Publications

This chapter presents the \u201cAgam Program for the Development of Visual Thinking\u201d designed to actualize this vision. First, we relate to definitions of visual thinking and to the importance of developing visual thinking at a young age. Then, we describe the Agam Program for developing visual thinking and a visual language. We present the programs aims and content, the way they are contextualized in the teaching units for kindergarten teachers and school teachers, and the accessories kit that accompanies the program. We discuss the unique pedagogical approach \u201cvisual pedagogy\u201d of the program and refer to the potential benefits of the program. The programs potential is to develop the childrens visual thinking, problem solution skills and creativity. For this description we use authentic examples from four teaching units: Circle, Square, Patterns and Numerical Intuition. Later, we include research findings illustrating that childrens visual thinking can be developed through of the Agam Program.

This study investigates secondary school mathematics teachers attention to potential teaching situations that encourage argumentation. A group of 17 seventh grade teachers were asked to choose three tasks from a textbook they all use in teaching, which, in their view, have the potential to encourage argumentation, and then to justify their choices. Analysis of the teachers responses revealed categories that fall into three dimensions of attention: (1) Attention to the mathematics in which the argumentation is embedded, focusing on three aspects: the mathematics inherent in the task; the mathematics related to the teaching sequence of which the task is a part; and the meta-level principles of mathematics; (2) Attention to socio-cultural aspects related to argumentation; and (3) Attention to students ways of thinking which might be revealed by the task. Analysis of each response revealed four types of combinations of dimensions of attention: a. Responses attending to all three dimensions; b. Responses attending solely to the mathematics inherent in the task; c. Responses attending only to the socio-cultural dimension; and d. Responses refers to none of these dimensions. Analysis also found that responses of the same teacher were of the same type of combination. The findings were interpreted in light of theory and practice and suggestions for additional research emerged.

In the course of the last few years, we have investigated shifts of knowledge among different settings in inquiry-based mathematics classrooms: the individual, the small group and the whole class community. The different theoretical perspectives we used for analysing group work and for analysing whole class discussions, and the empirical data, led us to hypothesize links between shifts of knowledge and students creative reasoning. Therefore, the goal of the current study is to investigate creative reasoning within the shifts of knowledge in an inquiry-based classroom. Specifically, we ask: What is the role of creative mathematical reasoning in the shifts of knowledge between the knowledge agents and their followers in the classroom? To this end, we analysed a whole class discussion and the subsequent work of a small group. Our findings show that creative reasoning has a role in researchers characterization of shifts of knowledge in the classroom. In particular, we found that the students who expressed creative reasoning all had followers and thus became knowledge agents, while students contributions that were not characterized as creative were not always followed up. Finally, in cases where both, the knowledge agent and the follower expressed creative ideas, we named these ideas milestones.

The present study focuses on students' problems solving processes of one activity from a course, especially designed for third-grade mathematically talented students. The course was designed to foster students' mathematical creativity and reasoning in a problem-solving context. It included 28 meetings over the course of one academic year and interwove problem solving in dyads or small groups, peer argumentation, and teacher-led discussion. We identified the types of solutions, the kinds of reasoning and the kinds of verbal and non-verbal actions (gestures, drawings, folding etc...) used. We show how gestures and other non-verbal actions were interwoven with childrens verbal peer argumentation and led them to new insights on the concept of area.

In this report, we highlight the epistemic actions and concomitant discursive shifts of four students as they reinvent the fundamental idea and technique in Euler's method. We use this case to further the theoretical and methodological coordination of the Abstraction in Context (AiC) approach, with its associated model commonly used for the analysis of processes of constructing knowledge by individuals, and small groups and the Documenting Collective Activity (DCA) approach, with its methodology commonly used for identifying norma-tive ways of reasoning with groups of students. In this report, we show students' first steps towards re-inventing Euler's method and explicate the theoretical and meth-odological commonalities of AiC and DCA.

Knowledge shifts are essential in the learning process in the mathematics classroom. Our goal in this study is to better understand the mechanisms of such knowledge shifts, and the roles of the individuals (students and teacher) in realizing them. To achieve this goal, we combined two approaches/methodologies that are usually carried out separately: the Abstraction in Context approach with the RBC+C model commonly used for the analysis of processes of constructing knowledge by individuals and small groups of students; and the Documenting Collective Activity approach with its methodology commonly used for establishing normative ways of reasoning in classrooms. This combination revealed that some students functioned as \u201cknowledge agents,\u201d meaning that they were active in shifts of knowledge among individuals in a small group, or from one group to another, or from their group to the whole class or within the whole class. The analysis also showed that the teacher adopted the role of an orchestrator of the learning process and assumed responsibility for providing a learning environment that affords argumentation and interaction. This enables normative ways of reasoning to be established and enables students to be active and become knowledge agents.

We present the second stage of a study within the context of geometry, whose aim is to investigate relationships between and influence of visualization, the concept images of students concerning geometrical concepts and their definition, and students' ability to prove. We focus on links between the understanding of the definition's role in concluding the geometrical concept attributes and proofs that deal with these attributes. We exemplify this stage in our research, by means of examples, which reveal that the difficulties students have in understanding the geometric concepts' definitions affect the understanding of the proving process and hence the ability to prove.

The present study focuses on one activity of a whole course, especially designed for third-grade gifted and talented students. The course was designed to foster students' mathematical creativity and reasoning in a problem-solving context. It included 28 meetings over the course of one academic year and interwove problem solving in dyads or small groups, peer argumentation, and teacher-led discussion. The activities developed for this course relied on five design principles: (a) creation of problems with multiple solutions, (b) creation of collaborative learning situations, (c) stimulation of socio-cognitive conflict, (d) provision of tools for checking hypotheses, and (e) opportunity for reflection upon and evaluation of solutions. In the paper we describe how students from three successive years of the course solved and justified their solutions to tasks purposefully designed according to the above principles. We go on to explore how this design, especially the stimulation of socio-cognitive conflict, promote students' understanding of the area concept , in particular the fact that geometrical figures can have the same area without being congruent. The necessity to create multiple solutions to a given problem situation, as well as the encouragement of using multiple channels of argumentation, led to the co-construction of new ideas in geometry, and the emergence of deductive reasoning.

We present a design research on learning beginning algebra in an environment where spreadsheets were available at all times but the decision about using them or not, and how, in any particular situation was left to the students. Students activity is analyzed in Kierans framework of generational, transformational and global/meta-level activity, and compared to the designers intentions. We do this by focusing on the activity of one student in four sessions spread over several months and discussing the activity of 51 additional students in view of the analysis of the focus student. We show that the environment enables a number of different entries into algebra and as such supports students in becoming autonomous learners of algebra, and in making the shift from arithmetic to algebra via generational and global/meta-level activity before dealing with the more technical transformational activities.

We show that a meticulous design can encourage students in dyads to shift from informal reasoning (visual, inquiry-based) to reasoning moved by logical necessity (abductive and deductive). We describe a case study in which one dyad solves a series activities purposely designed. We show that argumentation first relies on intuition, and then intertwines the activities of conjecturing and checking the conjectures though the use of different gestures.

Classrooms provide extremely varied settings in which learning may take place, including teacher-led conversations, small group unguided discussions, individual problem solving or computer supported collaborative learning (CSCL). Transformation of Knowledge through Classroom Interaction examines and evaluates different ways which have been used to support students learning in classrooms, using mathematics and science as a model to examine how different types of interactions contribute to students participation in classroom activity, and their understanding of concepts and their practical applications. The contributions in this book offer rich descriptions and ways of understanding how learning occurs in both traditional and non-traditional settings. Combining theoretical perspectives with practical applications, the book includes discussions of: the roles of dialogue and argumentation in constructing knowledge the role of guidance in constructing knowledge abstracting processes in mathematics and science classrooms the effect of environment, media and technology on learning processes methodologies for tracing transformation of knowledge in classroom interaction. Bringing together a broad range of contributions from leading international researchers, this book makes an important contribution to the field of classroom learning, and will appeal to all those engaged in academic research in education.

We show how the RBC model for abstraction in context can be used to follow the emergence of a learners knowledge constructs and to identify in detail the learners partially correct constructs (PaCCs). These PaCCs are used to explain the learners inconsistent answers and provide added insight into processes of knowledge construction. The research process is illustrated by means of an example from elementary probability. We thus demonstrate the analytic power of the RBC model for abstraction in context.

Processes of abstraction of a group of students working together in a classroom on tasks from a unit on probability are analyzed with the aim of identifying mechanisms for consolidating recent knowledge constructs. Three such mechanisms are identified by means of indicative epistemic actions: Consolidating during building-with the construct, consolidating during reflecting on the construct, and consolidating during processes of constructing further constructs. While the first two mechanisms have been reported in previous research studies, in different settings, the third one is presented here for the first time.

This study concerns teachers' actions and roles while interacting with students in knowledge construction activities. We aimed at tracing how teachers led discussions with students, how they triggered explanations and how they helped integrating them in coherent arguments. We identified some patterns in two teachers who taught the same sequence of activities, with the same didactical aims of constructing probability concepts. We show that these patterns were recurrent and characterized the interactions of each teacher. The patterns concerned how teachers trigger explanations and expand them to turn claims to arguments.

This paper focuses on how teachers guide construction of knowledge in classrooms. We suggest that guidance hinges on the kind of dialogue teachers choose to engage students in. We propose several classroom dialogue types relevant for the construction of knowledge and suggest that critical dialogue is particularly effective for knowledge construction. We describe a lesson on probability conducted in a Grade 8 classroom in order to illustrate how a teacher chooses dialogue types, and to what extent she attends during dialogue to epistemic actions, which are constitutive of knowledge construction.

The goal of this chapter is to shed light on the development of mathematics curricula integrating interactive computerized learning environments. Rather than describe and analyze one of the components in isolation from the others, we will try to give a comprehensive picture of the compound and long-term activity of curriculum development.

Since there is no universally accepted research paradigm in mathematics education, theories and terminology tend to multiply. It is therefore one of the tasks of the
research community to critically compare theories that deal with closely related issues and have similar aims. The setting of a research forum at PME conferences is one of the few opportunities where attempts at such comparison can be undertaken in public by a large group of researchers. [1st paragraph]

We propose an approach to the theoretical and empirical identification processes of abstraction in context. Although our outlook is theoretical, our thinking about abstraction emerges from the analysis of interview data. We consider abstraction an activity of vertically reorganizing previously constructed mathematics into a new mathematical structure. We use the term activity to emphasize that abstraction is a process with a history; it may capitalize on tools and other artifacts, and it occurs in a particular social setting. We present the core of a model for the genesis of abstraction. The principal components of the model are three dynamically nested epistemic actions: constructing, recognizing, and building-with. To study abstraction is to identify these epistemic actions of students participating in an activity of abstraction.

Visualization is a central component in mathematical activity [1±4]. `Visualization generally refers to the ability to represent, transform, generate, communicate, document, and re¯ect on visual information ([5], p. 75). As such it is crucial to the learning of geometrical concepts. A visual image, by virtue of its concreteness, `is an essential factor for creating the feeling of self-evidence and immediacy ([6], p.101). In this paper, we propose to illustrate that: visualization can be much more than the intuitive support of higher level reasoning, it also may constitute the essence of rigorous mathematics; and visualization can be central not only in areas which are obviously associated with visual images (such as geometry), but also in formal symbolic argu- ments (such as high school algebra). First, we present a problem (borrowed from a verbal communication with Professor I. Weinzweig), which was tried out in teacher courses in various countries and with several colleagues. What started as an in-service teacher course activity, became a source of interesting data. Second, we present the data and analyse it. The analysis uncovers some `mechanisms of visualization towards the building of a mathematical generalization. Finally, we summarize the ®ndings and exemplify how pedagogy can harness them to guide the design and use of beginning algebra problems.

The emergent perspective (Yackel and Cobb, 1996) is a powerful theory for describing cognitive development within classrooms. Yackel and Cobb have shown that the formation of social and sociomathematical norms, and opportunities for learning are intertwined. The present study is an attempt to extend the range of application of the emergent perspective to middle high school classrooms. The learning environments we consider are rich in the sense that (i) the tasks in which students are engaged are open-ended problem-situations (ii) the activities around the tasks are multiphased, consisting of small group collaboration on problem solving, reporting and reflection in a classroom forum with the teacher (iii) the tools used are multi-representational software. We identify here some practices rooted in such rich environments from which several sociomathematical norms stemmed. The present study shows that socio-mathematical norms do not rise from verbal interactions only, but also from computer manipulations as communicative non-verbal actions.

As Hershkowitz suggests in the epilogue, the contributors to this volume advance several related agendas for mathematics education. First, the authors help us better understand the wide range and influence of spatial reasoning and geometry in mathematics. The research presented here suggests that instead of the current arrangement of years of arithmetic with occasional small helpings of geometry, geometry and spatial reasoning can and should be incorporated as a central feature of a general mathematics education: geometry for all. Second, the contributors emphasize the diversity and range of student thinking encompassed by spatial reasoning and geometry. Not only are existing theories called into question, but several fruitful avenues for new theoretical development in mathematics education are suggested. Third, contributors explore how the development of spatial thinking is tied to tools, ranging from modest (but powerful) ones like Polydrons™ to mechanical curve-drawing devices and the new notational forms made possible by computer-based technologies. Taken together, the research suggests renewed curricular focus on geometry and space: Geometry is not only central to reform in mathematics curricula and the instructional focus on learning with understanding, but with its inherent (and in these studies, enhanced) emphasis on conjecture, argumentation, deductive proof, and reflection, is also central to a solid general education and, as many in this volume note, to good habits of mind.

There are two main \u201cclassic\u201d aspects of teaching and learning geometry: viewing geometry as the science of space and viewing it as a logical structure, where geometry is the environment in which the learner can get a feeling for mathematical structure (Freudenthal, 1973). At a more advanced stage, this geometry environment acquires a broader sense, without the necessity of a real environment as a basis.There is a consensus that these two aspects are linked because some levels of geometry as the science of space are needed for learning geometry as a logical structure. This point of viewone that sees the different phases of learning geometry as a developmental processis intrinsic to most of the theoretical work, research, and instruction that is done in geometry and is the thread that connects the different sections in this chapter.The various phases of geometry learning raise different kinds of psychological questions. If our concern is geometry as the science of space in general, the initial questions are broad, such as:How do children perceive their surroundings?What kinds of codes are used in processing visual information?The questions become narrower if we confine ourselves to visualization; for example:What kinds of visual abilities are needed for geometry learning? In particular, how do children create documentation of their surroundings and how do they interpret this documentation; that is, how do children describe (verbally or visually) the three-dimensional world, and how do they interpret such a description?

Common sense has many aspects and is developed by a variety of experiences in and out of school. Number sense is one aspect of common sense that we rightly expect schooling to improve. But does it? Given a problem, do students pay any attention to the meaning of the numbers in the data or in the solution they obtain? In a study devoted to estimation and reasonableness of results, we found that sixth- and seventh-grade students either do not have a reasonably developed number sense or, if they have it, do not apply it to simple tasks in a mathematical context.

This article discusses an approach in which algebra and geometry are inter woven in a series of problems that develop one from the other, forming an assignment of the kind described in Bruckheimer and Hershkowitz (1977). The two main concepts in this activity are the algebraic concept of the function and the geometric concept of the " family of quadrilaterals. " The geometric concept serves as the "real physical world" (Usiskin 1980) in which the student can develop the concept of function. Yet the algebraic way of think ing, necessarily involved in the concept of function and its application in the geo metrical reality, stimulates the dynamic process that progressively extends geo metrical ideas. This assignment has been used with ninth-grade students of above-average abil ity and with teachers in in-service work shops. It has been particularly effective in eliciting the type of mathematical activity we believe should be at the core of the teaching of mathematics.

This paper describes some segments of a large comprehensive oject, which is responsible for the mathematics instruction in most of the 7th, 8th, and 9th grade classes is Israel. The rationale guiding the project 1.. that develo)- ment, implementation, evaluation, feedback, and research take plate in interlocking an ongoing cycles. These cycles aim to improve "conditions", "means", processes and products of learning mathematics in the ;elevant population. Research is planned to effect, and vo be used in, th- 'lvelopment and implementation stages. Adopting the view that "any changes in curriculum and instruction must be through the minds, motives, and activities of teachers" (Shulman, 1979), a major part of our implementation and research activities are directed at the teacher.

מטרה: בדיקת דרכי התמודדותם של תלמידים עם בעיות מתמטיות שאינן ניתנות לפתרון אלגוריתמי שגור. נבדקים: 290 תלמידי כתות ה'- ט'. שיטה וכלי מחקר: הילדים נתבקשו לפתור חידה מתמטית, אשר אינה ניתנת לפתרון בדרך אלגוריתמית מקובלת, ולנמק את הפתרון. מן הממצאים: העימות עם בעיה מתמטית שאינה ניתנת לפתרון אלגוריתמי שגור הביאה לתחושת קונפליקט בקרב התלמידים הפותרים. חלקם התעקש על השימוש באלגוריתם המוטעה, חלקם ניסה דרכים חלופיות, וחלקם העדיף לא לענות על הבעיה. נראה כי עימות מסוג זה יחדד בעתיד את ביקורתם של התלמידים לגבי מידת ההגיון והרציונליות בפתרון בעיות.