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
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.