Dina Tirosh

• Tel Aviv University

This brief history of SEMT provides details of the life of the Symposium on Elementary Mathematics Teaching (SEMT), which, as its name suggests, focuses on the teaching of mathematics to children within the range of 5 to 12 years of age. The first SEMT symposium was in 1991, with the participation of some 39 educators and researchers, and since the...

This paper focuses on the definitions and the mis-out and mis-in examples of rational numbers that four prospective elementary teachers presented while working on rational number assignments. The participants were first asked to respond, individually, to an Individual Rational Number Assignment, consisting of items aiming at detecting their persona...

This paper reports on five secondary school mathematics prospective teachers’ conceptions of extreme point . The analysis of the data addressed students’ definitions, examples, and evaluation of given examples, with special attention to the related domain. Written assignments and individual interviews uncover salient, erroneous concept images regar...

This paper reports on students’ conceptions of minima points. Written assignments and individual interviews uncovered salient, concept images, as well as erroneous mis-out examples that mistakenly regard examples as non-examples and mis-in examples that mistakenly grant non-examples the status of examples. We used Tall and Vinner’s theoretical fram...

This paper aims to broaden the scope of the construct concept image by addressing the mathematically oriented notion of “solution of an algebraic equation.” The notion solution of an algebraic equation is not mathematical per se, but it is most prevalent in mathematical communications and in mathematics education. The paper describes five episodes...

This paper reports on concept images of 38 secondary school mathematics prospective teachers, regarding the evenness of numbers. Written assignments, individual interviews, and lesson transcripts uncover salient, erroneous concept images of even numbers as numbers that are two times “something” (i.e., 2i is an even number), or to reject the evennes...

Considering that young children spend a great deal of time outside of school, there is a need to increase the expertise of adults when it comes to engaging young children with numerical activities. This study presents the Cognitive Affective Mathematics Adult Education framework used to design such a course for interested adults. Focusing on promot...

Taking into consideration that young children engage with mathematics outside of the preschool classroom, this study focuses on adults’ awareness of numerical ideas raised by young children without and with adult involvement. Ninety adults in Israel, the majority of whom were parents or grandparents of children aged 3–6 years, participated in this...

This study focuses on adults who are neither preschool teachers nor professional caregivers and investigates their beliefs regarding the importance of engaging young children with numerical activities. It also examines the types of numerical activities adults report having observed children engaging with, as well as the types of activities they pro...

In this article, we describe a case study that was conducted within a study aiming to diagnose grade 5 students’ concept images of parallelograms. The theoretical framework that we adopted for this study was that of concept definition–concept image as reported by Tall and Vinner (Educational Studies in Mathematics 12:151–169, 1981), a theory that i...

This study investigates preschool teachers’ implementations and children’s engagement with copying repeating patterns, as well as their descriptions when comparing patterns. Children were presented with an AB-structured pattern in the form of a strand of beads and were asked to construct a similar strand of beads but using colors different from tha...

Taking a socio-cultural perspective, in this study we explored the challenges toddlers might face as they practice 1–1 correspondence in the playful context of setting a table, and how different individuals may participate in this playful activity. Findings indicated that toddlers’ competence in carrying out one-to-one correspondence may be related...

This study investigates 27 preschool teachers’ verbalization of various aspects of pattern structure as well as their knowledge of pattern structure while solving patterning activities. Aspects of structure that are investigated include the unit of repeat, its length and the amount of times it is repeated, and whether or not the pattern ends in a c...

This paper raises the possibility of enhancing prospective and practicing teachers’ awareness of example use in classrooms by using theories to analyze an authentic case. The study was carried out in the context of a university course where participants analyzed an excerpt from a tenth-grade geometry class. A qualitative analysis of participants’ c...

Patterning activities in preschool are considered one way for enhancing young children’s appreciation for structure. Preschool teachers, however, are not always aware of the mathematics behind these activities. This paper describes one part of a professional development program that employs the use of tasks for children to promote preschool teacher...

This paper synthesises research from three separate studies, analysing how different representations of a mathematical concept may affect young children’s engagement with mathematical activities. Children between five and seven years old engaged in counting objects, identifying triangles and completing repeating patterns. The implementation of thre...

Several studies have investigated children’s engagement with repeating pattern tasks, but few have related to patterns with ABA as the minimal unit of repeat. This study focuses on identifying intuitive characteristics of children’s work as they engage with repeating pattern tasks of different structures, focusing on the differences between ABA pat...

In Israel, there is a mandatory preschool mathematics curriculum. Yet, most preschool teachers have little preparation for teaching mathematics to young chil dren. In this chapter, we present the Cognitive Affective Mathematics Teacher Education (CAMTE) framework that was developed at Tel Aviv University and used to investigate different elements o...

This chapter describes the program for prospective secondary school mathematics teachers held at Tel Aviv University. At the university, prospective teachers already have a bachelor's degree in mathematics or a mathematically rich field of study. The main aim of the program is to promote their mathematical knowledge needed for teaching. Prospective...

This study examines three aspects of early childhood teachers’ patterning knowledge: identifying features, errors and appropriate continuations of repeating patterns. Fifty-one practicing early childhood teachers’ self-efficacy is investigated in relation to performance on patterning tasks. Results indicated that teachers held high self-efficacy be...

The use of cases as a pedagogical tool in teacher education is seen as one way of bringing practice closer to theory. This study describes the use of cases in a university course for secondary school prospective mathematics teachers. The study investigates participants’ views of cases taken from different sources and presented in different situatio...

This book gives insight in the vivid research area of early mathematics learning. The collection of selected papers mirror the research topics presented at the third POEM conference. Thematically, the volume reflects the importance of this relatively new field of research. Structurally, the book tries to guide the reader through a variety of resear...

This chapter describes the beliefs and attitudes of five preschool teachers towards involving families in promoting children’s numerical competencies, such as saying number words in a sequence to ten. The backgrounds of the children in each class, along with the teachers’ educational and social backgrounds, form the context of the study and are imp...

This paper describes kindergarten children’s engagement with two patterning activities. The first activity includes two tasks in which children are asked to choose possible ways for extending two different repeating patterns and the second activity calls for comparing different pairs of repeating patterns. Children’s recognition of the unit of repe...

Patterning activities, specifically those related to repeating patterns, may encourage young children’s appreciation for underlying structures. This paper investigates preschool teachers’ knowledge and self-efficacy for defining, drawing, and continuing repeating patterns. Results indicated that teachers were able to draw and continue various repea...

This chapter describes an integrated program in Israel for 3-year-old children and their caregivers. For the caregivers, the aim of the program was to increase their mathematical and pedagogical knowledge for teaching geometric concepts. For the children, the aim of the program was to introduce geometry into the different spaces of the classroom, a...

This chapter focuses on methodological issues related to investigating preschool teachers’ self-efficacy for teaching geometry. The first issue discussed is the specificity, as opposed to the generality, of self-efficacy and the need to design instruments which are sensitive to this aspect of self-efficacy. Specificity may be related to content, in...

This study explores two number composition and decomposition activities from a numeracy perspective. Both activities have the same mathematical structure but each employs different tools and contexts. Twenty kindergarten children engaged individually with these activities. Verbal utterances as well as actions of the child and interviewer were recor...

This study investigates practicing early-years teachers’ concept images and concept definitions for triangles, circles, and cylinders. Teachers were requested to define each figure and then to identify various examples and non-examples of the figure. Teachers’ use of correct and precise mathematical language and reference to critical and non-critic...

The issue of sustaining and scaling up professional development for mathematics teachers raises several fundamental issues for researchers. This commentary addresses various definitions for sustainability and scaling up and how these definitions may affect the design of programs as well as the design of research. We consider four of the papers in t...

This paper demonstrates how professional development which focuses on task design principles can impact on what a preschool teacher may learn from implementing that task which in turn may impact on changes made to specific task features. Specific design principles for preschool mathematics tasks are presented and exemplified

This chapter presents the Cognitive Affective Mathematics Teacher Education (CAMTE) framework, a framework used in planning and implementing professional development for teachers. The CAMTE framework takes into consideration teachers’ knowledge as well as self-efficacy beliefs to teach mathematics. The context of counting and enumeration is used to...

This article describes a study that investigates preschool teachers’ knowledge of their young students’ number conceptions and the teachers’ related self-efficacy beliefs. It also presents and illustrates elements of a professional development program designed explicitly to promote this knowledge among preschool teachers. Results indicated that tea...

A major thrust in science education research has been the study of students’ conceptions and reasoning. Many have pointed out the persistence of misconceptions, naïve conceptions, alternative conceptions, intuitive conceptions, and preconceptions. Studies have covered a wide range of subject areas in physics, in chemistry, and in biology (Thijs and...

Cílem studie je zkoumat geometrické znalosti a vnímanou vlastní účinnost v geometrii (self-efficacy) u dětí z mateřské školy, včetně dětí zneužívaných a zanedbávaných. Bylo provedeno 141 individuálních rozhovorů s dětmi ve věku 5-6 let, z nichž bylo 69 diagnostikováno místním sociálním odborem jako zneužívané a zanedbávané. Výsledky ukazují, že obě...

This article reports on young children’s self-efficacy beliefs and their corresponding performance of mathematical and nonmathematical tasks typically encountered in kindergarten. Participants included 132 kindergarten children aged 5–6 years old. Among the participants, 69 children were identified by the social welfare department as being abused a...

This paper explores the use of video as a tool for promoting inquiry among preschool teachers and didacticians. In this case, the didacticians are teacher educators who are also mathematics education researchers. Preschool teachers recorded themselves with video implementing number and geometry tasks with children and shared these recordings with o...

This chapter is concerned with developing teachers’ knowledge for teaching mathematics in preschool. Like Alan Schoenfeld, we are concerned with teachers, in this case preschool teachers, knowing school mathematics in depth and in breadth. Like Günter Törner, one of the founders of theMAVI (Mathematical Views) conference, we are concerned with the...

This paper focuses on the first session of a professional development course revolving around the topic of mathematical statements and their appropriate proving methods. It analyzes the interactive development of the teachers’ knowledge by focusing on the relation between the mathematical statements, the instructor, and the teachers. Different role...

Policy documents and researchers agree that proofs and proving should become common mathematical practice in school mathematics. Towards this end, teachers are encouraged to implement proving activities in their classrooms. This article suggests a tool that may help teachers to integrate proofs and proving in their practice – the six-cell matrix. I...

In light of recent reform recommendations, teachers are expected to turn proofs and proving into an ongoing component of their classroom practice. Two questions emerging from this requirement are: Is the mathematical knowledge of high school teachers sufficient to prove various kinds of statements? Does teachers’ knowledge allow them to determine t...

Recently, there have been increased calls for enhancing preschool children’s mathematics knowledge along with increasing calls for preparing preschool teachers to meet the demands of new preschool mathematics curricula. This article describes the professional development program Starting Right: Mathematics in Preschools. Focusing on four episodes t...

This special issue comes at a time when the issue of early childhood mathematics has come to the fore. At a recent 2009 Conference of European Research in Mathematics Education, a new working group in Early Years Mathematics was established in response to increased calls for research regarding mathematics learning and mathematics teacher education...

This chapter describes how a theory of Deborah Ball and her colleagues, embedded in the realm of teacher knowledge, was combined with a theory of David Tall and Shlomo Vinner, embedded in the realm of mathematics knowledge, to develop kindergarten teachers’ knowledge for teaching geometrical concepts. The chapter describes the separate theories and...

This paper reviews studies on teachers’ professional learning of teaching proof and proving. From them we conceptualise three essential components of successful teaching: teachers’ knowledge of proof, proof practices and beliefs about proof. With respect to each component, we examine research studies of primary and secondary teachers. We also discu...

It is obvious that mathematically, triangles and circles are different. But are they different psychologically? Let’s try an experiment. Draw a circle. Now draw another circle. Now draw another circle. In what ways are the circles that you drew different? Perhaps they are different sizes. Perhaps they are different colors. However, the symmetry of...

Who says that preschool teachers need to teach geometry (or for that matter any mathematics)?!? In the past, researchers held the opinion that young children have little knowledge of mathematics and should not begin learning mathematics before beginning formal schooling in elementary school (Bereiter & Engelmann, 1966; Thorndike, 1922). Recently, a...

After having discussed elements of task design, after having reviewed examples of tasks in national guidelines, after having analyzed in great depth two different geometrical tasks, we present in this chapter a variety of tasks which teachers reported implementing in their preschool classes. As you review these tasks, you may ask yourself: What are...

In order for us to discuss with you, the reader, how geometrical concepts are developed, we need to establish a common language and a common background. This chapter provides terminology and theories on which the other sections and chapters of this book rest. It begins by presenting theories related to concept formation in general, proceeds to theo...

Professional development for practicing teachers may vary in duration, form, and content. A program attended by teachers may range from a one-day summer meeting followed by eight workshops during the year, to a semester course given on a weekly basis (Tsamir, 2008), to an intensive two-year program (Graven, 2004). The program may take the form of u...

On the one hand, the task above was designed to be implemented in a group so that the child has a chance to show his or her friends what was found and to explain this finding. On the other hand, Nadine is an impatient child. It is hard for her to wait for her turn. Or, maybe she is an enthusiastic child. She also wants to join in and the teacher at...

Are you surprised by the children’s judgments? Are you surprised by their justifications? As discussed in the previous chapter, young children mostly operate at the first van Hile level, relying on visual reasoning, taking in the whole shape when identifying examples and nonexamples of geometrical shapes.

As we stated in the beginning of this book, there is a need at the beginning of any dialogue to establish a common language and a common background between participants. This chapter intends to fulfill that purpose by providing the terminology and theory on which the others chapters in this part of the book rest. It begins by taking a look at acade...

In Part Two of this book, we discussed at length different aspects of tasks that ought to be considered when designing and implementing geometrical tasks with children. Many of those aspects may be applied to tasks that are intended to be implemented with both prospective and practicing preschool teachers. In addition to those issues raised previou...

According to reform documents, teachers are expected to teach proofs and proving in school mathematics. Research results indicate that high school students prefer verbal proofs to other formats. We found it interesting and important to examine the position of secondary school teachers with regard to verbal proofs. Fifty high school teachers were as...

The study explores the intuitive methods different-aged students use to determine whether a given set is finite or infinite; it also examines the relationship between these methods and the accepted, mathematical definition of infinite sets. It was found that two main methods were intuitively used by students: examining the finite/infinite nature of...

The Teaching-In-Context theory describes the kinds of decision-making in which teachers are engaged in the act of teaching. Testing the relevance of this theory in various situations could significantly contribute to determining its realm of application. Törner, Rolka, Rösken and Sriraman’s chapter testifies to its applicability in a context that h...

Some mathematical statements can be validated by a supportive example or refuted by a counterexample. Our study investigated secondary school teachers' knowledge of such proofs. Fifty practising secondary school teachers were first asked to validate/refute six elementary number theory statements, then to suggest justifications that students might g...

This article focuses on elementary school teachers’ preferences for mathematically based (MB) and practically based (PB) explanations. Using the context of even and odd numbers, it explores the types of explanations teachers generate on their own as well as the types of explanations they prefer after reviewing various explanations. It also investig...

Engaging students with multiple solution problems is considered good practice. Solutions to problems consist of the outcomes of the problem as well as the methods employed to reach these outcomes. In this study we analyze the results obtained from two groups of kindergarten children who engaged in one task, the Create an Equal Number Task. This tas...

This study proposes a framework for research which takes into account three aspects of sociomathematical norms: teachers’ endorsed norms, teachers’ and students’ enacted norms, and students’ perceived norms. We investigate these aspects of sociomathematical norms in two elementary school classrooms in relation to mathematically based and practicall...

Calls for reform in mathematics education around the world state that proofs should be part of school mathematics at all levels. Turning these calls into a reality falls on teachers’ shoulders. This paper focuses on one secondary school teacher's reactions to students’ suggested proofs and justifications in elementary number theory. To determine wh...

This study investigates the effect on student performance in drawing their attention to relevant task variables, focusing on accuracy of responses and reaction times. We chose this methodology in order to better understand how such interventions affect the reasoning process. The study employs a geometry task in which the irrelevant salient variable...

Teacher education programmes at tertiary educational institutions traditionally comprise three key strands–disciplinary studies, educational studies, and teaching practice (Comiti & Ball, 1996). The aim of these strands is to develop an integrated competence in student teachers and is often referred to as teacher knowledge. Winsløw and Durrand-Guer...

In this paper, we describe how the combination of two theories, each embedded in a different realm, may contribute to evaluating teachers’ knowledge. One is Shulman’s theory, embedded in general, teacher education, and the other is Fischbein’s theory, addressing learners’ mathematical conceptions and misconceptions. We first briefly describe each o...

In this paper we examine the possibility of differentiating between two types of nonexamples. The first type, intuitive nonexamples, consists of nonexamples which are intuitively accepted as such. That is, children immediately identify them as nonexamples. The second type, non-intuitive nonexamples, consists of nonexamples that bear a significant s...

This study investigates two sixth grade students’ dilemmas regarding the parity of zero. Both students originally claimed that zero was neither even nor odd. Interviews revealed a conflict between students’ formal definitions of even numbers and their concept images of even numbers, zero, and division. These images were supported by practically bas...

It has been observed that students react in similar ways to mathematics and science tasks that differ with regard either to their content area and/or to the type of reasoning required, but share some common, external features. Based on these observations, the Intuitive Rules Theory was proposed. In this present study the framework of this theory wa...

One theoretical framework which addresses students’ conceptions and reasoning processes in mathematics and science education is the intuitive rules theory. According to this theory, students’ reasoning is affected by intuitive rules when they solve a wide variety of conceptually non-related mathematical and scientific tasks that share some common e...

This paper presents a cross-cultural study on the intuitive rules theory. The study was conducted in Australia (with aboriginal children) in Taiwan and in Israel. Our findings indicate that Taiwanese and Australian Aboriginal students, much like Israeli ones, provided incorrect responses, most of which were in line with the intuitive rules. Also, d...

This paper is an initial investigation of teachers’ and students’ preferences for mathematically-based (MB) and practically-based (PB) explanations and the relationship between those preferences and sociomathematical norms. The study focuses on one fifth grade teacher and two of her students and discusses three issues. The first issue concerns stud...

Solving problems by reduction is an important issue in mathematics and science education in general (both in high school and in college or university) and particularly in computer science education. Developing reductive think-ing patterns is an important goal in any scientific discipline, yet reduction is not an easy subject to cope with. Still, th...

Mathematics education is a relatively new research domain. As such, it struggles with fundamental issues of defining its identity, aims, orientations and methods (e.g., Schoenfeld, 2002; Sierpinska & Kilpatrick, 1998). The special issue: ''the con-ceptual change approach to mathematics learning and teaching'' is an attempt to explore the promises f...

Many teaching and learning theories assume that knowledge about children’s mathematical and scientific thinking could significantly improve mathematics and science education. This assumption has driven intensive research on students’ conceptions and reasoning in mathematics and science education. Most of this research has been content-specific, and...

In this chapter we describe a study in which we explore secondary school students’ adherence to the perform-the-operation belief in the cases of division by zero. Our aims were: (1) to examine whether secondary school students identify expressions involving division by zero as undefined or tend to perform the division operation, (2) to study the ju...

This chapter summarizes general factors influencing change, and discusses matters which need to be considered by those working to achieve change in mathematics teaching practices. In the first section, we discuss two major sources of impetus for teacher change: (1) values and beliefs, and (2) technological advances. In the next two sections we expl...

In our work we have observed that students tend to react similarly to a wide variety of conceptually unrelated situations in science and mathematics. We suggest that many students’ responses, which the literature describes as specific alternative conceptions, could be interpreted as evolving from a small number of intuitive rules. In this article w...

This article presents the intuitive rules theory, relating to students’ responses to different tasks in science and mathematics. We argue that many alternative conceptions apparently related to different mathematical and scientific domains originate in a small number of intuitive rules: “More A-More B”, “Same A-Same B” and “everything can be divide...

In this final chapter, we consider some of the many research issues that need attention in the advancement of our discipline. Specifically, we have identified the following questions as worthy of consideration: 1. What role can research play in illuminating the multidisciplinary debates on the powerful mathematical ideas required for the 21st cent...

Without Abstract

In this chapter we illustrate how the explanatory and predictive power of the Intuitive Rules Theory can be used to plan instructional sequences that help students overcome the negative effects of intuitive rules on their work in mathematics and science. We then describe a research-based seminar we developed for raising mathematics and science teac...

In this article I present and discuss an attempt to promote development of prospective elementary teachers' own subject-matter knowledge of division of fractions as well as their awareness of the nature and the likely sources of related common misconceptions held by children. My data indicate that before the mathematics methods course described her...