Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Calculus paper for Monthly
Replies: 0

 david klein Posts: 125 Registered: 12/6/04
Calculus paper for Monthly
Posted: May 22, 1997 5:12 PM

Dear Fellow Math Teachers,

has given me permission to share his article, "What do we do about
calculus? First, do no harm," with this group.

David Klein
Math Dept.
Calif. State University, Northridge
____________________
What do we do about calculus? First, do no harm.

In the spring of 1994, the Dean of our Engineering School paid the
expenses of four speakers to tell the Mathematics Department how
to teach calculus in a modern way. To him, a modern way was intensive
use of computers. The real goal was to have us teach the same
amount of calculus but with fewer credits, so that more of the
time of engineering students could be spent taking engineering
courses. A joint committee was set up to look at what has
been taught and what needed to be taught. The conclusion was that with
heavy use of computers it would take more time to teach the same
material rather than less, so nothing came of the push to cut
the number of credits.
I asked the first speaker about proofs. He replied that calculus was
not the place to do proofs. Proofs should start in the junior year of
college, primarily for students who are mathematics majors.
There was an education meeting held at the University of Chicago.
could express quantitative relations using letters (variables) and
students could learn how to do proofs, why couldn't our calculus
students also do this? While we have a higher percentage of students
taking calculus now than we did forty five years ago when I was a
young college student, we do not begin to have 50% of the
age cohort taking calculus. Saying that we have so many
more students taking calculus that we cannot possibly expect
them to be able to do this is looking at the wrong comparison
group. In addition to proofs in geometry, there are other
proofs in Japanese middle school books, such as a proof that
the square root of 2 is irrational. See [8].
The second speaker talked about differential equations, and
began with this equation:

x' = x^2 - t

with an initial condition. Once this was put up on an overhead,
I worked out the solution. The speaker said this was a differential
equation that could not be solved [exactly], and I let this go by
without saying anything. After talking on other topics, the speaker
came back to this, her favorite equation, and put up an overhead, which
seemed to show a pole. She said that a pole was there and that a
colleague had shown this. This equation is just a Riccati equation,
so can be linearized, and the linear equation solved. In the
present case, the linear equation is the Airy equation, so a solution
is easy to find. Riccati equations are important in control theory, and
there were electrical engineers in the room, so I did not want them
to think that the mathematicians did not know what was happening.
I asked if they then solved the equation to explain where the
pole comes from. She repeated what she had said earlier about a
colleague having proven that the equation could not be solved. I
said that it depends on what is meant by solved. In the present case,
the solution is easily found in terms of Airy functions, which are
Bessel functions in a slight disguise. The pole that was claimed to
exist comes from the smallest zero in the denominator.
There is a tendency to downplay the role of infinite series in
calculus and in differential equations. The usual argument for
differential equations is that it is hard or impossible to see
the long range behavior from a power series. In the present case,
it is the smallest zero that is in question. The other argument
given against power series solutions of differential equations is
that few differential equations have solutions that can be
written in the form of a nice series. It is one of those miracles
of nature that many very important problems lead to just
those differential equations that can be solved in series with
nice coefficients. These series have the property that the term
ratio of the coefficients is a rational function of n, and are
called hypergeometric series. A course in calculus is not
the place to study hypergeometric series in detail, but the
most important one, the binomial theorem, should be there.
The ratio test for convergence is as popular with students as
it is because it is easy to compute the limit of a term ratio
that is a rational function of n, and many of the elementary
functions studied in calculus have power series of this type.
Students should start to be led in the direction of seeing
that this class of functions is important.
The third speaker was someone I have known for years, so I
asked some questions in an e-mail before he arrived. One
was about differentiating x^n. This can be done in several
different ways. The traditional one in our texts was to
quote the binomial theorem to get started. This used to be
a standard topic in algebra. One of the new calculus books
does it this way, and refers the reader to any high school
algebra book for a proof of the finite binomial theorem. I called
and asked one of the authors if he had looked at any high
school algebra or precalculus books recently. He said no. He
should, for the binomial theorem is no longer the staple it
once was. For example, the precalculus book written by the
faculty of The North Carolina School of Science and Mathematics [1]
does not have either the binomial theorem or the geometric series.
In response to my question to readers of the e-mail discussion
group calc-reform, someone replied that most of his students
had taken calculus in high school. If there is anything students
remember from high school, it is the formula for the derivative
of x^n, so he does not give a derivation. Many people who are
supporting the current reform efforts do not like formulas, and
so do not want to use the binomial theorem to differentiate
x^n. One solution to this problem is to use another formula.
(x+h)^n - x^n
--------------
h
it is possible to write
y^n - x^n
-----------
y - x
or even
(qx)^n - x^n q^n-1
-------------- = x^(n-1)* -----
qx - x q-1
and use the sum of a finite geometric series. This arguemnt also
works when n is rational. In the course of changing variables
to see this, you give an introduction to the chain rule and to the
simple form of l'Hospital's rule.
However, some of those who do not like formulas even object to
the formula for the sum of a geometric series, so there is a way
to differentiate x^n without using any formula. Just observe
that
(x+h)^n=(x+h)(x+h)...(x+h)
and observe that x^n appears once. The next term,
h*x^(n-1), appears once for each factor, so n times. Every other
term in the expansion has at least two factors of h. This way
the student can understand why
(x+h)^n = x^n + n*h*x^(n-1) + terms that involve h^2 or higher powers of h.
Contrast this with the treatment in [7]. The formula for the expansion
of (x+h)^n is stated when n=2,3,4,5. Then the authors write
"we can say that (x+h)^n = x^n + n x^(n-1)*h + terms involving h^2
and higher powers of h". There is a big difference between "we
can say that" and "we see why". Mathematics should be an
open subject, where students do not take such simple facts because
"we can say that" or because the computer algebra system gives such a
formula.
Another of my concerns can be illustrated by a problem in the
same book, but other books could have been used equally well. This
deals with when something has been shown to be true.
Consider problem 48 on page 365 of [7]. This has three parts.
The first is to use Riemann sums to evalute the integral from 1 to 2
of ln x . The second is to evaluate this integral using anti-derivatives.
done in the text, but from 2 to 3. Both of these parts are fine,
except it would have been better not to use "evaluate" in the first
part, but "approximate", and it would have been better for the students
to have been asked to do an integral that had not been done in the
text in the second part. However, it is the third part that bothers
the Fundamental Theorem of Calculus". This has not "verified" the
Fundamental Theorem of Calculus, but has illustrated that the approximation
in the first part gives an approximation to the exact value obtained
in the second. The first definition of "verify" in the dictionary
at my desk is: "To prove to be true". Words mean things and they
are important. Meanings should not be changed without very good reasons.
The fourth speaker tried
to show us how a computer algebra system could be used in a lecture
setting. One of his main examples was Simpson's rule. He
set up the problem, got three linear equations in three variables,
and said that these were far too complicated to solve at the board.
He then displayed the solution via a computer algebra system. I
asked him why he felt it was necessary to do the interpolation at
the points a, a+h, and a+2h, when you could do the interpolation at
-h, 0, and h, or even -1, 0, and 1. Then the equations fall apart and
it is very easy to do the algebra by hand. There is an important
mathematical lesson taught when doing this: you can adapt the
coordinate system to the problem at hand. That lesson needs to be
learned whether you do calculations by hand or by computer algebra.

guidelines that should be considered when teaching, writing a book,
or developing a curriculum:

* Do not lie to your students but don't tell them the full truth.
* Some results in mathematics are more important than others and this
should be reflected in texts and in class.
* Mathematics is not a secret guild where something is true because
I say it is or because a computer algebra system says it is. When
something simple and important is studied, reasons should be given.
* Words are important and their meanings should not be changed without
very good reasons.

Examples where these have not been followed have already been given.
There are many more in newer books. Since first mentioning these,
I have decided to add one, which is very important for textbook writers
and curriculum developers to observe.

* Be careful that what you are doing does not lead others to make
changes that will hurt the long-term education of students.

In his book [6] on textbooks written for the TIMSS study, Geoffrey
Howson makes the following point: "The passing of the 1960s emphasis
on algebraic structure need not be regretted. What is sad is that
it has not been replaced by some other clear philosophical or pedagogical
structure more appropriate to school mathematics." He ends this
paragraph with "A first attempt to establish such a framework of `recurring
themes' has been made by Gardiner [4]. It is an idea which deserves
further consideration, development, and elaboration."
In the absence of such guidelines, textbook writers, curriculum
developers and test writers will look at the current curriculum
and try to provide material that will get students ready for later
courses. Thus, one frequently overlooked point is how changes being
made for one reason will impact in other ways. The newer calculus
books tend to be more qualitative, and this is starting to show
up on the AP Calculus exams.
For various reasons, which will not be listed here, the knowledge
of arithmetic and algebra that students starting calculus have has
fallen. As a response to this poorer knowledge of algebra, the
Harvard Consortium has tried to finesse the problem by emphasizing
the use of graphing calculators. Other reasons are given
for this, but a quotation from Tony Phillips suggests that this
was a major factor. After saying that students' manipulative skills
have become much weaker, Phillips continued with:
"And the HCC curriculum makes a great virtue out of this necessity.
By eliminating some of the symbolic manipulation from calculus, they
were able to make the course more accessible to students." This
was written in a newsletter from the Harvard Calculus Consortium.
The report from a committee looking at the future of the AP
Calculus exam reads like a description of the Harvard Calculus book.
This is a very poor idea. Let me explain why with an analogy. When
my son was in high school, the precalculus book they used was
[3]. This was a very nice book, and he learned a lot from
it. We could not use it for the corresponding course at
the University of Wisconsin. Our course went about twice as fast,
and the students who took it were in general not as good
at mathematics as those who took this in high school. Many
of the students who will use mathematics seriously are now taking
calculus in high school. They need to develop technical skills
beyond those of students who take calculus for the qualitative
ideas there. The past AP Calculus exams were reasonable exams.
My finals in first and second semester calculus were in general
a bit harder than the AB and BC exams. A few years ago, there
were a couple of students in calculus lectures who wanted to
transfer to California Institute of Technology. Exams were
given to these students, in chemistry, mathematics, and physics.
The mathematics exams over a three year period were sent to
me, to share with the students or for them to take. They
were a bit harder than the final exam in my second semester
calculus course, as is appropriate. They should be at the
level of exams for an honor section. The MAA has published
translations of university entrance exams for some Japanese
universities [12]. The one for students who want to study humanities
at Tokyo University is harder than the sophomore placement
exam for Cal. Tech. Thus, the level of the traditional AB and BC
advanced placement exams was not too high. It is likely that
the newer one will continue at an equally high level for a couple
of years, but from then on it needs to be watched carefully.
There are experienced high school teachers who feel the earlier
exams were harder than those given in the last few years, and
with a more qualitative exam it will be very easy to have
the level slip as students who take the exam have less
technical skill.
The message from the Calculus Reform programs that is being
heard is that students do not need to be able to do algebra well.
The message from the NCTM Reform is that students do not need
to know how to do arithmetical calculations well. Both of
these messages are different from those sent by the countries
that did best on TIMSS. See [2] and [10].
Technology has a place in mathematics instruction, but it
needs to be used carefully. Until much more is learned about
the drawbacks, it should not be pushed too much. In England, the
results on the age 11 exams in the summer of 1995 were so
poor that six months later a decision was made that
calculators were no longer to be used
in one of the two maths exams. A teacher in Milwaukee talked
at the Wisconsin Mathematics Council meeting in May 1995. He
said that he was probably the first one to use calculators in
high school in the Milwaukee area, and also the first to use
graphing calculators. However, he now has some serious doubts
First, students do not know enough yet to profit from their
use to the same extent that teachers could, with their greater
knowledge. Second, he looked at books from 30 years before
and found that many topics now in second year algebra were in
first year algebra then, topics from second year algebra then
are frequently done in precalculus now, and quite
a few things that were once done in precalculus are
not done in high school now. He did not mentioned specific
things, but conic sections comes to mind as something that
is frequently not done now, either in precalculus or in calculus.
The binomial theorem is another. Both belong somewhere, and
high school seems the right place. Unfortunately, NCTM put conic
sections down for decreased emphasis. Other countries do them and
many other things that we do not do.
We need to look seriously at what is being done in the rest of the
world, to see what our students could learn if they had a good mathematics
program. Then we need to develop one. The most important part
of this is not calculus, but elementary school. However, what
we do in calculus has an impact on the rest of our mathematics
program, so we need to be very careful about what we do and how we
talk about what we are doing. The medical advice of "Do no harm"
is good advice for us as well.

[1] B. B. Barrett et al, Contemporary Precalculus Through Applications:
Functions, data analysis and matrices, Janson, Dedham, MA, 1992.
[2] Albert Beaton et al, Mathematics Achievement in the Middle School
Years, TIMSS International Study Center, Boston College, Chestnut
Hill, 1996.
[3] Robert Fisher and Allen Ziebur, Integrated Algebra and Trigonometry
With Analytic Geometry, Second Edition, Prentice-Hall, Englewood
Cliffs, N.J., 1967.
[4] Anthony Gardiner, Recurring Themes in School Mathematics, Birmingham
Univ., UK Mathematics Foundation, Birmingham, 1992.
[5] Geoffrey Howson, Mathematics Textbooks: A Comparative Study of
Grade 8 Texts, TIMSS Monograph No. 3, Pacific Educational Press,
Univ. of British Columbia, Vancouver, 1995.
[6] Deborah Hughes-Hallett, Andrew M. Gleason, et al, Calculus,
Wiles, New York, 1994.
[7] Kunihiko Kodaira, editor, Japanese Grade 9 Mathematics, 1984,
Project, Chicago, 1992.
[8] Tatsuro Miwa, Mathematics in Junior and Senior High School in
Japan: Present State and Prospects, in [11], 172-190.
[9] William Schmidt, Curtis McKnight and Senta Raizen, A Splintered
Vision: An Investigation of U.S. Science and Mathematics Education,
Kluwer, Dordrecht, Boston, London, 1997.
[10] Izaak Wirszup and Robert Streit, Developments in School Mathematics
Education Around the World, Proc. of UXSMP International
Conference on Mathematics Education, Univ. Chicago, March 1985,
NCTM, Reston 1987.
[11] Ling-Erl Eileen T. Wu, Japanese University Entrance Examination
Papers in Mathematics, Mathematical Association of America, Washington,
D.C., 1993.

Dept. of Mathematics