Department of Mathematics
UNIVERSITY OF ROCHESTER
Rochester, New York 14627
Ralph A. Raimi
29 February 2004
Dr. Alfred S. Posamentier
Dean, School of Education - THE CITY COLLEGE of NEW YORK
Convent Ave. & 138th Street - R 3/203
New York, NY 10031
Dear Dean Posamentier:
Dick Askey reproduced part of a letter you wrote him asking him to
ask his colleagues to make some constructive suggestions you can use in your
present work in the new New York State math standards. At one point in your letter you gave as an example of a current pedagogical question:
For example, I believe that in general, we do not train our youngsters
to think proportionally....
I've been thinking about this point off and on for some years now,
since "proportional reasoning" is a phrase I never heard in all my life before
being exposed, in my old age, to professional math-educators. I don't know
how "constructive" I can be, in relation to a skill I believe should not be
phrased in these terms at all. If you read 19th century arithmetic books you
will see that students learned to recite answers to problems of the sort you
have in mind, using a certain mind-numbing sequence of phrases involving a
rate, and input and an output, though in different settings one had to recog-
nize certain things (like price) as rates and certain other things (like dozens)
as inputs, or sometimes outputs. Most such problems were taught under the
rubric of "the rule of three."
When I was a child I was able to do such problems easily enough --
sometimes -- but at other times, e.g. the problem of two hoses filling a tank,
I was paralyzed by the boxes (chalked rectangles) into which my teacher put
the key numbers on the blackboard, and I could not remember whether to add
or multiply, or where to put 1/8, or maybe 8, if hose #1 took 8 hours, etc.
In other words, the attempt to teach me "proportional reasoning" collapsed
when the problem required the invention of a rate. I did not overcome my
shyness about such problems (I had to draw pictures, rather than boxes with
numerals, and then couldn't explain what I had done) until I learned some
algebra and was able to label every quantity in sight with a letter, writing
down all relations I could think of, that the data gave me, and then solving
for the quantity asked for. What was proportional to what no longer needed
to concern me; the relations dictated by the problem led to airtight equations
about the meaning of which I no longer had to think. It was, to me, a
Since learning about the current obsession with "proportional
reasoning" I have decided that the language of functions, input and output, is
the easiest way to understand such problems. After all, Proportional is the
description of only one class of functions, and all science is the quest for
analogous relations. What is there against the use of letters and equations
from the very beginning, when such real-life problems are first attacked?
First learn the number system itself, then observe that in the real world there
are many relations expressed by formulas, in which an input and an output
are related by a scale factor, or rate. Write the relation and solve.
Next stage (Temperature C and F) is slightly different, for now it is
temperature differences measured in C and F that are related by a scale
factor. Later stages: concentrations, pressures, etc. And of course, in the
fullness of time, powers, roots, cosines, logarithms,
My advice, which will likely not be regarded by everyone as construc-
tive, is to abandon the phrase "proportional reasoning" in favor of "linear
relation", and to teach such problems in algebraic terminology with emphasis
on the idea of 'rate'. R = pN means R is receipts in dollars, N means the
number of units sold, and p (price, in dollars per item) is the rate. C - 0 =
(5/9)(F-32) has a similar scale factor (5/9), one that has no standard name
such as "price", but of course it is "degree change C per degree change F".
Both equations have graphs worth studying, both for solving problems and
for clues as to how the student could have arrived at drawing the graph from
knowledge of the data in the problem or in the encyclopedia, or both.
When it comes to filling the tank with two hoses of different diameters
things get a bit harder, but trying for such relationships cannot be evaded,
whatever language you use.
If someone were to come to me and ask for help in explaining "propor-
tional reasoning", I would simply know no other reasonable way to do it,
and I would insist on language that expresses the idea better than that phrase
does, so much so that I believe the phrase itself gets in the way.
Your more general question, about what things are most important in
the improvement of school mathematics, has the same answer in large part. I
can only repeat the recommendation of the 1923 MAA report, which said
school mathematics should ultimately be founded on the idea of "function".
Obviously, one would first have to have a domain, and so one begins with
"number" itself; but as soon as possible the value of "number" as an idea
must be elucidated by the notion of function.
Equations like d = rt, which in traditional school math appear early,
confuse the issue by their appearance of particularity (here the rate is called
"rate", rendering the word suspect in other connections), unless they can be
given as part of an idea with more than one example. Geometry and proof
make another story, and should come much later anyhow; but it is certainly
worth noting that arc length, area and volume are functions whose domains
contain geometric objects, or measurements thereof.
In calling particular attention to "proportional reasoning" in a sentence
mentioning the SAT, you name a peculiarly schoolroom anxiety, which I
believe inhibits teachers' proper teaching of what would render this class of
problems more comprehensible than they usually are painted, whether by
"traditional" exposition or by "discovery". Exams such as SAT, and even
more so the Math A and other currently controversial standardized exams,
make so much of linear relationships that the schools have become
accustomed to thinking of these few problems as a main stumbling
block to student
These problems exhibit such a stumbling block, sure, but it is one that
will not be overcome by explicit attention to exactly those problems, which
appear year after year on test after test in so routine a manner that teachers
are impelled to have students merely learn to memorize the solutions within a
limited repertoire of disguises, the repertoire known as "proportional
reasoning problems". Alternatively, in some of the recent reform
such problems are solved -- and only in the simplest cases -- by going back to
first principles and counting, in disregard of the past few thousand years'
development of mathematics.
I am well aware of the history of "The New Math" of the 1960s, and
the excessive attention given by the reformers of the time to abstract notions
and language. In introducing "function" and in particular "linear function"
early in K-12 I am by no means advocating the tedious learning of an arcane
vocabulary, "domain" and "range", "number" vs "numeral" and all that. The
word "function" itself need hardly appear in K-6, but the idea should be
present from its first appearance, and for linear relationships the word "rate"
should be learned and used in contexts far from that invoking speed and
distance. And graphs all the way.
Ralph A. Raimi