Anyway I am quite surprised and a little shocked to find a Scheme programmer caring about concrete syntax :-)
That's an excellent point about subjectivity, John! I spent enough time in the Scheme culture to easily read Scheme code, with all the ridiculous extra parentheses and prefix notation, and I also spent enough time in the math culture to want to see code like a,b ≡ p,q ∧ p,q ≡ r,s ⇒ a,b ≡ r,s It's subjective, not objective, but I'm really happy you're with me on the objective point: | And there's no Post Modernist talk about how proofs are social | constructs, it's up to every community to decide what a proof is. Yes, this is one of the very appealing things about formalization, that the question of the correctness of a proof becomes an objective one. --- I found the paper online and I'll be interested to see the details. Excellent, John. I wrote the to the middle school math coordinator: Ms. Farrand, here's a longer letter about Stylianides's article The Notion of Proof in the Context of Elementary School Mathematics, published in Educational Studies in Mathematics, 2007 http://www.jstor.org/stable/27822668 . Stylianides makes this astonishing claim about a nice proof that a third-grade girl gave of odd + odd = even: p 15: First, Betsy's argument cannot count as proof, because it was not accepted as such by the classroom community. The only sensible action would have been for (p. 16) the teacher [the author's advisor Deborah Ball] to ratify Betsy's argument as a proof at the end of the episode, explaining to the students why this argument qualified as proof I give evidence now that the author does not explain clearly what a proof is or why they're important. p 1: Proofs should be incorporated into the mathematical experiences of even elementary students (e.g., Ball and Bass...) Hyman Bass is a good mathematician, so we should take this seriously. The article says nothing about why we should teach proofs. I say we should teach real proofs because math is all about proofs, and some students (like my son) refuse to memorize stuff that hasn't been explained, and require real proofs to continue. p 2: students' transition from elementary to secondary school mathematics is abrupt and is linked to a 'didactical break' represented by the introduction of the new requirement for proof. I say this is irrelevant. I taught Calculus, MV Calculus, ODE & Linear Algebra for many years at universities. And I say high school Geometry is the one math class ever get where they must understand mathematical proofs. I was told over & over at universities, ``Don't give proofs. If you're at the blackboard writing down a proof, you're just doing it for yourself. Nobody's listening.'' p 4: Most students first --- and sometimes only --- encounter proof in high school courses on Euclidean geometry, and, when this happens, proof seems alien and unfamiliar to them... Here the author agrees with my point: proofs are not an essential part of math education. So why is he writing about proofs at all? Section 1 ends with the misleading claim Furthermore, the process of accepting an argument as proof relies heavily on the social mechanisms of the mathematical community (cf. social dimension) (e.g., Ernest, 1998; Tymoczko, 1986/1998). Mathematicians are in complete agreement about what a proof is. But proofs are often extremely complicated and skip many steps. So `accepting an argument as proof' then turns into a political circus. The author repeats versions of his misleading claim many times. So let me explain more. All mathematicians agree that a mathematical proof is something that could eventually be written as an axiomatic proof using the accepted set theory axioms (ZFC: the Zermelo-Frankel axioms plus the axiom of choice). These axiomatic proofs could then be checked for correctness by a computer program. These axiomatic proofs are very similar to the rigorous Hilbert proofs that my son learned. The ETHS book follows standard practice by giving ``picture-proofs'' instead of axiomatic proofs, and the ETHS axioms are not strong enough to prove much. In practice, mathematicians never give axiomatic set theory proofs, as they would be too complicated, and they often give ``picture-proofs''. So my objection to the ETHS course isn't ``picture-proofs'' per se, but that the students are not being taught what an axiomatic proof is, nor what axioms they are actually using. Most of the students wouldn't even care, as this course is their one exposure to proofs, so the less proofs the better. Some students however will want real proofs like the Hilbert proofs my son learned. A mathematical community can work faster and define `hot areas' by agreeing to skip steps. This is a dangerous practice, and has nothing to do with teaching children. None of this `knowledge construction' stuff has any relevance to teaching junior level math classes for college math majors. That's the important point, because I've taught my son a great deal of Algebra & Geometry from junior level math courses. -- Best, Bill ------------------------------------------------------------------------------ Live Security Virtual Conference Exclusive live event will cover all the ways today's security and threat landscape has changed and how IT managers can respond. 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