Is a careless error really careless?

In a 1982 study by John Clement, 150 freshman engineering students were given the following problem:

Write an equation using the variables S and P to represent the following 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.

37% of the students answered incorrectly and wrote: 6S=P.

Why did they make that error?   Take some time and see if you can figure out why.

At first, Clement thought the mistake was due to carelessness. These were engineering students and this was a simple algebra problem.  They must have just been careless, right?

Clement wanted to check his hypothesis.  Using clinical interviews, he asked students to think aloud as they solved the problem.  At first, he thought he was correct in assuming it was a careless error– it appeared that students were using what he called a “word-order-matching approach.”  They incorrectly mapped the order of the key words in the problem directly to the order of the symbols in the equation.

However, as he continued his analysis of the interviews, he realized there was another explanation.   Students who incorrectly answered the questions were doing something that made sense to them.   Their intuition was to place the multiplier next to the letter associated with the larger group.   Although incorrect, it was meaningful to students.

The correct answer, S=6P, was not meaningful to them.  It did not describe the situation in the problem.  Professors are not multiplying.  It did not make sense.

Yet it makes sense to us and to the students who could answer it correctly.   Clement explains that we understand it because of 2 reasons:

1. We see the variables as representing numbers rather than objects

2. We are able to invent a hypothetical operation on the variables that creates an equivalence.  We know the equation represents what would happen if you made the group of professors six times larger than it really is.  Then the number of professors would be equal to the number of students.

The problem doesn’t seem so simple now, does it?   If we attributed students’ mistakes on it to a careless error, we wouldn’t diagnose the problem correctly and if we don’t diagnose the problem correctly, we have little chance of figuring out how to remediate it.

This happens a lot in classrooms.  We see a student make what looks like a careless error and we tell him to check his work assuming he just didn’t think carefully.  Then the student changes the answer because “check your work” is teacher code for “you got the answer wrong.”  Sometimes the error is a case of not being careful.  Sometimes, like here, it’s not.

Instead of assuming it’s a careless mistake, assume that the incorrent answer makes sense to the student and try to figure out why.  Then try to figure out why the correct answer makes sense to you.  The contrast may help uncover the understanding you have that they still need to develop.