ON SWITCHING TO OTHER SECTIONS OF MATH 75, MY TEACHING PHILOSOPHY, ETC
---------Warren Smith--------------------------------------------------

Some people have the idea I'm not a very good teacher, or 
my homeworks are too hard, or I don't follow the book closely enough,
or something, and maybe they'd be happier with ANOTHER math 75 section
(there are many of them).  Well, you can try to switch sections,
and different instructors do have different styles.
But I just want to warn you, even if you do that, you may
discover that it is not working out and now you've wasted time
and now are just in even more hot water.  There is no quick fix.
There is no easy way to learn calculus.  It takes a lot of work.

The payoff is, at the end of a year of calculus, you will feel 
much more empowered.  It will be a quantum leap in your ability
to do math.  It will be like you are no longer the slave of math,
you are the master of math.  You won't be trying to memorize formulas
from nowhere, you'll be able to create formulas of your own, en masse.
You'll be able to solve problems that just seemed totally 
impossible before calculus.   Not only that,
but you'll look back and you'll say to yourself "I can't
believe I was that stupid back in the old days.  This is so easy now."
It is like learning to add and multiply.  Once, those seemed hard. Now 
they seem easy.

Furthermore, you just may discover that I'm a better teacher than
you think, and there is a reason I'm doing things the way I am,
and it's going to pay off later in the course.  And it just might pay
off on the final exam too.

THE BOOK: 
If you worry I do not follow the book enough:
there is at least one instructor (not me) who said on
the first day of class "I am not going to pay any attention to the
book. Get used to it."  So I am not the worst book-follower around.
But I happen to know that Prof. Boris Datskovsky  (in charge of everything)
*does* like the book (he selected it) so if you want to follow the 
book a lot, his section could be a good choice.  Also, let me say that
this book has a poor pre-calculus review in chapter 1, ignoring 
things like the sum-symbol, solving triangles via law of sines and
cosines, elimination techniques for solving equations, etc., so
I had to go outside of (this) book to review that.  I like the book
better in chapter 2 so I plan to follow it more then.

THE WORK: 
If you worry my homeworks are too hard:
Well, unfortunately that same B.Datskovsky told me he favors assigning
20-30 homework problems PER DAY!  That's more than I assign...
B.D. thinks the only way to learn calculus is to do a lot of problems.
Actually, everybody thinks that.

I agree with him... but I think it takes a mixture of easy and hard
problems, and just flooding you with drudge work, alone, does not do the job.
You have to learn to think deeply and make multistage plans involving 
several concepts, where it is not immediately obvious (to somebody at
your level, anyhow) what that plan should be.  And if you don't
succeed, you have to have the experience of that failure, coupled 
with later experience of finding out how you should have done it,
to learn.  That calls for hard problems, corrected by hand.  
Such as my "homework" problems.  Some people still have the idea they
can just start writing the answer as soon as they see the problem -
but they can't - and their solution soon degenerates into random
scrawlings followed by a big question mark.  That won't work.  
Work out different plans on OTHER sheets of paper, and then, 
when you find one that works, ONLY
THEN copy it onto the homework you plan to turn in.

You also have to learn to think shallowly and solve quick problems
that just make sure you know ONE concept.  That calls for things like
the COW problems.  

THE TEACHING: 
Well, I am an inexperienced teacher compared to most
of them, and I think, unfortunately, that's showed in some cases.
But, (1) I'm learning fast.  (2) It is not as simple as "X is a better
teacher than Y".  Really, X is going to be better sometimes on some
subjects and worse on others.  I'm quite confident I'm
not worse than the others, I'm better than the others, on some things
some times. 

THE GRADING:
Don't get the idea that there is any "easy A" to be had in math75.
At the end we instructors are all going to get together to
figure out the "curves" for the grades, which will be different
in different sections, depending how hard we think they were,
but believe me, there is not going to be any section 
with a big pile of A's, and there are going to be a lot
of students (based on past experience) who fail the course.
If you complain to, say, B.Datskovsky about my grading, that won't
work.  He fails plenty of people, and he has been known to check
out the tests and homeworks from sections which seemed to be
getting too high grades, and WHAM, decreasing all their grades,
AND disciplining that instructor for being a too-easy grader.
He's a tough guy.  We only are going to "curve" the grades so far.
We have absolute standards - if you want to pass the course, you'll
have to know X,Y,Z.  If 90% of the students don't know X,Y,Z, then 90%
of them will fail, and we don't care if that 90% failure rate feels
bad - if this happens we'll still fail them.

So the only way out is to work hard.

OTHER MISCONCEPTIONS:
Some of you have ideas about how I should be teaching the course,
but the trouble is you don't know the subject yet, so your ideas
(in some cases) are wrong.  

EXAMPLE: NUMBERS vs LETTERS:
Somebody thought I should be using numbers a lot, not letters.
Wrong.  You have to get used to manipulating formulas not
manipulating numbers.  It's a higher level of thinking.
With this kind of thinking, every move you make has the strength of
ten:  If you make a formula, then you can use it to do 10 calculations 
with numbers later.  So you have to get comfortable with manipulating
formulas, so I'm trying to do a lot of such calculations in class.

It's like learning to walk.  At one point you thought a lot about
"where should I put my left foot now?"  It was hard.  You fell over a lot.
Now you don't even think about that.  You think about higher level
questions like "should I walk to the post-office now?"
So you have to master the mechanics (where to put foot; how to
manipulate numbers; how to manipulate formulas reliably) but then you
have to go to higher level thinking (where to walk next?; What should
I be trying to get my formula to turn into next?  How can I devise
a sanity check suited to the present situation?  What do I need
to look up in a book?  What is my overall plan to solve this?).

I've been trying to go through mechanics in a lot of detail, but I
warn you, later on I'm going, more often, to take the attitude you should
know the mechanics by now.

And later on in calculus, we are going to reach even higher levels of
thinking: it won't just be manipulating formulas, it will be
creating them, sort of.  We will have new tools for doing that.
Then it will be like every move you make has the strength of 100.

EXAMPLE: REGURGITATION vs PLANNING:
Some people feel it is "unfair" if I pose them a problem
which I have not gone thru before in exactly similar way, so that
the problem can be solved by just doing what I did, over again
(with different numbers perhaps).
Wrong.  The goal here is for you to learn to think for yourselves
and learn to solve new kinds of problems - even ones which perhaps have never
been posed before by anyone!  There are simply too many kinds of possible
problems for me to go through them all ahead of time.  You have
to view formulas NOT as recipes that are memorized for one purpose
and one purpose only. Instead they should be viewed as MULTIPURPOSE
TOOLS which you combine and use for a huge number of possible purposes.

Think of it this way. Suppose I were teaching you carpentry.
If I teach you how to use a hammer and nails, glue, and a saw, and then
say "exercise: build a doghouse" was this "unfair"?  Would it ONLY
be "fair" if I ask you "saw a plank 1.5 feet long" because in
class we had gone through the procedure for planks 1 foot and 3 feet
long?  And would it be "unfair" to ask you to figure out how to
combine several steps of sawing and nailing and gluing to accomplish
something?  If this were your notion of "fairness" you'd never
be able to accomplish anything very interesting.  You should not
limit yourself in that way.

EXAMPLE: CALCULUS should be just like PREcalculus?
Wrong.  Calculus really is different and marks a transition
in the power of your mathematical thinking.  Historically,
almost all the precalculus math you learned was known
to the ancient Greeks by 300BC.  This includes trigonometry, pi,
sines, cosines and their properties, solving quadratics, arithmetic,
areas and volumes of basic shapes, and algebraic manipulation.  
Then almost nothing happened for 2000 years, then suddenly calculus 
appeared and started a revolution.

I was quite internally upset when somebody, who said she was a math major,
told me she'd taken a lot of precalculus classes, and my class was
not like them, hence was bad.  If she continues to be a math major,
and thus hopefully goes well beyond calculus I, she will hopefully 
discover that math is becoming far too rich to be treated in the kind
of rote way that a lot of precalculus classes are taught.  Also
hopefully, she will discover she likes that.