Learning Algebra with Models and Reasoning

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.