Reasoning about steepness: strategies of fifth through eighth grade students when solving steepness tasks

Abstract

This study investigated fifth through eighth grade students' reasoning about steepness. The study explored whether research participants' success rate was affected by grade, numerical complexity, and structure of the steepness problem. Numerical complexity was determined by the numerical values of the rise and the run related to lines, while structure was defined by whether or not the rise-run triangle formed beneath the line with the x- and y-axes was explicit in the problem scenarios. The sample included 205 students in grades 5 through 8. As a mixed methods study, this research included both qualitative and quantitative data obtained from the research instrument, the Steepness Questionnaire. The instrument contained eleven numerical comparison problems about steepness of spiders' webs, nine of which required participants to explain their reasoning. Problem level analysis of success was performed using general estimating equations for logistic regression. Repeated measures analysis was used to determine whether numerical complexity and problem structure affected participants' success on steepness problems. Strategies were classified as Irrelevant Data, Univariate, Angular, and Multivariate. The Multivariate strategy was further divided into sub-strategies, Qualitative, Additive, and Multiplicative, which enabled the researcher to differentiate between participants' proportional and non-proportional solution methods. Logistic general estimating equations were used to investigate factors affecting frequency of use of each solution strategy. There was significant evidence that grade, numerical complexity, and problem structure affected research participants' success rate with steepness problems. There was a gradual increase in success across grade and a decrease in success with increasing complexity levels. Participants had higher success on problems where the rise-run triangle was explicit than on problems where it was non-explicit. Strategies also changed significantly across grades. Through the grades, there was a slight decrease in the use of Irrelevant Data, a substantial decrease in the Univariate strategy, and an increase in the use of the Multivariate Qualitative and the Multiplicative sub-strategies. Numerical complexity and problem structure had minimal effect on strategy use.