# Visual Transformations in Symbolic Elementary Algebra.

1993

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Educational Theory, Policy, and Practice

David Kirshner

## Abstract

Competence in algebra requires knowledge of parsing and transformations. The parsing component specifies the structure of algebraic expressions based on conventional operation hierarchies. The transformational component specifies the real number properties used to transform algebraic expressions into equivalent forms. In standard models of mathematical cognition, both components are conceived as strictly propositional domains (e.g., Anderson, 1983a). Kirshner (1989b) demonstrated that parsing knowledge initially is apprehended in a visual (non-propositional) modality. This dissertation extends that visual analysis beyond parsing to the transformational component. It is proposed that transformations range on a continuum from highly propositional (e.g., x(y$\sp{-1}) = ({\rm x \over y}$, x$\sp2$ - y$\sp2$ = (x - y)(x + y)) to highly visual (e.g., (x$\sp{\rm y})\sp{\rm z}$ = x$\sp{\rm yz}$, (xy)$\sp{\rm n}$ = x$\sp{\rm n}$y$\sp{\rm n}$. It is hypothesized that visual rules are: (1) easier to apprehend initially, but (2) less easily constrained to their proper contexts of application. Thus common errors like ${a+b\over a+c} = {b\over c}$ and ${\root n\of {a+b}} = {\root n\of a} + {\root n \of b}$ are analyzed as overgeneralizations of visual rules like ${ab \over ac} = {b\over c}$ and ${\root n \of {AB}}$ = ${\root n\of a}{\root n\of b}$, respectively. Two groups of algebra neophytes were taught a mixture of visual and nonvisual rules. One group was taught using ordinary algebraic notation: the other using a syntactic tree notation which distorts the visual structure of ordinary notation, forcing propositional level learning for all rules. Each group was evaluated using recognition tasks, which were applications of rules that had been taught, and rejection tasks, to which no rule applied though one rule nearly applied. Recognition tasks assess the students' initial rule acquisition. Rejection items invite overgeneralization of rules. In tree notation the two rule types were equally difficult to recognize, but in ordinary notation visual rules were significantly easier to recognize than propositional rules. For rejection tasks in tree notation, visual and propositional items were equally difficult. In ordinary notation the visual items tended to be more difficult to constrain than propositional items; though because of basement effects these differences were significant only in some cases. The visual salience construct is more fully analyzed in the Conclusions section.

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