Learning Algebra with Models and Reasoning

Abstract

In traditional mathematics classrooms, the use of representations is often restricted to the “show your work” instruction, perhaps involving a diagram or graph. As such, models and representations may have a more limited use. Mathematicians however, often use rough diagrams, gestures, etc. to think about and discuss mathematical ideas as they evolve. Hence in a problem-based classroom, encouraging students to make use of such models and representations as tools to think with can have a powerful effect. Such a use is very different from the assumption that models, such as in the form of physical classroom manipulatives, are tools to support struggling students. Rather, this chapter takes the stance that a vigorous use of models and reasoning can be highly mathematical, and effective for all students.

Many current Canadian curricula, as well as the Principles and standards for school mathematics (NCTM, Principles and standards for school mathematics. Author, Reston, 2000), have both Representation as well as Reasoning listed as learning processes. The use of models and modelling can support both of these processes.

As well as providing some background, this chapter provides a specific classroom example, which may be useful in grade 9 or 10 classrooms. The lesson has been field tested in both grade levels, as well as in relatively more and less high level courses. Without exception, students have been able to discover the rules for factoring a simple quadratics for themselves. As well as generating a useful factoring method, students are left with a conceptual understanding of how and why the method works, which may be helpful in terms of not having to memorize a mysterious procedure.

A secondary benefit of using models in learning may be that they may be a structure that help teachers move more and more towards encouraging increased student autonomy in learning. Once students have some effective tools with which to think, they may be able to become increasingly independent as learners.