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."