Is Mathematics a Foreign Language?


Frank Allen was President of the National Council of Teachers of
Mathematics about thirty years ago, and is today one of its most prominent
critics. He agrees, however, that the failure in today's mathematics
teaching in the schools is not a localized phenomenon, but has features in
common with failure of school education in other fields as well. In
particular he identifies as a root cause of poor mathematics learning in
school the failure of school education (and home and street education more
generally) to teach children the proper use of the English language. For
purposes of this discussion I call it the English language, but only
because that is the standard language of this country; what I say applies
mutatis mutandis to other countries as well. That is, when I say (for
example) that students who have learned to speak English well can
therefore learn school mathematics the more easily, I mean to refer to any
civilized language.

There is a movement on in our departments of English in the
universities, to teach English composition via a theory called "discourse
analysis." I have only read one manifesto on this matter, written by a
man who holds appointments in both the department of English and the
School of Education in a university I shall not name here; but the thing
was so badly written that while I read it more than once I was unable to
make much sense of it. Generally speaking, it seemed to say that different
walks of life use different modes of discourse, and that a writer should
make allowance for this.

I know that it has been a truism for centuries that one must know
his audience before putting pen to paper, but somehow this observation has
here been elevated into a theory whose corollary is that Freshman
Composition as it was known to a previous generation is passé. You have
to write for chemists in chemistrese, according to the doctrine of
Discourse Analysis, and presumably for chiropractors in chiropractese, and
so on and so on, "each after his kind."

This theory cannot possibly appeal to chemists and mathematicians,
who understand better than those ignorant of such matters that chemistry
and mathematics are written in English. It does apparently appeal to some
elements in the theoretical pedagogy business, however, for (as I imagine)
without some such notion there might be no expertise attributable to their
own kind. Nor is it mere theory; even at the University of Rochester they
ask students to behave as if made sense. Thus, I received a freshman
student last year who asked my advice:

She had been required by her freshman English teacher to turn in a
paper written in the "discourse" of the subject she planned to major in.
In her case this was mathematics, and she was puzzled. She was taking a
calculus course, but didn't feel entitled to reproduce a page of a
calculus book for her paper, or a problem solution of the sort she
produces for math exams, because she was not sure that sort of thing
constituted "discourse".

I encouraged her to do exactly that, however, i.e. to turn in a
problem solution, complete with a careful statement of the problem and a
careful statement of how it was formulated so as to be amenable to the
methods of calculus, giving, as she went along, explicit reference to the
full statements, perhaps as appendices, of all theorems invoked to get
from one statement to another. I explained that her paper should be
written in sentences and paragraphs, but that she need not be afraid of
using symbols as ordinary English words when that's what they meant. I
also pointed out to her that all mathematics is written this way, though
apparently abbreviated because of the expenses of typesetting and
printing, time constraints during examinations, and -- believe it or not
-- ease of reading. All this was new to her, and interesting. The
abbreviations I referred to often make a page of mathematics look unlike
the English of Thomas Hardy, but in fact there is no difference except in
subject matter -- and skill.

What I did *not* credit in this student's professor's instructions
was the notion that her "discourse" should be different in principle from
what her classmate, the potential history major, probably was supposed to
submit in answering the same assignment. Do historians write a language
not understandable to mathematicians without special training in the
"discourse"? I think not. I agree that technical work in history may
well invoke references not recognizable to outsiders, philosophical
debates that might take a non-historian a year or a lifetime to
assimilate; but all this is mere abbreviation, and quite comparable to
what mathematicians do in their own papers. Nobody claims that a layman
can read the Proceedings of the American Mathematical Society, and while a
layman can probably come closer -- or thinks he can -- when he tries the
journals of history or clinical psychology, it is not because he
understands their *language* better, but because he already knows a bit
more of the subject-matter background. If technical papers in psychology
or history are understandable at all it is because they are written in
English, like good mathematics. And while good English prose is only a
necessary, not sufficient, condition for the value of the communication,
the effort to formulate one's ideas in such prose itself comes close to
assuring the sufficiency, given that the ideas were valid to begin with.
It is extraordinarily difficult to put bad doctrine into good English
without being found out.

If a professor of English is to do a good job teaching freshman
composition, he should not attempt to second-guess the needs of
specialties he knows little of. Whether his scholarly expertise is the
poetry of Spenser or the novels of Ross MacDonald, he should also know
enough to decipher standard English expository prose, and to teach his
charges to produce it. In due course, I believe, they will carry the
lesson to their own fields without him. But if he won't teach standard
expository prose he should be out of a job.

This does not mean that he must teach his exposition students to
write about Spenser or MacDonald, if they do not know anything in that
direction, still less that they should write about Peano axioms or
non-Euclidean geometry if they know nothing in that direction. There is
plenty to be taught, evaluated and improved in the prose of students
writing on subjects familiar to us all. Exercise in any of this, complete
with a demand for paragraphs, periods and proper diction, will better
serve the mathematics student than the false instruction that discourse in
mathematics is not really discourse in English.

What this professor of English and pedagogy fails most to understand
is that the root of children's difficulty with mathematics, beginning at
the kindergarten level, is the disjunction of mathematics, beginning with
counting and arithmetic, from good English rhetoric. It is a great
misfortune that the metaphor, "Mathematics is the language of science,"
has been taken literally by so many, who, ignorant of science and
mathematics both, imagine a profundity where there is only misconstruction.

Within the profession of the teaching of mathematics the
relationship of language to matter is also sometimes misunderstood.
"Manipulatives" are fine, and cutting out circles and squares, and
measuring perimeters and carving pies, and it might appear that these
excellent activities, these useful exercises, form a discourse of their
own. Not so. Teachers of school mathematics should understand that, like
so much else in life, little if any of this will enter the permanent
understanding until it is put into language -- language that can be
carried across the street and into the future. Let a garbled version of
even the simplest idea become part of one's language and the idea will
surely be garbled as well.

Ralph A. Raimi
1995