## 2007 Conference Proceedings

Proceeding

#### Abstract of proposed session

In this article, I write about my research on five preservice secondary teachers’ (PST) understanding and sense making of representational quantities associated with magnetic color cubes and tiles. Data came from individual interviews during which I asked PST problems guided by five main tasks: prime and composite numbers, summation of counting numbers, odd numbers, even numbers, and polynomial expressions in x and y. My work drew upon an analysis framework (Behr et. al, 1994) supported by a unit coordination construct (Steffe, 1988) associated with linear and areal quantities inherent in the nature of figures produced by these PST. Linear quantities can be thought of as generated via linear measurement units (e.g., inches, centimeters, units) whereas areal quantities are generated via areal measurement units (e.g., square inches, square centimeters, square units, etc.) I used thematic analysis supported by constant comparison and retrospective analysis to explain my theories and hypotheses concerning PST’s representational quantities. I developed a data analysis framework which I named “Relational Notation” to describe these PST’s understanding of linear and areal units. PST also treated the quantitative multiplication and addition operations as some kind of functions, mappings, when expressing the area of their growing rectangles made of magnetic color cubes and tiles as sums and products. Their behavior necessitated the existence of another component for my data analysis framework which I called “Mapping Structures”

#### Keywords

Pre-service secondary teachers, Representational quantities, Linear quantities, Linear measurement units

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Paper 1: Preservice Teachers’ Mapping Structures Acting on Representational Quantities

In this article, I write about my research on five preservice secondary teachers’ (PST) understanding and sense making of representational quantities associated with magnetic color cubes and tiles. Data came from individual interviews during which I asked PST problems guided by five main tasks: prime and composite numbers, summation of counting numbers, odd numbers, even numbers, and polynomial expressions in x and y. My work drew upon an analysis framework (Behr et. al, 1994) supported by a unit coordination construct (Steffe, 1988) associated with linear and areal quantities inherent in the nature of figures produced by these PST. Linear quantities can be thought of as generated via linear measurement units (e.g., inches, centimeters, units) whereas areal quantities are generated via areal measurement units (e.g., square inches, square centimeters, square units, etc.) I used thematic analysis supported by constant comparison and retrospective analysis to explain my theories and hypotheses concerning PST’s representational quantities. I developed a data analysis framework which I named “Relational Notation” to describe these PST’s understanding of linear and areal units. PST also treated the quantitative multiplication and addition operations as some kind of functions, mappings, when expressing the area of their growing rectangles made of magnetic color cubes and tiles as sums and products. Their behavior necessitated the existence of another component for my data analysis framework which I called “Mapping Structures”