First off, having been a math major end to end (bachelors, doctorate)
and having been exposed to people in mathematics education (those who
teach the future math teachers), let me say this: make sure the math is
correct! This should go without saying, but there are actually books
that make it to print with incorrect content, and my impression is that
teachers (and certainly those who teach the teachers) will prefer a
boring but correct book to an interesting but incorrect book. I
definitely would, on the theory that it's easier for me as an instructor
to motivate and explain what was in a "dry" book than it is for me to
deprogram someone who "learned" a bit of nonsense from an "official" source.
That said, and noting that there are some pretty good (and successful)
K-12 books out there (some from my colleagues here), I tend to agree
with the other comments in this thread.
Gunnar Lindholm wrote:
3. On reflection, most maths books are impenetrable because they go from
the general to the particular. My belief is that paedagogically the
reverse works better: Use a concrete example as an introduction to an
abstract concept.
Very good suggestion.
I second (third? fourth?) this. I'm not sure how much of the problem is
"general first, then specific" and how much is what I would call an
"axiomatic" approach (assume this because I'm telling you to, then
derive this). It's a valid way to teach math to a mathematician, but at
the K-12 level my suspicion is that it stifles the development of any
ability to *discover* mathematical relationships (and maybe develops an
unhealthy reliance on the deus ex machina delivery of an axiom/theorem
to cover any situation). Starting with a specific (and, where possible,
realistic) example is a good way to motivate learning.
4. Answer the question "why". "Why should I care what 2x+3x is?"
Also a very good (and, sadly, oft overlooked) suggestion.
Here are a couple more suggestions, or variations on identified themes:
6. Try to diversify the sources of the examples. Some things will be
fairly obvious: for instance, if you're teaching numeric bases, the use
of binary and hex (and octal if you still have a CDC mainframe :-) sort
of jump out. Others may not be as obvious (e.g., prime numbers and
prime factorization are the backbone of digital encryption schemes).
One of the problems I saw as a student Way Back When was that most text
book examples that were not "pure math" seemed to come from physics.
The one that sticks in my mind is a block sliding down an inclined
plane. If I see one coming, I'm getting out of the way. The math
doesn't seem too engaging.
7. Spend a little time every so often on the subject of when something
*doesn't* work (or why it's a bad way to do things). Again, some times
this is obvious (multiplying/dividing both sides of a relation by the
same thing? better be sure it's not zero) and sometimes it isn't. It
might be useful to talk to teachers about frequently encountered errors
on homework assignments -- there may be some patterns to them that
indicate common misunderstandings.
Good luck. Any improvement in math education here in the States is
worth doing.
/Paul
