On the Use of Technology to Represent and Explore Mathematical Objects or Problems Dynamically
Santos-Trigo, Manuel, Mathematics and Computer Education
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Questions and answers are more important than a professor's repetition of what can be found in books or what can be put in lecture notes, duplicated, and passed out to students [4, p. 97].
The development of mathematical ideas can be traced and explained in terms of the type of representations used to deal with problems or mathematical objects. Indeed, "much of the history of mathematics is about creating and refining representational systems, and much of the teaching of mathematics is about students learning to work with them and solve problems with them" (Lesh, Landau, & Hamilton, 1993, cited in Goldin, & Shteingold, 2001, p 4). Similarly, Schoenfeld (1985) suggests that students' construction or understanding of mathematical ideas and their ways to solve problems depend on the types of representations they use to recognize and examine mathematical relations. For example, the number 46656 can be represented as 2^sup 6^ × 3^sup 6^, 36^sup 3^, 216^sup 2^ and, 12 × 888 + 12 × 3000. Each representation makes visible mathematical properties that involve prime factorization, perfect cube, perfect square, or multiple of 12. Hence, the significant development and use of computational tools, such as Computer Algebra Systems (CAS), dynamic software, and spreadsheets may facilitate students' ways to represent and operate mathematical objects embedded in problems. As a result, the use of the tools can help students identify and examine mathematical properties and diverse representations of the same object or problem.
How can students use particular tools to represent and solve mathematical problems? What types of representations of problems may be useful for students to examine mathematical relations? To what extent does the students' use of a particular tool shape and favor a particular way of thinking about problems? Discussing these type of questions may shed light on the extent to which students' use of dynamic geometry software, like Cabri or Geometer's Sketchpad, helps them identify and explore mathematical relations or conjectures.
A relevant feature in using dynamic software is that in order for students to represent mathematical objects or problems, they need to think of the problems in terms of mathematical properties. For instance, to draw a rectangle, students need to identify properties associated with this figure, such as pairs of parallel or perpendicular sides to help them choose proper commands to represent the figure. With the use of the software, students can also build dynamic geometric configurations formed by simple figures (lines, segments, triangles, perpendicular bisectors, etc.) that may be used as points of departure to identify mathematical conjectures or relationships.
From this perspective, dynamic representations of problems seem to be an important tool that allows students to reconstruct or develop some mathematical relations. That is, by moving objects within the representations, students may have the opportunity to observe relations that lead them to make conjectures and to present various arguments to support them. Presenting an argument is a way for students to remove or overcome doubts about the validity of a conjecture. Indeed, "a person ceases to consider an assertion to be a conjecture and views it to be a fact once the person becomes certain of its truth" (Harel, in press).
How can students construct a particular geometric configuration that can be used as a starting point to formulate conjectures or mathematical relationships? There may be different ways for students to assemble a particular configuration that lead them to identify relationships. For instance, students can start by drawing some initial objects, like points, segments, lines, triangles, rectangles, circles, etc., and then use them to build a configuration. Throughout the construction of that configuration, students can direct their attention to the behavior of some elements while moving objects within the configuration to observe invariants, loci, or mathematical relationships. …