Precalculus students' problems in understanding variables, an intervention and its effect

Date

2018-01-22

Authors

Wyeth, Margaret H.

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Abstract

The results of a qualitative analysis of 170 precalculus students' interpretations of mathematical variables, constituted the foundation for a teaching intervention in a precalculus course at the University of Victoria. Some serious misconceptions of variables were identified. The possible effects of the intervention were investigated in a retrospective analysis of students' mathematics course grades. Students' interpretations of variables were extrapolated from their written explanations of answers to three algebra questions and from interview responses (N = 17). The subjects seldom interpreted variables as representing generalized sets of numbers or as co-variants. Their interpretations of variables were context-dependent, and generally inappropriate. In simplifications and equation solving most subjects appeared to use arbitrary rules to manipulate non-numeric symbols. When forced to consider numerical interpretations many described the variables as single numbers occurring in different instances. Some subjects appeared to substitute instances of variable use for the generalized number interpretation of variables, and patterns across instances for variable change. The interpretation of the variable as a single value in multiple instances can account for responses ranging from denial that variables change values to apparently correct descriptions of variable change. Some students interpreted letters as concrete objects or as units. The intervention, which was incorporated into the researcher's precalculus course lectures, consisted of making explicit the contextual interpretations of mathematical variables as single, generalized, or co-varying numbers, and of expressions as actions or as variable objects. Student response to the intervention content was very positive. The effect of the intervention was investigated quantitatively using log-linear models of the distributions of students' precalculus grades and, more important, their subsequent calculus grades. The models controlled for student changes over time, for instructor effects, and for differences in class composition based on students' year classifications. For students continuing to calculus there was a possible association between the intervention and better calculus grades (N = 166, p = 0.0008) but the confound of year standing prevented conclusions being drawn for their precalculus grades. For the subjects who did not continue on to calculus ( N = 524), there was no association between grade distributions and the experimental and control groups.

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Keywords

Calculus, Study and teaching, Variables (Mathematics)

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