Publications

In an extensive survey on learning mathematics administered to 470 second graders from 17 classes throughout Israel, two problems were included that required students to write explanations (justifications) to their solutions. The purposes of these items were (a) to investigate the ability of second graders to provide explanations to their mathematical productions; (b) to characterize these explanations; and (c) to establish possible factors that may affect the characteristics of their explanations as well as their avoidance to offer them. We discuss a number of theoretical aspects of explanatory processes and explanations and we analyze and discuss students’ written responses sequentially from several points of view. We include some teaching implications from the conclusions.

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 goals of the current study are to investigate if and how the genre of the mathematics teaching-learning discourse in the whole class setting has some longitudinal effect on individual students' knowledge as expressed in a post-test. Specifically, we observed elements of creativity in the whole class discourse and their echoes in students' responses. With this aim we analysed probability learning in two eighth grade classes. We present here data analysis from one lesson and one corresponding post-test item from each class. The differences found in the analysis of students' responses to the post-test item reflect differences in the genre of the whole class discussions brought about by the two teachers. While one teacher opened opportunities for students to express their creative explanations, the other one did not.

Understanding how students construct abstract mathematical knowledge is a central concern of research in mathematics education. Abstraction in Context (AiC) is a theoretical framework for studying students’ processes of constructing abstract mathematical knowledge as it occurs in a context that includes specific mathematical, curricular and social components as well as a particular learning environment. The emergence of constructs that are new to a student is described and analyzed, according to AiC, by means of a model with three observable epistemic actions: Recognizing, Building-with and Constructing–the RBC-model. While being part of the theoretical framework, the RBC-model also serves as the main methodological tool of AiC.In the first section of this chapter, we give an outline of the theoretical aspects of AiC as background to the description of the elements of our methodology in the second section, and their application to a specific example in the third section. In the concluding section, we close the circle by exhibiting the strong relationship of theory and methodology in AiC as it is mediated by the RBC-model.

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 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 flow of theoretical and methodological paradigms which determine the frames for research work in science and mathematics learning has become rich and more and more sophisticated. However, it seems that more than in the past, researchers today do not feel obliged to and/or satisfied with sticking to one methodological paradigm. Research trends in our area are nowadays characterized by flexibility and creativity in combining research methods and methodological tools, which fit the researchers’ theoretical framework and meet their goals and needs to explain and answer some ‘big questions’ emerging from their explorations (e.g. see the chapter by Saxe et al. in this volume). In this chapter I first discuss issues concerning the contour lines between the theoretical framework and the methods and methodological tools within the same research work. I argue that in more and more research work these boundaries are flexible and even a bit vague in the sense that the same scheme or model may serve as a theoretical framework in one piece of research, as a methodological tool in a second one, and as both of them in a third piece of research. I will discuss these issues via two examples, which illustrate dynamic relationships between theory and methodology in two different research domains of mathematics learning. In each example, the questions will emerge from analyses of the above relationships in a few research papers concerning a particular topic, looking at differences and similarities between theoretical and methodological frameworks. The first example is taken from research on argumentation in mathematics learning, and the second example is taken from research using the RBC+C model for abstraction in context.

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 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.

We show how the RBC model for abstraction in context can be used to follow the emergence of a learner’s knowledge constructs and to identify in detail the learner’s partially correct constructs (PaCCs). These PaCCs are used to explain the learner’s 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.

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.

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. 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 girls carrying out an algebraic proof. The analysis reveals how two kinds of knowledge emerge in the students: Knowledge of algebraic structures and knowledge about algebra as a tool for proof.

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.

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.

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.

Orly’s doctoral research was devoted to studying the processes of justification and proving in geometry, used by the pupils she taught in the 9th and 10th grade. After two years of research her findings lead her in a new direction.

There are two main “classic” 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 view–one that sees the different phases of learning geometry as a developmental process–is 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?

Examines the role of visualization within the process of geometrical concept attainment for students in grades five through eight, preservice elementary teachers, and inservice senior elementary teachers. Investigates the "bitrian" and "biquad" examples, numbers of critical attributes, elements in triangles, judgment in quadrilateral examples, and elements in altitude examples. (YP)

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.

The guiding idea of this paper is well expressed by Ginsburg (1977): "Children make mistakes because they use faulty rules.... The faulty rules have sensible origins. Children's mistaken procedures are in fact good rules badly applied or distorted to some degree (p. 110).... Errors result from organized strategies and rules. Children's behavior is meaningful, not capricious" (p. 129). In a recent review paper, Radatz (1979) suggested a classification of errors and their possible causes, giving examples from different mathematical topics. We shall analyze some mistaken procedures used by students when adding fractions. The addition tasks given to our sample belong to the first and second levels of the cognitive taxonomy, namely, knowledge and comprehension with the corresponding thought processes-recognition, recall, and algorithmic thinking. Why is adding fractions such a hard problem? It is suggested that "students are not viewing the fractions to be added as representing quantities but see them as four separate whole numbers to be combined in some fashion" (Carpenter, Coburn, Reys, & Wilson, 1976, p. 138). We would like to suggest possible reasons that are more specific and demonstrate some general principles of cognitive processes. The discussion complements some of the work reviewed by Radatz