
Re: Maths pedagaogy
Posted:
Mar 17, 2013 1:09 PM


On Sun, 17 Mar 2013 16:21:57 +0200, Kaba <kaba@nowhere.com> wrote:
>17.3.2013 14:25, pepstein5@gmail.com wrote: >> Maths texts and lectures often refer to observations as being "easy to check", >> "trivial" or "obvious." > >I find two extreme situations for why someone uses such weasel words. > >1) The writer is an expert, and is bored of going around the same >argument for himself for the thousandth time. The claim is probably correct. > >2) The writer is a novice, and does not have the energy to go into >details which detract him from the main point he is trying to make. >There is a high risk of the claim being incorrect, or of that the claim >is correct, but has a tedious proof.
There are better reasons. Not that the writer doesn't have the energy to write out the trivial proof, but that it's "obvious" that the book will be a better book, easier to read(!), if the triviality is omitted.
To give a silly example, suppose we're proving something, and we need the fact that the sum of two even integers is even. Which of the following seems best?
(i) Since the sum of two even integers is even, ...
(ii) Since 2a + 2b = 2(a+b), the sum of two even integers is even. Hence...
(iii) Lemma. The sum of two even integers is an even integer. Proof: Suppose that n and m are even integers.
It follows that n and m are integers, so that n + m is an integer, since the integers are closed under addition. (Here we are using the standard notation n + m for the sum of n and m.)
Since n is an even integer, n = 2a for some integer a. Similarly, m = 2b for some integer b. (Here we are using the standard notation 2a for the product of 2 and a.)
Now, since we have established that n = 2a and m = 2b, the Subtitutability of Equals for Equals (Lemma 0.0 above) shows that
n + m = 2a + m.
Similarly,
2a + m = 2a + 2b.
Now the transitivity of equality shows that
n + m = 2a + 2b.
We now apply the distributive property to deduce that
2a + 2b = 2(a+b).
Again applying the transitiviity of equality, we have shown that
n + m = 2(a+b).
But again, the integers are closed under addition. Since a and b are integers, this shows that a + b is an integer. Thus n + m is equal to the product of 2 and an integer (since n + m = 2(a+b), as shown above), and so by definition n + m is an even integer.
\qed
(Recall our convention that the symbol \qed denotes the end of a proof.)
Ok, that's a little silly. But I hope the point is clear. If one includes all the details then the book will be much too long, and more important, the book will be much harder to read! Leaving the honestly trivial parts to the reader makes it easier for the reader to see what the main _points_ to the argument are.
The difference between the two books you mention below is not that one uses "weasel words" and one does not. One cannot include all the details, so there is always a choice to be made regarding which details to omit. The difference between the two books is just that in one case the author's decision on this issue matches your preference much better than in the other book (possibly because of your background in one area versus the other). There _are_ _many_ details left to the reader in the book you say you like!
> >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. 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; it should take about the >same space as stating it trivial. If it takes more than a few sentences, >then it is not trivial. I find the advice in Strunk & White (Elements of >Style) relevant: "Do no inject opinion." > >A contrasting example is to take any book from John Lee (Introduction to >Topological Manifolds, Introduction to Smooth Manifolds, Riemannian >Manifolds). These are masterpieces to learn from. No weaselwords, >precise, and minimum amount of errors of any kind. It shows that the >author is interested on transmitting knowledge as efficiently as >possible, and also knows how to do that. > >To me, the use of weasel words reflect a lack of effort; that the writer >isn't interested on giving the reader the best learning experience he >can. They make a book confusing to read, and indeed, I have sometimes >missed important points this way. You can afford to be careless when >writing to experts, but not when you are writing to students (readers of >the book). > >In my opinion, weasel words do not undermine the readers confidence. To >the contrary: they contaminate the reader with a false sense of >security, opinions of what is easy and what is not. What actually >happens to me is that, if the claim is not immediately obvious, I skip >checking that claim to get back to the flow of the text. > >Related, there is this effect which I call the Stockholm Syndrome for >Mathematicians :) This happens when people read a book which leave large >gaps in their proofs, and force the reader to fill them. From the >helplessness of the start of not understanding, because information is >missing, the reader works through the proofs, and increasingly builds >confidence in himself. After having mastered the book this way, his >emotions have gone through a rollercoaster of frustration to a feeling >of control. And suddenly those positive feelings are projected to the >book; since I know this well, the book must be great. But it's not the >book; it's the massive work that was done to recover the details and >essential techniques. I hope future writers avoid writing their books >this way; it's abuse in disguise.

