The Nature of Mathematics: A Heuristic Inquiry

摘要

What is mathematics? What does it mean to be a mathematician? What should students understand about the nature of mathematical knowledge and inquiry? Research in the field of mathematics education has found that students often have naive views about the nature of mathematics. Some believe that mathematics is a body of unchanging knowledge, a collection of arbitrary rules and procedures that must be memorized. Mathematics is seen as an impersonal and uncreative subject. To combat the naive view, we need a humanistic vision and explicit goals for what we hope students understand about the nature of mathematics. The goal of this dissertation was to begin a systematic inquiry into the nature of mathematics by identifying humanistic characteristics of mathematics that may serve as goals for student understanding, and to tell real-life stories to illuminate those characteristics. Using the methodological framework of heuristic inquiry, the researcher identified such characteristics by collaborating with a professional mathematician, by co-teaching an undergraduate transition-to-proof course, and being open to mathematics wherever it appeared in life. The results of this study are the IDEA Framework for the Nature of Pure Mathematics and ten corresponding stories that illuminate the characteristics of the framework. The IDEA framework consists of four foundational characteristics: Our mathematical ideas and practices are part of our Identity; mathematical ideas and knowledge are Dynamic and forever refined; mathematical inquiry is an emotional Exploration of ideas; and mathematical ideas and knowledge are socially vetted through Argumentation. The stories that are told to illustrate the IDEA framework capture various experiences of the researcher, from conversations with his son to emotional classroom discussions between undergraduates in a transition-to-proof course. The researcher draws several implications for teaching and research. He argues that the IDEA framework should be tested in future research for its effectiveness as an aid in designing instruction that fosters humanistic conceptions of the nature of mathematics in the minds of students. He calls for a cultural renewal of undergraduate mathematics instruction, and he questions the focus on logic and set theory within transition-to-proof courses. Some instructional alternatives are presented. The final recommendation is that nature of mathematics become a subject in its own right for both students and teachers. If students and teachers are to revise their beliefs about the nature of mathematics, then they must have the opportunities to reflect on what they believe about mathematics and be confronted with experiences that challenge those beliefs.