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:

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>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