Connecting proportionality and slope: middle school students' reasoning about steepness
Cheng, Diana S.
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This study investigates the relationship between middle school students' proportional reasoning abilities and their understanding of steepness, since steepness may be a key developmental understanding that students need in order to understand slope in algebra. This study uses mixed methods, with a large-scale survey and one-on-one interviews with students. The large-scale survey involves two tests: an adapted version of the Ratio and Proportion Test administered in England for the Concepts in Secondary Mathematics & Science (CSMS) and Increasing Competence & Confidence in Algebra and Multiplicative Structures (ICCAMS) projects, and a steepness test which I created and for which I established content validity and test-retest reliability. The problems on the steepness test involve a comparison of two roofs, two staircases, or two lines. The problems vary by structural difficulty based upon the values of the slopes involved. Analysis of 413 research participants' survey data indicate that approximately 25% of the variability in students' scores on the Steepness test is explained by their performance on the Ratio and Proportion Test. Linear regression has shown that there is a positive correlation between participants' proportional reasoning abilities and their abilities to solve steepness problems. Using paired t-tests, there is evidence of significant difference in students' performance on the three contexts. The difficulties of contexts in the order of difficulty from hardest to easiest are: stairs, roofs, lines. Analyses of interviews with middle school students solving steepness problems indicate that they do use different strategies to solve problems based upon their proportional reasoning levels. Participants who attained higher levels of proportional reasoning used quantitative considerations of two measurements, norming, and rates and ratios to determine relative steepness more frequently than participants who attained lower levels of proportional reasoning. It was concluded that participants' abilities to solve steepness problems are related to their abilities to reason proportionally. The findings of this research contribute to literature on early algebraic reasoning that explores ways in which algebraic topics such as slope can be made accessible to students prior to their formal studies of algebra.
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