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Topic: Importance of Algebra, Prof. Askey
Replies: 5   Last Post: May 31, 1997 5:25 PM

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david klein

Posts: 125
Registered: 12/6/04
Importance of Algebra, Prof. Askey
Posted: May 27, 1997 10:25 AM
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Dear AP Calculus Teachers,

A discussion has evolved in this forum on the need for basic techniques
of algebra in order to learn calculus and to pursue higher mathematics.
Richard Sisley, and no doubt many others, maintain that less attention
should be given to basic algebra skills because of the availability of
computer algebra systems to students.

Richard Sisley even claims that such skills as factoring quadratic
polynomials over the integers need no longer be taught.

The following message comes from Richard (Dick) Askey, John Bascom Professor
of Mathematics, at the University of Wisconsin, Madison (one of the
premier institutions for mathematics in the world). Dick is a fellow of
the Indian Academy of Sciences, a fellow of The American Academy of Arts
and Sciences, and currently serves on the American Mathematical Society
committee to advise NCTM about a revised version of their Standards.

Dick has kindly given examples of algebra problems he would like good
students to be able to solve. The last one he gives in his message below
is particularly interesting because he suggests using a computer algebra
system for the initial stages of a problem. The completion of the problem
is up to you with whatever tools you choose.

EXERCISES HE PROVIDES (especially the last one in his message).

I also strongly encourage any math teacher to take advantage of Dick's
generous offer and request a copy of the algebra notes he and Prof Wu
wrote. I have a copy. The notes are clear, concise, and beautifully
written. They are valuable for any teacher of mathematics.

David Klein
Math Dept.
California State University, Northridge

Dear David,
Pass this on to the AP-calc group.
I have been getting some of the messages about what algebra students
should learn in high school. Hung-Hsi Wu at Univ. of California-Berkeley
and I have written a set of notes about what we think students should
have learned in high school mathematics before they start calculus.
I can send a copy to people who are interested in seeing these notes.
Send a message asking for them to I may have
to stop sending if there are more than 100 requests, and will have
to have more printed in any case, so it may be a little while
before these are send. Send a US postal address.
One thing which has not been mentioned in the messages sent to
me is work with equations with parameters. This is very important
in both mathematics and physics, but little is done with it in high
school mathematics according to what students tell me and from what
I read in school texts. One very nice problem is to find the rational
points on the unit circle this way. Consider the unit circle and
the straight line through (-1,0) with slope t. To find the points
of intersection, replace y^2 in x^2 + y^2 = 1 by y = t(x+1) to get
x^2 + t^2(x+1)^2 =1. At this point, I ask students if this is a
hard or an easy problem to do. They should be able to say it is
easy, for it is clearly a quadratic equation and all of these can
be solved via the quadratic formula. However, very few students
are able to do see this. They do not understand what polynomials are,
or even recognize quadratic polynomials when the quadratic is
not given in the form ax^2+bx+c or as something like 6x^2+4x-5.
Students need much more experience seeing polynomials in different
forms. To show that all rational points are found this way takes
a little more work, but finding the formulas for Pythagorean triples,
which is what is being done here, is enough to interest most students.
There were questions about whether or not higher degree polynomials
ever arise which one needs to factor. Here is an important example of
one. You all know that binomial coefficients count things. One
thing they count is a refinement of the number of ways of putting
0s and 1s into n places. C(n,k)=n!/k!(n-k)! counts the number of
ways of putting k 0s and n-k 1s into n spots. This is a refinement
since the sum on k from 0 to n of C(n,k)=2^n, which is the number
of ways of putting zeros and ones into n spots. There is a further
refinement. Consider the following numbers

0011 0
0101 1
0110 2
1001 2
1010 3
1100 4

This is a listing of the ways of putting 2 zeros and 2 ones into
four spots. The numbers on the right give the number of inversions
needed to switch 0s and 1s to get back to the original pattern of
zeros on the left and 1s on the right. Form a generating function
of the numbers in the right hand column. Write this as
1q^0 + 1q^1 + 2q^2 + 1q^3 + 1q^4 = 1 + q + 2q^2 + q^3 +q^4.
When q=1, this is 6, as it should be. However, the binomial coefficient
C(4,2) is not only 6, it is better written as 4!/2!2!, so we want
to write the fourth degree polynomial above in a similar form.
This can be done by factoring it first. One way to do this is to
write it as
1+q+q^2 +q^2+q^3+q^4 = 1+q+q^2 +q^2(1+q+q^2) = (1+q+q^2)(1+q^2).
This is analogous to 3*2 rather than 4*3/1*2. To get the better form,
sum the geometric series to get (1-q^3)(1-q^4)/(1-q)(1-q^2), and
then put in the extra factors to get (1-q)(1-q^2)(1-q^3)(1-q^4)/
(1-q)(1-q^2)(1-q)(1-q^2), which starts to look like 4!/2!2!. To
make it look even more like this, divide each factor by 1-q and expand
via the sum of a finite geometric series. If n!_q=1(1+q)...(1+q+...+q^(n-1)),
then this is

This is probably new to all or almost all of you. The idea of having
products like those found above is old, and a version of this was
found in the early part of the nineteenth century. The q-extensions
of the binomial coefficients are called Gaussian binomial coefficients.
The interpretation of the coefficients in the expansion of these
coefficients as polynomials in q that was given above was found by
MacMahon about 1915. There is a version of this due to Schutzenberger
in 1953 in the form of a non-commutative version of the binomial
theorem. Consider (x+y)^n and expand it, but use the strange multiplication
rule that yx=qxy, qx=xq, qy=yq. The result is sum of C_q(n,k)x^(n-k)y^k,
where the Gaussian binomial coefficient has what should be the obvious
extension of C(n,k), using the case n=4, k=2 as a model. It is a nice
exercise to show this is true by induction.
Detailed ways of counting which lead to such beautiful formulas will
certainly have important uses, and this now does. There are objects
called quantum groups, which first arose in some work in mathematical
physics, and now occur in many areas of mathematics. This non-commutative
version of the binomial theorem is central in this subject. Toward
the end of his active career, George Polya wrote a couple of papers
about inversions and Gaussian polynomials. These were written after
Schutzenberger introduced the noncommutative version of the binomial
theorem, but Polya was unaware of it. His version was more complicated
to understand. Even so, he suggested that this would be a nice topic
for high school mathematics. The commentator on these two papers in
Polya's "Collected Papers" expressed reservations about this. I have
done this in a high school mathematics club, and the students could
follow and help with some of the work, and found it interesting.
One thing which came up in the argument above was the finite
geometric series. Students need to know this, and very few of them
do when they get to college. The first day in first semester calculus,
I ask the students to turn a repeating decimal into a fraction, say
.454545... At most 20% are able to do this even in an honor section,
and only about half of these know how to do it by calling the number
x, and multiplying by the appropriate power of 10. The other half
now do it by pattern matching, saying it is 45/99, but have no idea
why. The first method is important, for it leads to the sum of
the general finite geometric series. The other method is a dead end,
since it leads nowhere. Unfortunately, there were a number of letters
in Mathematics Teacher praising a student Joey for discovering this
pattern matching, and none pointing out that the method is a dead end,
and so should not be encouraged.
There is a lot of symbolic manipulation which students should learn
in algebra. The book "Algebra" by I.M. Gelfand and A. Shen has some
very nice problems. This is published by Birkhauser-Boston, and the
paperback version costs under $20. Both Tom Romberg, who chaired
the committee which wrote the NCTM Standards, and Hyman Bass, chair
of the Mathematical Sciences Education Board, have written that
this book "corresponds to the NCTM Standards". I can not make that
claim, since I do not understand many things in the "Standards", but
claim that it is a very good book, full of interesting problems
and explanations for why things work as they do.
Here is one problem from this book, along with a solution showing
how one can do the factoring,
and a couple of things which follow from the factoring. It is a cubic,
and the argument is not obvious, but one learns things from it.

x^3 + y^3 + z^3 - 3xyz.

I will skip a screen so that those of you who want to work on this
can stop reading, and scroll down once you have either solved it or
decided you spent enough time.


Observe that this polynomial vanishes when x=y=z. Set z=x+a, y=x+b.
Then the polynomial is

x^3 + x^3 + 3x^2b + 3xb^2 + b^3 + x^3 + 3x^2a + 3xa^2 + a^3
- 3x(x^2 + ax + bx + ab)
= 3x(a^2-ab+b^2) + a^3 + b^3 =(3x + a + b)(a^2 - ab + b^2)

One thing which follows from this is the arithmetic-geometric
mean for three numbers. For,
x^2+y^2+z^2-xy-xz-yz =1/2[(x-y)^2 + (y-z)^2 + (z-x)^2]
which is non-negative. The general solution of a cubic also follows,
since the quadratic just given above can be factored into two linear
factors, when treated as a function of x. Consider the general
cubic equation.
We can assume a=1 by division, and then that b=0 by shifting x by
a constant to remove the x^2 term. Thus we can consider
x^3 + cx + d =0 = x^3 - 3yzx + y^3 + z^3.
Set -3yz=c and y^3+z^3=d, and solve for y and z, by finding y from the
first equation, putting it in the second and solving the resulting
equation which is a quadratic in y^3.
Can all students do this? Probably not given the weak background
they have. Should many students see this? Yes. If we look at the
results of the eighth grade TIMSS test, we see that we have 25% of
our students in the bottom 25% of the students internationally.
However, we only have 18% of our students in the top 25% internationally,
and only 5% of our students in the top 10%. This seems to say that
we are cheating our better students more than our poorer students,
if we measure by what we should expect them to accomplish.
Contrast this with results from Singapore, where only 1% of the
students are in the bottom 25% and 45% are in the top 10%. Yes, all
students can learn mathematics, and real mathematics, not the watered
down version we have been giving to our students.
For those of you who like to use computer algebra systems, try
the following problem. Let

f(x) = -8(1-(1-x)^(1/2))^3

Compute f(f(x)) and ask the computer algebra system to simplify it.
Then graph it and get a surprise. Finally, try to prove that
the surprise is true. Something which has been outlined in this
message will help.
Dick Askey
Dept. of Mathematics
Univ. of Wisconsin-Madison
480 Lincoln Drive
Madison, WI 53706

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