abstraction in context - an introduction tommy dreyfus, tel aviv university, israel merga 31,...

Abstraction in Context -an introduction

Tommy Dreyfus, Tel Aviv University, IsraelMERGA 31, Brisbane, AUS June 30, 2008Research supported by the Israel Science Foundation under grants 973/02 and 1166/05

13 May 2008 Auckland, NZ: LOGOS 2

The complexity of (research in) mathematics education Even a seemingly simple event in a mathematics

classroom is a complex issue Different researchers have different interests and think in

different theoretical frameworks about such events Their focus may be (some but not all of) cognitive, social,

cultural, affective, beliefs, design, learning environment, …

As researchers, we have to be aware that we always deal with some aspects of a problem or situation and ignore others


The focus Historically:

A curriculum development program Design-research-design cycles ‘Rich’ activities What remains (is consolidated)?

The focus is on cognitive processes, especially abstraction, emergence of new knowledge constructs

The learning environment is considered as context within which these processes take place

We propose a framework that allows us to analyse such processes at the micro-level


Abstraction in Context (AiC) Approach developed over the past ten years with

Rina Hershkowitz, Baruch Schwarz and others Abstraction is a process of interweaving earlier

constructs and leading to a construct that is new for the learner

Abstraction is an activity of vertical [Freudenthal, Treffers & Goffree] reorganisation of knowledge, within mathematics and by mathematical means

Vygotsky, Davydov, …


Abstraction in Context

Processes of abstraction take place in context learning context (classroom, available tools incl ICT) historical (prior experience and learning) social context (peers, teacher) curricular (task sequence)

More on context below, if time permits


The nested epistemic actions model of abstraction in context

This is the name of our tool for analysis The name expresses that

epistemic actions form the main tool of analysis epistemic actions are dynamically nested we attribute great importance to context


Epistemic actions

Epistemic Actions are observable mental actions by means of which knowledge is constructed

(Pontecorvo & Girardet, 1993) We found the following three epistemic actions

useful for the analysis of processes of abstraction: Recognizing Building-With Constructing


Recognizing (a previous construct)

The 're-cognition' of previously encountered mental constructs that are inherent in a given mathematical situation


Building-with (previous constructs)

The combination of mental constructs in order to achieve a given goal

Goals: solving a problem understanding and explaining a situation reflecting on a process


Constructing (a new construct)

‘Cognizing’ novel constructs Assembling and integrating previous constructs

by vertical mathematization to produce a new construct

Constructs include Methods Concepts Strategies

Process may be slow or sudden


Dynamic Nesting

In processes of abstraction, the epistemic actions are dynamically nested: R-actions are nested in B-actions: you cannot build-

with a construct unless you have first recognized it Similarly, R-actions and B-actions are always nested in

C-actions; in fact, C-actions consist of (alternating) R and B actions

C-actions at different levels may be nested in each other since I may need a certain construct in order to reach another one


The genesis of an abstraction

Processes of abstraction have three stages The need for a new construct The emergence of a new construct The Consolidation of the new construct

The second stage is the central one, and so far I have mainly related to this stage

I will now briefly relate to the other two stages


Stage 1: The need for a new construct This need is inherent in the design but it is relative to the

context: The student population Their prior knowledge and experience Available tools such as computer tools Habits of collaboration

Our research, so far, has concentrated on the second and third stages of processes of abstraction; we have taken the need for granted – provided by the instructional design. We plan research on the first stage in the near future.


Stage 3: Consolidation

Consolidation is a long-term process Consolidation is likely to occur during problem-

solving and reflection activities Consolidation contributes to awareness of one’s

use of the constructs and to flexible problem solving


Mechanisms of consolidation The analysis of the work of students in sequences of

activities over several lessons has allowed us to identify several mechanism of consolidation

The most interesting of these is the consolidation of a previous construct during the process of constructing a further one, with the earlier one serving as an element in constructing the new one

For the other mechanisms, as well as for example, I refer to the literature (Schwarz, Hershkowitz & Dreyfus, 2008)


The role of context


Context

Computer tools may be a component of the context. In a recent paper, we analyzed the influence of a computer tool on construction of knowledge (Kidron & Dreyfus, 2008). More research in this direction is planned.

Another important aspect of context is social context. For example, in MERJ (Hershkowitz et al. 2007), we analyzed the social construction of knowledge by student groups in classrooms.


Social context Hershkowitz et al. investigated processes by

which two groups of individual students (three students each) construct shared knowledge and consolidate it.

We identified an 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 allowed them to continue together the constructing of further knowledge in the same topic.


Sample topics (student age/authors) of published AiC-based studies Rate of change as a function (14/HDS) Algebra as a tool for justification (12/DHS) The power of a countably infinite set (16/TD) Elementary probability concepts (13/RDH, …) Function transformation (17/OM) Bifurcations in a dynamical system (adult/DK) Limits (adult/K) Finite arithmetic structures (adult/S)


Thank you!

[email protected]

abstraction in context - an introduction tommy dreyfus, tel aviv university, israel merga 31,...

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