Publications

The current study is part of a comprehensive research on linking visualization, students construction of geometrical concepts and their definitions, and students ability to prove. The aim of the current study is to investigate the effect of learners understanding of definitions of geometrical concepts on their understanding of the essence of geometrical proofs and their ways of proving. By \u201cunderstanding\u201d (geometrical definitions and/or proofs), we mean knowing the definitions and/or proofs meaning and their role within the logic structure of geometry and also the ability to define and/or prove in \u201cgeometrically correct\u201d ways. Ninety grade 11 students from an Arab high school in Israel participated in the comprehensive study in geometry of which the current study forms a part. Research tasks for the investigation of the current studys aim were designed, constructed, and used in a questionnaire to the whole research population and in interviews with about 10% of the population. The findings point clearly to effects of elements within the students understanding of definitions on their understanding of proofs and on their ability to prove. These elements are as follows: (a) The difficulty to internalize that an incomplete definition is an incorrect definition and may lead to an incomplete proof. (b) The difficulty to deal with non-economical definitions forms the basis to non-economical proofs. (c) Students have difficulties to accept equivalent definitions to the same geometrical concept. (d) The students lack of understanding the origin of a constructive definition of geometrical concept.

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.

The goals of the study were to design and investigate a teaching-learning environment that encourage freedom and autonomy of pre-service teachers in constructing their own new geometrical concept, and to analyze the dialectic process of the concept construction in the designed environment. A dialectic process of the participants defining activity emerged from the necessity to resolve the tensions between hypothesizing the concepts examples and the appropriate critical attributes. Such a process created a vivid learning trajectory, in which learners examined logically the ways in which the examples, the critical attributes and the definition match. In this way, the three elements of the mathematical concept cannot play a passive role, or be neglected, and it appeared that prototypical examples were not created, so that no example is more dominant than others. The groups constructed different concepts, but with full harmony between its definition, example space, and concept-critical attributes.

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.

We use the notion Partially Correct Constructs (PaCCs) for students constructs that partially match the mathematical principles underlying the learning context. A frequent expression of partial construction of mathematical principles is that a students words or actions provide an inaccurate or misleading picture of the students knowledge. In this study, we analyze the learning process of a grade 8 student, who learns a topic in elementary probability. The student successfully accomplishes a sequence of several tasks without apparent difficulty. When working on a further task, which seems to require nothing beyond his proven competencies, he encounters difficulties. Using the epistemic actions of the RBC model for abstraction in context as tracers, we analyze his knowledge constructing processes while working on the previous tasks, and identify some of his constructs as PaCCs that are concealed in these processes and explain his later difficulties. In addition, our research points to the complexity of the knowledge structures students are expected to deal with in their attempts to learn an elementary mathematical topic with understanding.

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.

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.

To what extent can instructional design be based on principles for instilling a culture of problem solving and conceptual learning? This is the main focus of the study described in this paper, in which third grade students participated in a one-year course designed to foster problem solving and mathematical reasoning. The design relied on five principles: (a) encouragement to produce multiple solutions; (b) creating collaborative situations; (c) socio-cognitive conflicts; (d) providing tools for checking hypotheses; and (e) inviting students to reflect on solutions. We describe how a problem solving task designed according to the above principles, promoted students' understanding of the area concept. We show that the design afforded the surfacing of multiple solutions and justifications in various modalities (including gestures) and initiated peer argumentation, leading to deep learning of the area concept.

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.

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.

The transition from arithmetic to algebra in general, and the use of symbolic generalizations in particular, are a major challenge for beginning algebra students. In this article, we describe and analyze students' learning in a "computer intensive environment" designed ad hoc and implemented in two seventh grade classrooms throughout two consecutive school years. In particular, this article focuses on the description and analysis of how students initial generalizations (which relied on computerized tools that enabled different students' to work with different strategies) shifted to recursive and explicit symbolic generalizations.

Amodel for processes of abstraction, based on epistemic actions, has been proposed elsewhere. Here we apply this model to processes in which groups of individual students construct shared knowledge and consolidate it. The data emphasise the interactive flow of knowledge from one student to the others in the group, until they reach a shared knowledge a common basis of knowledge which allows them to continue the construction of further knowledge in the same topic together.

Studies of knowledge constructing often focus on the analysis of a single episode, without considering enough the history of the learners, or the future learners' trajectories with regard to the concepts learned. This paper presents an example of knowledge constructing within the context of peer learning. We show how the design of the task and the tools available to the students afford the constructing of conceptual knowledge (the phenomenon of exponential growth and variation, as it is expressed in its numerical and graphical representations). We trace the constructing of knowledge through a series of dyadic sessions for a few months in a classroom environment. We show that knowledge is constructed cumulatively, each activity allowing for the consolidating of previous constructs. This pattern indicates the nature of the processes involved: knowledge constructing and consolidating are dialectical processes, developing over time, when new constructs stem from old ones already consolidated, which gain consolidation through the new construction, creating a new abstract entity. We also discuss the potential of the tool the students used (a spreadsheet program) to such processes of learning mathematics.

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.

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.Le but de cette recherche est de montrer que des activités dinvestigation conçues pour créer des contradictions entre des conjectures, prédictions ou doutes émis par des élèves poussent ces derniers à formuler des explications à caractère déductif. Trois activités dinvestigation ont été élaborées afin damener les élèves à émettre des conjectures, à explorer les situations à létude grâce au rôle médiatisant dun logiciel de géométrie dynamique, à tirer des conclusions et à les expliquer. Les activités ont été mises au point selon un processus de conception-recherche-conception.Les élèves qui participaient à chacune de ces activités étaient regroupés deux par deux. Les activités se sont dabord déroulées dans le cadre dentrevues semi-structurées, puis en classe. Les entrevues nous ont permis dobserver lévolution des conjectures émises par les élèves ainsi que le processus suivant lequel ces derniers produisent des explications dans des situations où ils font face à des contradictions ou à des incertitudes. Les transcriptions des entrevues et des rapports portant sur les activités réalisées en classe par chaque paire délèves ont fourni à la fois une perspective quantitative et des exemples qualitatifs. Ces données nous ont permis détablir dans quelle mesure les élèves ont dû résoudre des contradictions ou des incertitudes, et dobtenir une grande variété dexplications que nous avons regroupées en catégories.Les activités ont été analysées selon trois aspects différents quoique interdépendants. Tout dabord, nous nous sommes intéressés à la structure épistémologique des activités: nous avons répertorié toutes les façons possibles daborder la tâche, sans en privilégier une en particulier. Deuxièmement, nous nous sommes penchés sur les caractéristiques didactiques des activités, y compris le rôle de loutil que constitue le logiciel de géométrie dynamique, et ce en tenant compte des intentions du concepteur, qui favorisent certaines possibilités sur le plan épistémologique. Un examen plus attentif de ces intentions a orienté le troisième volet de notre analyse, qui portait sur laspect cognitif sous-tendant les actions des élèves (conjectures ou explications) et leurs analyses. Les résultats obtenus ont montré que, en général, les élèves exercent des choix qui les amènent à faire face à des contradictions ou à des incertitudes. Ces choix suscitent chez eux le besoin de formuler des explications dont la catégorisation, dans le cadre de notre recherche, a mis en évidence le caractère largement déductif.

Artifacts both mediate our interaction with the world and are objects in the world that we reflect on. As computer-based artifacts are generally intermingled with multiple praxes, studying their use in praxis uncovers processes in which individuals, the community, and tools are involved. In this article, we examine a now common computer-based artifact in mathematics classrooms, the representative. This artifact is often in continual transformation in the course of action during school activities. We document how several praxes with representatives mediate the construction of meaning. We show that the ambiguity of computer representatives regarding the examples and concepts they are meant to represent boost this construction. The construction of meaning of mathematical functions is described as a process that occurs through social interaction and the interweaving of the ambiguous computer-based artifacts. We show that this construction depends heavily on intentional design of activities by the teacher, leading to the creation of states of intersubjectivity.

Prototypes are used as references to cope with new examples in concept learning. Prototypes can, however, he detrimental to concept learning, as shown in the use of linear functions for learning the function concept. This research characterizes students' function concept images that arise in an interactive environment based on multirepresentational software. We capitalize on a 20-year curricular program to contrast the concepts students develop in this environment with those developed in traditional environments. We show that students who are learning functions in the interactive environment (a) often use prototypic functions (linear and quadratic) but do not consider them as exclusive, (b) use prototypes as levers to handle a variety of other examples, (c) articulate justifications often accounting for context, and (d) understand functions' attributes.

The goal of this paper is to show some mutual relationships between design of activities aimed to put students in geometrical situations where they feel the need for proof and cognitive research on students' actions in such activities. We exhibit an example of the development process of such an activity in several cycles of design, experimentation and analysis, by describing the work of two pairs of students in two different cycles of the development. We also analyze the task as well as the work of the students. Finally, we discuss principles of activity design for leading students to feel the need for and to produce proofs by deductive reasoning.

Presents several examples of investigations which demonstrate the following: (1) algebra is connected to other mathematical topics; (2) algebra is an effective tool for representing patterns and relationships; and (3) algebra is an effective tool for analyzing patterns. (DDR)

מטרה: בדיקת דרכי ורמות החשיבה הוויזואלית אשר הפעילו תלמידים בעת שימוש בתוכנת LOCI להנחלת מושגים בהנדסה. נבדקים: קבוצה קטנה של תלמידי כיתה ט' וכיתה שלמה של תלמידי י'. שיטה וכלי מחקר: התנסות בעבודה בתוכנת LOCI במהלך ארבעה שעורים ,הדרכה בשימוש בתוכנה בעזרת דפי עבודה ובעזרת מורה הנוכח במקום, שאלון הבודק את הבנת התלמידים בעזרת שאלות בהנדסה, בהן מנמק הנבדק את תשובותיו. מן הממצאים: דרכי החשיבה הנפוצות ביותר בקרב התלמידים סווגו כמקומיות (local) או כגלובאליות .רוב התלמידים שינו את רמת החשיבה בה השתמשו משאלה לשאלה.

In Friedlander et al (1989) we analyzed the mathematical behavior of seventh graders in generalization and justification processes. The analysis of the data presented there and additional data led us to focus our attention on the interplay between aspects of the mathematical structure of the problem situations we designed and the spectrum of observed student behaviors in these problem situations. Our aim is to tackle this issue by analyzing some epistemological aspects of problem situations in the first part of this paper. In the second part, we analyze the "traces" of the mathematical structure of the problems on student behavior.

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)

There are certain advantages which can be obtained by a deductive discovery approach to the learning of some topics in the regular mathematics curriculum. Deductive discovery is the term used for learning by discovery within a deductive structure. It is to be distinguished from inductive discovery, which tends to be diffuse, and from the traditional deductive method which usually gives the result at the beginning of the investigation. The example used to illustrate the approach is the 'discovery' of the graph and the zeros of the quadratic function. The mathematical skeleton of the approach is given, together with a description of its application in the classroom. In this particular example, the resulting mathematics is considerably different from that usually associated with this topic, and is also arguably more elegant than the usual approach found in textbooks.

מטרה: בדיקת הגורמים המשפיעים על רכישת מושגי יסוד גאומטרים בקרב ילדים, ע"י בדיקת מושגים אלו בקרב מורים בבתי-ספר יסודיים. נבדקים: 142 סטודנטים לחינוך, 25 מורים וותיקים. שיטה: לנבדקים הועבר אותו שאלון בו נעשה שימוש במחקרים קודמים לבחון את המושגים הגאומטריים של ילדים. מן הממצאים: א) לא נמצאו הבדלים גדולים בין המורים לבין תלמידים שנחקרו במחקרים קודמים, בהבנת מושגי יסוד בגיאומטריה. חלק גדול מהמורים עשו אותם טעויות שעשו התלמידים. ב) המורים לא נעזרו בהגדרות שסופקו להם בשאלון. ג) זוהו מספר גורמים שכנראה משפיעים על יצירת מושגים גאומטריים גם בקרב תלמידים וגם בקרב מורים.

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.

מטרה: לבנות מודל לרכישת מושגים מסויימים אשר יש להם מרכיבים ויזואלים חזקים. נבדקים: 550 תלמידים מבתי"ס ט"ט, כיתות ז', ח', ט'.שיטה: שאלון, כלים סטטיסטיים לתיקוף היררכיות בלמידה.ממצאים: זויות כהות אשר להן צד אנכי ניתנות לזיהוי ביתר קלות מאחרות, וכנ"ל לגבי זוית ישרה. כ-%74 מהתלמידים תופסים את הרעיון של המשולש הישר. רוב התלמידים הבינו את הרעיון של גובה המשולש.

מן התוכן: הצגת מודל יחודי לפיתוח תכניות למודים, אשר יסודו בהפעלה אינטנסיבית של התכנית, קבלת משוב על ההפעלה ויצירה מחודשת. כהדגמה לתהליך זה, מוצג פרוייקט המתבצע במסגרת קבוצת המתמטיקה במחלקה להוראת המדעים לטעוני טיפוח בחטיבות הביניים. תהליך פיתוח התכנית מורכב ממספר מעגלים, המכונים כל אחד "מיקרו פיתוח", והמאופיינים בחמישה שלבים: תצפית, הגדרת הקושי, ברירת הדרך הנראית יעילה לטיפול בקושי, יצירה, הפעלה בכיתה וחוזר חלילה. בסופו של דבר, מצטרפים כל המיקרו-פיתוחים זה לזה ליצירת ה"סליל", שהוא תהליך הפיתוח החדשני של תכנית הלמודים. בעזרת תהליך גמיש מעין זה, ניתן להחליף סעיפים בחומר הקיים, ולחולל שינויים דינאמיים בתכנית הלמודים

Three different geometric constructions of the parabola are described.