Research in mathematics education has been no less productive, but implementation of that research often leads to difficult questions such as “how much technology is appropriate,” and “in which situations is a given teaching method most effective.” In response, this paper combines personal observations and education research into a model of mathematical learning. The result is in the spirit of the models mentioned above, in that it can be used to guide the development of curricular and instructional reform.
Before presenting this model, however, let me offer this qualifier. Good teaching begins with a genuine concern for students and an enthusiasm for the subject. Any benefits derived from this model are in addition to that concern and enthusiasm, for I believe that nothing can ever or should ever replace the invaluable and mutually beneficial teacher-student relationship.
This section briefly reviews the research results in mathematics education and applied psychology that most apply to this paper. This is far from exhaustive and no effort is made to justify the conclusions in this section. Interested readers are referred to the references for more information.
Decades of research in education suggest that students utilize individual learning styles (Felder, 1996). Instruction should therefore be multifaceted to accommodate the variety of learning styles. The literature in support of this assertion is vast and includes textbooks, learning style inventories, and resources for classroom implementation (e.g., Dunn and Dunn, 1993).
Moreover, decades of research in applied psychology suggest that problem solving is best accomplished with a strategy-building approach. Indeed, studies of individual differences in skill acquisition that suggest that the fastest learners are those who develop strategies for concept formation (Eyring, Johnson, and Francis, 1993). Thus, any model of mathematical learning must include strategy building as a learning style.
This results in 4 types of learners:
· Concrete, reflective: Those who build on previous experience.
· Concrete, active: Those who learn by trial and error.
· Abstract, reflective: Those who learn from detailed explanations.
· Abstract, active: Those who learn by developing individual strategies
Although other models also apply to mathematics, there is evidence that differentiating into learning styles may be more important than the individual style descriptions themselves (Felder, 1996).
Finally, let us label and describe the undesirable “memorize and regurgitate” method of learning. Heuristic reasoning is a thought process in which a set of patterns and their associated actions are memorized, so that when a new concept is introduced, the closest pattern determines the action taken (Pearl, 1984). Unfortunately, the criteria used to determine closeness are often inappropriate and frequently lead to incorrect results.
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.
Space does not permit me to elaborate on each role, but let me point out one that I feel should not be neglected. 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.
Sumber: Jeff Knisley,Department of Mathematics,East Tennessee StateUniversity
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