Visualizing number sequences: Secondary preservice mathematics teachers’ constructions of figurate numbers using magnetic color cubes |
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| Institution: | 1. University of Alicante, Spain;2. University of Leuven, Belgium;1. School of Kinesiology, Auburn University, 301 Wire Road, Auburn, AL 36849, USA;2. Maryland Exercise and Robotics Center of Excellence, Veterans’ Health Administration, 10 North Greene Street, Baltimore, MD 21201, USA;3. Department of Psychology, University of Maryland, 1147 Biology/Psychology Building, College Park, MD 20742, USA |
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| Abstract: | 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, 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. |
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| Keywords: | Summation formula Sequence Figurate numbers Representation Connection Visualization Quantitative reasoning Mapping structures Rectangle Area |
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