K–12
Mathematics
Education: How Much Common Ground Is There?
By Anthony Ralston
From FOCUS, January 2006
In the August/September 2005 issue of FOCUS there
was a brief
summary [1] of a document entitled “Finding Common Ground in K-12
Mathematcs
Education” (hereafter CG), whose full text may be found at
http://www.maa.org/common-ground
[2]. The authors of CG are two research mathematicians, three
mathematics
educators and the convener of the group, who is a senior vice-president
and
math and science policy advisor for a major American technology
corporation and
who has a Ph. D. in applied mathematics.
There has been much controversy about American school
mathematics in, at least, the past 15 years. The players have been,
roughly
speaking, the research mathematics community on one side, and the
mathematics
education community on the other. Thus, trying to find common ground
between
these two communities would appear to be a valuable exercise. And,
indeed, much
that is in CG will seem unexceptionable to almost all readers of FOCUS.
But creating
a document of consensus among six individuals who represent various
points of
view on the matter being discussed is fraught with difficulties.
One difficulty is that the attempt to be
unexceptionable too
easily results in blandness. Another is that, although the authors have
certainly wished to avoid ambiguity, they have not always succeeded. A
final
difficulty is that when, occasionally, there is a definite
recommendation, a
group of six — any six — is just not enough to assure that
there will
not be significant disagreement in the communities they are addressing.
The
authors of CG have not avoided these pitfalls.
There is no need to say much about the blandness of some
of
the statements in CG since it is inevitable that there will be some in
a
document like this. But statements like “All students must have a solid
grounding in mathematics to function effectively in today’s world”,
“Students
must be able to formulate and solve problems”, and “Teaching
mathematics
effectively depends on a solid understanding of the material” would
perhaps
better have been omitted or, preceded by “Since”, they could have been
in each
case attached to the sentence that follows.
I am sure the authors of CG strove mightily to avoid
ambiguity. Here are two examples when I think they have not succeeded
(at least
for me).
(i) “Certain procedures and algorithms in
mathematics are so
basic and have such wide application that they should be practiced to
the point
of automaticity.” But, without examples, what can this mean? The only
example
given is this one: “Computational fluency in whole number arithmetic is
vital.”
What other procedures and algorithms, if any, should be automatic? And
what
about “computational fluency in whole number arithmetic”? Does this
mean, for
example, that students should be expected to be fluent with the
traditional
algorithm for long division? At most a small fraction of students have
ever
become “fluent” with this algorithm. And with calculators so easily
available,
what expectation can there be that more than a small fraction of
students will
become fluent in the future? And why just “whole number arithmetic”? Is
arithmetic with decimal numbers less important than whole number
arithmetic? Certainly
not in the workplace.
(ii) “Calculators can have a useful role even in
the lower
grades, but they must be used carefully, so as not to impede the
acquisition of
fluency of basic facts and computational procedures.” Since some of the
authors
have in the past opposed any use of calculators in K–6, this is a step
forward.
But what is the second portion (“but …”) supposed to imply? If only
that
calculators should not be used mindlessly or for one-digit arithmetic,
then
this is a triviality not worth saying. If something more than this,
then what? One
suspects that, in order to accede to the first portion of this
sentence, some
of the authors insisted on the ambiguous second portion. This is a
standard
problem with consensus documents.
Some of the above might be viewed as mere quibbling
although
I think it is more than this. In any case, the examples below are of
issues
that will surely elicit disagreement with CG among a substantial number
of
readers of FOCUS.
(i) “By the time they leave high school, a
majority of
students should have studied calculus.” Leave aside the fact that this—
or
anything close to it — cannot be achieved in any foreseeable future.
Leave
aside also the fact that many students who now study calculus in high
school
come away from it with little understanding and little more than an
ability to
perform mechanically various algorithms, all of which can be done
better on a
calculator. But, anyhow, why would you wish half the students to have
studied
calculus? Too much of the mathematics community has failed to come to
terms
with the fact that discrete mathematics is (almost?) as good an
entrée to
college mathematics as calculus. Not to recognize this in a document
such as
this is to arouse the suspicion that too many of the authors are living
in the
past. If they had said “…a majority of students should have studied
first year
college mathematics”, that would at least have been a defensible
aspiration. I
would still not have agreed with it on the grounds of unattainability
but, at
least, the document would have sounded like it had had input from some
younger
mathematicians.
(ii) “Students should be able to use the basic algorithms
of
whole number arithmetic fluently, and they should understand how and
why the
algorithms work.” This statement, no doubt, is to forestall people like
me who
have advocated abandoning traditional instruction in paper-and-pencil
arithmetic [3]. But it sounds like voices from another century (the
20th!) to
expect that most students will become fluent in the traditional
algorithms when
it is obvious that, outside of school, many (almost all?) students will
use
calculators to do their arithmetic homework, no matter how much their
teachers
inveigh against it. And is there any chance that a significant number
of students
will “understand how and why the algorithms work”? <>
(iii) “The arithmetic of fractions is important as a
foundation for algebra.” Many of you may think this statement is
innocuous but
I don’t. No one doubts that any non-trivial study of algebra must
involve
arithmetic with algebraic fractions. But while students should learn
about
reciprocals and the conversion of fractions to decimals and vice versa
before
college, it does not follow that prior study of the arithmetic
of
numerical fractions, even if still remembered by the time algebra is
studied,
is a good or necessary prelude to this. Indeed, the addition,
subtraction and,
particularly, the division of algebraic fractions [4] is rather easier
than the
same operations for numerical fractions. So what if students come to
algebra
without knowing the arithmetic of numerical fractions? Just teach it as
part of
the algebra course. Not only are the algorithms generally easier but
the more
mature high school students will learn them more rapidly than middle
school
students. Then, if you wish, apply the algebraic algorithms to
numbers.<>It
is, I believe, almost surely futile at this time to
attempt to find significant agreement between the research mathematics
and
mathematics education communities on the major issues confronting
American
school mathematics education. The disagreements on various matters —
curriculum
and technology being perhaps the most profound and obvious — are just
too deep
at this time to allow any non-trivial consensus.
<>
Before such consensus can be reasonably attempted there will
have to be, at least, a level of respect in both communities for the
other that
will mean that inevitable disagreements need not erupt into shouting
matches. The
CG document evinces such respect but it is far from universal among
research
mathematicians or mathematics educators. Mathematics educators must
accept that
professional mathematicians, research and otherwise, through their
experience
and insights, have the potential to offer much to school mathematics
education.
Research mathematicians need to understand that college and university
mathematics educators generally, as well as many secondary school
mathematics
teachers, know and understand school mathematics. And research
mathematicians
will have to accept that the mathematics education community generally
knows
considerably more than they do about appropriate pedagogy for school
mathematics.
References
- Pearson, M., Finding Common Ground in K-12 Mathematics, FOCUS,
Vol. 25,
No. 6,
2005, p. 40.
- Ball, D. L., Ferrini-Mundy, J., Kilpatrick, J., Milgram, R. J.,
Schmid, W.,
Schaar, R, Reaching for Common Ground in K-12 Mathematics Education,
http://www.maa.org/common-ground;
also in Notices of the AMS, Vol. 52, No. 9, October 2005, pp.
1055-1058.
- Ralston, A., Let’s Abolish Pencil-and-Paper Arithmetic, http://www.doc.ac.ic.uk/~ar9/abolpub.htm.
- Ralston, A., The Case Against Long Division, http://www.doc.ic.ac.uk/~ar9/LDApaper2.html.
Response
As the convener of the team of research mathematics and
mathematics educators who are the authors of Finding Common Ground in K-12
Mathematics Education, (FCG) I felt it was necessary to comment
on the above-mentioned
piece. Speaking for the other authors, I must thank Anthony Ralston
for his in-depth analysis of our document. We will certainly consider
his remarks as we continue and expand
our work. They are very helpful.
However, I must comment on the last two paragraphs in his piece. As for
finding “significant agreement,” FCG
is an existence proof that such agreement can be developed from mutual
understanding starting with a good diversity of expert opinions. When I
convened the group, many confided that
they thought the group would agree on very little; after defining terms
and working on the issues, however, the group agreed on almost
everything. Will research mathematicians and mathematics
educators agree on everything? No, they
will not. Not all research mathematicians
(or mathematics educators) agree on everything, but it is the dialogue
and development of what they can agree upon that is the key. It is my
belief that there is enough
significant agreement that as a group, we can move forward in educating
our youth in mathematics, which is a crisis area for the United States
and is one that cannot wait to be solved.
Finally, from what I have seen, there is a lot
more respect between both communities than I was led to believe when I
started this
work. I have witnessed a great deal of cooperation and understanding to
solve the common problems of K-12
mathematics education. It is interesting that on the same page as the
two paragraphs, there is an advertisement for the Institute of Advanced
Studies’ Park City
Mathematics Institute. If you look at the participants and organizers,
you get a glimpse of the broad spectrum of participation
around the education theme of “Knowledge for Teaching Mathematics.”
Thus the small group I led does not represent
the only ongoing discussion aimed at bringing the community together to
find areas of agreement and to approach disagreement amicably and
respectfully. Such conversations are ongoing and
expanding. The community must continue to move beyond
questions of respect to get the job done.
-Richard Schaar, January 2006