As a contemporary mathematics education researcher, Richard Lesh is know for describing what has been known as models and modeling perspectives in regard to mathematical problem solving, learning, and teaching (Lesh & Doerr, 2003). Models are defined as “purposeful mathematical descriptions of situations, embedded within particular systems of practice that feature an epistemology of model fit and revision” (Lesh & Lehrer, 2003). What modeling involves is a series of tests for fitness on models developed by the students as they think mathematically about a presented problem situation. This is all drawn from the work of other cognitive theorists (Dienes and Vygotsky included) who believe that we learn by interpreting our experiences. Lesh suggests that students go beyond the surface of their experiences to organize and transform information as well as to look for patterns in their experiences in order to predict (Lesh & Doerr, 2003).
While some theorists treat the ability of students to interpret or describe problem solving situations as already existing in the minds of the learner, Lesh’s notion of models and modeling perspectives realizes that students must be exposed to a variety of ways to communicate the mathematics being explored (Cramer, 2003). Included in the forms of communication are the spoken and written language, symbols, diagrams, metaphors, and computer-based simulations (Cramer, 2003; Johnson & Lesh, 2003). This is also related to what Wertsch (1985) described as “mediated activity” as an extension of Vygotsky’s social formation of learning which has become an important component of learning as different forms of media will emphasize different aspects of a problem situation and the conceptual systems within the problem.
The rationale behind the work of Lesh is the need to prepare students so that they have the mathematical skills required by business and other facets of society. What Lesh has witnessed is that the skills now needed have changed for many reasons, one of which is the transformation of information technology. According to Lesh, creating and using models is an important goal for instruction as students interpret experiences (Lesh & Doerr, 2003). What we need are students that can communicate and generate problem-solving options. A combination of mathematical computation through learning and practice along with applications of those models should be the goal of educators to improve the understanding of their students.
Lesh recommends a case study approach in instruction that provides simulations of “real life” problem-solving situations (Lesh & Zawojewski, 2007). He suggests that multiple approaches to the same problem be encouraged so that students learn to develop and communicate models for solving problems. Lesh is a proponent of teachers advocating ten big ideas within a course that the instructor would like for students to understand through problem solving.
It is not with too much of an assumption in...