Michelle Manes (email@example.com) wrote: > ... > High school (maybe even middle school) is a good time to start > formalizing the argument process, but I would argue that the > two column proof is not the way to go... It doesn't really lead to > andy of the 3 positive aspects of proof, and it's not the way > "real mathematicians" write proofs. > > Two column proofs are fairly sterile, they leave little room for > creativity (and hence the joy that comes from problem solving). > Often students are simply asked to prove a result -- so they > already know it's true because they're beings asked to prove it. > A better way is to have them *find* a result through experimentation, > and then want to prove it because they're already convinced > themselves that it's true. And two-column proofs rarely lead > directly to another result.
I think the two-column proof was created to emphasize the need to justify (or be prepared to justify) every assertion and show a logical *train* of thought.
A problem I see with teaching most young people the two-column method is that they would interpret it too literally to understand what they are doing. They would think that the most important part of proving is filling in two columns with stuff. Some would think that using two-columns is the only valid way of *finding* a proof. This error could be addressed by the teacher. Still I would expect some students not to understand. They would miss that the chief work in finding a proof is rooting around, trying to find connections via scratchwork and experimentation. They would not view the two-column write-up as a means of intelligibly communicating their proof.
As for how matheticians write proofs: Brevity is the soul of the mathematics' publishing industry. But I think most mathematicians still operate in a bimodal (assertion/justification) way. If a justification is too obscure or is important, it is included although not in a separate column ("By Theorem 3.14, the circumference...", or "...which can be deduced from Gauss' Lemma"). We also operate in this way when we read mathematics. If a justification is not supplied in the text, we try to supply it ourselves. We fill in the second column so to speak.
Finally I would argue that two column proofs, being essentially the same as proofs, do lead to another result (at least not rarely). I do believe that they rarely lead young people to another result. Those students tend to be too worried about the trees to be able to see their way out of one forest and into the next.
From archive Thu Feb 25 10:02 PS 1993 In-reply-to: <firstname.lastname@example.org> Subject: Re: Two-column Proofs From: Keith Grove - Dover Sherborn HS <email@example.com> Date: Feb 25 10:02 PS 1993 Message-ID: <firstname.lastname@example.org> Epigone-indent: 1
I am a student of Dover-Sherborn high, and a member of Mister Groves sophmore geometry class and we have done equal numbers of two column proofs and paragraph style proofs, and I have found paragraph style the most usefull. The paragraph style encourages us to write down all we know, but the two column proof only needs or asks for a certain thing. The paragraph lets you write something not so specific but will help you along in your completion of the paragraph style proof. Also the paragraph style will make all those insignifigant (supposedly) in your head, and when the next proof comes you will remember them, when they are more important than they were in the last proof. As a matter of fact the two column proof is not letting me retain all those "insignifigant" facts. If right now you asked me to do a proof I did not know how to do, but it involved basics of triangles, rectangles and what not, and you asked me to do it in paragraph. If you asked me to do it in two columns I probably wouldn't even get the first step let alone completion.