What Will Be the Effect of a Standards -based
Education on College Students?
This was the title of a panel that I spoke on at the symposium Math
Reform Goes To College, sponsored by the Mathematical Sciences Education
Board, at the National Academy of Sciences last year. None of the
panelists had an answer, of course, since no answer exists right now.
But the question raised a lot of issues, and it is those issues that I
would like to address here.
Just what are the Standards?
The National Council of Teachers of Mathematics' Curriculum and
Evaluation Standards , published in 1989, took some time to be
noticed by mathematicians not directly involved in K-12 education, and
probably still is not familiar to most of us. The Curriculum and
Evaluation Standards was followed by the Professional
Standards and about a dozen short volumes in an addenda series.
Soon to appear are the Assessment Standards and a Framework
for Constructing a Vision of Algebra . The Standards , as these
have collectively been called, is exerting a profound influence on K-12
education in the United States. As those of us not involved directly in
K-12 have become aware of the Standards , we have tended to divide
into two camps: those who defend the Standards and those who
I defend the Standards . Here is my version of what they say: The
mathematics that children learn in school will be meaningful. It will
make sense, it will be creative, it will not be severed from other human
activities, technology will be used appropriately, and there will be
real mathematical content.
Here are the accusations against the Standards : None of the above
will happen. Basic skills will be lost, appreciation for algorithm will
be lost, in K-12 the sense of the subject's direction will be lost,
appreciation of formal systems will be lost, appreciation of abstraction
will be lost, mathematical content will be lost, the best and the
brightest will be lost, and no one will graduate from high school
knowing significant mathematics.
Many opponents of the Standards add that in fact this has already
I agree that basic skills, appreciation for algorithm, sense of the
subject's direction, appreciation of formal systems and abstraction, and
mathematical content are missing in many of my university students, and
have been for a long time. I don'tthink we can blame the
Standards for this. They simply haven't been around long enough.
While back in 1989 there were some curriculum projects compatible with
the Standards already in existence, they had been around for only a
couple of years. No child currently past 9th grade has had the
possibility of what is known as a Standards -based education, and
very few children in 9th grade or earlier have spent the bulk of their
mathematical education in what are known as Standards -based
classrooms. Just because the NCTM formally adopted them doesn't mean
the average school district or classroom teacher has changed very much
of what she or he does. No, the students in my freshman classes (some of
whom are seniors who have taken freshman math courses many times in
order to graduate) are products of the old system.
The fact that people of good will and intelligence can disagree so
vehemently about the Standards tells us that they are, in fact, quite
ambiguous. In some sense what you read into them is what you get, and
just because someone says that what they do is Standards -based
doesn't mean it is. I believe the Standards provides an
opportunity for real mathematics to be done in the classroom. Others
believe the Standards will drive real mathematics out of the
classroom. That is the overriding issue in the debate over the
What are the controversial issues raised by the
The Standards emphasizes applications. When I read the
Standards it is clear to me that "applications" includes
applications to other areas of mathematics, and I have heard this view
defended quite eloquently by K-12 teachers. But when other people read
the Standards it is clear to them that, as an award-winning high
school teacher once told me, "Meaningful mathematics is something that a
child can apply in his or her daily life." What do we mean by
applications, and how large a part should they play in the
The Standards coincides with and strengthens another movement in
education, the concern with equity. It's becoming indefensible to teach
a course called Business Math in high school which is just arithmetic.
But when you decide that no child of remotely normal intelligence is to
be thrown on the mathematical junk heap, you have to ask how to teach
all children, and what to teach, and this turns out to lead to some very
hard questions. What worked for those of us reading this is known not
to work for the majority of children - that's why we literally have
thousands of students in my university taking essentially high school
mathematics (half of them in a course misnamed College Algebra) before
they can take a university math course required for graduation. We are
not unusual in this. How are we going to teach algebra and geometry
successfully and, these days, probability and statistics and a little
finite combinatorics, to all children, starting in kindergarten and
going through high school? And what is the algebra and geometry and
probability and statistics and a little finite combinatorics that we
believe all children should learn? That we believe all future college
students should learn? That we believe all future engineers and
scientists and mathematicians and economists and etc.- the heavy users
of mathematics @ Drexel should learn? These are hard questions. I and many
others think that the best answer is to strengthen the curriculum in
interesting directions. But others disagree (read on for process vs.
The Standards calls for the development of mathematical
reasoning. But this means very different things to different people.
Some very thoughtful high school teachers are throwing out theorem
proving in geometry classes - the Standards doesn't tell them to
do so, but they feel they have permission if they see fit. On the other
hand, all teachers who use the Standards, from kindergarten on,
demand that children defend arguments and explain their reasons. What
should mathematical reasoning mean at which level of mathematics? When
should formal proofs be introduced, and how?
Less emphasis on algorithms
The Standards calls for less emphasis on standard algorithms.
This doesn't necessarily mean no emphasis, although some people have
interpreted it that way, including mathematicians. (The issue isn't
algorithm vs. no algorithm, but standard algorithm vs. a child's
discovered algorithm.) How much emphasis on standard algorithms is
There is the issue of process vs. content. I see the Standards as
allowing thinking, if you will, into the mathematical curriculum @ Drexel
that's what seems to be meant by process @ Drexel hence allowing for more
interesting content. A mathematics supervisor I know believes
approvingly that process is so much more important than content as to
make content almost not matter @ Drexel the purpose of the mathematics
curriculum for her is to teach kids how to think. How much time do we
devote to exploring the process of doing mathematics, and how does this
effect the content we teach? This is not an easy question.
There is something else about the Standards which isn't always
publicly talked about, but which probably leads to a lot of the
resentment among mathematicians, and that is that very few
mathematicians were involved in their creation. People don't talk about
it because when you do you get very angry responses of, "oh yeah, just
how do you define 'mathematician,' you elitist mathematician you." But
it's a fact that very few of the people involved in writing any of the
Standards have published a mathematics paper, have a Ph.D. in
mathematics, or teach upper division mathematics courses. While every
member of the NCTM (the vast majority of whom are K-12 teachers) was
given a chance to respond to drafts, classroom realities made it hard to
involve K-12 teachers on the Standards committees as well. The
Standards-writing committees were predominantly Ph.D.s in
education who do not teach K-12 classes. I'm not sure what effect this
had on the actual documents, but it certainly has had its political
consequences, both in mathematics departments, and to some extent also
in K-12 classrooms.
The questions the Standards raise are important ones. Whether or
not the mathematics community has been involved in the past, we need to
be involved in the future. Each community with a major interest in
mathematics education @ Drexel the K-12 community, the mathematics educators,
the mathematicians @ Drexel has its own perspective, and all of these
perspectives are needed if things are going to improve the way they
What should we do next?
Now we come to the punch line, which is that nobody's got everything
right. We will not improve mathematics education by impugning each
other's professional integrity, by sneering at each other's professions,
by assuming that we're right and everyone else is wrong. Yet that's
exactly what's been happening: K-12 teachers are denigrated by college
folks as people to be talked at or even down to; mathematicians are
considered unspeakably elitist, arrogant, and naive by non-
mathematicians; and folks from education departments are considered
dull, obfuscating drones by mathematicians and irrelevant by K-12
teachers. Needless to say, this is not helpful. There are smart people
everywhere, and these smart people need to work across disciplinary turf
and prejudicial lines.
I started off saying that I defend the Standards. This doesn't
mean that I agree with everything that supposedly supports the
Standards @ Drexel I couldn't, because there are too many contradictory
claims @ Drexel or that I consider them etched in stone for all time. Yes, I
believe that there are some trends which could be dangerous in
mathematics education @ Drexel there are always dangerous trends in education,
although one person's danger is another person's opportunity. I
absolutely believe that mathematicians need to be listened to in K-12
education, and that our role is considerably more than simply nodding
our heads in agreement with whatever other people have cooked up. But
if we want to be listened to, we have to listen. We have to admit that,
whether or not we agree with their conclusions, some folks who are not
mathematicians know some things we can learn from.
And we have to watch our rhetoric, or we'll continue to ensure that the
mathematics community has no voice in what is going on in K-12
Judith Roitman is a professor of mathematics at the University of
This article was taken from FOCUS, Vol. 15, Num. 2. Ap '95 pgs 9-10.