Proportionality and proportional reasoning play pivotal roles in the foundation of algebra and higher-level mathematics study. Proportionality is a mathematical structure that models the relationship within contextual situations in which two quantities, x and y, change together in ways that the rate between the quantities stays the same, such as speed or density. Proportional reasoning involves the psychological underpinnings that facilitate the interpretation, sense making, and operational flexibility necessary for working with proportion related situations. The development of these understandings and reasoning processes is both mathematically and psychologically complex. Although there has been much research surrounding the ways children come to understand proportionality and reason proportionally (e.g. Lamon, 2007; Lesh, et al., 1987; Lobato et al. 2010; Post et al., 1988), there is a need for research into the ways that these concepts and reasoning processes emerge in older students and adults (e.g. Lamon, 2007; Mesa, Wladis, & Watkins, 2014; Sitomer et al., 2012). This study explored the relationships between understandings of proportionality and proportional reasoning processes in community college mathematics students, and the teaching and learning activities that support their construction in post-secondary developmental mathematics students. The study employed design experiment methodology that included two two-week teaching experiments (Cobb et al., 2003; Cobb & Steffe, 1983/2011; Gravemeijer & van Eerde, 2009). The findings showed that the understanding and interpretation of rate relationships are central to a connected understanding of proportionality and flexible proportional reasoning processes. This key understanding was characteristic of college-level mathematics students, and successfully constructed by developmental mathematics students through the teaching experiment. The interpretation of a y = mx functional relationship in proportional contexts served to stabilize the understandings and reasoning processes of developmental students and facilitated reasoning processes similar to those of college algebra students. These results provide evidence that a non-traditional approach to the treatment of proportionality in developmental mathematics contexts can effectively build connected and meaningful understandings that will support student success in college-level mathematics courses. The two teaching experiments allowed for observation based modifications to a Hypothetical Learning Trajectory (Simon, 1995) consistent with the tenants of design study.
University of Minnesota Ph.D. dissertation.May 2015. Major: Education, Curriculum and Instruction. Advisors: Kathleen Cramer, Thomas Post. 1 computer file (PDF); xi, 286 pages.
Understandings of Proportionality as a Mathematical Structure and Psychological Aspects of Proportional Reasoning in Community College Mathematics Students.
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