Learning Equations using a Computerised Balance Model: A Popperian Approach to Learning Symbolic Algebra

Abstract : This study investigates, using a perspective based on the work of Karl Popper, how students aged 10-15 can learn about simple linear equations, with particular reference to the use of a computerised balance model of an equation. Popperian epistemology implies a conjectural view of knowledge, in which rigour is dependent on the potential for intersubjective criticism. A Popperian approach to psychology is advocated, in which "understanding" is viewed as problem-solving rather than sense-making, imagining or reenactment; and learning occurs through trial-and-improvement of strategic theories in response to concerns, rather than through the development of context-free modes of thought. From this perspective, explanatory constructs from research into learning algebra such as "letter interpretations" and "equation metaphors" are seen as recontextualised meta-algebraic theories rather than as slowly maturing "underlying" algebraic cognitive structures. A Popperian reinterpretation of the research literature into the problem of learning algebra enables the development of an instrument to detect learning in a range of principal algebraic concerns - representation, interpretation, transformation and utilisation. A computer program called EQUATION is also constructed, which acts as a research tool to explore the educational limitations of the balance model of an equation. Fieldwork is carried out to test conjectures relating to the program, involving around 100 students. Analysis involves reconciliation of classwork learning and pre-post testing. It is argued that a concern for symbolic algebra can be initiated firstly by using the balance model to promote formal operations on equations and secondly by encouraging the formulation of equations to find an unknown number in a word problem. In addition, by providing progressive challenge and feedback on the effects of operations, it is possible for students to create, test and improve strategic theories for a number of transformation and representation problems.
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James Aczel. Learning Equations using a Computerised Balance Model: A Popperian Approach to Learning Symbolic Algebra. 1998. ⟨hal-00190199⟩

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