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The Role of the Proof in Math
The notion of proof has long played a key role in the study of mathematics. It is in my opinion the role of proof that separates mathematics from the sciences and other fields of study. It is the existence of proofs that give mathematicians the confidence that their work is credible and thus allows them to continue to build upon prior work without the need to second guess what has previously been accomplished.
Based upon this observation, it becomes natural to ask the questions pertaining to the use of proof in learning and understanding mathematics. If the concept of proof is so important to the field of mathematics, then is it possible that by writing proofs and studying proofs that an individual will be better equipped to understand the mathematics for which the proofs pertain? And if this is possible then when should a person be first exposed to proofs and at what level? In this paper I will give my views pertaining to these questions, as well as, a few more of my views pertaining to some other topics related to these questions.
Before discussing the virtues of proofs as a means of learning and understanding mathematics, I feel that it is first necessary to begin with a brief discussion of the functions of proof within mathematics. Following I will give a list of the functions of proof that I have comprised from three sources (Hanna , Knuth , Tucker ):
1.verification, the act of arguing that a statement is true
2.explanation,providing reasons for why a statement is true, which in turn
may lead to understanding
3.systematization,organizing statements and definitions into a system of
axioms, lemmas, theorems, etc.
4.discovery,creating knowledge and new results
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...ducation, V178 N1, pp. 35-45
Hanna, Gila (2000), “Proof, Explanation and Exploration: An Overview,” Educational Studies in Mathematics, V44, pp. 5-23
Knuth, Eric (2002), “Secondary School Mathematics Teachers Conceptions of Proof,” Journal for Research in Mathematics Education, V33, pp.379-405
Lester, Frank K. (1975), “Developmental Aspects of Children’s Ability to Understand Mathematical Proof,” Journal for Research in Mathematics Education, V6 N1, pp. 14-25
Selden, Annie and Selden, John (2003), “Validations of Proofs Considered as Texts: Can Undergraduates Tell Whether an Argument Proves a Theorem?,” Journal for Research in Mathematics Education, V34 N1, pp. 4-36
Tucker, Thomas (1999), “On the Role of Proof in Calculus Courses,” Contemporary Issues in Mathematics Education, MSRI Publications, Cambridge University Press, Cambridge, UK
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