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Topic: more on reform/response to Richard
Replies: 2   Last Post: May 28, 1997 7:24 PM

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jerry rosen

Posts: 244
Registered: 12/6/04
more on reform/response to Richard
Posted: May 24, 1997 5:25 AM
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I probably should have mentioned that my interest in mathematics came
about in HS through the efforts of several HS math teachers.

When I was in 9th grade I was truant and in danger of being left back.
Fortunately, at that time, NY had good assessment and it forced most kids
to keep their acts together.

My parent's got me a math tutor who was an unemployed HS math teacher. At
the first lesson he showed up with a copy of Birkoff and Ma Clane's
Modern Algebra - I still have it. He told me he would show me how to
excel in the rote stuff and that one needed to learn how to do that
before learning more interesting things. After I started getting A's on
9th grade algebra exams I started teaching myself a little modern
algebra. In HS I had excellent teachers who knew how to balance rote
stuff with conceptual and theoretical stuff - but without a doubt the
rote stuff was primary.

I was having a discussion with a colleague of mine about how in the "old
days" we had to learn interpolation for evaluating logs. He was
complaining about how dull this was. It was dull and it was very useful!!
The way we evaluated logs was we used the three basic log properties to
"beat a big number down to size" and then interpolate.
As a result, we had a good understanding of the definition of log as well
as it's fundamental properties.

These days the majority of calculus students don't know logs or
exponential functions - the reform calculus books avoid the exponentials
like the plague - I guess they know what isn't happening in HS's. Of
course they don't know the properties either. Almost none of them know
the difference between the log and ln buttons of their calculators and
are clueless to what information these buttons give. The same holds for
trig. and inverse trig. functions.

As a result two important and related concepts are a total mystery to
most students - composition and inverse functions. Now I do not view the
chain rule as symbol manipulation and anyone who teaches college math
knows that one of the big stumbling blocks is composition of functions.
Because these students have at most a calculator intro to the
transcendental functions they really can't grasp composition or inverses
in calculus. Most linear algebra texts are reform oriented these days and
so composition is downplayed there too. The fact that matrix
multiplication comes from composition of linear maps is unknown to most
linear algebra students. Of course they may know what LU decomposition is

So what we see is that an over use of calculators in HS and a de-emphasis
on rote properties of transcendental functions leads to major problems
for math, science, engineering, CS and business students later on.

Again we see how reformers miss key issues. Rote stuff is not only good
because the student is learning useful procedures, but much of it results
in a deeper conceptual understanding of important ideas. As I said
previously, rote stuff also gets the students in the habit of studying in
a way which will serve them well in college and in their careers.

Richard mentioned that his student's used calculators and/or computers to
generate conjectures which could be proved by induction. This is good. If
his HS student's can do induction proofs in HS he has accomplished a very
useful thing and my proverbial hat goes off to him - albeit, I don't see
these student's at CSUN. None of my student's knows induction or the
binomial theorem before I or someone else (David is another trouble maker
who teaches these things to "virgin" minds).

However, I doubt that a machine is needed to generate enough conjectures
to get a student started on induction proofs. In my 20 years teaching
math and doing math research I have never needed a machine to come up
with a conjecture for an induction proof. Of course, there are
complicated conjectures where having a machine would be helpful. But I
seriously doubt this is needed at the HS level. Later on, when a math,
science, CS or engineering student has developed a certain amount of
mathematical dexterity he/she will have little trouble using machines to
help them formulate conjectures. But the sad reality of most (almost all)
reform math is that it will never lead students to develop proof
strategies and at best they will be OK at making guesses but at a loss
to prove anything or to generalize or to explain to others.

There is nothing good about any reform math program I have seen. I
suspect Richard, like my math tutor, is good at math, enjoys doing it and
can teach it to others - I would conjecture he would be just effective,
if not more, 20 years ago in the NYC school system which had standards
and assessment and where it was taken for granted that students had to
learn to progress.


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