Date: Mar 17, 2013 6:55 PM
Author: Kaba
Subject: Re: Maths pedagaogy

17.3.2013 21:52, quasi wrote:
> (3) The writer is a wise, experienced teacher who has a clear
> understanding of his likely readership and knows which lemmas
> would be trivial for his readers to prove, and
> * While the author needs the statement as a lemma, it's offered
> to the reader as an easy exercise.
> * To set the expected level of difficulty of the text by omitting
> trivialities (which would be less trivial if the reader was less
> experienced or less able).
> * (repeating from (2)), doesn't want to detract from the main
> points by watering it down with trivial details.

I'll reply on this topic to David Ullrich, and to you on the rest of
what follows.

>> I'll concentrate on the type 1 writers; the type 2 writers
>> hopefully improve on their writing as time passes.
>> Speaking of books in particular, whose main purpose is to teach,
>> one quality metric for me is to count the density of weasel words
>> in the text.

> Better metrics are:
> * The readability and elegance of the writing.
> * Intelligent, natural choices for notation and terminology.
> * The right motivation for key concepts.
> * A well chosen mix exercises, some easy, some medium, some
> challenging, chosen so that working through most the
> exercises builds the students' power over the subject and
> confirms their mastery of the concepts.
> * Clear, well-worded proofs, not cluttered with trivial details,
> and with the key parts said "just right".

These are important too. They become detectable especially when they are
not present. Since the OP asked about opinions on weasel words in
particular, I am concentrating on them.

> > An unfortunate example is Lang's Algebra, where everything is
>> obvious, easy and trivial. This is almost always contradictory.
>> If it really is trivial, then why not write it down;

> Why not offer it as an in-place exercise for the reader?

In-place exercises are nice, when they can be carried in the head. Lee's
books have plenty of these, and it's here where you appreciate that
those really can be solved in head in a minute or two. In other words,
he has tried them himself:)

In my opinion, proofs, once started, should always be precise and
complete, without hand-waving.

>> it should take about the same space as stating it trivial.
> Right, but Lang expects the readers to participate.

Possibly, but whether it's a good idea I disagree on. In general, a book
should not expect or ask anything "extra" from the reader. It might be
that developing your own proofs is the best way to learn, but people
have different amounts of time to invest on given topic. I wished a
given book maximized the benefit of reading that book.

> Just like some math teachers who, when teaching lessons, ask
> (mostly) easy questions as they go along. This serves a dual
> purpose. Firstly, it reassures the teacher that the students
> are "with it". Secondly, it gives the students an opportunity
> to be part of the development. Most of the questions should
> be easy ones with quick answers, so as not to slow down the
> lesson too much.

Heh:) Aside: I don't know whether this is a cultural or a psychological
thing, but at least here in Finland if the teacher (professor, exercise
assistant) asks anything from the class, then he'll usually face
silence. This has to do with group size; once the size drops below
certain threshold, say 6, then people start participating spontaneously
again. When there is more, people mostly want to listen passively and
not be bothered by any interruptions in the flow. I wonder whether this
happens in other countries too?

> In your indictment of Lang's Algebra, let's see some examples.
> Show us statements from that text for which, assuming the
> student
> * has the assumed prerequisites for the text
> * has successfully understood (and done exercises to prove
> it) for the material up to that point in the text
> have proofs left to the reader, but which, in your opinion,
> would actually not be trivial for the average reader, as
> qualified above.

Here is an example from Lang's Algebra, page 572, Proposition 1.1. This
is a chapter on bilinear forms. Notation is as follows. The _|_ is for
the orthogonal sum, that is, a direct sum of vector spaces such that the
vector spaces are orthogonal to each other. I will denote the bilinear
form by <u, v>. The radical of the bilinear form in vector space U is
denoted by rad(U). We are only interested on bilinear forms that are
either symmetric, alternating, or hermitian; in these cases the left-
and right-radicals coincide.

Lang's proof

Let E be a vector space over the field k, and let g be a form of one of
the three above types [symmetric, alternating, or hermitian]. Suppose
that E is expressed as an orthogonal sum,

E = E_1 _|_ ... _|_ E_m.

Then g is non-degenerate on E if and only if it is non-degenerate on
each E_i.


Elements v, w of E can be written uniquely

v = sum_{i = 1}^m v_i,
w = sum_{i = 1}^m w_i,

with v_i, w_i in E_i. Then

<v, w> = sum_{i = 1}^m <v_i, m_i>,

and <v, w> = 0 if <v_i, w_i> = 0 for each i = 1, ..., m. From this our
assertion is obvious. QED.


Whoever reads this post can measure the time on how long it takes to
understand Lang's underlying logic here. I myself am unable to decode
the last two rows of the proof, which should be the core of the proof.

In addition to problems with clarity, Lang does not actually use g
anywhere; he uses a dot instead. Such problems show up when you don't
polish, i.e. read through, your writing carefully.

Since I had hard time with Lang's proof, I ended making up my own proof;
that proof follows next. Perhaps these can be compared for clarity.

My proof

Let V be a bilinear space, where the bilinear form is either symmetric,
alternating, or hermitian. Let U, W subset V be subspaces of V, such
that V = U _|_ W. Then V is non-degenerate if and only if U and W are


Assume U and W are non-degenerate. Let v in rad(V). Then v = v_U + v_W,
for some v_U in U, and v_W in W. Let u in U. Then

<u, v> = <u, v_U> + <u, v_W>
= <u, v_U>
= 0,

since v in rad(V). Therefore v_U in rad(U). Since U is non-degenerate,
v_U = 0. By the same argument for W, v_W = 0. Therefore v = 0, and V is
non-degenerate. Assume V is non-degenerate. Let u in rad(U), and v in V.

<u, v> = <u, v_U> + <u, v_W>
= <u, v_U>
= 0,

since u in rad(U). Therefore u in rad(V). Since V is non-degenerate,
u = 0, and U is non-degenerate. By the same argument for W, W is
non-degenerate. QED.