Linear algebra: Challenging problems for students
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The study was conducted with mathematics teacher-candidates. The responses of these students to four problems and interview data conducted with eight students were used to identify a student's way of thinking.
During qualitative analysis of the responses, Sierpinska's framework for thinking modes was adopted by the researcher. Based on the author's ten years' experience in teaching and research, it presents problems of varying difficulty--ranging from elementary to some which may baffle even professional mathematicians.
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Our approach to whole class discussion relies on both the students and the instructor playing active roles. The Typical Day page provides five goals for productive whole class discussions and detailed suggestions of how to foster it. A central tenet of RME is that mathematics is first and foremost a human activity.
The notion of guided reinvention emphasizes the active role an instructor plays in utilizing student ideas and justifications to move forward the mathematical development of the class. The notion of emergent models emphasizes that classroom endeavors should support students in developing models of their mathematical activity that can in turn be used as models for subsequent mathematical activity.
In keeping with these heuristics, we have developed task sequences that are based on realistic starting points and are designed to support students in making progress toward a set of mathematical learning goals.
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Research-Based Curriculum These units are a product of our research over the last decade in the teaching and learning of linear algebra, which has been grounded in the design-based research paradigm of classroom-based teaching experiments [9,10].