-being in school-aged children and that could prevent the onset or impact of math anxiety. Another key point for further research is to find effective interventions to reduce the negative impact of math anxiety on mathematics achievement. Research suggests that cognitive-behavioural therapy treatment protocols can be effective in reducing anxiety symptoms in school children (Fisak, Richard, & Mann, 2011; Neil & Christensen, 2009). For example, the FRIENDS for Life programme ( Barrett & Pahl, 2006) involves teachers as facilitators to run group sessions as a routine component of the class activity ( Lowry-Webster, Barrett, & Dadds, 2001) and has shown effectiveness in randomized controlled trials ( Fisak et al., 2011; Stallard et al., 2014). As for specific interventions in math anxiety, Supekar, Iuculano, Chen, and Menon (2015) demonstrated that an intensive eight-week, one-on-one cognitive tutoring programme reduces math anxiety in children and improves maths skills through desensitization. In another study, Park, Ramirez, and Beilock (2014) employed an expressive writing technique aimed at reducing the number of intrusive thoughts of individuals with high math anxiety in order to improve their maths performance. Further studies are needed to replicate and extend these promising lines of research for the treatment of math anxiety. In our view, distinguishing between the different reasons why students fail in mathematics (high math anxiety vs. developmental dyscalculia) is crucial in order to provide them with specific, individualized strategies for coping with math anxiety. Again, individual protective factors such as intrinsic motivation and resilience, and contextual information, such as home environment and interactions with teachers, should be considered as mediating factors. To conclude, both the potentially complex interplay between emotion and cognition and the influence of genetics and

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contextual factors need to be considered to better understand the relation between math anxiety and maths performance.

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9 PROBING THE NATURE OF DEFICITS IN MATH ANXIETY Drawing connections between attention and numerical cognition Orly Rubinsten, Hili Eidlin Levy and Lital Daches Cohen

Math anxiety is characterized by an excessive and sometimes unreasonable fear of numerical related activities (e.g., homework in mathematics), as well as a fear of situations wherein individuals are concerned about possible negative evaluation (e.g., by teachers or classmates) of their numerical performance (e.g., during math exams) ( Richardson & Suinn, 1972; see also Dowker, Sarkar, & Looi, 2016; Maloney & Beilock, 2012; Suárez-Pellicioni, Núñez-Peña, & Colomé, 2016). Ashcraft and colleagues (e.g., Ashcraft & Krause, 2007) presented one of the leading theories explaining math anxiety. They suggested that it is “the actual doing of math” (Ashcraft & Kirk, 2001, p. 235) that acts as an obstacle to mathematical achievements, over and above the real mathematical ability or disability (Ashcraft & Krause, 2007). That is, individuals with math anxiety do not necessarily suffer from math-learning disabilities (or dyscalculia). Rather, they suffer from math anxiety-induced ruminations (i.e., repetitive thinking about negative personal concerns and/or about the implications and causes of a negative mood; Nolen-Hoeksema, Wisco, & Lyubomirsky, 2008), which jeopardize cognitive resources such as working memory. These insufficient cognitive resources (e.g., working memory) lead to lower performance on numerical tasks (see also Beilock, 2010; Maloney & Beilock, 2012; but see Carey, Hill, Devine, & Szücs, 2016 who suggested a bidirectional link between math anxiety and math performance). Indeed, for example, in a study conducted by Ramirez, Chang, Maloney, Levine, and Beilock (2016), math anxious children with the highest working memory capacity showed the lowest math performance (compared to those with a lower working memory capacity). Against this background, our suggestion is that some of the cognitive traits associated with general anxiety, such as the tendency to display attentional bias toward negative information ( Bar-Haim, Lamy, Pergamin, Bakermans-Kranenburg, & Van Ijzendoorn, 2007; Muris, Rapee, Meesters, Schouten, & Geers, 2003) which

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leads to anxiety-induced ruminations, are involved in not only general anxiety but also math anxiety. Specifically, selective attention to threat ref lects the adaptive neurocognitive function of protecting us from danger ( Bar-Haim et al., 2010; Cisler & Koster, 2010; Robinson, Charney, Overstreet, Vytal, & Grillon, 2012). However, dysfunction in these adaptive attentional processes results in excessive attention to threat stimuli, a condition called “threat related attentional bias”. Threat related attentional bias has been linked to deficient neurocognitive processes that underlie the etiology of anxiety related disorders ( Bar-Haim et al., 2007; Bishop, 2009; Cisler & Koster, 2010; Luijten et al., 2012). That is, it has been clearly shown that there is a significant contribution of general (trait) anxiety and state anxiety to threat perception abnormalities (e.g., Muris et al., 2003). Here, we wish to suggest a theoretical model (see Figure 9.1 at the end of the chapter, acting as a summary) in which environmental factors interact with intrinsic elements, particularly with the child’s attentional resources, to increase the risk of developing math anxiety. Such a hypothesis goes hand in hand with findings showing that anxiety increases the inf luence of the stimulus-driven attentional system (see Eysenck, Derakshan, Santos, & Calvo, 2007). Thus, children with a predisposition to (general) anxiety, including maladaptive attentional resources (specifically threat related attentional bias, as typically appears in general anxiety), are proposed here to be at particular risk for developing math anxiety if they also experience environmental risk factors for math anxiety for example, in school (as has been shown, e.g., in Beilock, Gunderson, Ramirez, & Levine, 2010; Gunderson, Ramirez, Levine, & Beilock, 2012), during extra-curricular activities (as has been shown, e.g., in Berkowitz et al., 2015), and within the child’s family (as has been shown, e.g., in Daches Cohen, & Rubinsten, 2017; Gunderson et al., 2012; Maloney, Ramirez, Gunderson, Levine, & Beilock, 2015; for protective parental effects see Gunderson et al., 2017). We propose that these environmental and intrinsic (attentional bias) factors interact and lead to math anxiety as well as inf luence the individual’s acquisition of numerical skills (see the third column from the left in Figure 9.1). Attentional bias involves a content-specific aspect (for a meta-analysis see Pergamin-Hight, Naim, Bakermans-Kranenburg, van IJzendoorn, & Bar-Haim, 2015). Such content is induced by specific threats and unpleasant feelings that are created through personal learning and memory. It has been shown that social environmental experiences shape responses to distress (e.g., Michl, McLaughlin, Shepherd, & Nolen-Hoeksema, 2013). Consequently, contents that lead to attentional bias are distinctively relevant to anxiety type and the individual’s past experience (e.g., Bar-Haim et al., 2007; Beck & Clark, 1997; Mogg & Bradley, 1998; Öhman, 1996). If a pedagogical environment requires, for example, the processing of learned symbolic numerical information (e.g., Arabic numerals such as “7” “123”, or numerical words such as “seven” and symbolic arithmetic such as 43 + 25 =), then only these specific stimuli are more likely to be associated with stress and unpleasant feelings over time (as in, e.g., Rubinsten, Bialik, & Solar, 2012; Rubinsten & Tannock, 2010).

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Accordingly, in the first part of the review we demonstrate the following two points: (1) In some cases math anxiety is linked, at least during initial developmental stages, to symptoms that are known to be involved with general anxiety. As a result, people who are intrinsically vulnerable hold cognitive traits of anxiety (maladaptive threat related attentional bias) which can constitute an independent risk factor for math anxiety. (2) Math anxiety is associated with distinctive patterns of processing (threat related attentional bias) of mainly personally relevant threat information that was previously experienced, learned, and memorized, leading these individuals to obtain negative attitudes towards mathematics. In the second part of the chapter, we show that math anxiety is linked mainly to learned numerical information (but not to innate or core knowledge), suggesting a cognitive profile of threat related attentional bias. The theoretical model (Figure 9.1) that we present highlights the complex pathways toward the development of math anxiety, but we focus mainly on the interplay between environmental and intrinsic factors (left column in Figure 9.1). Some components of our model are still hypothetical. For example, different pathways and factor mixes can result in math anxiety, and any one risk factor can result in a variety of outcomes (e.g., not just math anxiety or different levels of math anxiety). While recognizing the limitations of the model with its restricted focus, we tried to draw together ample evidence to suggest, in some cases of math anxiety, an interplay between intrinsic (attentional resources) and environmental factors (such as parental and teacher inf luences). We propose to view math anxiety as stimulus- and situation-specific anxiety (as also suggested by Young, Wu, & Menon, 2012), and suggest that threat related attentional bias may play a partial role in the initial development of math anxiety. Considering the universal nature of math anxiety, it is important to examine how such attentional processes come to be established and whether threat related attentional bias interacts with (negative) environmental factors to increase the risk of developing math anxiety.

What is math anxiety? Math anxiety: General description Numerical development has been the focus of a continuing theoretical debate concerning the origins of cognition and how it develops throughout one’s lifetime. Approximately 10% of the population have low numerical skills, a condition called developmental dyscalculia (DD) or mathematical learning disability (MLD) ( Kaufmann et al., 2013; Rubinsten & Henik, 2009; Szűcs & Goswami, 2013). Such numerical difficulties result in reduced educational and employment achievements, and in increased health costs ( Duncan et al., 2007; Parsons & Bynner, 2005; Reyna, Nelson, Han, & Dieckmann, 2009; Woloshin, Schwartz, Moncur, Gabriel, & Tosteson, 2001). Also, numerical skills have been

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associated with socioeconomic status ( Ritchie & Bates, 2013) and mortgage default (Gerardi, Goette, & Meier, 2013). Some argue that, in Western society, poor numeracy is a greater handicap than poor literacy (e.g., Estrada, MartinHryniewicz, Peek, Collins, & Byrd, 2004; Rivera-Batiz, 1992). Learning arithmetic or mathematics is also complicated for many people who have negative attitudes toward math ( Dowker, Bennett, & Smith, 2012) as well as a persistent negative reaction to mathematics (henceforth, math anxiety; Richardson & Suinn, 1972) ranging from mild discomfort to extreme avoidance (Ashcraft & Ridley, 2005; Hembree, 1990; Ma & Xu, 2004a, 2004b; and see Maloney & Beilock, 2012). Math anxiety may include feelings of tension ( Richardson & Suinn, 1972), low self-confidence in one’s ability to learn mathematics ( Jain, 2009), or a decline in working memory (Ashcraft & Kirk, 2001), inhibitory control, and fact retrieval ( Passolunghi, Caviola, De Agostini, Perin, & Mammarella, 2016), as well as in counting abilities (Maloney, Risko, Ansari, & Fugelsang, 2010) and in the precision of the mental representations of numerical magnitudes (Maloney, Ansari, & Fugelsang, 2011). In general, math anxious individuals show poor cognitive abilities in mathematics (e.g., Rubinsten & Tannock, 2010). Math anxiety was also found to have a possible genetic root ( Wang et al., 2014), with genetic factors accounting for about 40% of the variation in math anxiety. In addition, math anxiety seems to involve a specific neural basis ( Young et al., 2012), which was found even when only anticipating mathematical problems ( Lyons & Beilock, 2011). Finally, although there are exceptions, most studies of math anxiety have reported higher levels of math anxiety in females than in males (e.g., Devine, Fawcett, Szucs, & Dowker, 2012; Hill et al., 2016) (for a meta-analysis see ElseQuest, Hyde, & Linn, 2010). Given the universal and heterogenic nature of math anxiety, a systematic description of the vulnerability factors that contribute to the development of math anxiety is vital. Such information will be of particular value in informing the design of preventive interventions as well as specific assessment tools.

Math anxiety: Assessment Typically, clinicians and scientists use explicit tools such as the Math Anxiety Rating Scale (MARS: Richardson & Suinn, 1972), the Abbreviated Math Anxiety Scale (AMAS: Hopko, Mahadevan, Bare, & Hunt, 2003; for validation of the AMAS in Polish see Cipora, Szczygieł, Willmes, & Nuerk, 2015; and in Italian: Caviola, Primi, Chiesi, & Mammarella, 2017; Primi, Busdraghi, Tomasetto, Morsanyi, & Chiesi, 2014), the math anxiety questionnaire ( Wigfield & Meece, 1988; for a German version see H. Krinzinger et al., 2007), the modified Abbreviated Math Anxiety Scale (mAMAS: Carey, Hill, Devine, & Szűcs, 2017), or the revised Math Anxiety Rating Scale (MARS-R: Alexander & Martray, 1989; Hopko, 2003) to diagnose math anxiety. Such explicit tools typically assess accessible self-representations. Explicit questionnaires have been the

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primary method for obtaining information on symptoms of math anxiety and other psychopathology in school settings, in part because of their convenience, standardization, and good psychometric properties (as suggested, for instance, in cases of ADHD; Pelham, Fabiano, & Massetti, 2005). However, women, for example, have consistently been found to score higher than men on self-report measures of trait anxiety (e.g., Costa, Terracciano, & McCrae, 2001; Egloff & Schmukle, 2004; Feingold, 1994), possibly resulting from gender differences in anxiety that are not due to anxiety per se (see rightmost column in Figure 9.1). Implicit measures, on the other hand, typically assess inaccessible cognitive structures that are processed automatically (as in the emotional Stoop task: Algom, Chajut, & Lev, 2004; or in the dot probe task: MacLeod, Mathews, & Tata, 1986). It has been shown that affective traits can be activated automatically and inf luence emotional, cognitive, and behavioral processes (e.g., Giner-Sorolla, Garcia, & Bargh, 1999) even in the case of math anxiety ( Rubinsten et al., 2012; Suárez-Pellicioni, Núñez-Peña, & Colomé, 2015; Tomasetto, Galdi, & Cadinu, 2012). That is, affective processing begins involuntarily upon seeing a salient affective word or picture (see Rubinsten, 2015) and, evidently, gender differences may be eliminated ( Egloff & Schmukle, 2004) (see bottom of second column in Figure 9.1). Thus, a systematic description of the factors that enhance the development of math anxiety may enable the development of implicit assessment tools that aim to measure specific intrinsic cognitive aspects that contribute to the development and maintenance of math anxiety. Such assessment tools may be more effective than explicit questionnaires (as in the case of gender differences).

The link with general anxiety Math anxiety has been found to be positively, albeit moderately, correlated with general, state, and trait anxiety (Ashcraft & Moore, 2009; Hembree, 1990) and with math performance (Owens, Stevenson, Norgate, & Hadwin, 2008; but see Wu, Amin, Barth, Malcarne, & Menon, 2012 who did not find a significant correlation between trait anxiety and math achievement in second and third graders). In a sample of young adolescent 12-year-old twins, Wang and colleagues (2015) found that general anxiety was minimally correlated with math-task performance and highly correlated with math anxiety. No similar findings were found in a second sample of college students. There is insufficient evidence, as yet, to determine whether general anxiety is a cause or effect of math anxiety related problems. However, longitudinal studies are beginning to examine these complex relationships. For example, in the developmental cross-sectional model, Cargnelutti, Tomasetto, and Passolunghi (2017a) found that general anxiety (examined by a questionnaire completed by teachers, i.e., Anxiety-Observed) affected math achievements in both second as well as third grades, whereas a significant direct role of math anxiety emerged only in Grade 3. Similarly, and with high relevance to the current theoretical model, their longitudinal path model revealed that (general) Anxiety-Observed

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was a significant predictor of math achievements and was stronger in Grade 2 than in Grade 3. Genetic studies may also help determine whether general anxiety does in fact emerge first and cause the development of math anxiety. One such study suggests that 9% of the total variance in math anxiety results from genes related to general anxiety ( Wang et al., 2014). These findings may support the current hypothesis that intrinsic factors associated with general anxiety have a major role in math anxiety (see Figure 9.1 top left side), as suggested, for example, by Liberman and Öst (2016) for cases of specific phobia. That is, we suggest that the cognitive and intrinsic symptoms associated with general anxiety, such as the tendency to ruminate over negative thoughts and stressful situations ( Donaldson, Lam, & Mathews, 2007; Young & Dietrich, 2015) or the tendency to display attentional bias toward negative information ( Bar-Haim et al., 2007), are involved not only in general anxiety but also in math anxiety. Namely, compared to non-anxious individuals, (general) anxious individuals are more likely to show a preference to attend to threatening stimuli over non-threatening stimuli in their environment (attentional bias) (see Van Bockstaele et al., 2014). It has been shown that high levels of general and state anxiety are significantly related to increased threat perception and lower threat thresholds and that both general and state anxieties account for a unique proportion of the variance in threat perception abnormalities (Muris et al., 2003). This tendency of (general) anxious individuals to excessively attend to threat stimuli has been demonstrated with different attention tasks and in different types of anxiety ( Bar-Haim et al., 2007). We ( Rubinsten, Eidlin, Wohl, & Akibli, 2015) and Suárez-Pellicioni and colleagues (2015) were the first to document attentional bias in adults with math anxiety, which led us to argue that math anxiety may be the result of maladaptive characteristics that are involved in general anxiety. Accordingly, children with a predisposition to general anxiety, including maladaptive threat related attentional bias and a tendency to view the world as threatening, are proposed here to be at risk for the development of math anxiety. These maladaptive factors lead these individuals to experience the world, but mainly the specific part of the world that is associated during development with negative valence (e.g., learned numerical information), as threatening (see Figure 9.1 top left side). Our model also proposes that, for children with a predisposition to maladaptive threat related attentional bias, numerically related adverse pedagogical and social events increase the risk of developing and perpetuating math anxiety through their impact on negative beliefs and thoughts related to numerical tasks and skills (i.e., negative beliefs act as a mediating variable; see Figure 9.1 second column from the left).

Negative beliefs in math anxiety It has been shown that math anxious individuals hold beliefs whereby when they enter a mathematical situation they are at risk of behaving in an unskilled way,

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leading to negative consequences. As a result, they may view numerical situations as highly threatening. In their two-year longitudinal study, Gunderson and colleagues (2017) found that children who start school (first grade) with lower levels of math achievement are more likely to develop math anxiety and less adaptive motivational frameworks (i.e., believing that intelligence is malleable versus fixed) (see also Gunderson et al., 2017; Hoffman, 2010; Park, Gunderson, Tsukayama, Levine, & Beilock, 2016). Consequently, it might be argued that during and prior to numerical tasks, high levels of negative beliefs as well as distractions from the numerical task (i.e., through ruminations) may contribute to the experience of math anxiety and are therefore expected to impair numerical performance. Please note that we also wish to argue that even if initial numerical performance is lower compared to that of peers (see, e.g., Gunderson et al., 2017), and if there may be some legitimacy to the child’s reduced evaluations, a tendency to evaluate even more negatively is likely to contribute to math anxiety. In an attempt to explain the causal mechanisms at work, we propose that adverse pedagogical and social environments, particularly encompassing teachers (e.g., Beilock et al., 2010; Gunderson et al., 2012) and parents (e.g., Daches Cohen & Rubinsten, 2017; Gunderson et al., 2012; Maloney et al., 2015), in combination with maladaptive mechanisms of threat related attentional bias, may lead to math anxiety through their creation of negative beliefs. Research into the mediating pathways between adverse pedagogical or environmental experiences and negative beliefs is in its infancy. Park and colleagues (2016), however, recently found that the more first- and second-grade teachers emphasized that children should demonstrate competence in the classroom, the more students believed at the end of the school year that ability is stable and inf lexible. Indeed, in their review paper, Gunderson and collegues (2012) clearly showed that parents and teachers significantly shape children’s math attitudes. Therefore, there is now good evidence that negative social experiences, particularly with teachers and parents, play a role in the development and maintenance of negative beliefs. Thus, negative beliefs are acquired through learning and experience and are specifically related to numerical information and situations. We suggest that, particularly for children with anxiety predisposition who use maladaptive threat related attentional bias and who also experience a range of adverse pedagogical outcomes (left side in Figure 9.1), such a maladaptive attentional focus on negative experiences should predict increases in negative beliefs. These negative beliefs act as mediators and predict math anxiety and low mathematical proficiency (second column from the left in Figure 9.1).

Attentional bias in math anxiety: A link with learned symbolic or with more innate non-symbolic numerical information? By reviewing the scientific literature, we wish to suggest the idea that when faced mainly with numerical information that has been associated over the years

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with negative valence, math anxious (compared to non-anxious) individuals tend to worry and to bias their attention toward the learned numerical situation. As a result, math anxiety may be conceptualized as a situation (i.e., trait) specific anxiety that manifests itself in mathematics-related contents linked during development with unpleasant feelings and generating threat related attentional bias. Indeed, and as mentioned above, Cargnelutti and colleagues (2017a) found that while general anxiety affected math achievements in both second as well as third grade, a significant direct role of math anxiety emerged only in Grade 3. This finding supports the hypothesis that math anxiety may be the result of an accretion of negative experiences during math education (see Figure 9.1 bottom left).

What are the contents that are learned during math education and may be linked with negative valence? Evidence shows that humans have two different mechanisms for processing different types of numerical representations. One cognitive system is for processing non-symbolic numerical representations (e.g., groups of dots that represent quantities), termed here the non-symbolic system (e.g., Landerl, Fussenegger, Moll, & Willburger, 2009; Mazzocco, Feigenson, & Halberda, 2011; Piazza et al., 2010). This non-symbolic system is considered to have evolutionary roots ( Dehaene, Dehaene-Lambertz, & Cohen, 1998; Hauser, Tsao, Garcia, & Spelke, 2003), relies mostly on visuospatial cognitive abilities (Anobile, Turi, Cicchini, & Burr, 2015; Gallistel & Gelman, 2000), and involves approximate quantity processing (e.g., 100 dots that are presented brief ly can be estimated as 90, and not exactly as 100) ( Barth, Beckmann, & Spelke, 2008; Lyons, Ansari, & Beilock, 2015). Estimation of numbers or quantities is related to the strategy employed when a stimulus comprises a large number of items is presented brief ly ( Pavese & Umiltà, 1998). Another numerical cognitive system involves the processing of symbols that represent numbers (e.g., “1”, “7”, “ten”), termed here the symbolic system (e.g., De Smedt, Verschaffel, & Ghesquière, 2009; Holloway & Ansari, 2009; Kolkman, Kroesbergen, & Leseman, 2013; Lonnemann, Linkersdörfer, Hasselhorn, & Lindberg, 2011; Sasanguie, Van den Bussche, & Reynvoet, 2012; Vanbinst, Ansari, Ghesquière, & De Smedt, 2016; Vanbinst, Ghesquière, & De Smedt, 2012). This symbolic system involves an accurate and exact processing of numbers (e.g., the written number “100” will be considered exactly “100” and not “90” – as possible in the non-symbolic system) ( De Smedt, Noël, Gilmore, & Ansari, 2013) and is inf luenced by language and culture ( Bender et al., 2014; Stanislas Dehaene, 2007; Gunderson, Spaepen, & Levine, 2015; Nys, Ventura, Fernandes, Querido, & Leybaert, 2013; Obersteiner, Reiss, & Ufer, 2013). If a pedagogical environment requires, for example, the processing of symbolic numerical information, then these stimuli are more likely to be associated with stress and unpleasant feelings over time. In such cases, some students may show attentional bias toward symbolic numerical information. Therefore, to argue for threat related attentional bias in math anxiety, it is relevant to discuss the

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link between symbolic and non-symbolic numerical information and advanced mathematical skills. Generally, it is suggested that education enables a shift from informal approximate (non-symbolic) representations of numerosity to symbolic and exact representations of numbers ( Dehaene, Izard, Spelke, & Pica, 2008; Siegler & Mu, 2008). However, a detailed review of the literature suggests that the link between symbolic and non-symbolic numerical information and numerical skills is controversial and includes two different lines of findings. Thus, current predictions about attentional bias to symbolic vs. non-symbolic numerical information in math anxiety are not straightforward.

The first line of findings: Bidirectional links between symbolic and non-symbolic numerical information, suggesting attentional bias in math anxiety for both formats Some evidence shows a relationship between non-symbolic numerical abilities and formal mathematics or arithmetic (e.g., Halberda, Mazzocco, & Feigenson, 2008; Libertus, Odic, & Halberda, 2012). Indeed, some studies have shown that symbolic and non-symbolic numerical processing enhance one another over the course of math education (e.g., Piazza, Pica, Izard, Spelke, & Dehaene, 2013) and that the approximate numerical system possibly acts as a foundation for numerical skills (Szkudlarek & Brannon, 2017). Also, it has recently been suggested that the association between non-symbolic numerical comparisons and mathematical competence might be explained by more general non-numerical cognitive abilities, such as inhibitory control (Fuhs & McNeil, 2013; Gilmore et al., 2013; Keller & Libertus, 2015). These findings may suggest that non-symbolic numerical processing gives people a number sense, which is a central aspect of advanced (symbolic/learned) mathematical competence. In line with this, the learning of symbolic numerical and non-symbolic numerical information should be linked throughout the lifespan. Therefore, and based on the suggested impact of specific learned content on attention bias in anxiety (e.g., Bar-Haim et al., 2007; Beck & Clark, 1997; Mogg & Bradley, 1998; Öhman, 1996), it may be argued that for some children during mathematical experiences, both symbolic and non-symbolic numerical information may be linked with unpleasant or even threatening meanings. Moreover, non-symbolic numerical information is more likely to be linked with unpleasant feelings in children with neurocognitive deficiencies in the processing of non-symbolic numerical information. This would lead to the hypothesis that math anxiety will generate attentional bias toward both symbolic and non-symbolic numerical information throughout one’s school years. Indeed, findings show that even children as young as first grade report different levels of math anxiety, which are inversely related to their math achievements (Ramirez, Gunderson, Levine, & Beilock, 2013). Young and colleagues (2012) examined the neurodevelopmental basis of math anxiety by investigating brain response and connectivity during arithmetic problem solving in 7- to 9-year-old children.

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They showed that, in children as young as 7–9 years of age, math anxiety is associated with hyperactivity and abnormal connectivity of the amygdala, a brain region associated with processing negative emotions. In addition, they found reduced activity in brain regions known to support working memory and numerical processing (e.g., the dorsolateral prefrontal cortex and posterior parietal lobe) in math anxious children. Recently, Cargnelutti, Tomasetto, and Passolunghi (2017b) found that specific components of both verbal working memory (i.e., digit span) and approximate addition mediated the relationship between general anxiety and math performance. Ashkenazi and Danan (2017) found a significant negative link between spatial working memory and math anxiety. All of these findings may suggest (and yet not strongly support) that math anxiety could result from deficient development even in very basic (e.g., Maloney et al., 2010) and core numerical knowledge (Maloney et al., 2011), such as non-symbolic numerical information (e.g., dots) (Cantlon, Platt, & Brannon, 2009; Feigenson, Dehaene, & Spelke, 2004). Yet, to the best of our knowledge, only one study showed a direct link between non-symbolic numerical information and math anxiety. Specifically, Lindskog, Winman, and Poom (2017) found that people with high levels of math anxiety show poorer functioning in non-symbolic numerical comparisons compared to those with low levels of math anxiety.

The second line of findings: No links were found between symbolic and non-symbolic numerical information, suggesting attentional bias in math anxiety for learned numerical symbols only The second line of findings shows that literacy affects symbolic but not nonsymbolic numerical processing (Zebian & Ansari, 2012). Also, a recent review ( De Smedt et al., 2013), as well as meta-analyses (Chen & Li, 2014; Fazio, Bailey, Thompson, & Siegler, 2014; Schneider et al., 2015) and scientific investigations (Sasanguie, Defever, Maertens, & Reynvoet, 2014), showed that the processes measured by non-symbolic numerical tasks in scientific labs are not critical for school-relevant mathematics and presented a weak link between non-symbolic numerical processing and mathematical skills (see also Halberda et al., 2008; Libertus, Feigenson, & Halberda, 2013; Sasanguie et al., 2014). It has been suggested that learned numerical symbols do not acquire their numerical meanings from the non-symbolic numerical system (e.g., Reynvoet & Sasanguie, 2016) and that the exact symbolic system is behaviorally distinct from the approximate non-symbolic one ( Lyons, Ansari, & Beilock, 2012; Sasanguie, De Smedt, & Reynvoet, 2017). In addition, the link between non-symbolic numerical information and mathematics education beyond the elementary school years is still under debate ( Rips, Bloomfield, & Asmuth, 2008; Szűcs, Soltész, & Goswami, 2009). These findings may indicate that education mainly modulates symbolic numerical representations but not non-symbolic numerical magnitudes. This may suggest another hypothesis, that math anxiety will not show attentional bias

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toward non-symbolic numerical information. This is because it is less likely that non-symbolic numerical information will be associated with negative feelings during development, as it is not relevant for school-like arithmetic skills. Indeed, most of the findings support the strong link between math anxiety and learned symbolic information as well as the involvement of working memory (ruminations). For example, it has been shown that mathematics anxious individuals, relative to non-mathematics anxious individuals, showed a deficit in the counting but not in the subitizing range (Maloney et al., 2010). Counting is considered to be an exact learned symbolic skill (Cordes, Gelman, Gallistel, & Whalen, 2001; Gallistel & Gelman, 1992). Furthermore, working memory was found to mediate this group difference. Similarly, math anxious participants showed a decline in the accuracy of mental representations of learned symbolic but not of innate non-symbolic numerical magnitudes (Maloney et al., 2011; for replication, see Dietrich, Huber, Moeller, & Klein, 2015). Similarly, Douglas and LeFevre (2018) found no link between math anxiety and innate spatial abilities. Mammarella and colleagues (2015) found that children with math anxiety were impaired in the backward words task (i.e., verbal working memory task), which requires maintenance of information in the phonological loop ( Berch, 2008; Engle, Tuholski, Laughlin, & Conway, 1999). In summary, it seems that there is a strong link between mainly learned symbolic numerical information and math anxiety (e.g., Dietrich et al., 2015; Maloney et al., 2011), possibly due to accumulation of negative experiences with these learned symbols (Cargnelutti et al., 2017a; Gunderson et al., 2017), which leads to threat related attentional bias toward learned symbolic information ( Rubinsten et al., 2015).

Summary and description of the theoretical model There is strong evidence that math anxious people tend to experience a range of adverse math experiences involving teachers (e.g., Beilock et al., 2010; Gunderson et al., 2012) and parents (Maloney et al., 2015; e.g., Daches Cohen & Rubinsten, 2017; Gunderson et al., 2012), but mainly in relation to learned symbolic numerical information (e.g., Dietrich et al., 2015; Maloney et al., 2011). This leads to the suggestion that math anxiety may involve maladaptive threat related attentional bias towards learned numerical information. As can be seen in Figure 9.1, we propose a theoretical model by which adverse learning experiences (with learned numerical information) are proposed to result in math anxiety through negative beliefs and ruminations only for those people who are intrinsically vulnerable (those who, for example, show a tendency toward excessively attending to threat stimuli, as in the case of general anxiety). The model illustrates intrinsic and environmental risk factors, as well as moderating (marked by dark gray boxes) and mediating processes (marked by light gray boxes), that are proposed to have an impact on the development of math anxiety. Although some of the linkages are still not fully proven, there is

Intrinsic Factors

Peers

Teachers

Parents

Linking learned symbolic numerical information with negative valence

Maladaptive tendency to display attentional bias toward negative information (as in general anxiety)

The left column of the model is the main focus of the review paper:

Negative beliefs

Working memory is occupied by negative thoughts (ruminations)

Math anxiety

(e.g. dot probe, emotional Stroop):

Implicit Diagnosis

Specific Math Related Factors

Levels of math proficiency

Levels of math anxiety

Culture

Age

Gender

Math anxiety

(questionnaires):

Explicit Diagnosis

Diagnostic Threshold

FIGURE 9.1 A model of the development of math anxiety: Environmental factors interact with intrinsic elements, and particularly with the child’s attentional resources, to increase the risk of developing math anxiety. Dark gray boxes = moderating processes; light gray boxes = mediating processes

Environmental Factors

Cultural and Personal Factors

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increasing evidence to support a model in which environmental factors interact with intrinsic elements, and particularly with the child’s attentional resources, to increase the risk of developing math anxiety. Thus, children with a predisposition toward general anxiety, including mainly maladaptive threat related attentional bias and an innate tendency to experience the world as threatening, are proposed to be at particular risk for the development of math anxiety if they also experience environmental risk factors for math anxiety (e.g., involving teachers, parents, and peers). Such negative environments can link learned (symbolic) numerical information with negative valence, and in turn contribute to math anxiety through negative beliefs about math proficiencies. Specifically, we propose that adverse math-learning experiences, for those who are intrinsically vulnerable, result in maladaptive beliefs related to the self and to numerical abilities. These negative beliefs mediate the way to math anxiety. That is, through their pedagogical and social experiences, young people come to believe that they are mathematically deficient, with little ability to control the outcomes of mathematical situations. The theoretical model suggests that a maladaptive range of attentional biases contributes to beliefs about the person’s numerical abilities and to ruminations. These negative beliefs and ruminations lead to the behavioral, somatic, and emotional responses of (math) anxiety. We suggest that increased negative attention to numerical situations can lead to expectations that one will perform incorrectly and that the outcome will be poor. These expectations, in turn, will contribute to the experience of math anxiety (second column from the left in Figure 9.1). In numerical tasks, high levels of negative beliefs and distraction from the numerical task (through ruminations) lead to math anxiety but are also expected to impair numerical performance (third column from the left in Figure 9.1). We wish to argue that even if children’ numerical performances are less skilled than those of their peers (e.g., as in cases of developmental dyscalculia) and there may be some validity to the low self-evaluations, an inclination to evaluate even more negatively is likely to contribute to math anxiety (see, e.g., Gunderson et al., 2017). Avoidance of numerical activities in the future may be a behavioral manifestation of math anxiety (Ashcraft, 2002), which, in turn, reduces positive opportunities to acquire numerical skills and limits the development of good math proficiency ( Krinzinger, Kaufmann, & Willmes, 2009). In this way a vicious cycle might be established, leading to increasing levels of math anxiety. Eventually, the diagnostic criteria for math anxiety will be met when math anxiety is visible (i.e., depending on the level of math anxiety) and possibly combined with increasing math impairment (third column from the left in Figure 9.1). In turn, explicit (but not implicit) assessment will be affected by variables such as the individual’s age, gender, and culture (right side of Figure 9.1).

Conclusions The model that we described here highlights pathways to the development of math anxiety and the interplay between environmental and intrinsic factors. As described

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in this chapter, evidence has emerged in recent years to explain how risk factors can interact to amplify the potential development of math anxiety. There remain many unresolved questions. For example, different risk factors may be more or less influential at different ages. Also, any one risk factor can result in various outcomes. As an example, some cases of math anxiety may be the result of extreme environmental effects with no initial intrinsic factors (e.g., in cases of very strong environmental pressure to succeed in math with no initial symptoms of general anxiety). While recognizing the limitations of the model, we indicate several optional interacting risk factors that may have an effect on the development of math anxiety. We suggest, in some cases, an interplay between intrinsic factors (attentional resources) and environmental factors (such as parental and teacher influence). Future longitudinal studies will need to clarify the strengths and weaknesses of the model. Such information will have important value in informing the development of focused assessments for math anxiety and in the formation of preventive interventions.

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10 GENDER STEREOTYPES, ANXIETY, AND MATH OUTCOMES IN ADULTS AND CHILDREN Carlo Tomasetto

Whenever African American students perform an explicit scholastic or intellectual task, they face the threat of confirming or being judged by a negative societal stereotype – a suspicion – about their group’s intellectual ability and competence. This threat is not borne by people not stereotyped in this way. And the self-threat it causes – through a variety of mechanisms – may interfere with the intellectual functioning of these students. [Steele & Aronson, 1995, p. 797]

Introduction Being a member of a stigmatized social group is stressful. A long-standing tradition of research at the intersection between sociology (e.g., Goffman, 1963) and psychology (see Major & O’Brien, 2005) has outlined that individuals who belong to disadvantaged groups are at risk of experiencing a variety of negative life outcomes, including poorer health (Hatzenbuehler, Phelan, & Link, 2013) and reduced psychological well-being (Schmitt, Branscombe, Postmes, & Garcia, 2014), that stem from the physiological and psychological responses to the unpleasant feeling of being the target of prejudice and negative stereotypes. In 1995, Steele and Aronson outlined for first time that the same mechanisms may account for other important consequences: Namely, the finding that members of disadvantaged groups actually tend to underperform on tasks that assess competencies that their groups are assumed to lack, as compared to their nonstereotyped counterparts. Steele and Aronson termed this predicament stereotype threat. In the subsequent decades, stereotype threat became one of the most widely studied topics in social psychology. The case of women’s tendency to underperform on math tasks and withdraw from maths-related fields – consistent with the societal stereotype stating that women are not good at math – has since been the subject of most of the studies on the causes and consequences of stereotype threat (see Spencer, Logel, & Davies, 2016).

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Linking anxiety to stereotype threat Anxiety as a mediator of stereotype-related performance drops In their seminal work in 1995, Steele and Aronson defined stereotype threat as the negative situation encountered by members of disadvantaged groups due to the anticipated concern that one’s (poor) performance in a stereotype-relevant domain may inadvertently confirm an existing negative stereotype about one’s group. When facing such a concern, stereotyped individuals activate a variety of stress responses at the physiological (e.g., autonomic activation), cognitive (e.g., negative thoughts, vigilance for threatening cues), and emotional (e.g., anxiety) levels. Self-monitoring and suppression processes are then activated to buffer these negative states. However, intensive monitoring and volitional suppression efforts also entail unintended consequences. On the one hand, these processes overload working memory resources, ultimately disrupting performance on controlled and complex tasks that recruit executive control. On the other hand, undue vigilance interferes with the completion of tasks relying on over-learned skills, for which automatized procedures are more effective than controlled ones (Schmader, Johns, & Forbes, 2008). Within the complex cascade of mechanisms outlined above, anxiety has been intuitively suggested as a pivotal mediator of the link between negative stereotyping and poor performance. However, until the early 2000s, no single piece of evidence was provided to show that anxiety may be involved in the process, despite the fact that an increasing number studies had already demonstrated that performance decrements were consistently associated with exposure to negative stereotypes among the members of a variety of stigmatized groups. In the original work by Steele and Aronson (1995), for example, African American students systematically obtained significantly lower scores on diverse measures of verbal intelligence when the diagnostic purpose of the tests (Studies 1–3) or the participant’s racial identity (Study 4) was stressed, thus making the stereotype salient and threatening. Across the four studies, numerous ancillary measures were also obtained, which suggested that threatened participants were more likely to strive to disconfirm the stereotype, blame the unfair nature of the task, or adopt selfhandicapping strategies to justify a possible failure (e.g., by reporting having slept some two hours less than non-threatened participants the night before the test). However, no difference in reported anxiety (Study 2) or perceived stress (Study 3) emerged between threatened and non-threatened students. Four years later Spencer, Steele, and Quinn (1999) demonstrated for the first time that stereotype threat was also responsible for a decrease in women’s performance on standardized math tests. Again, even though a widely used selfreport measure of anxiety was included in one of their experiments (Study 3) as a potential mediator, they also failed to detect clear evidence that women under stereotype threat were more likely to feel anxious than those for which the threat was alleviated. For instance, Spencer and colleagues observed that women for whom the stereotype was made less relevant (i.e., those forewarned

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that the test produced no difference between women and men) tended to report a slightly lower level of anxiety during the testing session, but the observed difference was small and did not reach significance. Subsequent studies focusing on gender stereotyping of math, in which a measure of anxiety was also included, yielded strikingly similar results (e.g., Aronson et al., 1999; Keller & Dauenheimer, 2003).

Disentangling the role of emotionality, worry, and physiological arousal In sum, no evidence supporting the role of anxiety as a determinant of stereotypeinduced performance drops was provided until the early 2000s. Different findings emerged, though, in subsequent years, when alternative approaches were adopted, in particular by separating the effects of affective, cognitive, and physiological components of anxiety. Cadinu and colleagues, for example, asked female university students to list any thought that came to their mind immediately before solving each of seven difficult math problems (Cadinu, Maass, Rosabianca, & Kiesner, 2005). Thought listing is an effective method for tapping into the cognitive component of anxiety, rather than into the emotional component. The cognitive component is particularly relevant to the stereotype mechanism because verbal rumination – as well as the production of irrelevant thoughts – occupies verbal working memory resources, therefore reducing the availability of such resources for the maintenance and elaboration of task-relevant information. As predicted, Cadinu et al.’s study indicated that negative thoughts increased throughout the test, from the first to the seventh problem, and that women under stereotype threat reported a higher number of negative mathsrelated thoughts than those for whom the threat was alleviated. More importantly, the number of negative thoughts produced mediated the disruptive effect of stereotype threat activation on math performance. Similar results were also obtained by Beilock, Rydell, and McConnell (2007), who showed that this relation was due to a consumption of verbal working memory resources. Important evidence also comes from other studies, in which physiological indices, rather than behavioral or self-reported measures of anxiety, were used to assess arousal and emotional activation. Osborne (2007), for example, observed that performance of college students on math items drawn from the Graduate Record Examinations test strictly followed the prediction of the stereotype threat model, with female students having their performance disrupted when reminded of the stereotypes that women do worse than men in math, as compared to those reassured that the stereotype was not relevant to the test. More importantly, by monitoring participants’ physiological parameters during the test, Osborne also observed that female participants in the high stereotype threat condition displayed significant evidence of physiological reactivity, such as increased skin conductance, higher surface skin temperature, and increased diastolic blood pressure.

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The role of negative emotional activation under stereotype threat has been further clarified with neuroimaging techniques. By using functional magnetic resonance imaging (fMRI), for example, Krendl, Richeson, Kelley, and Heatherton (2008) monitored both the activation of neural regions involved in mathematical cognition and executive control, such as the inferior prefrontal cortex, left inferior parietal cortex, and bilateral angular gyrus, and of regions associated with emotional regulation and social cognition, such as the ventral anterior cingulate cortex. Results illustrated that female participants who were reminded of the negative gender stereotype about math displayed reduced activation of the brain areas relevant to task completion (i.e., those associated with numerical processing and control), and increased activation of areas implied in the elaboration of emotional and social stimuli. Overall, the evidence summarized above suggests that variations in stereotypeinduced anxiety may be relatively difficult to detect via self-report measures, and therefore physiological indices may be more adequate in capturing emotional responses under stereotype threat. It is indeed plausible that at least some individuals under stereotype threat do not consciously perceive themselves to be worried or under stress merely because their social identity is made salient, especially when their social identity is not blatantly stigmatized (such as in the case of women). Moreover, whereas individuals with high levels of math anxiety, or lower self-efficacy in math, may be perfectly aware that performing a math task may causes them fear or apprehension, individuals under stereotype threat may paradoxically be perfectly at ease with the task at hand. Women who are highly competent in math, and probably disavow the gender stereotype about women’s math ability, are nevertheless susceptible to stereotype threat because the fear of confirming or being judged in the light of the societal stereotype stems from the mere fact that the stereotype exists (Aronson et al., 1999; Schmader et al., 2008). Finally, stereotype threat may be triggered by very subtle cues, such as being the only woman in a math class (e.g., Huguet & Regner, 2007), or being examined by a male vs. a female experimenter (Stone & McWhinnie, 2008). In all these cases, it is quite unlikely that women under stereotype threat will be consciously aware of an increased feeling of apprehension related to the task. And yet, stress responses may operate in the background and ultimately damage performance regardless of the individual’s awareness of the ongoing threat.

More than simple mediation: Alternative roles of anxiety in the stereotype threat model Beyond being suggestive of a mismatch between subjective self-reports of anxiety and physiological measures, the mixed results regarding anxiety as a potential mediator of performance outcomes may also be indicative of the complexity of the relations that link stereotype threat susceptibility to performance outcomes. In the framework of the stereotype threat model, anxiety is typically conceived as a situational response to threatening cues activated by either the nature of the

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task (i.e., the task taps into an ability for which one’s group is negatively stereotyped), or the relevance of one’s social identity within the testing context (i.e., membership into the stigmatized group is made salient). In other words, stereotype threat research has typically assessed transitory and context-specific negative emotional responses elicited by identity-threatening cues (i.e., state anxiety). However, anxiety also encompasses dispositional and non-transitory tendencies to appraise a variety of contests and situations as potentially threatening (i.e., trait anxiety; Endler & Kocovski, 2001). Even more important, inter-individual variability exists in both the general tendency to appraise situations as threatening (i.e., general anxiety), and in the peculiar sensitivity to display negative emotional reactions in specific contexts or activities, such as assessment situations (i.e., test anxiety), or single academic domains (e.g., math anxiety). Given the multidimensional nature of anxiety, it is quite conceivable that dispositional aspects of anxiety, and not just context-driven transitory feelings of apprehension, may be involved in individuals’ response to potentially stereotype-threatening cues. In line with this reasoning, Tempel and Neumann (2014) administered a word-problem math test to a sample of female university students in two alternative conditions of stereotype threat (i.e., by merely presenting the task as a math test) or stereotype removal (i.e., by reassuring participants that the test was gender-fair). Results showed that women who had scored higher at a previous assessment of trait test anxiety were not affected by the stereotype threat manipulation and consistently performed at a low level in both conditions. By contrast, female students with low trait test anxiety performed significantly lower under stereotype threat – at the level of their highly anxious counterparts – but had their performance restored when stereotype threat was removed. In other terms, anxiety as a dispositional trait may act as a moderator, rather than a mediator, of vulnerability to stereotype threat. Even more complex relations between anxiety and stereotype threat emerged from a study by Gerstenberg, Imhoff, and Schmitt (2012). Across three studies, they found that female university students with a strong explicit math self-concept (i.e., who rated themselves as good at math), but with a weak implicit math selfconcept (i.e., with only weak associations between the concept of self and the concept of math) were particularly susceptible to stereotype threat when gender identity was made salient prior to test taking. Importantly, when assessed for their current worries immediately before the math test – but after the stereotype threat manipulation – these participants displayed the highest level of intrusive troublesome thoughts (e.g., “I ask myself whether my performance will be good enough”) when stereotype threat was activated, and increased worry was found to mediate the stereotype threat effect. Finally, recent works have included anxiety within more complex sequential mediational models, in which anxiety is conceived as either triggered by other first-order mediators of stereotype threat (e.g., achievement goals; Brodish & Devine, 2009), or as the cause of subsequent processes (e.g., mind-wandering; Mrazek, Chin, Schmader, Hartson, Smallwood, & Schooler, 2011) that ultimately disrupt math performance.

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When stereotypes are positive: Performance boost or choking under pressure? It is quite evident that negative stereotypes are detrimental to performance, so should we assume that positive stereotypes are beneficial? In part, the answer is yes: positive stereotypes are helpful, at least under some circumstances. First of all, most descriptive stereotypes are essentially dual in nature, meaning that the inferiority of a group on a given task is benchmarked against another group’s alleged superiority. As a case in point, women’s ability in math is stereotypically judged as low in comparison with another group (usually, men), which is stereotyped as more competent. Therefore, when members of the advantaged group are evaluated on a stereotype-relevant task, and their social identity is made salient, they may take advantage from a favorable downward social comparison with the negatively stereotyped outgroup ( Walton & Cohen, 2003). In other terms, by comparing themselves with a devalued group, they may expect to be judged by others in positive terms and boost their sense of self-efficacy on the task, which may increase their ability to concentrate, preserve a positive attitude in the presence of frustrations, and ultimately achieve a better performance. This process may account for several findings from the earliest studies on stereotype threat, showing that members of the advantaged group performed slightly better in conditions that were stereotype-threatening for the disadvantaged outgroup (e.g., when social identity was made salient), as compared to when the threat to the outgroup was removed (e.g., Aronson et al., 1999; Shih, Pittinsky, & Ambady, 1999; Spencer et al., 1999). In an early meta-analysis, Walton and Cohen (2003) – who termed this phenomenon as stereotype lift – observed that such an effect was relatively small in size as compared to the corresponding stereotype threat effect for the disadvantaged groups, and only tended to appear in a subset of the studies conducted at the time. As regards the role of anxiety in explaining stereotype lift, however, very little evidence is available, as most of the meditational analyses were only conducted on the stereotype-threatened – and not the stereotype-advantaged – groups. A second circumstance under which positive stereotypes have been found to exert beneficial effects is when individuals who may be subject to stereotype threat as members of a devalued social group may simultaneously also benefit from a positive social identity as members of other groups that are positively stereotyped with regard to the same ability. An example is the case of Asian American women, who can alternatively be negatively stereotyped as weak at math – as women – or very skilled at math – as Asian. In line with this reasoning, Shih and colleagues found that – at least in a US sample (Study 1) – Asian American female university students performed better on a difficult math task when asked to report their (positive) ethnic identity prior to the test, compared to when they had to report their gender identity or no social identity (Shih et al., 1999). In a similar vein, individuals may be inf luenced by whether a given social identity implies a positive or a negative stereotype, depending on the label

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attached to a given task. A mental rotation task, for example, may be reasonably labeled as a geometry test – for which a woman would be negatively stereotyped as lacking relevant abilities – or as a “perspective taking” task – for which the same woman may benefit from a positive stereotype regarding the ability to put herself into others’ shoes. By using this paradigm, Wraga, Helt, Jacobs, and Sullivan (2007) demonstrated that women’s mental rotation performance varied as a function of the positive or negative stereotype activated ( Wraga et al., 2007). More importantly from our point of view, Wraga and colleagues also collected online f RMI measures of activation of cortical regions associated with spatial and social-emotional processing during the task completion. They showed that areas associated with emotional self-regulation and social knowledge (rostralventral anterior cingulate cortex and right orbital gyrus) were less strongly activated when the positive stereotype was made salient, and that this difference in brain activity mediated the observed difference in women’s spatial performance in response to stereotype threat. In addition to the findings reported above, evidence also exists showing that individuals for whom positive stereotypes exists as to a given ability, may unexpectedly fail – rather than having their performance improved – when the positive stereotype becomes salient. Cheryan and Bodenhausen (2000), for example, observed results opposite of those of Shih and colleagues (1999) when observing Asian American women take a difficult math test. Specifically, Cheryan and Bodenhausen (2000) found that their female participants’ performances dropped when the (positive) Asian identity was made salient, and that this effect was mediated by their reported inability to concentrate during the task. Although the two opposite phenomena may be difficult to reconcile, the possibility to drop following a positive stereotype activation parallels a widely studied phenomenon in social and cognitive psychology, termed choking under pressure ( Baumeister, 1984). The unexpected underperformance of individuals who are supposed to do well on a task – either because of their abilities or self-confidence, or because of a positive stereotype attached to their group – may be explained in terms that are very similar to those invoked to account for stereotype threat: an extra pressure arises – in this case, due to the fear of not confirming a positive expectation – that triggers negative emotions, worries, and physiological stress reactions, which ultimately consume working memory resources (Ashcraft and Kirk, 2001; Beilock and Carr, 2005). Consistent with this account, several studies have also found unexpected negative effects of positive stereotypes and suggest that when the pressure to succeed is excessive, the stereotype lift effect may reduce and even reverse, thus giving rise to choking under pressure. Unlike Shih and colleagues (1999), who subtly activated either gender or ethnic group membership by merely asking participants to answer questions regarding these aspects of their identity, Cheryan and Bodenhausen (2000) brought up the stereotype much more directly, by explicitly mentioning the positive or negative stereotype to participants (i.e., Asians are good at math vs. women are bad at math). Rosenthal and Crisp (2007) reminded

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their male participants of one or both of their available positive identities (i.e., being male and attending prestigious courses). Results revealed that male students who studied math, for whom the task was more relevant, had their performance damaged in the double-positive-stereotype condition, thus suggesting that an exceedingly positive expectation may have exceeded their ability to cope with the pressure to succeed. It is indeed critical to determine in advance when positive stereotypes may be advantageous or exceedingly demanding. Individual dispositions may play an important role, and stable tendencies to appraise events or situations as threatening (i.e., state anxiety) may moderate individuals’ susceptibility to both positive and negative pressures induced by relevant stereotypes (Mattarella-Micke, Mateo, Kozak, Foster, & Beilock, 2011).

The link between stereotype threat and anxiety in young children: Do these dissociate, or is this still not proved? Compared to the large amount of studies on stereotype threat with adults, relatively few studies have sought to demonstrate whether, when, and possibly how children are susceptible to stereotype-induced variations in their cognitive achievements. Overall, a meta-analyses including 47 studies found evidence for a small-size negative effect of stereotype threat on girls’ math performance, in spite of mixed findings and some evidence of publication bias (Flore & Wicherts, 2015). In their pioneering work focused on children, Ambady and colleagues reported that 5-year-old Asian American girls displayed evidence of both stereotype threat – when their gender identity was activated – and stereotype lift effects – when their ethnic identity was evoked (Ambady, Shih, Kim, & Pittinsky, 2001). Subsequent work with elementary school children suggested that – as with adult women – stereotype threat appears only when tasks are highly demanding (Neuville & Croizet, 2007) and operates regardless of personal endorsement of existing societal stereotypes (Huguet & Régner, 2009). However, few studies of stereotype threat among children have adopted additional measures to disentangle the role of age-specific mediators, and only one study included a measure of anxiety (McKown & Weinstein, 2003). In this case, the study pertained to the negative impact of racial stereotypes on intellectual performance of African American children, and consistent with previous studies with adults, self-reported anxiety did not emerge as a significant mediator of the observed effects. As regards gender stereotypes and mathematics, a mediational role was observed for implicit gender stereotypes, even in the absence of explicit stereotype awareness, among 6-year-old girls (Galdi, Cadinu, & Tomasetto, 2014). In other terms, girls’ performance under stereotype threat was found to depend on the salience of automatic associations between female gender and language rather than math. These findings indicate that among young children, stereotype-induced performance drops may appear even in the absence

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of awareness of any ongoing threat, suggesting that anxiety-driven mechanisms may be less relevant to stereotype threat among children than among adults. However, research with young children is still scarce, and no study to date has included alternative and more sensitive measures of anxiety to provide a more compelling test of its role. The lack of research tapping into gender stereotypes about math and anxiety in children is unfortunate, as numerous relations have already emerged between implicit or explicit gender stereotypes and a number of maths-related outcomes, including not only actual performance (as in stereotype threat studies) but also subjective perception of math ability (e.g., Passolunghi, Rueda-Ferreira, & Tomasetto, 2014), identification with math (Cvencek, Meltzoff, & Greenwald, 2011), and desire to enroll in math-related courses (Steffens, Jelenec, & Noack, 2010).

Conclusions Research on stereotype threat has captured the interest of a large number of researchers throughout the last two decades; and yet, the role of some key mechanisms linking individuals’ social identity to their abilities to perform at an optimal levels in a variety of domains is still only partially clarified ( Pennington, Heim, Levy, & Larkin, 2016). As regards the role of anxiety, convincing evidence has been produced illustrating the role of its cognitive (e.g., worry, rumination) and physiological components (e.g., autonomic activation and neural correlates) in determining performance drops and – in some cases – performance boosts in response to negative or positive stereotypes. However, numerous studies conducted thus far have failed to determine a clear role for anxiety, especially when anxiety had been assessed by means of self-report measures, or when the focus has been on the emotional component of anxiety. Another important limitation of the vast majority of the studies reviewed above is that attention has been mainly devoted to transitory anxiety states experienced in the assessment context. By contrast, only few studies have attempted to clarify whether and how high levels of dispositional anxiety – either as a general stable disposition, or as a specific tendency to negatively react to a specific domain of activity, such as mathematics – contribute to increasing or decreasing individuals’ vulnerabilities to stereotypes attached to their social groups. Finally, the role of anxiety has been surprisingly overlooked in research on stereotype threat with young children. It is hoped that future research will address this gap, as the mechanisms and the boundary conditions that inf luence susceptibility to stereotype threat in adult women and in young girls may not necessary overlap (see Galdi et al., 2014). Clarifying all these gaps will indeed help us to design more effective interventions aimed at reducing performance drops stemming from negative societal stereotypes, as well as those originating from positive stereotypes, as in the case of the phenomenon of choking under pressure.

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11 THE ROLE OF PARENTS’ AND TEACHERS’ MATH ANXIETY IN CHILDREN’S MATH LEARNING AND ATTITUDES Julianne B. Herts, Sian L. Beilock and Susan C. Levine

Children of parents and teachers who experience fear or discomfort about math, termed math anxiety, (Ashcraft, 2002), tend to underperform on tests of math achievement as compared to their peers ( Beilock, Gunderson, Ramirez, & Levine, 2010; Berkowitz et al., 2015). This relation between adult math anxiety and child math achievement has broad implications, as math anxiety is prevalent and associated with lower math scores across the globe (Foley et al., 2017). For example, within the United States, research with community college students has reported that nearly half of the students sampled experience moderate, quite a bit, or high levels of math anxiety (Sprute & Beilock, 2016). Further, the negative relation of math anxiety to math achievement is a global phenomenon. In 63 of 64 countries participating in the Programme for International Student Assessment (PISA), students with higher levels of math anxiety scored lower on the PISA math test (Foley et al., 2017). Moreover, counterintuitively, the negative effects of math anxiety tend to be greatest for individuals with the greatest potential to achieve at high levels in mathematics (Beilock & Carr, 2005; Ramirez, Chang, Maloney, Levine, & Beilock, 2016). Given the prevalence of math anxiety and its relation to low math achievement, it is important to consider its etiology. In this chapter, we explore one factor that has the potential to contribute to children’s math anxiety – intergenerational effects of math anxiety on both children’s math achievement and their math anxiety and other negative attitudes about math. Identifying the nature of such intergenerational effects holds promise for developing interventions that have the potential to disrupt this negative cycle of low math achievement and negative math attitudes. By way of background, we begin by brief ly reviewing how individuals’ math anxiety relate to their math performance, a topic that has been the focus of most studies on math anxiety. This part of our chapter shows that existing studies have provided an understanding of the mechanisms through which math anxiety and low math performance are related in individual children and adults.

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We then turn to the new work that is emerging about intergenerational effects of adult math anxiety on children’s math learning and attitudes. This part of our chapter is organized around the following questions: 1) 2)

3)

4)

How and why does parent and teacher math anxiety relate to children’s math performance and math anxiety? How can existing interventions designed to reduce anxiety or improve performance be harnessed to mitigate the negative relation between adults’ math anxiety and children’s math performance and attitudes? How can we use existing research findings to inform the development of effective interventions that break the negative intergenerational effects of adult math anxiety on children’s math learning and attitudes? What are some important avenues for future research on the topic of intergenerational effects of adult math anxiety?

How does one’s math anxiety relate to math performance? Adults At the individual level, math anxiety is related to poor math performance and avoidance of math (Ashcraft, 2002; Foley et al., 2017), with high math anxious individuals taking fewer math courses and entering careers in science, technology, engineering, and math (STEM) at lower rates than low math anxious individuals (Hembree, 1990). The fact that people with math anxiety tend to perform lower in math raises three possible explanations for this relation. Math anxiety may cause low math achievement, low math achievement may cause math anxiety, or math anxiety and math achievement may be bidirecationally related. Of course, it is also possible that a third factor may lead to both high math anxiety and low achievement (e.g., poor executive functioning). Additionally, it is possible that the causal relations differ for different individuals, a topic we return to later. Investigations of the nuanced relation between math anxiety and math performance have often found math anxiety to have the greatest negative impact when math tasks are challenging (Ashcraft & Krause, 2007; Ashcraft & Kirk, 2001), though there is also evidence to suggest that math anxiety is related to lower performance on basic foundational math skills related to success in mathematics ( Holloway & Ansari, 2009). For example, math anxious adults perform at lower levels on tasks requiring them to quickly identify the larger of two numerals of similar magnitude (Maloney, Ansari, & Fugelsang, 2011). They also show lower performance than non-math anxious adults in estimating the number of elements within a range of 5–9 displayed brief ly on a screen (Maloney, Risko, Ansari, & Fugelsang, 2010). These findings may indicate that math anxiety may lead to basic math deficits, or that basic math deficits can lead to the development of math anxiety.

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One finding that is important to consider when studying causal mechanisms relating math anxiety and math achievement is that math anxiety is not just a proxy for low math achievement (Beilock & Carr, 2005). That is, although math anxiety is more prevalent in people with low math achievement, math anxiety exists at all levels of math ability (Foley et al., 2017). Moreover, math anxiety takes its greatest toll on people who have high potential to do well in math – those with higher working memory ( Beilock & Carr, 2005; Ramirez et al., 2016). This may be due to the fact that math anxiety can deplete the working memory resources that high working memory individuals would otherwise utilize to solve math problems ( Ramirez et al., 2016). The idea that math anxiety can deplete working memory resources, leading to lower math performance, is supported by evidence that math anxious adults have particular difficulty with complex math problems. For example, high math anxious participants performed worse than low math anxious participants on mental addition that involved carry operations, and thus taxed working memory, but not on simpler addition tasks (Ashcraft & Krause, 2007). Ashcraft and Kirk (2001) similarly found that giving high math anxious individuals a secondary task, like holding a letter in mind, led to decreased performance on simple arithmetic problems. This evidence suggests that math anxiety is particularly associated with lower performance on tasks that place high demands on working memory. Furthermore, experimental evidence from Beilock and Carr (2005) revealed that when adults were placed under pressure when solving math problems, high working memory individuals experienced decreased performance on math problems that placed a high demand on working memory. Low working memory individuals, on the other hand, did not show decreased performance under pressure. Anxiety’s negative impact on working memory thus leads to the counterintuitive finding that it is most detrimental for individuals with high working memory, who typically rely heavily on working memory-intensive strategies to solve math problems and arguably have the greatest potential to excel in mathematics. These findings, considered together, suggest a potential bidirectional relation between math anxiety and low math skills. The cognitive and emotional components of math anxiety could well reinforce each other, with poor math ability and high math anxiety becoming a vicious cycle. Math anxiety may lead individuals to avoid math and to have less working memory to devote to solving math problems. As a result, these individuals may learn less math, perform poorly in math, and become even more anxious. Alternatively, people may become math anxious because of problems they are having with math, and as a result avoid math, learn less math, and become more math anxious. This vicious cycle may have different origins for different people, with some individuals onboarding this high math anxiety-low math achievement cycle due to low math ability, and others onboarding this vicious cycle due to high math anxiety. For still others, both factors may contribute to the vicious cycle.

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Below, we turn to the topic of how children’s math anxiety and math achievement relate, which provides some important insights into the way the vicious cycle, described above, typically gets started.

Children Similar to adults, as early as first grade, children’s math anxiety is negatively related to their math performance (Ramirez, et al., 2016). Moreover, in primary and secondary school, children with high working memory are most negatively affected by math anxiety, as is the case with adults. This effect has been found internationally for secondary school students, with data from the 2012 PISA test revealing that math anxiety has its largest effect on fifteen-year-old students in the highest quantile of performance (Foley et al., 2017 ), suggesting that anxiety takes the largest performance toll on students with high working memory capacity. This pattern has been found in younger children as well. Even in elementary school, students with high working memory capacity show a negative relation between math anxiety and math performance ( Ramirez et al., 2016). These high working memory-math anxious children are less likely to apply sophisticated math strategies (e.g., decomposition) to math problem-solving than their high working memory-low math anxious peers. Instead, they adopt more rudimentary strategies such as finger counting, which tends to be error prone. In their study of children in China, Wang and Shah (2014) found similar results. They asked third- and fourth-grade students to perform addition problems. After completing some problems in a low-pressure environment, students were placed under pressure by being filmed and told that their performance would later be judged by education experts in the United States. The addition problems were of three levels of difficulty. Wang and Shah found that, when under pressure, high working memory students adopted rudimentary problem-solving strategies on the difficult problems when this was possible. As a result, they performed like low working memory students. However, when it came to challenging problems for which there was no obvious easy shortcut available, the high working memory students outperformed the low working memory students when under pressure. Wang and Shah take this to mean that the performance of high working memory-math anxious students particularly suffers on difficult problems where it is possible to adopt an alternative, less efficient strategy. Their findings also suggest that when this is not possible, these students are able to apply more sophisticated strategies. It seems, therefore, that task difficulty, working memory, pressure, and availability of alternative strategies can all impact the relation between math anxiety and math performance. Developmental research also provides evidence that the relation of math anxiety and math achievement is bidirectional early in life, with some indication

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that low math achievement more strongly leads to high math anxiety than vice versa (Gunderson, Park, Maloney, Beilock and Levine, 2018). By assessing math achievement and math anxiety in first- and second-graders at the beginning and end of the school year and analyzing results using path analysis, Gunderson et al., (2018) found that math anxiety at the beginning of first and second grade predicted math achievement at the end of these grades. Further, math achievement at the beginning of first and second grades predicated math anxiety at the end of these grades. Moreover, this analysis provides evidence that low math achievement predicted later high math anxiety more strongly than vice versa. Thus, the vicious cycle between math anxiety and poor math ability appears to begin in childhood.

How and why does parent and teacher math anxiety relate to children’s math performance and math anxiety? The relation of parent math anxiety and child math achievement Just as high math anxious individuals tend to underperform at math compared to low math anxious individuals, the children of high math anxious parents tend to underperform in math as compared to their peers with low math anxious parents (Berkowitz, 2015). For example, in first grade, Schaeffer, Rozek, Berkowitz, Levine, & Beilock (2018) found an achievement gap associated with parent math anxiety. By the end of first grade, the children of high math anxious parents underperformed in math as compared to the children of low math anxious parents. The correlation of parents’ and children’s math anxiety raises a number of possibilities with respect to cause, including genetic, environmental, and geneenvironment interactions that need to be examined. As discussed below, recent findings suggest that parent-child interactions around math may play some role in the intergenerational relationship between high parent math anxiety and low child math achievement.

The relation of teacher math anxiety and child math achievement Other evidence of intergenerational relations of adult math anxiety and child math achievement comes from studies of teachers and their students. Teachers vary in math anxiety, making it possible to measure the relation of teacher math anxiety to student learning outcomes. Indeed, college students who major in elementary education have higher math anxiety than students who choose other college majors ( Hembree, 1990). Many colleges have few math requirements for early education majors, making the major attractive to those who wish to avoid math. Classroom teachers generally do not specialize until junior high or middle school in the United States, so the role of teaching math to preschool and elementary school students most often falls to the general

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classroom teacher, who may be math anxious and did not pursue a career specific to math. Evidence of the relation between teacher math anxiety and child math performance comes from a study showing that girls with high math anxious female teachers have lower math achievement at the end of school year than those with low math anxious teachers ( Beilock et al., 2010). This effect was found to be mediated by girls’ explicit endorsement of the gender stereotype that boys, and not girls, are good at math. The mechanism by which some girls, and not others, come to endorse this stereotype is not clear. Perhaps high anxious female teachers confirm gender stereotypes about mathematics for girls, but it is not clear how this plays out in the classroom, or what individual factors determine whether or not a child with a math anxious teacher endorses this stereotype. The impact of teacher math anxiety on child math achievement needs to be further investigated, with larger sample sizes and with research into the role of parent math anxiety on child math performance. Because teachers and parents are important role models for children, it is possible that both significantly affect the development of children’s attitudes toward math and that there are interactive effects. Presently, little is known about how the math anxiety of teachers and parents jointly relate to children’s math achievement. While it is possible that having a non-math anxious parent or teacher could be a protective factor for children, it is also possible that having a math anxious parent or teacher could be a significant risk factor for children.

Potential mechanisms: instruction, attitudes, and stereotypes New research concerning math anxiety indicates that math anxious parents may be offering poor instructional support to children, laying the foundation for similar studies of teachers. There is also strong evidence suggesting that parents’ and teachers’ beliefs and attitudes can inf luence their children’s performance ( Park, Gunderson, Tsukayama, Levine, & Beilock, 2016; Gunderson, Donnellan, Robins, & Trzesniewski, 2018; Schaeffer et al., 2018). High and low math anxious teachers and parents may differ from one another in terms of the quality/quantity of math instruction they provide and/or the attitudes they convey to children about math. Both of these factors could impact children’s math achievement. Recent research into parental homework help offers hints as to how math anxious adults may be affecting their children’s math achievement. When high math anxious parents report helping more frequently with homework, this help may backfire as their children show less growth over the course of the school year than children of high math anxious parents who report helping less with homework (Maloney, Ramirez, Gunderson, Levine, & Beilock, 2015). Importantly, this relation holds even controlling for children’s math achievement levels from the beginning of the school year. These correlational findings suggest that high

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math anxious parents may be passing on their math anxiety to their children and/or providing poor support during homework interactions. Either or both of these factors could hinder the math learning of their children. Given the Maloney et al. findings that the homework help of math anxious parents may not be effective, it is important to carefully look at the math interactions of adults and children, and examine how these interactions vary as a function of adults’ math anxiety.

Differences in math instruction related to math anxiety Recent research suggests that adult math anxiety may harm child math performance due to differences in the quality of math support provided to children between high and low math anxious adults. The possibility that math anxious parents may be offering ineffective support when helping their children with their homework is supported by a pilot study by Eason et al. (2017), which found parents’ anxiety about teaching math to their young children to be significantly correlated with parents’ own math anxiety, suggesting that parents who do not feel confident about math also do not feel confident in their ability to offer their young children appropriate instruction in math. Parents’ math anxiety may play out in homework interactions, when parents’ lack of confidence or ability is on display in a highpressure situation in the home, as well as in other kinds of math interactions. To further investigate whether high and low math anxious parents differ in the support that they provide to their children during math interactions, we conducted a pilot study in which we videotaped parents using a math iPad application that provides age-appropriate math problems for children and parents to solve together. This videotaped sample was as part of a larger study called Bedtime Learning Together. Based on the videotapes, we rated the quality of parents’ math instruction to their first-grade children on a scale designed to evaluate specific components of parents’ math support. The scale we used was a modified version of the Reformed Teacher Observation Protocol (RTOP; Sawada et al., 2002), which was designed to measure the quality of teaching in classrooms. We selected six items from the RTOP that applied to the context of working one-on-one with a child to teach a concept. These items were altered to refer to parents and children, as opposed to teachers and students. Each parent was scored from 0 to 4 on six items, which concerned: 1) respecting child’s prior knowledge, 2) providing opportunities for exploration, 3) promoting conceptual understanding, 4) parent’s understanding the math content, 5) promoting critical thinking, and 6) being patient. For example, parents who asked their child how to solve the problem, or encouraged the child to think about it before jumping in, scored higher on providing opportunities for exploration than those who took the pencil from the child’s hand and began solving the problem themselves.

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Score on Modified RTOP

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Parent Math Anxiety FIGURE 11.1 Relation between parent math anxiety and instruction quality on modified Reformed Teacher Observation Protocol.

The scores for each item from two raters were averaged together, and then all items were standardized and subsequently averaged across parent-child interactions on two math problems. Preliminary results indicate that parents’ overall instruction quality scores were negatively related to parent math anxiety (see Figure 11.1). This effect held when other potentially informative variables were entered into the model. That is, when parent math ability, as measured by a Woodcock Johnson Math Fluency test, and child math ability, as measured by the Woodcock Johnson Applied Problems, were added to the regression equation, the negative relation of parent math anxiety and RTOP ratings remained. Thus, questionnaire and objective instructional quality ratings indicate that math anxious parents do a worse job at providing math support to their children. Their suboptimal help may contribute to the observed achievement gap between the children of high and low math anxious parents ( Berkowitz et al., 2015). Similar analyses of teachers’ classroom instruction would shed light on whether their instruction differs as a function of math anxiety as well. It is also possible that math anxious adults pass on their own anxiety while teaching children math, leading their children to dislike and subsequently avoid math. Soni and Kumari’s (2015) study of fifth- through tenth-grade students in India indicates that by late elementary school, parent math anxiety was related to children’s math anxiety, which, in turn, was strongly related to children’s math achievement.

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Relation between adult math anxiety and beliefs about math Math instruction, both at school and in the home, may also convey important information to children about adult beliefs about math, including what it takes to be good at math. A few studies are beginning to show that math anxiety may involve a set of beliefs about math that can negatively affect children’s math achievement. Parents and teachers communicate to children about the importance and value of math as well as the types of people who are good at math, and children can pick up on these beliefs. For example, parents’ math anxiety has been found to relate to their expectations and values regarding math for their children such that high math anxious parents hold lower expectations and values regarding math for their children than low math anxious parents (Schaeffer et al., 2018). These findings are important as numerous studies have shown that parents’ expectations and valuing of learning affects students’ learning ( Rozek, Svoboda, Harackiewicz, Hulleman, & Hyde, 2017). Thus, math anxiety, and the associated beliefs about math that travel with this anxiety can negatively affect children’s math achievement and attitudes in numerous ways. Parents and teachers frequently communicate their expectations, values, and judgments to their children, and children’s perceptions of adults’ beliefs can impact their attitudes as well as their achievement. Just as high math anxious parents hold lower expectations and values regarding math for children than low math anxious parents, this may also be the case for high math anxious teachers. Gender stereotypes about math may make these beliefs even more detrimental for girls than for boys. This is an important avenue for future study, as negative messaging around math, around the difficulty of math, and about students’ abilities to do math could have negative consequences for children’s learning and their math self-concepts.

Gender stereotypes and math anxiety Parents’ and teacher’s beliefs about traits like gender may also play a role in children’s development of math anxiety. Given that math anxious teachers are more highly related to the math achievement of girls, and that this is mediated by girls’ beliefs about stereotypes, it is important to consider the role that stereotypes about math ability may play in determining the inter-generational impact of math anxiety ( Beilock et al., 2010). Moreover, a recent pilot study with adults indicated a relation between math anxiety and implicit gender stereotypes about math (Cvencek, Meltzoff, & Levine, unpublished). Importantly, most early education teachers are female. (National Education Association, 2003). Moreover, as mentioned above, elementary school majors have higher math anxiety than college graduates majoring in other disciplines (Hembree, 1990). Given that the vast majority of children are taught math by a female teacher in elementary school, and teachers tend to be math anxious, the stereotype that women are less capable than men at math may be reinforced in

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the classroom (Beilock, Gunderson, Ramirez, & Levine, 2010; Cvencek et al., 2011). In fact, we have found that 1st and 2nd grade girls with math anxious female teachers are particularly likely to believe the stereotype that boys are good at math and girls are good at reading (Beilock et al., 2010). Consistent with this finding, research suggests that children endorse gender stereotypes about STEM fields by 6 years of age. For example, Bian, Leslie, and Cimpian (2017) found that though five-year-old girls and boys tend to link brilliance to their own gender, by 6- to 7-years of age, girls are less likely than boys to report that members of their own gender are very smart. Girls were also less likely to select a game for very smart people than boys. This work has important implications for STEM fields, as sectors like math and engineering are often associated with genius or brilliance ( Bian et al., 2017). The association of these traits with males may be discouraging girls and women from pursuing STEM fields and preventing them from feeling a sense of belonging and competence in math class. Girls who do endorse explicit gender stereotypes about math may demonstrate lower math scores for a variety of reasons. It could be that these girls come to identify with math less, and therefore avoid math insofar as this is possible, or do not try as hard at math as they do at other subjects, because they believe themselves destined to fail (Cvencek et al., 2011). It could also be that these girls are particularly likely to experience stereotype threat – pressure that their performance will be consistent with a stereotype about their gender in this case – during math tests (Steele & Aronson, 2005; Spencer, Steele & Quinn, 1999). Women may feel stereotype threat during math class or math exams, with the worry that if they perform poorly they will be providing evidence in support of a negative stereotype about their group. Similar effects have been found in children. Ambady, Shah, Kim, and Pittinsky (2001) found that Asian American female students aged 5–7 and 11–13 showed greater math performance than controls when their ethnic identity was made saliant (through coloring a picture of Asian children or answering a survey about ethnicity) and worse math performance when their gender identity was made salient (by coloring a picture of girl with a doll or answering a survey about gender), revealing the role that stereotypes can play in performance even at a very young age. Eight to ten-year-old girls, on the other hand, showed improved math performance as compared to both control and ethnic identity activation when their gender identity was activated. The authors speculate that this may be because eight- to ten-year-old children are likely to feel positively about their own gender. It would be beneficial to investigate how math gender stereotypes are transmitted in the classroom, in the media, or at home, as well as how these stereotypes may interact with math anxiety in the classroom. Parents’ and teacher’s beliefs about traits like gender may also play a role in children’s development of math anxiety. It is important to consider the role that stereotypes about math ability may play in determining the inter-generational impact of math anxiety (Beilock et al., 2010). Moreover, a recent pilot study with adults indicated a relation between math anxiety and implicit gender stereotypes about math (Cvencek, Meltzoff, & Levine, unpublished). Perhaps gender stereotypes can be

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combatted in the classroom to improve student math performance by mitigating the impact of stereotype threat. Existing ways of combatting stereotype threat may be of use here. For example, there are some indications that increasing students’ sense of belonging can be an effective way to combat stereotype threat ( Walton & Cohen, 2011). Fostering a sense of group belonging has even been found to change the math attitudes and behavior of preschool children. Master, Cheryan, and Meltzoff (2017) told four- to five-year-old children that they were part of either a green group or an orange group. Children wore a t-shirt with their group’s color, sat a table with a table cloth of the group’s color, and put up a little f lag of their group’s color. The children were then told that their group did a math or spatial task. As compared to when they did the tasks with no group affiliation, the children worked on the tasks longer, were more accurate, and felt a greater sense of self-efficacy and interest in the task when they were part of a color group. Even this very minimal group was able to change the attitudes and behavior of the children involved, indicating that small nudges could go a long way toward creating a sense of belonging and identification with math. Similar interventions making students feel that they identify with math and belong in an environment with math could increase children’s self-efficacy and improve their performance. It remains to be seen how they could be adapted for the home.

How can existing interventions designed to reduce anxiety or improve performance be harnessed to mitigate the negative relation between adults’ math anxiety and children’s math performance and attitudes? Structured math interventions for families Interventions designed to break the link between adult math anxiety and child math achievement have focused on changing parent behavior or beliefs. The Bedtime Learning Together intervention was able to close the achievement gap between the first-grade children of high and low math anxious parents over the course of a school year ( Berkowitz et al., 2015). This intervention involved providing parents with a tablet equipped with a math application designed to be a low-pressure way for parents to solve math problems with their children. The app contained short math passages about interesting subjects, like very tiny frogs or the history of Groundhog’s Day, along with questions at varying levels of difficulty that parents answer with their children. First-grade children of high and low math anxious parents who had received this math app in the fall showed similar math growth over the school year. In contrast, children who received a similar app that asked reading comprehension questions instead of math questions, showed a persistent achievement gap between the children of high and low anxious parents. The mechanism by which the app was able to change child achievement outcomes is not clear. It is possible that the app made parents reevaluate their own

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feelings about math, having had the opportunity to engage with their children around math in a low-pressure environment. One factor that may contribute to the formation of math anxiety is having a negative experience with math at some point during the course of schooling ( Dowker, Sarkar, & Looi, 2016). Perhaps engaging with math again, with their children and without being evaluated, creates more positive connotations around math for math anxious parents, potentially encouraging them to engage in more math talk outside of the app context and/or to share more favourable attitudes about math with their children. It could also be that using this app with their children may have changed high math anxious parent’s expectations of their children’s math abilities and may have increased how much they value math for their children. Just as teachers’ expectations can inf luence children, so too can the expectations and beliefs of parents. For example, parents give more math input, including more challenging goals, to their sons than to their daughters during block play – a gender stereotyped play activity that is associated with math achievement (see for example Casey, Nuttall, & Pezaris, 1997). Additionally, Jacobs and Eccles (1992) found that mothers who had stronger gender stereotypes about math ability were found to rate their daughters’ math ability lower and their son’s math ability higher. Presumably, parents’ expectations of their sons’ and daughters’ abilities differ, and this difference in expectations, even if subconscious, may inf luence their math relevant input. By providing parents with a low-pressure opportunity to observe their children engage with math, Bedtime Learning Together may change parents’ expectations of what their children (of both genders) are capable of doing in terms of math, and thereby increase the quality or quantity of the input they provide to their children. Schaeffer et al. (2018) measured Bedtime Learning Together parents’ expectations and valuing of math for their children through a survey asking parents questions about how important math is for their child, how good their child is at math, and what they expect in terms of their child’s math performance in the future. They found that though the BLT math app intervention did not alter parents’ math anxiety, it did increase parents’ expectations and values regarding math for their children. Thus, the Bedtime Learning Together math app was able to break the link between parent math anxiety and low expectations and lower valuing of math for their children. Further, even after families stopped using the math app, its benefit to children’s math achievement persisted, and this was partially mediated by parents’ expectations and values regarding math for their children. This suggests that parents’ expectations and values about math for their children may be an important area in which parents differ by math anxiety level, and one where it is important to intervene. Interestingly, our preliminary data from Bedtime Learning Together hints that high math anxious parents are less effect than low math anxious parents at offering mathematical support when interacting with their children around math, even though the app helps them do this, as indexed by children’s math learning. This suggests further interventions that could be done to help math anxious parents change their expectations and value of math for their children by providing

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them with appropriate ways to scaffold to their children. Perhaps providing parents with simple tips, potentially with video examples of effective teaching practices could improve the quality of the interactions between high math anxious parents and their children. Other home- and school-based interventions, like storybooks incorporating math, may similarly be able to provide a low-pressure environment in which adults and children can have pleasant math interactions (Gibson, Gunderson, & Levine, under review; Carrazza & Levine, in progress). Over time, these low-pressure interactions around math may alter high math anxious parents’ perceptions of their children’s math potential, by giving them a chance to observe thier children’s mathematical thinking and enjoyment of math. This, in turn, might improve the quantity and quality of their math interactions with their children more generally, and thus lead to to higher math achievement in their children.

Utility value interventions for families Changing both parents’ beliefs about their children’s STEM potentials and children’s beliefs about themselves was the goal of another successful intervention focused on increasing families’ valuing of STEM (Harackiewicz, Rozek, Hulleman, & Hyde, 2012; Rozek et al., 2017). This intervention aimed to improve STEM outcomes for high school students in Wisconsin. Brochures about the utility of STEM were mailed to families with high school students. The brochures explained how math and science are important to a wide variety of jobs, and the usefulness of math and science even to students who do not plan to pursue STEM-centric careers. Parents were also given access to a website touting the value and benefits of STEM for all students and encouraged to talk to their children about the role that math and science play in their own lives. As compared to families in a control group that received no intervention, mothers in the utility value intervention group reported valuing math for their children to a greater extent (Harackiewicz, Rozek, Hulleman, & Hyde, 2012; Rozek et al., 2017). Additionally, the students in the intervention group took more STEM courses in their final years of high school than those in the control group and outperformed the control group on the ACT math test. This study suggests that even very brief interventions can change parents’ and children’s attitudes about math and that changes in math attitudes can have real consequences for math achievement. A study of a similar type with younger children may be likewise positive and potentially amplify the effects of the intervention given the incremental and hierarchical nature of math learning.

Successful interventions for adults Interventions that have proven to be successful in mitigating the relation between adult math anxiety and their own performance on math tests may also be of use in helping these adults have more high-quality math interactions with their children. If adults could become less anxious before teaching children, child outcomes may

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improve. This may be especially the case if the intervention mitigates the link between adult math anxiety, poor instruction, and low child math achievement. One intervention that has proven to be successful in improving adult math outcomes is expressive writing. Spending a few minutes before a math or science test writing about one’s feelings has been shown to improve test performance ( Ramirez & Beilock, 2011; Park, Ramirez, & Beilock, 2014). Writing about one’s feelings before taking a test has been found to improve performance before a math test in the lab, and before a final exam in a high school class. Participants who engaged in expressive writing outperformed those who either simply sat and waited for the test or wrote about their daily lives instead of their feelings. By off loading negative emotions via writing, students may be reducing anxiety during the test and freeing up working memory resources crucial to performing well at complex tasks. Examining whether similar interventions could support parent-child math interactions is an interesting open research question. Perhaps if parents and teachers could alleviate their own anxiety before engaging in math with their children, they would be able to provide better support to their children or students. Another promising intervention involves arousal reappraisal. Jamieson, Mendes, Blackstock, and Schmader (2010) found that simply telling students that high physiological arousal was indicative of imminent success led to improved scores on the math GRE, both in the lab and on the actual GRE exam. This study provides strong evidence that it is not just the physiological arousal associated with anxiety that harms performance, but also, importantly, one’s interpretation of said arousal. Participants who thought feeling nervous was a good sign were able to perform better than those who experienced their arousal as a negative emotion. Perhaps brief expressive writing or emotion reappraisal before a math lesson might help math anxious parents and teachers more calmly and effectively offer instruction to children. These interventions could be taught to pre-service and in-service teachers, and their effects on the quality of teaching and student achievement could be assessed. This direction is important given findings that show that without such interventions doing math is an unpleasant experience for math anxious individuals, and even the anticipation of doing math problems is predictive of activation of brain regions associated with pain in adults ( Lyons & Beilock, 2011).

How can we use existing research findings to inform the development of effective interventions that break the intergenerational negative effects of adult math anxiety on children’s math learning and attitudes? As existing interventions show, it is possible to break the intergenerational cycle of high math anxiety/low math performance by improving the quality of the math input that parents and teachers provide, and/or by targeting parent and teacher attitudes about math. Though there is not a great deal known about how specifically teacher math anxiety harms student performance, there has been

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a lot of research aimed at discovering best teaching practices in general (e.g., Adamson et al., 2003). It is possible that the most effective teachers use methods that mitigate the negative effects of student math anxiety on performance and prevent their own math anxiety from harming their students’ math attitudes and math achievement. Investigating the link between math anxiety and teachers’ mastery and performance goals is one fruitful avenue, but there are undoubtedly others as well.

Lessons from best practices in education One practice of effective teachers that may be difficult for math anxious adults is to let students struggle when they do not know an answer (Stigler & Hiebert, 1999). Future research should investigate whether math anxious parents or teachers may be especially prone to being intrusive, as their own negative feelings about math and fear of failing at math may make them reluctant to allow their children to experience what they remember about math – the struggles and negative emotions that accompanied failure. Additionally, math anxious parents and teachers may not want to take the time to think about the best way to explain a problem to a child. They may instead want to finish the interaction as quickly as possible, in keeping with their desire to avoid math, and perhaps a desire to protect their children from the negativity they feel about math. The fastest way to finish an assignment may be to simply tell a child the answer or how to get the answer, again avoiding the possibility of failure on the part of the child, but also not providing an opportunity for the child to learn or grow in their math knowledge by thinking about the problem and figuring out how to solve it. Intrusive instruction has been linked to poor student performance, not only at the individual teacher level, but also in cross-national studies. Stigler and Hiebert (1999) have found that in countries with higher math performance overall, teachers tend to provide students with more opportunities for exploration and failure. On the other hand, in relatively low math performing countries, like the United States, teachers are more likely to either give students the answer to a difficult problem or break the problem up in to a series of much easier procedures, without providing an explicit conceptual link between different procedures. Letting students struggle, and asking them questions instead of simply providing answers, is important to learning. In a similar vein, Richland (2015) found that teachers in higher achieving cultures such as Hong Kong and Japan used more gestures to link together different representations during eighth-grade math lessons than did teachers in the United States. Tools such as gestures, analogies, and rich language and discussions about math may be particularly helpful tools for teachers as well as for parents (Richland, 2015; Fisher, Hirsh-Pasek, Newcombe, and Golinkoff, 2013). One result of differences in teaching methodology is that many community college students in the United States believe that math is nothing more than a series of unrelated procedures, with no conceptual underpinning whatsoever (Stigler, Givvin, & Thompson, 2010). Because it is not possible to memorize every single math procedure, and different problem situations require different ways of operating on

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quantities, these students are likely to eventually do poorly in higher level math courses, lacking even a basic understanding of the purpose and conceptual underpinnings of math (Richland, Stigler & Holyoak, 2012).

Helping teachers and parents provide scaffolding for children Simply instructing parents to help their children with math more is likely not to work well, particularly for math anxious parents (Maloney et al., 2015). Further, simply telling math anxious parents that their math anxiety can negatively impact their child’s math performance is likely to be ineffective. It is clear from surveys that people with math anxiety are aware that they are anxious about math, yet their math anxiety still seems to take a negative toll on their math performance (Ashcraft & Kirk, 2001). Furthermore, pointing out to parents and teachers that they are math anxious, or even that their math anxiety may negatively impact their children or students, may have a negative effect on the adults’ feelings about math, and in fact make teachers or parents even more nervous about math, and especially nervous about teaching math. Parents and teachers may be better off if, instead of just being told to do more math with their children or students, they were provided with tips (e.g., ways to gesture, question, math language) and materials they can use with their children to support positive math interactions, as was done by Berkowitz and colleagues (2015) with the Bedtime Learning Together app. This may allow parents and teachers to effectively share math with their children and students in ways that they would not come up with on their own. Further, these scripted interactions may lead to spillover effects such that parents will engage in more productive math conversations with their children in other contexts, not just when engaging with the app (e.g., when setting the table, when playing games).

Generalizable tools for teachers and parents The literature around scaffolding in math lessons at schools may provide a positive framework to develop new interventions to help math anxious parents and teachers effectively teach math to children, ideally in lower-pressure situations as often as possible. Simple teaching techniques and ways to share math in a playful, meaningful way may contribute toward helping math anxious parents and teachers provide highquality support to their children. For example, Fisher, Hirsh-Pasek, Newcombe, and Golinkoff (2013) found that exposing children to non-cannonical examples of shapes through guided play can help them understand for themselves the essential features of shapes such as non-typical triangles. Similarly, doing common activities like jigsaw puzzles and block play with children is associated with children’s later spatial reasoning skills, which have been associated with math acheivement (Jirout & Newcombe, 2015; Levine, Ratliff, Huttenlocher, & Cannon, 2012). These activities can be lower-pressure than homework help, while still providing adults with ways to engage in math interactions with their children.

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Generalizable tools for teachers and parents: using gesture to improve instruction Studies of gesture and learning may be particularly relevant in terms of providing insight into how math interactions could be effectively scaffolded in school and at home. Children’s gestures may be able to help parents and teachers become aware of the specific math needs of each child, thus providing them with a better idea of how to provide appropriate supports. Indeed, sometimes students perform a correct gesture, even when they get a math problem wrong. Perry, Church, and Goldin-Meadow (1988) found that these gesture mismatches indicate a readiness to acquire new math knowledge. For example, in a study of mathematical equivalence, they found that students were likely to believe that an equals sign meant to add up all of the numbers before the equals sign, and put the sum on the other side, rather than understanding the concept of mathematical equivalence in a problem such as 7+3+2 = ___+2. However, students who made this error, but also gestured with their hands in such a way as to indicate grouping together only the numbers that needed to be added in order to correctly solve the problem, were more likely to learn the correct conceptualization of equivalence after instruction than those who had no speech and gesture mismatch. This mismatch between the verbal error and the correct gesture, therefore, is an indication of a readiness to learn a new concept. Similar results were found in younger children as well (Gunderson, Spaepen, Gibson, Goldin-Meadow, & Levine, 2015). Threeto five-year-old children who did not yet have a firm understanding of number were more accurate in labelling set sizes when they gave their answer with gestures rather than words. Moreover, children who produced gesture-speech mismatches were more likely to learn the next number word when provided with strong instruction (Gibson, Gunderson, Spaepen, Levine, & Goldin-Meadow, in press). Here again, children’s gestures revealed that they had access to information that they were unable to communicate verbally and that this information ref lected a readiness to learn. Thus, when teachers or parents fail to let students struggle or simply give them the correct answer, they are not only limiting children’s abilities to problem-solve for themselves, they are also missing out on valuable information about the child’s knowledge and capabilities that could have been revealed by an error. Encouraging parents and teachers to tune in to the gestures and mistakes of the child could be a first step toward helping them to develop more effective scaffolding. This could be particularly useful for intrusive parents or teachers, as communicating the need to let students’ struggle is essential to scaffolding learning.

Changing values and beliefs to combat math anxiety The utility value literature points toward another potentially beneficial type of intervention. Though value interventions have been done in the home (Harackiewicz, Rozek, Hulleman, & Hyde, 2012; Rozek et al., 2017), it is conceivable that a similar

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intervention in a school setting could be beneficial, especially given the role that teachers play in the intergenerational effects of math anxiety. Perhaps giving high school teachers, guidance counsellors, or principals information about the utility of math and science could change the way that teachers and students conceptualize math and lead to better learning outcomes. Of course, in order to scale this intervention up to the school level, the mechanism of its effectiveness must be considered. For example, it may have been the parents’ and/or students’ increased valuing of STEM subjects that led students in the intervention group to take more STEM classes and get better math test scores. Future studies should investigate whether helping parents and teachers to value math, and recognize its utility, even in the early grades, could lead to higher math performance among students, particularly those who are interacting with math anxious adults, both teachers and parents. Utility-value interventions could also be adjusted for younger children, encouraging parents and teachers to see the early utility of mathematics. Elementary school students are not ready to choose their own math courses or make serious decisions about their careers, but thinking about the use of math even at an early age may be very beneficial. Additionally, if teachers or principals were encouraged to think about the potential utility and value of math for each of their students in the early years, schools might do a better job supporting math learning and development. In addition to new interventions, further investigations into the social mechanisms by which high math anxious parents and teachers impact children’s math performance are also needed. For example, it would be beneficial to know how, and how much, parents and teachers are communicating to children about the value, fun, and usefulness of math, and how this affects children’s attitudes about math.

What are some important avenues for future research on the topic of intergenerational effects of adult math anxiety? In conclusion, the link between adult math anxiety and child math performance is pervasive and persistent. Parent and teacher math anxiety are associated with decreased child math performance. Recent research has begun to examine the mechanisms that drive this relation, hinting at a role of suboptimal instruction and negative attitudes about math offered to children by math anxious adults. Promising interventions have been developed to break the link between anxiety and performance, in both adults and children. Future research should focus on the mechanisms by which these interventions are successful and delve deeper into the means by which adult math anxiety impacts child math achievement. New interventions should be developed and tested, and, at the same time, interventions that have worked to ameliorate math anxiety in individuals should be modified and tested to address the intergenerational transmission of math anxiety. It is also important to test the effectiveness of these interventions not only in the laboratory but also in real-world learning environments where children are exposed to math, notably, at home and school.

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Ramirez, G., & Beilock, S. L. (2011). Writing about testing worries boosts exam performance in the classroom. Science, 331, 211–213. Ramirez, G., Chang, H., Maloney, E. A., Levine, S. C., & Beilock, S. L. (2016). On the relationship between math anxiety and math achievement in early elementary school: The role of problem solving strategies. Journal of Experimental Child Psychology, 141, 83–100. Richland, L. E. (2015). Linking gestures: Cross-cultural variation during instructional analogies. Cognition and Instruction, 33(4), 295–321. Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics. Educational Psychologist, 47(3), 189–203. Rozek, C. S., Svoboda, R. C., Harackiewicz, J. M., Hulleman, C. S., & Hyde, J. S. (2017). Utility value intervention with parents increases students’ STEM preparation and career pursuit. Proceedings of the National Academy of Sciences, USA, 114(5), 909–914. Sawada, D., Piburn, M. D., Judson, E., Turley, J., Falconer, K., Benford, R., & Bloom, I. (2002). Measuring reform practices in science and mathematics classrooms: The reformed teaching observation protocol. School science and mathematics, 102(6), 245–253. Schaeffer, M. W., Rozek, C., Berkowitz, T., Levine, S. C., & Beilock, S. L. (2018). Long-lasting change in children’s math achievement starts with parents. Journal of Experimental Psychology: General. Soni, A., & Kumari, S. (2015). The role of parental math attitude in their children’s math achievement. International Journal of Applied Sociology, 5, 159–163. Spencer, S. J., Steele, C. M., & Quinn, D. M. (1999). Stereotype threat and women’s math performance. Journal of Experimental Social Psychology, 35, 4–28. Sprute, L., & Beilock, S. L. (2016). Math anxiety in community college students. MathAMATYC Educator, 7, 39–45. Steele, C. M., & Aronson, J. (2005). Stereotypes and the fragility of academic competence, motivation, and self-concept. In A. J. Elliot & C. S. Dweck (Eds.), Handbook of competence and motivation, (pp. 436–455). New York: The Guildford Press. Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York: Free Press. Stigler, J. W., Givvin, K. B., & Thompson, B. J. (2010). What community college developmental mathematics students understand about mathematics. MathAMATYC Educator, 1(3), 4–16. Walton, G. M., & Cohen, G. L. (2011). A brief social-belonging intervention improves academic and health outcomes of minority students. Science, 331, 1447–1451. Wang, Z., & Shah, P. (2014). The effect of pressure on high-and low-working-memory students: An elaboration of the choking under pressure hypothesis. British Journal of Educational Psychology, 84(2), 226–238.

CONCLUDING REMARKS Irene C. Mammarella, Sara Caviola and Ann Dowker

Math is often considered the “bugbear” among school subjects, not only in relation to achievement (or school grade), but also for the negative feelings associated with its experience. This negative halo affects both the protagonists of the learning (i.e., the children), and all the other actors involved in its process (i.e., teachers and parents). This has led, in recent years, to an increasing recognition of the importance of these negative emotional factors, labeled with the term “math anxiety.” Several studies have indeed focused on the specific cognitive deficits that characterized children with mathematical difficulties, including those labeled as children with developmental dyscalculia. Within this framework, researchers often overlook the emotional aspects involved in mathematical learning. To present a complete framework of the individual differences involved in young children’s low mathematics performance in this volume, we provided a fresh perspective in which these negative emotions related to math were specifically taken into account. In particular, the book raised some important questions about the drivers of math anxiety from different perspectives. In recent years, math anxiety has become a subject of increasing interest, not only in research but also in both educational and clinical settings, especially with regard to its consequences in limiting people’s mastery of mathematics. One of the negative consequences of mathematics anxiety is that highly math-anxious students tend to avoid mathematical situations (Ashcraft, 2002). Therefore, it is not surprising that highly math-anxious individuals are also more likely to avoid career paths that require quantitative skills (Ashcraft, 2002; Hembree, 1990). Thus, math anxiety can have serious long-term consequences, adversely inf luencing an individual’s choice of career, type of occupation, and career progression in adulthood (Ashcraft & Ridley, 2005; Beasley, Long, & Natalia, 2001; Hembree, 1990; Ho et al., 2000). Moreover, these consequences may have a negative impact not only on the individual but also on society,

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as low mathematical attainment has considerable adverse social and economic effects on a country. For example, estimates of the cost of low numeracy to the United Kingdom’s economy have ranged from £2.4 billion a year (KPMG, 2008) to more than £20 billion a year (National Numeracy, 2015), with the lower of the figures already representing a very significant cost to society. Even when math anxiety does not cause or enhance actual numeracy difficulties, it can have a negative impact by reducing capacity in occupational areas that involve or depend on mathematical knowledge. The importance of math anxiety has motivated us to edit a book on this topic. In various chapters of this book the relationship between math anxiety with mathematics performance has been addressed (see Chapters 4, 8, 9, and 11), and consideration has been given to the ways in which working memory may be involved in both math anxiety, test anxiety, and mathematics achievement (see Chapters 6 and 7). Not only cognitive processes related to math anxiety, but also contextual factors, such as the relationship of math anxiety to social stereotypes, especially with regard to gender, have been analyzed (Chapter 10), without forgetting how to measure math anxiety and psychophysiological aspects and the neural correlates of math anxiety (Chapter 2 and 3). Hence, we have tried to offer a comprehensive review of most of the factors related to math anxiety. However, there are still several aspects of math anxiety that need to be better understood. We seek to summarize some of them here below. What have we learned from previous research? First, we know that math anxiety tends to increase with age during childhood and adolescence. Most research suggests that severe math anxiety is uncommon in young children, though some studies indicate that math anxiety may develop even among early primary school children (Petronzi, 2016; Wu, Barth, Amin, Malcarne, & Menon, 2012). This apparent increase in math anxiety with age is to some degree consistent with other findings indicating that other forms of anxiety increase with age. In particular, it has been generally found that the onset of clinical anxiety disorders peaks in early adolescence (Kessler, Chiu, Demler, & Walters, 2005), although it is possible that such disorders in younger children are under-diagnosed due to lack of clear and appropriate diagnostic methods (Egger & Angold, 2006). It may also be that children only develop math anxiety after repeated negative experiences with mathematics. Attitudes toward mathematics other than anxiety tend to deteriorate as children get older (Ma & Kishor, 1997; Dowker, 2005; Mata, Monteiro, & Peixoto, 2012). For example, Blatchford (1996) found that two-thirds of 11-year-olds rate mathematics as their favorite subject, but that few 16-year-olds do. However, as most of these studies are cross-sectional rather than longitudinal, it is hard to establish the direction of causation with math anxiety. Thus, more longitudinal studies are needed if we are to better understand how math anxiety develops over time. Another interesting avenue worth exploring is the relationship between math anxiety and other forms of anxiety (i.e., general anxiety and test anxiety). The

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experience of anxiety seems to be separable into at least two major components, worry and emotionality, where worry refers to the cognitive elements of the anxiety experience, such as negative expectations about oneself, and potential consequences of failure (Morris, Davis, & Hutchings, 1981). Emotionality refers to nervousness and tension and to the respective autonomic reactions felt by the individual. These two components seem to be consistent in different forms of anxiety though they differ with regard to the “object” of the anxiety. For instance, general anxiety can be defined as a disposition to be anxious or worried about future events, past behaviors, and personal competence, which occurs across domains (Spence, 1997). Test anxiety can be defined as a multidimensional construct involving cognitive, bodily affective and behavioral responses associated with concerns about possible negative consequences of failure in evaluative situations, especially academic examinations and similar evaluations (Zeidner, 1998). By definition, test anxiety is considered to be related to school evaluations and it is believed to have a domain-general negative effect on academic tasks such as reading, writing, and mathematics (Putwain, 2008). In fact, test anxiety and math anxiety have repeatedly been found to co-occur in individuals, with studies typically reporting moderate correlations between the two constructs, while general anxiety tends to be less related than test anxiety to math anxiety (Hembree, 1990). Again, test and mathematics anxiety have similar relationships to student ability, gender, and ethnicity, indicating some overlap between the two constructs (Hembree, 1990; Kazelskis et al., 2000). Studies have also shown that both test and mathematics anxiety are negatively associated with academic performance (Hembree, 1990; Carey, Hill, Devine, & Szűcs, 2017). With regard to this point, most of the studies so far have focused either on the relationship between test anxiety and academic achievement or on the relationship between math anxiety and mathematics, while general anxiety has been only rarely studied in relation to academic achievement, and, when considered in studies at all, is most usually included as a control variable (Devine, Fawcett, Szűcs, & Dowker, 2012; Hill et al., 2016; Carey et al., 2017). Thus, only a limited amount of research has investigated the relationships between the different forms of anxiety. For instance, Carey and colleagues (2017) assessed different forms of anxiety by conducting a latent profile analysis on 1720 students in grade 4, or in grades 7 and 8. They identified four profiles in grade 4, ranging from low to high anxiety. This four-group solution also emerged in older students, but the profiles appeared more specific in this case, and were described as low anxiety, general anxiety, academic anxiety (i.e., math anxiety and test anxiety), and high anxiety. Mammarella, Donolato, Caviola, and Giofrè (2018) also performed a latent profile analysis on 664 children from grades 3 to 6, revealing three main profiles. Around 12% of students expressed a low-risk profile, meaning that they had very low scores on a variety of anxiety measures; around 66% showed an average risk, exhibiting average scores on a variety of anxiety measures, and the remaining 22% revealed a high-risk profile, exhibiting high scores on different forms of anxiety measures. Interestingly, the average-risk profile was characterized by

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higher levels of general anxiety and math anxiety than test anxiety, but in all cases the scores ranged around the average; and the high-risk profile featured higher levels of general anxiety and test anxiety than math anxiety, although all anxiety scores were in the high range. These findings seem to support the hypothesis that general anxiety could be a risk factor for the onset of other, more specific academic-related forms of anxiety (math anxiety and test anxiety). At the same time, these findings do not exclude the possibility that repeatedly negative academic experiences could enhance the possibility of developing such forms of more specific anxiety related to academic evaluation (i.e., test anxiety) or mathematics (i.e., math anxiety). Once again, both the studies of Carey et al. (2017) and Mammarella et al. (2018) were cross-sectional, and only longitudinal research can truly clarify whether the forms of anxiety seen in younger children are precursors to those identified in older students. Another issue to take into account in future studies concerns the role of individual resources in the development and maintenance of math anxiety symptoms. Individual resources could be considered as attributes and strengths that help individuals foster competence and sustain and promote successful development in different domains of functioning (i.e., individual, school, family) (Dekovic, 1999). Mathematics anxiety seems to be particularly related to self-rating with regard to mathematics. People who think that they are bad at mathematics are more likely to be anxious. Most studies indicate a negative relationship between mathematics self-concept and math anxiety (Hembree, 1990; Pajares & Miller, 1994; Jain & Dowson, 2009; Hoffman, 2010). By contrast, a meta-analysis considering more than two hundred research studies highlighted a positive relation between self-concept and achievement (Hansford & Hattie, 1982). In addition, Ahmed et al. (Ahmed, Minnaert, Kuyper, & van der Werf, 2012) carried out a longitudinal study of 495 seventh-grade students who completed self-report measures of both anxiety and self-concept three times over a school year. Structural equation modeling suggested that each characteristic inf luenced the other over time, but that the effect of self-concept on subsequent math anxiety was significantly greater than the effect of anxiety on subsequent self-concept. Although the results should be interpreted with some caution, because the longitudinal study was carried out only over one school year, it provides evidence that the relationship between math anxiety and mathematics self-concept is reciprocal. In a similar vein, Mammarella et al. (2018) showed that children with a low-risk profile for anxiety reported greater feelings of competence and a stronger academic self-concept than children with average- or high-risk profiles for developing anxiety. Within the framework of individual resources, a closely related construct is self-efficacy. Self-efficacy does not have precisely the same meaning as selfconcept, as the former includes beliefs about the ability to improve in mathematics and take control of one’s learning, whereas the latter only includes beliefs about one’s current performance. Some studies have shown an inverse relationship between self-efficacy and math anxiety (Cooper & Robinson, 1991; Lee,

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2009). In addition, motivation, and specifically intrinsic motivation, could be considered an individual resource that may protect against math anxiety. In particular, Wang et al. (2015) observed an inverted-U relation between math anxiety and math performance only in students showing a high intrinsic motivation, while students with low motivation revealed a negative linear relation between math anxiety and math performance. Thus, future studies should consider the ways in which individual resources interact with math anxiety in different developmental periods and the ways in which these interactions might determine different profiles of academic performance. In particular, a better understanding of how all these variables mediate the presence of math anxiety in both children and adults could contribute to the development of more effective intervention programs. The challenge of understanding how math anxiety effects mathematics is considerable and defined, in part, by the numerous sources of factors involved. Indeed, math anxiety can be seen as a complex psychological construct whose development depends on many variables. Among these, an important role is played by contextual factors, as described by Beilock and colleagues (Beilock, Gunderson, Ramirez, & Levine, 2010; see Chapter 11). They note the power of the learning context, asserting that the presence of math anxiety in teachers can inf luence the development of math anxiety in children, specifically affecting girls’ math achievement – and by consequence inf luencing girls’ gender-related beliefs about who is good at math. Similarly, the home environment plays an important role not only in developing mathematics skills (LeFevre et al., 2009) but also for precipitating the onset of math anxiety (Maloney, Ramirez, Gunderson, Levine, & Beilock, 2015). Thus, adults’ math anxieties and beliefs that math ability is a stable trait (Mueller & Dweck, 1998) may have significant impacts on children’s development of math attitudes. In addition, a large body of research has shown that parents’ and teachers’ gender stereotypes, beliefs, and expectations regarding children’s math aptitude affect children’s subsequent math attitudes and achievement in a way that perpetuates gender-stereotypical roles ( Jacobs & Eccles, 1992; Midgley, Feldlaufer, & Eccles, 1989). However, as previously mentioned, personal factors could play an important role and, in some cases, act as individual resources: a high motivation to learn new mathematical concepts, a good self-concept, and self-efficacy should, in some way, contrast or delay the onset of math anxiety. Thus, a comprehensive model of math anxiety should consider the interplay among all these aspects, by also bearing in mind the effect of other variables such as gender and age of the child. In addition to contextual and individual factors, we cannot forget cognitive factors. In previous chapters of this book we have reviewed the role of showing an intrinsic tendency to display attentional bias toward negative information (Rubinsten, Eidlin, Wohl, & Akibli, 2015, Chapter 9) in the onset of math anxiety, such as the crucial role of working memory (Ashcraft & Kirk, 2001), and how ruminations and negative thoughts can occupy and overload working memory resources. Finally, physiological measures such as heart rate, skin conductance, and cortisol secretion as a

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response to stressful situations (Hellhammer, Wüst, & Kudielka, 2009) are parts of the experience of math anxiety. Mathematics anxiety involves all of these aspects, and thus our biggest need for further studies may involve the ways in which the various aspects relate to each other. Of course, it is not easy to consider so many variables in the frame of a single study, but researchers should continue merging some of these factors in order to understand math anxiety more concretely, by drawing its implications for everyday life. This could also be helpful for developing effective interventions. Hembree (1990), in his meta-analysis, pointed out that cognitive-behavioral interventions developed for the treatment of test anxiety or general anxiety were effective in reducing or eliminating the effects of math anxiety on mathematics performance. The interventions reported merely focused on changes in classroom curricula, relaxation therapy, or group counseling that resulted to be ineffective. Later studies relying on more transitory disruption due to the threat of choking under pressure, focused on relieving the cognitive symptoms of anxiety, particularly those that are supposed to relieve working memory resources, and have shown promising results (Park, Ramirez, & Beilock, 2014; Ramirez & Beilock, 2011; Supekar, Iuculano, Chen, & Menon, 2015). Ramirez and Beilock (2011) showed how a 10-minute writing intervention by college students enhanced their performance in a forthcoming mathematics examination. Students were asked to write about their negative thoughts and ruminations about math; expressive writing improved their performance on high-stakes tests than those who did not write or wrote about a different topic. Similar results were shown from Park and colleagues (2014): they demonstrated that writing about worries before completing a test improves their performance. Asking university students to write spontaneously about their emotions regarding an upcoming test lightened the gap between low- and high math anxiety students. The authors concluded that expressive writing helped in focusing subjects’ attention during the maths task by reducing the interfering role of worries and negative thoughts. Expressive writing may serve to “off load” worries from working memory and attentional resources, therefore relieving the distracting effects of worry on cognition and consequently improving the performance on the upsetting task. Taken together, these findings offer initial experimental support that math anxiety can be reduced with tailored interventions, at least in adult populations. Although the adverse consequences of math anxiety are extensively described in the literature, to date there have been only few intervention studies for remediating math anxiety in young children, as described by Passolunghi and colleagues in Chapter 6. Supekar and colleagues (Supekar, et al., 2015) tested an intervention program based on well-validated studies of behavioral exposurebased therapy for anxiety disorders (Abramowitz, Deacon, & Whiteside, 2011, Wolitzky-Taylor, Horowitz, Powers, & Telch, 2008) on third graders. They found that an intensive cognitive tutoring program designed to improve mathematical skills (Fuchs et al., 2013) alleviated math anxiety in primary school-aged

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children by virtue of mere exposure to distressing stimuli. However, a limitation of this study was that the findings could not be extended to children with high scores in math anxiety, since the high- and low math anxiety groups of this study were formed using a median split of a normal sample (Sokolowski & Necka, 2016). Similarly, Verkijika and De Wet (2015) aimed to reduce the negative impact of math anxiety on mathematics performance using a brain-computer interface (BCI) mathematics educational game. They trained children from primary and secondary schools to manage and control their anxiety levels with the BCI neuro-feedback while they were playing a math game (2-session intervention). Results indicated that positive emotions enhance mathematical learning by increasing the perseverance, resilience, and strategic f lexibility through recruitment of cognitive resources. Unfortunately, even this study suffers from several limitations, among which is the absence of any control group. As previously mentioned, only few studies have been developed thus far to assess interventions for remediating or preventing math anxiety in young students. Although limited, these data on interventions seem to suggest that students can be taught with simple self-instruction strategies and activities to handle their negative thoughts and feelings. Likewise, as highlighted in Chapter 11, students’ math anxiety can also be deeply related to school systems, socioeconomic status, or parental background (Casad, Hale, & Wachs, 2015; Maloney, et al., 2015). Research indicated that teachers and parents who are afraid of math can negatively influence children’ attitudes and beliefs about the subject, thus increasing their likelihood of developing math anxiety. By gaining more knowledge about the interplay among the intervening variables related to math anxiety, different kinds of interventions programs could be realized and tested for dealing with mathematics anxiety. So, interventions may also be addressed to teacher/parental figures in order to provide them with the right skills and strategies for prompting positive attitudes (i.e., appraising an achieved goal rather than emphasizing failure) in creating a successful learning atmosphere (Cargnelutti, Tomasetto, & Passolunghi, 2017). Suppose, for example that for younger children the home environment shows a stronger relation between math anxiety and math-related performances than individual resources. In this scenario, parent training could be suggested as a specific intervention program for improving the home environment and, consequently, reducing math anxiety in the child. Teacher training could also be developed, making both parents and teachers aware of the crucial effect of context in the onset of math anxiety, and giving them coping strategies for mitigating negative effects. These trainings could suggest how parents, educators, and education policymakers should interpret and hence respond to high math-anxious students. The use of effective instructional techniques, such as focusing on what students can do, encouraging multiple outcomes, and being sensitive to past histories of failure, can positively change the attitude towards maths (Furner & Duffy, 2002). Effective interventions should thus aim not only to improve mathematical skills or cognitive resourses but also to foster positive attitudes toward mathematics, which may, in turn, for example, promote engagement with mathematics,

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encourage persistence and resilience despite difficulty with the subject, and mitigate the negative effect of math anxiety. In addition, the efficacy of cognitive training in children should be tested: training aimed to reduce attentional bias, training for reducing interfering thoughts and rumination, and training for improving mathematics abilities that indirectly could enhance self-concept and self-efficacy. Finally, interventions seeking to improve a child’s motivation and individual resources could be developed. Comparing the efficacy of different kinds of intervention is a crucial point in the field of math anxiety. Educational and clinical implications are immediately clear, and reducing math anxiety is the final indirect goal of the studies in this field. To achieve this goal, further research is needed. In conclusion, although we have acquired substantial knowledge in recent years concerning math anxiety, more studies identifying methods of reducing the impact of math anxiety on students’ performance are still needed. Thus, research with stronger educational implications will be welcome in this perspective.

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INDEX

Page numbers in italic indicate a figure and page numbers in bold indicate a table on the corresponding page. Abbreviated Math Anxiety Scale (AMAS) 23, 24, 26, 45, 159 ability see math ability academic performance: academic anxiety and 136; general anxiety and 131; test anxiety and 213; see also math performance adolescents, math anxiety measurement in 22 –23 adults: math anxiety in 190 –207, 215; math anxiety measurement in 22 –23; math performance in 191–193; successful interventions for 202 –203 affective dimension 21–22 , 62 , 67, 68 African American students 179, 185 age-related changes, in test and math anxiety 135 –137 AMAS see Abbreviated Math Anxiety Scale amygdala 13 –14, 54, 81– 82 ANS see approximate number system anxiety: definition of 110 –111, 127; genetic factors in 147; introduction to 42; neurological pathways for 137; number 1, 2 , 21, 62; performance 62; performance and 179 –180; physiological response to 42; stereotype threat and 179 –186; trait-state theory of 127–128; see also general anxiety; math anxiety; test anxiety

Anxiety Toward Mathematics Scale (ATMS) 43, 45 apprehension 93 approximate number system (ANS) 106 –108, 113 arithmetic calculation skills 108 –109 arousal reappraisal 204 Asian Americans 183, 199 assessment see math anxiety measurement ATMS see Anxiety Toward Mathematics Scale attentional bias 156 –158, 161, 162 –166, 168, 215 attentional control theory (ACT) 128, 129, 132 attitudes, toward mathematics 66 – 67, 82 , 86 – 89, 95 –96, 159, 161–162 , 211, 212 , 217–218 autonomic measures 43 – 48, 47, 55 autonomic nervous system (ANS) 43, 44 avoidance 12 , 16, 79, 83 – 84, 88, 96 –97, 141, 144, 192 , 211 Bedtime Learning Together 196, 201–202 , 205 behavioral measures 28 –29 blood-oxygenation-level dependent (BOLD) contrast 52 blood pressure 43 body temperature 29

Index

boys see males brain-computer interface (BCI) 217 brain mechanisms 81– 82 , 137 brain studies 13 –16, 29 –30, 48 –56, 81– 82 , 181 British Abilities Scales Basic Number Skills test 66 calculation skills 108 –109, 113 CAMS see Children’s Anxiety in Math Scale central executive 108 –109, 128 Child Math Anxiety Questionnaire (CMAQ) 24 children: adult math anxiety and 190 –207; age-related changes in test and math anxiety in 135 –137; coping strategies of 89 –90; developmental changes in math anxiety and performance in 67– 69; development of math anxiety in 22 , 77–97, 156 –158, 166, 167, 168 –169, 212 , 213; early education classroom experiences for 77–79; feedback on math anxiety from 86 – 88; with learning difficulties 111, 143 –145, 147; math anxiety questionnaires for 26 –27, 54; math performance in 193 –194; numeracy experiences of young 86 –97, 87; selfratings by 66 – 67; stereotype threat and anxiety in 185 –186; working memory in 126 –137 Children’s Anxiety in Math Scale (CAMS) 24 Children’s Mathematics Anxiety Scale UK (CMAS-UK) 85 – 86 choking under pressure 112 , 184, 216 classroom environment 77–79, 148, 200 CMAQ see Child Math Anxiety Questionnaire CMAS-UK see Children’s Mathematics Anxiety Scale - UK cognitive abilities 159, 215 cognitive-behavioural therapy 149, 216 cognitive construct, math anxiety as 4 –9 cognitive control 53, 54, 56 cognitive dimension 21–22 , 62 cognitive processing, of numerical representations 163 –166 cognitive resources 156 cognitive theories 128 –129, 132 cognitive traits 156 –157, 158 comparison 96 competition 91, 96 conf lict sustained potential 51

223

contextual factors 168 –169, 215 continua model, of working memory 105, 106 coping strategies 89 –90, 149 correct-response negativity (CRN) 50 –51 cortisol 29, 43 – 48, 47, 55 CRN see correct-response negativity crying 93 cultural differences 69 –70 default-mode network (DMN) 53 developmental changes, in math anxiety and performance 67– 69 developmental dyscalculia 72 , 143 –145, 147–149, 156, 158 difficulty 89 disadvantaged social groups 176, 179 disruption account 7 distance effect 107 DMN see default-mode network dorsolateral prefrontal cortex (dlPFC) 47 dual-task interference effect 6 –7 dyscalculia see developmental dyscalculia dyslexia 147 early education: classroom experiences in 77–79; math anxiety development in 82 – 84 East Asia 69, 70, 77, 204 educational psychology 92 –93 education best practices 204 –205 electroencephalography 29 –30, 48 electrophysiology 48 –52 , 55 –56 elementary school teachers 10, 194; see also teachers emotionality 62 , 71, 180 –181, 213 emotional regulation 53 –56, 81– 82 , 181, 184 emotional responses 80 – 81, 86 – 89 emotional stress 142 entity theory of intelligence 12 , 15 environmental factors 146 –148, 157, 162 , 167, 168 –169, 215, 217 episodic buffer 104, 128 error-related negativity (ERN) 50 –51 event-related brain potentials (ERPs) 48, 51, 56 executive function 108 –109, 128 –129 expressive writing technique 115 –116, 149, 204, 216 failure 88, 96, 127 families: structured math interventions for 200 –202; utility value interventions for 202; see also parents

224

Index

fear 88, 96 females: general anxiety in 70, 160; math anxiety in 10 –12, 70 –71, 142–143, 148, 159; math performance in 70 –71, 142–143; stereotype threat and 178 –186; see also gender Fennema-Sherman Mathematics Attitudes Scales - Short Form (FSMAS-SF) 24 FRIENDS for Life programme 149 functional MRI (fMRI) 52 –55, 56, 181

for math anxiety 15 –16, 115 –117, 149, 200 –203, 216 –218; research for developing effective 203 –207; structured 200 –202; utility value 202; value 206 –207; for young children 216 –217 intrinsic factors 149, 157–158, 160 –161, 166, 167, 168 –169 intrinsic motivation 95, 149, 215 intrusive instruction 204

Galvanic Skin Response (GSR) 2 gender: general anxiety and 12 , 70, 160; math anxiety and 10 –12 , 70 –71, 142 –143, 148, 159, 198 –200; math performance and 70 –71, 142 –143; stereotypes 178 –186, 198 –200 general anxiety 3; cognitive traits associated with 156 –157; gender differences in 12 , 70, 160; genetic factors in 147, 161; math anxiety and 13, 158, 160 –161, 212 –214; performance and 113, 114, 129, 131 genetics: of general anxiety 147, 161; of math anxiety 12 –15, 146 –148, 159 gesture 204, 206 girls see females Graduate Record Examination (GRE) 180, 204 GSR see Galvanic Skin Response guilt 79

Latent Profile Analysis (LPA) 33 –34 learning disorders 143 –145, 147, 148, 156, 158 literacy 146 longitudinal studies 135 –137 low maths-anxious individuals (LMAs) 45, 49 –56

heart rate 29, 43, 45, 47, 47, 55 helplessness 142 high maths-anxious individuals (HMAs) 45, 49 –56 home environment 148, 149, 215, 217 homework 92 , 195, 196 hypothalamus-pituitary-adrenal (HPA) axis 43, 44 incremental theory of intelligence 12 , 15 individual resources 66 – 67, 82 – 83, 214 –215 inhibition function 8, 128 instruction 195 –198, 204 –206, 217 intelligence: entity theory of 12 , 15; hierarchy 91; incremental theory of 12 , 15; motivation and 92 –93; socialcognitive theory of 12 intergenerational effects 190 –207 interpretation account, of math anxiety 15 –16 interventions: for adults 202 –203; cognitive-behavioural 149, 216;

magnetic resonance imaging 29 magnetic resonance imaging (MRI) 52 –55, 181 males: general anxiety in 70, 160; math anxiety in 11–12 , 70 –71, 142 –143, 148, 159; math performance in 70 –71, 142 –143; see also gender mAMAS see Modified Abbreviated Math Anxiety Scale MAQ see Math Anxiety Questionnaire MARS see Mathematics Anxiety Rating Scale MARS-E see Mathematics Anxiety Rating Scale for Elementary School Students MAS see Mathematics Anxiety Scale Maslow’s hierarchy of needs 92 , 92 math ability 105 –110; approximate number system 106 –108; calculation skills 108 –109; comparison of 96; components of 105 –106; genetic factors in 146 –147; high sense of 95; low sense of 95; mathematical problem-solving 109 –110 math achievement 3, 66, 69 –70, 81– 82 , 86 – 88; parental math anxiety and 194, 195; see also academic performance; math performance math anxiety: acquisition, development, and maintenance of 77–97; in adults 190 –207, 215; age of development of 22; assessment of 2 – 4; attentional bias in 156 –158, 162 –166, 168; changing values and beliefs to combat 206 –207; characteristics of 127, 156; as cognitive construct 4 –9; cognitive theories of

Index

128 –129; correlations with 3 – 4, 3; cultural and national differences in 69 –70; definition of 1, 42 , 62 , 111, 127, 141; developmental changes in 67– 69; developmental perspective on 133 –134; development of 22 , 77–97, 156 –158, 166, 167, 168 –169, 212 , 214; diagnostic criteria for 31–32 , 168; dyscalculia and 72; emotional responses and 80 – 81; environmental aspects of 146 –148, 157, 162 , 168 –169, 215, 217; factors inf luencing 79 – 80, 82 – 84, 117; feedback from young children on 86 –97; future research on 207; gender and 11–12 , 70 –71, 142 –143, 148, 159, 178 –186, 198 –200; general anxiety and 158, 160 –161, 212 –214; genetic factors in 12 –15, 146 –148, 159; impacts of 141, 144, 145, 211–212; individual resources and 66 – 67, 82 – 83, 214 –215; interest in 211–212; intergenerational effects of 190 –207; interpretation account of 15 –16; interventions for 15 –16, 115 –117, 149, 200 –203, 216 –218; ladder of 79, 80; longitudinal change in 135 –137; math difficulties and 143 –145, 148 –150; math instruction and 195 –197; models of 1–16; negative beliefs in 161–162; as neuro-biological construct 12 –15; optimum level of 71; overview of 20, 158 –159; in parents 10 –11, 190 –191, 194 –207, 215; performance and 32 –34, 62 –72 , 80 – 82 , 84, 111–115, 132 –134, 143 –144, 148 –150, 156, 190, 191–194, 212 , 213; as personality construct 2 – 4; prevalence of 30 –32 , 126; psychophysiological correlates of 42 –57; reasons for 78; research on 1–2; situational 32 ; societal impacts of 211 –212; as sociocultural construct 10 –11; stereotype threat and 178 –186, 198 –200; structure of 20 –22; in teachers 10, 190 –191, 194 –195, 198–200, 203 –204, 207, 215; test anxiety and 137, 212 –214; working memory and 103 –118, 126 –137, 145, 156, 192 , 193 math anxiety measurement 20 –35, 113, 159 –160; Abbreviated Math Anxiety Scale 23, 26; in adults and adolescents 22 –23; autonomic measures and cortisol levels 43 – 48, 47; behavioral, physiological, and neuroscientific methods 28 –30; broad perspective on 32 –35; in children 26 –27, 84 – 86; electrophysiology 48 –52; examples

225

of instruments 24 –25; fMRI 52 –55; implicit measures 42 –57; paper-andpencil self-descriptive instruments 22 –28; physiological measures 42 –57, 215 –216; practical usefulness of 32 –35; self-report measures 22 –28, 42 , 84 – 85, 159 –160; trends in 27–28 Math Anxiety Questionnaire (MAQ) 25, 113 Math Anxiety Scale for Children (MASC) 85 math avoidance 12 , 16, 79, 83 – 84, 88, 96 –97, 141, 144, 192 , 211 math competency 7–9, 16, 79 – 80, 103, 158 –159 math curriculum 77 math difficulties 143 –145, 148 –150, 158 mathematical learning disability 158; see also developmental dyscalculia mathematical problem-solving 109 –110 Mathematics Anxiety Rating Scale (MARS) 2 – 4, 21, 22 –23, 43, 68, 159 Mathematics Anxiety Rating Scale for Elementary School Students (MARS-E) 26, 68, 84 – 85, 113 Mathematics Anxiety Rating Scale Revised (MARS-R) 22 –23, 26 Mathematics Anxiety Scale (MAS) 43 Mathematics Anxiety Scale for Young Children 25 Mathematics Anxiety Scale - Revised (MAS-R) 25 Mathematics Anxiety Scale - UK (MAS-UK) 21, 24 Mathematics Attitude and Anxiety Questionnaire (MAQ) 66, 68 math experts, insights from 93 –97 math instruction 195 –198, 204 –206, 217 math performance 31–34, 54; in adults 191–193; in children 193 –194; cultural and national differences in 69 –70; deficits in 64 – 65; developmental changes in 67– 69; developmental perspective on 133 –134; gender and 70 –71, 142 –143, 198 –200; math anxiety and 62 –72 , 80 – 82 , 84, 111–115, 132 –134, 143 –144, 148 –150, 156, 190, 191–194, 212 , 213; parent math anxiety and 194 –207; positive stereotypes and 183 –185; stereotype threat and 179 –182; teacher math anxiety and 194 –195; test anxiety and 126, 129 –132 , 213; working memory and 111–115, 129 –134, 192 , 193 math self-concept 33

226

Index

memory 51; see also working memory Mindset Theory 92 –93 Modified Abbreviated Math Anxiety Scale (mAMAS) 25 motivation 9, 33, 83, 92 –93, 149, 215 multiple mediation model 114 national differences 69 –70 negative attitudes, toward mathematics 66 – 67, 82, 88 – 89, 95 –96, 159, 161–162 , 211, 212 negative information, attentional bias toward 156 –158, 161, 162 –166, 168, 215 negative public experience 93 –94 negative stereotypes 178 –186, 199 negative thoughts 143, 156, 157, 180, 215 neuro-biological construct, math anxiety as 12 –15 neuroimaging 52 –56, 181 neurons 48 neuroscientific measures 29 –30 no-attempt error 79, 83 – 84 non-symbolic numerical representations 163 –166, 168 number anxiety 1, 2 , 21, 62 number sense 5 – 6, 106 –107 numeracy: ability in 95; coping strategies for 89 –90; definition of 97n1; difficulty of 94, 158 –159; emotional responses to 86 – 89; genetic factors in 146; in public 93 –94; teachers and 90 –91 numerical representations, cognitive processing of 163 –166 numerical skills 158 –159 P2 amplitude 48 –50, 56 P3b amplitude 50, 56 parasympathetic nervous system 43, 44 parents: beliefs about math of 198 –200; generalizable tools for 205 –206; inf luence of 90, 95 –96, 117, 148, 157, 162 , 166, 194, 195, 215, 217; math anxiety in 10 –11, 190 –191, 194, 195 –207, 215; math interventions and 200 –202; support from 92 peers 11, 82 , 90, 91, 96 performance anxiety 62 , 179 –180 performance deficits 64 – 65 personality construct, math anxiety as 2– 4 PET see processing efficiency theory phonological loop 108 –109, 128

physiological arousal 180 –181 physiological measures 29, 42–57, 215 –216 PISA see Programme for Internal Student Assessment positive attitudes, toward mathematics 66, 82 , 86 – 88, 217–218 positive stereotypes 183 –185, 199 prejudice 176 pressure 93 –94 problem-answer associations 84 problem-solving 109 –110 processing efficiency theory (PET) 5, 128, 129, 132 Programme for Internal Student Assessment (PISA) 23, 77, 142 , 190 psychophysiological correlations 42 –57 reaction time (RT) assessments 28 –29 Reading Span task (RSPAN) 46 reduced competency 7–9 Reformed Teacher Observation Protocol (RTOP) 196 remedial mathematics 15 resilience 149 ridicule 82 ruminations 7– 8, 63, 143, 156, 157, 168, 180, 186, 215, 216, 218 scaffolding 205, 206 Scale for Assessing Math Anxiety in Secondary education (SAMAS) 25 Scale for Early Mathematics Anxiety (SEMA) 25, 54 self-concept 67, 132 , 149, 214, 218 self-confidence 9, 66, 159 self-efficacy 67, 82 – 83, 141, 214 –215, 218 self-esteem 78 self-image 82 self-math overlap 33 self-monitoring 179 self-ratings 66 – 67, 214 self-report measures 22 –28, 42 , 84 – 85, 159 –160 shame 79, 88 shifting 128 shyness 78 Single Item Math Anxiety Scale (SIMA) 23, 24 situational stress 130 size effect 107 skin conductance 45, 47, 47, 55 sMARS 2 , 21, 22 –23, 46 social cognitive theory 82 – 83 social identity 183 –184

Index

societal impacts, of math anxiety 103, 141, 211–212 sociocultural construct, math anxiety as 10 –11 socioeconomic status 159 sociology 176 standardized tests 15 state anxiety 32 , 127–128, 137, 157, 161 STEM fields 199, 202 , 207 stereotypes 198 –200 stereotype threat 12 , 178 –186, 199 –200 stress: autonomic response to 44; emotional 142; performance under 112 , 184, 216; situational 130 stress research 29 Stroop task 8, 45, 50 –51, 56n1 success 86 – 88 symbolic numerical representations 163 –166, 168 sympathetic nervous system 43, 44 task completion 83 Taylor Scale of Manifest Anxiety 2 , 22 teachers: best practices for 204 –205; female 198 –199; generalizable tools for 205 –206; inf luence of 90 –91, 94 –95, 117, 148, 162 , 166, 194 –195, 203 –204, 215, 217; math anxiety in 10, 190 –191, 194 –195, 198 –200, 203 –204, 207, 215; training for 217; value interventions and 206 –207 teaching methodologies 204 –206 tension 159 test anxiety 4, 5, 9, 21, 63, 70, 82; cognitive theories of 128 –129; definition of 127, 213; developmental

227

perspective on 133 –134; longitudinal change in 135 –137; math anxiety and 137, 212 –214; performance and 126, 129 –132 , 213; prevalence of 126; working memory and 129 –132 , 134 –135 threat detection 14 threat related attentional bias 157, 158, 161, 162 –166, 168 threat response 43 time pressure 29 trait anxiety 32 , 127–128, 131, 157, 161 trait-state theory 127–128 tutoring 116, 149, 216 –217 twin studies 13, 146, 147, 160 updating 128 utility value interventions 202 , 206 –207 visceral pain 14 visuospatial sketchpad 128 voxel 52 , 57n2 word problems 109 –110 working memory 5 – 8, 16, 46, 63 – 65, 68 – 69, 82 , 84; age-related changes in 136; cognitive theories of 129; continua model of 105, 106; math ability and 105 –110; math anxiety and 103 –118, 126 –137, 145, 156, 192 , 193; overview of 104 –105, 104, 128 –129; performance and 111–115, 129 –134, 192 , 193; test anxiety and 129 –132 , 134 –135; in young children 126 –137 worry 5, 7– 8, 62, 67, 71, 88, 127, 180 –181, 213