Towards a theoretical synthesis of
research in the early learning of
algebra
James Aczel
The Open University, UK
[email protected]
Introduction
Can diverse research findings relating to
students’ learning of algebra be
synthesised?
Maybe not…
But here are some thoughts anyway ;-)
Popperian psychological
perspective
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Background – defence of democracy,
critique of authoritarianism & positivism
An early philosophical advocate of
intersubjectivity (World 3)
Against the “bucket theory of the mind”
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role of linguistic & social conventions
centrality of creative conjectural processes
Against behaviourism
Popperian psychological
perspective (continued)
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BVSR (Campbell, 1960)
psychological selection pressures
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intertwining action, context & theory
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“concerns” - the fuel that powers learning
“strategic theories”
problem-solving schema
P1  TT  EE  P2
C1
ST
C2
World 3
World 2
Cognitive readiness and
cognitive obstacles
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A particular set of tasks
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2
2
2
“Add 4 onto 3n”; “multiply n + 5 by 4”
“If cakes cost c pence each and buns b pence each, and 4 cakes
are bought and 3 buns are bought, what does
4c + 3b stand for?”
Categorising the treatment of letters
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2
Küchemann - “letter evaluated”, “letter not used”, “letter as object”,
“letter as specific unknown”, etc.
Why do these categories work?
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unique hierarchy of difficulty – related to student maturation?
but presentation, language & experience matter (O’Reilly, 1990)
deep-rooted conceptions?
Popperian view: they are strategies devised by students for such
tasks (tasks detached from the students’ mathematical concerns?)
Interpretations, images,
meanings and metaphors
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Meta-algebraic theories
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by-products
not static structures
not “underlying”
not “fundamental”
Value to researchers
Value to students
Should meta-algebraic theories be taught?
The student-professor problem
and reflection on meaning
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Six times as many students as professors
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Strategies described by Clement (1982)
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6S = P
«translate the situation syntactically»
«put the number next to the largest group»
«find an operation»
«find likely equations and test»
What concerns generate these strategies?
The importance of concerns
“Can we develop a school algebra culture in
which pupils find a need for algebraic
symbolism to explore their mathematical
ideas?” (Sutherland, 1991)