As mentioned in the introduction, the value of this 4-stage model of mathematical learning is that it can be used as a guide to implementing reform methods and curriculum. For example, we can use this model to describe and explore the role of the teacher in a reformed mathematics course.
To begin with, synthesis is a creative act, and thus, not all students will be able to synthesize with a given concept. Moreover, appropriate allegories are based on a student’s cultural background, and as a result, new allegories must be developed continually. Finally, some concepts require more allegorization, integration, and analysis than others. Simply put, this model does not allow us to reduce mathematical learning to an automated process with 4 regimented steps.
As a result, there must be an intermediary—i.e., a teacher—who develops allegories for the students, who determines how much allegorization, integration, and analysis should be used in presenting a concept, and who insures that students learn to think critically about each concept. And once students can think critically, the teacher will need to synthesize for many of the students by presenting problem-solving strategies and creating new allegories.
To be more specific, this model suggests the following roles for the teacher in each of the 4 stages of concept acquisition:
- Allegorization: Teacher is a storyteller.
- Integration: Teacher is a guide
- Analysis: Teacher is an expert
- Synthesis: Teacher is a coach.
Students who have talent are too often bored or even stifled in our educational system. If we accept that a coach is someone who applies discipline and structure to creativity, then clearly these are students who need to be coached. In particular, teachers need to insure that synthesizers realize that there is creativity in mathematics, and they need to show that such creativity is both enjoyable and rewarding.
Reference: Knisley J, A Four-Stage Model of Mathematical Learning, Department of Mathematics, East Tennessee State University, [Online] Available form: http://faculty.etsu.edu/knisleyj/calculus/learn.htm