Visualizing number sequences: Secondary preservice mathematics teachers’ constructions of figurate numbers using magnetic color cubes

This study is about preservice secondary mathematics teachers’ visualization of summation formulas modeled by magnetic color cubes representations. The theoretical framework for this research draws from studies on quantitative reasoning (Smith and Thompson, 2008 and Thompson, 1995) and quantitative transformations (Schwartz, 1988). Data consist of videotaped qualitative interviews during which preservice mathematics teachers were asked to construct growing rectangles representing summation formulas. Data analysis is based on analytic induction and constant comparison methodology. Preservice teachers provided a diversity of additive and multiplicative visualizations. Results indicate that quantitative reasoning and mapping structures are fundamental constructs in establishing additive and multiplicative visualizations, hence constructing summation formulas meaningfully. Preservice teachers often had difficulties in explaining the relationships between the same-valued linear and areal quantities. They also established the rectangle condition as the essence of multiplicative visualization.