Date: Thu, 3 Oct 1996 From: Ray Mines Subject: Re: What makes math hard? I am also taking an English class: Dante. A couple of years ago I got into Joyce's *Ulysses*. I found all sorts of mathematics and wrote a paper with the teacher of the course. The paper will soon come out in the *James Joyce Quarterly*. We are now lookig at Dante's influence on Joyce. Lots of fun. I have assigned that problem out of Rosen's book and then spent weeks trying to get my students to understand what the problem is saying. You have hit my interest right in the center. Much better than Robin Hood. We have a two semester course devoted to discrete mathematics. Frank Williams and I are trying to do something about the first course, called Math 279. Our handouts say that the goal of 279 is to learn how to read and write mathematics. I discovered while on an exchange at UConn that the students do not know how to read the text. Why don't they read the text? It is not because the students are lazy, but it is because they do not know how to read. Math 279 is a course in vocabulary building and at the end of the course the students do not know the meanings of any of the terms that were discussed. Frank and I are using some notes written by the fellow in Maine that had a letter in the Notices about two years ago. He has a method for teaching how to do proofs that he claims works for elementary ed students. It seems to work in 279, but we have been unable to get the students to learn the definitions. Yesterday in my combinatorics course I someone in the class present problem 12 of chapter 5 in brualdi's 2nd edition. I knew what was going to happen. after he began the proof I had him move to the other half of the board and set r = \sqrt 2 and k =3 and use these values in is proof. He wrote \sqrt 2! on the board and I asked how do we evaluate that. No one knew. I then asked the class to find the place in the text where Brualdi explains how to evaluate r \choose k when r is a real number. After 10 minutes someone noticed that it was on page 132. I asked her what is going on in that paragraph. Neither she nor any of the other students knew what it was about. So I had her read it aloud. I stopped her at the end of the first sentence and asked her to repeat the third word. Then I asked her what's going on. Finally someone says "a definition". We then had a long talk about how to read math textbooks. One student said she reads lying in bed. I handed out the attached TeX document and we decided that one should be sitting at a desk with paper and pencil. That one needed to copy each sentence from the text and figure out what it means. At the end we discussed the words "shade" and "shadow". The class decided that it would be a shame to use only one of them because the different meanings are important. I then said that words in mathematics have precise meanings and that they needed to know them so that they could use the words with confidence as they do with shade and shadow. enough of this! attachment: -------------------------------------------------------------------------- >From the preface to the 1954 edition of The foundations of statistics'', by Leonard J. Savage, second edition published by Dover. ... I therefore take the liberty of giving some pedagogical advice here and elsewhere that mathematically more mature readers will find superfluous and possibly irritating. In the first place, it cannot be too strongly emphasized that a long mathematical argument can be fully understood on first reading only when it is very elementary indeed, relative to the readers's mathematical knowledge. If one wants only the gist of it, he may read such material once only; but otherwise he must expect to read it at least once again. Serious reading of mathematics is best done sitting bolt upright on a hard chair at a desk. Pencil and paper are nearly indispensable; for there are always figures to be sketched and steps in the argument to be varified by calculation. In this book, as in many mathematical books, when exercises are indicated, it is absolutely essential that they be read and nearly essential that they be worked, because they constitute part of the exposition, the exercise form being adopted when it seems to the author best for conveying the particular information at hand. \bye