This article is composed of two parts. The first part of the article delves into Yehuda Rav's paper, "Why do we prove theorems?" According to Rav, who completed his undergraduate and graduate studies at Columbia University, proofs can be used as "methods, tools, strategies and concepts for solving problems." Rav identifies (or differentiates) proofs to be either derivation proofs("application of rules of logical inference") or conceptual proofs. A conceptual proof "does not have a precise mathematical definition, mathematicians would readily understand its overall structure and could verify the correctness."
The second part of the article discusses about proofs as being expressions of mathematical knowledge. Numerous aspects of proofs, such as cognitive, intuitive, reasoning, logical, etc., are identified by Hanna and Barbeau in order to teach mathematical methods and strategies by mathematics educators. Several examples are provided as case studies to show how proofs may be incorporated into a secondary level for improving problem solving techniques. The authors acknowledge that their outlines (suggestions) were not tested in actual classrooms.
The authors of this article seem to approach the issue of proofs from a theoretical point of view, which may be in line with Kilpatrick's line of thought for an effective research program.
The second part of the article discusses about proofs as being expressions of mathematical knowledge. Numerous aspects of proofs, such as cognitive, intuitive, reasoning, logical, etc., are identified by Hanna and Barbeau in order to teach mathematical methods and strategies by mathematics educators. Several examples are provided as case studies to show how proofs may be incorporated into a secondary level for improving problem solving techniques. The authors acknowledge that their outlines (suggestions) were not tested in actual classrooms.
The authors of this article seem to approach the issue of proofs from a theoretical point of view, which may be in line with Kilpatrick's line of thought for an effective research program.
"Proofs as being expressions of mathematical knowledge" gives rise to the image in my mind of a dictionary for mathematics. Our English dictionary is full of definitions, and relations, and so too would the mathematical one, except that its words would be proofs.
ReplyDeleteI think that many high school students are burdened with the thought that the mathematics that they practice in school is far-removed from what one might practice in 'every day' life, but also very far-removed from what actual mathematicians might practice. Hanna and Barbeau's want for incorporating proofs in secondary levels could help alleviate that while also, as they suggest, helping to improve problem-solving techniques
The concept of proofs being expressions of mathematical knowledge makes a lot of sense to me, as does Conrad's analogy to a dictionary for mathematics. I have always thought of proofs as having to be complicated and by introducing the concept earlier, it becomes more accessible and less daunting for students. After this weeks readings, I feel much more at ease with the application of proofs and I would love to see Hanna and Barbeau do some field research on the suggested use of proofs for problem solving classrooms, I believe the results would be fascinating.
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