Publications

Fractals describe many natural phenomena; their strong visual-figurative nature found its mathematical conceptualization in the concept of self-similarity. In the current study, we investigate how students construct (fully or partially) the self-similarity concept while recursively constructing the Sierpiński triangle, working in small group and whole class settings in an inquiry-based MA level mathematics education course. We follow shifts of knowledge from individuals to groups and/or to the whole class community during the process of constructing the self-similarity concept. Our theoretical and methodological approach is based on networking between Abstraction in Context and Documenting Collective Activity. We found that the knowledge constructing processes of different students varied, some thinking recursively about finite cases and others thinking more directly about the infinite case. Some students acted as knowledge agents, with shifts of knowledge occasionally occurring in chains. We also observed a tendency to report results of group discussions back to the plenum only partially and in a purified manner.

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.

Although visual thinking contributes to the development of basic skills in many domains, this subject is neglected in preschool and primary school. The Agam Program is a structured curriculum of 36 units that teaches visual skills, visual vocabulary and visual thinking to young children. Research shows that this approach has many positive effects for young children from diverse backgrounds, in areas such as mathematical thinking, school readiness, and general intelligence.

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 investigate students' knowledge construction and shifts of the constructed knowledge in a mathematical classroom. An early lesson of a differential equations course serves as a paradigmatic example. We use existing methodological tools for analyzing construction of knowledge by individuals and groups (abstraction in context) and for analyzing whole class discussions (documenting collective activity). We offer a way to adapt these methodological tools in order to coordinate analyses of the individual, the group and the collective in a mathematical classroom. The combination of both analyses allows us to follow the evolution of ideas from their construction in small groups to their becoming a normative way of reasoning during whole class discussion, or vice versa. Our overall goal is to exhibit the role played by individuals and groups in the class as well as by the class as a whole, in the knowledge constructing process.

We present the first stage of a "study in progress" whose aim is to link visualization, students construction of geometrical concepts and their definition, and students ability to prove. We exemplify this stage in our research (visualization and concept formation), by means of paradigmatic examples, which reveal visual and verbal processes related to construction processes of geometric figures and inclusion relationships between figures and their attributes. Our results confirm known findings (e. g., the position of a shape affects its identification and the related inclusion relationships) and point to findings in a new direction, like the effect of the question's representation on students responses concerning the inclusion relationships.

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 Kieran's 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.

The goal of this paper is to show that argumentation gains from being multimodal in learning geometry, especially at elementary levels. Grade 3 students participated in a year long course designed to foster mathematical reasoning. The course combined problem solving in dyads, peer argumentation and teacher-led discussions. We focus on one activity: identifying the types of solutions, the kinds of reasoning and the kinds of non-verbal actions (gestures, drawings, folding etc) used. We show how gestures and other non-verbal actions were interwoven with children's verbal peer argumentation and led them to new insights on the concept of area.

We present a view of knowledge construction processes, focusing on partially correct constructs. Motivated by unexpected and seemingly inconsistent quantitative data based on the written reports of students working on an elementary probability task, we analyze in detail the knowledge construction processes of a representative student. We show how the nested epistemic actions model for abstraction in context facilitates following the emergence of a learner's partially correct constructs (PaCCs). These PaCCs provide added insight into processes of knowledge construction. They are also used in order to analyze and explain students' thinking in situations where some of the students' answers were unexpected in light of their earlier answers or inconsistent with earlier answers. In particular, PaCCs are explanatory tools for correct answers based on (partially) faulty knowledge and for wrong answers based on largely correct knowledge.

This study reports on how a large number of second graders performed in two explanation tasks of different kinds. We found that: a) a considerable number of them were able to produce (full or partial) explanations, b) the number of "explainers" cluster in certain classes (suggesting that the practice of explaining maybe related to classroom norms) and c) the way explanations are envisioned and understood played an important role in children responses. Suggestions fur educational implications and further research are proposed.

This chapter is a follow-up to the chapter by Hershkowitz et al. (2002) in the fi rst edition of this handbook, which describes and analyzes the stages of the CompuMath Project, as a paradigm for research-intensive development and implementation of compound and long-term curricula for computerized environments.

This study is part of a large research and development project aimed at observing, describing and analyzing the learning processes of two seventh grade classes during a yearlong beginning algebra course in a computer intensive environment (CIE). The environment includes carefully designed algebra learning materials with a functional approach, and provides students with unconstrained freedom to use (or not use) computerized tools during the learning process at all times. This paper focuses on the qualitative and quantitative analyses of students' work on one problem, which serves as a window through which we learn about the ways students worked on problems throughout the year. The analyses reveal the nature of students' mathematical activity, and how such activity is related to both the instrumental views of the computerized tools that students develop and their freedom to use them. We describe and analyze the variety of approaches to symbolic generalizations, syntactic rules and equation solving and the many solution strategies pursued successfully by the students. On that basis, we discuss the strengths of the learning environment and the open questions and dilemmas it poses.

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.

We present the learning process of a pair of grade 8 students, who learn a topic in elementary probability. The students successfully accomplish a sequence of several tasks without apparent difficulty. When working on a further task, which seems to require only actions they have previously carried out well, they run into difficulty. In order to explain this difficulty, we analyze their learning process during the task sequence, and identify some partial knowledge.

We describe and analyze episodes taken from a long-term research project, whose main goal is to investigate the constructing and consolidating of knowledge in elemental)) probability. Specifically, we follow the constructing and consolidating of "shared knowledge" by a group of three students in one of the project classrooms. The RBC model is used as the main methodological tool. We found that the group constructed shared common basis of knowledge, which enable them to continue the constructing of a new knowledge. We also found that this knowledge flows from one student to the other, where many times each partner has her own way of constructing knowledge.

The ideas presented in this lecture are based on the observation of processes of construction and consolidation of knowledge by individual students learning in groups within classrooms along a sequence of activities. Whereas the uniformity of the basic elements used to describe the knowledge construction processes may be seen as inclusive, there is a lot of diversity in the different ways in which individual students combine these basic elements into their personal learning trajectories.

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]

The goal of this study is to show that inquiry activities in a dynamic geometry environment, intentionally designed to confront students with contradictions and uncertainties, push them towards explanations that include deductive elements. Three different but dependent aspects of the activities are characterized and analysed: The epistemological, which includes all possible inquiry paths; the didactic, which involves only those paths that reflect the intention of the designer; and the cognitive, which accounts for actual student actions (conjectures and explanations) and their analyses. The research conclusions are based on the interplay among these three aspects. The analysis of students investigations and the analysis of their explanations fulfil, to a broad extent, the design goals.

In designing mathematics learning with the mediation of computerized tools, one of the crucial questions to be considered is how much and in what way we would like the tool «to do the work » for the students. In problem situations where the solution is achieved mostly via graphical representatives, and algebraic models are mostly used as «keys » for obtaining graphical representatives on the screen, the algebraic representatives and their form seem minimized in importance, and students may tend to generate them from tables by mechanistic-algorithmic procedures. The above questions will be mainly demonstrated within the description and analysis of a case study involving a group of three 10th graders (about 16 years of age) working together to investigate and solve a problem situation on the topic of functions, having graphing calculators (TI-83 Plus) at their disposal. The role of contextual factors is highlighted by means of contrasts with the work of students on the same problem from another country.

Abstraction is defined as an activity of vertically reorganizing previously constructed mathematical knowledge into a new structure. In a previous study, the authors translated this definition into an operational model whose elements are 3 nested epistemic actions. This model was illustrated the model by means of a case study. In the present article, the authors validate and refine the model by analyzing additional, more complex case studies, extend the model to a richer context, namely pairs of collaborating peers, and investigate the distribution of the process of abstraction in the context of peer interaction. This is done by carrying out 2 parallel analyses of the protocols of the work of the student pairs, an analysis of the epistemic actions of abstraction as well as an analysis of the peer interaction. The parallel analyses led to the identification of types of social interaction that support processes 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.

We take abstraction to be an activity of vertically reorganising previously constructed mathematical knowledge into a new structure. Abstraction is thus a context dependent process. During group work, peer interaction is an important component of the context. In a previous publication, we proposed a model for processes of abstraction. The model is operational in that its components are observable epistemic actions. Here we use the model to analyse an interview with a pair of grade seven boys collaboratively constructing an algebraic proof The analysis of the interview reveals subtle links between the abstraction process and the peer interaction.

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.

In the following paper we describe a sequence of episodes in which subjects in different ages were involved in judgment of visual estimation of discrete quantities. The analysis of these episodes seems to demonstrate two opposing trends: In the first, situations of visual estimation judgment seem to push young children, (8 to 10 years old) towards proportional reasoning. In the second, the same situations may push subjects (10-11 years old children, student-teachers and teachers) away from a mathematical proportional reasoning towards what might be called "situational reasoning".

This study has two main goals: (1) to investigate the processes involved in visual estimation (part I of the study), and (2) to investigate the processes of judgment in visual estimation situations, which mostly involved proportional reasoning (part II). The study was conducted with 9-year old children in the third grade. Four strategies were expressed by the children in visual estimation situations. Exposure to a unit in the Agam project, designed to enhance visual estimation capabilities resulted in changes in the children's strategies. These changes reflected the processes by which children overcame their limited ability to process visual information. The development of proportional reasoning was investigated through a series of judgment situations. Although, as was expected, most of the children showed an additive behavior, these situations stimulated some children towards qualitative proportional reasoning, where easy/difficult considerations played an important role.

Four strategies were expressed by third grade children in visual estimation situations. After going through visual estimation activities, some changes in children strategies were found. These changes were in agreement with theories of visual information processing. Most of the children had difficulties in judging the quality of the estimate. They showed "additive" behavior without any traces of relative thinking. After a very directed intervention some of them moved towards relative thinking.

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?

מטרה: בדיקת היכולת הפרה-אלגברית של סטודנטים לעבור מגורמים כמותיים לאיכותיים ולהיפך, בהכללה והצדקה של מצבים. נבדקים: שני זוגות של תלמידים בכיתה שביעית, שרמתם במתמתיקה תוארה ע"י מוריהם כבינונית לאחד מהזוגות ומעל לממוצע לזוג האחר. שיטה: ראיונות, שנמשכו כמחצית השעה לכל אחד הוקלטו וצולמו. בראיונות התקיימו דיונים רבי-עוצמה בין שני התלמידים שבשני הזוגות, כאשר תפקיד המראיין היה לשאול ולספק מסגרת. מן הממצאים: באופן כללי התלמידים היו מסוגלים, לפחות חלקית, לפעול ברמה האיכותית בלי שפה אלגברית סימבולית. יש מצבים שבהם העדר כלים סימבוליים מפריע לביצוע ברמה גבוהה יותר. לימוד סימבולים וטכניקות אלגבריות מפתח את מודעות התלמיד לצורך בכלים אלגבריים.

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.

This paper describes a study undertaken as a small part of a large, comprehensive project. The project itself is devoted to the development, implementation, evaluation and evolution of a mathematics program for junior-high schools in Israel (the "Rehovot Program"). It started approximately fifteen years ago with the legal change of the educational system in Israel, called "reform," which, among other things, introduced junior-high schools and a new math curriculum for grades 7-9. At present, the "Rehovot Program" is followed by more than 60% of the relevant population.

מן התוכן: הצגת דוגמא של לימוד מתימטי, המראה כי ניתן לשלב הבטים של הגישה הלוגיסטית ללימוד מתימטיקה עם הבטים של הגישה הפסיכולוגית. בדוגמא זאת, העוסקת בגישה להוראת משוואות ריבועיות, מוצעת מסגרת, בה התלמיד מגיע לגילויים מתימטיים בעצמו במסגרת מבנה דדוקטיבי, שנוצר בחלקו ע"י התלמיד.