International Handbook of Research in Statistics Education [1 ed.] 9783319661933, 9783319661957

Citation preview

Springer International Handbooks of Education

Dani Ben-Zvi · Katie Makar Joan Garfield  Editors

International Handbook of Research in Statistics Education

Springer International Handbooks of Education

More information about this series at http://www.springer.com/series/6189

Dani Ben-Zvi • Katie Makar • Joan Garfield Editors

International Handbook of Research in Statistics Education

Editors Dani Ben-Zvi Faculty of Education The University of Haifa Haifa, Israel

Katie Makar School of Education University of Queensland St Lucia, QLD, Australia

Joan Garfield Department of Educational Psychology The University of Minnesota Minneapolis, MN, USA

ISSN 2197-1951     ISSN 2197-196X (electronic) Springer International Handbooks of Education ISBN 978-3-319-66193-3    ISBN 978-3-319-66195-7 (eBook) https://doi.org/10.1007/978-3-319-66195-7 Library of Congress Control Number: 2017959604 © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

This handbook is a collection of articles, grounded in science but equally dedicated to practice, on topics carefully chosen by the editors and written by leading authors in the field. At the same time, this book is much more than just a collection of articles. It is the realization of a decades-old vision for the future of a new subject, the flowering of seeds planted long ago and cultivated by generations of students and scholars. In the beginning, there was the soil and the seed. The soil was statistics, the science of learning from data, not a crop in itself, so much as an environment for raising a crop, using data to learn about a subject area. The seeds came from the subject area, in this case education, wanting to learn how we learn. When the seed found the right soil, there sprouted the new science of statistics education, driven by a commitment to teaching statistics well and led in the early years by such statistics textbooks as Snedecor and Cochran (1937), Hoel (1947), Hogg and Craig (1958), and Mosteller, Rourke, and Thomas (1961) and, more recently, in a second wave, by Freedman, Pisani, and Purvis (1978), Moore (1978), Anscombe (1981), Moore and McCabe (1989), and Scheaffer, Watkins, Witmer, and Gnanadesikan (1996). In parallel with the second wave, a new academic subject was born: using statistics to learn how to teach statistics. Like farming, teaching is a craft, not an innate talent you either have or don’t have, not a green thumb you are born with or without, but something you can learn from experience if you pay attention to data. Data from research in statistics education can and should guide how those of us who teach statistics can improve our craft of teaching statistics well. At the same time, we who teach statistics can and should guide those who use data for their study of how students learn. In that spirit, the authors who have written here have much to offer to those of us who teach the science of learning from data to use data to improve how we teach. Surely, statistics education stands at a junction of uniquely fruitful possibilities. The advance of statistics as a subject depends on the advance of statistics as a profession. The advance of statistics as a profession depends on the advance of teaching statistics as a calling. The advance of statistics teaching as a calling depends on teaching statistics as a craft, and the advance of teaching statistics as a v

vi

Foreword

craft depends on scientific research on how students learn statistics and how teachers who pay attention can best help their students learn. I conclude my preface with a concerning challenge and a brief for optimism. I start with my concerning challenge: a growing time lag. The cutting edge of statistics-the-subject is advanced by those who do the new research and teach it to their graduate students in their Ph.D. programs. But that reach is narrow—the person who teaches the newest ideas in genomics is not often the same person who teaches the latest in Markov chain Monte Carlo or mining of business data. The cutting edge of teaching statistics-the-subject is advanced mainly by the very few Ph.D. graduates from those research-oriented universities who choose to teach at the few liberal arts colleges where the lighter teaching load allows time for curricular innovation and carries an expectation of creating new courses. A cutting edge of research in statistics education, one edge among many, depends on those faculty from liberal arts colleges who write and give talks about curricular innovations that trickle down from the research universities. To exploit the pernicious metaphor of trickle down: innovations in statistics-the-subject are funded from the top, innovators in the K-16 curriculum get the fewer and smaller grants in statistics education, and the researchers in statistics education have all too often been consigned to feed off what’s left at the bottom. That was then. Fortunately, the growing emphasis on data science and assessment of effectiveness is changing these priorities. My point here is to illustrate a concern about a time lag. It takes time for the latest research to make its way into graduate courses, and thence to the undergraduate curriculum, and from there to articles and presentations that engage the attention of those education researchers who use data science to advance our understanding of how students learn data science. It’s not just the current time lag. I worry even more about a possible growing divergence between statistics-the-subject and statistics education research. Until recently, the overlap between statistics-the-subject and what education researchers learned about statistics as part of their Ph.D. programs could be regarded as the core of the subject. Now, statistics-the-subject expands rapidly in many directions. My concern: what should statistics education research regard as the core? Concern and challenge aside, I conclude with deep reasons for optimism. Research in statistics education is unique in that the target subject (statistics), whose teaching and learning is being studied, is at the same time the main conceptual and methodological approach to learning from the research data. Those who study the teaching and learning of chemistry do not rely primarily on molecules to understand how their students learn. Those who study the learning and teaching of astronomy or microbiology do not observe students through a telescope or microscope. Uniquely in statistics education, all three of the (1) subject itself, (2) those committed to teaching the subject well, and (3) those who use science to study the teaching and learning of the subject share a “common core,” the subject itself. Our history bears this out: there have been no “stat wars.” It is a deep pleasure to recommend this pioneering and peaceful volume to your attention. George W. Cobb Mount Holyoke College South Hadley, MA, USA

,

Foreword

vii

References Anscombe, F. J. (1981). Computing in statistical science through APL. New York: Springer. Freedman, D., Pisani, R., & Purves, R. (1978). Statistics (1st Ed.). New York, NY: W.W. Norton & Company. Hoel, P. G. (1947). Introduction to mathematical statistics (1st Ed.). Wiley. Hogg, R. V., & Craig, A. T. (1958). Introduction to mathematical statistics (1st Ed.). New York: Macmillan Publishing. Moore, D. S. (1978). Statistics: Concepts and controversies (1st Ed.). New York: W. H. Freeman and Company. Moore, D. S., & McCabe, G. P. (1989). Introduction to the practice of statistics (1st Ed.). New York: W. H. Freeman and Company. Mosteller, F., Rourke, R. E. K., & Thomas, G. B. (1961). Probability with statistical applications (1st Ed.). Reading, MA: Addison-Wesley. Scheaffer, R. L., Watkins, A., Witmer, J., & Gnanadesikan, M. (1996). Activity-based statistics: Instructor’s guide (1st Ed.). New York: Springer. Snedecor, G. W., & Cochran, W. G. (1937). Statistical methods (1st Ed.). Iowa University Press.

Preface

After many years of hard work and extensive collaboration, we are deeply honored to present to the statistics and mathematics education communities the first International Handbook of Research in Statistics Education. We have been working for several decades to bring visibility and credibility to this important discipline that supports the teaching and learning of statistics. This handbook not only reflects those efforts but is designed to further promote high-quality research and improvements in the teaching and learning of statistics. This book builds on our commitment over the past decade to explore ways to understand and develop students’ statistical literacy, reasoning, and thinking. Despite living and working in three different countries, we have collaborated together in person and via frequent Skype calls to produce a volume that reflects the current knowledge and ideas in statistics education. Initial conversations about the need for a handbook began at the 2011 gathering of the International Research Forum on Statistical Reasoning, Thinking and Literacy (SRTL) held in the Netherlands. Plans solidified and an initial editorial board meeting was held in conjunction with the 2013 SRTL meeting in the USA.  We created a structure that included three main sections of the book, each overseen by two coeditors. Together we shaped the scope and goals of a unique handbook that could provide a valuable foundation for educators and researchers. We are deeply indebted to the six editors who worked with us in helping this vision become a published book: Part I: Beth Chance and Allan Rossman Part II: Maxine Pfannkuch and Robert delMas Part III: Janet Ainley and Dave Pratt It has been a great experience to work on this book with such dedicated and topnotch editors and with the group of international scholars who collaborated on each chapter. Producing this book required ongoing reading, writing, discussing, and learning as well as a face-to-face meeting with many of the authors at ICOTS in the USA (July 12–13, 2014).

ix

x

Preface

We gratefully acknowledge the tremendously valuable assistance and feedback offered by all of the editors and chapter authors who reviewed the book chapters and from the entire SRTL community. We would also like to thank our colleagues in statistics education who acted as external reviewers for the chapters: Dor Abrahamson, Keren Aridor, Pip Arnold, Arthur Bakker, Stephanie Budgett, Gail Burrill, Helen Chick, Neville Davies, Adri Dierdorp, Andreas Eichler, Lyn English, Jill FieldingWells, Iddo Gal, Einat Gil, Randall Groth, Jennifer Kaplan, Sibel Kazak, Cliff Konold, Gillian Lancaster, Cindy Langrall, Aisling Leavy, Rich Lehrer, Marsha Lovett, Helen MacGillivray, Sandy Madden, Hana Manor Braham, Maria MeletiouMavrotheris, Jen Noll, Susan Peters, Robyn Pierce, Robyn Reaburn, Jackie Reid, Jim Ridgway, Luis Saldanha, Susanne Schnell, Mike Shaughnessy, Bob Stephenson, Jane Watson, Jeff Wilmer, Lucia Zapata-Cardona, and Andy Zieffler. It has been a positive and productive collaboration. It has also been a delight for the three of us to work together on this project, especially as one of us (Joan Garfield) retires from her faculty position. We are grateful to Springer Publishers, for providing a publishing venue for this book, and to Joseph Quatela, the editor who skillfully managed the publication on their behalf. Lastly, many thanks go to our spouses Hava Ben-Zvi, Sanjay Makar, and Michael Luxenberg and to our children—Noa, Nir, Dagan, and Michal Ben-Zvi, Keya Makar, and Harlan and Rebecca Luxenberg—as our primary sources of energy and support. Dani Ben-Zvi Katie Makar Joan Garfield

Haifa, Israel St. Lucia, QLD, Australia Minneapolis, MN, USA

Contents

Part I  Statistics, Statistics Education, and Statistics Education Research Beth Chance and Allan Rossman 1 What Is Statistics?������������������������������������������������������������������������������������    5 Christopher J. Wild, Jessica M. Utts, and Nicholas J. Horton 2 What Is Statistics Education? ����������������������������������������������������������������   37 Andrew Zieffler, Joan Garfield, and Elizabeth Fry 3 Statistics Education Research ����������������������������������������������������������������   71 Peter Petocz, Anna Reid, and Iddo Gal Part II  Major Contributions of Statistics Education Research Maxine Pfannkuch and Robert delMas 4 The Practice of Statistics ������������������������������������������������������������������������  105 Jane Watson, Noleine Fitzallen, Jill Fielding-Wells, and Sandra Madden 5 Reasoning About Data ����������������������������������������������������������������������������  139 Rolf Biehler, Daniel Frischemeier, Chris Reading, and J. Michael Shaughnessy 6 Research on Uncertainty ������������������������������������������������������������������������  193 Dave Pratt and Sibel Kazak 7 Introducing Children to Modeling Variability��������������������������������������  229 Richard Lehrer and Lyn English 8 Learning About Statistical Inference ����������������������������������������������������  261 Katie Makar and Andee Rubin 9 Statistics Learning Trajectories��������������������������������������������������������������  295 Pip Arnold, Jere Confrey, Ryan Seth Jones, Hollylynne S. Lee, and Maxine Pfannkuch xi

xii

Contents

10 Research on Statistics Teachers’ Cognitive and Affective Characteristics������������������������������������������������������������������  327 Randall Groth and Maria Meletiou-Mavrotheris Part III Contemporary Issues and Emerging Directions in Research on Learning and Teaching Statistics Janet Ainley and Dave Pratt 11 The Nature and Use of Theories in Statistics Education����������������������  359 Per Nilsson, Maike Schindler, and Arthur Bakker 12 Reimagining Curriculum Approaches ��������������������������������������������������  387 Maxine Pfannkuch 13 Challenge to the Established Curriculum: A Collection of Reflections����������������������������������������������������������������������  415 Robert Gould, Roger D. Peng, Frauke Kreuter, Randall Pruim, Jeff Witmer, and George W. Cobb 14 Building Capacity in Statistics Teacher Education������������������������������  433 João Pedro da Ponte and Jennifer Noll 15 Revolutions in Teaching and Learning Statistics: A Collection of Reflections����������������������������������������������������������������������  457 Robert Gould, Christopher J. Wild, James Baglin, Amelia McNamara, Jim Ridgway, and Kevin McConway 16 Design of Statistics Learning Environments������������������������������������������  473 Dani Ben-Zvi, Koeno Gravemeijer, and Janet Ainley Index�������������������������������������������������������������������������������������������������������������������� 503

Contributors

Janet Ainley  School of Education, University of Leicester, Leicester, UK Pip Arnold  Cognition Education, Auckland, New Zealand James Baglin  RMIT University, Melbourne, VIC, Australia Arthur Bakker  Freudenthal Institute, Utrecht University, Utrecht, The Netherlands Dani Ben-Zvi  Faculty of Education, The University of Haifa, Haifa, Israel Rolf Biehler  Institute of Mathematics, University of Paderborn, Paderborn, Germany Beth  Chance  Department of Statistics, California Polytechnic State University, San Luis Obispo, CA, USA George W. Cobb  Mt. Holyoke College, Hadley, MA, USA Jere Confrey  College of Education, North Carolina State University, Raleigh, NC, USA João  Pedro  da Ponte  Instituto de Educação, Universidade de Lisboa, Lisboa, Portugal Robert delMas  Department of Educational Psychology, University of Minnesota, Minneapolis, MN, USA Lyn  English  Faculty of Education, Queensland University of Technology, Brisbane, QLD, Australia Jill Fielding-Wells  School of Education, The University of Queensland, Brisbane, QLD, Australia Noleine  Fitzallen  Faculty of Education, University of Tasmania, Hobart, TAS, Australia Daniel  Frischemeier  Institute of Mathematics, University of Paderborn, Paderborn, Germany xiii

xiv

Contributors

Elizabeth Fry  Department of Educational Psychology, University of Minnesota, Minneapolis, MN, USA Iddo Gal  Department of Human Services, The University of Haifa, Haifa, Israel Joan  Garfield  Department of Educational Psychology, The University of Minnesota, Minneapolis, MN, USA Robert Gould  Department of Statistics, University of California, Los Angeles, CA, USA Koeno  Gravemeijer  Eindhoven Netherlands

University

of

Technology,

Eindhoven,

Randall Groth  Education Specialty Department, Salisbury University, Salisbury, MD, USA Nicholas J. Horton  Department of Mathematics and Statistics, Amherst College, Amherst, MA, USA Ryan  Seth  Jones  College of Education, Middle Tennessee State University, Murfreesboro, TN, USA Sibel Kazak  Faculty of Education, Pamukkale University, Denizli, Turkey Frauke Kreuter  University of Maryland, College Park, MD, USA University of Mannheim, Mannheim, Germany Hollylynne S. Lee  College of Education, North Carolina State University, Raleigh, NC, USA Richard  Lehrer  Department of Teaching and Learning, Peabody College, Vanderbilt University, Nashville, TN, USA Sandra Madden  Teacher Education and Curriculum Studies Department, University of Massachusetts Amherst, Amherst, MA, USA Katie Makar  School of Education, The University of Queensland, Brisbane, QLD, Australia Kevin  McConway  School of Mathematics and Statistics, The Open University, Milton Keynes, UK Amelia McNamara  Smith College, Northampton, MA, USA Maria  Meletiou-Mavrotheris  School of Humanities, Social and Education Sciences, European University Cyprus, Egkomi, Cyprus Per Nilsson  School of Science and Technology, Örebro University, Örebro, Sweden Jennifer Noll  Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR, USA

Contributors

xv

Roger D. Peng  Department of Biostatistics, Johns Hopkins University, Baltimore, MD, USA Peter Petocz  Department of Statistics, Macquarie University, North Ryde, NSW, Australia Maxine Pfannkuch  Department of Statistics, The University of Auckland, Auckland, New Zealand Dave Pratt  Institute of Education, University College London, London, UK Randall Pruim  Department of Mathematics and Statistics, Calvin College, Grand Rapids, MI, USA Chris Reading  SiMERR National Research Centre, University of New England, Armidale, NSW, Australia Anna Reid  Sydney Conservatorium of Music, University of Sydney, Sydney, NSW, Australia Jim Ridgway  School of Education, Durham University, Durham, UK Allan Rossman  Department of Statistics, California Polytechnic State University, San Luis Obispo, CA, USA Andee Rubin  TERC, Cambridge, MA, USA Maike  Schindler  Faculty of Human Sciences, University of Cologne, Cologne, Germany J. Michael Shaughnessy  The Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR, USA Jessica M. Utts  Department of Statistics, University of California—Irvine, Irvine, CA, USA Jane Watson  Faculty of Education, University of Tasmania, Hobart, TAS, Australia Christopher  J.  Wild  Department of Statistics, The University of Auckland, Auckland, New Zealand Jeff Witmer  College of Arts and Sciences, Oberlin College, Oberlin, OH, USA Andrew  Zieffler  Department of Educational Psychology, The University of Minnesota, Minneapolis, MN, USA

Introduction

Statistics education has come of age. This unique discipline has emerged from a diverse set of foundations. Over the past 50 years, it has grown and developed its own identity. Every discipline needs a recognized body of research to establish its credibility as a legitimate field of knowledge and study. It is now time for this relatively new field of statistics education to have a research handbook that provides a collection and synthesis of the body of knowledge that supports the teaching and learning of statistics. Statistics has become one of the most central topics of study in the modern world of information and big data. The dramatic increase in demand for learning statistics in all disciplines is accompanied by tremendous growth in research in statistics education. Increasingly, educators at all levels are teaching more topics and courses in quantitative reasoning, data analysis, and data science at lower and lower grade levels within mathematics and science and across other content areas. However, despite the growth in statistics education, research has continually revealed many challenges in helping learners develop statistical literacy, reasoning, and thinking. New curricula and technology tools show promise in facilitating the achievement of these desired outcomes. New research in the field can critically inform college instructors, classroom teachers, curriculum designers, researchers in mathematics and statistics education, policymakers, and newcomers to the field of statistics education. This International Handbook of Research in Statistics Education aims to provide a solid foundation for such studies. Like statistics itself, statistics education research is by nature interdisciplinary, with its practices and principles developed from many different fields. In addition, current problems and methods of statistical practice in the changing world need to be shared with educators who are teaching today’s students. We see the foundations of the knowledge and research findings presented in this handbook as based primarily on three areas of work that can be represented by the contributions of three extraordinary individuals who have made major contributions to knowledge. They are John Tukey (1915–2000), Amos Tversky (1937–1996), and Jean Piaget (1886–1980). xvii

xviii

Introduction

Tukey helped move the practice of statistics into a new era of exploring data. In the 1970s, the reinterpretation of statistics into separate practices comprising exploratory data analysis (EDA) and confirmatory data analysis (CDA, inferential statistics) (Tukey, 1977) freed certain kinds of data analysis from ties to probabilitybased models, so that the analysis of data could begin to acquire status as an independent intellectual activity. The introduction of simple data tools, such as stem and leaf plots and boxplots, paved the way for students at all levels to analyze real data interactively without having to spend hours on the underlying theory, calculations, and complicated procedures. The work of Tukey and his colleagues was spread to students at all levels and led to new curriculum at the primary and secondary school. Computers and new pedagogies would later complete the “data revolution” in statistics education. Tversky studied and enlightened the world about the ways people think and reason. He documented how often people misunderstand randomness and probability, leading them to use faulty heuristics when reasoning about samples. The work of Tversky and his colleagues (e.g., Kahneman, Slovic, & Tversky, 1982) has led to recognition of the new ways to build learning trajectories on existing foundations, of challenging faulty heuristics and biases through simulations and visual explorations, and of carefully assessing reasoning and thinking. Finally, the work of Piaget (e.g., Inhelder & Piaget, 1958; Piaget & Inhelder, 1962) provided models of how to carefully study individual children as they understand the world, as well as how they reason about chance and probability (Piaget & Inhelder, 1975). His methods for studying students in depth have paved the way for current researchers who carefully observe and study the thinking of children, including how they think about inference and chance events. Although their work was developed in the last century, we believe that much of the current research in statistics education has been built on the insights and knowledge that these three brilliant and creative thinkers provided—and statistics education may be unique in connecting the fruits of their studies. Research in statistics education includes studies of how people think about data and chance, the errors they systematically make that affect their inferences and judgments, the use and impact of new tools and learning environments, and the use of rich, qualitative data through observation interviews and teaching experiments. The work in this volume represents a collaboration amid a diverse set of professionals, including leading educators, researchers, and statisticians from around the world. Our goal was to provide a resource that connects the practice of statistics to the teaching and learning of the subject in light of current and future challenges. The chapter authors and part editors contributed in the development, writing, and editing of this book. Just as the discipline of statistics education is built on diverse scientific areas, the writers of these chapters come from the departments of statistics, mathematics education, psychology, and technology education and, in some cases, new programs in statistics education. We have organized the book into three main parts that encompass the breadth and depth of research on teaching and learning statistics across educational levels and settings. As such, the authors and editors strove to work in collaboration to link the main sections of the book with the diversity of ideas articulated throughout this handbook.

Introduction

xix

Part I: Statistics, Statistics Education, and Statistics Education Research Part I of this handbook describes the interplay among the disciplines of statistics, statistics education, and statistics education research. Part II: Major Contributions of Statistics Education Research This part focuses on major contributions of statistics education research that relate to teaching, learning, and understanding statistics. It includes summaries of classic work as well as current work to show the progress and contrasting perspectives on main themes. Gaps in the research knowledge base are also identified. Part III: Contemporary Issues and Emerging Directions The focus of Part III of the handbook is on looking forward and examining emerging areas of statistics education research and their implications. Much of this section discusses more theoretical than empirical findings as the topics often have little research published so far or may be anticipated to be “on the horizon”. The collaboration that led to the production of this handbook aims to provide a resource that can be utilized by all people interested in the latest international research on teaching and learning statistics. We are proud to be part of this new discipline that has come of age, and we look forward to seeing new advances in the teaching and learning of statistics to students at all levels. Dani Ben-Zvi The University of Haifa, Haifa, Israel Katie Makar The University of Queensland, St Lucia QLD, Australia Joan Garfield The University of Minnesota, Minneapolis MN, USA

References Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. New York: Basic Books. Kahneman, D., Slovic, P., Tversky, A. (1982). Judgment under uncertainty: Heuristics and biases. Cambridge, England: Cambridge University Press. Piaget, J., & Inhelder, B. (1962). The psychology of the child. New York: Basic Books. Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children. UK: Routledge & Kegan Paul Ltd. Tukey, J. W. (1977). Exploratory data analysis. Reading, MA: Addison-Wesley.

Part I

Statistics, Statistics Education, and Statistics Education Research Beth Chance and Allan Rossman

Introduction to Part I Our goal with Part I of the International Handbook of Research in Statistics Education is to set the stage for the articles and topics that form the bulk of the collection. We do this with three chapters that overview the history, core components, and future of the fields of statistics, statistics education, and statistics education research. In other words, Part I provides a careful examination of the interplay of the last three words of the handbook’s title. Consistently enough, all three chapters are written by a team of three coauthors, experienced and prominent statisticians, statistics educators, and statistics education researchers representing four countries. The goal of Part I is to give the reader an understanding of how statistics education has developed and grown into a discipline of its own, with an eye to future needs and research questions. We hope that this part of the handbook will help to establish common ground and encourage interaction among three groups that are integral to statistics education: statisticians, teachers of statistics, and statistics education researchers. These groups have much to offer each other. For example, teachers can improve their students’ learning of statistics not only by understanding the discipline but also by applying the findings of education researchers and adapting them for effective teaching. Similarly, education researchers can engage in more meaningful and impactful research studies by not only knowing the discipline but also noting practices of effective teachers. We fear that even when members of these groups attend the same conference, such as ICOTS (the International Conference on Teaching Statistics), they may tend to focus on their own sessions and not engage with members of other groups; we therefore hope that the three chapters in this part lead to increased communication and consultation. Data scientists comprise a fourth group to be invited to the conversation, and these chapters touch on the role of data science in statistics education and its future. To help understand the nature of statistics education research, we believe it is important to first understand the nature of statistics itself and how it differs from

2

B. Chance and A. Rossman

other academic disciplines, which helps inform what should be taught and how. To address this very broad topic in the opening chapter, we turned to three accomplished statisticians who have also been very involved with curriculum development and other statistics education efforts at tertiary and secondary levels. Chris Wild has served as president of the International Association for Statistics Education and has provided substantial input into the K-12 statistics curriculum for New Zealand. Jessica Utts has been president of the American Statistical Association and has written well-received textbooks for introductory statistics at the undergraduate level. Nick Horton chaired the ASA committee that revised guidelines for undergraduate programs in statistics and also served on the committee that revised the GAISE (Guidelines for Assessment and Instruction in Statistics Education) report for introductory undergraduate statistics. All three authors have played a role in identifying the needs and agenda for statistics education for informed consumers of quantitative information. The first chapter, written by Wild, Utts, and Horton, discusses the nature of statistics as a scientific discipline, recounting some of its history and identifying ­characteristics that exemplify statistical thinking and draw distinctions with mathematical thinking. The chapter also considers why statistics is important and relevant in a wide variety of professional fields as well as for all educated citizens. Several examples highlight how statistical thinking is relevant in everyday life and necessary for advancing knowledge. The chapter concludes with discussion of statistics as a still-­evolving and vibrant discipline in our contemporary world where data abound. Noting the growth of statistics as an academic major and as a career choice, along with the emergence of data science as a closely related field and popular career option, the final section dares to make some predictions about where the discipline of statistics is heading in the coming decades. Having addressed the question of what is statistics, the second chapter moves on to consider the question: What is Statistics Education? The three authors share an association with the first doctoral program in the USA to specialize in statistics education. Joan Garfield is one of the leading statistics education researchers of the past 25 years, whose research and writings have had a substantial impact on the teaching and learning of statistics at all levels. Joan played a leading role in developing and shaping the Ph.D. program in Statistics Education at the University of Minnesota. Two of her former students in that program have joined her in authoring Chap. 2. Andy Zieffler was Joan’s first student in this program and has gone on to lead curriculum development and other statistics education projects. Elizabeth Fry is one of Joan’s most recent students, whose research focuses on developing high-­ quality assessments of curricular innovation in statistics education. Chapter 2, written by Zieffler, Garfield, and Fry, begins by providing a brief history about the teaching and learning of statistics, first grounded in mathematics and science education and later broadened to include the teaching of statistics for all students at all levels. The authors identify milestones that led to statistics education establishing itself as a viable academic discipline, separate from statistics and from mathematics education. These milestones include the establishment of various professional associations, conferences, and journals devoted exclusively to statistics

Part I  Statistics, Statistics Education, and Statistics Education Research

3

education. Statistics education encompasses a wide breadth of aspects and topics that will appear in later chapters in this volume. Zieffler et al. discuss several of these, including cognitive and noncognitive instructional goals, pedagogical approaches, use of technology, teacher preparation, and graduate programs in statistics education. As much as possible in a single chapter, these issues are discussed for various education levels and from an international perspective. As statistics education has evolved and distinguished itself, so has the related research literature. The final chapter in Part I provides perspectives on different forms of that research, and how that research has been positively influenced from several converging disciplines. The author team includes two recent editors of the Statistics Education Research Journal: Peter Petocz and Iddo Gal, and a third expert in both qualitative and quantitative approaches to education research: Anna Reid. As one piece of this chapter, Petocz, Reid, and Gal performed their own research study to examine the current landscape of statistics education research by conducting a qualitative study of articles published in various outlets for statistics education research in the past few years. Their goals are to describe who performs this kind of research, what kinds of questions are being addressed with this research, and the methods and conceptual schemes used to conduct the research. Petocz et al. distinguish between what they refer to as small-r and large-R research, the former addressing local problems in a particular context and the latter investigating larger issues that generalize more broadly. After analyzing the results of their analysis, they conclude the chapter by suggesting some directions for future development of the discipline of statistics education. This first part of the handbook addresses very far-reaching themes as it strives to provide a context for the more focused chapters that comprise the other parts of the handbook. We expect these opening three chapters to provide the reader with valuable perspectives on the challenging questions of what constitutes statistics, statistics education, and statistics education research. We trust that these chapters will succeed in whetting the reader’s appetite to discover the research findings in statistics education described in remaining handbook chapters, understand their implications, and look towards the future of statistics education.

Chapter 1

What Is Statistics? Christopher J. Wild, Jessica M. Utts, and Nicholas J. Horton

Abstract  What is statistics? We attempt to answer this question as it relates to grounding research in statistics education. We discuss the nature of statistics as the science of learning from data, its history and traditions, what characterizes statistical thinking and how it differs from mathematics, connections with computing and data science, why learning statistics is essential, and what is most important. Finally, we attempt to gaze into the future, drawing upon what is known about the fast-­ growing demand for statistical skills and the portents of where the discipline is heading, especially those arising from data science and the promises and problems of big data. Keywords Discipline of statistics • Statistical thinking • Value of statistics • Statistical fundamentals • Decision-making • Trends in statistical practice • Data science • Computational thinking

1.1  Introduction In this, the opening chapter of the International Handbook on Research in Statistics Education, we ask the question, “What is statistics?” This question is not considered in isolation, however, but in the context of grounding research in statistics education. Educational endeavor in statistics can be divided very broadly into “What?” and “How?” The “What?” deals with the nature and extent of the discipline to be conveyed, whereas any consideration of “How?” (including “When?”) brings into C.J. Wild (*) Department of Statistics, The University of Auckland, Auckland, New Zealand e-mail: [email protected] J.M. Utts Department of Statistics, University of California—Irvine, Irvine, CA, USA e-mail: [email protected] N.J. Horton Department of Mathematics and Statistics, Amherst College, Amherst, MA, USA e-mail: [email protected] © Springer International Publishing AG 2018 D. Ben-Zvi et al. (eds.), International Handbook of Research in Statistics Education, Springer International Handbooks of Education, https://doi.org/10.1007/978-3-319-66195-7_1

5

6

C.J. Wild et al.

play a host of additional factors, such as cognition and learning theories, audience and readiness, attitudes, cultural issues, social interactions with teachers and other students, and learning, teaching and assessment strategies and systems. This chapter discusses the nature of statistics as it relates to the teaching of statistics. The chapter has three main sections. Section 1.2 discusses the nature of statistics, including a brief historical overview and discussion of statistical thinking and differences with mathematics. Section 1.3 follows this up with a discussion of why learning statistics is important. Section 1.4 concludes the chapter with a discussion of the growing demand for statistical skills and a look at where the discipline is heading. A thread pervading the chapter is changing conceptions of the nature of statistics over time with an increasing emphasis recently on broad, as opposed to narrow, conceptions of what statistics is. We emphasize broader conceptions because we believe they best address emerging areas of need and because we do not want researchers to feel constrained when it comes to deciding what constitutes fair game as targets for research.

1.2  The Nature of Statistics “Statistics”—as defined by the American Statistical Association (ASA)—“is the science of learning from data, and of measuring, controlling and communicating uncertainty.” Although not every statistician would agree with this description, it is an inclusive starting point with a solid pedigree. It encompasses and concisely encapsulates the “wider view” of Marquardt (1987) and Wild (1994), the “greater statistics” of Chambers (1993), the “wider field” of Bartholomew (1995), the broader vision advocated by Brown and Kass (2009), and the sets of definitions given in opening pages of Hahn and Doganaksoy (2012) and Fienberg (2014). Figure 1.1 gives a model of the statistical-inquiry cycle from Wild and Pfannkuch (1999). This partial, rudimentary “map” hints at the diversity of domains that contribute to “learning from data.” The ASA description of statistics given above covers all elements seen in this diagram and more. Although statisticians have wrestled with every aspect of this cycle, particular attention has been given by statistical theory-and-methods thinkers and researchers to different elements at different times. For at least the last half century, the main focus has been on the use of probabilistic models in the analysis and conclusion stages and, to a lesser extent, on sampling designs and experimental designs in the plan stage. But a wider view is needed to chart the way of statistics education into the future. The disciplines of statistics and, more specifically, statistics education are, by their very nature, in the “future” business. The mission of statistics education is to provide conceptual frameworks (structured ways of thinking) and practical skills to better equip our students for their future lives in a fast-changing world. Because the data universe is expanding and changing so fast, educators need to focus more on looking forward than looking back. We must also look back, of course, but predominantly so that we can plunder our history’s storehouses of wisdom to better chart

1  What Is Statistics?

7

Fig. 1.1  The statistical-inquiry cycle

pathways into the future. For educational purposes, statistics needs to be defined by the ends it pursues rather than the means statisticians have most often used to pursue them in the past. Changing capabilities, like those provided by advancing ­technology, can change the preferred means for pursuing goals over time, but the fundamental goals themselves will remain the same. The big-picture definition that we opened with “keeps our eyes on the ball” by placing at the center of our universe the fundamental human need to be able to learn about how our world operates using data, all the while acknowledging sources and levels of uncertainty: Statisticians develop new methodologies in the context of a specific substantive problem, but they also step back and integrate what they have learned into a more general framework using statistical principles and thinking. Then, they can carry their ideas into new areas and apply variations in innovative ways. (Fienberg, 2014, p. 6)

At their core, most disciplines think and learn about some particular aspects of life and the world, be it the physical nature of the universe, living organisms, or how economies or societies function. Statistics is a meta-discipline in that it thinks about how to think about turning data into real-world insights. Statistics as a meta-­ discipline advances when the methodological lessons and principles from a particular piece of work are abstracted and incorporated into a theoretical scaffold that enables them to be used on many other problems in many other places.

8

C.J. Wild et al.

1.2.1  History of Statistics This section outlines the evolution of major threads that became interwoven to make statistics what it is today, forming the basis of the ways in which we gather, think about, and learn using data. These threads include the realization of the need for data, randomness and probability as a foundation for statistical modeling and dealing with uncertainty, theories underpinning principled approaches to data collection and analysis, and graphics for exploration and presentation of messages in data. Although the collection of forms of census data goes back into antiquity, rulers “were interested in keeping track of their people, money and key events (such as wars and the flooding of the Nile) but little else in the way of quantitative assessment of the world at large” (Scheaffer, 2001, para. 3). The statistical analysis of data is usually traced back to the work of John Graunt (e.g., his 1662 book Natural and Political Observations). For example, Graunt concluded that the plague was caused by person-to-person infection rather than the competing theory of “infectious air” based on the pattern of infections through time. Graunt and other “political arithmeticians” from across Western Europe were influenced during the Renaissance by the rise of science based on observation of the natural world. And they “thought as we think today … they reasoned about their data” (Kendall, 1960, p. 448). They estimated, predicted, and learned from the data—they did not just describe or collect facts—and they promoted the notion that state policy should be informed by the use of data rather than by the authority of church and nobility (Porter, 1986). But the political arithmetician’s uses of statistics lacked formal methodological techniques for gathering and analyzing data. Methods for sample surveys and census taking were in their infancy well into the nineteenth century (Fienberg, 2014). Another fundamental thread involved in building modern statistics was the foundation of theories of probability, as laid down by Pascal (1623–1662) and later Bernoulli (1654–1705), which were developed to understand games of chance. The big conceptual steps from that toward the application of probability to inferences from data were taken by Bayes in 1764 and Laplace (1749–1827) by inverting probability analyses, i.e., using knowledge about probabilities of events or data given parameters to make inferences about parameters given data: The science that held sway above all others around 1800 was astronomy, and the great mathematicians of the day made their scientific contributions in that area. Legendre (least squares), Gauss (normal theory of errors), and Laplace (least squares and the central limit theorem) all were motivated by problems in astronomy. (Scheaffer, 2001, para. 6)

These ideas were later applied to social data by Quetelet (1796–1874), who with notions such as the “average man” was trying to arrive at general laws governing human action, analogous to the laws of physics. This was after the French Revolution when there was a subtle shift in thinking of statistics as a science of the state with the statists, as they were known, conducting surveys of trade, industrial progress, labor, poverty, education, sanitation, and crime (Porter, 1986). Another thread in the development of statistics involves statistical graphics (see Friendly, 2008). The first major figure is William Playfair (1759−1823), credited

1  What Is Statistics?

9

with inventing line charts, bar charts, and the pie chart. Friendly (2008) characterizes the period from 1850 to 1900 as the “golden age of statistical graphics” (p. 2). This is the era of John Snow’s famous “dot map” in which he plotted the locations of cholera deaths as dots on a map which then implicated water from the Broad Street pump as a likely cause; of Minard’s famous graph showing losses of soldiers in Napoleon’s march on Moscow and subsequent retreat; of Florence Nightingale’s coxcomb plot used to persuade of the need for better military field hospitals; and of the advent of most of the graphic forms we still use for conveying geographically linked information on maps, including such things as flow diagrams of traffic patterns, of grids of related graphs, of contour plots of three-dimensional tables, population pyramids, scatterplots, and many more. The Royal Statistical Society began in 1834 as the London Statistical Society (LSS), and the American Statistical Association was formed in 1839 by five men interested in improving the US census (Horton, 2015; Utts, 2015b). Influential founders of the LSS (Pullinger, 2014, pp.  825−827) included Adolphe Quetelet, Charles Babbage (inventor of the computer), and Thomas Malthus (famous for his theories about population growth). The first female LSS member was Florence Nightingale, who joined in 1858 (she also became a member of the ASA, as did Alexander Graham Bell, Herman Hollerith, Andrew Carnegie, and Martin Van Buren). These early members of LSS and ASA were remarkable for representing such a very wide variety of real-world areas of activity (scientific, economic, political, and social) and their influence in society: Near the end of the nineteenth century, the roots of a theory of statistics emerge from the work of Francis Galton and Francis Ysidro Edgeworth and from that of Karl Pearson and George Udny Yule somewhat later. These scientists came to statistics from biology, economics, and social science more broadly, and they developed more formal statistical methods that could be used not just within their fields of interest but across the spectrum of the sciences. (Fienberg, 2014, p. 4)

Another wave of activity into the 1920s was initiated by the concerns of William Gosset, reaching its culmination in the insights of Ronald Fisher with the development of experimental design, analysis of variance, maximum likelihood estimation, and refinement of significance testing. This was followed by the collaboration of Egon Pearson and Jerzy Neyman in the 1930s, giving rise to hypothesis testing and confidence intervals. At about the same time came Bruno de Finetti’s seminal work on subjective Bayesian inference and Harold Jeffreys’s work on “objective” Bayesian inference so that by 1940 we had most of the basics of the theories of the “modern statistics” of the twentieth century. World War II was also a time of great progress as a result of drafting many young, mathematically gifted people into positions where they had to find timely answers to problems related to the war effort. Many of them stayed in the field of statistics swelling the profession. We also draw particular attention to John Tukey’s introduction of “exploratory data analysis” in the 1970s; this is an approach to data analysis that involves applying a variety of exploratory techniques, many of them visual, to gain insight into a dataset and uncover underlying structure and exceptions.

10

C.J. Wild et al.

Short histories of statistics include Fienberg (2014, Section 3); Scheaffer (2001), who emphasized how mathematicians were funded or employed and the influence this had on what they thought about and developed; and Pfannkuch and Wild (2004) who described the development of statistical thinking. Lengthier accounts are given by Fienberg (1992) and the books by Porter (1986), Stigler (1986, 2016), and Hacking (1990). Key references about the history of statistics education include Vere-Jones (1995), Scheaffer (2001), Holmes (2003), and Forbes (2014); see also Chap. 2. The current scope and intellectual content of statistics is the result of evolutionary processes involving both slow progress and leaps forward due to the insights of intellectual giants and visionaries, all affected by the intellectual climate of their day and the recognized challenges in the world in which they lived. But it has not reached some fixed and final state. It continues to evolve and grow in response to new challenges and opportunities in the changing environment in which we now live.

1.2.2  Statistical Thinking Statisticians need to be able to think in several ways: statistically, mathematically, and computationally. The thinking modes used in data analysis differ from those used in working with mathematical derivations, which in turn differ from those used for writing computational code. Although there are very strong internal connections within each of these thinking modes, there are relatively weak connections among them. Here we will concentrate on “statistical thinking” in the sense of the most distinctively statistical parts of the thinking that goes on in solving real-world problems using data. In statistics, however, we sometimes talk about “solving real-world (or practical) problems” far too loosely. For the general public, “solving a real-world problem” involves taking action so that the problem either goes away or is at least reduced (e.g., unemployment levels are reduced). We need to better distinguish between satisfying “a need to act” and “a need to know.” Figuring out how to act to solve a problem will typically require acquiring more knowledge. This is where statistical inquiry can be useful. It addresses “a need to know.” So when statisticians talk about solving a real-world problem, we are generally talking about solving a (real-world) knowledge-deficit or understanding-deficit problem. 1.2.2.1  Statistical Thinking in Statistical Inquiry Wild and Pfannkuch (1999) investigated the nature of statistical thinking in this sense using available literature, interviews with practicing statisticians, and interviews with students performing statistical-inquiry activities and presented models for different “dimensions” of statistical thinking.

1  What Is Statistics?

11

Dimension 1 in Wild and Pfannkuch’s description of statistical thinking is the PPDAC model (Fig. 1.1) of the inquiry cycle. The basic PPDAC model was due to and later published by MacKay and Oldford (2000). There are also other essentially equivalent descriptions of the statistical-inquiry cycle. The inquiry cycle has connections with standard descriptions of the scientific method but is more flexible, omitting the latter’s strong emphasis on being hypothesis driven and having (scientific) theory formulation as its ultimate objective. The PPDAC inquiry cycle reminds us of the major steps involved in carrying out a statistical inquiry. It is the setting in which statistical thinking takes place. The initial “P” in PPDAC spotlights the problem (or question) crystallization phase. In the early stages, the problem is typically poorly defined. People start with very vague ideas about what their problems are, what they need to understand, and why. The problem step is about trying to turn these vague feelings into much more precise informational goals, some very specific questions that should be able to be answered using data. Arriving at useful questions that can realistically be answered using statistical data always involves a lot of hard thinking and often a lot of hard preparatory work. Statistics education research says little about this, but the PhD thesis of Arnold (2013) makes a very good start. The plan step is then about deciding what people/objects/entities to obtain data on, what things we should “measure,” and how we are going to do all of this. The data step is about data acquisition, storage, and wrangling (reorganizing the data using various transformations, merging data from different sources, and cleansing the data in preparation for analysis). The analysis step which follows and the conclusions step are about making sense of it all and then abstracting and communicating what has been learned. There is always a back-and-forth involving doing analysis, tentatively forming conclusions, and doing more analysis. In fact there is back-and-forth between the major steps whenever something new gets learned in a subsequent step that leads to modifying an earlier decision (Konold & Pollatsek, 2002). Any substantial learning from data involves extrapolating from what you can see in the data you have to how it might relate to some wider universe. PPDAC focuses on data gathered for a purpose using planned processes, processes that are chosen on statistical grounds to justify certain types of extrapolation. Much of the current buzz about the widespread availability and potential of data (including “big data”) relates to exploiting opportunistic (happenstance or “found”) data—data that just happen to be available in electronic form because they have accumulated for other reasons, such as the result of the administrative processes of business or government, audit trails of internet activity, or billing data from medical procedures: In a very real sense, we have walked into the theatre half way through the movie and have then to pick up the story. … For opportunistic data there is no extrapolation that is justified by a data-collection process specifically designed to facilitate that extrapolation. The best we can do is to try to forensically reconstruct what this data is and how it came to be (its ‘provenance’). What entities were ‘measures’ taken on? What measures have been employed and how? By what processes did some things get to be recorded and others not? What distortions might this cause? It is all about trying to gauge the extent to which we can

12

C.J. Wild et al. generalize from patterns in the data to the way we think it will be in populations or processes that we care about. (Wild, 2017, p. 34)

In particular, we are on the lookout for biases that could lead us to false conclusions. Dimension 2 of Wild and Pfannkuch’s model lists types of thinking, broken down into general types and types fundamental to statistics. The general types are strategic, seeking explanations, constructing and using models, and applying techniques (solving problems by mapping them on to problem archetypes). The types fundamental to statistics listed are recognition of the need for data, transnumeration (changing data representations in search of those that trigger understanding), consideration of variation and its sources, reasoning using statistical models, and integrating the statistical and the contextual (information, knowledge, conceptions). Something that is not highlighted here is the inductive nature of statistical inference—extrapolation from data on a part to reach conclusions about a whole (wider reality). Dimension 3 is the interrogative cycle, a continually operating high-frequency cycle of generating (possible informational requirements, explanations, or plans of attack), seeking (information and ideas), interpreting these, criticizing them against reference points, and judging whether to accept, reject, or tentatively entertain them. Grolemund and Wickham (2014) dig much deeper into this dimension bringing in important ideas from the cognitive literature. Dimension 4 consists of a list of personal qualities, or dispositions, successful practitioners bring to their problem solving: skepticism, imagination, curiosity and awareness, a propensity to seek deeper meaning, being logical, engagement, and perseverance. This is amplified in Hahn and Doganaksoy’s chapter “Characteristics of Successful Statisticians” (2012, Chapter 6). 1.2.2.2  Statistical Thinking for Beginners Although it only scratches the surface, the above still underscores the richness and complexity of thinking involved in real-world statistical problem solving and provides a useful set of reference points against which researchers and teachers can triangulate educational experiences (“Where is … being addressed?”). It is, however, far too complex for most students, particularly beginners. In discussing Wild and Pfannkuch, Moore (1999) asked, “What shall we teach beginners?” He suggested: … we can start by mapping more detailed structures for the ‘Data, Analysis, Conclusions’ portion of the investigative cycle, that is, for conceptual content currently central to elementary instruction. Here is an example of such a structure: When you first examine a set of data, (1) begin by graphing the data and interpreting what you see; (2) look for overall patterns and for striking deviations from those patterns, and seek explanations in the problem context; (3) based on examination of the data, choose

1  What Is Statistics?

13

appropriate numerical descriptions of specific aspects; (4) if the overall pattern is sufficiently regular, seek a compact mathematical model for that pattern. (Moore, 1999, p. 251)

Moore (1998) offered the following for basic critique, which complements his 1999 list of strategies with “Data beat anecdotes” and the largely metacognitive questions, “Is this the right question? Does the answer make sense? Can you read a graph? Do you have filters for quantitative nonsense?” (p. 1258). There are great advantages in short, snappy lists as starting points. Chance’s (2002) seven habits (p. 4) bring in much of Moore’s lists, and the section headings are even “snappier”: “Start from the beginning. Understand the statistical process as a whole. Always be skeptical. Think about the variables involved. Always relate the data to the context. Understand (and believe) the relevance of statistics. Think beyond the textbook.” Grolemund and Wickham (2014, Section 5) give similar lists for more advanced students. Brown and Kass (2009) state, “when faced with a problem statement and a set of data, naïve students immediately tried to find a suitable statistical technique (e.g., chi-squared test, t-test), whereas the experts began by identifying the scientific question” (p. 123). They highlighted three “principles of statistical thinking”: 1. Statistical models of regularity and variability in data may be used to express knowledge and uncertainty about a signal in the presence of noise, via inductive reasoning. (p. 109) 2. Statistical methods may be analyzed to determine how well they are likely to perform. (p. 109) 3. Computational considerations help determine the way statistical problems are formalized. (p. 122) We conclude with the very specialized definition of Snee (1990), which is widely used in quality improvement for business and organizations: I define statistical thinking as thought processes, which recognize that variation is all around us and present in everything we do, all work is a series of interconnected processes, and identifying, characterizing, quantifying, controlling, and reducing variation provide opportunities for improvement. (p. 118)

1.2.3  Relationship with Mathematics Although definitions that characterize statistics as a branch of mathematics still linger in some dictionaries, the separate and distinct nature of statistics as a discipline is now established. “Statistical thinking,” as Moore (1998) said, “is a general, fundamental, and independent mode of reasoning about data, variation, and chance” (p. 1257). “Statistics at its best provides methodology for dealing empirically with complicated and uncertain information, in a way that is both useful and scientifically valid” (Chambers, 1993, p. 184).

14

C.J. Wild et al.

“Statistics is a methodological discipline. It exists not for itself but rather to offer to other fields of study a coherent set of ideas and tools for dealing with data” (Cobb & Moore, 1997, p. 801). To accomplish those ends it presses into service any tools that are of help. Mathematics contains many very useful tools (as does computing). Just as physics attempts to understand the physical universe and presses mathematics into service wherever it can help, so too statistics attempts to turn data into ­real-­world insights and presses mathematics into service wherever it can help. And whereas in mathematics, mathematical structures can exist and be of enormous interest for their own sake, in statistics, mathematical structures are merely a means to an end (see also Box, 1990, paragraph 2; De Veaux & Velleman, 2008). A consequence is, adapting a famous quotation from John Tukey, whereas a mathematician prefers an exact answer to an approximate question, an applied statistician prefers an approximate answer to an exact question. The focus of the discipline of statistics, and in particular the role of context, is also distinct. “Statistics is not just about the methodology in a particular application domain; it also focuses on how to go from the particular to the general and back to the particular again” (Fienberg, 2014, p. 6): Although mathematicians often rely on applied context both for motivation and as a source of problems for research, the ultimate focus in mathematical thinking is on abstract patterns: the context is part of the irrelevant detail that must be boiled off over the flame of abstraction in order to reveal the previously hidden crystal of pure structure. In mathematics, context obscures structure. Like mathematicians, data analysts also look for patterns, but ultimately, in data analysis, whether the patterns have meaning, and whether they have any value, depends on how the threads of those patterns interweave with the complementary threads of the story line. In data analysis, context provides meaning. (Cobb & Moore, 1997, p. 803; our emphasis)

There is a constant “interplay between pattern and context” (Cobb & Moore, 1997). As for statistical investigations for real-world problems, the ultimate learning is new knowledge about the context domain—we have gone “from the particular to the general” (to enable us to use methods stored in our statistical repository) “and back to the particular again” (to extract the real-world learnings). We will now turn our attention to the role of theory in statistics. When most statisticians speak of statistical theory, they are thinking of mathematical theories comprising “statistical models” and principled ways of reasoning with and drawing conclusions using such models. Statistical models, which play a core role in most analyses, are mathematical models that include chance or random elements incorporated in probability theory terms. Perhaps the simplest example of such a model is y = μ + ε where we think in terms of y being an attempt to measure a quantity of interest μ in the process of which we incur a random error ε (which might be modeled as having a normal distribution, say). In the simple linear model, y = β0 + β1x + ε, the mean value of y depends linearly on the value of an explanatory variable x rather than being a constant. Random terms (cf. ε) model the unpredictable part of a process and give us ways of incorporating and working with

1  What Is Statistics?

15

uncertainties. “Statistical theory” in this sense is largely synonymous with “mathematical statistics.” Probability theory is a body of mathematical theory which was originally motivated by games of chance. More recently it has been motivated much more by the needs of statistical modeling, which takes abstracted ideas about randomness, forms mathematical structures that encode these ideas and makes deductions about the behavior of these structures. Statistical modelers use these structures as some of the building blocks that they can use in constructing their models, as with the random error term in the very simple model above. Recent work by Pfannkuch et al. (2016) draws on interviews with stochastic modeling practitioners to explore probability modeling from a statistical education perspective. The paper offers a new set of (conceptual) models of this activity. Their SWAMTU model is basically a cycle (with some feedback). It has nodes problem Situation → Want (to know) → Assumptions → Model → Test → Use. Models are always derived from a set of mathematical assumptions so that assumption checking against data is, or should be, a core part of their construction and use. As well as being used in statistical analysis, they are commonly used to try to answer “what if” questions (e.g., “What would happen if the supermarket added another checkout operator?”). Although there is much that is distinct about statistical problem solving, there is also much that is in common with mathematical problem solving so that statistics education researchers can learn a lot from work in mathematics education research and classic works such as Schoenfeld (1985). When most statisticians hear “theory,” they think “statistical theory” as described above, mathematical theories that underpin many important practices in the analysis and plan stages of PPDAC. But “theory” is also applicable whenever we form abstracted or generalized explanations of how things work. Consequently, there is also theory about other elements of PPDAC, often described using tools like flow charts and concept maps (cf. Fig.  1.1). For example, Grolemund and Wickham (2014) propose a theoretical model for the data analysis process by comparing it to the cognitive process of the human mind called “sensemaking” involving updating schemas (mental models of the world) in light of new information. In recent years there has also been a shift in the “balance of power” from overtly mathematical approaches to data analysis toward computationally intensive approaches (e.g., using computer simulation-based approaches including bootstrapping and randomization tests, flexible trend smoothers, and classification algorithms). Here the underlying models make much weaker assumptions and cannot be described in terms of simple equations. So, although “the practice of statistics requires mathematics for the development of its underlying theory, statistics is distinct from mathematics and requires many nonmathematical skills” (American Statistical Association Undergraduate Guidelines Workgroup, 2014, p.  8). These skills (required also by many other disciplines) include basic scientific thinking, computational/algorithmic thinking, graphical/visualization thinking (Nolan & Perrett, 2016), and communication skills.

16

C.J. Wild et al.

So, “… how is it then that statistics came to be seen as a branch of mathematics? It makes no more sense to us than considering chemical engineering as a branch of mathematics” (Madigan & Gelman, 2009, p. 114). The large majority of senior statisticians of the last half century began their academic careers as mathematics majors. Originally, computational capabilities were extremely limited, and mathematical solutions to simplified problems and mathematical approximations were hugely important (Cobb, 2015). The academic statisticians also worked in environments where the reward systems overwhelmingly favored mathematical developments. The wake-up call from “big data” and “data science” is helping nudge statistics back toward its earlier, and much more holistic, roots in broad scientific inference (Breiman, 2001).

1.3  Why Learning Statistics Is More Important Than Ever In today’s data-rich world, all educated people need to understand statistical ideas and conclusions, to enrich both their professional and personal lives. The widespread availability of interesting and complex data sets and increasingly easy access to user-friendly visualization and analysis software mean that anyone can play with data to ask and answer interesting questions. For example, Wild’s Visual Inference Tools (https://www.stat.auckland.ac.nz/~wild/VIT/) and iNZight software (https:// www.stat.auckland.ac.nz/~wild/iNZight) allow anyone to explore data sets of their own choosing. The CODAP (Common Online Data Analysis Platform, https://concord.org/projects/codap) provides a straightforward platform for web-based data analysis, as does iNZight Lite (http://lite.docker.stat.auckland.ac.nz/), and commercial solutions created by TuvaLabs (https://tuvalabs.com), and Tableau (https:// www.tableau.com). Statistical methods are used in almost all knowledge areas and increasingly are used by businesses, governments, health practitioners, other professionals, and individuals to make better decisions. Conclusions and advice based on statistical methods abound in the media. Some of the thinking used for decision-making based on quantitative data carries over into decision-making involving uncertainty in daily life even when quantitative data are not available. For these reasons, probably no academic subject is more useful to both working professionals and informed citizens on a daily basis than statistics. The rapid development of data science and expansion of choices for what to teach in statistics courses provides challenges for statistics educators in determining learning goals, and opportunities for statistics education researchers to explore what instructional methods can best achieve those goals. For example, in articles that appeared almost simultaneously, both Cobb (2015) and Ridgway (2015) argued that we need a major overhaul of the university statistics curriculum, particularly the introductory course and the undergraduate curriculum. Similar arguments can be made for greater inclusion of data skills at the primary and secondary school levels.

1  What Is Statistics?

17

To ignore the impact of the widespread availability of data and user-friendly software to play with data would lead to marginalization of statistics within the expanding world of data science. An important and engaging component of the current data revolution is the existence of “data traces” of people’s day to day activities, captured in social networks, personal logging devices, and environmental sensors. Teaching students how to examine their own personal data in constructive ways can enhance the attractiveness of learning data skills. These advances offer people new ways to become agents and advocates empowered to use data to improve the world around them in relation to situations of special relevance to their own lives (Wilkerson, 2017, personal communication). In this section we provide some additional motivation for why everyone would benefit from studying statistics and some examples of what could be useful for various constituencies. Statistics educators could then make sure to emphasize the usefulness of statistics when they teach, targeted to their respective audiences. Statistics education researchers study how we can teach students to use statistical reasoning throughout their lives to ask and answer questions relevant to them. The ideal type and amount of statistical knowledge and competencies needed by an individual depends on whether the person will eventually be working with data as a professional researcher (a producer of statistical studies), interpreting statistical results for others (a professional user of statistics), or simply needing to understand how to use data and interpret statistical information in life (an educated consumer of data and statistics). Professional users include health workers (who need to understand results of medical studies and translate them into information for patients), financial advisors (who need to understand trends and variability in economic data), and politicians (who need to understand scientific data as it relates to public policy, as well as how to conduct and understand surveys and polls). Producers of statistical studies are likely to take several courses in statistical methods, and will not be the focus of this chapter (see Sect. 1.4 for more discussion). Educated consumers include pretty much everyone else in a modern society. They need to understand how and what valid conclusions can be made from statistical studies and how statistical thinking can be used as a tool for answering questions and making decisions, with or without quantitative data.

1.3.1  What Professional Users of Statistics Need to Know Many professionals do not need to know how to carry out their own research studies, but they do need to know how to interpret and question results of statistical studies and explain them to patients and customers. In business applications, professionals such as marketing managers may need to understand the results generated by statisticians within their own companies. In this section we provide a few examples of why professional users of statistics need to understand basic statistical ideas

18

C.J. Wild et al.

beyond what is needed for the general consumer. Commonly used statistical methods differ somewhat across disciplines, but there are some basic ideas that apply to almost all of them. One of the most fundamental concepts is the importance of variability and a distribution of values. The first example illustrates how that concept is important for financial advisors and their clients.

1.3.2  E  xample 1: How Much Should You Save? Income and Expense Volatility In 2015 the financial giant JP Morgan Chase announced the establishment of a research institute to utilize the massive and proprietary set of financial data it owns to answer questions about consumer finances. The inaugural report, published in May 2015 (Farrell & Greig, 2015), examined financial data for 100,000 individuals randomly selected from a 2.5 million person subset of Chase customers who met specific criteria for bank and credit card use. One of the most important and publicized findings (e.g., Applebaum, 2015) was that household income and expenditures both vary widely from month to month and not necessarily in the same direction. For instance, the report stated that “41% of individuals experienced fluctuations in income of more than 30% on a month-to-month basis” (p. 8) and “a full 60% of people experienced average monthly changes in consumption of greater than 30%” (p. 9). Additionally, the report found that changes in income and consumption don’t occur in tandem, so it isn’t that consumers are spending more in months when they earn more. Why is this important information? Financial planners routinely advise clients to have accessible savings equivalent to 3–6 months of income. But in some cases, that may not be enough because of the volatility in both income and expenditures and the possibility that they can occur in opposite directions. One of three main findings of the report was “the typical individual did not have a sufficient financial buffer to weather the degree of income and consumption volatility that we observed in our data.” (p. 15). A concept related to variability is that few individuals are “typical.” In the previous example, individual consumers should know whether the warnings in the report are likely to apply to them based on how secure their income is, what major expenditures they are likely to encounter, and what resources they have to weather financial storms. Knowing that the “typical” or “average” consumer needs to stash away 3–6 months of income as savings could lead to questions about how each individual’s circumstances might lead to recommendations that differ from that advice. Baldi and Utts (2015) discussed topics and examples that are important for future health practitioners to learn in their introductory, and possibly only, statistics course. One central concept is that of natural variability and the role it plays in defining disease and “abnormal” health measurements. The next example, adapted from Baldi and Utts, illustrates how knowledge about variability and a distribution of

1  What Is Statistics?

19

Borderline high 36.5% "Normal"

High

28.5%

35% 200 216 240 Cholesterol (mg/dl)

Fig. 1.2  Cholesterol values for women aged 45–59; mean ≈ 216 mg/dL (standard deviation ≈ 42 mg/dL)

values can help physicians and their patients put medical test results in perspective.

1.3.3  Example 2: High Cholesterol for (Almost) All Medical guidelines routinely change based on research results, and statistical studies often lead pharmaceutical companies and regulatory agencies to change their advice to medical practitioners about what constitutes pathology. According to the United States National Institutes of Health, high cholesterol is defined as having total blood cholesterol of 240 mg/dL or above, and elevated or borderline high cholesterol is defined as between 200 and 240 mg/dL (http://www.nhlbi.nih.gov/health/ health-topics/topics/hbc). Suppose you are diagnosed with high or borderline high cholesterol and your doctor recommends that you take statin drugs to lower it. You might be interested in knowing what percentage of the population is in the same situation. Using data from the World Health Organization (Lawes, Vander Hoorn, Law, & Rodgers, 2004), we can model the total cholesterol levels of women aged 45–59 years old in the United States using a normal distribution with mean of about 216  mg/dL and standard deviation of about 42  mg/dL.  Figure  1.2 illustrates this distribution. As shown in the figure, about 35% of women in this age group have high cholesterol, and an additional 36.5% have borderline high values. That means that only about 28.5% do not have this problem! Should more than 70% of middle-­ aged women be taking statin drugs? Given that there are risks associated with high cholesterol and side effects associated with taking statin drugs, this is a discussion for individuals to have with their doctors. But it would be helpful for both of them

20

C.J. Wild et al.

to understand that more than a majority of the population fall into these cholesterol risk categories. Additional statistical reasoning tools (beyond the scope of this example) are needed to help physicians and consumers understand the trade-off in risks associated with taking statin drugs or not. But this example illustrates the importance of understanding the concept of a distribution of values, and how it relates to decisions individuals need to make.

1.3.4  W  hat Educated Consumers of Data and Statistics Need to Know Most students who take a statistics course will never use formal statistical procedures in their professional lives, but quite often they will encounter situations for which they could utilize data and statistical information in their personal lives to make informed decisions. Teachers of introductory statistics should provide instruction that will help people utilize that information. In the papers by Ridgway (2015), Baldi and Utts (2015), Utts (2003), and Utts (2010), more than a dozen important topics are described, with multiple examples. A few of them were covered in the previous section on what professional users of statistics need to know. Here, we list more of them and explain why they are important for consumers. Examples to illustrate each of these can be found in the references mentioned here, as well as in textbooks with a focus on statistical literacy such as Cohn and Cope (2011), Hand (2014), Moore and Notz (2016), and Utts (2015a). A resource for current examples is the website http://www.stats.org, a joint venture of the American Statistical Association and Sense About Science USA. 1.3.4.1  Unwarranted Conclusions Based on Observational Studies The media have improved in the interpretation of observational studies in recent years, but reports implying causation based on observational studies are still quite common. Citizens should learn to recognize observational studies and know that cause and effect conclusions cannot be made based on them. Here are some recent examples of misleading headlines based on observational studies or meta-analyses of them: • “6 cups a day? Coffee lovers less likely to die, study finds” (NBC News, 2012) • “Citrus fruits lower women’s stroke risk” (Live Science, 2012) • “Walk faster and you just might live longer” (NBC News, 2011) In many cases those conducting the research cautioned against making a causal conclusion, but the headlines are what people read and remember. Students and citizens should always question whether a causal conclusion is warranted and should know how to answer that question.

1  What Is Statistics?

21

1.3.4.2  Statistical Significance Versus Practical Importance Ideally, university students would learn the correct interpretation of p-values in an introductory statistics course. (See Nuzzo, 2014, for a nontechnical explanation of p-values.) But the most common misinterpretation, that the p-value measures the probability that chance alone can explain observed results, is difficult to overcome. So students at least should learn the importance of distinguishing between statistical significance and practical importance. For example, in the previously referenced article, Nuzzo (2014) discusses a study claiming that those who meet online have happier marriages (p