[Textbook-l] Re: Textbooks

[Toby Bartels]

Jimmy Wales wrote in small part:

>One thing about textbooks is that they need to be self-contained and
>*ordered* in a way that an encyclopedia is not.  This depends, in
>part, on the subject matter, but take math as an example -- the
>concepts of trigonometry need to be presented more or less in a
>particular order, or the student will be lost.

Take a look at an advanced (upper level undergrad or higher) math text.
If it was recently published, then there's a pretty good chance
that you'll find, among the introductory material,
a description of possible course outlines (omitting some material).
With some luck, you'll even find a chart of dependence,
explaining which chapters are needed for which other chapters.
We can link to similar information from the front page of our textbook,
thus allowing teachers (or students studying alone)
to choose an order to go through our various modules.
We shouldn't impose any more order than is logically necessary --
which isn't to deny that /some/ order is logically necessary.

Even in elementary school mathematics,
where fractions build on division builds on multiplication etc,
there are two tracks being followed (in US schools):
a development of arithmetic (and later algebra),
and a largely independent development of plane geometry.
(Sometimes geometry will borrow from arithmetic or algebra,
but not the reverse until you get into trigonometry).

The trickier course is when there are multiple possible orders.
In calculus, differentiation and integration are logically independent,
but the fundamental theorem of calculus requires both of them.
Since you need the FTC to do /practical calculations/ with integration,
most courses (again, in US schools) today do differentiation first,
allowing the FTC to be presented early in the integration material.
But Caltech (at least 10 years ago), taking a more theoretical approach,
does things in the reverse (more old-fahsioned) order,
defining integrals directly after the preliminary work with limits,
but saving examination of calculation techniques for later.
The advanced undergraduate analysis course
usually puts differentiation before (Riemann) integration,
(since the new material to cover beyond calculus is easy,
while the Riemann integral in all its rigorous glory is a nasty beast),
while the graduate analysis course puts (Lebesgue) integration first
(since almost all of the new material on differentiation
depends on the background of Lebesgue measure theory).
So the order of a subject will, in many cases, depend on your purpose.
One textbook might set that purpose by choosing its level
(the advanced undergrad course, or instead the grad course),
while another might prefer to leave that up to the teacher,
as in a calculus textbook with separate modules on the main idea
(either order) and calculation techniques (differentiation first).

One can even get into POV battles about appropriate orders,
and we should strive to remain neutral about those.
For example, there's a new movement in universities these days
to teach the Henstock integral instead of the Riemann integral
in the advanced undergraduate course, since the theory is similar
but Henstock's is (like Lebesgue's) more its powerful.
Of course, people argue about /which/ to teach,
but we can simply allow both and let the reader choose
(and let the writers' interests decide when each is written);
that's not what I'm worried about.

The complicated bit is what this says about the /graduate/ course,
which if we follow the usual pattern is a completely different book.
The new movement uses Henstock to define the Lebesgue /integral/,
then applies that to set theory to define Lebesgue /measure/;
the traditional course defines Lebesgue /measure/ using infinite series,
then applies that to functions to define the Lebesgue /integral/.
So when studying Lebesgue's theories for the first time,
does the integral depend on the measure or the other way around?
Well, it could be either, depending on /which/ prerequisites
you already know from your undergraduate course.
As long as only one development appears in the textbook,
then we can rightly say that one module depends on the other.
But once both developments appear (as must be allowed for NPOV),
then this will change, and we have to watch for that.

The simplest course might be separate modules,
one for either of the different developments of definitions,
and another for the further devlopment of the concept.
But this makes our modules much smaller than a traditional chapter!
I think that this (small modules) will generally be a good thing.
Aside from the flexibility that it offers
for NPOV on various pedagogical disagreements,
it also makes it easy to save the more advanced material
for when it will be needed (or to omit it entirely),
rather than telling people that (say) the end of Chapter 6
will not be studied until we need it for Chapter 10.
(Of course, right now we have the technical difficulty
that some people can't edit pages that are longer than 30 kB,
but we should still look towards a technical solution for that.)
Many books divide chapters into sections that approximate this.


-- Toby