Ralph Raimi, who recently passed away, was involved in his retirement years of fighting the battles commonly referred to as “the math wars”. I am happy to say I knew him and corresponded with him frequently about the various battles in math education.

I’m also happy to say that his website is still accessible, and it contains much of his valuable writings on various subjects. One that recently caught my eye is a letter he wrote to the Dean of Education at CUNY, Alfred Posamentier. Dr. Raimi takes apart the focus of “understanding” proportions. His letter synthesizes a lot of what’s wrong with reformist/progressivist math ed arguments that involve “understanding”, and comes to the conclusion that much of what is perceived as confusion is alleviated with the use of letters to represent values (via formulas) and eventually a proper teaching of algebra when the time is appropriate.

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.

The letter can be accessed here.