1. Kaput (in press) Representations, Inscriptions, Descriptions and Learning: A Kaleidoscope of Windows Theory of Representations Mathematics as a Representational Domain Intro to SimCalc An Example from SimCalc

2. Representations Three parts to a representation –What is being represented (the velocity of a car) –The physical representation (the scribbles, pictures, graphs, etc) –The person who reads (and makes meaning from) the representation Representation seems to be mostly a function of humans (and probably other animals). Manipulating them is a large part of human cognition. Example: Language

3. Mathematics and Representation Mathematics is, in part, a representational language. –E.g., “3” stands for the number of some objects, a position on a number line, and an element of a number set (the integers). Mathematics typically emphasizes three main types of representational systems –Numerical (3,5,30.65, 45%, 2/3, etc) –Graphical (lines, pie, functions, etc). –String (e.g., 2x + 3y = 24)

4. SimCalc - Linking Representations with each other and with phenomena SimCalc is one approach to learning mathematics. However, it seeks to go beyond helping students manipulate symbols in the “big three” (numeric, graphic, string) –Putting phenomena and situations first. Recalling that the other three representations must be representing something, but what they represent is often hidden from students. –Making distinctions between representations (graphs, numbers, charts), and descriptions (e.g. the velocity or position of a car). –Links between representations, descriptions, and phenomena

5. Example Situation / Phenomena –Elevator going up and down. Represented by animated icons on the screen Descriptions –Rate of elevator –Position of elevator –Acceleration Representations (for each description) –Tables –Graphs –Animated Elevators