CARMA Seminar

"On simple groups acting on trees"
Mr Chris Banks

Location: V205 Mathematics Building
Date: 4:00 pm, Wed, 23rd Apr 2014

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Selected quotation

"Should we teach mathematical proofs in the high school? In my opinion, the answer is yes...Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics' contribution to general culture." George Polya (1981). Mathematical discovery: On understanding, learning, and teaching problem solving (Combined Edition), New York, Wiley & Sons (p. 2-126) "A mathematical deduction appears to Descartes as a chain of conclusions, a sequence of successive steps. What is needed for the validity of deduction is intuitive insight at each step which shows that the conclusion attained by that step evidently flows and necessarily follows from formerly acquired knowledge (acquired directly by intuition or indirectly by previous steps) ... I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning." (ibid, p. 2-128) This "quasi-experimental" approach to proof can help to de-emphasis a focus on rigor and formality for its own sake, and to instead support the view expressed by Hadamard when he stated "The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it" (J. Hadamard, in E. Borel, Lecons sur la theorie des fonctions, 3rd ed. 1928, quoted in Polya, (1981), (p. 2/127). "intuition comes to us much earlier and with much less outside influence than formal arguments which we cannot really understand unless we have reached a relatively high level of logical experience and sophistication. Therefore, I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning." (ibid, p. 2-128)

"In the first place, the beginner must be convinced that proofs deserve to be studied, that they have a purpose, that they are interesting." (ibid, p. 2-128)

"The purpose of a legal proof is to remove a doubt, but this is also the most obvious and natural purpose of a mathematical proof. We are in doubt about a clearly stated mathematical assertion, we do not know whether it is true or false. Then we have a problem: to remove the doubt, we should either prove that assertion or disprove it." (ibid, p. 2-129)

(Polya quotes are thanks to Laurie Edwards)


Membership to CARMA offers many benefits and is available by invitation to all University of Newcastle academic staff. Associate membership, also by invitation, is available to external researchers and practitioners for three-year renewable terms. Associate members are expected to visit CARMA with some frequency, typically for a total of three to four weeks in a year, and to be involved in one or more ongoing research projects with CARMA members. CARMA is able to assist with the travel and living costs of such visits.