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Saturday, June 1, 2013

Making Sense of Variables

Some of us have no trouble with writing expressions or equations for word problems.  But we are in the minority.  Earlier today I was reading from a rather old (1980s) study on this very difficulty.  One might have hoped that in the 30 plus years since then, a pedagogical solution to the problem would have been found, but any algebra teacher can tell you that making sense of word problems remains traumatic for many children.
The following question (taken from Translation difficulties in Learning Mathematics, American Mathematical Monthly, v88 n4 p286-90 Apr 1981) was given to 47 (non-science) algebra students and 150 calculus students.
Write an equation for the statement, ‘There are six times as many students as professors at this university.’ Use S for the number of students and P for the number of professors. 
I was not surprised that more than half (57%) of the algebra students got this question wrong. But a large minority (37%) of the calculus students also had an incorrect answer.   According to the authors, the most common mistake was 6S = P.  When asked to explain their thinking, some students drew a picture like this:

The mathematics teacher might be frustrated that the requirement (or warning? or advice?) “Use S for the number of students…” was not followed.   Otherwise, the drawing should look like this:
The number of students = 6.
The number of professors   = 1
Obviously, 6 1.  However,

Replace “The number of students” with S and “The number of professors” with P to obtain:
Make ‘S’ the subject of the equation by multiplying both sides by P.
Finally, obtain, S = 6P. 
Is this too much work for something that’s obvious? Perhaps the joke is on us:
The professor is giving a lecture and has made an assertion as part of his presentation. A student, not understanding the basis for the assertion asks why it is true. The professor responds that "It is obvious." Then the professor steps back, stares at the board and ponders for several minutes. Then he turns and walks out of the lecture hall. He is absent for a fairly long time  … Finally, just before the class is scheduled to end the professor reappears, and announces "Yes, it is obvious."