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Replies: 6   Last Post: Jun 17, 1997 11:25 AM

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Posts: 10,730
Registered: 12/3/04
Posted: Jun 14, 1997 4:14 PM


Two items for your consideration:

Item 1. On the 1996 AB Advanced Placement Exam the first Free Response
Question gave the graph of the derivative of a function and asked some
questions about the function itself. The students were expected to glean
information about the function from only the graph of its derivative. Fully
32.5 percent of the (~105,000) students writing the exam received no points
for this question (none, zip, zero).

Item 2. I ran across an ad for DERIVE, a symbol manipulation software
package, the other day. The left page extolled the virtues of the software
and on the right was a listing of all the things DERIVE could do. The list
was two columns running about 2/3 of the page's length. This list went from
simple addition to multiple integration. I estimate that a typical student
takes about six years to learn how to do what was listed on that 2/3 of a

There has been of late in these news groups a lot of discussion of the place
of symbolic manipulation in the calculus curriculum and the mathematics
curriculum in general. Some of it tied to criticism of the "reform calculus"
movement. The gist was that all sorts of horrible things would result if much
more time were not spent teaching students how to do algebra (by which is
meant: how to manipulate symbols).

The question referred to in item 1 above is pretty close to a pure reform
calculus question: interpreting a graph and writing the conclusions. I
thought it was a great question and asked things very fundamental to some of
the big ideas of calculus. A student who understood what he or she had been
studying all year, should surely have been able to answer this question.
There was no computation , no manipulation of symbols to do; in fact there
was none possible. I suspect that students who had trouble with this
question, could have answered the same questions had they been given the
expression for the derivative to manipulate. And that's the point: they had
in front of them everything the manipulation could have told them and were
unable to interpret it.

On the other hand there was no way to use a calculator to do the problem
either. However, if the students had spend some quality time with their
calculators during the year investigating the relationships between the
graphs of functions and their derivatives, I suspect they would have had no
trouble with this question. The point is that the calculator in a high school
or college classroom should be a tool for investigation, not just a shortcut
to the answer.

Symbol manipulation is neither mindless nor unnecessary. It's what makes
mathematics work. Students need to be able to use, interpret, read, write,
decode and otherwise handle symbols. You really can't do mathematics without
that ability. BUT there is a lot more to mathematics than that. Along with
the symbols are the numbers, the graphs and the communication. Much more time
needs to be spent with these rather than with the pure manipulation of

Machines can do the manipulation. Let them. People still need to understand
what the machines are doing, what the symbols mean, how they apply to the
problem under consideration and so on. Too many, far too many, students think
that mathematics is only symbol manipulation. Why? Because that's what they
spend so many years doing in math class. Because that's what shows up on the
math tests. Because that's what shows up on the standardized tests.

Allow me to coin a word. The word is "algemetic" (al/ ge/ me/ tic rhymes
with "arithmetic"). "Algemetic" means anything the TI-92, Derive,
Mathematica, Maple or MathCad can do. "Algemetic" is not all there is to
algebra. It is what students spend years learning.

Some instruction must include having students manipulate variables with
pencil and paper. The goal is not to get them to be great "algemeticians,"
but rather to enable them to know what "algemetic" can and should be done to
solve a particular problem or investigate a particular situation. To know
when, where and what is appropriate is more important than the ability to do
it by hand. In other words they have to know when to do it, but they don't
have to be good at it.

Now before you all jump on that, consider:

(A) YOU balance your check book with a calculator, don't you? Admit it. Even
though you probably have several degrees in mathematics and can do arithmetic
more accurately and quicker than most, you still use a calculator when it
really counts (it's your money).

(B) When scientific calculators came out I was still required for (too) many
years to teach how to compute with logarithms. I can still see the students
sitting there adding up logarithms with their scientific calculators.

We are still in the same continuum. People need more than ever to understand
mathematics, particularly algebra and the calculus. But when it comes to
"algemetics" what is the problem with letting the machines do it? They are
better, faster and more accurate. If someone needs to do a computation or
to solve an equation or to factor or to graph or to find a derivative or to
find an antiderivative or even to add a few integers, what's so bad about
pushing a few buttons?

The Reform Calculus ideas, the Rule of Three -- numerical, analytical,
graphical and communication and technology (see they needed a calculator
there) -- should be applied not just to calculus but to all mathematics
starting in Kindergarten. Put the emphasis, spend the time, on teaching the
mathematics not the "algemetics." Concentrate on the ideas, concentrate on
the relationships (numerical, analytic, graphical) between the ideas and be
able to communicate those relationships. Use technology not as a quick way to
the answer, but as a tool for investigation. Use technology to go wider and
deeper into the problem; don't use it to drive straight through to the
answer. Use it as a way to keep focused on the mathematics by avoiding the
"algemetics." Use the technology correctly and to its fullest extent -- not
just to add logarithms.

The DERIVE ad: Six years of work reduced to less than a page! Think of all
that time being used to really get into mathematics; all the time to
investigate, solve problems, think, discuss and write about what's happening
-- sure beats factoring x^2-4 for the hundredth time.

That's MHO.

Lin McMullin
Ballston Spa, NY


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