Rina Hershkowitz

Abstraction in Context: Epistemic Actions

ion in Context: Epistemic Actions Rina Hershkowitz, Weizmann Institute of Science, Rehovot, Israel Baruch B. Schwarz, Hebrew University of Jerusalem, Israel Tommy Dreyfus, Holon Academic Institute of Technology, Israel 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.

The Emergent Perspective in Rich Learning Environments: Some Roles of Tools and Activities in the Construction of Sociomathematical Norms

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.

Abstracting Processes, from Individuals' Constructing of Knowledge to a Group's "Shared Knowledge"

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.

Relative and Absolute Thinking in Visual Estimation Processes.

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 childrens 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.

The Nested Epistemic Actions Model for Abstraction in Context: Theory as Methodological Tool and Methodological Tool as Theory

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.

Production and Transformation of Computer Artifacts Toward Construction of Meaning in Mathematics

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.

Analyses of Activity Design in Geometry in the Light of Student Actions.

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.Sommaire exécutifLe but de cette recherche est de montrer que des activités d’investigation 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 d’investigation ont été élaborées afin d’amener les élèves à émettre des conjectures, à explorer les situations à l’étude grâce au rôle médiatisant d’un 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 d’abord déroulées dans le cadre d’entrevues semi-structurées, puis en classe. Les entrevues nous ont permis d’observer l’évolution 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 d’élèves ont fourni à la fois une perspective quantitative et des exemples qualitatifs. Ces données nous ont permis d’établir dans quelle mesure les élèves ont dû résoudre des contradictions ou des incertitudes, et d’obtenir une grande variété d’explications que nous avons regroupées en catégories.Les activités ont été analysées selon trois aspects différents quoique interdépendants. Tout d’abord, nous nous sommes intéressés à la structure épistémologique des activités: nous avons répertorié toutes les façons possibles d’aborder la tâche, sans en privilégier une en particulier. Deuxièmement, nous nous sommes penchés sur les caractéristiques didactiques des activités, y compris le rôle de l’outil 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 l’aspect 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.

Deductive discovery approach to mathematics learning or In the footsteps of the quadratic function

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.

PME and the International Community of Mathematics Education

The International Group for the Psychology of Mathematics Education (PME) was founded in 1976 in Karlsruhe (Germany), during the ICME-3 Congress. Since 1977, the PME group has met every year somewhere in the world, since then, and has developed into one of the most interesting international groups in the field of educational research. In this paper, after a short introduction, we draw some main features of the unique essence of the PME as a research group. We focus on and analyse the change and development of the group’s research over the past 40 years, and exemplify these changes and developments by tracing on a few main research lines. Based on specifics of PME research, we describe the more comprehensive lines of PME research, its change and progress in the past four decades.