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Topic: Precalculus
Replies: 6   Last Post: Feb 27, 2002 12:45 PM

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Deane Yang

Posts: 13
Registered: 12/8/04
Posted: Feb 11, 2002 11:17 PM
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In my earlier post I mentioned precalculus and pre-precalculus,
but never really defined these terms, forgetting that what
we mean at Polytechnic is different from what it means in the
high schools and most universities.

Precalculus at most institutions is indeed more properly called
Analytic Geometry or Advanced High School Algebra. It covers
what I consider to be largely arcane and useless topics in
algebra and co-ordinate geometry and is of no help to learning
calculus or anything else.

We follow the lead of the Harvard Curriculum. To us Precalculus
and Calculus is a unified set of courses devoted to the study
of real-valued functions on the real line. We use the Harvard
precalculus book for the pre-precalculus and precalculus courses.

Even though these courses are "remedial", we do not simply
reteach the high school courses. I don't know about other schools,
but this approach is a proven failure at ours. The students are
bored to death by the lectures and just do not take the course seriously
at all. So although we certainly review exponentials, logarithms,
trigonometry, the focus of the courses is on really understanding
what the concept of function is and how to use it effectively.
Again, we are always focused on teaching mathematics as a USEFUL
SKILL. The need for "understanding" is completely driven by this.

Over the years, I have always noticed that when students are weak
in calculus, the root cause is not their lack of understanding of
what a derivative or integral is but their lack of understanding
what a function is. So now when I help students, I almost start
by asking them to tell me what a function is. Usually, I just
get a lot of inarticulate noises ending with "I know what it is,
but I don't know how to say it". Others tell me it is a formula.
Some try to recite from memory a precise mathematical definition
involving ordered pairs. Some mutter something about the vertical
line rule.

So here's my big beef. Mathematicians have to learn that a
precise mathematical definition is NOT a useful working definition.
At the same time students MUST be taught to articulate what they
think they know.

And here is the definition we drill into our students at Polytechnic:
A function is a box that eats and spits out numbers. Whenever you feed it
a number, it spits one out. Different functions do different things.
For example, one might be a very stubborn one; no matter what
number you feed it, it always spits out a 3. That's commonly
called the constant function 3. Another function simply spits
out the same number you feed it; that's commonly called "y = x".

The key point we want students to understand is that a function
is not an object in the usual sense; rather, it is a process.
Once you establish this point, it can be constantly reinforced
throughout the precalculus and calculus curriculum and helps
motivate everything you do. On the other hand, if you do not
constantly remind students of this, then the concepts of derivative
and integral remain mysterious to students. They then simply think
of derivatives and integrals as being symbol manipulation processes
that are for some unclear reason useful to mathematicians.



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