** **

** Department of Mathematics
UNIVERSITY OF ROCHESTER**

*Ralph A. Raimi
Professor Emeritus
*

29 February 2004

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

liberation.

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
performance.

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
curricula,

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"

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.

Sincerely yours,

Ralph A. Raimi