This study examines 8th grade students_ coordination of quantitative units arising from word problems that can be solved via a set of equations that are reducible to a single equation with a single unknown. Along with Unit-Coordination, Quantitative Unit Conservation also emerges as a necessary construct in dealing with such problems. We base our analysis within a framework of quantitative reasoning (Thompson, 1988;1989;1993;1995) and a theory of children_s units-coordination with different levels of units (Steffe 1994) that both encompass and are extended by these two constructs. Our data consist of videotaped classroom lessons, student interviews, and teacher interviews. Ongoing analyses of these data were conducted during the teaching sequence. A retrospective analysis using constant comparison methodology was then undertaken during which the classroom video, related student interviews, and teacher interviews were revisited many times in order to generate a thematic analysis. Our results indicate that the identification and coordination of the units involved in the problem situation are critical aspects of quantitative reasoning and need to be emphasized in the teachinglearning process. We also concluded that unit coordination and unit conservation are cognitive prerequisites for constructing a meaningful algebraic equation when reasoning quantitatively about a situation.

This study examines eighth grade students' use of a representational metaphor (cups and tiles) for writing and solving equations in one unknown. Within this study, we focused on the obstacles and difficulties that students experienced when using this metaphor, with particular emphasis on the operations that can be meaningfully represented through this metaphor. We base our analysis within a framework of referential relationships of meanings (Kaput 1991;Kaput, Blanton, and Moreno, et al. 2008). Our data consist of videotaped classroom lessons, student interviews, and teacher interviews. Ongoing analyses of these data were conducted during the teaching sequence. A retrospective analysis, using constant comparison methodology, was then undertaken in order to generate a thematic analysis. Our results indicate that addition and (implied) multiplication operations only are the most meaningful with these representational models. Students also very naturally came up with a notation of their own in making sense of equations involving multiplication and addition. However, only one student was able to construct a "family of meanings" when negative quantities were involved. We conclude that quantitative unit coordination and conservation are necessary constructs for overcoming the cognitive dissonance (between the two representations-drawn pictures and the algebraic equation) experienced by students and teacher.

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This study investigates prospective secondary mathematics teachers' visual representations of polynomial and rational inequalities, and graphs of exponential and logarithmic functions with GeoGebra Dynamic Software. Five prospective teachers in a university in the United States participated in this research study, which was situated within a framework of productive disposition and visual representations in pre-calculus. The main result was that the role of GeoGebra as a cognitive tool fostered the research participants' productive disposition, despite recurrent mismatches between the algebraic and visualized formalisms. Moreover, participants exhibiting dynamic productive disposition seemed to understand and make better sense of the conceptual underpinnings of the mathematical content they explored in contrast to those embracing static productive disposition.Visual representations play a crucial role in understanding and making sense of mathematics. Students' favorite mathematics textbooks and teachers are often ones that use a variety of colorful figures, diagrams, pictures, and graphs

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