History of Mathematical Instructional Controversy in the United States
since 1960s
Richard Askey
Some background needs to be given before getting to the new math period
in the 1960s.
Lee Shulman's Presidential Address to the American Educational Research
Association (1986) started with the contrast between what knowledge was
expected of teachers about 1875 with what was expected in 1985.
Content was the main focus in 1875, 19 of 20 sections in California
exams, while in 1985 content was almost completely missing. Here is
one mental arithmetic question which was on an exam in California in
1875.
Divide 88 in two parts that are to each other as 2/3 is to 4/5.
As part of a diagnostic test given at the start of a course in
arithmetic for prospective elementary school teachers I gave this
question. Of the 26 students, one solved the problem with algebra, a
second student solved it by guessing an answer and checking it. The
other 24 were not able to solve the problem.
Neither of the two solutions are what we hope teachers will be able to
do, solve the problem using arithmetic.
One item which will or has been distributed is an article on rational
numbers which appeared in the Wisconsin Teacher of Mathematics. Two
weeks after this article appeared, the newsletter from the Wisconsin
Mathematics Council had an article about the Presidential Award winning
elementary school mathematics teacher, an award give every two years,
one per state. The winner was the author of this article on rational
numbers. See how many mathematical errors you can find in the article.
As you can see, content has not been a high priority in some circles.
We skip to about 1920. Two important reports appeared about this time.
One was from the National Educational Association, which at that time
was an organization of school administrators. They felt that the large
number of new students who were starting to attend high school were not
capable of doing the rigorous work which earlier students had been
doing, algebra and geometry being two examples. They recommend a
course on general mathematics be introduced in high school. Data on
the number of students taking this course shows up about 1920.
Initially this was to be a course with some substance, but it rapidly
became a remedial course with a lot of consumer arithmetic. The
subject changed over the years but the level did not change.
The other report was from a committee from the Mathematical Association
of America. Among other things it looked seriously at what other
countries were teaching students at various grade levels, and
recommended that algebra be taught in grade 8 rather than grade 9 as
had been traditional.
One can see who won the battle between these two positions, we are only
now starting to approach 50% of our 8th grade students taking algebra.
However, for better or worse, consumer mathematics has disappeared.
Skip again to the post World War II era. There was general
dissatisfaction with the knowledge of far too many students who went to
college. A project started at the Univ. of Illinois, and it involved a
good high school teacher, and a logician in the Mathematics Department,
among others.
Shortly after this the US Government put a lot of money into
mathematics education, partly funding the School Mathematics Study
Group to write prototypes of new textbooks, partly for training
teachers in what was felt to be the way of mathematicians doing
mathematics, and some for conferences. Some of the programs built on
what was called New Mathematics were far too formal. New Math programs
were to be for high school and a little further down, and for the top
25% of students. However, elementary school books based on what was
felt to be the ideas of modern mathematics were used. We spent a year
just outside of Amsterdam. When we got back and our son started sixth
grade, he said: "We did sets in first grade, second grade, third grade,
fourth grade, did not do them last year, but are now doing them again.
Do we have to do this every year?" Of course I said that while sets
are important, one should not spend much or any time on them in
elementary school. A little later a set of books was being adopted in
the Madison, Wisconsin school district. One of the books circulated in
the Math Dept and I had it for a couple of days. Opening it almost at
random, I found a page in which addition of rational numbers was done.
A rational number was a pair (a,b). Two such numbers were added as
follows: (a,b)+(c,d)=(a+c,b+d). This made no sense. It does when I
finally found that (a,b) is not a/b, but a-b where a and b are
"rational numbers of arithmetic", nonnegative rationals. What was
being introduced was the negative rationals.
There was a public letter from over 50 mathematicians about some of the
directions being taken. It appeared in The American Mathematical
Monthly and in Mathematics Teacher.
There was a reply from Edward Begel. Both are still worth reading.
By sometime in the 1970s it was clear that this reform had failed,
although it had introduced some useful topics in school mathematics.
The replacement was called "Back to the Basics", which emphasized an
important topic in school mathematics which had been neglected in the
New Math, computational ability. By 1990 the National Council of
Mathematics had started to call for a program based on problem solving.
They published a document which had almost no impact. Then came the
1989 NCTM Curriculum and Evaluation Standards which had a major impact.
The story behind this is interesting, and has not been fully told.
One of a number of parts which led to controversy was the inclusion of
lists calling for Increased Attention to certain things and Decreased
Attention for others.
Listed for decreased attention were such things as manipulating
symbols, synthetic geometry, memorizing almost anything and technical
skills. Conic sections were listed in the decreased attention list.
This is an example of what can happen when something which is already
very weak is put down for decreased attention. In TIMSS-95, the
advanced math part for some students in the last year of high school,
one question was: What is the following curve:
4x^2-9y^2=36, a circle, an ellipse, a parabola or a hyperbola. About
1/4 of the US students wrote it was each one of these, between 23.x to
25.y percent.
In essence these were completely random guesses. All of them should
know this is not a circle, most should know it is not a parabola since
the word "parabola"
is used to describe y = kx^2 as soon as students learn to graph y =
x^2.
To give you an idea about the dropping of essential skills, here is
part of what was written about some computational work with fractions.
"The mastery of a small number of basic facts with common fractions
(e.g., 1/4 + 1/4 = 1/2, 3/4 + 1/2 = 1 3/4; and 1/2 * 1/2 = 1/4) and
with decimals (e.g., 0.1 + 0.1 = 0.2 and 0.1 * 0.1 = 0.01) contributes
to students' readiness to learn estimation and for concept development
and problem solving. This proficiency in the addition, subtraction,
and multiplication of fractions and mixed numbers should be limited to
those with simple denominators that can be visualized concretely or
pictorially and are apt to occur in real-world setting: such
computation promotes conceptual understanding of the operations. This
is not to suggest, however, that valuable instruction time should be
devoted to exercises like 17/24 + 5/18 or 5 3/4 * 4 1/4, which are
much harder to visualize and unlikely to occur in real-life situations.
Division of fractions should be approached conceptually. An
understanding of what happens when one divides by a fractional number
(less than or greater than 1) is essential."
Need I write more? Programs based on such a premise are likely to get
quite a few people up in arms.
The next controversy started in California. In 1992 California adopted
a new framework. About six or seven years later Tom Romberg, who
chaired the committee which wrote the
1989 Standards, mentioned in a talk that he wrote the committee writing
the California 1992 Framework that they had forgotten to include
mathematical content. A large fraction dealt with process and methods
of teaching. Unfortunately, Romberg's comments to the California group
were private until he mentioned them about 10 years ago, and when they
were circulated it was too late to stop the so called "Math Wars".
These started in California, with one group in Palo Alto, where
Stanford is, and a second group in southern California, mostly in San
Diego. The second group set up a website, Mathematically Correct, and
some parents from around the US used it. There were other groups of
parents who had similar concerns, in Illinois, somewhat later in New
York City, and the western states of Washington and Utah developed
groups later.
The peak of the controversy probably came when the US Department of
Education set up an Expert Panel to evaluate school math programs.
This was in the later part of the 1990s. The report was published in
1999, and one of the programs listed as promising was MathLand. This
was an elementary school program which was used by many California
schools.
I was one of 92 reviewers of programs, and the Expert Panel used the
reviews and also reviews of evidence of success to decide on the
programs to rank as exemplary or promising.
10 were initially listed, five in each category, and later two were
added to the promising list.
Of the reviewers, one other had a Ph.D. in mathematics, and he had been
doing educational administration work in California for something like
25 years. I knew when the guidelines were explained at the training
session that this review would cause problems, but never in my worst
nightmare did I think that MathLand would be listed. I was not alone
in concern after the results were announced.
A few of us got letters and emails from parents and teachers asking
what could be done about the problem of districts restricting their
choice of books to those ranked in this review. To aid these people a
letter was written and funding secured to publish it as a full page in
the Washington Post, asking that the recommendations be withdrawn.
Needless to say, there was an immediate response from NCTM saying that
the process was fine and that this letter was an attack on some members
of the Expert Panel who were members of NCTM. What was not known
publicly at that time was that the one mathematician on the Expert
Panel either voted against or abstained from voting for each of the
12 programs. This came out in a talk he gave when being given the
Mathematical Association of America's highest honor for professional
service. More can be said about this review in the discussion, but let
me conclude my comments on this by saying that when I got copies of the
48 reviews of the 12 programs listed and read them, none of the
reviewers mentioned a single error in their review. A program can be
promising and even exemplary and have a number of errors, but if the
reviewers did not mention any to help the publishers and developers to
improve their books, you can forget this review.
The next important event was the publication of Liping Ma's book
"Knowing and Teaching Elementary Mathematics".
This provided an existence proof that it was possible to have a good
enough group of elementary school teachers who could teach mathematics
well, and included comments from teachers about how they could answer
some important mathematical questions related to teaching. In
particular, one of the questions used (all of the four having been
developed and first used by Deborah Ball) dealt with division of 1 3/4
by 1/2. The Chinese teachers described this as multiplying by the
inverse. US teachers had traditionally said "invert and multiply".
Neither is perfect, but the Chinese description has mathematical
meaning, the US one does not. The fact that both sides of the Math
Wars praised this book gave some hope that the divide could be bridged.
The next thing which came out was in 2000. NCTM published PSSM
(Principles and Standards for School Mathematics).
This was an improvement over the 1989 Standards, but there were still
problems. Division of fractions was at least mentioned as something
students should learn something about. Unfortunately, when an analogy
was made to whole number arithmetic, the analogy was to division as
successive subtraction rather than as an inverse to multiplication. I
keep asking how one divides 1/5 by !/3 by successive subtraction.
There is a second problem which I have never written about other than a
private note to one of the writers. Recall that in the 1989 Standards,
synthetic geometry was put down for decreases attention. One important
result in high school geometry is that the medians of a triangle
intersect at a point which is 2/3 of the way from a vertex to the
opposite side. In both versions of Standards, NCTM has pushed multiple
ways of solving problems. This is an ideal problem to do in more than
one way. There are beautiful synthetic proofs which use important
ideas, and students need to see important ideas used in different
settings. There is a proof via vectors, but without an emphasis on
this form of geometry, it would be a little too hard for almost all
students. What was done was to give an algebraic proof, starting with
writing that if the vertices are given in general position, the
resulting equations are too complicated to solve by hand. The vertices
are taken at (0,0), (2a,0), and (2b,2c). The factors of 2 are too
clever by half, and were only included so that fractions only show up
at the last step. The 2/3 part of the theorem is unlikely to be
noticed by all except the very best students, and teachers if they do
not know the theorem with this as part of the statement. The synthetic
proofs I know have the 2/3 come up naturally, as do vector proofs. If
one takes the vertices in general position, one only needs to write one
equation of a line and think what symmetric point is likely to be on
this line. The obvious extension from the two dimensional case of the
midpoint of a line segment satisfies the equation of the line and you
are done. This proof extends immediately to three dimensions once one
learns how write equations for a line using a point and a direction.
To do this with special choices for the vertices leads to computational
nightmares. I mention this since one new geometry text proves this
theorem using exactly the same notation used in PSSM. It is very
important to do mathematics well in books like this.
Next, there were two very important developments which showed that it
was possible to bridge the gap. Six people, two of them here, three
math educators and three mathematicians, were able to reach common
grounds on many areas in school mathematics. Their short report can be
read on the web, at http://www.maa.org/common-ground/cg-report2005.pdf
The other was a year by year program sketched by NCTM from preschool to
grade 8. While there are parts of it which I think could be improved,
it seems very workable. Both of these raise hopes that the
controversies of the last 20 years can be put to rest.
Balanced against that is a draft which NCTM put out for public comment
dealing with high school mathematics. As it stands, it will move us
back a lot, which is unfortunate. However, the report will not be
released until next spring so there is hope that changes can be made.
Finally, and probably most importantly, we have the report from the
National Mathematics Advisory Panel, which is a major focus of this
meeting. However, this report, which I think is very good, with the
usual caveat that there are a few things which I think should be
changed and that in certain areas the research base was not good enough
to allow the committee to make the recommendations I would have liked
to see, has not be received in the same way by all groups. Some of the
complaints probably came because people only read the small book, and
did not look seriously at the reports from the Task Groups. However,
there are people whose views would not be changed by reading the full
report, so it is still up in the air how influential this report will
be. I am optimistic, and pleased that this meeting has been set up to
hear comments on the report and to learn about some of the reactions to
it from people in China.