This is MY thinking, shared, perhaps, by
nobody. But it may prove useful.
I cheat!
That is what I call it - I learn how to
do math from a teacher, and
then I try to find another way to do it, one not approved by the
teacher. WHY???
What is mathematics, anyway?
You will see a lot of different definitions in the books but the one
which appeals most to me it this:
Mathematics
is a game mathematicians play with numbers and patterns.
What a crazy definition! But I believe it is basically
true. Nobody will contest the numbers and patterns part. Why a game?
Well, games are arbitrary sets of pieces and rules which people adopt
to work with in obtaining some kind of goal of value to them. This is
what mathematics is! The pieces are numbers and lines and sets and all
the other things we use in math. And the rules are very arbitrary
whether you believe it or not. In fact, sometimes the greatest advances
in math occur when someone takes an accepted rule (like the parallel
postulate) and changes it. Then mathematicians who have the guts will
explore the new set of rules to see where it takes them. Sometimes the
results are astounding!
But it is still a game. In fact, it is FUN! This does NOT mean it is
worthless. Some of the most important things humans do is play games.
Mathematics is one of them - consider how incredibly useful mathematics
is in the real world, and I am not limiting myself to arithmetic!
So, why do I say I cheat? Because it IS a game, and I play by whatever
rules I need to play by, but I do not blindly accept arbitrary rules
which limit my play.
So, from time to time in these tutorials I will point out that my
methods are a little odd, perhaps, but they work, and you just may find
them useful. More than that, I want to encourage you all to look at the
big picture - LEARN THE RULES, then see if they make sense in the
context of the math you are doing, and just try, now and then, to
"cheat" a little to see if there is another way to do the math – try to
find shortcuts. You will be amazed at what you can learn from the
process.
An Additional Note:
For years professional math teachers have been emphasizing the need to
base math instruction in "real world" problems and situations. I have
no doubt about the usefulness of such approaches in learning basic
concepts and ideas. But recent research has actually shown that
teaching an entire subject this way is no more useful than teaching
basic algebraic rules and procedures - sometimes worse! Often students
just need to know HOW to do something, not how to manipulate rods or
shapes to get an analog to work. So I usually just work with the
definitions, rules and procedures, not models.
This does NOT train in basic problem solving skills. I believe these
skills are obtained not by instruction but by hard work, by struggling
with problems and being led through in recognizing situations where
past experience can be applied. (see my Problems
of the Week) The "REAL WORLD" approach may work better here, I am
not sure. All I know is that learning math can be hard work but it can
also be fun - attitude is everything!
