Abstraction in Context an introduction
Tommy Dreyfus, Tel Aviv University, Israel
MERGA 31, Brisbane, AUS June 30, 2008
Research supported by the Israel Science Foundation under
grants 973/02 and 1166/05
The complexity of (research in)
mathematics education
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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
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The focus
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Historically:
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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
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Abstraction in Context (AiC)
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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, …
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Abstraction in Context
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Processes of abstraction take place in context
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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
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The nested epistemic actions model
of abstraction in context
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This is the name of our tool for analysis
The name expresses that
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epistemic actions form the main tool of analysis
epistemic actions are dynamically nested
we attribute great importance to context
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Epistemic actions
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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:
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Recognizing
Building-With
Constructing
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Recognizing (a previous construct)
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The 're-cognition' of previously encountered
mental constructs that are inherent in a given
mathematical situation
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Building-with (previous constructs)
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The combination of mental constructs in order to
achieve a given goal
Goals:
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solving a problem
understanding and explaining a situation
reflecting on a process
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Constructing (a new construct)
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‘Cognizing’ novel constructs
Assembling and integrating previous constructs
by vertical mathematization to produce a new
construct
Constructs include
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Methods
Concepts
Strategies
Process may be slow or sudden
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Dynamic Nesting
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In processes of abstraction, the epistemic actions
are dynamically nested:
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R-actions are nested in B-actions: you cannot buildwith 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
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The genesis of an abstraction
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Processes of abstraction have three stages
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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
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Stage 1: The need for a new construct
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This need is inherent in the design but it is relative to the
context:
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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.
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Stage 3: Consolidation
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Consolidation is a long-term process
Consolidation is likely to occur during problemsolving and reflection activities
Consolidation contributes to awareness of one’s
use of the constructs and to flexible problem
solving
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Mechanisms of consolidation
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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)
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The role of context
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Context
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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.
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Social context
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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.
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Sample topics (student age/authors) of
published AiC-based studies
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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)
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Thank you!
tommyd@post.tau.ac.il
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