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The relationship between students' mathematical thinking types and representation preferences in definite integral problems

Students’ cognitive differences in problem solving have been the focus of much research. One classification of these differences is related to whether visualisation is used. Krutetskii (1976) categorised mathematical thinking into three different types. In addition to the analytic and geometric preferences, he drew attention to the existence of harmonic processes, which use both analytic and geometric reasoning. Presmeg (1985), in her study on the visualisation process and learner difficulties, used Krutetskii’s (1976) classification, and concluded that students struggled more in managing visual processes than in managing analytic ones. Like mathematical thinking differences, multiple representation preferences are important when considering individual differences. Choosing an appropriate representation is an important step to successful problem solving. Sevimli and Delice (2011) claimed that students have difficulties when choosing an appropriate representation. The reason for these difficulties is the differences in the students’ thinking. Therefore, it is necessary to explore the relationship between preferred representation types and differences in mathematical problem-solving processes. The main research question of this study is related to the effect of input representations in the problem statement on students’ choices. This study identifies students’ representation preferences. The participants were 37 first year mathematics education students, who were selected by using a purposeful sampling technique. In this research, two different tests were used for two different purposes. The Mathematical Process Instrument (MPI) developed by Presmeg (1985) was used to determine the type of the participants’ mathematical thinking: visual, analytic or harmonic. The second data collection tool was a Representation Preferences Test (RPT) which was developed by the researchers. The RPT was designed to determine participants’ tendencies to use different representations for the definite integral. The test consisted of nine items, each of which stated a question in one of three representations (numerical, graphical or algebraic). The participants were expected to identify the representation type(s) which they believed would facilitate the process of solving a given definite integral problem. When tests were administered, participants were not allowed to make any calculations. Hence, the RPT provided a measure of students’ representation preferences. Each participant’s representation preferences for each question were