## LSU Historical Dissertations and Theses

1994

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Educational Theory, Policy, and Practice

David Kirshner

#### Abstract

The purposes of this study were to investigate theories that explain why common errors of the type ($a \pm b)\sp{c} = a\sp{c} \pm b\sp{c}$ and $\root c \of {a \pm b} = \root c \of {a} \pm \root c \of {b}$ occur in algebra problem solving by novices; and to develop and assess techniques for remediating these errors. The meaning theory of learning (ML), procedural learning theory (PL), and implicit structure learning theory (ISL) are alternative frameworks for the explanation of the errors. The ML theory hypothesizes that experts have rich semantic connections to the procedures and symbols of algebra, but novices lack such connections (Ausubel, Novak & Hanesian, 1978; Brownell, 1947; Wearne & Hiebert, 1985). The PL theory hypothesizes that adept problem solvers have technical proficiency in memorizing and applying mechanical rules (Anderson, 1983; Lewis, Milson, & Anderson, 1987; Matz, 1980). The ISL theory hypothesizes that students enter the classroom with nascent abstract rule structures on which to build a more mature "grammar of algebra" through inductive processes (Bolio, 1989; Drouhard, 1988; Kirshner, 1987). In order to obtain some measure of the relative efficacy of these theories for remedial purposes, three brief educational treatments have been designed to reflect the three frameworks for learning. An analysis of variance for repeated measures was used to assess the effectiveness of the treatments in reducing the occurrences of the ($a \pm b)\sp{c} = a\sp{c} \pm b\sp{c}$ and $\root c\of {a \pm b} = \root c \of {a} \pm \root c\of {b}$ errors. Forty students participated in the study. They were enrolled in four intact developmental intermediate algebra classes at Southern University in Baton Rouge. The study used a pretest-posttest-retention test, control group design with three treatments--ML, PL, and ISL--and one control (C) which receives no special instruction concerning the errors. Results indicate that no significant difference was found in the number of errors between the groups on the post and retention tests. However, there was a significant difference between the mean scores of the pretest and the posttest. These results do not provide support for one theory over another in reducing the error types mentioned above, but do indicate a small decrease in the error rate for distributivity overgeneralization for all treatment groups.

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