Misconceptions in Science Education : Help Me Understand [1 ed.] 9781527514843, 9781443893893

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Misconceptions in Science Education Help Me Understand

Misconceptions in Science Education Help Me Understand By

Ilana Ronen

Misconceptions in Science Education: Help Me Understand By Ilana Ronen This book first published 2017 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2017 by Ilana Ronen All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-4438-9389-7 ISBN (13): 978-1-4438-9389-3

To my ever-present parents and inspiring family, that keep on showing me the power of empathy, tolerance, and acceptance.

TABLE OF CONTENTS

Acknowledgments ................................................................... xi Epigraph ................................................................................. xv List of Illustrations, Figures and Tables................................ xix Introduction .............................................................................. 1 Misconceptions in Science Education: Help Me Understand Section 1: Misconceptions: A Barrier to Learning or a Learning Opportunity? Chapter One ........................................................................... 20 Misconceptions in Science Education 1.1. Misconceptions as a Barrier in Science Education 1.2. Misconceptions and Teachers’ Instruction 1.3. Addressing Misconceptions during Learning 1.4. Limits of the "classical" Conceptual Change Approaches References Chapter Two ........................................................................... 45 Intuitive Rules and Misunderstanding of Sciences 2.1. The Intuitive Rules Theoretical Framework 2.2. The Intuitive Rule “same A – same B” References

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Table of Contents

Section 2: The Intuitive Rule “same A – same B”: A Pedagogical Tool Chapter Three ......................................................................... 60 “I Conserve Area, Do I Realize that Volume is Not Necessarily Conserved?” 3.1. Conservation Tasks: Conserved and Non-conserved Quantities 3.2. The Intuitive Rule “same A-same B” in Conservation Tasks 3.3. Method 3.4. Key Findings 3.5. Concluding Remarks 3.6. Educational Implication References Chapter Four .......................................................................... 94 Potential Approaches to Overcoming Incorrect Responses in Conservation Tasks 4.1. Problem Solving and Cognitive Conflict 4.2. Authentic Tasks 4.2.1. Methods 4.2.2. Key Findings 4.3. Extreme Tasks 4.3.1. Methods 4.3.2. Key Findings 4.4. Concluding Remarks References

Misconceptions in Science Education: Help Me Understand

ix

Chapter Five ......................................................................... 140 Teachers Facing Misconceptions 5.1. The Role of Teachers’ Awareness in Students’ Misconceptions 5.2. Teachers’ Instructional Strategies for Addressing Students’ Misconceptions 5.3. Pre-service Teachers and Science Misconceptions 5.4. Effective Preparation of Pre-service Teachers in Teacher Education 5.5. Pre-service Teachers’ Incorrect Responses in Conservation Tasks 5.6. Methods 5.7. Key Findings References Section 3: Empathic Space: Shifting Pedagogy from Content-centered Teaching to Learning-centered Teaching Chapter Six ........................................................................... 182 Empathy and Education 6.1. Empathy in Multiple Lenses 6.2. Empathy Levels 6.3. Empathy in an Educational Lens 6.4. Empathy and Academic Achievement 6.5 Pre-service Teachers’ Feelings and Thoughts as Teachers References Chapter Seven ...................................................................... 219 Incorrect Responses: Improve my Empathy Awareness 7.1 Pre-service Teachers’ Incorrect Use of the Intuitive Rule “same A – same B”

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Table of Contents

7.2 The Pre-service Teachers’ Turning Point: Awareness of Empathy 7.3 Incorrect Responses in Accordance with the Intuitive Rule “same A – same B” as a Pedagogical Tool References Chapter Eight ....................................................................... 233 Putting It Together: Misconceptions Help Me Understand 8.1. Using Misconceptions as a Motivating Factor for Learning 8.1.1. Misconceptions in Science and Math are Common and Universal 8.1.2. Affordance of the Intuitive Rule “same A – same B” as a Pedagogical Tool 8.1.3. The Role of Empathy Awareness in Improving Learning Interaction 8.2. Educational Implications 8.3. Face Future Research 8.4. Key Insights References

ACKNOWLEDGMENTS

This book brings together conclusions and insights regarding a study on misconceptions and the source of incorrect responses in science and mathematics, and focuses primarily on the intuitive rule “same A - same B” and its incorrect application in conservation tasks. The intuitive rule “same A - same B” is one of three intuitive rules that have been extensively studied and formulated by Prof. Ruth Stavy and Prof. Dina Tirosh, Tel Aviv University, Israel. I wish to express my most sincere gratitude and appreciation to Prof. Stavy and Prof. Tirosh, who introduced me to the wondrous concept of intuitive rules and its implications for science and mathematics education, which became the basis for the studies presented in this book. This concept, like many ingenious ideas, appears at first blush selfevident, but in fact, is a complex notion that calls into question existing approaches and demands reexamination and a measure of skepticism, both of which are vital to learning and teaching.

xii

Acknowledgments

At the center of the book lies the intuitive rule “same A – same B.” The decision to focus on this rule is based on its unique role as expressed in conservation tasks - an increase in incorrect “same A - same B” responses with the development of the conservation scheme. This surprising phenomenon reinforces the notion that the source of the incorrect responses is related neither to knowledge (about the subject of conservation) nor to cognitive ability (such as the ability to acquire an understanding of conservation), but rather to perceptual-given, intuitive or sensory factors. The application of this intuitive rule as a pedagogical tool will make it possible to deal with incorrect responses given by learners from varied age groups (from kindergarten to pre-service teachers), and thus contribute to the work of the pre-service teachers that understand the importance of discussing the limitations of its application. I would like to thank my students (pre-service teachers) who helped with their ideas and discussions, every year for five years, in the context of the seminar course on “Misconceptions in the teaching of science.” The students’ hesitation at the start of the process was quickly transformed into increasing enthusiasm for the work and the learning, as they delved into the study that uncovered incorrect responses

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xiii

in their field of experience. The students became partners to the research, presenting up-to-date, research-oriented points of view. We shared knowledge, thoughts, concerns, dilemmas, evidence from the field, all of which became authentic and objective components that contributed to a better understanding of the findings. I thank them for their willingness to participate in the study, for rising to the challenge posed by an action research seminar and for their application of the insights produced by the research in the field of education. I would like to express my appreciation and thanks to the teachers for their honest and courageous sharing of dilemmas that engaged them in their teaching, and for their questions: Should I have been teaching differently? What could I have done to make the subject clearer? My gratitude goes to Prof. Shlomo Beck for reading the various parts of the book and for his very important and helpful comments. It is my hope that the message that this book proposes will find representation in the teacher-training process both in terms of content and affect. I further hope that the advantages presented by the use of the intuitive rule “same A – same B” as a pedagogical tool in teacher training will contribute to

xiv

Acknowledgments

improving the quality of the relationship of those involved in it thanks to empathic effort and support, alongside recognition of and respect for the feelings of learners as part of the practical-reflective teaching-learning process.

EPIGRAPH

Once when I was six years old I saw a magnificent picture in a book, called True Stories from Nature, about the primeval forest. It was a picture of a Boa constrictor in the act of swallowing an animal. Here is a copy of the drawing.

In the book it said: "Boa constrictors swallow their prey whole, without chewing it. After that they are not able to move, and they sleep through the six months that they need for digestion." I pondered deeply, then, over the adventures of the jungle. And after some work with a colored pencil I

xvi

Epigraph

succeeded in making my first drawing. My Drawing Number One. It looked something like this:

I showed my masterpiece to the grown-ups, and asked them whether the drawing frightened them. But they answered: "Frighten? Why should anyone be frightened by a hat?" My drawing was not a picture of a hat. It was a picture of a Boa constrictor digesting an elephant. But since the grown-ups were not able to understand it, I made another drawing: I drew the inside of a Boa constrictor, so that the grown-ups could see it clearly. They always need to have things explained. My Drawing Number Two looked like this:

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xvii

The grown-ups' response, this time, was to advise me to lay aside my drawings of Boa constrictors, whether from the inside or the outside, and devote myself instead to geography, history, arithmetic, and grammar. That is why, at the age of six, I gave up what might have been a magnificent career as a painter. I had been disheartened by the failure of my Drawing Number One and my Drawing Number Two. Grown-ups never understand anything by themselves, and it is tiresome for children to be always and forever explaining things to them. So then I chose another profession, and learned to pilot airplanes.… I have lived a great deal among grown-ups. I have seen them intimately, close at hand. And that hasn't much improved my opinion of them. Whenever I met one of them who seemed to me at all clear-sighted, I tried the experiment of showing him my Drawing Number One, which I have always kept. I would try to find out, so, if this was a person of true understanding. But, whoever it was, he, or she, would always say: "That is a hat." Then I would never talk to that person about Boa constrictors, or primeval forests, or stars. I would bring myself down to his level. I would talk to him about bridge, and golf, and politics, and neckties. And the grown-up would be greatly pleased to have met such a sensible man.

xviii

Epigraph

The Little Prince, Written and illustrated by Antoine de Saint Exupéry Retrieved April, 17, 2017, from, http://www.angelfire.com/hi/littleprince/frames.html

LIST OF ILLUSTRATIONS, FIGURES AND TABLES

Title

Page

Illustration 1

The “Derived rectangle” task

68

Illustration 2

The “String loops” task

69

Table 1

Possible response types

70

Illustration 3

The “Turning a rectangle into a

71

cylinder” task Illustration 4

The “Filling cylinders” task

72

Table 2

Possible response types

73

Figure 1

Distribution by grade of students

75

who conserve area and “conserve” perimeter in the “Derived rectangular” task Figure 2

Distribution by grade of students

76

who conserve perimeter and “conserve” area in the “String loops” task Figure 3

Distribution by grade of students who conserve area and “conserve” volume in the “Turning a rectangle into a cylinder” task

81

xx

Figure 4

List of Illustrations, Figures and Tables

Distribution by grade of students

82

who conserve volume and “conserve” area in the “Filling cylinders” task Illustration 1

The “Derived rectangle” task

102

Illustration 2

The “Turning a rectangle into a

102

cylinder” task Table 1

Distribution of students’

108

performance and responses to the perimeter in the “Greeting card" task by grade (N=60) Table 2

Distribution of students’

110

performance and responses to the volume in the “Candy roll” task by grade (N=60) Table 3

Types of tasks and sequences

122

presented to the Grade 9 students (N=40) Table 4

Percentages of correct responses regarding the non-conserved parameter in: extreme tasks and regular tasks (analogous task and different task) (N=40)

124

Misconceptions in Science Education: Help Me Understand

xxi

Illustration 1

The “Derived rectangle” task

156

Illustration 2

The “String loops” task

158

Illustration 3

The “Turning a rectangle into a

159

cylinder” task Table 1

Percentage of pre-service teachers’

163

incorrect responses regarding the non-conserved quantity: perimeter in the “Derived rectangle” task, the area in the “String loop” task, and the volume in the “Turning a rectangle into a cylinder” task (N=40) Figure 1

Schematic illustration of generation

190

and modulation of empathy

Figure 2

Generation and modulation of

208

empathy: A meta-cognitive feedback loop Figure 1

Generation and modulation of

250

empathy: A meta-cognitive feedback loop Figure 2

Misconceptions as a motivating factor for learning

262

INTRODUCTION MISCONCEPTIONS IN SCIENCE EDUCATION: HELP ME UNDERSTAND

Or: How do we make sense of our world? In discussions held with pre-service teachers as part of a seminar course about “misconceptions in science education,” the question of how the students' worldview is shaped in regard to science-related phenomena, often comes up. During the discussion, two main ideas emerge, one of which relates to the development of technology, which facilitates the students’ accessibility to information, making it much more available to them. On the other hand, however, it does not offer a solution to students’ misconceptions that are inconsistent with the accepted scientific theories. Moreover, it is also suggested that because students have become accustomed to receiving immediate answers to any question or subject - by clicking a button on the keyboard - this impacts the immediate, intuitive way students answer questions - sometimes leading to incorrect answers.

2

Introduction

The second idea is derived from the first, and focuses on changing the status of the teacher as an exclusive source of knowledge, as well as on the implications of this change on the pre-service teachers’ sense of self-confidence as teachers, on just how much a teacher is needed in teaching, and hence, on the changing role of the teacher in the teaching-learning process. The pre-service teachers sometimes stand helpless and devoid of any tools to deal with the situation. Indeed, in light of studies that point to the complexity of the learning process (e.g., Berkovich & Eyal, 2015; Blackmore, 2010; Muijs & Harris, 2003); the change in the teacher’s role in the era of knowledge and technological availability (e.g., Murphy, 2005; Spillane, 2012); and the importance of training preservice teachers to deal with misconceptions (e.g., GomezZwiep, 2008; Halim & Meerah, 2002; Meyer, 2004), there seems to be a real basis for this feeling, and it points to the importance of relating to this essential issue within the framework of teacher training. The need to cope with this challenge led to the search for a different perspective on misconceptions in a way that might also affect the role of the teacher in the complicated teachinglearning process. This combination pointed to the use of the intuitive rule “same A - same B” as a pedagogic tool. The

Misconceptions in Science Education: Help me understand

3

researchers Stavy and Tirosh (2000), who undertook an indepth

study

of

incorrect

responses

in

science

and

mathematics, described in the literature as misconceptions, maintain that a considerable proportion of the incorrect responses described in the literature as misconceptions result from the use of a limited number of intuitive rules (“more A more B,” “same A - same B,” and “everything can be divided”), leading to an erroneous intuitive response. The current study is based on these ideas and suggests that incorrect responses of this kind should be treated as an opportunity for learning rather than as a barrier or obstacle, and suggests making use of the potential contribution that lies in using the intuitive rule “same A - same B” as a pedagogical tool in teacher training. This contribution, which goes beyond predictability for incorrect responses (Stavy & Tirosh, 2000), can also be expressed in possible ways of dealing with incorrect responses, and in exposing teachers to a learning experience that inspires a sense of empathy for their learners. Activity of this kind as part of the educational work of teachers can help to reduce teachers' sense of helplessness when faced with their students’ misconceptions, and on the other hand, inspires teachers to constantly examine their ability to operate

Introduction

4

effectively in situations of ambiguity and uncertainty in different learning contexts when they are exposed to misconceptions. The current study cautiously suggests that misconceptions should be treated as a challenging opportunity to better understand learners’ incorrect responses; to address incorrect responses by using the intuitive rule “same A - same B” during learning; and to gain a better understanding of the role of

feelings

and

empathy

awareness

during

learning

interactions. Doing this may turn misconceptions into a means to understand and improve learning. Thus, two central ideas underlie the book: The first befriend

misconceptions

-

is

related

to

approaching

misconceptions as an opportunity for learning rather than as an obstacle to make them part of the teaching-learning process. The second - comprehending emotion - is related to the changing role of the teacher and suggests a more vigorous application of affective aspects in response to the challenges of teaching-learning in the information age.

Befriend Misconceptions The idea of treating misconceptions as an opportunity for learning rather than as an obstacle combines the findings of

Misconceptions in Science Education: Help me understand

5

studies that point to teachers’ meager knowledge regarding their students' misconceptions (Gomez-Zwiep, 2008), and as a result, their tendency to ignore incorrect responses during the learning process and to consider them a specific difficulty of learners, one that causes teachers to feel helpless: “I don’t know what else to do; after all, I've taught it and the students seemed to understand. How is it that they are giving incorrect responses?”

Indeed, ignorance of the source of the misconception is a barrier to learning, but focusing on them is a challenge that can lead to the advancement of learning. In this study, an effort was made to understand the phenomenon of misconceptions by placing the spotlight on the learners - kindergarten to junior high school students and pre-service teachers who major in mathematics and science with an eye to investigating not only the perspective of the learners, both the younger ones and the pre-service teachers, but also to try to understand the perspective of the pre-service teachers as teachers. The book proposes a change in the attitude toward misconceptions, that is to see them as an educational event that can advance learning rather than as one that limits it, thus helping teachers and students “make friends” with incorrect responses as part of the teaching-

Introduction

6

learning process. This can be done by training an ability to delay the natural need to obtain a single 'correct' answer (Land & Hannafin, 1996), to encourage discussion on different points of view, and to foster tolerance for ambiguous and uncertain

situations

when

questions

remain

open

to

discussion, despite the sense of unease that accompanies these situations. To advance this process, the teacher should try to understand the possible source of incorrect responses. Indeed, since misconceptions appear to be a barrier to learning, understanding what lies behind them might reinforce the process of knowledge building. Some of the ideas proposed in this book are consistent with studies that point to the importance of exposing teachers already at the early stage of learning - in elementary school - to incorrect perceptions and to look for ways that contribute to dealing positively with them (e.g., Gomez-Zwiep, 2008). However, students often avoid seeing them, while teachers treat them as gaps in knowledge that will be filled during learning, although research findings contradict this current assumption (e.g., Allen & Coole, 2012; Gomez-Zwiep, 2008). Despite the attempt to understand the possible source of the misconceptions, most of the studies relate to a specific content area (for example, electric circuit, force, energy, evolution),

Misconceptions in Science Education: Help me understand

7

and thus lack a comprehensive, overall and broad view to the problem (Fensham, 2001). A more overall and broad point of view is offered by Stavy and Tirosh (2000), who address misconceptions from another perspective, based on a theoretical

framework

that

can

interpret

important

misconceptions in science and mathematics as evolving from some general intuitive rules. They maintain that in many cases, students give answers that are not based on a single correct or incorrect perception, but rather that their answers vary, based on visual information related to a specific aspect of the task, from which they often erroneously infer for another aspect (Stavy & Tirosh, 1996, 2000). For example, young children (aged 4 - 5) claim that two glasses of sugar water of different size are equally sweet because both glasses have the same amount of sugar (one teaspoon of sugar was placed in different amounts of water - one cup filled with water and the other only half filled with water). The visual information regarding the equal amounts of sugar led to an incorrect answer regarding the degree of the water’s sweetness. Similarly, children often claim that a taller child must be older (when comparing two children of the same age who differ in height; a similar answer is given even if the older child is shorter than the younger one). The visual

Introduction

8

information regarding the children’s height leads to an incorrect answer regarding their age. According to the researchers, in the first case, the children's answer was consistent with the intuitive rule “same A - same B” (same amounts of sugar - same water’s sweetness); in the second case, the children's answer was consistent with the intuitive rule “more A - more B” (the taller - the older). The studies presented in this book are based on the intuitive rule “same A - same B” (Stavy & Tirosh, 2000) that serves as a pedagogical tool whose application contributes to understanding the possible source of students' incorrect responses to conservation tasks; to examining the limits of the application of this intuitive rule in conservation tasks; and to find ways to deal with the incorrect responses. According to this approach, the proposal is to examine different learners’ perspectives, to try to investigate the source of the incorrect answer and to accordingly plan and implement the instruction. Alongside the cognitive pursuit of the content aspect of the incorrect responses, the present study also emphasizes the importance of the affective-emotional aspect, the quality of the relationship with the learners and teacher - learner interactions, as a factor that contributes to learning.

Misconceptions in Science Education: Help me understand

9

Comprehending Emotion The perception of the significance of the teacher’s role in fostering access to an empathic space that promotes the teaching-learning

process

is

gaining

an

increasingly

prominent place in recent years. It is suggested that a teacher who recognizes the importance of different points of view even if they do not comply with the scientifically accepted answers - will allow different answers, correct and incorrect, to be heard, based on an awareness of the importance of the learning process; and will design tools to combine reflection about the thought processes and thereby promote tolerant and respectful dialogue, which contribute to a gradual process of structuring knowledge on the subject being studied. The core of the study is based on the intuitive rule “same A - same B” (Stavy & Tirosh, 2000), which serves in it as a pedagogical tool for learning and teaching. However, the book offers an additional aspect: the potential contribution of the empathic space in teaching, with teachers focusing not only on the subject matter, investigating the source of the incorrect response and finding ways to deal with it, but also on the affective-experiential aspect, which is directed at the learners' feelings following an incorrect response, and its application accordingly. Thus, the study explores how

10

Introduction

misconceptions are able not only to offer an indicator for problems in learning and instruction, but by means of the suggested approach also create learning environments that can potentially promote both academic and affective changes. The chapters of the book investigate the behavior of preservice teachers in the wake of their new feelings, their insights and suggestions for future action. These require constant monitoring of students’ responses, respect for different points of view and the ability to receive answers that do not conform to accepted scientific concepts, alongside tolerance for ambiguous situations in which it is not clear what prompted the learner to make their response, and creating a reflective process shared by all learners. These are made possible by the quality of a relationship that involves an empathic effort and a sympathetic attitude that can provide conditions of security and confidence. This ability of teachers empowers the active process of teaching-learning, which is based on the structuring of knowledge and takes into account both incorrect responses and the feelings that attend them.

The Structure of the Book The current study focuses on three main aspects of misconceptions based on a longitudinal study that investigated

Misconceptions in Science Education: Help me understand

11

students’ and pre-service teachers’ incorrect responses to conservation tasks. The book comprises three sections based on research, each advances the reader’s understanding of misconceptions and of the intuitive rule “same A - same B” as a pedagogical tool in science and mathematics education: 1. Misconceptions: A barrier to learning or a learning opportunity? (Chapters one and two); 2. The intuitive rule “same A - same B”: A pedagogical tool (Chapters three, four and five); 3. Empathic space: Shifting pedagogy from contentcentered

teaching

to

learning-centered

teaching

(Chapters six, seven and eight). Each aspect further deepens our insight and understanding of the role of misconceptions in learning and instruction, as described briefly below.

Section One Misconceptions: A Barrier to Learning or a Learning Opportunity? The first section presents misconceptions based on the research literature on the subject, presents the features of misconceptions,

discusses

the

educational

impact

of

misconceptions in science and mathematics during learning and the complexity of handling misconceptions in teaching

Introduction

12

and the perspectives of teachers and pre-service teachers. The effort to identify a common source for the misconceptions led the researchers Ruth Stavy and Dina Tirosh (1996, 2000) to suggest a different perspective on misconceptions that formulates three intuitive rules, one of which is the intuitive rule “same A - same B.” Chapter one elaborates on the context of misconceptions and their educational impact during learning. This helps to explain some of the current challenges and opportunities facing educators, since misconceptions can act as a barrier to learners’ progress at all stages of education. Chapter two presents the intuitive rules and suggests a theoretical framework defined by Stavy and Tirosh (1996, 2000) that can be used to interpret the incorrect responses that many students and adults give to science and mathematics problems.

Section Two The Intuitive Rule “same A - same B”: A Pedagogical Tool The second section describes the range of students` incorrect responses to conversation tasks in line with the intuitive rule “same A - same B”; and aims to address students’ incorrect responses by using authentic tasks and

Misconceptions in Science Education: Help me understand

13

extreme tasks; and to confront the pre-service teachers’ actual measurements with their formal incorrect responses, consistent with the intuitive rule “same A - same B,” to each of the conservation tasks. Thus, the intuitive rule “same A same B” as a pedagogical tool offers a practical solution to addressing the complex nature of misconceptions, in dealing with conservation tasks in the context of pre-service teacher training as well as in teaching-learning processes. The chapters describe the range of incorrect responses to conversation tasks as examined among elementary and junior high school students, as well as among pre-service teachers majoring in mathematics and science. It further describes possible ways to investigate the contribution of authentic tasks and extreme tasks to deal with incorrect answers given by learners. Chapter three argues that students’ incorrect responses to conservation tasks, focusing on the non-conserved quantity of area, perimeter and volume develop with age, are based on the intuitive rule “same A - same B,” and have been empirically tested. Chapter four emphasizes the possible pivotal role of the intuitive rule “same A - same B” in incorrect responses relating to conservation tasks. The use of authentic tasks and extreme tasks is suggested as a potential approach to overcome incorrect responses in conservation tasks, based on

Introduction

14

students’ incorrect responses that are consistent with the intuitive rule “same A - same B.” Chapter five discusses the considerable potential for conceptual and emotional change that lies in pre-service teachers’

self-awareness

when

dealing

with

students’

incorrect responses. This awareness is developed through affective training of pre-service teachers in teacher education, using the intuitive rule “same A - same B” as a pedagogical tool.

Section Three Empathic Space: Shifting Pedagogy from Contentcentered Teaching to Learning-centered Teaching Unlike the first two sections, the third section focuses on the importance of the affective aspect and the empathic space in the training process of pre-service teachers and students. In this way, it elaborates on the impact of incorrect responses on pre-service students’ empathy awareness during instruction and on the strategies designed to address such incorrect responses. Hence, the pre-service teachers directly encounter an experience that evokes an authentic sense of empathy with their students. Chapter six discusses recent research on empathy and the emotional aspects of education, including the improvement of

Misconceptions in Science Education: Help me understand

15

empathy development in teacher candidates, and the potentially positive correlation between pre-service teachers’ empathy and indices of academic achievement. Chapter seven returns to the key concept - cognitive (intellectual) and affective (emotional) development - key tenets of education, while discussing the pre-service teachers’ incorrect responses to conservation tasks and the ‘aha’ moment they experienced, which resulted in their empathic concern. This resonates with Hein and Singer (2008) who maintain that a prerequisite for successful social interaction and mental well-being is empathy, i.e. the ability to share the other's feelings Chapter eight ties these concepts together: the benefits of incorrect responses consistent with the intuitive rule “same A - same B,” address the complex challenge of providing practical instruction for pre-service teachers based on both intellectual aspects and empathy awareness.

References Fensham, P. (2001). Science content as problematic - Issues for research. In H. Behrendt, H. Dahncke, R. Duit, W. Gräber, M. Komorek, A. Kross, & P. Reiska, (Eds.), Research in science education - Past, present, and future

Introduction

16

(pp.

27-41).

Dordrecht,

the

Netherland:

Kluwer

Academic Publishers. Allen, M., & Coole, H. (2012). Experimenter confirmation bias and the correction of science misconceptions. Journal of Science Teacher Education, 23(4), 387–405. Blackmore, J. (2010). Preparing leaders to work with emotions in culturally diverse educational communities. Reflective Diary of Educational Administration, 48, 642658. Berkovich I., & Eyal O. (2015). Educational leaders and emotions: An international review of empirical evidence 1992-2012, Review of Educational Research, 85, (1), 129-167. Gomez-Zwiep, S. (2008). Elementary teachers’ understanding of students’ science misconceptions: Implications for practice and teacher education. Journal of Science Teacher Education, 19, 437-454. Halim, L., & Meerah, S. M. (2002). Science trainee teachers’ pedagogical content knowledge and its influence on physics teaching. Research in Science and Technological Education, 20, 215-225. Land, S.M., & Hannafin, M.J. (1996). A conceptual framework for the development of theories-in-action with

Misconceptions in Science Education: Help me understand

open-ended

learning

environments.

17

Educational

Technology Research and Development, 44 (3), 37-53. Meyer, H. (2004). Novice and expert teachers’ conceptions of learners’ prior knowledge. Science Education, 88, 970983. Muijs D., & Harris A. (2003). Teacher leadership Improvement through empowerment? An overview of the literature. Educational Management and Administration, 31(4), 437-448. Murphy, J. (2005). Connecting teacher’s leadership and school improvement. Thousand Oaks, CA: Corwin Press. Spillane, J. P. (2012). Distributed leadership. San Francisco, CA: John Wiley. Google Scholar. Stavy, R., & Tirosh, D. (1996). Intuitive rules in science and mathematics: The case of “more of A - more of B.” International Journal of Science Education, 18, 653-667. —. (2000). How students (mis)understand science and mathematics: Intuitive rules. New York: Teachers College Press.

SECTION 1 MISCONCEPTIONS: A BARRIER TO LEARNING OR A LEARNING OPPORTUNITY?

CHAPTER ONE MISCONCEPTIONS IN SCIENCE EDUCATION

Einstein and Infeld (1938) argue that scientific theories are the creation of the human mind, with its freely invented ideas and concepts attempting to form a picture of reality and to establish its connections with the wide world of sense impressions. However, there is clear evidence (e.g., Abimbola, 1988; Caramazza, McCloskey & Green, 1980; Champagne, Gunstone & Klopfer, 1983; Fensham, 2001; Hashweh, 1988; Gilbert & Swift, 1985) that students develop frameworks of belief about natural phenomena that conflict with

accepted

scientific

understanding

-

known

as

misconceptions - although these will probably change over time (Ross, Lakin & Callaghan, 2004). In the literature, misconceptions correspond to the concepts that have peculiar interpretations and meanings in students` articulations that are scientifically inaccurate (Bahar, 2003). Misconceptions are also referred to as naïve beliefs (Caramazza, McCloskey & Green, 1980; Champagne, Gunstone & Klopfer, 1983), preconceptions (Hashweh, 1988), erroneous ideas (Fisher,

Misconceptions in Science Education

21

1985), underlying sources of error (Fisher & Lipson, 1986), personal models of reality (Hashweh, 1988), private versions of science (McClelland, 1984), common sense concepts (Halloun & Hestenes, 1985), and children’s science (Gilbert, Watt & Osborne, 1982). Thus, during the learning process it is difficult to teach children new ideas until we know the existing ideas they hold - these often appear to conflict with accepted scientific ideas and are described also as ‘alternative frameworks of belief’ (Driver et al, 1994). Thus, alternative conception is another term that refers to experience-based explanations constructed by a learner, to make a range of natural phenomena and objects intelligible. This latter concept, which is preferred by some researchers (e.g., Abimbola, 1988; Fensham, 2001; Gilbert & Swift, 1985; Wandersee, Mintzes & Novak, 1994), also confers intellectual respect on the learner holding those ideas. Notably, there may be many grains of truth in these ‘alternative ideas’ and careful attention should be paid to what children say about their ideas in various contents in science and mathematics. Since students hold alternative ideas that are not always compatible with those accepted in the sciences, it was suggested that students should restructure their specific

22

Chapter One

conceptions during the learning process to make them conform to currently accept scientific ideas (Ross, Lakin & Callaghan, 2004). This chapter presents the commonly accepted notion that misconceptions are a barrier to science education and impede teaching.

It

discusses

the

educational

impact

of

misconceptions in learning science and mathematics, points to the challenge of addressing misconceptions during learning and describes the limitations of approaches involving conceptual change.

1.1 Misconceptions as a Barrier in Science Education It is well established that science misconceptions can act as a barrier to learners` progress in all stages of education. In such situations, learners` implicit knowledge of particular science topics before learning, sometimes conflicts with scientific theories and the content that they learn during science lessons (e.g., Bar & Travis, 1991; Duit & Treagust, 2003; Eryilmaz, 2002; Smith, diSessa & Roschelle, 1993). Essentially, it has been well documented that students have well-defined views of the world, based on their encounters with the natural world, before ever entering formal education

Misconceptions in Science Education

(Bar

&

Travis,

1991;

Eryilmaz,

23

2002).

Hence,

misconceptions, known also as alternative conceptions (e.g., Abimbola, 1988; Fensham, 2001; Gilbert & Swift, 1985; Wandersee, Mintzes & Novak, 1994), are part of a larger knowledge system that involves many interrelated concepts that students use, to make sense of their experiences (GomezZwiep, 2008; Eryilmaz, 2002) and they are based on their effective knowledge, among others. We can actually relate to misconceptions as to extensions of effective knowledge that function productively within a specific context. Hence, these misconceptions become apparent when students attempt to use their knowledge beyond the context, in which the knowledge

functions

effectively

(Smith,

diSessa,

&

Roschelle, 1993). Indeed, empirical evidence has shown that children

have

qualitative

differences

in

his

or

her

understanding of science that are often inconsistent with what the teacher intended through his or her instruction (Bar, 1989; Bar, Zinn, Goldmuntz & Sneider, 1994; Pine, Messer & Johan, 2001; Tao & Gunstone, 1999; Trend, 2001). However, misconceptions are more than misunderstandings about a concept during instruction, as can be seen by the NRC-National Research Council categorization (1997 pp. 28), which includes the following: Preconceived notions - popular

24

Chapter One

conceptions rooted in everyday experiences relates to students` views of heat, energy, gravity, among others; nonscientific beliefs -

learned from sources other than

scientific education, such as religious or mythical teachings dealing with abbreviated history of the earth and its life forms; conceptual misunderstandings - arise during learning when students do not confront conflicts resulting from their preconceived notions; vernacular misconceptions - arise from the use of words that have different meanings in everyday life and in scientific context (e.g., work, heat, force); and factual misconceptions - retained unchallenged since early age into adulthood (e.g., `lightning never strikes twice in the same place`). Misconceptions sometimes include a combination of several features at the same time and regarding the same concept. In addition, research findings consistently show that misconceptions appear within all age groups and across all areas of science (NRC, 1997) and they are deeply rooted, often remaining even after instruction (Eryilmaz, 2002). Nevertheless, since misconceptions are often integrated with other knowledge, they may include aspects of both expert and novice understandings and may be useful in constructing accurate scientific understandings during instruction (Gomez-

Misconceptions in Science Education

25

Zwiep, 2008). Thus, teachers in Elementary School play a vital role as the first line of defense against the continued existence of misconceptions, and help encouraging the construction of science knowledge (Allen & Coole, 2012; Gomez-Zwiep, 2008).

1.2 Misconceptions and Teachers’ Instruction Research on students' and teachers' conceptions and their roles in teaching and learning science has become one of the most important domains of science education research on teaching and learning during the past four decades. Starting in the 1970s with the investigation of students' pre-instructional conceptions on various science content domains, such as the electric circuit, force, energy, and evolution, the analysis of students’ understanding across most science domains has been comprehensively documented in the bibliography by Duit (2002).

This sheds light on teachers` instruction as an

essential issue in science education, focusing on Elementary School teachers. This issue has been investigated thoroughly. Methods based on the idea that pupils build up, or construct, ideas about their world, are often called constructivist approaches. As Ross and co-authors argue, if we want pupils to understand and use scientific ideas, their existing beliefs

26

Chapter One

need to be challenged or extended. We cannot always replace their naïve ideas, but we can encourage pupils to use the scientific ones, when appropriate, and show them the inconsistencies in many of their existing ideas (Ross, Lakin & Callaghan, 2004). These can already be used by Elementary School teachers. Indeed, this issue was investigated thoroughly by GomezZwiep (2008), who demonstrates what Elementary School teachers know about students’ science misconceptions and how teachers address students’ misconceptions in instruction. The results, based on interviewing 30 Elementary School teachers from California, pointed to that most teachers described misconceptions as gaps in knowledge that need to be filled and as something that results from external sources, rather than originating in the student’s own thinking. Indeed, it is suggested that misconceptions originate from three sources: misconceptions develop from stories passed on to students from their parents, friends, or television and movies; misconceptions are poor explanations, rather than ones that contradict accepted scientific theory; or misconceptions develop when the concept is beyond the developmental level of the student. Hence, the results suggest that although most of the teachers are aware of misconceptions, they do not

Misconceptions in Science Education

27

understand how they develop or fully appreciate their impact on their instruction. The results of these studies (e.g., Gomez-Zwiep, 2008; Halim & Meerah, 2002; Meyer, 2004) suggest that teachers are not prepared to confront science misconceptions as they arise in their classrooms, even if the teachers recognize that such misconceptions exist. Beyond that, some of them underappreciate

the

learning

gained

from

personal

experiences. A gap remains between what research has revealed about misconceptions and knowledge of how this research is applied in the classroom. Indeed, teachers` flawed understanding of the term and the connection that some of them made between a misconception and a misunderstanding, may lead them to underestimate how deeply rooted a misconception can be in student thinking. So, how can teachers move toward this understanding? Findings suggest that, although there is a tremendous amount of literature regarding student misconceptions, it has not filtered down to the everyday world of the classroom (Gomez-Zwiep, 2008). The evidence from this study indicates that, while aware that misconceptions exist, teachers are not attentive to their impact on instruction. Seeing that misconceptions have been shown to be prevalent and

28

Chapter One

predictable and can interfere with the processing of new information, teachers need to be aware of the instructional implications

and

the

strategies

designed

to

address

misconceptions. Exposing teachers to student misconceptions in their teacher preparation course through example and definition is not enough to ensure that they will be adequately prepared to address them in their own class (Gomez-Zwiep, 2008; Halim & Meerah, 2002; Meyer, 2004). Examples of several approaches that attempt to deal with misconceptions during learning are described in the following: dialogue and experimentation (Ross, Lakin & Callaghan, 2004); conceptual change strategies including Prediction-Observation-Explanation (POE) (Bahar, 2003; White & Gunstone, 1992); Dual Situated Learning Model (DSLM) (She, 2004a, 2004b); Open Learning Environments (OLEs) (Hannafin, Land & Oliver, 1999; Land & Hannafin, 1996) and multiple analogical models (Harrison & Treagust, 2000).

1.3 Addressing Misconceptions during Learning There is a significant body of research on instructional strategies shown to be effective at dealing with student misconceptions (e.g., Ausubel, 1968; Guzzetti, 2000; Posner,

Misconceptions in Science Education

29

Strike, Hewson & Gertzog, 1982). The research-based strategies have demonstrated some success in addressing misconceptions by expanding student thinking through dialogue and experimentation. For example, elicitation questions can be used during instruction to probe children’s understanding. It is worthwhile getting your trainees to try the questions themselves - if they do happen all to agree and you are happy with their responses, they can be asked to predict what pupils of various ages might say. If they disagree, it is a chance to sort out their own misconceptions (Ross, Lakin & Callaghan, 2004). These activities are selected to specifically confront the misconception by presenting unexpected results not previously considered by the learner. Although these strategies often involve some form of activity, various instructional technique methods have been shown to be successful in addressing misconceptions (Eryilmaz, 2002; Guzzetti, 2000; Tsai, 2003). Several strategies and techniques are used for externalizing ideas and modifying misconceptions in students’ cognitive structure. These strategies (e.g., word association tests, clinical interviews, interviews about instances and events, Prediction-Observation and Explanation, group discussions, and conceptual change texts) can be called conceptual change

30

Chapter One

strategies (Bahar, 2003; Wandersee, Mintzes & Novak, 1994). Some of these are Prediction-Observation and Explanation (POE) steps, which primarily are used to learn how to use the information students acquire to interpret

events and

experiences (White & Gunstone, 1992). With this technique, students need to do three tasks. First, students must predict the outcome of some events and the prediction must be justified; then, they describe what they see is happening; and finally, students must reconcile any conflict between the prediction and the observation. Bahar (2003) suggests that this powerful technique should be used more commonly for externalizing and modifying misconceptions. Another model, aimed to help students to restructure their science concepts, is suggested by She (2004a, 2004b). Based on Piaget’s theory of disequilibrium She (2004a, 2004b) developed the Dual Situated Learning Model (DSLM), a constructivist strategy model suggesting six major stages: examine the attributes of the science concept; probe students’ misconception about a particular concept; analyze for mental sets which the students lack; design dual situated learning events; instruct with dual situated learning events; and instruct with challenging situated learning events. Unlike positivism, which advocates that knowledge is an objective and existing

Misconceptions in Science Education

31

truth, the constructivist strategy advocates that knowledge is not an objective reality, but subjective, and is affected by the learner’s mind in the process of cognition. Knowledge is therefore obtained by actively organizing a set of generalized and rationalized experiences, in which the meaning of the experiences is constructed by individual learners. Another

theory

of

learning,

the

Open

Learning

Environments (OLEs), affect with concrete manipulation, focusing not only on how to overcome incorrect responses but also on how learning tasks can be designed (Hannafin, Land & Oliver, 1999). Indeed, this is a theory for situations where divergent thinking and multiple perspectives are valued over a single `correct` perspective (Land & Hannafin, 1996). The theory is based on social learning and is intended to foster critical

thinking,

inquiry-oriented

and

heuristic-based

learning. Thus, the open learning environments theory is appropriate for exploring fuzzy, ill-defined and ill-structured problems. Some of the values upon which this theory is based include: personal inquiry - promote learning through individual experience and personal theories; divergent thinking - encourage multiple perspectives; self-directed learning - provide learner autonomy with metacognitive support; hands- on concrete individual experience - support

32

Chapter One

involving relevant realistic problems; and provide tools and resources to aid the learners` efforts at learning (Hannafin, Land & Oliver, 1999). Moreover, Harrison and Treagust (2000) add another point of view by using multiple analogical models to explore students’ understanding of higher secondary chemistry concepts. Their results indicated that when analogical models are presented in a systematic way and when capable students are given ample opportunity to explore the models, their understanding of abstract concepts can be enhanced. Their study also confirmed the importance of motivation factors, such as “active learning” and “willingness to change,” as these play essential roles during the process of conceptual change. Regardless of the strategy, most methods include initiating some type of cognitive conflict within the learner, between his or her expectations, based on a misconception, and the actual observations presented. Hence, the authors believe that it will help students achieve conceptual changes and then reconstruct new scientific concepts. Indeed, when discussing what instructional strategies might be used to help students address their misconceptions, teachers were generally optimistic about their ability to mediate a misconception. They suggest

Misconceptions in Science Education

33

integrating videos, books, field trips, and eliciting prior knowledge, while hands-on experimentation (Gomez-Zwiep, 2008; Windschitl, 2002) was the most frequently mentioned method of moving students toward a more scientific understanding of a concept. Although the teachers advocated constructivist strategies that allow students to make sense of their misconceptions, such as hands-on activities and experiments, they would later fall back on the traditional teaching view that, if you tell a student a concept, they will internalize and comprehend it (Gomez-Zwiep, 2008). Thus, although conceptual change approaches have played a significant role in research on teaching and learning, as well as in instructional design, their influence on students` responses is limited.

1.4 Limits of the “classical” Conceptual Change Approaches Research has revealed that the conceptual change approaches of the 1980s and the early 1990s are not necessarily superior to more traditional approaches of teaching and learning science (Duit, 2002). In addition, Limon (2001) pointed out that these approaches put too much emphasis on sudden insights, facilitated especially by

34

Chapter One

cognitive conflict, while Fensham (2001) addressed this limitation by stating: “Another weakness in the range of alternative conceptions is that the focus in most of the studies is on isolated concepts of science, rather than on the contexts and processes of conceptualization and nominalization that led to their invention in science.” (Fensham, 2001, pp. 30).

However, it also becomes evident that actual practice is far from what conceptual change perspectives propose and that change, using this practice, continues to be a rather difficult and long-lasting process (Treagust & Duit, 2009). Moreover, research has shown that a single instructional method (like addressing students’ pre-instructional conceptions) usually does not lead to better outcomes per se, while quality of instruction is always due to a certain orchestration (Oser & Baeriswyl, 2001) of various instructional methods and strategies. Thus, learning science should be viewed as a gradual process involving "meta-conceptual" awareness of the students. In other words, students will be able to learn science concepts and principles only if they are aware about the shift of their initial meta-conceptual views towards the metaconceptual perspectives of science knowledge (Amin, Smith

Misconceptions in Science Education

35

& Wiser, 2014). Metacognition is seen by Georghiades (2000) as a potential mediator in improvement of conceptual change learning with primary school children, especially in terms of their inability to transfer their conceptions from one domain to another and the short durability of their conceptions, both of which give rise to problems faced by classroom practitioners. His model of learning draws upon three overlapping areas: conceptual change sets the epistemological background; transfer and durability of scientific conceptions are the problems to be addressed; and metacognition as the potential mediator for improving learning (Georghiades, 2000). In sum, cognitive psychologists were developing a new framework for understanding concepts and conceptual change. It was content-based and domain specific; it assumed that knowledge was organized in deeply similar ways to scientific theories, and it granted young children the same kind of concepts and conceptual organization as adults. It could explain how young children’s thinking could be, at the same time, so similar and so different from adults’ and scientists’ thinking, by proposing that concepts can change radically, while the format of representation stays the same (Amin, Smith & Wiser, 2014).

36

Chapter One

Hence, conceptual change strategies may only be efficient if they are embedded in a supporting learning environment that includes many additional features, such as specially organized instruction based on models of teaching (Hannafin, Land & Oliver, 1999; Land & Hannafin, 1996; Treagust & Duit, 2009). On the other hand, there is the possibility to address misconceptions from another perspective, based on a theoretical

framework

that

can

interpret

important

misconceptions in science and mathematics as evolving from general intuitive rules. Indeed, Stavy and Tirosh (1996; 2000) suggest another point of view regarding incorrect responses to variant tasks in sciences, which can be explained in line with the intuitive rules, as discussed in detail in their book “How students (mis-)understand science and mathematics: Intuitive rules” (2000). The intuitive rules idea, and specifically the intuitive rule “same A - same B”, is the leading theme in the current study, which will be discussed further in Chapters Two and Three.

References Abimbola, I. O. (1988). The problem of terminology in the study of student conceptions in science. Science Education, 72, 175-184.

Misconceptions in Science Education

37

Allen, M., & Coole, H. (2012). Experimenter confirmation bias and the correction of science misconceptions. Journal of Science Teacher Education, 23(4), 387-405. Amin, T.G., Smith, C., & Wiser, M. (2014). Student conceptions and conceptual change: Three overlapping phases of research. In N. Lederman, & S. Abell (Eds), Handbook of research on science education, (Vol. 2). New York: Routledge. Ausubel, D.P. (1968). Educational psychology: A cognitive view. New York: Holt, Rinehart and Winston. Bahar, M. (2003). Misconceptions in biology education and conceptual change strategies. Educational Sciences: Theory and Practice, 3 (1), 55-64. Bar, V. (1989). Children’s views about the water cycle. Science Education, 73, 481-500. Bar, V., & Travis, A. S. (1991). Children’s views concerning phase changes. Journal of Research in Science Teaching, 28, 363-382. Bar, V., Zinn, B., Goldmuntz, R., & Sneider, C. (1994). Children’s concepts about weight and free fall. Science Education, 78, 149-169. Caramazza, A., McCloskey, M., & Green, B. (1980). Curvilinear motion in the absence of external forces:

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Chapter One

naive beliefs about the motion of objects. Science, 210, 1139- 1141. Champagne, A., Gunstone, R., & Klopfer, L. (1983). Naive knowledge and science learning. Research in Science and Technological Education, 1 (2), 173-183. Driver, R., Squires, A., Rushworth, P., & Wood-Robinson, V. (1994). Making sense of secondary science: Research into children’s ideas. London: Routledge Duit, R. (2002). Bridging the gap between research on students’ conceptions and school practice. In P. Hewson, M.F. Rowe, M. Nieswandt, J. Lemberger, R. Tytler, M. Jiminéz-Aleixandre, M.G. Hennessey, R. Duit, & M. Beeth, A retrospective of research on students’ conceptions and its applications in educational practice. A symposium presented at the Annual Meeting of the National Association for Research in Science Teaching, New Orleans. Duit, R., & Treagust, D. (2003). Conceptual change: A powerful framework for improving science teaching and learning. International Journal of Science Education, 25 (6), 671-688. Einstein, A., & Infeld, L. (1938). The evolution of physics. New York: Simon and Schuster.

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Eryilmaz, A. (2002). Effects of conceptual assignments and conceptual

change

discussions

on

students’

misconceptions and achievement regarding force and motion. Journal of Research in Science Teaching, 39, 1001-1015. Fensham, P. (2001). Science content as problematic - Issues for research. In H. Behrendt, H., Dahncke, R., Duit, W., Gräber, M., Komorek, A., Kross, A., & P. Reiska (Eds.), Research in science education - past, present, and future (pp.

27-41).

Dordrecht,

the

Netherland:

Kluwer

Academic Publishers. Fisher, K. (1985). A misconception in biology: Amino acids and translation. Journal of Research in Science Teaching, 21, 53-62. Fisher, K., & Lipson, J. (1986). Twenty questions about students’ errors. Journal of Research in Science Teaching, 23, 783-803. Georghiades, P. (2000). Beyond conceptual change learning in science education: focusing on transfer, durability and metacognition. Educational Research, 42(2), 119-139. Gilbert, J., Osborne, R., & Fensham, P. (1982). Children’s science and its consequences for teaching. Science Education, 66, 623-633.

40

Chapter One

Gilbert, J., & Swift, D. (1985). Towards a Lakatosian analysis of the Piagetian and alternative conceptions research programs. Science Education, 69, 681-696. Halim, L., & Meerah, S. M. (2002). Science trainee teachers’ pedagogical content knowledge and its influence on physics teaching. Research in Science and Technological Education, 20, 215-225. Halloun, I.A., & Hestenes, D. (1985). Common sense concepts about motion. American Journal of Physics, 53 (11) DOI http://dx.doi.org/10.1119/1.14031 Hannafin, M., Land, S., & Oliver, K. (1999). Open learning environments: Foundations, methods, and models. In C.M. Reigeluth (Ed.), Instructional design theories and models: A new paradigm of instructional theory (pp. 115140). Mahwah, NJ: Lawrence Erlbaum Associates, Inc. Harrison A. G., & Treagust, D. F. (2000). Learning about atoms, molecules, and chemical bonds: a case study of multiple-model use in grade 11 chemistry. Science Education, 84 (3), 352-381. Hashweh, M. Z. (1988). Descriptive studies of students’ conceptions in science. Journal of Research in Science Teaching, 25, 121-134.

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Gomez-Zwiep, S. (2008). Elementary teachers’ understanding of students’ science misconceptions: Implications for practice and teacher education. Journal of Science Teacher Education, 19, 437-454. Guzzetti, B. (2000). Learning counter-intuitive science concepts: What have we learned from over a decade of research? Reading and Writing Quarterly, 16, 89-98. Land, S.M., & Hannafin, M.J. (1996). A conceptual framework for the development of theories-in-action with open-ended

learning

environments.

Educational

Technology Research and Development, 44 (3), 37-53. Limon, M. (2001). On the cognitive conflict as an instructional strategy for conceptual change: a critical appraisal. Learning and Instruction, 11, 357-380. McClelland, J. (1984). Alternative frameworks: Interpretation of evidence. International Journal of Science Education, 6, 1-6. Meyer, H. (2004). Novice and expert teachers’ conceptions of learners’ prior knowledge. Science Education, 88, 970– 983. National Research Council [NRC], (1997). Science teaching reconsidered: A handbook. Washington, DC: The National Academies Press, doi: 10.17226/5287

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Chapter One

Oser, F.K., & Baeriswyl, F.J. (2001). Choreographies of teaching:

Bridging instruction to learning.

In V.

Richardson (Eds.), Handbook of research on teaching (4th ed., pp. 1031-1065). Washington DC: American Educational Research Association. Pine, K., Messer, D., & St. John, K. (2001). Children’s misconceptions in primary science: A survey of teachers’ views. Research in Science and Technological Education, 19, 79-96. Posner, G. J., Strike, K. A., Hewson, P. W., & Gertzog, W. A. (1982). Accommodation of a scientific conception: Towards a theory of conceptual change. Science Education, 66, 211-277. Ross, K., Lakin, E., & Callaghan, P. (2004). Teaching secondary science. (Second edition) London: David Fulton. ISBN 1843121441 She, H.C. (2004a). Facilitating change in ninth grade students, understanding of dissolution and diffusion through DSLM instruction. Research in Science Education, 34 (4), 503-525. —. (2004b). Fostering “Radical” conceptual change through Dual Situated Learning Model. Journal of Research in Science Teaching, 41 (2), 142-164.

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Smith, J.P., diSessa, A.A., & Roschelle, J. (1993). Misconception reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3, 115-163. Stavy, R., & Tirosh, D. (1996). The role of intuitive rules in science and mathematics education. European Journal of Teacher Education, 19, 109-119. —. (2000). How students (mis)understand science and mathematics: Intuitive rules. New York: Teachers College Press. Tao, P., & Gunstone, R. (1999). The process of conceptual change in force and motion during computer supported physics instruction. Journal of Research in Science Teaching, 36, 859-882. Treagust, D.F., & Duit M, R. (2009). Multiple perspectives of conceptual change in science and the challenges ahead. Journal of Science and Mathematics Education in Southeast Asia, 32 (2), 89-104. Trend, R. D. (2001). Deep time framework: A preliminary study. Journal of Research in Science Teaching, 38, 191221. Tsai, C. (2003). Using a conflict map and an instructional tool to change student alternative conceptions in simple series

44

Chapter One

electric-circuits.

International

Journal

of

Science

Education, 25, 307-327. Wandersee, J. H., Mintzes, J. J., & Novak, J. D. (1994). Research in Alternative Conceptions in Science: Part II Learning. In G. L. Dorothy (Ed.), Handbook of research on science teaching and learning (pp. 177-210). New York: Macmillan Publishing Company. White, R., & Gunstone, R. (1992). Probing understanding. London: The Falmer Press. Windschitl, M. (2002). Framing constructivism in practice as the negotiation of dilemmas: An analysis of the conceptual, pedagogical, cultural, and political challenges facing teachers. Review of Educational Research, 72, 131–175.

CHAPTER TWO INTUITIVE RULES AND MISUNDERSTANDING OF SCIENCES

This chapter discusses the concept of intuitive rules as a theoretical framework that can be used to interpret the misconceptions, which many students and adults have about science and mathematics. It starts with examples, pointing to scientific misunderstandings in various content areas, including everyday experiences, demonstrates the use of the intuitive rule “more A - more B,” and finally presents the way the intuitive rule “same A - same B” is employed in comparison tasks, based on Stavy and Tirosh’s theory (1996, 2000). As shown in Chapter One, much research has been conducted on misconceptions held by students in mathematics and science education, with a focus on both identifying specific misconceptions and understanding their sources. Most of this research is content-specific and it is suggested that in the process of learning science or mathematics, students should restructure their specific conceptions to make

46

Chapter Two

them conform to currently accepted scientific ideas. However, studies indicate that in many cases students’ responses contradict each other and do not correspond to a single correct or incorrect concept regarding a specific content area (Clough & Driver, 1985; diSessa, 1993; Stavy & Tirosh, 1996, 2000; Tirosh, 1990). Moreover, there are inconsistencies in many of the students` existing ideas (Ross, Lakin & Callaghan, 2004), students are often unaware of their theories-in-use and might avoid a discussion of the conflict situation, hoping to suppress the conflict, and yet to appear competent (Argyris, Putnam & Smith, 1985). This discrepancy in students’ responses to different tasks in the same content area, calls into question the assumption that students have a certain perception, whether correct or incorrect, that guides them in their responses. For example, let us take a look at the following task: You have two identical boxes, one is full of sand and one is empty. The boxes are held at the same height above the ground. If we drop the boxes at the same time, will they hit the ground at the same time? As physics explains, free-fall speed is independent of a body’s weight and relates only to the height from which it was dropped, and is equal for all bodies falling from the same height (neglecting air resistance). Yet, research shows that

Intuitive Rules and Misunderstanding of Sciences

47

many students incorrectly argue: "Greater weight means they fall at a greater speed.” This response is described in the literature as a misconception in the specific content-related field. However, Stavy and Tirosh (1996, 2000) argue that many incorrect responses are not specific to a particular content, and extensive employment of the intuitive rule "more A - more B" can be demonstrated for a variety of content areas. For example, in mathematics: “longer horns - bigger angle”; in chemistry: “more sugar-water - more sweetness”; and in physics: “heavier - falls faster.” The structure common to all these tasks is, that each task has two items (or systems), which differ in terms of a prominent external feature. The student is asked to compare the two items with respect to another feature or quantity. The commonly observed response to comparison tasks of this kind is an answer having the following structure: "more of the prominent external feature more of the asked-about feature.” Stavy and Tirosh (2000) argue that incorrect responses such as "heavier - faster,” which are misconceptions, derive from the external characteristics of the task and employ a small number of intuitive rules. I clearly recall an incident from my own life when intuitive thinking led me to draw a mistaken conclusion. I first encountered the powerful and

48

Chapter Two

misleading effect of intuitive thinking when I was about eight years old, although I knew nothing then about intuition or thinking skills. My mother asked me to go to the nearby grocery to buy pasta, adding: "Please buy the thin pasta we usually eat." When the salesman asked me, "Do you want pasta number 8 or number 7?" I immediately responded, “7 please,” without hesitation. When I got back home, my mother said "I see you bought the thicker pasta." I still remember that moment. It never occurred to me that a larger number could indicate thinner pasta. I was obviously convinced that the larger the number the thicker the pasta, and vice versa. Indeed, it was one of those “aha” moments when it dawned on me that a larger number did not necessarily mean greater thickness, like thinking that I could be taller than a classmate, who was older than me. Thus, during an in-depth research focusing on students` incorrect responses in science and mathematics tasks, Stavy and Tirosh (1996, 2000) identify three intuitive rules - two of which relate to comparison tasks (“more A - more B”, “same A - same B”) and to successive division tasks (“everything can be divided”) - suggesting a theoretical framework that can be used to interpret misconceptions many students and adults have about science and mathematics. Their work suggests that

Intuitive Rules and Misunderstanding of Sciences

49

many incorrect responses, which the literature describes as misconceptions

or

alternative

conceptions,

could

be

interpreted as evolving from a small number of common, general intuitive rules.

2.1 The Intuitive Rules Theoretical Framework Based on extensive mathematics and science research, Ruth Stavy and Dina Tirosh (1996, 2000) indicate how learners react in similar ways to scientifically unrelated situations in scientific and mathematics tasks that have some common, external features. The finding that students react in similar ways to a wide variety of conceptually non-related mathematical and scientific tasks led the researchers to explain these inconsistencies in students` responses by the intuitive rules framework. Based on their observations, they argued that many alternative conceptions, apparently related to specific mathematical and scientific domains, are actually instances of intuitive rules. Such incorrect responses are well known in physics (e.g., in mechanics, electricity, light, and heat), chemistry (e.g. in material structure, concentration, and density) (Benson, Wittrock & Baur, 1993; Stavy & Stachel, 1985), biology (e.g., in growth, photosynthesis, and reproduction) (Treagust, 1988), Mathematics

(e.g., in

50

Chapter Two

perimeter, surface area, and functions) (Dembo, Levin & Siegler, 1997; Fischbein, 1987; Tirosh, 1990), and even in interpreting kinematic graphs (Eshach, 2014). As mentioned before, Stavy and Tirosh (2000) have identified three intuitive rules, two of which focused on comparison tasks: “more A - more B”, “same A - same B.” Responses of the “more A - more B”- kind are common in mathematical and scientific comparison tasks including in classic Piagetian conservation tasks. In all these tasks a participant is presented with two objects that differ in a salient quantity A (A1 > A2), and is asked to compare the objects with respect to another quantity B (B1 < B2, or B1=B2). They have suggested that students` responses are induced by an irrelevant salient quantitative feature of the task (A) and not necessarily by their ideas about the task-specific content. The argument that these rules are intuitive responses, stems from the nature of the responses. The typical features of these responses are that they are stated confidently, they are immediate, obvious and comprehensive - all characteristics of an intuitive response (Fischbein, 1987; Kahneman, 2011). Indeed, the intuitive rules can be applied as a conceptual framework for understanding students’ incorrect responses in a variety of subjects in science education. Moreover, they

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pointed out the predictable power of the intuitive rules that can serve as useful teaching strategies grounded in this framework and may be used to improve students` understanding of scientific and mathematics content. In addition, a cross-cultural study, conducted in Australia and Taiwan, indicates that the intuitive rules have universal features, as Taiwanese and Australian Aboriginal students, much like Israeli ones, respond incorrectly to conservation tasks in line with the intuitive rules (Stavy, Babai, Tsamir & Tirosh, 2006). As a matter of fact, the intuitive rules, which can be applied in a variety of subjects in science and mathematics, suggest a possible response for the weakness indicated by Fensham (2001) in the range of alternative conceptions. It states that the focus in most of the studies is on isolated concepts of science, rather than on the contexts and processes of conceptualization that led to their invention in science. Additionally, a psychological perspective on the intuitive rules can be based on the renowned psychologist, Daniel Kahneman (2011), for his work on the psychology of judgment and decision-making. Kahneman argues that we can think about our brains as having two systems – a super-fast, automatic, and intuition-based ‘system 1’ (S1) and a much

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slower, effortful, reasoning-based, ‘system 2’ (S2). Most of our daily decisions are produced by S1, are automatic and are based on habits. They require little attention or effort. Through experience, S1 allows us to become experts who can make fast, intuitive and mostly good decisions. Thus, S1 is the source of intuitions and emotional responses, and actually kicks in first, without us having any capacity to stop it. S2 is the deliberative system, making calculations or doing any non-obvious thinking, and Kahneman describes it as ‘lazy’, i.e., S2 won’t bother to get motored if S1 appears to have found an adequate answer. Thus, these intuitive incorrect responses may be considered as evolved from the automatic, super-fast and intuition-based ‘system 1’ (S1) instead of the effortful, reasoning-based, and much slower ‘system 2’ (S2). Notably, intuitive responses, such as these, are based on everyday experiences and are often correct, because a visual difference of a quantitative feature usually helps to correctly predict the right answer, when compared with a different size, e.g., a larger body occupies larger space. However, this intuitive rule sometimes leads to incorrect judgment as in classic Piagetian conservation tasks, which improve with age, development and learning. And since already at a young age, we encounter everyday incidents in which the intuitive rule

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“more A - more B” leads to incorrect responses (as in my example above: higher number - thicker pasta), we learn to approach the matter more cautiously. However, as was hypothesized, the incorrect use of the intuitive rule “same A - same B” in conservation tasks surprisingly increases with age and with the development of conservation ability, as will be discussed in detail in Chapter Three. That is why this study focuses on the intuitive rule “same A - same B” and how it can be used as a pedagogical tool in science and mathematics education.

2.2 The Intuitive Rule “Same A - Same B” The intuitive rule “same A - same B” is expressed in comparison tasks (similar to the intuitive rule “more A - more B”), but can be activated by situations in which A1=A2 while B1 B2. In such situations, a substantial number of students, who were asked to compare the quantity B1 and B2, incorrectly argued that B1=B2 since A1=A2. For example, children aged 4-14 were presented with two cups of water, one cup was full of water and the other cup was half full. One teaspoon of sugar was mixed into each of the two cups. The children were asked to compare the sweetness of the resulting solutions. Young children (aged 4-8) argued that the solutions

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were equally sweet (B1=B2) since each cup contained one teaspoon of sugar (A1=A2). Such behavior, which is usually .

called alternative conception and is explained as children’s difficulty in coping with inverse ratio in the context of intensive quantities, can be viewed as an instance of the intuitive rule “same A - same B”, i.e., same amount of sugar same sweetness (Stavy & Tirosh, 2000). In this case, the equality of quantity A (one teaspoon of sugar) was perceptually given, another example of using the intuitive rule “same A - same B”, in which the equality of quantity A is not perceptually given but logically deduced, is described in detail in Chapter Three, demonstrating a possible explanation to overgeneralization of conservation of quantities (e.g., perimeter, area or volume) in formal mathematical tasks.

References Argyris, C., Putnam, R., & Smith, M.D. (1985). Action science: Concepts, methods, and skills for research and intervention. San Francisco: Jossy-Bass. Benson, D. L., Wittrock, M. C., & Baur, M. E. (1993). Students’ preconceptions of the nature of gases. Journal of Research in Science Teaching, 30, 587–597.

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Clough, E.E., & Driver, R. (1985). Secondary students` conceptions of the conduction of heat: Bringing together scientific and personal views. The Physical Educator, 20, 176-182. Dembo, Y., Levin, I., & Siegler, R.S. (1997). Comparison of the geometric reasoning of students attending Israeli ultraorthodox and rain stream schools. Developmental Psychology, 33, 92-103. diSessa, A.A. (1993). Phenomenology and evolution of intuition, In D. Gentner, & A.I. Stevens (Eds.), Mental models (pp. 15-33). New Jersey: Lawrence Erlbaum Hillsdale Eshach, H. (2014). The use of intuitive rules in interpreting students’ difficulties in reading and creating kinematic graphs. Canadian Journal of Physics, 92(1): 1-8, 10.1139/cjp-2013-0369 Fensham, P. (2001). Science content as problematic - Issues for research. In H. Behrendt, H. Dahncke, R. Duit, W. Gräber, M. Komorek, A. Kross, & P. Reiska, (Eds.), Research in science education - past, present, and future (pp. 27-41). Dordrecht, the Netherland: Kluwer Academic Publishers.

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Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. D. Holland: Reidel Publishing Company. Kahneman, D. (2011). Thinking, fast and slow. New York: Farrar, Straus and Giroux. Ross, K., Lakin, E., & Callaghan, P. (2004). Teaching secondary science. (Second edition) London: David Fulton. ISBN 1843121441 Stavy, R., Babai, R., Tsamir, P., & Tirosh, D. (2006). Are intuitive rules universal? International Journal of Science and Mathematics Education, 4(3), 417-436. Stavy, R., & Stachel, D. (1985). Children’s conceptions of change in the state of matter: From ‘‘solid’’ and ‘‘liquid’’. Archives de Psychologies, 53, 331-344. Stavy, R., & Tirosh, D. (1996). Intuitive rules in Science and mathematics: The case of “more of A - more of B”. International Journal of Science Education, 18, 653-667. —. (2000). How students (mis)understand science and mathematics: Intuitive rules. New York: Teachers College Press. Tirosh, D. (1990). Inconsistencies in students` mathematical constructs. Focus on Learning Problems in Mathematics, 12, 111-129.

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Treagust, D.F. (1988). Development and use of diagnostic tests to evaluate students’ misconceptions in science. International Journal of Science Education, 10 (2), 159169. doi: 10.1080/0950069880100204

SECTION 2 THE INTUITIVE RULE “SAME A–SAME B”: A PEDAGOGICAL TOOL

CHAPTER THREE “I CONSERVE AREA, DO I REALIZE THAT VOLUME IS NOT NECESSARILY CONSERVED?”

As noted already in Chapter Two, this chapter discusses the intuitive rule “same A - same B,” with a focus on quantitative research, including the incorrect responses to conservation

tasks

of

four

hundred

students

(from

kindergarten to Grade 9). Such tasks draw attention to the non-conserved quantity following a specific manipulation, i.e., when one quantity is conserved, while another quantity is not conserved under the same manipulation, it results in the overgeneralization of conservation. Despite the ongoing substantial interest in conservation, there are some basic questions regarding the phenomenon, which remain unanswered. Principal among these is the question dealing with the non-conserved quantities: Does a child, who conserves a certain quantity following a specific transformation, realize that another quantity is not necessarily conserved? Does a child, who conserves a quantity, realize when this specific quantity is not conserved?

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3.1 Conservation Tasks: Conserved and Non-Conserved Quantities Piaget and Inhelder (1969) have identified the belief in the conservation of matter as a principal criterion for the period of concrete operational thought. Their findings indicated that, at about 7-8 years of age, the child typically begins to construct a systematic set of beliefs regarding the invariance of material properties such as quantity, length, area, volume and weight over various transformations of shape and position. Based on Piaget`s theory, conservation relates to the understanding that a quantitative relationship between two objects (or systems) is conserved (is unchanged) following a transformation in one of the objects (or systems). As was summarized by Flavell: “It was an act of creative inspiration, when Piaget hit upon the idea that a wide variety of cognitive areas - numbers, quantity, time, etc. - are in certain crucial respects mastered per a common procedure: to discover what values do or do not remain invariant (are or are not conserved) in the course of any given kind of change or transformation; only when this is done is the way paved for further operations…” (Flavell, 1963, pp. 415).

Hence, per Piaget, conserving means differentiating between variant and invariant (conserved) variables.

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A very interesting feature of transformations of shape is that they may alter certain quantities of the object, while leaving other quantities of that object unchanged. A good example of this is the effect of the “extending” or “elongating” transformation, on the area of two identical closed string loops. Thus, the real mystery about conservation acquisition is not how people arrive at conserving, but rather how they arrive at distinguishing between transformations which alter a quantity, from those, which do not. Indeed, it has been documented in the literature that a considerable number of students incorrectly conserve quantities, which are not conserved in a specific conservation task (Dembo, Levin & Siegler, 1997; Hirstein, 1981; Mehler, 1980; Russell, 1976; Shultz, Dover & Amsel, 1979; Stavy & Tirosh, 2000; Strauss, 1977; Walter, 1970). Per these researchers many students who correctly conserve area, for example, incorrectly respond that shapes with the same area have the same perimeters, or vice versa, students who correctly conserve perimeter incorrectly respond that shapes with the same perimeters have the same area. Such incorrect responses, which result in overgeneralization of conservation, were considered as a misunderstanding of the relationship between area and perimeter (Dembo, Levin & Siegler, 1997), or by the inability to

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distinguish conserving from non-conserving transformations in conservation problems (Shultz, Dover & Amsel, 1979). Each research focused on a specific subject - area and perimeter. A possible source for these common incorrect responses is suggested by Stavy and Tirosh (2000), who explain student responses in various content domains as resulting from a general and common intuitive rules framework: “everything can be divided” relate to successive division tasks and “more A more B” as well as “same A - same B” relate to comparison tasks. The following researches in this study are focused on the intuitive rule “same A - same B” in conservation tasks and its in-depth implication to mathematics and science teacher education (Tsamir, Tirosh, Stavy & Ronen, 2001).

3.2 The Intuitive Rule “same A - same B” in Conservation Tasks Previous studies (Livne, 1996) have shown that directly (perceptually) given equality induces the use of the intuitive rule “same A - same B.” Stavy and Tirosh (2000) hypothesized that this intuitive rule might also be induced in conservation tasks, in which the equality in quantity A is not directly given in the task but could be logically deduced. For example, consider the following task: you have two identical

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string loops, and we create two differently shaped rectangles. Are the perimeters of the resulting rectangles equal? Are the areas of the resulting rectangles equal? Surely, the question about the perimeter of the rectangles is a classical conservation problem, but what about area, which is not conserved? Would a child, who conserves perimeter, realize that area is not necessarily conserved? Following Piaget’s theory, children, who correctly conserved one quantity perimeter, would realize that another quantity - area, is not conserved. However, in line with the intuitive rule “same A same B”, which is an effective tool for predicting children responses, the hypothesis is that a child, who conserves one quantity, would argue that another quantity is conserved as well. Since conservation is developed with age (Piaget, 1968), the assumption was that this incorrect response, based on the intuitive rule “same A - same B”, will increase with age as well (Stavy & Tirosh, 2000). To test this hypothesis, the researcher presented 400 children aged 5-15 with several conservation tasks. Each task deals with two systems (or objects), in which a transformation was done, resulting in conservation of one quantity, while changing another. The

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children were then asked to compare the two systems in relation to both quantities.

3.3 Method Participants Four hundred students from kindergarten to Grade 9 - forty students from each grade level, participated in this part of the research. The students from kindergarten to Grade 6 belonged to an Elementary School, and the students from Grade 6 to Grade 9 belonged to a Middle School. Both schools are in an average-high socio-economic status city. The students were randomly selected, keeping an equal number of boys and girls. Although some have suggested focusing on the social interaction that would be expected from a collaborative approach to the construction of knowledge (Attard, 2012), in this study, the tasks have been designed for individual solution rather than directed at a group. This was done in the context of monitoring individual student responses and reasoning.

Research Tool This was a quantitative study based on a questionnaire specially designed for this study, the aim of which was to test the students’ performance, responses and reasoning to formal

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conservation tasks focusing on perimeter, area and volume. Each student was interviewed by the researcher, for about 30 minutes. The interviews were audio-recorded and transcribed by the researcher under an expert revision. Data were collected by counting correct and incorrect responses and analyzing students’ reasoning by the emerged categories, based on logical conservation scheme as follows, e.g., students` responses to the ‘derived rectangle task’: identity (originally, they are the same rectangles), compensation (one rectangle is long and narrow, the other is short and wide), or reversibility (the transformation on the derived rectangle can be reversed).

The Tasks Each student was presented with two pairs of conservation tasks, the first pair (study 1) relates to area and perimeter - the “Derived rectangle” task and the “String loops” task. The second pair (study 2) relates to the area and the volume “Turning a rectangle into a cylinder” task and “Filling cylinders” task. To avoid possible confusion between the concepts – the area, the perimeter and the volume, the researcher ask about each quantity by using the following terms: the space on the table, instead of the area, e.g., “Does rectangle 1 take up more/the same/less space on the table than

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rectangle 2?”; the length of the string needed to go around, instead of the perimeter - e.g., “Is the length of the string needed to go around the perimeter of rectangle 1

longer

than/equal to/shorter than that of rectangle 2?”; the amount of lentils, instead of the volume, e.g., “Is the amount of lentils needed to fill cylinder 1 greater than/equal to/less than that of cylinder 2?”. Since this wording doesn’t read smoothly, we refer to it for short as area, perimeter and volume.

Study 1 Study 1 deals with the area and the perimeter and comprises two tasks: The “Derived rectangle” task and the “String loops” task. Task 1: The “Derived rectangle” task: in this task, the area is conserved while the perimeter is changed, following a specific transformation. The student was presented with two identical rectangular (15 cm X 10 cm) sheets of paper (rectangle 1 and rectangle 2). One rectangle (rectangle 2) was divided to obtain two long and narrow rectangles that were reorganized, resulting in a double length rectangle (see Illustration 1). The student was then asked to compare the areas of rectangle 1 and rectangle 2:

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“Does rectangle 1 take up more/the same/less space on the table than rectangle 2?”

Subsequently, the student justified his/her response. The student was then asked to compare the perimeter of the two rectangles: “Is the length of the string needed to go around the perimeter of rectangle 1 longer than/equal to/shorter than that of rectangle 2?” Again, the student justified his/her response. Illustration 1: The “Derived rectangle” task

2 1

Task 2: The “String loops” task: in this task, the perimeter is conserved while the area is changed, following a specific transformation. The student was presented with two identical string loops. The interviewer changed the identical string

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loops into two differently shaped rectangles, rectangle 1 and rectangle 2 (Illustration 2). The student was then asked to compare the perimeters of rectangle 1 and rectangle 2: “Is the length of the string needed to go around the perimeter of rectangle 1 longer than/equal to/shorter than that of rectangle 2?” Subsequently, the student justified his/her response. The student was then asked to compare the areas of the two rectangles: “Does rectangle 1 take up more/the same/less space on the table than rectangle 2?” Again, the student justified his/her response. Table 1 details students` optional responses in line with Piaget’s theory and with the intuitive rule framework to each one of the described tasks. Illustration 2: The “String loops” task

1 2

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Table 1: Possible response types Student response in line with Piaget

“Derived rectangles” task

Area Equal to* The intuitive rule: “same A-same B”

Equal to*

Perimeter Smaller than* Equal to

“String loops” task

Area Larger than* Equal to

Perimeter Equal to* Equal to*

*Correct response

Study 2 The second pair of tasks relates to area and volume and comprises two tasks: The “Turning a rectangle into a cylinder” task and the “Filling cylinders” task. Task 1: The “Turning a rectangle into a cylinder” task: In this task, the area is conserved, while the volume is changed, following a specific transformation.

Each student was

presented with two identical rectangular sheets of paper (18 cm X 14 cm.), with one sheet rotated 90° (Illustration 3). The student was asked to compare the areas of rectangle 1 and rectangle 2. “Does rectangle 1 take up more/the same/less space on the table than rectangle 2?”, and to justify his/her response. At the next stage, each sheet was folded to create a cylinder. The student was then asked to compare the volumes

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of the resulting two cylinders “Is the amount of lentils needed to fill cylinder 1 greater than/equal to/less than that of cylinder 2?”, and to justify his/her response. Here again, Table 2 details students` optional responses in line with Piaget’s theory and in line with the intuitive rule framework to each one of the described tasks. Illustration 3: The “Turning a rectangle into a cylinder” task

1

90q

Task 2: The “Filling cylinders” task: In this task, the volume is conserved, while the area is changed, following a specific transformation (Illustration 4). Again, the student was asked to compare the conserved quantity and the non-

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conserved quantity in each one of the tasks. “Is the amount of lentils needed to fill cylinder 1 greater than/equal to/less than that of cylinder 2?” Subsequently the student justified his/her response. The student was then asked to imagine the rectangles that may be obtained by opening the cylinders, having emptied them of lentils. Finally, the student was asked to compare the areas of the two rectangles: “Does rectangle 1 take up more/the same/less space on the table than rectangle 2?” Again, the student justified his/her response. Illustration 4: The “Filling cylinders” task

1

2

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Table 2 details students` optional responses in line with Piaget’s theory and with the intuitive rules framework to each one of the described tasks. Table 2: Possible response types Student response in line with Piaget

The Intuitive Rule; “same A-same B” *Correct response

“Turning a rectangular to a cylinder” task Area Volume Equal to* Greater than* Equal to* Equal to

“Filling cylinders” task Area Larger than* Equal to

Volume Equal to* Equal to*

3.4 Key Findings The tasks in both studies will be discussed as follows: first we will relate to the conserved quantities in both tasks (e.g., the area in the “Derived rectangle” task and perimeter in the “String loops” task), and then we will focus on the nonconserved quantities in both tasks (e.g., the perimeter in the “Derived rectangle” task and the area in the “String loops” task).

Study 1 Study 1 deals with the area and the perimeter in relation to the tasks: “Derived rectangle” task, in which the area is

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conserved (while the perimeter is changed), and the “String loops” task, in which the perimeter is conserved (while the area is changed).

Area and Perimeter are conserved Area. The question dealing with the area in the “Derived rectangle” task is a classical Piagetian conservation question. Clearly, the areas of the two rectangles are equal, and as data show, from Grade 2 on, most (over 60%) of the students correctly conserved the area (see the area line in Figure 1). Most students based their judgment on identity (originally, they are the same rectangles), compensation (one rectangle is long and narrow, the other is short and wide), or reversibility (the transformation on the derived rectangle can be reversed). These findings are similar to those reported by Piaget and other researchers (Hoffer & Hoffer, 1992; Lunzer, 1968; Piaget, Inhelder & Szeminska, 1960).

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Perimeter. Similar results, which are in accordance with Piaget (Piaget, Inhelder & Szeminska, 1960), were observed for the perimeter question in the “String loops” task (see the perimeter line in Figure 2). The perimeters of both rectangles, created from identical string loops, are obviously equal but only from Grade 2 on, most students correctly conserved the perimeter based on logical operations such as identity (same string loops) and compensation (one rectangle is long and narrow, the other is short and wide). The results indicate that, as was widely documented in the literature, area and perimeter conservation develops with age.

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As mentioned before, per Piaget, conserving means differentiation between variant and invariant (conserved) variables. What will students’ responses be to the question dealing with the same, but non-conserved, variable? Would a student who conserves the area, for example, in the “Derived rectangle” task, correctly respond, following Piaget, that the perimeter is not conserved in the task, or would he incorrectly respond, following the intuitive rule ‘same A same B’, that the perimeter is ‘conserved’ as well?

Perimeter and Area are not conserved Perimeter. In the second part of the “Derived rectangle” task, the students were asked to compare another variable

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involved in the transformation, namely, the perimeter of the rectangles. Indeed, the perimeter of rectangle 1 is smaller than that of rectangle 2. As can be seen in Figure 1, there is a gradual increase, from kindergarten to Grade 5, in the percentages of students who incorrectly claimed that the perimeter of cylinder 1 is equal to that of cylinder 2. From Grade 4 on, more than half of the students incorrectly ‘conserved’ the perimeter. The students based their responses on identity: “The rectangles are identical but differently shaped.”

Some of the older students said: “The same area – the same perimeter.”

Here again, these results clearly indicate that students who conserved the area, ‘conserved’ the perimeter as well. Area. In the second part of the “String loops” task the students were asked to compare another variable involved in this manipulation, namely, the area of the created rectangles, predicting whether the rectangle areas are equal or not. In fact, the area of rectangle 1 is larger than that of rectangle 2. Yet, as can be seen in Figure 2, a gradual increase with age was observed in the incorrect responses. In fact, from Grade 5 on, over 80% of the students incorrectly claimed that the

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areas of the two rectangles are equal. Some of them explained: “The areas are equal since the rectangles were created from identical strings.”

Another explanation, based on compensation, was: “The areas are equal because rectangle 2 is larger but thinner than rectangle 1.”

Some students, mostly the older ones, even explicitly said, “The same perimeter – the same area.”

These results suggest that students, who conserved the perimeter, ‘conserved’ the area as well. Regarding the area and the perimeter related to these two tasks, there are several possible response types, some of which follow either Piaget or the ‘intuitive rule’. Table 1 indicates that per both Piaget and the ‘Intuitive Rule’, a student is expected to conserve the area in the “Derived rectangle” task and the perimeter in the “String loops” task. Indeed, from Grade 5 on, most students correctly conserved the areas in the “Derived rectangle” task and the perimeter in the “String loops” task. Thus, as was shown in two different tasks, relating to conservation of the area and the perimeter,

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our results are in line with Piaget`s theory. On the other hand, responses to the very same, but non-conserved, variables contradict Piaget’s theory. So far, concerning the conserved quantities, the results are in line with both Piaget and the ‘intuitive rule’ approaches. But when dealing with the ‘non-conserved’ quantities there is a difference between the two approaches. As data indicate (Figures 1, 2), there is a gradual increase with age in percentages of students` responses in line with the intuitive rule: students who incorrectly ‘conserve’ the perimeter in the “Derived rectangle” task and ‘conserve’ the area in the “String loops” task. The percentage of students who correctly follow Piaget, claiming that the perimeter in the “Derived rectangle” task and the area in the “String loops” task is not conserved, is negligible. These findings contradict the expected responses following Piaget’s theory and support the explanation based on the “intuitive rule” approach (Stavy & Tirosh, 2000). As the data illustrate, this incorrect response, in line with the intuitive rule “same A-same B”, gradually increases with age, showing no indication of decrement (Figures 1, 2). This incorrect response, has intuitive characteristics such as immediate use, self-evidence, great

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confidence and perseverance (Fischbein, 1987; Kahneman, 2011) despite perceptual input and formal education. Similar student and adult responses: “the same perimeter the same area”, or “the same area - the same perimeter”, were reported in mathematics education literature (e.g., Dembo, Levin & Siegler, 1997; Hirstein, 1981; Mehler, 1980; Russell, 1976; Shultz, Dover & Amsel, 1979; Spitz, Borys & Webster, 1982; Walter, 1970). The next study relates to the same participants, but is focused on other quantities, namely the area and the volume.

Study 2 Area and Volume are Conserved Study 2 deals with the area and the volume and comprises two tasks: The “Turning a rectangle into a cylinder” task and the “Filling cylinders” task. Area. In the “Turning a rectangle into a cylinder” task, the area is conserved, following a specific transformation, while the volume is changed. The question dealing with the area in the “Turning a rectangle into a cylinder” task is a classical Piagetian conservation question. Clearly, the areas of the two rectangles in the task are equal. Indeed, from Grade 2 on,

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most students correctly conserved the area (see area line in Figure 3). Most students based their judgment on identity (“It is the same rectangle”), compensation (“Rectangle 2 is tall and narrow while rectangle 1 is low and wide”), or reversibility (“You can turn it back and see that it is the same”). These expected findings are similar to those reported by Piaget and other researchers (Piaget, Inhelder & Szeminska, 1960; Shultz, Dover & Amsel, 1979; Lunzer, 1968; Russell, 1976).

Volume. In the “Filling cylinders” task, the volume is conserved, following a specific transformation, while the area is changed. Similar results were observed in relation to the

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volume question in the “Filling cylinders” task (see the volume line in Figure 4). From Grade 2 on, most students correctly conserved the volume based on logical operations such as identity (“The same amount of lentils is needed to fill up each cylinder”) and compensation (“Cylinder 1 is tall and narrow while cylinder 2 is short and wide”). Similar results were reported by Piaget and others (Piaget, 1968; Pinard & Chasse, 1977; Spitz, Borys & Webstrar, 1982).

The results indicate that, as was largely documented in the literature, area and volume conservation develops with age. Piaget (Piaget, 1968; Piaget, Inhelder & Szeminska 1960)

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explains the change in student responses with age as resulting from the development of the logical conservation scheme (identity, compensation, reversibility, additivity). What were student responses to the question dealing with the same, but non-conserved, variable? Volume and Area are Not Conserved Volume. In the second part of the “Turning a rectangle into a cylinder” task, the students were asked to compare another variable involved in the manipulation, namely, the volume of the cylinders. Indeed, the volume of cylinder 1 was larger than that of cylinder 2. However, as can be seen in Figure 3, there was a gradual increase, from kindergarten to Grade 5, in the percentages of students who incorrectly claimed that the volume of cylinder 1 was equal to that of cylinder 2 (‘conservation’ of volume). From Grade 5 on, almost all students (over 85%) incorrectly claimed that the volume of the two cylinders was equal. The students based their responses on identity: “The cylinders were made of identical rectangles,”

Or on compensation: “One cylinder is taller but thinner than the other.”

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Some of the older students said: “The same area - the same volume”.

Here again, these results clearly indicate that students who conserved the area, ‘conserved’ the volume as well. Area. In the second part of the “Filling cylinders” task the students were asked to compare another variable involved in this manipulation, namely, the area of the opened cylinders, predicting whether the area is equal or not. In fact, the area of the opened cylinder 1 was larger than that of opened cylinder 2. As can be seen in Figure 4, a gradual increase with age was observed in the incorrect responses. From Grade 6 on, most of the students (over 60%) incorrectly claimed that the area of opened cylinder 1 is equal to that of cylinder 2 (“conservation” of the area). Some of them explained: “The area is equal since the cylinders hold the same amount of lentils.”

Another explanation was based on compensation: “The area is equal because one cylinder is tall and thin while the other one is low and wide.”

Some students, mostly the older ones, even explicitly said, “The same volume - the same area”.

“I Conserve Area, Do I Realize that Volume is Not Necessarily Conserved?”

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These results clearly show that students who conserved the volume ‘conserved’ the area as well. Regarding the area and the volume related to these two tasks, there are several possible response types, some of which follow either Piaget or the intuitive rules. Table 2 indicates, both per Piaget and the intuitive rule, that a student is expected to conserve the area in the “Turning a rectangle into a cylinder” task and the volume in the “Filling cylinders” task. Indeed, as shown in the two different tasks dealing with area and volume conservation, our results are in line with Piaget`s theory. On the other hand, student responses to the very same but non-conserved variables contradict the expected reaction following Piaget`s theory. Per Piaget, students who have reached the stage where they conserve each of the two variables, should realize that either one of them (the area or the volume) might change in different manipulations. However, per the intuitive rule “same A same B” (Stavy & Tirosh, 2000), a student who conserves, for example, the area in the “Turning a rectangle into a cylinder” task will incorrectly ‘conserve’ the volume as well. Similarly, a student who conserves the volume in the “Filling cylinders” task will incorrectly ‘conserve’ the area as well.

Chapter Three

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As the data show, there is a gradual increase by age in percentage of students` responses in line with the intuitive rule - students who incorrectly ‘conserve’ the area in the “Filling cylinders” task (Figure 4) and ‘conserve’ the volume in the “Turning rectangle into a cylinder” task (Figure 3). The percentage of students who correctly follow Piaget, claiming that the area in the “Filling cylinders” task and the volume in the “Turning rectangle into a cylinder” task are not conserved, is negligible. Such evident findings, commonly related as misconceptions or alternative conceptions, contradict the expected reaction following Piaget`s theory and support the intuitive rule “same A - same B” presumption (Stavy & Tirosh, 2000).

3.5 Concluding Remarks Evolution with age of students’ incorrect responses to conservation tasks, focusing on the non-conserved quantity regarding area, perimeter and volume is presented in this part of the research. In each one of these tasks the two objects to be compared were equal with respect to one quantity A (A1=A2), equality, which was logically deduced based on the conservation scheme, but differed with respect to another quantity B (B1> B2 or B1 B2 or B1