The issue of mathematics teaching and its impact on learners' attainments in this subject has continuously been on

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- Dorit Patkin
- Avikam Gazit

*Table of contents : CoverTitleCopyrightContentsPrefaceAcknow ledgementsNotes on ContributorsPART 1What Should We Expect from Somebody Who Teaches Mathematics in Elementary School?Pre-Service Mathematics Teachers’ Attitudes toward Integration of Humor in Mathematics LessonsGeometric Thinking Levels of Pre-Service and In-Service Mathematics Teachers at Various Stages of Their EducationPART 2A Multicultural View of Mathematics Male-Teachers at Israeli Elementary SchoolsDo “Those Who Understand” Teach? Mathematics Teachers’ Professional ImageElementary School Mathematics Pre-Service Teachers’ Perception of Their Professional ImageRoot CanalEpilog*

The Narrative of Mathematics Teachers

The Narrative of Mathematics Teachers Elementary School Mathematics Teachers’ Features of Education, Knowledge, Teaching and Personality Edited by

Dorit Patkin Avikam Gazit

leiden | boston

Cover illustration: Photograph by Sara Gazit All chapters in this book have undergone peer review. The Library of Congress Cataloging-in-Publication Data is available online at http://catalog.loc.gov

Typeface for the Latin, Greek, and Cyrillic scripts: “Brill”. See and download: brill.com/brill-typeface. isbn 978-90-04-38400-2 (paperback) isbn 978-90-04-38401-9 (hardback) isbn 978-90-04-38406-4 (e-book) Copyright 2018 by Koninklijke Brill NV, Leiden, The Netherlands, except where stated otherwise. Koninklijke Brill NV incorporates the imprints Brill, Brill Hes & De Graaf, Brill Nijhoff, Brill Rodopi, Brill Sense, Hotei Publishing, mentis Verlag, Verlag Ferdinand Schöningh and Wilhelm Fink Verlag. All rights reserved. No part of this publication may be reproduced, translated, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission from the publisher. Authorization to photocopy items for internal or personal use is granted by Koninklijke Brill NV provided that the appropriate fees are paid directly to The Copyright Clearance Center, 222 Rosewood Drive, Suite 910, Danvers, MA 01923, USA. Fees are subject to change. This book is printed on acid-free paper and produced in a sustainable manner.

Contents Preface vii Acknowledgements xi Notes on Contributors xii

PART 1 Teachers’ Education and Teachers’ Knowledge 1 What Should We Expect from Somebody Who Teaches Mathematics in Elementary School? 3 Shlomo Vinner

2 Pre-Service Mathematics Teachers’ Attitudes toward Integration of Humor in Mathematics Lessons 16 Avikam Gazit

3 Geometric Thinking Levels of Pre-Service and In-Service Mathematics Teachers at Various Stages of Their Education 30 Dorit Patkin and Ruthi Barkai

PART 2 Teaching and Teachers’ Personality 4 A Multicultural View of Mathematics Male-Teachers at Israeli Elementary Schools 51 Eti Gilad and Shosh Millet

5 Do “Those Who Understand” Teach? Mathematics Teachers’ Professional Image 70 Dorit Patkin and Avikam Gazit

6 Elementary School Mathematics Pre-Service Teachers’ Perception of Their Professional Image 92 Nili Mendelson

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Root Canal 119 Avikam Gazit

Epilog 120 Dorit Patkin and Avikam Gazit

Preface The teaching profession is not perceived as prestigious and it is commonly believed that some teachers are forced to engage in this profession or choose it due to lack of other alternatives. Shulman (1986) presents a cynical quotation of Bernard Shaw: “Those who can – do and those who can’t – teach”. He slightly changed it into the slogan of “Tomorrow’s Teachers” reform in the United States: “Those who can – do and those who understand – teach”. Shulman emphasized three types of knowledge that teachers need: Professional knowledge, pedagogical knowledge and didactic-curricular knowledge (Shulman, 1987). Teaching is not a profession which is suitable for anyone. In education, there is a continuing challenge which aims to directly link teachers’ properties and characteristics to an effective education and teaching. This link is very important especially when it concerns the education of young children in elementary school (Thomason & La Paro, 2013). The question to be asked then is what about teachers’ personality? Studies which have examined the factors that affect students’ attainments, illustrate that teachers’ quality has a significant impact. Hattie (2003) conducted a meta-analytic study of factors that affect students’ academic attainments and found that the differences between teachers accounted for 30% of the variance in their students’ scores. A longitudinal study of Rockoff (2004) involved monitoring teachers for 10 years. The study found that the differences between those teachers accounted for 23% of the variance in their students’ attainments. There is a an educational and somewhat philosophical discussion about teachers’ centrality and the components of the teaching profession, according for example to Shulman’s (1987) approach. However, there is no validated scientific platform for answering who is the good, ideal or appropriate teacher. Wilson and Young (2005) offer three approaches to the determination of good teachers: scholar teachers, professional teachers and moral teachers. Scholar teachers are well educated, intellectual and apply a rich language. Professional teachers possess subject matter knowledge and are versed in a variety of teaching and learning methods. Moral teachers embrace caring values, believing that their students have emotional and intellectual skills. The third approach of caring teachers has become a leading approach in recent years due to the caring ideas of Noddings (2012). Many changes were introduced to teacher education during the last two decades (Beach & Bagley, 2013). The significant change in teachers’ education is not the acquisition of knowledge. Rather, it implies turning them into active partners to the knowledge processing and adaptation to the students. The implementation of ICT technology in the classroom and new methods of

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students’ assessment have changed the interaction between teachers, students and learning material. These changes have probably affected mathematics teaching and mathematics teacher education programs. The National Council of Teachers of Mathematics (NCTM, 2000) emphasizes six principles which address overarching themes: – Equity. Excellence in mathematics education requires equity – high expectations and strong support for all students. – Curriculum. A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well-articulated across the grades. – Teaching. Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them in learning it well. – Learning. Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. – Assessment. Assessment should support the learning of important mathematics and provide useful information to both teachers and students. – Technology. Technology is essential in teaching and learning mathematics; it impacts the mathematics that is taught and enhances students’ learning. These themes entail three main dilemmas in the professional education of mathematics teachers: 1. The gap between content knowledge of mathematics subject matter and pedagogical knowledge needed to teach heterogeneous student population generally (Jasmine & Singer-Gabella, 2011). 2. The gap between the goals of mathematics instruction, whereby students should view mathematics knowledge as challenging and interesting, and the teachers’ feelings and beliefs, which do not transcend this purpose and mission level (Vinner, 2011). 3. The gap between the real nature of mathematics and common perceptions about mathematics esoteric and alienated body of knowledge that only a privileged few are able to cope with its requirements (Dudley, 2010). Elementary school mathematics teacher education is designed with an emphasis on the elements of knowledge, personality and teaching methods. Unfortunately, not always there is a balance between these elements, and as a result both teachers and students lose out. This book aims to present chapters written by prominent scholars of mathematics education, referring to the following components of teachers’ image: education, knowledge, teaching and personality. Part 1 of the book deals with teachers’ education and knowledge components. The first chapter, written by Shlomo Vinner, discusses the realistic expectations

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from elementary school mathematics teachers. Vinner recommends that teacher education programs refrain from dealing with complex and unnecessary mathematics issues which Vygotsky (1986) argues are beyond teachers’ zone of proximal development (ZPD). Vinner suggests that in addition to the mathematics discipline, attention should be paid to other aspects of elementary school mathematics teaching. Chapter 2, written by Avikam Gazit, presents a study which explored the attitudes of elementary school mathematics preservice female teachers towards the integration of humor in class. Humor plays an important role in reducing anxiety and improving cognitive capabilities. Most of the participants concurred there is room for humor in class and expressed their willingness to introduce it in their lessons. Chapter 3, written by Dorit Patkin and Ruthi Barkai, presents a study of van Hiele geometric thinking levels of mathematics in-service and pre-service elementary school teachers. The study illustrates that many of the participants are versed in the topic of triangles and quadrilaterals on the highest thinking level. Conversely, the percentage of those versed in the topic of circles on the third level is lower. All the participants manifested a lack of mastery on the two highest levels as far as solids were concerned. Part 2 of the book deals with teachers’ components of teaching and personality. The first chapter in this part (Chapter 4), written by Eti Gilad and Shosh Millet, compares the motives and role perception of elementary school mathematics male-teachers who belong to different cultures: Israeli-born, Ethiopian immigrants, former USSA immigrants and Bedouins. The Israeliborn and former USSR immigrant teachers choose to engage in mathematics teaching on the basis of internal motives, such as sense of vocation, ambition to nurture the future generation or a wish to continue in a known state of learning. On the other hand, the internal motives of Ethiopian immigrants and Bedouins are integrated with external motives, such as wages, working conditions and social rewards. Chapter 5, by Dorit Patkin & Avikam Gazit, describes a study which examines the professional and personal self-image of elementary school mathematics teachers. According to the research findings, teachers maintain they embody most of the ideal properties while rejecting characteristics which are considered as unsuitable for teaching. The study found almost no significant differences between experienced and novice teachers. The last chapter (Chapter 6) is written by Nili Mendelson. It discusses a study which investigates the professional image of elementary school mathematics pre-service teachers in comparison with teachers of other disciplines’ image. The professional image was examined by means of a picture/metaphor series. The two figures with whom the participants identified to the greatest extent were the conductor and the animal keeper. At the same time, there was a sweeping rejection of the judge and animal trainer figures.

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References Beach, D., & Bagley, C. (2013). Changing professional discourses in teacher education policy back towards a training paradigm: A comparative study. European Journal of Teacher Education, 36(4), 379–392. Dudley, U. (2010). What is mathematics for? AMS Notices, 57(7), 608–613. Hattie, J. (2003, October). Teachers make a difference: What is the research evidence? Paper presented at the Australian Council for Educational Research Annual Conference on Building Teacher Quality, Melbourne, Australia. Jasmin, Y., & Singer-Gabella, M. (2011). Learning to teach in the figured world of reform mathematics: Negotiating new models of identity. Journal of Teacher Education, 62(1), 8–22. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Noddings, N. (2012). The caring relation in teaching. Oxford Review of Education, 38(6), 771–778. Rockoff, J. E. (2004, May). The impact of individual teachers on student achievement: Evidence from panel data. American Economic Review: Papers and Proceedings, 94(2), 247–252. Shulman, L. S. (1986). Paradigms and research program for the study of teaching. In M. C. Wittock (Ed.), Handbook of research on teaching (3rd ed., p. 153). New York, NY: Macmillan. Shulman, L. S. (1987). Knowledge and teaching: Foundation of new reform. Harvard Education Review, 57, 1–22. Thomason, A. C., & La Paro, K. M. (2013). Teachers’ commitment to the field and teacher-child interactions in center-based child care for toddlers and three-yearolds. Early Childhood Education Journal, 41, 227–234. Vinner, S. (2011, August). What should we expect from somebody who teaches mathematics in elementary school. In J. Novotna & H. Moraova (Eds.), The proceeding of the international Symposium on Elementary Math Teaching at Charles University (SEMT II). Prague. Vygotsky, L. (1986). Thought and language (English translation). Cambridge, MA: MIT Press. Wilson, S. M., & Youngs, P. (2005). Research on accountability processes in teacher education. In M. Cochran-Smith & K. M. Zeichner (Eds.), Studying teacher education: The report of AREA panel on research and teacher education (pp. 591–643). Mahwah, NJ: Lawrence Erlbaum Associates.

Acknowledgements The editors wish to express their heartfelt gratitude to all those who were involved in and contributed to the publication of the book. First and foremost, to all the writers who donated their knowledge and experience in writing the various chapters and to Sara Gazit for the extremely beautiful photographs, including the photograph on the cover of the book. Our thanks go to Nira Ben-Ari, the linguistic editor, who relentlessly invested time and thought in the verbal design of the book, bringing it to its final shape. We wish to express our thanks to Brill | Sense, to Peter de Liefde and Michel Lokhorst who assisted us in the preparation of the book. We are also indebted to the MOFET Institute for Research, Curriculum and Program Development for Teacher Educators that published the Hebrew version of the book and the Kibbutzim College of Education, Technology and the Arts that funded the publication of the English version. Personal Note by Dorit Patkin I contacted MOFET Institute with the idea of the book with which I had been preoccupied when my mother, the late Ahuva Sperling, was on her deathbed. She motivated and stimulated me in my academic studies, the choice of mathematics teaching as my profession for life and in my attainments in this field. She was an educator of the highest rank, a preschool-teacher who taught and educated numerous generations of children, including myself and my own children. I dedicate this book to her with my endless love.

Notes on Contributors

Editors Avikam Gazit Lecturer of mathematics education at the Kibbutzim College of Education, Technology and the Arts as well as faculty member of the Department of Education and Psychology at the Open University. Highly experienced in teaching methods of problem solution and in the integration of creativity, humor and history of mathematics teaching. Has published four books about thinking challenges, two books about the history of mathematics and a book of poetry. Dorit Patkin Head of the M.Ed. program in Mathematics education and a lecturer of mathematics education at the Kibbutzim College of Education, Technology and the Arts in Tel Aviv, tutor of in-service and pre-service teachers. Has a vast experience in teaching methods and in coping with mathematics students’ mistakes and misconceptions. Has published books engaging in geometry teaching and selected topics for high school students as well as a book about mathematical language and stories.

Authors Ruthi Barkai Head of the mathematics education department and a lecturer at the Kibbutzim College of Education, Technology and the Arts as well as in the Department of Mathematics, Science and Technology Education, School of Education, Tel Aviv University. Her areas of interest and research include: mathematics teacher education (from pre-school to high school), perception of mathematics concepts, proofs and justifications in mathematics teaching. Eti Gilad Lecturer in the Department of Education and the Department of Education Systems Administration at Achva College of Education as well as lecturer in the School of Education, Bar-Ilan University. Her research fields are: gender and feminism, change processes, educational entrepreneurship, leadership, multicultural education, teacher education, teachers’ professional development.

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Nili Mendelson Educational-organizational consultant. Manages the Department of Internship and Induction to Teaching at the School of Education, Haifa University. Lecturer in the M.Ed. pathway of Management and Organization of Education systems and in the pathway of teaching and learning – developing teachers for tutoring roles at the Gordon Academic College of Education. Her interest and research areas are: teacher education, novice teachers, teachers’ professional development, adults education, change processes, creative thinking, interpersonal communication, communication styles. Shosh Millet Former head of Achva College of Education (until 2006). At present is a lecturer in the M.Ed. pathway at Achva College of Education and coordinator of the “Growing Dialog” at MOFET Institute. Her interest and research areas include: novice teachers, knowledge of the worthy teacher, multiculturalism in teacher education, qualitative research, tutoring and instruction, unique teacher education programs for Ethiopian immigrants and excelling students, gender and mathematics literacy. Shlomo Vinner Professor of mathematics and mathematics education. In the past served as head of the Department of Sciences Teaching at the Hebrew University of Jerusalem and head of the Sciences Teaching Program at Ben-Gurion University of the Negev. At present serves as head of the M.Ed. degree pathway of elementary school mathematics education at Achva College of Education. His research areas include: mathematics concepts development, mathematics problem solution and value-oriented aspects of mathematics teaching.

PART 1 Teachers’ Education and Teachers’ Knowledge

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ChAPTER 1

What Should We Expect from Somebody Who Teaches Mathematics in Elementary School? Shlomo Vinner

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I would prefer to deal with the conference theme (The mathematical knowledge needed for teaching in elementary schools) in a broader context. The conference title restricts the knowledge to the domain of mathematics. I think that additional domains of knowledge are relevant to the elementary teachers. Thus, I would prefer a different conference title, such as: (I) The knowledge needed for teaching mathematics in the elementary schools. Since knowledge is, usually, restricted to cognitive issues and I think that there are additional issues which are relevant to teaching at the elementary level, I would like to address my talk to the following title: (II) What should we expect from somebody who teaches mathematics in elementary schools? However, it is possible that the intention of the conference organizers was to exclude aspects which are not mathematical knowledge from the conference agenda, assuming that these aspects should be addressed elsewhere. Therefore, I would like to start my talk by claiming that it is wrong to discuss mathematical knowledge needed for teaching in elementary schools without taking into account additional aspects of teaching mathematics. I believe that, all over the world, teacher training at this stage has adopted the approach that content knowledge and pedagogical content knowledge should be taught to those who prepare themselves to teach at the elementary level, as well as to teachers who come to us for further studies after serving for a while in the schools. In addition to the content knowledge and the pedagogical content knowledge, there is a consensus that elementary teachers should know something about children’s mathematical thinking. They should be aware of the causes for typical mistakes and should be able to understand children’s original ideas about doing mathematics, whether these ideas are correct or incorrect. This is in fact the message of Ball, Hill, and Bass (2005).

© Univerzita Karlova, Pedagogická fakulta, 2011 | DOI:10.1163/9789004384064_001

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I believe that in general, the majority of people who are involved with mathematics teacher training will agree with the above, but when it comes to details, the number of opinions is almost equal to the number of people who are involved in the domain. There are quite obvious reasons for that: As to the content knowledge – the content knowledge depends on the mathematics curriculum which is supposed to be covered at the elementary level. Different countries may have different curricula. Even in the same country the curriculum keeps changing over the years. It turns out that mathematics curriculum people have not reached an agreement about questions like: should we or should we not teach combinatorics or probability at the elementary level, or at what grade are we supposed to teach fractions or negative numbers? During the last fifty years, the elementary mathematical curriculum has been overloaded with some mathematical topics which are beyond the zone of proximal development (Vygotsky, 1986) of the elementary pupils. In addition to that, the dominant approach of the mathematical education community is that mathematical rules and procedures should be explained to the pupils. However, the real mathematical explanations are, sometimes, beyond the pupils’ zone of proximal development. Thus, some mathematics educators came up with alternative explanations which, supposedly, would be understood by the pupils. Unfortunately, some of these explanations are quite ridiculous. Their contribution to mathematics education at this level is rather negative. In this case, should we or should we not eliminate these mathematical topics from the curriculum? Should we or should we not give up the principle that every mathematical rule should be explained? At this point we can see that content knowledge decisions sometimes depend on pedagogical content knowledge available at a given moment. As claimed above, the number of opinions about these issues is almost equal to the number of people who are involved in the domain. As to the pedagogical content knowledge – different mathematics educators with different backgrounds or culture often have different opinions about the clarity and the efficacy of certain “real life models” which are supposed to explain why some mathematical operations are as they are. The following Internet excerpt (American Educator/fall, 1999) about multiplying negative numbers can clearly support my claim: Question: Are there any good websites or other resources to help explain negative times negative numbers? Reactions: Ds is going to through The Key to Algebra, Book-1 and it uses a football field explanation … She does not know much about football and it is confusing.

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Dd said her math book used football as well (Scott Foresman). She knows very little about football and feels his pain. I think he just memorizes. As to children mathematical thinking – there is a huge literature about it (many thousands of pages), how much of it and what exactly are we supposed to present to our teachers and prospective teachers? I am going to address shortly these issues but my main emphasis will be on other four questions which, in my opinion, are quite important (if not crucial) for the elementary teachers’ work: – Why do we teach mathematics? – What is mathematics? – In what ways does the teaching of mathematics serve the ultimate goal of education? An additional issue to all the above but an essential one is the question: To what extent do the elementary teachers have the necessary background to study what we expect them to now so that they will be able to implement the tasks that the education system presents to them? I would like to discuss first the last question since the answers to it determine the answers to the previous questions. 1.1 Typical Profiles of Elementary Teachers Elementary teachers have many profiles. They have many profiles even when speaking about one country. Since this is an international conference it seems that I was supposed to speak about several countries. In order to do it, I was supposed to rely on an early survey in these countries, and to present different profiles with their different distributions. Since there is no such a survey, I have chosen to present to you some anecdotal profiles of elementary teachers in my country and in the USA which were picked up from my own experience and from the mathematics education literature. Being concerned about the international relevance of my talk I would like to suggest the following justification for presenting such a local picture: a. There might be some similarities between the above two countries and other countries, and therefore, what I am saying might be relevant to other countries as well. b. The situation in other countries might be different than the situation in the above two countries. In this case, it is interesting to learn about other countries like tourists who visit other countries in order to see different landscapes. My anecdotal impression of talking to elementary school teachers (especially in grades 1–3) in my country and in the U.S.A., is that they decided to become

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elementary school teachers because they like the human interaction with little children. Being involved with the children’s intellectual and emotional development gives them a lot of satisfaction. Usually, they do not score very high on the college entrance examinations. For me, mentioning it is quite problematic. It is problematic because I cannot ignore the fact that this claim is expressed by some people with arrogance. It is also problematic because it suggests forming hierarchies of people by comparing their scores on college entrance exams. I myself do not believe in measuring people. The title of a canonical book on this issue is The Mismeasure of Man (Gould, 1981). To the educational community, I would like to suggest even a stronger title: The Immeasurable Man. Modern psychologists speak about many kinds of intelligence: emotional intelligence, social intelligence, and more (Gardner, 1993; Goleman, 1995). These kinds of intelligence are not less important to the success of teachers than their cognitive intelligence. Yet, there do not exist tests to measure these kinds of intelligence. It is not that I do not understand the need to evaluate the ability to study a certain domain when someone wants to study it at a college. However, since mathematics is an important component in college entrance exams, the fact that prospective teachers do not score high on college entrance exams may predict that they will have some difficulties studying certain mathematical topics. Supervision and in-service courses reveal quite often the mathematical weakness of some elementary teachers. For example, I am told by colleagues who teach remedial courses to these teachers that a significant number of them have difficulties in solving word problems like the following: 1. David holds 5/8 of the shares of a certain factory. He gives his son Daniel 2/3 of his shares. What part of the factory shares is owned by Daniel after this transaction? 2. Barbecuing meat causes it to lose 1/5 of its weight. What was the original weight of a piece of meat if after barbecuing its weight was 300 grams? In a study (Guberman, 2007), based on an adaptation of van Hiele four geometric levels to arithmetic, it was found that 63% of the pre-service teachers were below the third level in the beginning of the college arithmetic course. Only 4% of these students improved their location in the fourth level hierarchy at the end of the arithmetic course. Similar results were obtained in a study by Pandiscio and Knight (2011) which examined the van Hiele level of geometric understanding of pre-service mathematics teachers, both before and after taking the geometry course required by their teacher preparation program. Results indicate that prior to the course, pre-service teachers do not possess a level of understanding at or above that expected of their target students … the magnitude of the gains (obtained by the end of the course) was not enough to raise the sample population’s van Hiele level to that expected of their future K-12 students.

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There are more studies in which similar results were obtained but I am not mentioning them here because of space restrictions. I would like to conclude this section about elementary teacher profiles with a quotation from Ball, Hill, and Bass paper (2005): “many U.S. teachers lack sound mathematical understanding and skills … Mathematical knowledge of most adult Americans is often as weak, and often weaker”. 1.2 Some Recommendations From what I have said till now it follows that it is impossible to suggest a uniform list of mathematical topics that prospective teachers should study while preparing themselves to become teachers at the elementary level. However, I would like to suggest three pedagogical principles which should lead curriculum designers and teacher-educators at the colleges of education. Ausubel’s leading principle: “If I had to reduce all of educational psychology to just one principle, I would say this: The most important single factor influencing learning is what the learner already knows. Ascertain this and teach him accordingly” (Ausubel, 1968, p. vi). In other words, when instruction is designed, the starting point of the student should determine it. This implies that if the mathematical background of the student is poor we should first improve it and only later, move on to more advanced topics. The zone of proximal development principle (Vygotsky, 1986). Adapting the zone of proximal development principle to our situation implies that we should not try to teach our students topics which are beyond their intellectual ability. It is worthwhile mentioning that the notion of the zone of proximal development is quite vague. Namely, even if we know “what the learner already knows” we might have difficulties predicting whether the learner is able to cope with a given topic which presumably belongs to the learner’s ZPD. For instance, assuming the learner is familiar with the concept of rational numbers. Will he or she be able to learn meaningfully the concept of irrational numbers? The suitable pace of teaching. There is a general tendency to overload syllabi and then, because of the unwritten obligation to cover them, the pace of teaching is too fast for the decisive majority of the learners. However, ignoring the above principles leads to meaningless learning. Meaningless learning expresses itself very often in what I call pseudo conceptual and pseudo analytical behaviors (Vinner, 1997). I will elaborate on it later on. At this stage I would like to claim that teaching something which we do not really understand is disastrous. However, this is the case with many elementary teachers. Coming back to the prologue, my recommendations are the following:

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As to mathematical content knowledge (1.1), the above three principles can help us to determine a list of mathematical topics which can be presented to pre-service and in-service elementary teachers in different social and cultural settings. As explained earlier, these principles cannot lead to a uniform universal curriculum. Giving up a uniform universal curriculum is unacceptable by the majority of influential people involved in national systems of education. Unfortunately, because of the comparative international surveys in science and mathematics, education has become an international competition. Educators and educational policy makers can argue about the advantages and the disadvantages of this fact. However, if as a result of this competition a uniform mathematical curriculum for elementary pupils will be suggested, which will imply also a uniform mathematical curriculum for elementary teachers, it will not help mathematical education. We cannot overcome individual differences by uniform curriculum. On the contrary, individual differences should be taken care of by differential curricula. As to pedagogical content knowledge (1.2), I recommend that concrete models and representations should be used only if they are simple and clear. This is true for elementary teachers as well as for elementary pupils. And as to children mathematical thinking (1.3), since there is no canonical list for this topic I suggest that we should prefer clear, simple and straight forward texts to more sophisticated and complicated studies. For obvious reasons, I am not mentioning any specific texts. 1.3 Why Do We Teach Mathematics? I consider this question as a metacognitive one. Namely, elementary teachers who teach mathematics do not raise it and therefore do not have to answer it. They teach mathematics because it is part of the curriculum which they are supposed to teach. If the question is raised by an external agent there are ready made answers to the question. Usually, curriculum designers present a rationale for teaching the curriculum which they recommend. The most beautiful rhetoric for teaching mathematics which I know is the NCTM (2000, p. 1) rhetoric. We live in a mathematical world, whenever we decide on a purchase, choose insurance or health plan, or use a spreadsheet, we rely on mathematical understanding … The level of mathematical thinking and problem solving needed in the workplace has increased dramatically … Mathematical competence opens doors to productive future. A lack of mathematical competence closes those doors.

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This is not the place to elaborate in length how misleading these claims are. In short I will say only the following: No doubt mathematical knowledge is crucial to produce and maintain the most important aspects of our present life. This does not imply that the majority of people should know mathematics. Farming is also crucial to at least one aspect of our life – the food aspect, and yet, in developed countries, about 1% or 2% of the population can supply the needs of the entire population. In addition to this argument, if you are not convinced, I recommend to you to look around and to examine the mathematical knowledge of some high rank professionals that you know – medical doctors, lawyers, business administrators, and many others, not to mention politicians and mass communication people. Recently, an attempt to refute the above claim that the level of mathematical thinking and problem solving needed in the workplace has increased dramatically came from an unexpected source, a research mathematician, Underwood Dudley (2010), who sampled randomly from the yellow pages 8 categories of work places and found no evidence that algebra is required there, “even for training or license”. Another claim in the rhetoric which is supposed to justify the teaching of mathematics is the claim that mathematics is needed for everyday life. However, whenever I ask mathematics teachers who claim it for specific examples, the only examples they come up with are calculating tips in restaurants, calculating change (this concerns mainly taxi drivers) and cooking (calculating the amount of ingredients for n people, when the amount of ingredients for m people is given in the cook book). There might be other convincing arguments to study mathematics. Underwood Dudley claims that people should study mathematics in order to train their mind. However, there is no experimental evidence which supports the claim that, in non-mathematical domains, people who studied mathematics are better problem solvers than people who did not study mathematics. Another possible reason for studying mathematics is the application of certain mathematical chapters in sciences (physics, chemistry, biology, etc.). The question here is to what percentage of the population this claim is relevant and whether there are no other ways to reach this percentage rather than imposing mathematics on the entire population. Thus, if the above claims about the need to study mathematics are misleading, why do we teach mathematics and why do our students study mathematics in spite of all? You might suggest that the students believe in these claims although these claims are misleading. I suggest that the students have very good reasons to study mathematics. It is not the necessity of mathematics in their future professional life or in their everyday life. It is because of the selection role mathematics has in all stages of our education system. Mathematical achievements

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are required if you want to study in a prestigious place (whether this is a junior high school, a senior high school or university). A prestigious school increases your chance to get a good job. Confrey (1995) formulated it quite clearly: “In the vast majority of countries around the world, mathematics acts as a draconian filter to the pursuit of further technical and quantitative studies …”. Eventually, we have a convincing argument to study mathematics. Should we tell it to pre-service and in-service teachers? I believe we should. It is important for a soldier to know the real purpose of a battle in which he or she takes part. He or she should be convinced that there are good reasons for risking their life. Intentionally false rhetoric should be morally unaccepted. I am not claiming that there was a conspiracy to form a false rhetoric about the need to study mathematics. On the contrary, I think that the people who invented this rhetoric really believed in it bona fide. However, beliefs should be re-examined from time to time. The main thing is that teachers will have worthy goals for their endeavor. Is preparing students for crucial examination a worthy goal? I believe it is. Both students and teachers are victims of the same educational reality and as far as we can see, the chance to change this reality is very small. For a great part of the younger population, to continue their formal education (generally, not in a domain that requires mathematics) is an important goal. Pupils are expected to progress from the elementary level through the junior high level to the high school level and then to college and university. At crucial points of this journey, there are guards who examine them on mathematics. If the pupils pass the exams the guards let them move on. It is a worthy goal to help pupils complete this journey. Of course, there is much more to mathematics. There are intellectual values and educational values. Usually, because of the common way mathematics is taught, pupils are not exposed to it. I will come back to this point later on. 1.4 What Is Mathematics? This is again a meta-cognitive question. Generally speaking, people do not seek definitions for the notions they use. The meaning of the decisive majority of concepts in everyday thought is determined by means of examples and not by means of definitions. Some mathematicians, when being asked what mathematics is, prefer to give examples. Among them I can mention Courant and Robins (1948). Their book is full of mathematical examples. They probably believed that people, who were not mathematics majors but who were eager to know what mathematics is, would be able to understand the mathematical chapters which were presented in the book. Another book which deals with this question was written by Hersh (1998). This is a philosophical book which presents mathematics as a human endeavor. However, there is no attempt in this book to define mathematics as a generalization

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of specific examples. I believe that (in contrast to other concepts like poetry, art, etc.) it is possible to suggest a definition of mathematics which is a generalization of specific examples. It is true that in order to understand this definition one should have at least a Bachelor degree in mathematics. For this reason and also because of space restrictions I will not present it here. It will be presented elsewhere. However, the best place to look for simple and general definitions is the dictionary. The Webster’s Ninth New Collegiate Dictionary suggests that Mathematics is “the science of numbers and their operations, interrelations … and of space configurations and their structure …”. The mathematics which is characterized by this definition includes only arithmetic and geometry. It is not clear whether school algebra is included. Elementary school teachers are supposed to know these branches of mathematics. If you ask them what mathematics is, will they be able to give a similar definition to the one given by the dictionary? Out of 120 teachers whom I asked no one gave such an answer. Thus, we can assume that their idea of mathematics is determined by the mathematical experience they had in school and by the mathematics they teach. If you try to generalize this experience you may suggest that mathematics is a collection of procedures to be used in order to solve some typical questions given in some crucial exams (final course exams, psychometric exams, SAT etc.). When you present this view to mathematics teachers they notice the negative connotation of this statement and they try to reject it. However, the arguments they suggest in order to reject it are usually the following: i. Mathematics is not only for exams, it is also for real life situations. ii. Mathematics teaches you to think. When being asked to specify, in most of the cases, they use their right to remain silent. My question is whether we should tell the elementary teachers what really mathematics is? But before answering this question we should find out whether these teachers have the required mathematical and intellectual background to understand the answer. 1.4.1

In What Ways Does the Teaching of Mathematics Serve the Ultimate Goal of Education? Unfortunately, thousands of pages in educational philosophy have been written about the ultimate goal of education. It has also been the theme of hundreds of educational conferences. I say “unfortunately” because the too many trees prevent us to see the forest. Therefore, I would like to suggest a simple answer to this question. The ultimate goal of education is an educated person. This is, of course, circular. In order to avoid it, I would say that an educated person is a thoughtful person. “Thoughtful” in English is ambiguous. The above Merriam-Webster dictionary suggests the following:

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a. characterized by careful reasoned thinking … b. given to heedful anticipation of the needs and wants of others. In other words, “thoughtful” also means “considerate”. This can be tied to what is called in moral thinking, the golden rule. There are plenty of versions for this rule which come from various cultures and religions. One Jewish version of it is: “What you hate – do not do to other people.” In order to follow this rule, you should first apply your careful reasoned thinking. Namely, you should carefully analyze everyday situations and determine whether acting in a certain way in these situations will be unpleasant or even harmful to other people. Then you should control yourself and abstain from acting in such a way. Earlier I mentioned that mathematics education, the way it is taught in the majority of schools, focuses mainly on mathematical procedures by means of which typical questions in typical exams can be solved. Mathematical procedures have negligible importance in everyday life and in the majority of work places. However, procedures in general, play crucial role in everyday life and in all work places. By “procedure” I mean a sequence of actions which should be carried out one after the other. Crossing streets, driving, shopping, turning on dishwashers, dryers, DVD players (etc., etc.) are all associated with procedures. This is just an accidental choice out of an infinite list of procedures. Thus, respecting procedures as well as carrying them out precisely and carefully can be recommended as an educational value. Note that not following certain procedures is against the law. One example out of infinitely many: Crossing an intersection in a red light while driving a car. Not following other procedures can result in an economic damage. Again, one example out of infinitely many: Not turning off all the lights and electrical instruments when leaving home. Note also that many procedures in everyday life were formed in order to serve the golden rule. For instance: procedures related to behavior on lines, procedures related to pedestrians and drivers and procedures related to littering and recycling. Teachers, while teaching mathematical procedures can point to the pupils at procedures in everyday life and speak about the importance of following these procedures precisely and carefully, the same way as required in mathematics. By doing this, teachers add educational value discussions to their traditional role which is to cover the syllabus. Within the traditional role, the teacher is an instrument of the syllabus. By adding educational discussions to the syllabus, the syllabus becomes an educational instrument. There are many other contexts in the mathematical curriculum where educational value discussions can be integrated. However, once more, because of space restrictions I am not mentioning them here. Some of them are mentioned elsewhere (Vinner, 2007, 2008).

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The bottom line of this section brings me back to the beginning of my talk (the typical profile of the elementary teachers and to the recommendations related to it). While there are serious doubts about the feasibility of presenting to pre-service and in-service teachers some of the topics mentioned above, there should not be any doubt about asking them to work toward the ultimate goal of education, namely, asking them to be educators. Unfortunately, in some societies, teachers (and especially elementary teachers) are blamed for not having satisfactory knowledge in science or mathematics. However, if we focus on educational aspects of the teacher’s work then the above accusation becomes minor. Moreover, a counter accusation should be raised against parents who do not care so much about the education of their children but care a lot about their mathematical achievements; not because these achievements are really important for the future life of the children (as adults, 90% of them belong to the population which does not use mathematics, hates mathematics and does not know mathematics), but because mathematical achievements are required for further academic or technical studies. And as to the elementary teachers – imposing on them mathematical demands which are beyond their mathematical abilities or, if you wish, beyond their ZPD, will bear negative results as explained in the next section. 1.4.2

The Pseudo-Analytical and Pseudo-Conceptual Behaviors as a Reaction to Exaggerated Intellectual Demands In an early work of mine I explained in length what I mean by these two notions (pseudo-analytic and pseudo-conceptual behaviors). Here I will say very shortly that these behaviors are produced by people who try to show that they know a certain topic but as a matter of fact they do not know it (in some cases, people really believe they know while they do not know. In other cases, people know they do not know but they pretend they know). Here are two anecdotes: (I) The supervision of mathematics education in our country has decided that all elementary teachers should know certain chapters in probability. Thus, teachers who did not have this knowledge were invited to participate in a compulsory course in which some elementary concepts in probability were introduced to them. At the end of the course, among other questions, they were asked to solve the following question: There are 16 cards in a box. Each card is in an envelope. All the numbers between 1 and 16 (1 and 16 included) appear on the cards (one number per card). Describe an event the probability of which is 1/2. Non-negligent number of the teachers suggested that a right answer to this question was to pull out of the box the card which has 8 on it. When being

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asked to justify their answer, they said: Because 8/16 is 1/2. Superficially, it looks as a convincing analytical argument. I classify it as pseudo analytical behavior. (II) An elementary teacher taught how to calculate certain addition exercises by means of Cuisenaire rods. Then she gave some addition exercises to her class as homework assignment. One pupil solved all the exercises using his own (mathematically correct) strategy and got the correct answers for all the exercises. However, the teacher marked all his answers as wrong. When the child’s mother came to argue with the teacher about her judgment, the teacher’s response was that the use of the Cuisenaire rods is essential part of the final result. Here, the teacher did not distinguish between the pedagogical content and the mathematical content. I consider the failure to understand the conceptual difference between the two as a special case of pseudo conceptual understanding. 1.5 Epilogue I would like to conclude my talk with two comments. The first one is related to the elementary teachers. Observing them in their classes indicates that in most cases they are dedicated people. They do their best to teach mathematics. Sometimes, their best is not good enough mathematically. However, it is useless and pointless to request more than their best. The second comment relates to us, the mathematics education community. My recommendation to focus on aspects of the elementary teacher performance which are not content knowledge or pedagogical content knowledge may look to some of us as a threat. Improving mathematical achievements is considered by many of us as our ultimate goal. Do I recommend considering other educational values as the ultimate goal of education? As a matter of fact, I do, but it does not matter. There is nothing to worry about. The different education system (local, national and international) will not give up mathematical achievements as a draconian filter for further studying. Hence, improving mathematical achievements will still get the financial support that many of us look for, and mathematics education research will continue to focus on mathematical achievements. My recommendation, therefore, is only to look at things differently and, from time to time, to remind ourselves the real goal of education – an educated adult.

Acknowledgement This chapter was originally published as Vinner, S. (2011). What should we expect from somebody who teaches mathematics in elementary schools? Scienta in Educatione, 2(2), 3–21. © Univerzita Karlova, Pedagogická fakulta. Reprinted with permission.

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References Ausubel, D. (1968). Educational psychology: A cognitive view. New York, NY: Holt, Rinehart and Winston. Ball, D. B., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching. American Educator, 29, 14–46. Confrey, J. (1995). Student voice in examining “splitting” as an approach to ratio, proportion, and fractions. In L. Meira & D. Carraher (Eds.), Proceedings of the nineteenth conference of the int ernational group for the psychology of mathematics education (Vol. 1, pp. 3–29). Recife: Universidade Federal de Pernambuco. Courant, R., & Robins, H. (1948). What is mathematics? New York, NY: Oxford University Press. Dudley, U. (2010). What is mathematics for? AMS Notices, 57(7), 608–613. Gardner, H. (1993). Multiple intelligences: The theory in practice. New York, NY: Basic Books. Goleman, D. (1995). Emotional intelligence. New York, NY: Bantam Books. Gould, S. J. (1981). The mismeasure of man. New York, NY: W. W. Norton & Company. Guberman, R. (2007). The relationship between the developmental level of arithmetic thinking of preservice teachers, and the ability expected from them in meaningful learning (Unpublished Ph.D. dissertation). Ben Gurion University of the Negev, Beersheba, Israel. [Hebrew with an English abstract]. Hersh, R. (1998). What is mathematics, really? London: Vintage Books. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Pandiscio, E. A., & Knight, K. C. (2011). An investigation into the van Hiele levels of understanding geometry of preservice mathematics teachers. Journal of Research in Education, 20(1), 45–52. Vinner, S. (1997). The pseudo-conceptual and the pseudo-analytical thought processes in mathematical learning. Educational Studies in Mathematics, 34, 97–129. Vinner, S. (2007). From solving equations to the meaning of life: Mathematics, rationality and values. ZDM, 39, 183–189. Vinner, S. (2008). Some missing dimensions in mathematics teacher education. In D. Tirosh & T. Wood (Eds.), The international handbook of mathematics teacher education: Tools and processes in mathematics teacher education (Vol. 2, pp. 305–320). Rotterdam, The Netherlands: Sense Publisher. Vygotsky, L. (1986). Thought and language (English translation). Cambridge, MA: MIT Press.

CHAPTER 2

Pre-Service Mathematics Teachers’ Attitudes toward Integration of Humor in Mathematics Lessons Avikam Gazit

1

Theoretical Background

When, if ever, have you had an opportunity to smile in a mathematics lesson? The mathematician Littlewood (1953, p. 24) wrote that: “a good mathematical joke is better, and better mathematics, than a dozen mediocre papers”. Perhaps he also referred to a puzzle which sometimes comprised an element of humor or an illogical situation with which one had to cope, leading to a smile. Take, for example, the following puzzle (Gazit, 1996): “A Chinese meets his neighbor and asks what the age of his children is. The neighbor replies: the product of multiplying the ages of my three sons is 36 and the sum of their ages is equal to the number of the house where we live. The Chinese thinks and says: one given is missing. The neighbor answers: right, my first born plays the violin. What is the age of my three sons?” The violin makes us smile. What is the relation between the fact that the first born plays the violin and the age of the three children? This is the beauty of mathematics, manifested by this kind of humor. The violin is of secondary importance to the problem solution and what is important is the datum about the first born – namely there are no twin first born sons. Why is that important? If we write down all the possible combinations of the three ages – numbers with a product of 36, we find out that two of these combinations yield the same amount – the number of the house. The three ages 9, 2, 2 add up to 13 and so do 6, 6, 1 (the other six possible combinations result in sums which are different one from the other). The neighbor, who knows the number of the house, hesitates between the two options (if the combination was one of the six others which yield different totals he would not say that one given was missing and would solve the puzzle immediately). This is the catch of the puzzle. The solution is, obviously, 9, 2, 2, because there is one first born. However, mathematics teaching does not provide, to say the least, situations of jokes which might make us smile. © Israeli Journal of Humor Research, 2013 | doi:10.1163/9789004384064_002

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An example which makes us smile but presents mathematics teaching in a grotesque and cynical way is discussed in the article by Merseth (1993). She criticizes the way mathematics is taught in the United States and includes in her paper a problem given to 3rd-graders in the Midwest: “A shepherd has a herd consisting of 125 sheep and 5 dogs. How old is the shepherd?” This very question leads to the first smile, namely what is the relation between the shepherd’s age and the number of sheep in the herd? Nevertheless, educational researchers (Merseth, 1993) indicate that three out of four pupils gave an answer. In her paper, Merseth brings a photocopied answer taken from the notebook of one of the pupils: 125+5 = 130, too old! 125-5=120, still old! 125:5=25, that’s it, the shepherd is 25 years old. This answer makes us smile again, though sadly, because of the farfrom-satisfactory situation of mathematics teaching in the United States. Far be it from us to think that the situation in Israel is better. Perusal of text books illustrates boring learning material which includes recurrent exercises and monotonous problems without any trace of interest. All the books are written in an extremely serious manner, although here and there one can find some allusion to riddles or mathematical anecdotes relating to daily reality or to the history of mathematics. An American magazine for mathematics teaching (McGEoch, 1994) displays in a humoristic-ridiculous way the development of mathematics teaching during the second half of the 20th century by illustrating typical problems of each decade: The 1960s – A farmer sells a sac of potatoes for $10. His expenses constitute 4/5 of the price. How much will he profit? The 1970s – A farmer sells a sac of potatoes for $10. His expenses constitute 4/5 of the price, namely $8. How much will he profit? The 1970s (new mathematics) – A farmer exchanged a set P of potatoes with set M of money. The cardinality of set M is equal to 10 and each element of M is worth $1. Draw 10 big dots representing the elements of M. The set C of production costs is composed of two big dots less than the set M. Represent C as a subset of M and answer the question: What is the cardinality of the set of profits. The 1980s – A farmer sells a sac of potatoes for $10.00. His production costs are$8.00 and his profit is $2.00. Underline the word potato and conduct a discussion with your group members. The 1990s – A male/female farmer sells a sac of potatoes for $10. His or her production costs are 0.8 of his or her proceeds. Draw on your calculator a graph of the proceeds versus the costs. Run the POTATO program in order to find out the profit. Discuss with your group members the results. Write a short report which analyzes the example, using terms from the field of economics.

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This example evokes a smile which probably hides our dissatisfaction with the so-called changes introduced in mathematics teaching in order to make it more interesting or/and meaningful. Starting from a technical way of learning in the 1960s, to the study of arithmetics by means of set theory in the 1970s and then to teaching methods like small groups teaching and up to the use of the calculator. Nevertheless, the queen is naked because this is the same queen in a different coat which is sometimes less dense and less challenging. Mathematics is not supposed to be a difficult and complex subject. The way of teaching mathematics, though, using uninteresting text books and with not the best qualified teachers, turns learning into a traumatic experience for many pupils. The mathematician, philosopher and sociologist, Bertrand Russell, wrote that mathematics is a subject whereby no one understands what this is about. Yet if one does know, they are not certain it is right. The Jewish-Hungarian mathematician Erdos, known as the home-less mathematician, used to travel from one colleague to another, staying in their houses and writing papers together. He drank a lot of coffee to stay alert. The book written about him (Hoffman, 2000) includes the following saying attributed to him: “A mathematician is a machine which turns coffee into mathematical phrases” – it is nice and makes you smile. So, one can smile also in a mathematics lesson, seasoning it with humor and anecdotes without undermining the educational rights. It can be done in various ways, e.g. in a context of a mathematical topic, a figure which contributed to mathematics, calculation method, interesting exercise, formula, a certain proof and so on and so forth. For instance, when dealing with the orthogonal coordinate system conceived by Descartes, we can tell a bit about the man who was quite an adventurer or relate to his philosophy which advocated: I think, therefore I am. It can also be presented in negative way: I don’t think; therefore, I do not exist. Then, we can tell the joke about Descartes who entered the local pub one evening. The bartender greeted him and asked: Shall I serve you the usual drink? Descartes answered: “I think not” and immediately disappeared … “I think not” means “I don’t think” and therefore logically, he does not exist …). Humor plays an important role in communication between people, starting at a very early age. Humoristic contents which evoke laughter change in the transition from infancy, throughout childhood and up to adolescence. Humor is essential in interpersonal relations within the various groups to which people belong. A sense of humor is perceived as a criterion for assessing popularity and adolescents with a sense of humor are considered as having a higher social status (McGee & Shevlin, 2009). According to Freud (1960), humor facilitates relations which are not threatening in contexts of sex or aggression. Moreover, it

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helps to release tensions and prevent stress. The use of humor is perceived as an indicator of a positive mood, decrease of anxiety and depression as well as improvement of cognitive capabilities (Herzog & Strevey, 2008). In a study conducted by Ford, Ford, Boxer, and Armstrong (2012), adult students who were exposed to humorous cartoons performed better on mathematics test comparing to control group students who were exposed to non-humorous poems or nothing at all. The researchers’ explanation is that exposure to humorous cartoons reduced anxiety while taking the test. Another study (Berk & Nanda, 2006) investigated differences between humorous and serious versions of the same test content among graduate biostatistics students. The results showed significant impact of humor on performance in descriptive statistics items, a finding explained by the reduction of anxiety. The classroom is a social system to which various teacher-pupil and pupil-pupil messages are channeled. During mathematics lessons, there is usually a relatively high heterogeneity between the pupils. Low-achieving pupils lose confidence and their self-image is diminished in addition to the low sense of self-efficacy. Mathematics teachers who are aware of the situation can use humor as a means of improving the class climate and promoting inter-personal communication. Friedman, Friedman, and Amoo (2002) suggested how to use humor in statistics course for creating positive learning environment and for improving communication between students and teachers. They included in their article some jokes and humor to be used according to the order of presentation of introductory statistics. Humor leads to fraternity and equality between group members regardless of their status. Regarding the use of humor in class, it was well expressed by the reforming educator Alexander Sutherland Neill (in Cohen, 1996): Humor indicates equality. Those teachers who do not bring humor into their class do it deliberately since humor unites all those present and eliminates the distance between teachers and pupils. If teachers joke with their pupils, making them laugh, they undermine the attitude of politeness pupils show them. The pupils realize that the teachers are human, God forbids. (p. 148) If, for example, mathematics teachers want to lift the spirit of pupils who usually encounter difficulties in learning the subject, they can tell the story about Einstein (Calaprice, 1996). During the 1940s, Einstein was a famous scientist in the United States and he received letters about varied subjects from different people. One female high school student wrote him a letter, asking his advice about her problems in mathematics. Einstein told her not to worry because his problems in

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learning mathematics were much more serious … This is one way of presenting situations which evoke smiles in order to calm pupils and reduce their anxiety. Another way is telling a joke associated with a specific topic or some figure from the world of mathematics, like the story about Descartes described earlier. There is a website (Chercaev & Chercaev, 2000) which deals with humor in mathematics. The introduction indicates that mathematical folklore can cause pleasure to both mathematicians and pupils since every joke contains a certain amount of truth or lie about the subject. Below are several examples of jokes which mathematics teachers might use when teaching the subject or perhaps in the context of the relevant topic. Several scientists were asked to reply to the question: what is the product of 2x2? The engineer took out his slide rule (an “ancient” means of calculation used prior to the age of calculators), slid it forward and backward until reaching the result: 3.99. The physicist used the appropriate formulae, inserted them into his PC and announced that the result is somewhere between 3.98 and 4.02. The mathematician reflected for a while and then said: “I don’t know the answer but I can definitely tell you that it exists! “The philosopher smiled, asking: “But what do you actually mean by 2x2? “The sociologist said: “I don’t know but it was nice talking about it”. The student of medicine said: “4”. All the others looked at him, astonished: “How do you know? And the student replied: “I remembered it …”. This joke embodies some sort of irony and ridicule about mathematics teaching which, in fact, requires pupils just to remember, while the other scientists try to adjust the solution to their structure of knowledge. Teachers should be sensitive to the state of mind in class and to the pupils’ characteristics and avoid presenting humor which might be difficult to understand, might offend or be rejected by several pupils. Warwick (2009) found in a small- scale experiment among first year undergraduate students studying computing a diversity of opinions about what constitutes humor. This diversity depended on ethnic, gender, age and social grouping. Some people find it emotionally hard to accept humor due to suspiciousness; others could be introvert people who did not get connected to humor. People are apprehensive that others would laugh at them and that they would be involved in this laughter in the presence of other significant people (Platt & Ruch, 2009). The relevant conclusion is that one needs to be cautious when using humor without knowing the other side – the message addressee. However, beyond that, humor plays an important cognitive role in the ability to bring about more attention and concentration by taking some time-off in order to share a smile for the continued learning. Furthermore, humor is a means of improving learners’ creativity. Humoristic ability attests to a high

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intellectual level combined with creativity. Humor has an element of wit which is synonymous with sharp mind, sophistication and ability to conceive bright verbal ideas which require some degree of creative thinking (Garner, 2006). Using humor for the explanation of terms and principles might stimulate learners to look for creative ways of solving mathematical assignments. Only positive numbers have a real square root whereas for negative numbers an imaginary root was defined as I – imaginary. The root of 4 = 2/-2 and the root of -4 = 2i. And the joke: Two mathematicians meet. One of them says to the other: “I had a frightening dream in which I was minus one sitting under a root sign”. “What did you do?” asks his friend. “I jumped out and shouted ay (i) …”. Pupils can be encouraged to create amusing situations or invent mathematical jokes and humoristic phrases which relate to mathematical topics. Pupils can be asked to write five minutes’ humoristic ideas about various mathematical issues. There are mathematical terms with double meaning like root, power, division, function etc. Allowing a humoristic linguistic celebration. And most importantly for teachers – humor decreases burnout, improves self-image and attributes added value to the teaching process. Studies show that using humor is one of the criteria by which pupils identify the figure of good teachers (Kuperman, 2006). To sum up, we emphasize the positive features of humor as removing barriers, increasing attention, improving thinking and creativity processes, serving as consolidating means in a group, in addition to enhancing the self-image of both learners and teachers. Humor brings about a more pleasant atmosphere in class, reduces anxieties and can promote motivation and interest in the teaching of mathematics – one of the goals of teaching this subject. This research aims to examine the attitudes of elementary school mathematics pre-service teachers towards the introduction of humor into mathematics lessons. 1.1 Research Questions What are the attitudes of elementary school mathematics pre-service teachers towards the introduction of humor into mathematics lessons?

2

Methodology

2.1 Research Population Thirty-two pre-service teachers in a teacher education college specializing in elementary school mathematics teaching.

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2.2 Research Instrument A questionnaire containing 20 assertions relating to the use of humor in mathematics class. The responses to each item range from 5 – fully agree to 1 – do not agree at all. The assertions covered five categories: – The nature of mathematics (assertions 4, 5) – Attitude towards the integration of humor into mathematics lessons (assertions 1, 8, 17, 19, 20) – The advantages of integrating humor into teaching (2, 7, 9, 10, 11, 15, 16, 18) – The disadvantages of integrating humor into teaching (assertions 3, 12, 14) – The attributes of humor (assertions 6, 13) 2.3 Research Limitations The small sample does not enable to make generalizations but it may show some trends to be studied in future with a larger sample.

3

Results

3.1 Results Analysis 3.1.1 The Nature of Mathematics Two assertions (4, 5) related to the nature of mathematics lessons and of mathematics itself. The majority of participants, with a mean of 4.1, agreed that mathematics lessons were characterized by a very serious atmosphere, whilst one participant did not agree, and one did not agree at all. Four participants indicated that they held no opinion (3) either way. The majority of participants, with a mean of 1.5, did not agree that the nature of mathematics did not make the use of humor possible, whilst only one participant did agree with this determination, and one did not express any opinion (3). 3.1.2 Attitude towards the Integration of Humor The three assertions relating to the integration of humor in the teaching of mathematics, not from the human aspect of the subject, but from the overall aspect relating to teaching, were 1, 8, and 20, where two were asked in negative form: 1. There is no room for integrating humor in mathematics lessons. 8. The teacher must not use humor in mathematics lessons. The participants, with a mean of 1.4, 1.2 respectively, almost unanimously did not agree with these two assertions; only one participant fully agreed (5) that humor should not be integrated into mathematics lessons.

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Integration of Humor in Mathematics Lessons

table 2.1 Distribution of responses according to the diffferent statements and mean values (N=32)

Assertions relating to use of humor

Agree Fully Do not No Mean Do not agree agree at all agree opinion 4 5 3 2 1

1. There is no room for integrating humor 2. Reduces anxiety 3. Lack of respect for the teacher 4. Serious atmosphere 5. The nature of mathematics* 6. Equal status with the teacher* 7. Improved atmosphere 8. Must not be used 9. Improved communication 10. Efffective teaching* 11. Encourages thinking 12. Interruptions during the lesson 13. Elimination of gap 14. Attention is not serious 15. Reducing attrition 16. Enhancing creativity 17. I’ll put in a sense of humor 18. Improving self-esteem 19. I’ll include jokes 20. Agreement with Littlewood statement (1953)

1.4

23

8

0

0

1

4.5 1.7

1 15

1 13

2 3

6 0

22 1

4.1 1.5 2.6

1 18 6

1 11 10

4 1 8

14 1 4

12 0 3

4.6 1.2 4.4 4.2 4.3 2.6

0 27 0 0 0 2

0 5 0 0 1 11

0 0 5 6 4 16

12 0 10 12 10 3

20 0 17 13 17 0

2.8 2.0 3.7 4.5 4.5 3.5 4.2 4.2

6 13 5 0 0 2 0 0

6 11 1 1 0 3 0 2

10 4 4 0 2 9 8 4

9 3 12 12 13 12 11 13

1 1 10 19 17 6 13 13

* For these assertions there were 31 responses from among the 32 participants

Assertion 20 examined the degree of agreement with the point of view of the English mathematician Littlewood (1953), regarding the preference of a good mathematical joke over a dozen mediocre exercises. The mean was 4.2, which indicated that the majority of participants agreed, except two that did not agree, and four that did not express any opinion.

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Two assertions related to the opinion regarding integration of humor into mathematics at the personal level. Assertion 17 investigated whether or not the participants would introduce a sense of humor into their lesson. A mean of 4.5 indicated a high degree of agreement. Thirty of the thirty-two participants, (93.75%) either agreed or fully agreed, and only two did not express an opinion. Assertion 19 was similar, but was more concentrated, and relevant to introducing humor into the classroom: “I will introduce mathematical jokes …”. A mean of 4.3 indicated a high degree of agreement, although lower than the assertion about introducing a degree of humor. Only 24 participants (81.25%) agreed, or fully agreed, and 8 expressed no opinion. 3.1.3 The Advantages of Integrating Humor into Teaching Six assertions related to the potential advantages of humor for mathematics lessons and the degree of agreement increased from 4.2 for effective teaching, through 4.3 for improving thinking, 4.4 for improved communication, 4.5 for reducing anxiety and enhancing creativity, up to 4.6. There was a complete agreement for an improved environment, with no one indicating that they had no opinion (3). One participant disagreed with the use of humor in order to encourage thinking and improve creativity. One participant did not agree, and one did not fully agree with the use of humor as a means of reducing anxiety. Regarding reduction of anxiety, improved communication, effective teaching and enhancement of thinking, there were 2, 5, 6 and 4 participants respectively that expressed no opinion. Two assertions related to advantageous potential of humor for improving the teacher’s personal qualities. Assertion 15 related to the reduction of attrition. A mean of 3.7 indicated agreement, with twenty-two participants (68.75%) agreeing, or fully agreeing, as compared to one who did not agree, and five participants who did not agree at all. This illustrated that there was a certain distribution of responses to this assertion, with six of the participants (18.75%) who did not agree with the assertion that using humor reduced attrition, and four participants expressed no opinion. Assertion 18 related to the improvement in the teachers’ self-esteem. The mean of 3.5 was similar to that for the reduction in attrition, where only just over half the participants (56.25%) agreed or fully agreed. Five participants did not agree, or did not agree at all, and a relatively higher number of participants – 9 (28.12%), did not express an opinion. 3.1.4 Disadvantages of Integrating Humor Three assertions expressed potential disadvantages of integrating humor. The participants expressed a maximum degree of disagreement with assertion 3 – lack of respect for the teacher – with a mean of 1.7, and only one participant

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fully agreed. Three participants did not express an opinion. Also for assertion 14, “Attentions not serious”, which was close to the item of lack of respect, there was a mean degree of disagreement of 2.0, with three participants agreeing, and one participant fully agreeing that humor may result in a non-serious attitude to a lesson. Three participants did not express an opinion, and if we relate to the degree of non–agreement, or non–agreement at all, it reaches 75%. Regarding the third disadvantage – “Use of humor may result in disturbance” (assertion 12), half the participants preferred not to express an opinion. This was the highest rate of “no opinion” amongst all the items, where the mean for the expression of no opinion the remaining assertions was 12.3%, an approximate mean of four responses per item. Perhaps the lack of teaching experience prevented half of the participants from responding, as this concerned practical teaching. 3.1.5 The Attributes of Humor Two assertions related to the attributes of humor, expressed in the theoretical background as a means of removing barriers and creating equality amongst the participants. The equality of the teacher to the pupils can be an advantage or disadvantage, depending on how one looks at it from a subjective point of view. One assertion related to the use of humor as a factor in the equalization of teachers’ status to that of the pupils (assertion 6). Here there was a distribution of responses amongst all the degrees of agreement with a mean of 2.6 that was between agreement and no expression of opinion. Ten participants did not agree, and six did not agree at all. Thus, the percentage of not agreeing that humor created an equal status between teachers and pupils reached 50%. Four participants agreed with the assertion, and three totally agreed. Eight participants (25) did not express an opinion and one did not respond at all. Assertion 13 was connected in a different way to the status of teachers and pupils, and related to the use of humor as a means of eliminating the gap between them. The mean obtained, 2.8, was similar but a little higher than the previous one about the status of the teacher – 2.6 – twelve participants (37.5%) did not agree, and ten did agree (31.25%), with ten not expressing an opinion. There was almost a uniform distribution of agreement, disagreement, and no opinion, despite the fact that six participants did not agree at all and one totally agreed with the assertion.

4

Discussion and Conclusions

The aim of the research was to examine, from different viewpoints, the opinions of pre-service teachers specializing in elementary school mathematics

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teaching regarding the integration of humor into mathematics lessons. The results of the responses to twenty assertions of the questionnaire presented a positive picture of accepting humor as an instrument for improving different teaching components, with the desire to use humor without fearing that its integration may result in negative behavior. The majority of participants agreed that mathematics lessons were characterized by an overly serious environment (81.1%) and at the same time did not agree that the nature of mathematics did not have room for humor (90.6%). If this was the case, then what was their opinion about the integration of humor into the teaching of mathematics? Most decisively, all the participants disagreed that humor should not be used in the teaching of mathematics, and only one fully agreed that there was no room for integrating humor into mathematics classes. When asked to relate to the claims made by the mathematician Littlewood (1953), regarding the preference of a good mathematics joke over a dozen mediocre papers, there was 81.1% agreement, and no disagreements. Where the integration of humor into the teaching of mathematics reached the personal level of the participants who were asked to express an opinion regarding the integration of humor into lessons, there was almost complete agreement – 93.7% – with the introduction of an element of humor into mathematics lessons. The introduction of an element of humor was not compulsory and not operative as it is something relative – a teacher’s comment, such as “what’s up?” can also be interpreted as the introduction of humor. But when the item related to the integration of a joke – something more defined and operative, than just an element of humor, then the percentage of agreement dropped to 75%. Although this was still the majority, there were still two that did not agree. If there was a high rate of agreement for integrating humor into the teaching of mathematics at the personal level, then the participants considered it as an advantage. Six advantages were presented to the participants: Reduction of anxiety, improved environment, improved communication, more effective teaching, enhancement of attention and improved creativity. The percentage of agreement with these advantages ranged from 100% for improved atmosphere; 96.9% for improved creativity with one disagreement; 87.5% for reduction of anxiety with two disagreements and two with no opinion; 84.4% for improved communication and enhancement of thinking, for which there was one disagreement; and 78.1% for more effective teaching. The attitudes of these pre-service teachers were consistent with the findings in the research literature concerning the contribution of humor to teaching. According to Wagner and Urios-Aparisi (2011), it is possible to logically and intuitively assume that humor creates a pleasant atmosphere in the classroom and reduces anxiety. Humor improves cognitive ability (Herzog & Strevey,

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2008), and can create for the pupil the desire to look for creative ways for solving mathematical tasks. The responses expressed by the participants demonstrated perhaps, in one way or another, a social wish or response to what was expected according to the literature. On the other hand, there was no pedagogical reason to support the integration of humor into teaching as would be expected from a method of teaching or evaluation. The items in the category of disadvantages can yield more information. 87.5% did not agree with the lack of respect, and only one agreed with it. Less than 75% disagreed with a non-serious approach, and one agreed. However, regarding disturbance during a lesson, 50% of the participants chose not to offer an opinion (3) and this might attest to an internal conflict between expressing an independent opinion and wishing to satisfy the researcher. Only 40.6% disagreed and three participants agreed that the integration of humor into a lesson created disturbance. It should be remembered that the participants learnt in a college in order to obtain a teaching qualification and only a few had any practical experience. Teachers indicated the lack of respect by the pupils as one of the factors leading to attrition, resulting in disturbance and a nonserious approach. Nevertheless, the participants did not have a problem of lack of respect as a result of the use of humor … Two advantages of humor related to the effect of reduced attrition and improved self-esteem on the teacher. The majority of participants – 68.75% – agreed that using humor reduced attrition and 56.25% agreed that humor improved the teacher’s self-esteem. There were five participants that expressed disagreement, and the rest offered no opinion. Maybe the range of responses was the result of the lack of experience, as was also the lack of expression for these two sensitive items during the education process. However, the overall point of view of the participants’ opinions was consistent with there commendations of the research literature, i.e. emphasizing the function of humor for reducing attrition, improving self-esteem, and providing added value to the process of teaching (McMahon, 1999; Minchew, 2001). As Neill (in Cohen, 1996) wrote, humor brings about equality of teachers and pupils, something which may threaten teachers. In the two items that examined the elimination of the gap between teachers and pupils and the equality between them, there was a range of answers that might have exacerbated the problem of an uncertain situation in which the results were unpredictable. Moreover, the responses did not clarify whether the situation of equality between teachers and pupils when humor was integrated was negative or positive. This could also be one of the reasons of this distribution. 31.25% of the participants expressed agreement about the elimination of the gap, while 37.5% did not agree and 31.35% did not express any opinion either way – an almost meaningless equal distribution of opinions. On the other hand, 50% of the participants agreed with the position

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on the equal status of pupils and teachers, and only 21.9% disagreed. An important conclusion to be drawn from the findings of this research was the positive attitudes of elementary school mathematics pre-service teachers regarding the integration of humor into mathematics teaching as well as the stronger need for this in teacher education process. In the search for articles that deal with humor and mathematics teaching from 2000 onwards, I could not find even one. On the other hand, there were articles that dealt with the integration of humor into teaching foreign languages or English (McMahon, 1999; Minchew, 2001; Wagner & Urios-Aparisi, 2011). The researchers show that the use of humor is one of the criteria by which pupils identified the character of a good teacher (Kuperman, 2006). Furthermore, humor has potential for improving the process of teaching and enabling the teacher to seem more human and less threatening (Torok, McMorris, & Wen-Chi, 2004). This is the character of mathematics teacher that is needed by the education system which is seeking its way between the noncomplimentary results of international examinations and school efficiency and growth indices in mathematics. In conclusion, we emphasize the positive advantages of humor as removing barriers, increasing attention, improving the process of thinking and creativity. It serves as an instrument for group formations and cohesion, while enhancing the self-confidence of both teachers and pupils. Humor creates a pleasant atmosphere in the classroom, reduces anxiety, and can improve motivation and interest in teaching mathematics – one of the goals of teaching the subject.

Acknowledgement This chapter was originally published as Gazit, A. (2013). Pre-service mathematics teachers’ attitudes toward integrating humor in math lessons. Israeli Journal of Humor Research, 3, 27–44. Reprinted with permission.

References Berk, R. A., & Nanda, J. (2006). A randomized trial of humor effects on test anxiety and test performance. Humor, 19(4), 425–454. Calaprice, A. (1996). The quoted Einstein. Princeton, NJ: Princeton University Press. Chercaev, A., & Chercaev, E. (2000). Mathematical humor. Retrieved from http://www.math.utah.edu/~cherk/mathjokes.html#topic4 Cohen, A. (1996). The book of quotations. Israel: Kineret Press. [in Hebrew]

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Ford, T. E., Ford, B. L., Boxer, C. F., & Armstrong, J. (2012). Effect of humor on state anxiety and math performance. Humor: International Journal of Humor Research, 25(1), 59–74. Freud, S. (1960). Jokes and their relation to the unconscious (J. Strachey, Ed. & Trans.). New York, NY: Norton. Friedman, H. H., Friedman, L. W., & Amoo, T. (2002). Using humor in the introductory statistics course. Journal of Statistics Education, 10(3). Retrieved from http://www.amstat.org/publications/jse/contents_2002.html Garner, R. L. (2006). Humor in pedagogy: How Ha-Ha can lead to Aha. College Teaching, 54(1), 177–180. Gazit, A. (1996). Thinking to the point: Puzzles and challenging brainteasers. Israel: Masada Press. [in Hebrew] Herzog, T. R., & Stervey, S. J. (2008). Contact with nature, sense of humor and psychology well-being. Environment and Behavior, 40(6), 747–776. Hoffman, P. (2000). The Men Who Loved Only Numbers. New York: Hyperion. Kuperman, A. (2006). The use of humor in the teaching of mathematics. Mispar Hazak – Magazine for math teaching at elementary school. Haifa University, 11, 14–20. [in Hebrew] Littlewood, J. E. (1953). A Mathematician’s Miscellany. London: Methuen & Co. Ltd. McGeoch, C. C. (1994). Does anybody know what time it is? American Mathematical Monthly, 101(5), 459–463. McGee, E., & Shevlin, M. (2009). Effect of humor on interpersonal attraction and mate selection. Journal of Psychology: Interdisciplinary and Applied, 143(1), 67–77. McMahon, M. (1999). Are we having fun yet? Humor in the English class. The English Journal, 88(4), 47–52. Merseth, K. (1993). How old is the shepherd? An essay about mathematics education. Phi Delta Kappan, 74(7), 548–554. Minchew, S. S. (2001). Teaching English with humor and fun. American Secondary Education, 30(11), 58–70. Platt, T., & Ruch, W. (2009). The emotions of gelotophobes: Shameful, fearful and joyless? Humor: International Journal of Humor Research, 22(1–2), 91–110. Torok, S. E., McMorris, R. F., & Wen-Chi, L. (2004). Is humor an appreciated teaching tool. College Teaching, 52(1), 14–20. Wagner, M., & Urios-Aparis, E. (2011). The use of humor in the foreign language classroom: Funny and effective? Journal of Humor Research, 24(2), 399–434. Warwick, J. (2009). An experiment relating humor to student attainment in Mathematics. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 19(4), 329–345.

CHAPTER 3

Geometric Thinking Levels of Pre-Service and In-Service Mathematics Teachers at Various Stages of Their Education Dorit Patkin and Ruthi Barkai

1

Theoretical Background

There are various theories dealing with the development of geometric thinking. One known theory was conceived by the Dutch wife and husband Dina van Hiele-Geldof and Pierre van Hiele (van Hiele, 1959, 1987). In 1959 the couple argued that there were five hierarchical levels. Due to doubts of mathematics educators, including Van-Hiele himself, as to the existence of the fifth level it is customary today to relate only to four levels: recognition (naming)or visualization level, analysis or description level, ordering or informal deduction level and rigor and deduction or formal deduction level (Burger & Shaughnessy, 1986; Gutierrez, 1992; Patkin & Levenberg, 2010; van Hiele, 1987). – Level I: Recognition or visualization: at this initial level, learners can identify geometric shapes and distinguish between them. Each of the concepts or the shapes is perceived as a whole, in the way it is seen. Learners are capable of distinguishing between similar shapes as well as name them. At this level, learners are unable to specify the properties of those shapes. – Level II: Analysis or description: at this level, learners can analyze properties of shapes but are unable to attribute properties of a particular item to the properties of the group to which it belongs. – Level III: Order or informal deduction: at this level learners identify a hierarchical order of connection between groups of different shapes according to their properties and definitions. However, at this level, they are incapable of proving claims related to the properties of the geometric shapes. – Level IV: Formal deduction and rigor: at this level learners understand the roles of basic concepts, axioms, definitions, theorems and proofs and their interrelations. They can use assumptions in order to prove theorems and understand the meaning of necessary and sufficient conditions. At this level learners are able to provide reasons and arguments or the various levels of the proving process. Moreover, they comprehend the importance © Hong Kong Educational Research Association, 2014 | doi:10.1163/9789004384064_003

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of discussing the proofs, the deduction from the particular to the general and even the need for a proof of any kind. Van Hiele (in Crowley, 1987) presents five properties of the model: sequential, advancement, intrinsic and extrinsic, linguistics and mismatch. Sequential: Similarly to most models which engage in development, one should undergo through the levels in a sequence. In order to succeed at a certain level, the strategies of previous levels should first be acquired. Advancement: Progress from one level to another depends more on contents and teaching methods than on age. The teaching methods should ascertain that learners do not skip or omit one level. Some of the teaching methods stimulate the progress and reinforce it whereas others delay or even prevent progress between two levels. Intrinsic and extrinsic: “The inherent objects at one level become the objects of study at the next level” (Crowley, 1987, p. 4 – free translation), i.e. a concept studied at a certain level ‘from above’ and generally speaking becomes the topic of study at the next level. For example: at the first level of van Hiele’s theory, the general matrix of the geometric shape is studied while at the second level the shape is already defined according to its properties and components. Linguistics: Each level has its own linguistic symbols which characterize it. At the first level the symbols are very simple and at higher levels the symbols are more complicated. For example: a square is the simplest linguistic symbol assigned to a regular quadrilateral. However, the names rectangle which is also a rhombus or a rhombus which is also a rectangle match the definition of a square. Only learners who are at the third level (order and informal and informal deduction) can understand it while learners who are at lower levels encounter difficulties in understanding it. Mismatch: Learners who are at a certain level will find it difficult to understand contents and vocabulary typical of higher levels. Hence, it will be difficult for them to monitor the processes which transpire at the high level. In order to comprehend the contents and the processes of the higher level, one should first understand and master all the contents and all linguistic symbols typical of their level. According to van Hiele’s theory, partial mastery of a certain level is a prerequisite, though insufficient for mastering a higher level. People cannot be versed in a certain level before having mastered all its previous ones; otherwise, they are referred to as “inconsistent”. In their theory, the van Hieles related to plane geometry only. Some studies have recently applied the theory of plane geometry to other branches of mathematics, such as solid geometry (Patkin, 2010; Patkin & Sarfaty, 2012) and arithmetic (Crowley, 1987; Guberman, 2008). It is important to point out that in most studies of thinking levels,

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difficulties generally focus on topics associated with plane geometry only, e.g. triangles and quadrilaterals (Halat & Sahin, 2008; Patkin, 1990; van Hiele, 1999) or studies which engaged only in difficulties of thinking levels in solid geometry (Gutierrez, 1992; Patkin, 2010; Patkin & Sarfaty, 2012). There are no studies which relate to the mastery of thinking levels in different issues as a comprehensive picture and to comparison of mastery of thinking levels between the subjects. Another characteristic of the van Hieles’ theory is that unlike other learning theories, particularly that of Piaget (1969), their theory is grounded in the assumption that moving from one thinking level to another depends on teaching or learning experiences rather than on age or biological maturity (Geddes, Fuys, Lovett, & Tischler, 1982; van Hiele, 1999). Studies indicate that pupils encounter difficulties at every age (Koester, 2003) as do in-service and preservice teachers (Halat & Sahin, 2008). A study of geometry and spatial thinking was conducted among kindergarten children and 6th-graders that had to perform the same assignment. The findings showed that in spite of the age gap there was only a minute difference in favor of the older children in performing the assignment (Clements & Battista, 1992). Another study explored “self-knowledge” of elementary school mathematics in-service teachers. It illustrated that while exposing their self-knowledge, in-service teachers manifested lack of mastery and comprehension of solid geometry. However, after getting acquainted with the van Hiele’s theory, including experiencing, being in situations which encouraged reflection on reflection (“meta-cognition”), they progressed in their thinking levels, demonstrating openness and wish to learn, cope and improve (Patkin & Sarfaty, 2012). According to van Hiele’s theory, based on the assumption that advancing from one thinking level to another is teaching-dependent, Crowley (1987) also argues that the activity type given to learners is meaningful. His study which investigated plane geometry considerations, showed that the compliance between learners’ level of comprehension and level of the assignments given to them is vital, if meaningful learning is to transpire. The present study focused on the first three levels of the van Hiele’s theory. Assuming that progress in geometric thinking levels (according to van Hiele) is teaching-dependent, we deemed it appropriate to examine whether generally speaking there are differences in geometric thinking levels of elementary school mathematics teachers. Namely, differences in the level of thinking between mathematics pre-service teachers in their first, third and fourth year of education as compared to the level of thinking of mathematics in-service teachers studying towards an M.Ed. degree in mathematics education as well as of academicians making a career change to mathematics teachers. Furthermore, we explored whether there were any differences in the participants’

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level of geometric thinking regarding three specific topics studied at elementary school (triangles and quadrilaterals, circles and solids). 1.1 Research Questions – Is there a difference in geometric thinking levels of mathematic pre-service and in-service teachers at different points during their education in geometry in general and in each of the three topics: triangles and quadrilaterals, circles and solids in particular? – At what thinking level do mathematics pre-service and in-service teachers at different points during their education master various topics of plane geometry (triangles and quadrilaterals, circles) and space geometry (solids)?

2

Methodology

2.1 Research Population The research population consisted of 142 mathematics in-service and pre-service teachers studying in an academic teacher education college. This population comprised five different groups of participants: 46 pre-service teachers in their first year of education for becoming elementary school mathematics teachers; 30 pre-service teachers in their third year of education; and 17 pre-service teachers in their fourth year of education. It should be mentioned that the pre-service teachers did not attend a geometry course during their second year of education; hence, second year pre-service teachers were not included in the present study. Moreover, the research population included 24 mathematics in-service teachers, studying towards their M.Ed. in elementary school mathematics education as well as 25 participants with a BA or MA degree in another discipline, making a career change to mathematics teaching at elementary school and junior high school. 2.2 Research Design 2.2.1 Research Instruments The research instruments included an attainments questionnaire with multiple choice questions, designed to determine the respondents’ attainments and thinking level according to van Hiele’s theory. The questionnaire comprises 45 close-ended questions. Five optional answers were given to each question and the respondents had to choose the correct one. The questionnaire consisted of 15 questions about solids and 30 questions related to various topics of plane geometry: Half of them (15 questions) dealt with triangles and quadrilaterals

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and the rest (15 questions) related to circles. Each five questions in a group of 15 represented a separate thinking level on a higher hierarchical order. The questions were presented in the following order: five questions at the first level dealing with triangles and quadrilaterals. Then five questions at the first level associated with circles and five questions about solids. Afterwards there were five additional questions about each of the topics representing the second thinking level and finally a group of 15 questions about the three topics, representing the third thinking level. The time allocated to answering the questionnaire was 45 minutes. All the questions were based on previous questionnaires, developed by Patkin (2010) and Patkin and Levenberg (2004, 2010), which have been processed and validated. The test reliability was 0.84. Validation of the content was determined by a logical-scientific analysis. The items were sent to five judges, all of them researchers in the field of mathematics education. These judges were requested to categorize the items into the different levels according to the van Hiele’s theory, indicating any items which were irrelevant or which did not comply with the appropriate criteria. Moreover, they were asked to point out unclear formulation which might have resulted in misunderstanding of the item. Assertions about which less than four judges concurred, were removed from the questionnaire. 2.2.2 Examples of Questions The questions at the first level included questions of identification and distinction. As mentioned above, at this initial level learners can identify geometric shapes and discern between them according to a drawing. Each of the concepts or the shapes is perceived as a whole, as it is seen. At this level, learners have not yet mastered the properties of those shapes. An example of a question at the first level dealing with the topic of circles: Below is a drawing of three circles: O, M and A.

Which of the following claims is correct? a. In circle A – MA is the diameter; b. In circle O – KE is the diameter;

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c. In circle A – MO is the diameter; d. In circle O – MO is the diameter; e. In circle M – KE is the diameter. In order to answer the question, learners should focus on each of the circles separately and identify whether the segment indicated in it is indeed a diameter. The questions at the second level relate to geometric properties. At this level learners are supposed to identify a certain shape, be aware that it has several properties and check the existence or non-existence of the properties with regards to the shape. An example of a question at the second level on the topic of solids: Which of the following claims is true regarding every prism? a. A prism envelop is entirely built of triangles; b. A prism has two parallel bases; c. The prism bases are rectangular; d. The prism envelope consists of regular polyhedron; e. Four sides meet at every prism vertex. The third level included questions relating to relationships of connection between the various shapes and solids and to drawing conclusions. At this level, learners can manipulate the properties as well as derive information about the properties of the shapes based on other properties presented to them. An example of a question at the third level on the topic of solids: Which of the following claims is true? a. If the solid has 8 vertices, it is necessarily a box; b. If the solid has 8 vertices, it is necessarily a cube; c. If the solid has 8 vertices, it is necessarily a pyramid; d. If the solid has 8 vertices, it is necessarily a regular geometric figure; e. All the above claims are incorrect. 2.2.3 Research Procedure At the beginning of the academic year, the questionnaire was administered to each of the different groups during various geometry courses which they attended according to their year of education (first, third and fourth year) and to the different pathways in which they studied (M.Ed., career change). In the first session of the course, the pre-service teachers were requested to respond anonymously to the questionnaire. They were explicitly told that the aim of the questionnaire was to map their knowledge in order to improve the teaching method of lecturers in those courses. The participants were told that no score would be given for responding to the questionnaire and that they did not have to write their name on it.

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2.2.4 Analysis Instruments The participants’ attainments at the various thinking levels were investigated according to the number of their correct answers, namely calculating the mean raw scores (as a percentage). Mastery of thinking levels relating to the various topics was determined according to the weighed scores. It is important to remember that mastering a certain level learners should be versed in each of its previous levels. The weighed scores were set according to a scale suggested by Usiskin (1982) and Patkin (1990), whereby at least four out of five correct answers at each level signified meeting the level criterion and accredited participants with one point. This reduced the chances of guessing and/or giving correct answers based on wrong thinking. a. Way of scoring the answers: 1. Giving at least four out of five correct answers at the first level awards one point. 2. Giving at least four out of five correct answers at the second level awards two points. 3. Giving at least four out of five correct answers at the third level awards four points. b. Way of weighing the scores Weighed score is comprised in the following way: complying with the criterion at the first level + complying with the criterion at the second level + complying with the criterion at the third level and so on. If a is the variable representing compliance with a criterion at any level, a can get the values 0 (fails to comply with the criterion) or 1 (complies with the criterion because the learner has given four out of five correct answers). In that case the weighed score can be represented in the following way: a.1+a.2+a.4 = weighed score. Hence, the scores range which relates to mastery of three out of first four levels of thinking in geometry, is between 0 – 7 for each topic (triangles and quadrilaterals – the first topic, circles – second topic and solids – the third topic) (Patkin, 1990, 2010). Thus, according to van Hiele’s theory, no learner can be at level X before having mastered level X-1. That is, learners must be versed in all previous levels of thinking; otherwise they are referred to as ‘inconsistent’. The weighed scores facilitate identification of learners’ levels of thinking as follows: learners who have not mastered the first level of thinking will get the score of 0. Learners who have mastered the first level of thinking will get the score of 1. Learners versed in the second level of thinking will get the score of 3 (1.1+1.2) and learners versed in the third level of thinking will get the score of 7 (1.1+1.2+1.4). The other scores represent ‘inconsistent’ learners in the examined topic.

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Geometric Thinking Levels Mathematics Teachers

3

Findings

The first research question focused on the difference in geometric thinking levels of mathematics pre-service and in-service teachers at different points during their education in geometry in general and in each of the three topics: triangles and quadrilaterals, circles and solids in particular. Table 3.1 illustrates the means in percent (M) and standard deviation (SD) of the participants’ raw scores in each of the five groups of the present study. As mentioned above, every correct answer accredited the participant with 1 point. Accordingly, every participant received a score between 0–45 (the questionnaire included 45 questions). It is important to point out that the scores were converted from a scale of 0–45 to a scale of 0–100. Thus, the mean scores (in percent) and the standard deviation of all the participants in each group was calculated. The findings analysis was performed by presenting frequencies and standard deviations since the number of participants was not big enough for checking significance. table 3.1 Mean raw scores (in percent) and standard deviation in van Hiele’s questionnaire

First year (N=46) Third year (N=30) Fourth year (N=17) M.Ed. (N=24) Career change (N=25)

M

SD

57 74 74 74 67

13.87 12.42 7.57 10.52 13.08

Table 3.1 shows that the participants’ mean raw scores in each of the different groups did not exceed 74. Furthermore, the mean scores in the first year was lower vis-à-vis the mean scores of the other participant groups. The findings indicate that the mean raw scores of third and fourth year pre-service teachers as well as of in-service teachers studying towards an M.Ed. degree was identical. As to the mean score of the career-changing academicians, the table illustrates that it was higher than that of the first-year pre-service teachers but lower than that of pre-service teachers in more advanced years and of inservice teachers studying towards an M.Ed. degree. Table 3.2 presents the mean scores (M) and standard deviation (SD) of preservice teachers’ raw scores for each of the investigated topics: triangle and

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quadrilaterals, circles and solids. Each topic included 15 questions and the mean score (in percent) and standard deviation of all the participants in that group were calculated. table 3.2. Mean of raw scores (in percent) and standard deviation classifĳied according to three topics

First year (N=46) Third year (N=30) Fourth year (N=17) M.Ed. (N=24) Career change (N=25)

Triangles and quadrilaterals

Circles

Solids

M

SD

M

SD

M

SD

2.70 2.17 1.30 1.84 2.39

68 83 84 83 79

2.74 2.45 2.21 1.72 2.60

56 61 68 68 67

2.04 2.60 2.08 3.37 2.21

46 80 69 71 56

Table 3.2 indicates that in each of the groups participating in the study, the attainments relating to triangles and quadrilaterals were higher than those for the two other topics (circles and solids). The findings show that the attainments of the first-year pre-service teachers were lower (a mean score of 68%) than the attainments of the participants in the other four groups (range of 79–84%). Concerning the topic of circles, the attainments of the first-year pre-service teachers were also the lowest (56%). However, the attainments of the participants in the other four groups were particularly low and ranged between 61–68% on average. Regarding solids, the first-year pre-service teachers had particularly low scores (46%). The score range of the other participants in the questions about solids was wider than in the other two topics (56–80%). The third-year pre-service teachers scored relatively high (80%) whereas the fourth-year pre-service teachers and teachers studying towards their M.Ed. scored within the range of 69 and 71 respectively. Moreover, the career-changing academicians did not attain high scores, receiving only 56% on average. We will now examine the differences in thinking levels of the various groups taking part in the present study for each of the three investigated topics. Table 3.3 presents the mean scores and standard deviations of the participants’ raw scores for each of the three topics included in the questionnaire, according to the different thinking levels. For each correct answer, the

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participants were accredited with 1 point. Hence, every participant got a score between 0–5 (each level had 5 questions about each of the different topics) and the mean score and standard deviation of all the participants in that group were calculated. Perusal of Table 3.3 illustrates that on the topic of triangles and quadrilaterals, participants in all the five groups gave at least three correct answers on average for each of the thinking levels. Conversely, with regards to the topic of circles, the mean of the correct answers at level 3 was lower than at the first and second level for this topic. A similar picture is obtained also for the topic of solids. The second research question explored at what thinking levels did mathematics pre-service and in-service teachers have mastered the three topics examined in the present study. Table 3.4 presents the percentage of participants responding correctly to at least 80% of the questions on each topic and at each level. Please note that at each of the levels there were five questions which focused on a certain topic (five questions on triangles and quadrilaterals, five on circles and five on solids). Table 3.4 shows the percentage of participants who responded correctly to at least four questions about each of the topics and at each of the levels. The findings indicate that on the topic of triangles and quadrilaterals, the first-year pre-service teachers and the in-service teachers studying for an M.Ed. degree gave a higher percentage of correct answers to at least four out of the five questions at the second level than the percentage of correct answers at the first level. That is, these participants knew to answer correctly questions associated with properties of triangles and quadrilaterals better than identifying and naming them. This was not the case for the findings of the other two geometric topics (circles and solids). The findings generally illustrate that the percentage of participants who gave at least four correct answers decreased as the level of difficulty of the questions increased (according to thinking levels). Table 3.5 displays the mastery of geometry thinking levels according to van Hiele’s theory for each of the five groups. As mentioned above, when calculating the weighed score in order to determine the participants’ thinking level, participants who had not mastered the first level got a score of 0 and if they have mastered the first level their score was 1. Participants versed in the second level should have mastered the first and second levels and, therefore, would get a score of 3. Participants versed in the third level get a score of 7 (mastery of all the previous levels as well as the present one). The other scores represented the “inconsistent” participants. Perusal of Table 3.5 indicates that most of the mathematics pre-service teachers at different points of their studies have mastered the topic of geometric

(N=25)

First year (N=46) Third year (N=30) Fourth year (N=17) M.Ed. (N=24) Career change

0.91

0.72 0.55 0.80 0.67

3.33

4.13 4.24 4.20 4.13

3.83 4.27 4.35 4.56 4.13

M 1.38 0.98 0.94 1.24 1.15

SD

level 3

3.04 4.03 1.03 3.94 0.80 3.76 1.01 3.58

1.24 1.12

SD

M

M

SD

level 2

level 1

Triangles and quadrilaterals

3.72 3.83 4.29 4.20 4.13

M

level 1

0.90 0.97 0.67 0.75 0.94

SD 2.83 3.10 3.59 3.52 3.71

M 1.39 1.30 1.37 1.02 1.02

SD

level 2

Circles

1.89 2.17 2.35 2.48 2.29

M 1.27 1.04 1.08 1.06 1.43

SD

level 3

table 3.3 Mean raw scores for each level (range 0–5) and standard deviation for each of the topics

2.78 4.30 4.06 4.08 3.42

M

level 1

0.88 0.90 0.87 1.23 0.81

SD

2.39 3.87 3.00 3.32 2.96

M

0.77 1.06 1.19 1.22 0.73

SD

level 2

Solids

1.67 3.80 3.29 3.20 2.00

M

1.00 1.33 1.07 1.55 1.25

SD

level 3

40 Patkin and Barkai

First year (N=46) Third year (N=30) Fourth year (N=17) M.Ed. (N=24) Career change (N=25)

39 87 94 84 83

0.49 0.34 0.24 0.37 0.37

65 87 82 92 79

0.48 0.34 0.32 0.27 0.41

SD 44 67 76 76 42

M 0.50 0.47 0.42 0.43 0.49

SD 59 57 88 80 75

M 0.49 0.50 0.32 0.4 0.43

SD 37 43 59 56 63

M 0.48 0.50 0.49 0.50 0.48

SD

M

M

SD

level 2

level 1

level 2

level 1

level 3

Circles

Triangles and quadrilaterals

9 7 12 12 25

M 0.28 0.25 0.32 0.32 0.43

SD

level 3

15 83 76 72 54

M

0.36 0.37 0.42 0.45 0.50

SD

level 1

7 67 29 48 17

M

0.25 0.47 0.46 0.50 0.37

SD

level 2

Solids

2 63 47 56 13

M

0.15 0.48 0.50 0.50 0.33

SD

level 3

table 3.4 No. of participants (%) who responded correctly to no less than 4 out of a group of 5 questions on the same topic and at same level

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0

0 26 13 0 0 8

Level

Weighed score First year (N=46) Third year (N=30) Fourth year (N=17) M.Ed. (N=24) Career change (N=25)

1 5 0 0 4 12

1

31 17 20 18 17 28

2

7 17 60 59 62 40

3 other 35 7 23 17 12

Inconsistent

Triangles and quadrilaterals

0 28 30 6 13 20

0 1 30 27 35 25 12

1 3 22 27 41 47 36

2 7 5 3 12 4 20

3

Circles

other 15 13 6 13 12

Inconsistent

table 3.5 Mastery (%) of thinking levels according to van Hiele’s theory – based on the weighed scores

0 83 14 24 25 40

0

1 15 23 29 20 44

1

3 0 0 0 0 0

2

7 0 0 0 0 0

3

Solids

Other 2 63 47 54 16

Inconsistent

42 Patkin and Barkai

Geometric Thinking Levels Mathematics Teachers

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shapes in a plane, i.e. triangles and quadrilaterals. Moreover, they were at the third level in accordance with van Hiele’s theory. The first-year pre-service teachers were less versed and a considerable number of them had mastered only the second level. More than one third of the first-year pre-service teachers (35%), were found to be inconsistent in their mastery. That is, they knew the properties of those geometric shapes but encountered difficulties in identifying them (first level) and consequently were considered as inconsistent. As to the topic of circles, most of the participants mastered only the second level and that applied to all the five groups. However, regarding solids, most of the participants were only at the first or even zero level. They were not versed in the higher levels of properties knowledge (second level) and the informal conclusion (third level). To sum up, the findings illustrate that the level of thinking of the pre-service teachers at the beginning of their first academic year and of the career changing academicians was lower than that of the participants learning in their 3rd or 4th year of education as well as of that of the in-service teachers. Moreover, the findings analysis shows that on the topic of triangles and quadrilaterals the participants in this study demonstrated mastery at the third level of thinking. Conversely, on the topic of circles, most of the participants were versed only in the second level and on the topic of solids they mastered the first level of thinking at the most.

4

Discussion and Conclusions

The present study investigated the difference in thinking levels of mathematics pre-service and in-service teachers in general with respect to three main topics in geometry studied in Israel, according to the mathematics curriculum at elementary school: triangles and quadrilaterals, circles and solids. Furthermore, the study explored the thinking level of mathematics pre-service teachers in the first, third and fourth year of education, teachers studying towards a M.Ed. degree and academicians making a career change to mathematics teachers with regards to the three different topics. The questions given to the participants related to identification shapes and solids (2-D and 3-D) (first level according to van Hiele), properties of those figures (second level according to van Hiele) as well as the ability to draw conclusions in an informal way (third level according to van Hiele). Studies of geometric capability of pre-service teachers indicated that most participants had usually mastered the first two levels and only few were versed in the third level (Gutierrez, Jaime, & Fortuny, 1991; Halat & Sahin, 2008; Patkin & Sarfaty, 2012).

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Findings of the present study corroborated the conclusions of the above. All the research participants had a full matriculation certificate, including mathematics and some of them learnt an advanced level of mathematics at high school. Nevertheless, in spite of their mathematics studies, including geometry, throughout their twelve years at elementary and high school, they were versed only in the first thinking level regarding circles and solids. They failed to master the two higher thinking levels (second and third levels), particularly on the topic of solids. It is noteworthy that with regards to triangles, they were versed in the third thinking level. The fact that in Israel, the topic of triangles and quadrilaterals is taught already from the 3rd grade and until the 12th grade can account for that. The topic of circles is taught only in the 6th grade, mainly with reference to naming and identification of the figures (at the first level according to van Hiele’s theory). Pupils who studied mathematics at an advanced level had learnt it once throughout high school at a formal level, without paying attention to previous knowledge, developing competences and basic skills, etc. The topic of solids was taught even less in the past. It was supposed to be learnt in the 6th grade. However, teachers who had no time to teach this subject, did not do it because it was not legally enforced. In high school, once more only pupils in advanced classes learnt it in higher grades, over a short period of time, at the technical level of memorization and procedures and mainly from the aspect of vectors. Due to the dissatisfaction with the pupils’ attainments in international mathematics tests such as TIMSS and PIZA as well as in national mathematics tests (Gonzales, Williams, Jocelyn, Roey, Kastberg, & Brenwald, 2009) and in order to improve pupils’ achievement, the Israeli education system decided to change the mathematics curriculum for elementary school and to put more focus on geometry (both plane geometry and solid geometry) (Ministry of Education, 2006). It also decided to change the junior high school curriculum, a change which is being implemented these days, particularly in the teaching of solids and circles (Ministry of Education, 2013). The topic of solids was taught in the past from the 6th grade and, later, only in the higher grades of high school within the framework of enhanced mathematics studies. This topic is now being studied in a spiral and gradual way from the 2nd grade until the end of junior high school. The objective was that pupils graduating the 9th grade would master at least the first thinking levels of this topic. Moreover, teaching the topic of circles was transformed although in fact the age group in which it was learnt for the first time had not been changed (as mentioned above, it was done in the 6th grade). In the past, after learning the topic of circles in the 6th grade, the topic was not learnt in junior

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high school (7th to 9th grades) and it was taught once more only in the 10th grade. Nowadays it is being learnt in all these three grades, aiming to deepen and preserve what has been learnt at elementary school; develop geometric thinking by integrating visualization aids (cutting, folding, models); and learn to solve every-day literacy problems. Thus, in the 10th grade, pupils who have chosen to specialize in mathematics within the framework of enhanced mathematics studies, will make a progress and master the topic also at higher levels of thinking. Teachers play an essential role in the promotion of pupils’ mathematics knowledge and the implementation of a new curriculum. Teachers with appropriate mathematics knowledge will be able to offer their pupils opportunities of a deep approach to learning (AFT, 2002). With reference to the different points of time in the participants’ education, the findings showed that the thinking level of first year pre-service teachers and that of the career-changing academicians was lower than the thinking level of third or fourth year pre-service teachers and that of in-service teachers. This might be explained by the fact that geometry teaching in teacher education college combines studies of required geometric contents, while emphasizing the learners’ necessary competences and skills, according to van Hiele. Experiencing in and focusing on didactic aspects, aiming to facilitate the learners’ progress in mastering thinking levels, led third and fourth year pre-service teachers to master higher thinking levels than career-changing academicians or first year pre-service teachers. These findings were in line with van Hiele’s theory, underscoring that the transition from one thinking level to another mainly depends on teaching or, more precisely, on teaching quality (Geddes, Fuys, Lovett, & Tischler, 1982; Halat & Sahin, 2008; van Hiele, 1999). Teaching which consists of memorization and repetition of the same contents without developing required skills and competences does not enhance the learners’ thinking levels. The findings for the third and fourth year pre-service teachers and the in-service teachers were similar. This is due to the fact that in-service teachers studied in teacher education colleges. Hence, during their studies, they were exposed to geometry in theory and to required skills and these improved their mastery of the higher thinking levels in geometry. Findings of the present study pointed out two aspects. The first aspect was that the factor which might improve geometry mastery level of pre-service and in-service teachers was their studies at the teacher education college. This is grounded in the fact that the mastery level of pre-service and in-service teachers in their advanced stages of training is higher than that of those who have just started their education process.

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The second aspect relates to the mathematics education given in Israeli colleges. It seems that the subjects, the teaching methods and the activities offered to pre-service teachers should be examined in a comprehensive way during the teaching of geometry and its different branches in teacher education colleges. All those engaged in teaching this subject should be better versed in it so that they have a sufficient knowledge base for teaching geometry to their elementary school pupils. As already mentioned, the type of activities given to the learners was meaningful and compliance between the learners’ comprehension level and their assignments level was crucial if we wanted to acquire a deep approach to learning (Crowley, 1987). Nevertheless, not enough is being done in some of the learning subjects and particularly as far as circles and solids are concerned. Consequently, it is necessary to offer teachers more learning hours and courses designed to help them master the high thinking levels of these subjects so that they can also teach it themselves. The similar findings relating to inservice teachers and third and fourth year pre-service teachers reinforced the need for expanding the knowledge of mathematics pre-service and in-service teachers, at each stage of their education and professional development in the different topics of geometry and, mainly, circle and solids. In these areas, all the participants demonstrated lack of mastery and weak attainments. To sum up it is important to reiterate that this chapter provides only a general picture of the various levels of thinking of in-service and pre-service teachers regarding different topics in geometry. Consequently, it is recommended conducting additional studies of difficulties and misconceptions embodied in each of the topics, whereby mastery of the levels of thinking is low. As a result, teacher-educators should give priority to teaching topics which are less familiar to the pre-service teachers (focusing less on familiar topics). Hence, in-service and pre-service teachers would master both topics – circles and solids – at the highest levels of thinking. Moreover, it is recommended expanding the research population and investigating larger groups. Thus, it will be possible to perform significance tests for the purpose of generalization and conclusion drawing. It is also recommended exploring whether in-service training courses and intervention programs can promote teachers and learners’ levels of thinking with regards to the various topics.

Acknowledgement This chapter was originally published as Patkin, D., & Barkai, R. (2014). Geometric thinking levels of pre- and in-service mathematics teachers at various stages of their education. Educational Research Journal, 29(1–2), 1–26. Reprinted with permission.

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References AFT. (2002). Principles for professional development, AFT’s guidelines for creating professional development programs that make a difference. Washington, DC: American Federation of Teachers. Retrieved from http://www.aft.org/pubs- reports/ downloads/teachers/PRINCIPLES.pdf Burger, W. F., & Shaughnessy, J. M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mzathematics Education, 17(1), 31–48. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In G. Douglas (Ed.), Handbook of research on mathematics teaching and learning (pp. 420–464). New York, NY: Macmillan. Crowley, M. L. (1987). The van Hiele model of the development of geometric thought. In M. M. Lindquist & A. P. Shulte (Eds.), Learning and teaching geometry, K-12. 1987 yearbook of NCTM (pp. 1–16). Reston, VA: National Council of Teachers of Mathematics. Geddes, D., Fuys, D., Lovett, J. C., & Tischler, R. (1982). An investigation of the van Hiele model of thinking in geometry among adolescents. Project Report, presented at NCTM 1982 Annual Meeting, Toronto, Canada. Gonzales, P., Williams, T., Jocelyn, L., Roey, S., Kastberg, D., & Brenwald, S. (2009). Highlights from TIMSS 2007: Mathematics and science achievement of U.S. fourth and eighth-grade students in an international context. Washington, DC: IES, National Center for Education Statistics. Retrieved from http://nces.ed.gov/pubs2009/ 2009001.pdf Guberman, R. (2008). A framework for characterizing the development of arithmetic thinking. In D. De Bock, B. D. Sondergaard, & C. C. L. Cheng (Eds.), Proceedings of ICME-11 – Topic study group 10, Research and development in the teaching and learning of number systems and arithmetic (pp. 113–121). Monterrey, Mexico. Gutierrez, A. (1992). Exploring the links between van Hiele levels and 3-dimensional geometry. Structural Topology, 18, 31–48. Gutierrez, A., Jaime, A., & Fortuny, J. M. (1991). An alternative paradigm to evaluate the acquisition of the van Hiele levels. Journal for Research in Mathematics Education, 22(3), 237–251. Halat, E., & Sahin, O. (2008). Van Hiele levels of pre- and in-service Turkish elementary school teachers and gender-related differences in geometry. The Mathematics Educator, 11(1–2), 143–158. Koester, B. A. (2003, April). Prisms and pyramids: Constructing three-dimensional models to build understanding. Teaching Children Mathematics, 9(8), 436–442. Ministry of Education, (2006). Mathematics curriculum for the 1st-6th grades in all the sectors. Jerusalem: Curricula Department, Ministry of Education, Culture and Sport. [in Hebrew]

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Ministry of Education. (2013). Mathematics curriculum for the 7th-9th grades in all the sectors. Jerusalem: Curricula Department, Ministry of Education. [in Hebrew] Patkin, D. (1990). The utilization of computers: Its influence on individualized learning, pair versus individualistic learning. On the perception and comprehension of concepts in Euclidean geometry at various cognitive levels within high school students (Doctoal dissertation). Tel-Aviv University, Israel. [in Hebrew] Patkin, D. (2010). The role of “personal knowledge” in solid geometry among primary school mathematics teachers. Journal of the Korea Society of Mathematical Education Series D: Research in Mathematical Education, 14(3), 263–279. Patkin, D., & Levenberg, I. (2004). Plane geometry – Part II for junior high schooland high school students. Israel: Rechgold. [in Hebrew] Patkin, D., & Levenberg, I. (2010). Plane geometry – Part I for junior high schooland high school students (2nd ed.). Israel: Author. [in Hebrew] Patkin, D., & Sarfaty, Y. (2012). The effect of solid geometry activities of pre-service elementary school mathematics teachers on concepts understanding and mastery of geometric thinking levels. Journal of the Korean Society of Mathematical Education. Series D: Research in Mathematical Education, 16(1), 31–50. Piaget, J. (1969). The intellectual development of the adolescent. In A. H. Esman (Ed.), The psychology of adolescence (1985). New York, NY: International Universities Press. Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry: Final report of the cognitive development and achievement in secondary school geometry project (ERIC Document Reproduction Service No. ED220288). Chicago, IL: University of Chicago. van Hiele, P. M. (1959). La pensee de L’Enfant et la Geometrie [The Geometric thinking of children]. Bulletin de association des Professeurs de Mathematiques de l’Enseignement Public, 198, 199–205. van Hiele, P. M. (1987). Van-Hiele levels: A method to facilitate the finding of levels of thinking in geometry by using the levels in arithmetic. Paper Presented at the Conference on Learning and Teaching Geometry: Issues for Research and Practice, Syracuse University, Syracuse, NY. van Hiele, P. M. (1999, February). Developing geometric thinking through activities that begin with play. Teaching Children Mathematic, 5(6), 310–316.

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PART 2 Teaching and Teachers’ Personality

∵

CHAPTER 4

A Multicultural View of Mathematics Male-Teachers at Israeli Elementary Schools Eti Gilad and Shosh Millet

1

Introduction

Feminization of the education system is increasingly expanding both in Israel and abroad. In Israel the average percentage of male-teachers at schools in the Jewish sector is 19%. The highest percentage of male-teachers work at secondary school (30%) and the lowest at elementary school (9%). In the Arab sector, the percentage of male-teachers amounts to 48% on average, 63% at secondary school and 43% at elementary school. In teacher education institutions, 18% of the pre-service teachers are men and these rates have been maintained on a similar level along the years (Ministry of Education, 2012). The small number of male-teachers is an issue of concern among policy makers and education researchers around the world. Many researchers (Gilad & Millet, 2010) argue that men should fulfill a more considerable role in education in general and in elementary education in particular. This will provide a balanced response to the feminine environment and eliminate the gender stereotype through alternate models of role models among male learners. Men should function as educational role models with the purpose of reducing disciplinary events and improving boys’ academic attainments (Connell, 2001). The need for male role models at school is particularly prominent at elementary school ages and it has become even greater with the increase in the number of single parent households. Furthermore, male teachers are necessary in order to promote gender equality in the education system. The requirement to narrow the gender gap between male and female teachers in the education system is in line with the need for multicultural education, designed to offer equal educational opportunities to learners from a different ethnic, racial, cultural and social origin. Schools should reflect the communities in which they operate (Banks & Banks, 2009; Gilad & Millet, 2010; Fernandez, Castro, Otero, Foltz, & Lorenzo, 2006; Ministry of Education, 2012). This study aims to explore the characteristics of the motives of choice and the teaching perception of men from different cultures in Israel who have opted to engage in elementary school mathematics teaching. © The author and IJLTER.org, 2014 | DOI:10.1163/9789004384064_004

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Theoretical Background

2.1 Motives for Choosing the Teaching Profession Numerous studies explored students’ motives for choosing the teaching profession as a career. Generally speaking, researchers tend to divide these motives into extrinsic (wages, stable occupation, status and prestige, convenient working hours and holidays, influence of family members and teachers); intrinsic (natural aptitude for teaching, academic capability, love for children, pleasure derived from teaching); altruistic (wish to contribute to society, help children/adolescents, wish to be social change agents in the community) (DeCorse & Vogtle, 1997; Papanastasiou & Papanastasiou, 1997; Seng Yong, 1995; Su, 1996; Zembylas & Papanastasiou, 2005). The social and cultural context can considerably impact students’ motives for choosing the teaching profession as a career. For example, factors associated with wages were the most influential in choosing teaching as a profession in Zimbabwe (Chivore, 1988). Similarly, Seng Yong (1995) investigated the motives for choosing the teaching profession as a career in Brunei Darussalam. The students he interviewed attributed the greatest importance to external factors (e.g. lack of other options of learning subjects, influence of others, wages, stable occupation and convenient hours). Moreover, Seng Yong (1995) points out that, … the motives for choosing the teaching profession are greatly affected by the status of this profession … in industrial or developed countries teachers no longer have the status they enjoyed in the past and teaching is not perceived as a distinguished career or a means for social mobility. (p. 278) A study conducted by a Cypriot researchers (Papanastasiou & Papanastasiou, 1997) investigated the motives of American versus Cypriot students for choosing the teaching profession. The study found that the American students considered the internal motives as the most important whereas external motives such as benefits and professional prestige were deemed most important by their Cypriot counterparts. Differences in motives for choosing the teaching profession, stemming from a different cultural-occupational context, are also prevalent among students in the same country who belong to different ethnic groups (Ford & Grantham, 2003). For example, Su (1996) interviewed Caucasian and minority group students in California. About one third of the minority group students (as compared to none of the Caucasian students) viewed themselves as social change agents.

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2.2 The Education System from a Multicultural Point of View Various cultures are encompassed in the Israeli society. This demographic differentiation requires a flexible system which complies with the cultural mosaics and the Ministry of Education policy (Millet, Golan, & Dikman, 2012). 2.2.1 The Arab and Bedouin Education in Israel Education in the Arab sector has undergone changes since the establishment of the State of Israel. During the first years of statehood, there were only a very few schools in the entire Arab sector and only a small number of people studied in higher education institutions. Gradually, more and more kindergartens and schools were established, catering to learners from the 1st–12th grades and offering varied options on the secondary school levels – agricultural, technological-vocational and theoretical pathways. Moreover, higher education in Jewish and Arab institutions became available to the Arab population in Israel. The place of girls in the Arab education system has been expanding since the establishment of the state. However, only in the last decade women enrolled in higher education institutions (Abu-Saad, 2005; Pessate-Schubert, 2003; Zeydan, Alian, & Thorn, 2007). Compared to the entire Arab sector, the Bedouins are viewed as a group which has adhered to customs and traditions of the Arab-Muslim culture for the longest period of time. The Statistical Yearbook of the Negev Bedouin (Center of Bedouin Studies & Development of the Negev, 2010) illustrates that, in the past, due to the nomadic way of life, hardly any schools developed within the Bedouin community. During the period of the British Mandate (1921–1948), the first schools were set up among the major tribes and only sons of rich sheiks studied there. With the establishment of the State of Israel, only 150 Bedouin children went to school and the number of educated people in the Bedouin sector was extremely small. In 1981, the Minister of Education and Culture appointed an “Education Authority for the Negev Bedouins”. It was in charge of municipal management of those schools which operated outside the jurisdiction of any local and regional authority. The objective was to narrow the gaps between the Bedouin education system in permanent settlements and in concentrations of nomad Bedouin population – constituting over one third of the entire Bedouin population in the Negev. Out of the teachers working in the Negev Bedouin sector, about 30% are not local. In the past, Arab teachers used to come to the south without any experience and after acquiring minimal competences they would go back to their place of residence in the north, entailing a very high teacher mobility. However, in recent years, this trend has been undergoing a change due to the fact that Bedouin women have been joining the teaching profession (Ministry of Education, 2012).

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2.2.2 Education and Ethiopian Immigrants in Israel The process of absorbing Ethiopian immigrants in the various educational frameworks faced many difficulties. Studies mainly conducted by Szold Institute and the Institute for the Nurturing of Education, Hebrew University of Jerusalem, point out the following complications: reading comprehension, social problems between Ethiopian children and children who have lived in Israel for many years; problems in the learning process, e.g. independent work, initiative, imagination, concentration, request for help, teachers’ attitude. Moreover, there is a rather high percentage of 1st graders who have problems of learning and acquiring language, thinking, learning and communication skills as well as attitude towards time, environment and authority (Ben-Ezer, 2002). The entirety of changes and processes of the information revolution and technological development have obliged the education system to apply renewed and creative thinking relating to the place and role of teacher education programs. These programs are designed to prepare the heterogeneous learner population for professional training and integration in an educated and advanced society (Cochran-Smith, 2000). The concept of pluralism and education for multiculturalism embodied in the training formats is manifested by the development of unique education programs for Ethiopians. The fundamental assumption underlying the need to initiate an Ethiopian teacher education pathway is offering a population from a difference background, culture and socio-economic status an equal opportunity for studying, acquiring a profession and joining the labor market (Millet & Gilad, 2004). Furthermore, in recent years, we have witnessed a high demand for educating Ethiopian pre-service teachers. Out of approximately 130.000 teachers working in the education system from kindergarten to secondary school, only about 150 Ethiopian teachers were integrated, 62 of them in kindergartens (Ministry of Education, 2012). 2.2.3 Education of Immigrants from the Former Soviet Union Since the beginning of the 1990s, with the renewed immigration waves from the former Soviet Union, immigrants’ absorption has been defined as a national challenge and normative value in the Israeli society. This placed the education system as a framework committed by its very essence to play a major role in the success of this process. Recently, both elementary and secondary schools have successfully absorbed teachers from the former Soviet Union who have a high self-image as mathematics teachers. In order to help these teachers join the labor market, the Ministry of Education initiated courses for immigrant-teachers. In these courses the teachers study the culture and history of the People of Israel and the State of Israel. In addition, they

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acquire teaching methods as well as pedagogy associated with their subject of specialization in the former Soviet Union, mathematics in our case (Patkin & Gesser, 2002). Immigrant teachers from the former Soviet Union attest that mathematics teaching in Israel differs from mathematics teaching in their country of origin. They emphasized that discipline in class was different and they believed it was a prerequisite for promoting learners’ attainments and managing lessons in the best way (Levenberg & Patkin, 2001). Michael and Shimoni (1994) characterized the immigrant-teachers from the former Soviet Union by the following features: (a) personal details indicating high academic education and previous experience in teaching; (b) pedagogical approaches which highlight frontal teaching, memorization, strict discipline as well as mainly enhancing the attainments of the prominent learners in class; (c) difficulty to be integrated in free and open frameworks in the educational space. Recently, after a 2-decade absorption process, the emerging picture is of a reality whereby teachers from the former Soviet Union succeed in mathematics teaching and function at school, creating a highly positive image among their Israeli-born colleagues, the children and the parents (Rosenbaum-Tamari, 2004). 2.3 Mathematics Education at Elementary School Mathematics is a complex subject and stereotypically perceived as difficult and challenging. Mathematics teachers are required already at elementary school to be versed not only in mathematical knowledge but also to have pedagogical content knowledge and be acquainted with learners’ ways of thinking (Casas, Catarreira, Gonzalez, Lopes, & Caralho, 2012; NCTM, 1989, 1991). Mathematical foundations are necessary for everyone in modern society, even if they do not become mathematicians or scientists (Harari, 1992). Hence, children should learn and understand mathematics already at elementary school. The teachers who are responsible for the way the pupils’ mathematical knowledge is built at the beginning of their way, should be specialized in teaching the subject of mathematics. Furthermore, making mathematics lessons at elementary school attractive and creative, combined with games and humor, is essential to a successful lesson and promotion of attainments. This contradicts the common and perceived opinion of mathematics teachers who find themselves in a race to meet the standards of the School Effectiveness and Growth Indices (Ministry of Education, 2009). In Israel, mathematics has recently become one of the central issues on the public agenda. This is mainly due to the fact that scores of international tests show that Israeli pupils are not ranked high among the countries who take these tests. Teachers

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undoubtedly have strong influence on the way pupils learn mathematics and teacher education has definitely an essential role in the pupils’ ability to learn (Casas et al., 2012; NCTM, 1991). The Ministry of Education (2009) maintains that the key to promoting mathematics education resides in the quality of teachers. Consequently and based on the Harari Report (“Tomorrow 98”), a mathematics professionalization program for elementary school was implemented in 2002 all over the country. The program aimed to enrich mathematical knowledge, introduce innovative mathematics teaching methods and thus improve academic achievements of the education system learners. Nesher and Hershkovitz (2004) present three main dilemmas in the professional education of mathematics teachers at school: a. the gap between pedagogical and content knowledge – the content knowledge of mathematics teachers at elementary school level is limited to what they have learnt in the past and to a small amount of knowledge acquired during the teacher education program. They are more specialized in elementary school pedagogy. b. the gap between teaching objectives and teacher’s feelings – teachers who attend in-service mathematics training courses engage more in mathematical knowledge rather than being involved in an experiential and successful mathematical learning. 3. the gap between mathematics and the elitist perceptions of the subject – mathematics is perceived as a subject destined for few people and only the gifted and good pupils are successful in mathematics teaching. Based on these beliefs, teachers encounter difficulties in promoting all children in class (Nesher, 2012). Another study (Ma & Singer-Gabella, 2011) illustrates that the induction of elementary school mathematics teachers into a pedagogical world re-shaped according to the reform in mathematics, is associated with building their professional identity. The study presents three cases out of a class of elementary school mathematics teacher education. In the described cases which occurred in the United States, the tutors offer new definitions to the role of “teacher” and “child”. The students discuss the new models of identity and mathematical terms and implement the role definition in various ways (Ma & Singer-Gabella, 2011). In addition to the above dilemmas and in order to comprehend the complex state of mathematics education, we should understand the different approaches to mathematics education at elementary school since it is the basis for the degree of success further on in the learning process. According to Yona (2011), the approaches to mathematics education are: the

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West-European approach, inquiry approach, discrete values approach and mediating approach. The West-European approach – underscores the principles of systematics, practice and memorization. The inquiry approach – highlights the principle of thinking and finding multiple ways for solving problems and mathematical exercises by learning out of pleasure. The discrete values approach – underscores the principle of finding structured models not in the natural environment of solutions, instead of a direct and simple learning. The mediating approach – emphasizes the principle of mediation whereby an interactive process takes place between pupils and teachers. The objectives according to this approach are: building thinking constructs, learning habits and developing the learners’ potential capabilities, developing mathematical literacy, consistency, awareness of processes, flexibility according to pupils’ needs and involvement of the children in the learning process. 2.4 Research Question What are the characteristics of the motives for choice and teaching perceptions of men from various cultures in Israel who have opted to engage in mathematics teaching at elementary school?

3

Methodology

3.1 Research Method The research method of this study is qualitative-interpretive of the case study type. A case study is used in teaching and learning research. One of the important advantages of a case study is its ability to provide insights about incidents in the contexts and physical sites where they transpire. Cultural and social incidents can be fully understood only if they are studied from the participants’ point of view and from the way those actively involved see them. Data collected from the participants can in fact be depicted as insufficient. However, they definitely facilitate comprehension and understanding of the thoughts and feelings of a small group as well as of their attitude and approach (Shkedi, 2005; Smolicz & Secombe, 1990). 3.2 Research Population The research population consists of eight male-teachers who teach mathematics at elementary school. The teachers represent different cultures, two teachers from each culture: Israeli-born, Ethiopians, immigrants from the former Soviet Union and Bedouins.

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Our decision to focus on in-service teachers is based on the assumption that they have experienced teaching for at least two years. They are capable of relating to the teaching and multicultural perception of themselves as being shaped out of the point of view of personal experience. The participants were between 28–62 years old and their seniority ranged between 2–19 years. Two teachers, one from the former Soviet Union and the other a Bedouin, have made a career change to teaching. One of the Israeli-born teachers is religious, defining himself as a teacher in a religious school. All the teachers work at the south of Israel and have been randomly chosen. 3.3 Research Instruments In order to identify the motives for choice and the professional perception of mathematics teaching at elementary school we used a semi-structured indepth interview. The interview questions focused on: (a) the teachers’ personal background, the motives for choosing their profession and their attitude towards the different motives leading them to make the decision to study teaching and education; (b) the teachers’ teaching perceptions and mainly what does ‘being a mathematics male-teacher at elementary school’ mean for them; (c) the teachers’ perception of mathematics teaching seen from the perspective of the culture from which they come. 3.4 Research Procedure Collecting the data about the motives for choice and professional perception of the male-teachers was performed through personal interviews. The interviews were conducted by the researchers, after randomly dividing the participants between them. This procedure was done in order to prevent bias. The interviews were in Hebrew and took place at an Academic College of Education. Each interview lasted about an hour. The interview built for the purpose of this study is based on the study of Millet (2001). 3.5 Data Processing The interviews were content analyzed as is customary in qualitative research (Shkedi, 2005). The categories were defined on the basis of previous studies which investigated the motives for choosing the teaching profession and perceptions of mathematics teaching at elementary school (Gilad & Millet, 2010). A qualitative-interpretive content analysis was performed on the open-ended interview questions. The categories were obtained after the researchers concurred about them at a level of agreement of no less than 67%. The analysis was performed on the following levels: the single interview level, the ethnic group level and the entire participant group level (Shkedi, 2005).

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Results

4.1

Perception of Mathematics Teaching and Motives for Choice of Mathematics Male-Teachers at Elementary School The interviews analysis illustrates features which were common to all the male-teachers investigated in this study. Moreover, the interviews show differences between the cultural groups with regard to the teachers’ motives for choice and their perception of being mathematics teachers at elementary school. 4.1.1 Results Common to All the Investigated Male-Teachers All the investigated mathematics male-teachers indicated more than one motive for choosing to teach mathematics at elementary school. They also specified varied perceptions. The entirety of the mentioned motives and perceptions can be divided into three categories: a. Extrinsic perceptions and motives: wages, stable occupation, occupational status, professional prestige as it is perceived in the Israeli society, influence of family members and teachers. b. Intrinsic perceptions and motives: love for mathematics, high academic competence in mathematics, natural gift for mathematics teaching, love for children, pleasure derived from mathematics teaching. c. Ideological-altruistic perceptions and motives: wish to contribute to society, wish to help learners cope with mathematics learning, reducing fear of this subject, wish to be value-oriented and social change agents in the community by demonstrating the beauty embodied in mathematics. Reference to the teaching method was also common to all the investigated teachers. They believe that different teaching approaches should be adopted according to different pupil populations in order to obtain high attainments, in spite of the obstacle of the School Effectiveness and Growth Indices and the standards thereof. That is, individual, group and whole class work as well as combination of various approaches prevalent in mathematics education, such as memorization and practice, inquiry and mediating approaches. 4.2 Results According to Cultures Below is a description of the perception of mathematics teaching and motives for choice among Bedouins, Ethiopians, immigrants from the former Soviet Union and Israeli-born teachers.

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4.2.1

Bedouin Teachers’ Perception of Mathematics Teaching at Elementary School and Motives for Choosing the Teaching Profession Among the Bedouin teachers, the role perception and motives for choosing the profession were both extrinsic and intrinsic. Stable occupation The perception and motives indicated in the Bedouin sector were the need for finding a stable occupation as well as limited career options for male high school graduates. For example, one of the teachers said: “I chose to become a mathematics teacher because I had no other option. I could not actualize my dream to study pharmacology abroad … and if I have decided to teach, then teaching mathematics is very distinguished”. Or a second teacher told: “I studied construction engineering and did not find myself and I switched to teaching … and then the career change is the nearest to the subject of mathematics”. Natural inclinations and wish to help others The two teachers pointed out that for them being a mathematics teacher implies being meaningful to the learners. One teacher said: “I view myself as someone who delivers a very valuable material to the pupils … there is an important part in consolidating a better future for my pupils so that they can cope with life … mathematics is a unique and special language … I like mathematics and I love teaching mathematics. Working as a teacher gives me great satisfaction, pleasure and motivation”. The second teacher explained: “For me, being a mathematics teacher means promoting the pupils and preparing them for academic studies … Mathematics constitutes an ‘entrance ticket’ to university … I teach them techniques for solving problems and exercises, we practice logical thinking and building a connection between comprehending mathematical text and the solution technique”. The words of the two Bedouin teachers attest to the wish to be a social leader and help others. The wish to be a socialization agent is illustrated much more emphatically by the second teacher: “… first of all I have the inclination to teach and, thus, to introduce changes for the better within society. I see that our society, the sector in which I live, needs good teachers and particularly teachers of mathematics since mathematics is the basis for everything … Moreover, it is highly considered in the Bedouin society”.

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4.2.2

Ethiopian Teachers’ Perception of Mathematics Teaching at Elementary School and Motives for Choosing the Teaching Profession as a Career Among the Ethiopian men the extrinsic perceptions and motives were prominent. Past teachers and home-class teachers as a role model The Ethiopian teachers indicated that when they chose the teaching profession they were influenced by the figure of their past teachers. One teacher told that: “As a new immigrant I had a mathematics teacher who was interesting and the subject was exciting. I asked myself if I could be like her. She made me fall in love with mathematics … I have a challenge to become a mathematics teacher … to influence my learners at elementary school”. A second teacher said that he was influenced by his father who encouraged him to become a teacher: “Father is a dominant figure at home and he inspired me to become a mathematics teacher. Father who worked in the municipality in Ethiopia understood the importance of education and the importance of being a mathematics teacher at elementary school which is the basis for success in life”. Pride for the Ethiopian ethnic group The Ethiopian teachers who participated in this study maintained that it was important to be mathematics teachers and, especially, at elementary school. One of teachers claimed: “I am going to be a role model for the young generation, for the community. They can be proud of themselves when they see Ethiopian teachers, mainly mathematics teachers … the children have something to strive for … they will also want to become mathematics teachers and it is important both for the Ethiopian society and the Israeli society”. The second teacher expressed himself in similar words: “I set a challenge to my pupils at school, a challenge to be like me, a mathematics teacher, a challenge which is feasible. If I succeeded so could they. You don’t have to be afraid of mathematics”. Being a mathematics teacher wins a lot of respect The Ethiopian mathematics teachers indicated that one of the motives for choosing teaching as a profession is respect, particularly for mathematics teaching. Among Ethiopian immigrants, in both Ethiopia and Israel, teaching has been perceived as a respectable profession and teachers have always been appreciated. Along these lines, one of the teachers said: “In Ethiopia I studied

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until the age of 15 and teachers there were very respected. I always thought that I would like to be teacher because teachers were respected by everyone. Teachers are highly respected, even more than the parents … Teachers are something special and mathematics teachers even more so”. Role perception by male-teachers Based on the experience they have accumulated during the teacher education program and in public and communal work, the Ethiopian teachers are quite confident of their success as teachers and home-class teachers in future. One of the teachers reiterated: “I have something to sell, I know my value. I see myself at the center of matters at school and am confident that I will succeed”. He added that: “In Ethiopia I was not acquainted with many Ethiopian female-teachers and most of the teachers were men. Also here in Israel I believe that a home-class teacher should be a man. When you teach mathematical contents the gender of the teacher is not important. But home-class teachers should be men”. 4.2.3

Perception of Mathematics Teaching at Elementary School and Motives for Choosing the Teaching Profession as a Career Among Teachers from the Former Soviet Union Among the men from the former Soviet Union the intrinsic perceptions and motives were prominent.

Mathematical excellence The teachers indicated the relation between being immigrants from the former Soviet Union and being mathematics teachers at elementary school. One of the teachers said: “After working as an engineer with a 30-year background in the electronics industry, I decided to make a career change and chose mathematics, a subject in which I can be integrated without having to acquire the lacking knowledge … my skills and learning habits allowed me to come to school with an authority of knowledge and ambition to encourage my pupils to grow higher and farther”. The second teacher underscored the importance mathematics has for life and the value of mathematics education: “I have the will to educate for excellence in mathematics and influence the future generation … understanding that mathematics is a main and essential instrument for pupils in order to be integrated into society. Mathematics enables logical thinking and creative thinking and persevering in mathematics studies is a key to entering the world of higher education”.

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Self-discipline in learning The teachers told that, for them, being mathematics teachers means navigating towards areas in which the pupils would probably develop throughout their adult lives. One of the teachers stated that: “In the course of my work I could contribute to my pupils from the vast experience which I had accumulated while working as an engineer and now as a mathematics teacher … As a learner who studied and was educated in the Soviet Union I highly consider practice in teaching processes … I require discipline from all the pupils and think that this will help them to study and make progress”. The second teacher too emphasized that the methods to which he had been exposed as a learner in the Soviet Union have a record of success: “In my opinion, the way to success is based on intensive and systematic work with a lot of self-discipline and responsibility for self-improvement … Pupils must understand that they come to school in order to study and not in order to disturb … to study and advance”. Self-image The results illustrate that teachers from the former Soviet Union maintain they are specialized in mathematics as far as area of knowledge and teaching methods are concerned. One teacher specified: “Being a mathematics teacher at elementary school is also a symbol of academic prestige because mathematics is the mother of all the learning and research subjects … mathematics is indefinite and unfathomable”. As male-teachers at elementary school and in the staff room they feel very ‘masculine’: “I know and am aware of my place at school … the female-teachers always ask my advice in the fields of mathematics and pedagogy”. The other teacher also relates to the issue of professional and gender prominence. In his opinion: “I feel like a king in the chicken coop …”. 4.2.4

Israeli-Born Teachers’ Perception of Mathematics Teaching at Elementary School and Motives for Choosing the Teaching Profession as a Career The Israeli-born teachers emphasized ideology as the main motive for choosing mathematics teaching as a profession, combined with their wish to influence society and be change agents in the community. Ideological motives The results indicate that all the Israeli-born teachers highlighted the ideological motive in their perception and choice of the teaching profession. They

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argued that the teaching profession in general and mathematics teaching at elementary school in particular would allow them to influence and change. According to one of the teachers: “First and foremost comes the need to change and influence and this should already start at elementary school … I don’t know if it is associated with my character or the society in which I grew up or the environment where I have lived all the years. But I am deeply concerned with what is transpiring in the western Israeli society of the years 2000 … I think that education and the subject of mathematics carry a great weight in changing and improving our society. As a future father I want my children to grow up in a better place”. This teacher told that when he was at school he had the opportunity to enjoy good education: “I chose mathematics teaching at elementary school mainly due to ideological reasons. I like working with children. I am not interested in money although I know that I will make money in another way (giving private lessons of mathematics and computers)”. Self-image The results show that the Israeli-born teachers did not settle for teaching but aspired to promote themselves to management level. They viewed teaching as a jumping-board for management positions in formal and informal frameworks. In light of the experience they acquired during the military service and their civilian life, they felt rather confident of their success as mathematics teachers and home-class teachers. One of the teachers restated that: “As a man I have advantages over femaleteachers. In the staff room at school I am a single male-teacher, not just a teacher but a mathematics teacher, so I am highly appreciated and have no competition. I am more confident in the company of women”. Another teacher expressed himself in a different way: “As a mathematics teacher, I feel that I am strong, authoritative, dominant … I think that a maleteacher is suitable to elementary school … I can better control”. A religious teacher described the relation between mathematics teaching and the Jewish bookcase. He maintained that: “Mathematics is connected to my spiritual religious world … learning the Gemara (literally ‘completion’, the second and supplementary part of the Talmud) which is a highly important layer in the Jewish bookcase, is entirely based on mathematical-logical thinking … study inquiry in the Gemara is similar in its ways of thinking to the inquiry of a mathematical problem. Moreover, the Gemara examines all the options for solving the issue while rejecting some of them based on logical thinking … I present Gemara issues in mathematics lessons, showing the pupils the features which are parallel to mathematical thinking”. Summary of the results is presented in Table 4.1.

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table 4.1 Prominent perceptions and motives of elementary school mathematics teachers from a multicultural perspective

Israeli-born

Intrinsic motives

Extrinsic motives

5

Bedouin

Ideology Love for the Change agents subject Natural aptitude for teaching Love for children Wages Stability

Ethiopian

Immigrants from the former Soviet Union

Respect for teachers Change agents

Education for excellence Knowledge and mastery of the subject

Role model Pride for the ethnic group

Discussion and Conclusions

This study focused on comprehending the educational, familial and social reality to which the men who chose elementary school mathematics teaching as their occupation were exposed. Moreover, the study demonstrated their perceptions and role. The research assumption was that the results were significant and beneficial to those educating pre-service teachers for elementary school mathematics teaching and to the professional development of the latter during their practice as male-teachers. This study was a case study investigating eight male-teachers from different cultures. The interviews analysis illustrated common features of all the research participants. It also showed the differences between the various cultural groups with regard to perception of teaching and the participants’ motives for choosing to teach mathematics at elementary school. The common features of all the investigated male-teachers are specified below. More than one motive for choosing to teach was demonstrated by all the investigated male-teachers. This result is in line with the professional literature dealing with male-teachers’ perception of teaching and the motives for choosing this profession (Papanastasiou & Papanastasiou, 1997; Su, 1996). Most of the male-teachers indicated that figures of male and female teachers from their past were some of the motives influencing their choice of the teaching profession. This result is corroborated by a study of novice teachers (Millet, 2001). All the investigated teachers stated the importance and prestige of teaching mathematics at elementary school. Furthermore, they pointed out

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the perception that various teaching methods should be adopted in accordance with various pupil populations in order to obtain high attainments (Yona, 2011). Moreover, the study illustrated unique results obtained from the different men groups from a cultural point of view: The social and cultural context can strongly influence teachers’ motives for choosing teaching as a profession and career as well as the role perception of mathematics teaching. Bedouin male-teachers – the role perception and motives specified by this group were both extrinsic and intrinsic. As the main motive, they mentioned the need for stable occupation and love for mathematics. They added that teaching mathematics at elementary school was their vocation in the Bedouin society. These results concur with the professional literature about minority groups and their options for choosing and finding an occupation (Zeydan, Alian, & Thorn, 2007). Ethiopian male-teachers – the Ethiopian teachers also indicated both extrinsic and intrinsic motives: influence of teachers from their past, perceiving mathematics teaching as a respectable profession, wish to serve as a role model in the ethnic group and be social change agents in the community. These results concur with a study conducted among Ethiopian pre-service teachers (Millet & Gilad, 2004). Male-teachers from the former Soviet Union – these teachers underscored the intrinsic perceptions and motives. The results corroborate studies of teachers from the former Soviet Union in general and mathematics teachers in particular (Levenberg & Patkin, 2001; Michael & Shimoni, 1994). Israeli-born male teachers – these teachers emphasized ideology as the main motive for choosing to teach mathematics at elementary school. These results are in line with a study conducted among Israeli-born, Bedouin and Ethiopian students (Gilad & Millet, 2010). This study has implications from several points of view: mathematics teaching from the cultural aspect, mathematics teaching from the gender aspect and mathematics teaching in teacher education, in-service training courses and in the field of education. The main research conclusions of this study are that the motives and perceptions of male-teachers choosing to teach mathematics at elementary school are varied, comprising extrinsic, intrinsic and ideological-altruistic motives and perceptions. Moreover, mathematics male-teachers from different cultures perceive the subject of mathematics in a different way. The Israeli society, which constitutes a meeting point of demographic and cultural differentiation, requires a flexible system in compliance with the fabric of cultures and the Ministry of Education policy.

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Based on the research findings, it is recommended planning teacher education and professional-mathematical development programs which highlight the valued-social aspect of the mathematics teaching profession. Such programs may convince men to choose this profession. As mentioned before, an increase in the number of male-teachers in the education system will bring about an educational and social balance in the class and at school, increasing teachers’ prestige in the Israeli society. Moreover, the researchers recommend investigating further the teaching-learning of elementary school mathematics male-teachers from various cultures. This may constitute a key to a change in the subject perception and its teaching as well as promote elementary school learners’ attainments.

Acknowledgement This chapter was originally published as Gilad, E., & Millet, S. (2014). A multicultural view of mathematics male-teachers at Israeli primary schools. International Journal of Learning, Teaching and Education Research, 7(1), 23–43. Reprinted with permission.

References Abu-Saad, I. (2005). Re-telling the history: The indigenous palestinian bedouin in Israel. AlterNative: An International Journal of Indigenous Scholarship, 1(1), 26–49. Banks, J. A., & Banks, C. A. M. (2009). Multicultural education: Issues and perspectives. New York, NY: John Wiley & Sons. Ben-Ezer, G. (2002). The ethiopian jewish exodus: Narratives of the journey to Israel via Sudan 1977–1985. New Brunswick, NJ: Transaction Publishers. Casas, L., Catarreira, S., Gonzalez, R., Lopes, V., & Caralho, S. (2012, April). Theory of concepts: Its application in the mathematics classroom. Paper presented at the 4th ATEE winter conference, University of Coimbra, Portugal. Center of Bedouin Studies & Development of the Negev. (2010). Statistical yearbook of the Negev Bedouin. Jerusalem: Center for Regional Development at Ben-Gurion University of the Negev in cooperation with Konrad Adenauer Foundation in Israel. Chivore, B. S. R. (1988). A review of factors that determine the attractiveness of teaching profession in Zimbabwe. International Review of Education, 34(1), 59–77. Cochran-Smith, M. (2000, January). Teacher education at the turn of the 21st century: Quo Vadis? Lecture delivered at Tel Aviv University. Connell, R. W. (2001). The man and the boys. Berkeley, CA: University of California Press.

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DeCorse, C. J. B., & Vogtle, S. P. (1997). In a complex voice: The contradictions of male teachers’ career choice and professional identity. Journal of Teacher Education, 48(1), 37–46. Fernandez, M. L., Castro, Y. R., Otero, M. C., Foltz, M. L., & Lorenzo, M. G. (2006). Sexism, vocational goals, and motivation as predictors of men’s and women’s career choice. Sex Roles, 55(3–4), 267–272. Ford, D. Y., & Grantham, T. C. (2003). Providing access for culturally diverse gifted students: From deficit to dynamic thinking. Theory Into Practice, 42(3), 217–225. Gilad, E., & Millet, S. (2010, July). Men in teaching: Motives for choice in different cultures. Proceedings of the 10th International Conference on Diversity in Organization, Communities and Nations, Belfast, Northern Ireland. Harari, H. (1992). Report of the supreme committee for scientific and technological education. Jerusalem: Ministry of Education. [in Hebrew] Levenberg, I., & Patkin, D. (2001). The new immigrant teacher as member of a mathematics staff. Proceedings of the 9th EARLI, Swiss Republic. Ma, J. Y., & Singer-Gabella, M. (2011). Learning to teach in the figured world of reform mathematics: Negotiating new models of identity. Journal of Teacher Education, 62(1), 8–12. Michael, A., & Shimoni, Z. (1994). A project of supporting and absorbing Israeli teachers and immigrant teachers: Similarity and differentiation. Dapim, 19, 90–95. Tel Aviv: MOFET Institute. [in Hebrew] Millet, S. (2001, August). “It’s an endless race in which I have to grow and help others grow …”. Changes in the development of the didactic knowledge and self knowledge of Novice teachers. Paper presented at the 9th European Conference – EARLI, University of Fribourg, Swiss Republic. Millet, S., & Gilad, E. (2004). Unique education program for Ethiopian pre-service teachers: Program components and perception of teachers’ figure (Research report). Tel Aviv: MOFET Institute. [in Hebrew] Millet, S., Golan, H., & Dikman, N. (2012, April). Exposing the multiculturalism approach of teacher educators at the MOFET institute. Paper presented at the 4th ATEE winter conference, University of Coimbra, Portugal. Ministry of Education. (2009). Special data processing: School effectiveness and Growth Indices. Jerusalem: Ministry of Education. [in Hebrew] Ministry of Education. (2012). Special data procession: Distribution of teachers according to the education stages. Jerusalem: Ministry of Education, Section of Marketing Systems and Clients Development Analysis. [in Hebrew] National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (1991). Professional standards for school mathematics. Reston, VA: Author.

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Nesher, P. (2012). On the diversity and multiplicity of theories in mathematics education. In E. Silver & C. Keitel-Kreidt (Eds.), Pursuing excellence in mathematics education. Cham: Springer. Nesher, P., & Hershkovitz, S. (2004). Professional education of primary school mathematics teachers. Tel Aviv: MOFET Institute. Retrieved May 01, 2012, from http://portal.macam.ac.il/ArticalePage.aspx?id=4991 [in Hebrew] Papanastasiou, C., & Papanastasiou, E. (1997). Factors that influence students to become teachers. Educational Research and Evaluation, 3(4), 305–316. Patkin, D., & Gesser, D. (2002). Mathematics teachers from the former Soviet Union: Experiences of the first days at school in Israel (Hachinuch Usvivo – Tel Aviv Yearbook). Tel Aviv: Kibbutzim College of Education. [in Hebrew] Pessate-Schubert, A. (2003). Changing from the margins: Bedouin women and higher education in Israel. Women’s Studies International Forum, 26(4), 285–298. Rosenbaum-Tamari, Y. (2004). Immigrants from the former Soviet Union: Motives for immigration and commitment to life in Israel. Special publication based on data obtained from monitoring the absorption of immigrants from the former Soviet Union (Vol. 1, pp. 137–143). Jerusalem: Ministry of Immigration Absorption, Department of Planning and Research. [in Hebrew] Seng Yong, B. C. (1995). Teacher trainees’ motives for entering into a teaching career in Brunei Darussalam. Teaching and Teacher Education, 11(3), 275–280. Shkedi, A. (2005). Multiple case narratives: A qualitative approach to studying multiple populations. Amsterdam: John Benjamins Publishing. Smolicz, J. J., & Secombe, M. J. (1990). Language as a core value of culture among Chinese students in Australia: A minor approach. Journal of Asian Pacific Communication, 1, 229–245. Su, Z. (1996). Why teach? Profiles and entry perspectives of minority students as becoming teachers. Journal of Research and Development in Education, 29(3), 117–133. Yona, S. (2011). View on various approaches to mathematics education. Tel Aviv: MOFET Institute. Retrieved June 10, 2012, from http://portal.macam.ac.il/Article Page.aspx?id=4628 [in Hebrew] Zembylas, M., & Papanastasiou, E. C. (2005). Modeling teacher empowerment: The role of job satisfaction. Educational Research and Evaluation, 11(5), 433–459. Zeydan, R., Alian, S., & Thorn, Z. (2007). Motives for choosing the teaching profession among pre-service teachers in the Arab sector. Dapim, 44, 123–143. [in Hebrew]

CHAPTER 5

Do “Those Who Understand” Teach? Mathematics Teachers’ Professional Image Dorit Patkin and Avikam Gazit

1

Theoretical Background

1.1 Perception of the Good Teacher In recent years, teachers’ status in Israel has been increasingly devaluated, leading also to the devaluation of those engaged in mathematics teaching. Teaching seems as a less prestigious profession in many countries, such as the U.S.A., Australia and several other western countries. Generally, the perception is that teaching practitioners are forced to work as teachers or choose it due to the lack of other alternatives. Shulman (1986) took the cynical phrase of the known playwright Bernard Shaw: “Those who can, do; those who can’t, teach”. He slightly changed this phrase, turning it into the slogan of the United States reform “Teachers of Tomorrow” introduced in the 1980s: “Those who can, do; those who understand, teach”. In that context, Shulman presents three types of knowledge which teachers need: professional, didactic-pedagogical and curricular knowledge (Shulman, 1987). The question raised, then, is: “What about teachers’ quality?” Studies of factors affecting students’ attainments found that teachers’ quality has a considerable impact. Hattie (2003) conducted a meta-analytical research of thousands of studies which investigated the effect of different factors on pupils’ academic attainments. He found that differences between teachers account for approximately 30% of the differentiation in their students’ scores. Rockoff (2004) conducted a longitudinal research in which teachers were monitored for 10 years. He showed that the differences between those teachers accounted for up to 23% of the differentiation in their students’ scores. However, in spite of teachers’ centrality and an educational-philosophical engagement in the teaching profession components (Shulman, 1987), there is no consolidated and validated scientific basis for the issue: “Who is the good, ideal, worthy or appropriate teacher?” Wilson and Youngs (2005) suggest three concepts for determining the good teacher: the educated teacher, the professional teacher and the value-oriented teacher. Educated teachers are intellectuals with extensive formal schooling © Rowman & Littlefield, 2016 | doi:10.1163/9789004384064_005

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and rich language. Professional teachers are versed in the area of knowledge as well as in teaching and learning methods. Value-oriented teachers are caring, believing that their students have emotional and intellectual skills. In recent years, the third concept of the value-oriented teachers has become a leading approach – the ethics of care approach (Noddings, 2002). Regarding caring teachers, Getzels and Smilansky (1983) conducted a study among 11th-grade students in Midwestern United States. The study dealt with the problems which preoccupy the students at school and its findings were interesting. The problem indicated by the highest percentage of students was unjust and non-caring teachers. Throughout the 2000s, teachers have been considered more humanist, namely capable of coping with differentiations between their students and with the changes transpiring in the postmodern era, the globalization age (Cochran-Smith, 2003). Teachers are required to be flexible and adjustable to the students, while demonstrating quick responses and decision-making related to unexpected teaching situations (DarlingHammond, 2007). Moreover, humanist teachers should be moral people, i.e. involved in numerous and meaningful life areas of learners. They should also be competent in critical reflection in order to educate students to be skeptical and ask questions (Darling-Hammond, French, & Garcia-Lopez, 2000). Although studies have shown that teachers are the dominant factor which affects attainments; and in spite of students’ responses to teachers’ attitudes towards them, very little has been done in order to effectively and systematically assess teachers’ practice. There are professional bodies, e.g. National Board for Professional Teaching Standard (NBPTS, 2007) operating in the United States. However, they specify only the prerequisites for being qualified, such as: a B.A. degree, 3-year experience and teaching certificate. Beyond that, there is no reference to the desirable and worthy teacher’s figure. School principals are supposedly in charge of teachers’ work but they hardly engage in formative assessment of the latter’s teaching, unless they are called upon to ‘extinguish the fire’ in case of serious and unusual events (Daley & Kim, 2010). In order to achieve a meaningful and reliable assessment of teachers’ work, Marshal (2001) suggests implementing a process of dynamic assessment by teacher teams. He proposes a series of instruments and operations for an extensive, multi-criterion assessment which integrates flexibility and creativity. In a study conducted by Arnon and Reichel (2007), pre-service teachers and novice teachers were asked about the qualities of the ideal teacher. 98% of the research participants indicated qualities associated with teachers’ personality whereas only 75% of them referred to the teachers’ knowledgerelated qualities. The qualities which received the highest percentage were “empathetic and attentive teacher” (91%). This quality was followed by three

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personality-associated qualities: “attitude towards the profession” (57%), “general personality qualities” (52%) and “teachers as leaders” (44%). Only then did participants indicate knowledge-related qualities, such as: “knowledge of the discipline” (43%) and “didactic knowledge of teaching methods” (42%). The two qualities which were attributed the lowest percentage were: “didactic knowledge of focusing on the individual student” (14%) and “possessing extensive formal schooling and being broad-minded” (12%). The desirable profile of the ideal teacher illustrates the gap between the desirable and the existing. The teaching profession has considerably become the default, the slogan “the good for teaching” seeming equally cynical as Bernard Shaw’s comment about teachers. 1.2 Perception of the Mathematics Teacher Vinner (2011) relates to the typical profile of elementary school teachers both in Israel and the United States in a qualitative-anecdotal way, based on his talks with teachers. These talks illustrate his impression that people have chosen to teach at elementary school because they love human interaction with children. Usually, these teachers did not score high on their university or college admittance tests. This is in line with the previous determination, namely that people use the teaching profession as a default. Vinner (2011) does not attribute much importance to the admittance test scores, seeing them as problematic and invalid. Knowledge is important but not enough for being a good mathematics teacher. McPhan, Morony, Pegg, Cooksey, and Lynch (2008) indicated international decrease of involvement in this discipline. Studies of this issue (Brown, Brown, & Bibby, 2008; McPhan et al., 2008; Watt, in Murray, 2011) asked students to point out why they run away from mathematics. Most of the reasons focused on mistrust of mathematics, lack of understanding of mathematics and, above all, dissatisfaction with the way of teaching mathematics. Good mathematics teachers use methods that are interesting and engaging to students. Students appreciate strategies that transcend text and workbook activities to include activities such as songs, games, simulations, and projects. They also appreciate teachers who spark love for the subject matter by capitalizing on students’ outside interests and students’ preferences for enjoyable, engaging activities (Guillaume & Kirtman, 2010). Due to social and technological changes of the 21st century, every citizen should understand mathematics and be able to implements its methods. Already at elementary school, mathematics teachers have the responsibility for providing formal schooling which is both suitable to learners’ further studies and their integration in everyday life of society.

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The Third International Mathematics and Science Study (TIMSS, 1996) found that teachers’ professional image was not high, particularly that of elementary school mathematics teachers, as compared to junior high school and high school teachers. Mendelson (2006) conducted a study among mathematics in-service teachers and pre-service teachers. When participants in this study were asked, to choose a profession with which they identify in their work, about 35% of them chose the figure of care-givers. Caring professions like nursing or social workers are professions with relatively low status. Consequently, the present study explores mathematics teachers’ self-perception from the point of view of their professional image. The study aims to add another dimension to studies of mathematics teachers’ world. It also aims to improve teachers’ training according to the participants’ main related issues.

2

Methodology

2.1 Research Aim The main purpose was to investigate the various professional characteristics of mathematics teachers from the point of view of experienced and novice teachers. The research questions were: 1. What is the professional image of mathematics teachers? 2. Is there a difference in the professional image between experienced teachers compared to novice teachers? 2.2 Research Population Sixty-one mathematics teachers, teaching at all age groups. Twenty-eight of them are young teachers with up to 6-year teaching experience, referred to in the present study as novice teachers and thirty-three experienced teachers with seven and more years of teaching experience. This is a convenience casual sample of teachers, who teach at various schools from 1st to 10th grades. 2.3 Research Instruments A professional image attitude questionnaire, consisting of 30 assertions and based on the questionnaire conceived by Avraham (1972) [Appendix A]. The original questionnaire had initially 38 assertions, the responses to which were rated according to Likert scale, ranging from ‘strongly agree’ – 1; ‘agree’ – 2; ‘neutral’ – 3; ‘disagree’ – 4; and up to ‘strongly disagree’ – 5. Eight assertions on which there was no full agreement between two experts in mathematics edu-

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cation were removed from the questionnaire. This left 30 assertions, most of which were formulated in the positive and some in the negative. Constructing the items and adapting them to mathematics teachers were done in compliance with the three terms of Cronbach and Gleser (1953), namely: all items were directly attributed to the investigated field; all items related to most aspects of the investigated field; and all items were formulated so that they were not biased in favor of one of the rating directions. In addition, two openended questions were given to the subjects. 2.3.1 Assertions’ Categories a. Teachers’ personal characteristics: assertions 9, 10, 12, 21, 24, 25. Five assertions presented positive characteristics and one assertion (10) referred to a negative characteristic (difficult to admit mistakes). b. Teachers’ approach towards students: assertions 1, 2, 6, 8, 23, 28, 29. Six assertions presented positive approaches and one assertion (6) – negative approach. c. Teachers’ attitudes towards teaching: assertions 4, 5, 11, 15, 16, 18, 25, 26. Three assertions presented anticipated behaviors (11, 25, 26), one assertion related to teachers’ personal responsibility (4) and the four other assertions represented pedagogically undesirable behaviors. d. Professional image as mathematics teachers: assertions 3, 7, 13, 17, 19, 22. Four assertions (3, 17, 19, 22) related to teachers’ perception of themselves whereas the remaining two assertions related to their degree of confidence in their ability to succeed as teachers. e. Teachers’ perception of the students’ attitudes towards them: assertions 14, 20, 30. Two open-ended questions: In the first question the participants were asked to indicate the competences required by all mathematics teachers. The second question dealt with the measures teachers should take in order to improve their teaching quality. 2.4 Analysis Methods Mean values and standard deviation for the rate of agreement of each of the thirty assertions in the questionnaire were calculated. A quantitative content analysis was used for the two open-ended questionnaires (31 and 32). The typical characteristics of the teachers’ skills (31) were categorized according to their frequency of appearance into four groups (50% and above, 20% to 25%, 8% to 13% and less than 8%). Answers to the question regarding the measures designed for

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improving the quality of teaching (32) were also divided according to their frequency of appearance (more than 40%, 10% to 15% and less than 8%). The items reliability was examined by means of the Alpha Cronbach Test and the significance of the differences between the two participant groups was calculated by means of t-tests for independent samples. 2.5 Ethical Considerations The research participants gave their consent to take part in the present study after the research aim had been clarified to them. The questionnaire was anonymous in order to follow the rules of ethics.

3

Findings

3.1

The First Research Question: Mathematics Teachers’ Professional Image According to the Five Categories 3.1.1 Teachers’ Personal Characteristics Table 5.1 displays mean values and standard deviation of assertions relating to teachers’ personal characteristics. The mean value of the five assertions representing positive characteristics is 1.54, namely rate of agreement between ‘agree’ and ‘strongly agree’. If we relate table 5.1 Teachers’ personal characteristics

Assertion No.

All participants Novice teachers (N=61) (N=28) Mean (S.D.) Mean (S.D.)

Experienced teachers (N=33) Mean (S.D.)

9. I demand a lot from myself 10. I fĳind it hard to admit my mistakes 12. I feel confĳident 21. I am empathetic and attentive 24. I have personal responsibility 27. I act according to behavioral code

1.6 (0.70) 4.4(0.70)

1.6 (0.77) 4.4(0.60)

1.6 (0.64) 4.4(0.74)

1.9(0.70) 1.5(0.56)

2.0(0.60) 1.4(0.50)

1.8(0.72) 1.6(0.54)

1.2(0.42)

1.2(0.36)

1.1(0.36)

1.5(0.70)

1.4(0.55)

1.5(0.66)

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to the assertion presenting a negative characteristic – “I find it hard to admit my mistakes” – with a mean of 4.4, then the rate of disagreement is between ‘disagree’ and ‘strongly disagree’. If we convert this value of negative statement to a value which fits positive statement, we obtain adjustment to 1.6 which is near 1.54. The Alpha Cronbach coefficient for these five assertions is 0.55. Moreover, we should take into consideration that a category relating to a certain aspect of the professional image might include assertions from different perspectives, thus reducing the correlation. The assertion with the highest mean rate of agreement relates to personal responsibility (1.2) and its standard deviation is also the lowest (0.42). Fifty-one participants indicated ‘strongly agree’, nine indicated ‘agree’ and only one participant indicated ‘neutral’, attesting to indecision towards one way or another. The two next assertions refer to behavioral code as well as to empathy and attention with a mean of 1.5 and standard deviation of 0.70 and 0.56 respectively. With regards to empathy, only two participants indicated ‘neutral’, without expressing any position while the rest were divided between ‘agree’ and ‘strongly agree’. On the other hand, only one participant indicated ‘neutral’ for the assertion relating to behavioral code but there were also two participants who indicated ‘disagree’. The assertions “I demand a lot from myself” and “I find it difficult to admit my mistakes” obtained the same mean rate of ‘agree’ and ‘disagree’ (1.6 and 4.4 respectively) which, by conversion to the opposite value, is identical to 1.6. Furthermore, these assertions have an identical standard deviation (0.70). Degree of distribution of the answers was identical after changing directions; five participants in each assertion indicated ‘neutral’ whereas one participant in each assertion expressed a “non-conformist” attitude, one participant indicated ‘disagree’ for “demand from himself” and one participant agreed that he found it hard to admit his mistakes. An assertion with a relatively high rate of agreement (or disagreement about a “negative” characteristic) was the one relating to self-confidence. The mean rate of agreement (1.9) stemmed from a relatively high number of participants (six) who refrained from taking any position, indicating ‘neutral’. Only one participant indicated ‘disagree’. 3.1.2 Teachers’ Approach towards Students Table 5.2 displays the mean values and standard deviation of assertions relating to teachers’ approach towards students. The mean value of the six assertions which express positive attitude (1, 2, 8, 23, 28, 29) is 1.72, namely rate of agreement closer to ‘agree’ than to ‘strongly agree’. The assertion relating to negative characteristic: “I can’t stand students

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Assertion No.

All participants (N=61) Mean (S.D.)

Novice teachers (N=28) Mean (S.D.)

Experienced teachers (N=33) Mean (S.D.)

1. I encourage independent thinking 2. I evoke trust in the pupil’s ability 6. I can’t stand students asking many questions 8. I accept my students as they are 23. I provide a service to the learners 28. I relate to the individual learner 29. I am capable of granting a lot to my students

1.6(0.73) 1.4(0.53) 4.7(0.49)

1.7(0.72) 1.3(0.45) 4.7(0.47)

1.6(0.74) 1.5(0.56) 4.7(0.51)

2.1(0.89) 1.8(0.94) 1.9(0.67) 1.5(0.56)

1.9(0.77) 1.7(0.92) 2.0(0.69) 1.6(0.55)

2.2(0.96) 2.0(0.96) 1.7(0.63) 1.5(0.57)

asking many questions” received a mean value of disagreement closer to ‘strongly disagree’ (4.7) or, if we reverse the scale, it is equal to 1.3. This assertion obtained the highest rate of agreement/disagreement in this group and its standard deviation is also extremely small (0.59). Only one participant indicated ‘neutral’ whereas forty-four participants indicated ‘strongly disagree’ and fifteen wrote ‘disagree’. The Alpha Cronbach coefficient for these items is 0.50. The assertion which was second in its rate of agreement was “I evoke trust in the pupil’s ability” with a mean value of 1.4 (standard deviation = 0.53). Only one participant indicated ‘neutral’ for this assertion. The assertion “I am capable of granting a lot to my students” was the third in the rate of agreement (1.5) (standard deviation=0.56). Two participants stated ‘neutral’ and the remainder were divided between ‘agree’ and ‘strongly agree’. The assertion “I encourage independent thinking” was the fourth in the rate of agreement – 1.6 (standard deviation=0.73). Two participants stated ‘disagree’ and three others wrote ‘neutral’. The assertion “provide a service to the learners” with a mean value of 1.8 (standard deviation=0.94) was fifth in the rating. One participant indicated ‘strongly disagree’, four participants wrote ‘disagree’ and five others claimed they were ‘neutral’. Ten participants stated they were ‘neutral’ for the following assertion, “I relate to the individual learner”, with a mean value of 1.9 (standard deviation=0.67). The assertion with the lowest rate of agreement in this category was “I accept my students as they are” with a mean value of 2.1 (standard deviation=0.89). Three participants indicated ‘disagree’.

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3.1.3 Teachers’ Attitudes toward Teaching Table 5.3 lists mean values and assertions relating to the criterion of teachers’ attitudes towards teaching. Table 5.3 Teachers’ attitude towards teaching

Assertion No.

All participants (N=61) Mean (S.D.)

Novice Experienced teachers teachers (N=33) (N=28) Mean (S.D.) Mean (S.D.)

4. I feel guilty for learners’ failure 5. I judge attainments and not current work 11. The teaching profession is a vocation for me 15. It is impossible to teach every student 16. I often solve exercises on the board 18. There are students that I fail to reach 25. I am an involved teacher 26. I am a teacher who leads

2.6(0.96) 3.5(0.86)

2.5(1.00) 3.4(0.90)

2.8(0.92) 3.6(0.82)

1.4(0.66)

1.3(0.47)

1.4(0.66)

4.1(1.06)

4.3(0.99)

3.9(1.10)

2.7(1.05) 3.5(1.09) 1.6(0.82) 1.8(0.73)

2.5(1.02) 3.6(1.04) 1.5(0.74) 1.8(0.80)

2.9(1.07) 3.4(1.13) 1.6(0.85) 1.8(0.67)

The Alpha Cronbach’s coefficient for these eight assertions is 0.51. Assertions 11, 25 and 26 present anticipated behaviors according to social desirability and the mean value of their rate of agreement is 1.6. That is, there is a rate of agreement ranging between ‘agree’ and ‘strongly agree’. The assertion with the highest rate of agreement among these three assertions was “The teaching profession is a vocation for me” with a mean value of 1.4 (standard deviation=0.66). The assertion “I am an involved teacher” was next, with a mean value of 1.6 (standard deviation=0.82) and then came the assertion “I am a teacher who leads” with a mean value of 1.8 (standard deviation=0.73). In each of the three assertions one participant wrote ‘disagree’ whereas three, ten, eight participants respectively indicated ‘neutral’. Assertions 4, 5, 15, 18, present behaviors which are undesirable according to pedagogy principles, and no agreement has been reached regarding them. As for the assertions: “I judge attainments and not current work” and “There are students that I fail to reach”, their mean value is 3.5 (standard deviation 0.86 and 1.09 respectively). This mean value attests to weak disagreement.

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However, if we check the distribution of the first assertion of these two, we will see that twenty-two participants wrote ‘neutral’ whereas more than half of the participants (27) marked ‘disagree’ and seven participants indicated ‘strongly disagree’. Only one participant expressed ‘strongly agree’ and six participants marked ‘agree’. The second assertion has the same mean value but the degree of distribution of the responses is wider. Thirteen participants indicated ‘strongly disagree’, eighteen wrote ‘disagree’ and sixteen stated ‘neutral’. On the other hand, twelve participants indicated ‘agree’ and one participant wrote ‘strongly agree’. The assertion with the highest mean value of disagreement among these four assertions is “It is impossible to teach every student” with a mean value of 4.1 (standard deviation=1.06). The mean value attests to disagreement; however, the distribution of respondents to this assertion was relatively wide. One participant stated ‘strongly agree’, five participants mentioned ‘agree’ and eleven wrote ‘neutral’. The assertion with the relatively highest percentage of agreement in this group is “I often solve exercises on the board” with a mean value of 2.7 (standard deviation=1.04). Twenty-five participants expressed ‘agree’ and six wrote ‘strongly agree’. Conversely, thirteen participants mentioned ‘disagree’ and three participants indicated ‘strongly disagree’. One assertion, “I feel guilty for learners’ failure” has a mean value of 2.6 (standard deviation=0.95). This mean value was obtained from a group of twenty-three participants who avoided taking a stand and marked ‘neutral’. Twenty-one participants, though, expressed ‘agree’ and seven others indicated ‘strongly agree’. On the other hand, eight participants mentioned ‘disagree’ and two participants indicated ‘strongly disagree’. 3.1.4 Professional Image as Mathematics Teachers Table 5.4 displays the mean values and standard deviations of assertions relating to professional image as mathematics teachers. The four assertions, 3, 17, 19, 22, relate to teachers’ perception of themselves as teachers whereas the other two assertions, 7 and 13, relate to teachers’ degree of confidence in their success, the difference between them being the formulation. The Alpha Cronbach coefficient for these assertions is 0.57 and if we disregard assertion 19: “Today I am a better teacher than in the previous year”, the coefficient will increase to 0.74. The mean value of the first four assertions is 2.1, attesting to agreement. The degree of distribution, though, is similar only in three of the assertions. In assertion 19: “Today I am a better teacher than in the previous year” there is a relatively wide degree of distribution of answers, most of the participants expressing ‘agree’. Nevertheless, five participants stated

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Table 5.4 Professional image as mathematics teachers

Assertion No.

All participants (N=61) Mean (S.D.)

Novice Experienced teachers teachers (N=33) (N=28) Mean (S.D.) Mean (S.D.)

3. I am satisfĳied with myself as a mathematics teacher 7. I sometimes wonder to what extent I succeed as a mathematics teacher 13. I sometimes doubt my success as a mathematics teacher 17. I have extensive knowledge in mathematics 19. Today I am a better teacher than in the previous year 22. I have wide pedagogical knowledge in mathematics teaching

1.9(0.77)

2.0(0.67)

1.7(0.75)

2.8(1.23)

2.5(1.35)

3.1(1.05)

3.3(1.24)

3.3(1.25)

3.4(1.25)

2.2(0.79)

2.2(0.76)

2.1(0.81)

2.3(1.13)

1.7(0.62)

2.6(1.25)

1.9(0.80)

2.3(0.87)

1.6(0.60)

‘strongly disagree’ and two others who wrote ‘disagree’. Ten participants stated ‘neutral’. In the remaining three assertions: “I am satisfied with myself as a mathematics teacher”, I have extensive knowledge in mathematics” and “I have wide pedagogical knowledge in mathematics teaching”, the degree of distribution is similar with eight, sixteen and ten participants respectively indicating ‘neutral’. In the first assertion, out of the three, one participant wrote ‘disagree’ while in each of the other two assertions one participant stated ‘strongly disagree’ and one who indicated ‘disagree’. In the two assertions relating to extent of success in teaching, the mean value is close to ‘neutral’ with a similar and relatively high degree of distribution in both (1.23, 1.24). As to the assertion: “I sometimes wonder to what extent I succeed as a mathematics teacher”, ten participants expressed ‘strongly agree’ and nineteen wrote ‘agree’, whereas fourteen participants indicated ‘disagree’ and six mentioned ‘strongly disagree’. For the assertion: “I sometimes hesitate about my success as a mathematics teacher”, five participants wrote ‘strongly agree’ and twelve responded ‘agree’, while thirteen participants said ‘strongly disagree’ and seventeen indicated ‘disagree’.

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3.1.5 Teachers’ Perception of the Students’ Attitudes toward Them Table 5.5 displays the mean values and standard deviation of items relating to teachers’ perception of the students’ attitudes towards them. Table 5.5 Teachers’ perception of the students’ attitudes towards them

Assertion No.

All participants Novice (N=61) teachers Mean (S.D.) (N=28 Mean (S.D.)

Experienced teachers (N=33) Mean (S.D.)

14. My students often tell me that I am a good mathematics teacher 20. My students like me 30. I believe I am a role model for my students

1.9(0.74)

2.0(0.70)

1.8(0.74)

1.7(0.74) 1.8(0.76)

1.7(0.69) 1.7(0.61)

1.7(0.77) 2.0(0.76)

The mean value of the three assertions among the entire teacher population is 1.8, attesting to a rate of agreement with a similar degree of distribution (0.74, 0.76). For each of these assertions there was one participant who expressed ‘disagree’ while the number of those stating ‘neutral’ was 10, 9 and 9 respectively. The Alpha Cronbach coefficient is 0.64, the highest of the five categories. However, we should take into account that in this category there were only three assertions, compared to other categories which consisted of 4–8 assertions. 3.2 Analysis of the Two Open-Ended Questions 3.2.1 Item 31: Skills Required by Mathematics Teachers Three out of the sixty-one research participants did not respond. Hence, the remaining fifty-eight participants gave a variety of qualities which in their opinion characterize the skills mathematics teachers should have. These skills were divided into a number of categories according to the frequency they were mentioned by the research participants. The first category consisted of two skills whose relative incidence was 50% and above. One skill related to mastery of varied teaching methods (56.9%) and a second skill related to having mathematical knowledge and being versed in learning materials, required by everyone engaged in mathematics teaching (50%). 31.8% indicated only the teaching methods, 25.9% indicated only the knowledge while 24.15% indicated both skills together. On the other hand, 17.2% of the participants did not state

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these skills as required skills. Among the experienced teachers, 54.8% mentioned both mathematical knowledge and mastery of teaching methods as compared to 63% of the novice teachers who indicated both skills. The second category, with a relative frequency of 20–25%, consisted of four additional skills: sensitivity and attention (24.1%), creativity (22.4%) as well as understanding children’s way of thinking and difficulties (20.6%). Among the experienced teachers, the four skills were indicated by 16.1%, 19.4%. 25.8% and 12.9% respectively while among the novice teachers, they were stated by 33.3%, 25.9%, 18.5% and 29.6% respectively. Only patience constituted a more required skill among the experienced teachers, which might be comprehensible based on their experience. The third category comprises five required skills at a relative frequency, ranging between 8%-13%: leadership and class management (12.1%), trusting the learners (10.3%), love of the subject and reducing fears (10.3%), empathy (8.6%) and consistency (8.6%). Among the experienced teachers, the skills were indicated by 6.5%, 6.5%, 6.5%, 9.7% and 9.7% respectively as compared to 18.5%, 14.8%, 14.8%, 7.4%, 7.4% of the novice teachers. The other twenty-four skills which were written obtained less than 8% and related to openness, motivation, self-confidence, love for children, charisma, inter-personal communication, wish to learn, sense of humor, professionalism (a general quality identified with several components mentioned before, like knowledge and teaching methods), optimism, inquisitiveness, respect and appreciation, relation with parents, giving, serenity, self-criticism, role model, acting skills, responsibility, support, integrity and expression capability. On average, each of the teachers indicated about three required skills (mean value of 3.2). Some indicated five and even six skills while others settled for one or two skills. 3.2.2

Item 32: What Measures Teachers Should Take in Order to Improve Their Teaching Quality Only fifty-five of the research participants responded to this open-ended question. Three participants wrote they did not have sufficient experience and, therefore, could not answer the question. Three other participants did not respond, without giving any explanation. The answers illustrate two prominent factors which might improve the teaching (more than 40% reported it). The first factor was improvement of teaching methods (43.6%) and the second was updating one’s knowledge and attending in-service training courses (41.8%). Among the experienced teachers (N=29), 58.6% indicated improvement of teaching methods as compared

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to 25.9% among the novice teachers (N=26). The need for updating one’s knowledge and attending in-service training courses was indicated by 51.7% of the experienced teachers versus 30.7% of the novice teachers. The third factor in the rating was reference to the individual learner (12.7%). This factor was indicated by the experienced teachers only. None of the novice teachers though stated this requirement for improving the teaching. The other factors were mentioned by less than 8% of the participants: making students love the subject, learning from one’s mistakes, improving class management competences, learning from colleagues, reducing the number of work hours, decreasing the workload, preparing oneself better for lessons, familiarizing oneself with learning disabilities, observing more discipline and homework preparation, developing creativity, being acquainted with links to the internet and being able to become more patient. On average, each research participant indicated only 1.3 factors which might improve their mathematics teaching. This mean value was lower than the mean value of skills required for mathematics teaching (3.2). This gap of about two qualities on average between what is required and what needs to be improved can be construed as positive and does not require any improvement. 3.3

The Second Research Question: Differences in the Professional Image between Experienced Teachers and Novice Teachers 3.3.1 Mathematics Teachers’ Personal Characteristics When referring to the two sub-groups, experienced teachers and novice teachers, minor differences were demonstrated in assertions relating to the sense of confidence. The experienced teachers reported it as high whereas the novice teachers reported higher empathy and attention. All the differences were found as non-significant in a t-test. 3.3.2 Mathematics Teachers’ Approach towards Their Students In this category, too, responses of teachers from both groups were similar. Assertions associated with teachers’ approach to individual learners, providing a service to the learners and accepting the students as they are, were the exception, with a gap of 0.3 on average between the two participant groups. The novice teachers declared they related more to the individual learner (1.7) as compared to experienced teachers (2.0 with a similar degree of distribution); accepting more the students as they are: 1.9 (standard deviation=0.77) versus 2.2 (standard deviation=0.96); and agree more about providing a service to the learners: 1.7 versus 2.0 among the experienced teachers, with a similar degree of distribution. All the differences were found as non-significant in a t-test.

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3.3.3 Teachers’ Attitude towards Teaching In three assertions, the answers show a gap in the mean value: 0.3 or 0.4. In the assertion relating to the fact that it is impossible to succeed with every student, the novice teachers presented a mean value of 4.3 (standard deviation=0.99), attesting to a state between ‘disagree’ and ‘strongly disagree’. The experienced teachers showed a mean value of 3.9 (standard deviation=1.10), attesting to lower disagreement with a relatively higher degree of distribution. In the assertion relating to the solution of exercises on the board, the percentage of agreement among the novice teachers was 2.5 versus 2.9 among the experienced teachers, with a similar and relatively high degree of distribution. In the assertion associated with feelings of guilt resulting from learners’ failure, the mean value of the novice teachers was 2.5 as opposed to 2.8 among the experienced teachers, with a relatively high degree of distribution. All the differences were found as non-significant in a t-test. 3.3.4 Professional Image as Mathematics Teachers In this category of professional and self-image as well as self-efficacy as mathematics teachers, the gaps in the mean value of agreement about the assertions were the highest. The largest gap was obtained for the assertion “Today I am a better teacher than in the previous year”. Among the novice teachers, the mean value of agreement was 1.7 (standard deviation=0.62), attesting to a state between ‘agree’ and ‘strongly agree’. On the other hand, among the experienced teachers, the mean value was 2.6 (standard deviation=1.25), with a relatively high degree of distribution. This finding illustrates that the experienced teachers did not improve from the previous year and this could apply perhaps to teachers who had reached the stagnation stage. Nevertheless, the experienced teachers concur that they have pedagogical knowledge in mathematics teaching and the mean value of agreement was 1.6 as opposed to 2.3 (standard deviation=0.87) among the novice teachers who showed a higher degree of distribution. This fact attests to heterogeneity in the perception of pedagogical knowledge. In the assertion “I sometimes wonder to what extent I succeed as a mathematics teacher”, the novice teachers were more hesitant and the mean value of agreement they expressed was 2.5 (standard deviation=1.35), with a relatively high degree of distribution. A mean value of 3.1 was obtained among the experienced teachers (standard deviation=1.05). In a similar assertion “I sometimes doubt my success as a mathematics teacher”, the mean value of the experienced teacher was 3.4, similar and only slightly higher, attesting to disagreement more than the item relating to doubts about the extent of success. The difference among the novice teachers is more meaningful and the mean

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value of 3.3 is similar to that of the experienced teachers with an identical and relatively higher degree of distribution in both groups (1.25). This distribution demonstrates heterogeneity regarding the doubts. The experienced teachers are more satisfied with themselves as mathematics teachers (mean value of 1.7 versus 2.0 among the novice teachers) with the same degree of distribution. Concerning the extensive professional knowledge there is hardly any difference between the groups, which is somewhat surprising (2.1 among the experienced teachers compared to 2.2 among the novice teachers, with a similar degree of distribution). All the differences were found as non-significant in a t-test. 3.3.5 Teachers’ Perception of Their Students’ Attitude towards Them The findings showed that teachers in both groups gave an identical report, with a mean value of 1.7, saying that their students like them. However, degree of distribution among the novice teachers was smaller than that of the experienced teachers. The novice teachers agree (2.0), less than the experienced teachers (1.7), that their students often tell them personally that they are better teachers. Furthermore, the novice teachers agree more that they believe they are a role model to their students (1.7 versus 2.0, with a small degree of distribution). All the differences were found as non-significant in a t-test.

4

Discussion and Conclusions

The present study aimed to examine mathematics teachers’ professional image, namely explore the way mathematics teachers depicted themselves in various aspects associated with teaching the subject. The first research question dealt with mathematics teachers’ professional image as presented by the participants themselves. The questions relating to teachers’ image were divided into five categories, comprising different aspects of teachers’ qualities and practice. The first assertions category referred directly to teachers’ perception of their personal characteristics. The sixty-one participants painted a rosy picture of teachers, almost the ideal teacher. In a study conducted by Reichel and Arnon (2005), 91% of the participants, pre-service and in-service teachers, indicated the characteristic “empathy and attentive” as the most prominent quality of the ideal teacher. Similarly, findings of the present study showed agreement about this characteristic. In the study of Avraham (1972), the quality “demand a lot from themselves” was the second in the teachers’ rating of qualities. The present study, too, found an average rate of agreement which was relatively high in both groups.

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The second assertions category related to teachers’ perception of their approach towards their students. Here, too, the picture was that of ideal teachers, treating their students according to educational papers, educational psychology and educational sociology. In the study of Avraham (1972), the quality “stimulating thought” obtained the highest rate of agreement among the teachers. Findings of the present study illustrated a high rate of agreement, with a negligible difference between the two groups. The third assertions category referred to the teachers’ attitudes towards teaching. In the present study there were hardly any differences between the two groups. The assertion with the highest rate of disagreement in this category was “it is impossible to teach every student”, for which teachers have an explanation, e.g. size of the class. The fourth assertions category related to the professional image as mathematics teachers. The rate of agreement to positive professional qualities, such as “I have wide professional content knowledge”, “I have wide pedagogical knowledge”, “I am satisfied with myself as mathematics teacher” or “Today I am a better teacher than in the previous year”, was higher, even more than for those assertions characterizing the first category, namely teachers’ personality. In the study of Arnon and Reichel (2007), 75% of the pre-service and inservice teachers attributed importance to knowledge among the ideal teacher’s qualities. In this assertions category, relatively wide gaps were shown between experienced and novice teachers. These gaps result from accumulated experience on the one hand and burnout on the other. This gap of almost a whole rating on this scale was obtained for the assertion: “Today I am a better teacher than in the previous year”. Most of the novice teachers stated ‘agree’ or ‘strongly agree’ (mean value of 1.7) while the mean value among the experienced teachers was 2.6, with a relatively high degree of distribution. In the issue of professional knowledge there was almost no difference between the two groups. There was a considerable difference, however, regarding pedagogical knowledge and experience accumulated with time, which the novice teachers still lacked. The lack of experience affected also the difference in the uncertainty the participants felt about their success as mathematics teachers. The fifth assertions category checked the way teachers perceived their students’ attitudes towards them. In the three assertions, the mean value and the standard deviation were similar. The average rate of agreement was close to agreement. The belief that teachers are a role model also obtained a mean value close to agreement. The two open-ended questions dealt with the skills which teachers need and the measures they should take in order to improve their teaching quality.

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The two prominent skills were teaching methods and mathematical knowledge. The next skills in the rating were sensitivity and attention, creativity, patience and students’ difficulties. The third-rated factors are sensitivity, attention and empathy which are not in line with the findings of Arnon and Reichel (2007), indicating it as the first place among pre-service and novice teachers. The quality of leadership obtained in the present study was only 12.1%, being a bit more noticeable among the novice teachers. Generally speaking, novice teachers consider they have to possess qualities which reflect on placing students at the center, e.g. the need for sensitivity and attention and comprehension of the students’ way of thinking and their difficulties. Conversely, experienced teachers attribute more importance to the improvement in teaching methods, updated knowledge and in-service training courses as well as acquisition of instruments for coping with individual learners. We believe that the differences stem from the gaps in experience between the investigated teachers. The experienced teachers indicated a professional need for being updated whereas the novice teachers placed students as the highest priority, before expanding the professional knowledge and enriching it. To sum up, a research limitation of the present study related to the sample which consisted of sixty-one mathematics teachers, namely not a representative sample. Nevertheless, the findings showed that the teachers’ professional image was rather high. The teachers who participated in the present study displayed a high rate of agreement in most of the required qualities discussed in the professional literature. For example: being versed in the area of knowledge, providing a service, assuming responsibility, being an involved teacher and acting according to a behavioral code (Reichel, 2010). On the other hand, they lack negative features like “find it hard to admit mistakes”, “cannot stand students asking many questions”. It is to be assumed that the teachers present themselves as ideal teachers, an illusion which constitutes an anticipated self (Avraham, 1972). Out of the items of the professional image questionnaire, the teachers responded consistently to the “social desirability”, the differences between experienced and novice teachers being prominent in items attributing importance to experience or mental burnout. In the open-ended questions, the teachers manifested a variety of skills appropriate for teachers’ professional image which they had portrayed. The same applied also to the issues in which teachers should improve. Teachers undoubtedly need mathematical knowledge and pedagogical knowledge. Yet, there are many other issues which had been mentioned above, such as sense of humor or charisma, which represent innate rather than acquired qualities.

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Based on the research findings, it is recommended being attentive to mathematics teachers’ professional image, reinforcing those required skills which can be acquired by in-service training courses throughout the teachers’ professional life or their teaching practice.

Acknowledgement This chapter was originally published as Patkin, D., & Gazit, A. (2016). Do “those who understand” teach? Mathematics teachers’ professional image. Teacher Education and Practice, 29(4), 660–681. Reprinted with permission.

References Arnon, S., & Reichel, N. (2007). Who is the ideal teacher? Am I? Similarity and difference in perception of students of education regarding the qualities of a good teacher and of their own qualities as teachers. Teachers & Teaching: Theory and practice, 13(5), 441–446. Avreham, A. (1972). Teachers self-image. In A. Avraham (Ed.), Teachers inner world (pp. 31--58). Tel Aviv: Otzar Hamoreh Publishing. [in Hebrew] Brown, M., Brown, P., & Bibby, T. (2008). I would rather die: Reasons given by 16-year olds for not continuing their study of mathematics. Research in Mathematics Education, 10(1), 3–18. Cochran-Smith, M. (2003). Teaching quality matters. Journal of Teacher Education, 4(3), 95–98. Cronbach, L. J., & Gleser, G. C. (1953). Assessing similarity among profiles. Psychological Bulletin, 50(6), 456–473. Daley, J., & Kim, L. (2010). A teacher evaluation system that works. Santa-Monika, CA: National Institute for Excellence in Teaching. Darling-Hammond, L. (2007). The story of Gloria as a future vision of the new teacher. Journal of Staff Development, 28(3), 25–26. Darling-Hammond, L., French, J., & Garcia-Lopez, S. P. (2000). Learning to teach for social justice. New-York, NY: Teachers College Press. Getzels, J. W., & Smilansky, J. (1983). Individual differences in pupil perceptions of school problems. British Journal of Educational Psychology, 53, 307–316. Guillaume, A. M., & Kirtman, L. (2010). Mathematics stories: Pre service teachers’ images and experiences as learners of mathematics. Issues, 19(1), 121–143. Hattie, J. (2003, October). Teachers make a difference: What is the research evidence? Paper presented at the Australian Council for Educational Research Annual Conference on Building Teacher Quality, Melbourne.

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Marshal, P. (2001, February). A study of primary ITT students’ attitude to mathematics. Paper presented iWork group 3 at CERME2 – Conference of European Research on Mathematics Education 2, Marienbad, The Czech Republic. McPhan, G., Morony, W., Pegg, J., Cooksey, R., & Lynch, T. (2008). Math-why not? Canberra: Department of Education, Employment and Workplace Relations. Mendelson, N. (2006). The math teacher’s self-image perception in relation to their professional development stages and the social status of the teaching profession (Unpublished Ph.D. dissertation). Haifa: University of Haifa. [Hebrew with an English abstract] Murray, S. (2011). Secondary students’ descriptions of “good” mathematics teachers. Australian Mathematics Teacher, 67(4), 14–21. NBPTS (National Boarding for Professional Teaching Standard). (2007). Eligibility and policies (Electronic version). Retrieved February 1, 2012, http://www.nbpts.org/ become_a_candidate/eligibility_policies Noddings, N. (2002). Educating moral people: A caring alternative to character education. New York, NY: Teachers College Press. Reichel, N. (2010). The ideal teacher. Hed-Hachinuch, 85(1), 60–63. [in Hebrew] Reichel, N., & Arnon, S. (2005). Three portraits of teachers in the view of students of teaching: The ideal teacher, the teacher of teachers and the image of the student him/herself as a teacher. Dapim, 40, 23–58. [in Hebrew] Rockoff, J. E. (2004, May). The impact of individual teachers on student achievement: Evidence from panel data. America Economic Review, 94(2), 247–252. Shulman, L. S. (1986). Paradigms and research program for the study of teaching. In C. Wittrock (Ed.), Handbook of research on teaching (3rd ed., pp. 3–36). New York, NY: Macmillan. Shulman, L. S. (1987). Knowledge and teaching: Foundation of new reform. Harvard Education Review, 57, 1–22. TIMSS. (1996). Quality assurance in data collection. In M. O. Martin & I. V. Mullis (Eds.), Third International Mathematics and Science Study. Chestnut-Hill, MA: International Study Center, Boston College. Vinner, S. (2011). What should we expect from somebody who teaches mathematics in elementary schools? Scienta in Educatione, 2(2), 3–21. Wilson, S. M., & Youngs, P. (2005). Research on accountability processes in teacher education. In M. Cochran-Smith & K. M. Zeichner (Eds.), Studying teacher education: The report of AREA panel on research and teacher education (pp. 591–643). Mahwah, NJ: Lawrence Erlbaum Associates.

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Appendix A Attitudes Questionnaire Dear colleague We kindly ask you to respond to the following assertions and indicate the degree of your agreement with the, on a scale of 1 to 5, the number 1 signifies ‘strongly agree’ and the number 5 – ‘strongly disagree’. Thank you for your assistance. Assertion No.

1.

I encourage independent thinking in my mathematics lesson pupils 2. I evoke trust in the pupil’s mathematic semantical ability 3. I am satisfĳied with myself as a mathematics teacher 4. I feel guilty for learners’ failure in mathematics tests 5. I judge attainments and not current work in mathematics lessons 6. I can’t stand students asking many questions in mathematics lessons 7. I sometimes wonder to what extent I succeed as a mathematics teacher 8. I accept my students in mathematics lessons as they are 9. I demand a lot from myself as a mathematics teacher 10. I fĳind it hard to admit my mistakes in mathematics lessons 11. The teaching profession is a vocation for me 12. I feel confĳident in almost every situation of teaching in mathematics lessons

2. 3. 4. 5. 1. strongly agree neutral disagree strongly disagree agree

Do “Those Who understand” Teach? Assertion No.

13. I sometimes doubt my success as a mathematics teacher 14. My students often tell me that I am a good mathematics teacher 15. It is impossible to teach mathematics to every student 16. I often solve exercises on the board 17. I have extensive knowledge in mathematics 18. There are students that I fail to reach in mathematics lessons 19. Today I am a better teacher than in the previous year 20. My students like me 21. I am empathetic and attentive to my pupils 22. I have wide pedagogical knowledge in mathematics teaching 23. I view myself as a service provider to the student 24. I have personal responsibility 25. I am an involved teacher 26. I am a teacher who leads 27. I act according to behavioral code 28. I relate to the individual learner in mathematics lessons 29. I am capable of granting a lot to my students 30. I believe I am a role model for my students 31. What are the skills required by all teachers who teach mathematics in elementary school? ____ 32. In my opinion, in order to improve my teaching quality, I have to …

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1. 2. 3. 4. 5. strongly agree neutral disagree strongly agree disagree

CHAPTER 6

Elementary School Mathematics Pre-Service Teachers’ Perception of Their Professional Image Nili Mendelson

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Introduction The secret of teaching means being versed in teachers’ local details and daily life. Teachers might be the richest and most effective source of knowledge about teaching. Those who strive to comprehend teaching, must address the teachers themselves at a certain point. (Shubert & Ayers, 1992, p. v)

The present study embraces this approach and addresses elementary school mathematics pre-service teachers [hereafter – “students”] so that they identify and examine their professional image from their own point of view. Various studies highlight teachers’ inner perspective (Ben-Peretz, Mendelson, & Kron, 2003; Kremer-Hayon, 1991; Staddart & Roehler, 1988) and emphasize the personal perspective as a meaningful factor in teaching. Acquaintance with the professional image of future teachers from their point of view, presents a personal perspective and opens a window to teachers’ inner world, the teaching profession and teachers’ professional orientation (Thomas & Beauchamp, 2011; Trumbul, 1987; Van Veel, Sleegers, Bergen, & Klaassen, 2001). Inbar (1996) investigated and analyzed thousands of metaphorical images collected from teachers and learners. He argued that an examination of such images enables exposure of the assumptions underpinning teachers’ basis of beliefs regarding various topics, such as education, learners and their role in class. Moreover, Fischman (2001) maintains that using visual information such as movies, paintings and illustrations might offer a new viewpoint for observing different topics in education and exposing covert points which cannot always be addressed by conventional ways. According to Combs, Blume, Newman, and Wass (1981), teachers’ self-professional image stems from the way they perceive themselves and their role in society. This image tends to directly affect their teaching strategies and conduct in class and indirectly impact learners’ attainments, promotion and development. During their first years of practice, when teachers are in the survival and © koninklijke brill nv, leiden, 2018 | doi:10.1163/9789004384064_006

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control-achieving stages, their professional insight is still limited. The teachers are inclined to adopt conformism and are less willing to introduce changes. As they progress in their professional career, they are more open to cope with changes. Based on these perceptions, the present study investigated the professional image of mathematics students in their own eyes. The participants were 60 students from two academic colleges of education in Israel who displayed their professional image by a series of metaphorical pictures. For the purpose of examining the professional image, an instrument based on visual metaphors was developed, making it possible to obtain rich information about the professional image. The pictures which comprised the instruments served as an artefact around which a conversation that demonstrated the participants’ opinion was conducted. Using the series of pictures/illustrations combined with the interpretation of the pictures through explanatory lenses, allowed the participants to expose to themselves and to others what they were thinking about their role as mathematics teachers already at their education stage. A function of this exposure is possible reflection and feedback about teaching processes which students are going to confront in future.

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Theoretical Background

Studies of mathematics usually focus on the varied problems which arise at school. These problems range from differences between boys and girls regarding academic attainments from reasons for failing in this subject (Fennema & Franke, 1992; Maqsud, 1997) and up to diversified teaching methods and models of professional development in mathematics (Malone & Taylor, 1992). Only a few studies engaged in mathematics teachers themselves. For example a study conducted by Levenberg (1998) that researched mathematics teachers’ adjustment to technological changed in the modern age or the study of Chissick (2004), which explored factors affecting teachers during reform processes at school. Prusk (2002) checked the impact of cultural elements on mathematics studies concerning differences between teachers from the former Soviet Union and Israeli-born teachers. However, like previous studies, her research does not investigate the beliefs and perceptions of the teachers themselves as to their profession and role in class. Using metaphors for exploring perceptions and beliefs is a prevalent way which is increasingly growing in the research of teachers’ thinking since metaphors are one of the fundamental instruments of human thought

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(Lakoff & Johnson, 1980; Thomas & Beauchamp, 2011) which constitute an alternative to common verbal instruments. A position paper about the promotion of mathematics teaching (MANOR – Mathematics Teachers Centre, 2004) stipulated that mathematics is a complex subject which requires conceptual knowledge and mastery of operations and manipulations. On the other hand, learners should be able to use this knowledge in various contexts by implementing mathematical models and applying judgments and strategies while demonstrating mathematical communication, reflective capability and organization and teamwork skills. In addition to a high score in mathematics which is a prerequisite for students interested in teaching this discipline, these skills might definitely indicate unique elements in the profile of students learning to teach mathematics as compared to those who intend teaching other subjects. This gives rise to the question whether the professional image of mathematics future teachers can illustrate a professional image unique to students learning to teach this discipline. 2.1

In-Service Teachers [Teachers] and Pre-Service Teachers [Students] in the Mirror of Metaphors In recent years, we have witnessed the expanding use of images and metaphors as diagnostic instruments in education qualitative research, aiming to examine teachers’ authentic perceptions and beliefs (Leavy, Mcsorely, & Bote, 2007; McGrath, 2006; Thomas & Beauchamp, 2011). These instruments constitute an alternative to the conventional verbal instruments (Inbar, 1996; Mendelson, 1997). Dooley (1998) indicates the use of metaphors for encouraging teachers to reflect upon their attitudes towards teaching as well as upon their beliefs associated with the teaching profession. According to her, this leads to the formation of a language which facilitates mediation between theory and practice. Expression by means of images and metaphors might add to the collected data another dimension which enables examination and analysis of contents typical of the training for the role stage, creating a reality at the same time. According to Wubbels (1992) and Goldstein (2005), the use of metaphors is one of the means of displaying students’ deep beliefs and perceptions which have been built over the years since they were learners. These beliefs sincerely illustrate what the students think about the teaching profession without expressing educational attitudes ‘preached’ to them during their education process but in which they do not believe. Several researchers have recently engaged in exploring metaphors and images in education, pointing out that teachers’ professional image is

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multi-hued. These studies specify a variety of images and metaphors from the world of nature (flora and fauna), the world of still life and the world of work (professionals). Weber and Mitchel (1995) indicate that different researchers attempted to describe teachers’ role by a metaphorical use of professionals, such as a gardener, sculptor, bank employee and even a revolutionary. Their book specifies researchers such as Socrates, Skinner, Eisner and others. Socrates, for example, simulated the female-teacher to a midwife whereas Skinner depicted the male-teacher as a technician. Eisner saw the teacher as a painter, Stenhouse viewed the teacher as a researcher while Green describes the teacher as a strategist. Unlike De-Castle (cited in Weber & Mitchel, 1995), Bullough (1991) chose to mention images which novice teachers tend to apply for describing the teaching profession. These teachers usually choose the caring mother which embodies the elements of giving, concern and empathy, an image which Noddings (1984) extensively describes in her book Caring. Furthermore, Bullough (1991) portrays additional images which came up, among them a male-teacher as a butterfly, a female-teacher as a police officer, a female-teacher as a chameleon and a female-teacher as a she-dog. Efron and Joseph (1994) describe the metaphor as a powerful instrument by means of which teachers can fully express the meaning of their practice in class. The researchers interviewed 26 teachers regarding their benefit from the inner process of finding their professional image by being given the opportunity to observe more clearly their role and the reality of their world of work. The study conducted by Efron and Joseph (1994) encompasses a variety of metaphors which describe teachers’ role and the teaching profession. It is noteworthy that the teachers who participated in the study defined themselves as career people rather than just ‘paycheck teachers’. They loved their profession and respected it. The metaphors used to depict themselves were: conductor, gardener, mother substitute, good friend, leader, dentist, skipper, passenger, guide, social worker as well as a breeze or butterfly. Two metaphors – a guide and a tutor – were chosen as representing the teacher’s image in a study conducted by Zimt and Dan (2001). The study explored students’ metaphors for the terms ‘teaching’, ‘learning’ and ‘assessment’ during their four years at college. The researchers maintain that the metaphors ‘guide’ and ‘tutor’ stand for perceptions of the teaching profession according to which guiding and tutoring the learners are the major part of teachers’ practice. These approaches differ from the conventional approaches like the one represented for example through the professor’s figure (Zimt & Dan, 2001), namely teachers are entrusted only with delivering knowledge. Zimt and Dan argue that, “perhaps the change is the result of a conceptual change which transpired in recent years regarding the perception of knowledge. Today, the constructivist

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approach is studied in many educational institutions and teaching methods implement new ways in line with its spirit” (p. 75). The participants in the study of Zimt and Dan (2001) chose visual metaphorical representations. For example, the boat and its surroundings (sea and coordinates) metaphorically representing the dynamic process which teachers and their learners undergo during the teaching and learning. Or the car which displays teaching as the driving leading the process. Visual metaphorical representations were also discussed in the study of Ben-Peretz et al. (2003) who used them through pictures of professionals. The metaphors were more like caricatures with intentionally exaggerated elements. The pictures are unique in that they do not directly threaten the observers and allow a ‘controlled space’ of images for describing teachers’ profession. The value of the pictures resides in the fact that they restrict the world of images and free metaphors (as many researchers portray) to the world of professionals. This restriction does indeed have a price manifested by limiting the differentiation, variety and rich expression embodied in the very presentation of free metaphors (Inbar, 1996). However, the inter-professional comparison yielded characteristics of teachers’ professional image and enhanced the distinctions regarding their professional functioning. Connelly and Clandinin (1999) argue that when teachers are requested to formulate their educational perceptions in a direct verbal manner, they frequently express what is expected of them (social desirability) rather than their inner belief which is unique to them. In her study of mathematics teachers’ professional image, Mendelson (2006) points out that the defense mechanism of an ‘expected’ professional image or social desirability presented as an ideal professional image might constitute a difficult hurdle for the researcher. Every question perceived by participants as a challenge to their self-professional image, will make them opt for a ‘prettified’ image and will prevent a real examination of teachers’ image in their own eyes. Using metaphors for exploring the professional image might overcome this obstacle. 2.2 The Status of Mathematics The unique status of mathematics stems, among others, from the fact that no one remains indifferent to it. People either love or hate mathematics, understand it or do not understand it. All people have their own special image about the subject (Furinghetty, 1993, cited in Picker & Berry, 2000). Most researchers who engage in characterizing the subject of mathematics, depict it as a collection of numbers, facts and procedures. In order to correctly and effectively cope with it, learners should know well the rules of mathematics and acquire suitable competences which will assist them to solve problems

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in an accurate and appropriate manner (Amit & Hillman, 1999; Karsenty & Vinner, 2000). According to Hill (1992), mathematics has a unique status of itself. She argues that there is a wide consensus as to the importance of the subject and its high status which contradicts the low social positioning of teaching in the hierarchy of professions. Similarly, Grossman and Stodolsky (1995) indicate the high status of mathematics at schools as well as the high status of teachers who teach this discipline. Their study highlights the centrality of mathematics in the school-based curriculum, and its weight in regional and national exams. Hill (1992) emphasizes the high intellectual requirements not only from learners of mathematics but also (and even more so) from the teachers of this discipline. The researcher specifies a list of competences necessary for mathematics teaching and learning. For example, development of thinking; research and discovery processes implemented while solving problems; and the ability to present hypotheses and use logical arguments. Lim and Ernest (1999) stipulate that two major myths portray mathematics as ‘difficult’ and ‘suitable for clever individuals only’. Researchers argue that the negative image of mathematics is the reason for the limited registration to mathematics studies in higher education institutions. Furthermore, they claim that due to this image, children and grownups lack more than once confidence and experience strong anxiety with regards to mathematics. Marshall (2001) found that attitude towards mathematics is related to the wish to be high achievers. In this context, Ernest (1996) maintains that anxiety which is increasingly intensified with each failure to attain correct answers, turns sometimes into a phobia about mathematics. Von Glasersfeld (1991) proposes a distinction of his own according to which the way mathematics and mathematics teaching aids are presented, results in learners’ prolonged aversion to numbers rather than to comprehension of the positive and practical aspect of the subject. Several researchers stipulate that past experience of mathematics’ learners is related to the picture they have about its nature (Andrews & Hatch, 2002; Lerman, 1990; Mcnamara, Jaworsky, Rowland, & Prestage, 2003; Thompson, 1984). Pajaras (1992), who also sees a relation between people’s beliefs system and their past experience regarding mathematics, depicts beliefs as filters through which the past experience is interpreted. Pajaras suggests conducting a study of teacher education and specialization in the subject while focusing on teachers’ beliefs since in his opinion these beliefs might enhance our insight about teachers’ conduct. Ponte (1994) concurs with this suggestion, declaring that, “one of the essential goals of education is to discuss the different beliefs and promote our awareness of them due to their major role in shaping the teaching practice” (p. 190). Like Ponte, Thompson (1992) mentions that

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teachers’ conduct and actions in class are grounded in their perceptions of the learnt subject and the teaching thereof. Chissick (2004) investigated the factors which impact mathematics teachers during change processes. According to her, various researchers widely concur that beliefs and perceptions play an important role in mathematics education. However, there are but a few studies which examine beliefs about mathematics or attitudes of the teachers themselves towards the subject. Consequently, the present study focuses on the way elementary school mathematics pre-service teachers (students) perceive their professional image and on the beliefs which shape this image. 2.3 Research Questions – What is the preferred professional image of the research participants who are elementary school mathematics pre-service teachers? – What are the professional images rejected by the research participants?

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Methodology

3.1 Research Instruments For the purposes of the present study a series of pictures/illustrations of various professions was chosen. These pictures reflect different ways of inter-personal professional functioning and embody qualities associated with teaching and educational worldviews. The series of pictures was developed by Ben-Peretz and Kron (see Appendix A). The pictures integrated exaggerated elements in a style which was typical of caricatures (Kumpunen, 1980; Perkins, 1975). As such, they might stimulate the observers (in our case the students participating in the study) to expose thoughts and emotions related to their function as mathematics teachers and to the teaching world in general. The pictures were displayed without a text and thus enabled a freer and wider expression than illustrations which came with a text and limited free interpretation. According to Kennedy (1982), the interaction between the observer and the picture is highly important. He claims that, “the observer of the picture will know to choose between the relevant and the irrelevant” (Kennedy, 1982, p. 604), something which the participants in the present study were requested to perform. 3.2 Research Population The research participants were 60 students from two academic colleges of education in Israel. In the first college 30 students were learning in two

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pathways: mathematics and sciences or mathematics and computer sciences. The 30 students from the second college were learning in three pathways: special education and specialization in mathematics, sciences and specialization in mathematics or communication and specialization in mathematics. Some of the participants were in their first year while the others were in their second year of studies. Distribution of the participants illustrated that in the first college the division between women and men was identical and so was the sectorial division between Jews and minorities, the average age being 24. In the second college all the participants were Israeli-born female students about 24 years old. They came from both the Jewish and Arab sectors. The participants indicated different and varied reasons for choosing to become mathematics teachers. Among them: love of the subject, influence of meaningful teachers from their past who constituted role models, a wish to create remedial experiences in learning this subject, inner satisfaction, wish to make an impact, childhood dream and viewing mathematics teaching as an ideal. 3.3 Research Procedure and Data Processing The research participants were recruited from the list of students the researcher received from the program coordinators at the colleges. The initial contact with the participants was established through the program coordinators who told the students about the study and received their consent to take part in it. At the second stage the researcher sent the participants an invitation by mail in order to coordinate the time and place for a group encounter. Then, a 1-hour meeting was organized with each group for responding to questionnaires. For that purpose, a spacious room was chosen in which the seven pictures could be spread on desks. Each participant was asked to answer the following questions: After looking at the seven pictures, please choose the picture which best represents your professional image as a teacher. 1. Please give arguments for your choice. 2. Based on your experience, what made you choose this very picture? 3. Is there a picture of any professional which is unsuited to represent your professional image as a teacher? 4. Is there another profession, not included in the seven pictures, which better represents your professional image as a teacher? 5. Do you have any comments regarding the pictures? The data obtained from the participants’ answers were content analyzed (Sabar Ben-Yehoshua, 2002). At the first stage, the data were collected according to the questions and thereafter the choices, rejections and the alternative choices were counted. The last stage was dedicated to analyzing the participants’

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explanations and interpretation of the pictures as well as to the building of categories and assertions based on the questionnaires.

4

Findings

The findings from the choices of all the participants in the two colleges (Figure 6.1) illustrated a professional image perception which focused on two out of the seven pictures in the series: 35% of the participants chose the ‘animal keeper’ and 33% of the participants chose the ‘conductor’. The other choices (in a descending order) were the following: 15% chose the ‘shopkeeper’, 10% chose the ‘puppeteer’, 5% chose the ‘entertainer’ and 2% chose the ‘judge’. The picture of the ‘animal trainer’ was not chosen at all.

figure 6.1 Distribution of the participants’ choice of picture.

4.1 The Professional Images Chosen by the Students 4.1.1 The Participants’ Reference to the ‘Animal Keeper’ Picture The ‘animal keeper’ picture was chosen by most of the participants, a total of 35%. Those who chose this picture for representing their professional image as teachers referred mostly to features which they believed characterized the ‘keeping’ teacher. For example, demonstration of concern and love for the learners as well as attention and much giving. From their point of view, mathematics teachers should deliver the material in an interesting way. However, first and foremost they should support the learners and encourage them when they encounter difficulties with learning the material and consequently fail in it. The participants underscore the need to be patient and supportive towards the learners and maintained that these qualities were essential for finding a way to help all pupils with difficulties, including generating a change in the subject perception:

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– from my point of view, being a teacher requires a certain approach: first you have to love the children … caring for the learners should encompass gentleness, kindness and a young head – mathematics teachers should join the profession being ready to give a lot – being a caring person, I see myself as a mathematics teacher who listens to my pupils, sharing their dilemmas … only not to be like the sciences teacher – I have chosen to become a mathematics teacher in order to be another teacher, eliminate fear of and anxiety about mathematics – I view my profession as a keeper-teacher in the sense of inculcating values to my pupils through attention and patience. As a mathematics teacher, I see my role as an integration of required knowledge delivery and care – teachers should be in contact with the learners and should embrace the phrase “Teach a child in the trade of his way” [Book of Proverbs, 22:6]; each pupil is a whole world – actually in mathematics you have to care for the pupils individually rather than deliver a frontal lesson, like history for example. The figure of the ‘animal keeper’ was hardly rejected by the students with the exception of three who maintained that as a result of the close care, the pupils were too dependent on the teachers: – the teacher in the ‘animal keeper’ picture is a key figure and the children depend on it to such extent that they cannot leave him or her; there is a too high level of dependence and the children’s independence is missing – a teacher who caresses too much will eventually be ridiculed by the pupils – caring is sometimes too gentle and might be perceived as non-authoritative. 4.1.2 The Participants’ Reference to the ‘Conductor’ Picture The ‘conductor’ picture was chosen by 33% of the research participants and it was ranked second after the ‘animal keeper’. The students described the ‘conductor’ as a person who projects confidence in his or her ability to lead the learners to success, demonstrates good control in managing the class and maintains interrelations and continuous dialogue with the learners: – I see conductors as leaders; they make the pupils act as they please and to do it willingly – teachers as conductors should project self-confidence; no one can intervene in their work and they are supposed to come ready for the lesson – the conductor should trace the way for working in harmony – as a mathematics teacher, I think that my role is to navigate the class like in an orchestra in order to create a good melody; teachers should have reigns in their hands.

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Most of the participants consider that a positive teacher-learner interaction is a prerequisite for the learners’ success and believe that ‘viewing each and every instrument with its uniqueness could compose the whole’: – similarly, to a conductor, teachers should achieve success for the orchestra, namely their concern is mainly addressed to the pupils – I believe that mathematics teachers must particularly walk one step at a time with their learners; it is important to direct all of them to attain comprehension – mathematics teachers should guide all the players-pupils according to everyone’s instrument, according to their special sound – as a teacher, I deem it important to begin at the starting point of the pupils’ knowledge; as a teacher my purpose is to navigate the class to desirable objectives and goals, while constantly relating to the nature of the class from the viewpoint of every pupil’s skills – the conductor can create harmony while emphasizing the differentiation between the learners. The figure of the ‘conductor’ was rejected by one student only. He argued that “the ‘conductor figure’ projects distancing … The pupils are distant and the conductor stands condescendingly on a stage and gives instructions from a distance”. 4.1.3 The Participants’ Reference to the ‘Shopkeeper’ Picture Fifteen percent of the participants who chose the ‘shopkeeper’ picture emphasized the positive aspect it represented. According to them, the ‘shopkeeper’ teacher should demonstrate a teaching ability through persuasion, good explanation ability, evoking the learners’ interest by the way the lessons are delivered. All those who chose the ‘shopkeeper’ attributed great importance to delivering the required knowledge to the learners: – I chose the ‘shopkeeper’ since I believe my role is to deliver knowledge to the pupils in a clear and distinct and also agreeable manner; when I am teaching the lesson I relate to the learners in a pleasant way – as a teacher, I want to sell to the pupils the knowledge which exists in this field – I maintain that the shopkeeper can offer the learners many opportunities for new knowledge and deliver it in different ways, through the pineapple or the closed tin (see the ‘shopkeeper’ picture). I can give (sell) every pupil what is suitable to him/her because there is a wide variety of teaching methods. Thus, I can match to all the pupils what is right for them

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– the teacher sells to the pupils the material. Once a sale has been transacted, the material is absorbed by construction and not inculcation and that is what I want to transpire in the lessons I will teach – the ‘shopkeeper’ picture embodies a message of giving, contributing; the teachers influences the pupils, making them improve and expand the existing knowledge. Those who chose the ‘shopkeeper’ particularly underscored the fact this was about mathematics. They emphasized their goal of clarifying to the learners the advantages of the subject and the importance they attributed to delivering the lessons in an interesting way and by varied methods. Some of them stressed that they wanted to make the pupils like the subject as they themselves had chosen to engage in it out of love: – many pupils have a negative image of the subject (a difficult subject, many fail in it). Therefore, it is very important to me to convince them that learning mathematics can also be fun – when I teach mathematics, I want first of all to ‘sell’ them the subject and make them like it. I want to avoid the feeling of disgust with mathematics which many pupils feel; many senses that it is a very difficult subject and it is not connected to daily problems – there are pupils who are quiet and others who are noisy; as a teacher I feel it is important to choose the best and the most interesting way for delivering the material. The ‘shopkeeper’ should convince the buyers (the learners) that learning mathematics is important and fun. The ‘shopkeeper’ picture was rejected by one student only. He protested that: “… pupils are not a merchandise … they have feelings … the communication here includes something very business-oriented, very materialistic … as if only selling is important”. 4.1.4 The Participants’ Reference to the ‘Puppeteer’ Picture The ‘puppeteer’ picture, chosen by 10% of the participants, was ranked fourth among the seven pictures. Those who chose this picture assumed that teachers’ role in the context of mathematics teaching comprised a proper activation of the pupils in the positive sense, namely by demonstrating authority for the purpose of guiding: – I am studying special education with specialization in mathematics. I see my professional self-image as an operator in a puppet theatre since for teaching special education learners I have to integrate many tangible aids and adjust to every pupil (puppet) his or her pace

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– the puppeteer controls the puppets and the way they are moving, so do teachers. In their image and way of teaching they affect pupils’ perception and their mathematical thinking – teachers should know to active the learners, control the class in the sense of class management; be strong and also know when to punish and when to reward. It is in their hands – I see a need to exercise authority; as a teacher I deem it important to activate the pupils rather than have them activate me. Please note that most of the participants’ references to the ‘puppeteer’ picture were negative and refrained from presenting teachers as authoritarian people who act towards the learners as they please: – I cannot accept that the pupils are like puppets on a string; that the teacher controls them, chooses for them and it is difficult for me to acknowledge that the pupils are driven only according to the teacher’s wishes – it horrifies me to consider the teacher as a ‘puppeteer’ – my goal in education is to shape a thinking person, having instruments for thinking, comprehension ability and self-judgement and not pupils who only do whatever they are told – I don’t want to be a teacher who absolutely controls her pupils. I prefer that the pupils have a freedom of speech, a freedom of thought and that they feel their independence during the lessons – teachers are not meant to ‘fish’ the pupils like marionettes; they are not supposed to control them and be disrespectful to them as if they lack a personality. 4.1.5 The Participants’ Reference to the ‘Entertainer’ Picture Three female-students who are 5% of the participants chose the ‘entertainer’ picture. Their arguments for doing that focused on the description of the teacher as creating an agreeable atmosphere in the lessons and integrating humor, trying to draw the pupils closer to the subject of mathematics: – the teacher should amuse and entertain the class, up to a certain limit of course; the more entertaining and experiential the material delivery is, the more attentive the pupils are. I believe that if I teach in this way, there are greater chances that the pupils internalize the mathematics material – a good teacher in my opinion delivers the mathematics material in an interesting and fascinating manner; integration of extensive illustrations and many entertaining stories – my choice of the ‘entertainer’ picture was due to the fact that the subject satisfies me and I feel glad to teach it; like an entertainer who feels happy all the time, so am I.

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Nevertheless, most of the participants’ references to the ‘entertainer’ picture were basically negative. The very comparison with an entertainer was perceived by the participants as disrespectful to the teacher: – I was very annoyed by the ‘entertainer’ picture; our status in the Arab society is respected. Why compare it to entertainers who slip on a banana peel just to make people laugh? – an entertainer does not represent a respectful learning environment – a teacher is not meant to be an entertainer-clown; there should be some respect for the teacher – an entertainer will downgrade teachers’ status; this might be appropriate for a democratic school. 4.1.6 The Participants’ Reference to the ‘Judge’ Picture The ‘judge’ picture received a high number of rejections, usually accompanied by anger and annoyance about the very attribution of such an image to teachers. Among the participating students, only two maintained that the ‘judge’ picture represented their professional image as future teachers. Those who chose the ‘judge’ picture emphasized the aspect of authority and dignity which according to them facilitated the delivery of the lessons: – I chose the ‘judge” because people treat judges with respect and take them seriously. Whatever judges say, is acknowledged; their words have a lot of power and more people listen to them – judges eventually decide everything. They can draw a conclusion even without hearing all the pupils. If teachers want to deliver a lesson properly, they have to make decisions and thus win the respect of the learners. As already mentioned, most of the references to the ‘judge’ image were negative and mainly disagreed with the presentation of teachers as condescending people who evoke distance and fear: – judge-teachers are not really interested in the people around them. They are the only ones who decide, they dictate the tone – judges have a status of power; they believe it is important that people are afraid of them. It is difficult to express an opinion around such teachers; pupils cannot speak in the presence of such teachers. Judge-teachers make learners feel a sense of discomfort. They sit on high chairs, towering over everyone and in addition drop a heavy gavel. – this is a dictatorial conduct and is not appropriate for teaching – judge-teachers are seen as figures that express their words and no one can undermine or appeal. No questions can be asked although this is the main element of learning; pupils should learn to ask questions, investigate. Around judge-teachers this is almost impossible.

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4.1.7 The Participants’ Reference to the ‘Animal Trainer’ Picture None of the students chose the ‘animal trainer’ picture. References to this picture were mainly negative. The students viewed it as an aggressive and to certain extent violent figure, unsuitable to engage in education since it implied that pupils were perceived as animals to be trained: – the teacher is not a trainer and the pupils are not animals – educating is not training; education involves inculcation of insights rather than using intimidation and imposing sanctions and threats – the ‘animal trainer’ like the ‘puppeteer’ image implies that pupils have no character of their own and everything depends on the teacher. The ‘animal trainer’ teacher exercises too much authority, power. 4.2 The Professional Images Rejected by the Students “No thesis is complete without its antithesis” said the philosopher Hegel (Fransella, 1995, p. 57). Consequently, in addition to choosing the preferred picture, all the students were requested to indicate which pictures evoked annoyance and rejection. It is noteworthy that the number of rejections was not limited to one as was the case of the preferred choice. Out of the seven pictures, most of the students rejected the pictures of the ‘animal trainer’, ‘puppeteer’ and ‘judge’, which represented authoritative figures characterized by control and forcefulness. The participants’ explanations illustrated that the very comparison of the teacher’s figure to a person who exercises control, forcefulness and excessive authority towards the learners entailed anger. Conversely, the ‘entertainer’ picture mainly evoked insult among students from the nonJewish sector. This was due to the very possibility of viewing teachers as entertainers who are meant to entertain the pupils instead of teaching them in an earnest and serious manner. Moreover, the ‘shopkeeper’ picture was rejected since the participants refused to consider their role as goods salespeople. The ‘animal keeper’ and ‘conductor’ pictures were rejected only by few students. One student doubted whether therapeutic behavior might create an impression of eliminating the distance between the teacher and the learners. On the other hand, participants who rejected the ‘conductor’ picture described them as exercising complete control over the orchestra (the learners) and mainly underscored their great distance from the pupils. 4.3 Alternative Options Comparing the alternative options suggested by the participants versus their initial choices of the pictures/illustrations indicated that those who chose the

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‘animal keeper’ chose many alternative figures that expressed care and concern, such as psychologist, doctor, counsellor, kindergarten teacher and even a farmer. Those who chose the ‘conductor’ chose professions which represent roles of an authoritative and leading nature, such as skipper, bus driver, team coach or social guide. One of the participants provided a unique explanation to the image of the teacher as a researcher. He described the teacher as someone ‘who is meant to investigate all the time the up-to-date knowledge and information, e.g. teaching methods and obtaining insights which might enhance the learning and improve the pupils’ attainments’.

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Discussion

The present study explored what the metaphors chosen by the participants illustrated about their perception of their professional image as mathematics teachers. One of the important stages in teachers’ professional development is the stage when they stop relating to the class as an entirety. They begin considering the learners as individuals and getting acquainted with the path of their personal growth (Nesher & Hershkovitz, 2004). Findings of the present study demonstrated it by the students’ very dominant choice of the ‘animal keeper’ picture as representing their professional image. This fact that 35% of the participants chose the ‘animal keeper’ picture as well as the variety of alternative choices of professions with a therapeutic orientation indicated a thinking process which the students learning in the college were undergoing. In the course of this process they gradually embraced the insight that teachers’ role was to focus more on pupils and their needs and less on teachers’ personal ‘gain’ (Adams, 1982). The ideal ideological ‘baggage’ acquired by the students during their preservice education at the college affects their professional image. This claim is grounded in the findings of Avdor (2001) who investigated the contents of the courses studied in colleges and universities. She showed that courses dealing with the ‘knowledge of relations’ (Webb & Blond, 1995) occupy a central place in the pre-service education program of the college. This approach is supported by Beck (2005) in his book Technicality as a Vision in Pre-Service Teacher Education. In the book he quotes the third item on the requirements list of the National Council for Accreditation of Teacher Education (NCATE), according to which, “Teachers are expected above

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all to act on the basis of their knowledge in a caring and professional way” (p. 35). The findings indicated that the students learning at the college (both male and female) viewed themselves first and foremost as ‘keeper’ teachers representing the humanistic approach (Rogers, 1969). They believe in teaching characterized by interaction of proximity to which Hargreaves (2000) refers as ‘emotional geography’. According to Hargreaves’ definition, the ‘animal keeper’ figure represents a progressivist approach in education. This approach advocates maintaining relations of caring and concern, that same caring that Noddings (1984, 1992) considers as most crucial for teachers’ practice. Conversely, the picture chosen in the second place (by 33% of the entire research population) was that of the ‘conductor’ teacher. This image underscored the role of mathematics teachers in delivering subject matter knowledge (SMK) as an essential part of pedagogical content knowledge (PCK) (Shulmann, 1986; Wilson, Shulman, & Richert, 1987). Based on the participants’ arguments for choosing the ‘conductor’ teacher, we learn that the students deem highly important the building of a learning environment which facilitates inculcation of knowledge, training the learners and leading them to high academic attainments. Those who chose the ‘conductor’ believed they were responsible for determining the teaching method and their pupils’ success in the various tests. The orchestra conductor was oriented at the main target, namely performing a complete and harmonious composition which will impress its audience by means of the players. Similarly, the mathematics pre-service teachers strove to lead their pupils to attainments and love of the subject (Levenberg & Becker, 2003). Conductors advocate a tight discipline and demand from the players to be obedient, meticulous and observe the laws (which are an essential condition in the communication setup between the conductor and the players). The same applies to the ‘conductor’ teachers who expect their learners to obey and acknowledge their authority while being meticulous about discipline and order. The third choice in size (15%) of the ‘shopkeeper’ teacher reflected a conservative-traditional approach. It viewed the teacher’s role as delivering and transmitting knowledge, whereas the learning was perceived as ‘knowledge acquisition’ (Beck, 1999; Richardson, 1996; Sfard, 1998). It was apparent that at the developmental stage where the research participants were (training for the role stage), some of them were still more focused on themselves and were concerned by the very possibility of failing to learn the required material. Their explanations for choosing the ‘shopkeeper’ emphasize that their knowledge delivery was essential. Moreover, they attributed importance to the fact that

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they were meant to do everything in their power to persuade, explain, improvise and evoke the pupils’ interest. As mentioned, the figures of the ‘puppeteer’, ‘animal trainer’ and ‘judge’ were rejected by most of the participants. They entailed emotional reactions of anger and annoyance for comparing the students to figures representing control, forcefulness and over-authority. The mathematics pre-service teachers have not yet accumulated experience in teaching at school (excluding few experiences required during their studies as practicum accompanied by an instructor/tutor). This leads to the question: What are the sources or factors which impact the students’ perception of their professional image? It is particularly interesting to explore the effect of the preservice education institution on the perception of teachers’ professional image particularly based on the study of Mendelson (2006) that found a significant relation between the pre-service education institution and the ‘perception of professional image’ variable. Various researchers explored the place of pre-service education as a factor impacting students in various disciplines. Doyle (1990) views teachers as products of their training, reflecting the goals of the pre-service teacher education institutions. The famous researcher Lortie (1975) opposed Doyle’ argument. According to him, pre-service teacher education has a limited effect on students from various areas, whereas previous impacts in life carry a greater weight. Students’ experiences as learners affect the perception of their professional image, mainly the influence of educational past figures such as teachers, parents and tutors. Students maintain that these figures constitute one of the key factors which affect their choice to teach mathematics. These explanations support similar findings of Flores and Day (2006). Furthermore, explanations of participants in the present study illustrated that some of them chose the teaching profession out of identification with meaningful teacher figures in their past. This point is indicated also in the study of Bullough and Gitlin (1995). For example, one of the students described a mathematics teacher she remembered from her time at school: ‘My mathematics teachers is my model … he is highly meaningful in my choice to become a mathematics teacher. He taught well and in an interesting way. He knew how to reach each and every pupil …’. It is noteworthy that this student chose the ‘animal keeper’ image. In his article, “Mainly intellectual and generous: On the fostering and training of worthy educators”, Aloni (2008) highlights the unique place of educators who accomplish their goals due to their moral sensitivity, empowering dialogue and humanistic approach. The high percentage of students who chose the ‘animal keeper’ image demonstrated the progressivist approach characteristic of many

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students at the college. Investigation of the gender-affiliation effect on this choice showed that it did not characterize only women but it existed also among men (40% of the males in the present study chose the ‘animal keeper’ image). A different viewpoint was demonstrated by other students who preferred erasing the memories of their teachers at school. Instead, they wished to create a remedial experience in mathematics for their pupils. This was manifested by one of the minority students who chose the ‘puppeteer’: In the past I had a teacher who behaved like a tough judge … he always shouted and harshly dominated the pupils. He did not care whether we understood the material or not. The main thing was to be quiet. And I loved mathematics … therefore today I see myself as a ‘puppeteer’, namely activating the entire class like a good guide, I will only guide them towards the studies … mathematics teachers should have all the competences suitable for teaching so that everybody understands … To sum up, teaching is multi-faceted. Hence, some participants found it hard to choose between the ‘animal keeper’ and the ‘conductor’. Although they were requested to choose only one image (which they did) they explained that they believed in an integrated perception of teachers’ role, mainly regarding the discipline they were teaching. They believed that relations with pupils were not less important than the learning materials. This was manifested by the words of one participant: ‘The mathematics teacher should both teach and correct … and do so patiently’. Pupils and their success in mathematics were central in the arguments of the present study research population. Hence, they implemented in fact the document written on 21.11.2006 by the Council of Higher Education concerning the issue ‘Guiding outlines for pre-service teacher education in Israeli higher education institutions’ (Ariav Committee, 2006). The document indicates among others that the ‘professional teachers act on the basis of disciplinary and didactic-pedagogical knowledge in a systematic and evidence-based manner regarding their own learning and that of their pupils’ (p. 2).

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Appendix A: The Pictures

Animal Trainer

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Shopkeeper

Puppeteer

Animal Keeper Judge

Conductor

Entertainer

Root Canal Avikam Gazit

Unpleasant Root canal With pain, From four molars Only two remain But it is not surprising Like an animal in the zoo: The root of four Is always two … (And for those Who ARE painstaking About the minus Of dual solution As required They will find it In the bank account – An undesired …)

© koninklijke brill nv, leideN, 2018 | DOI 10.1163/9789004384064_007

Epilog Dorit Patkin and Avikam Gazit

An end is always a beginning of something new which is built on what has been written in the previous pages. The book takes the reader on a fascinating journey which encompasses elementary school mathematics teachers’ world and examines it from different and diversified points of view. The journey passes through three chapters, each constituting a basic component in teachers’ world: teachers’ education and teachers’ knowledge, teaching and teachers’ personality. This journey facilitates enhancement and observation of those points, which are associated with the world of elementary school mathematics educators, elementary school mathematics teacher education, mathematics education or any other area connected to elementary school mathematics teaching. The expectations inspired by Shulman (1986) “Those who can, do and those who understand, teach” – are perhaps not fully materialized in elementary school mathematics teaching. However, navigating among the book chapters one can notice illuminating flickers of wish and hope for improving the teaching, starting from the teacher education stage and up to teachers’ consolidation as key figures in elementary school mathematics teaching.

© koninklijke brill nv, leiden, 2018 | doi 10.1163/9789004384064_008