> > Well I've been off the list for a while, but let me weigh in a little on > proofs (2-column & otherwise), with an apology to anyone whose ideas I'm > restating. (I also can't remember if I sent a version of this yesterday -- > I dimly remember trashing it, but if not, apologize.) > > 1. 2-column proofs seem to be an American abberation. They aren't done in > Europe or the Asian part of the Pacific rim.
I have to apologize to the list that I misunderstood the "2-column proof". You are right, I had never heard of that before and thought it simply meant short proof containing mostly mathematical notation, as oppose to the kind which contains mostly the human natural language flow. During the private exchange with Howard, I have learned that and I have explained to him what was my point. Which is, in mathematics classroom, it might be an important training for student to learn how to write (and thus to think) economical, but mathematically complete, sentences and paragraphs, using, whenever it is possible, mathematical notations and follow the grammer and sentence structure of mathematical language.
I have no problem with the claim that for some (or most) students, learning to write complete sentences, either for "proof" or during problem solving, it might be a good idea to involve mostly natural language to begin with, in order to make connection. But, eventually, they should be guided to learn to wirte them in mostly mathematical language. Because that can help them "to think mathematically" in an effective way. Acually, due to the not necessarily logical nature of natural human language and its limitation in expressing the mathemaitcal concept, most of the time, a few mathematics notations and good figures can express the mathematical thinking that thousand words can't.
And my concern is there are some students who, mathematics language is kind of natural to them, do not need to go through this "beginning stage". If the assessment want to test student's understanding by asking them to articulate or express in human natural language, especially by teachers who do not understand mathematics in depth, it will turn off those math smart's interest toward math.
I am not just imagining what could happen, this is what is now happening to my own son.
> > 2. Mathematicians don't use them. They don't use flow-proofs either. > Mathematicians try to write convincing arguments, period. Diagrams are > allowed, but you have to be careful how you use them. If flow-proofs help > kids write a good argument (and I suspect they do), let's use them as a > provisional step. If flow-proofs become another formal hoop to jump > through, no thanks. As for 2-column proofs, I can't believe they help > anybody, they are so *prissy* and devoid of ideas.
Yes, after Howards showed me what is "2-column proof", I have this thought too, worst of all, the example he showed me, those 2-column proof still contain mostly the "fragments" of natural language. Very annoying.
> > 3. Russian papers are so short because they leave out the details. I was > once told that they left out the details because of Stalin's paranoia > (sending technical information to the West and all), but that may be > apocryphal. When Russians publish in non-Russian journals their papers are > as long as anyone else's, since referees make them put the details in.
Well, there are different explanations. But I don't think it is only less details. No matter how it started, my feeling, and I am sure I am not alone in this, the result is that they got better training in expressing mostly the key information and key thinking flow, effectively.
It is not only details, it is that the modern papers here usually contains a lot of wasteful "talking", it often time becomes a test of reader's patience.
> > A really good hope-it-appears-soon paper on the notion of mathematical > proof is Keith Devlin's The Logical Structure of Computer-Aided > Mathematical Reasoning, which if you skip the technical parts is a nice > overview of just what it is that mathematicians are doing when we prove > theorems. Devlin is a mathematician at St. Mary's College in California, > and edits the Mathematical Association of America's publication FOCUS. For > a lot of really nice examples of how to present and write proofs at the > high school level (the way a mathematician would), see Cuoco et al's > Connected Geometry, modules available from Educational Development > Corporation in Newton, MA, a truly gorgeous work. > > Judy Roitman > > P.S. A similar discussion raged last year in the Geometry Forum > Pre-College list. > >