Focus on fixing bugs rather than overwriting procedures

Oftentimes students who make mistakes have a procedure for doing a particular piece of mathematics, and they always do that procedure correctly. It is just that their procedure is wrong in some way. It is far more helpful to fix what’s broken about their procedure than to repeat the instructions over again.

[From Diagnostic models for procedural bugs in basic mathematical skills, which has a staggering 1167 cites and well worth a read if you have access.]

One of the few tests of teachers to find a genuine correlate with teaching quality is the M.K.T. (Mathematical Knowledge for Teaching) designed by Deborah Ball.

Teaching depends on what other people think, not what you think.

Deborah Ball

One of the key aspects of the MKT is to test if teachers understand student errors. Here are some released question samples from 2008 (link to the entire set):


13. Mrs. Jackson is getting ready for the state assessment, and is planning mini-lessons for students focused on particular difficulties that they are having with adding columns of numbers. To target her instruction more effectively, she wants to work with groups of students who are making the same kind of error, so she looks at a recent quiz to see what they tend to do. She sees the following three student mistakes:
Which have the same kind of error? (Mark ONE answer.)

a) I and II
b) I and III
c) II and III
d) I, II, and III

15. Mrs. Jackson is getting ready for the state assessment, and is planning mini-lessons for students around particular difficulties that they are having with subtracting from large whole numbers. To target her instruction more effectively, she wants to work with groups of students who are making the same kind of error, so she looks at a recent quiz to see what they tend to do. She sees the following three student mistakes:

Which have the same kind of error? (Mark ONE answer.)
a) I and II
b) I and III
c) II and III
d) I, II, and III

18. At the close of a lesson on reflection symmetry in polygons, Ms. White gave her students several problems to do. She collected their answers and read through them after class. For the problem below, several of her students answered that the figure has two lines of symmetry and several answered that it has four.
How many lines of symmetry does this figure have?

Which of the following is the most likely reason for these incorrect answers? (Circle ONE answer.)
a) Students were not taught the definition of reflection symmetry.
b) Students were not taught the definition of a parallelogram.
c) Students confused lines of symmetry with edges of the polygon.
d) Students confused lines of symmetry with rotating half the figure onto the other half.

(META PROBLEM: These are the “discards” from the main test, which is kept private. These questions were not considered good statistically and so are imperfect in some way. It’s hence interesting to ask: why were these questions in particular discarded?)


With all this in mind, I’m trying to suss out what happened on a recent Algebra I test. Here are two problems, with instructions to simplify:

11. a + b + c + a + b + c + c

14. 4x + 5x + 7y – 2x

I had an enormous number of students getting 14 right but 11 wrong. If they did get 14 wrong it was because they overlooked that 2x was being subtracted rather than added, and very few made the mistake of writing what was being multiplied as an exponent. On the other hand, nearly half my students answered #11 with

a^2 + b^2 + c^3

instead of 2a + 2b + 3c. I have been baffling since over what happened. I think it’s possible earlier in their education they were exposed to statements like x \cdot x \cdot x = x^3 but never the equivalent with addition. Still, it seems like that wouldn’t explain all of it; there has to be something inexorably tempting about the mistake just like my order of operations problem from a few weeks back.

ADD: This post takes on the challenge of bug-fixing students directly.

9 Responses

  1. Do they know the Pythagorean Theorem? If so, their exposure to its usual formulation (a^2 + b^2 = c^2) could be “priming” them to think of a^2 + b^2 insted of 2a + 2b…?

  2. I wonder if there’s some sort of “where should I put the integer coefficients” thing going on as well. Possibly students would be able to answer

    1a + 1b + 1c + 1a + 1b + 1c + 1c

    more easily, because the questions primes them to put the integers in front of the variables.

  3. The reasons they make that mistake are pretty simple: they’re imperfect human beings who need to learn from their mistakes; the question is unusual; multiplicative and exponential reasoning are often confused.

    The comment above about priming is spot on, and a good way to help students address that error.

    • Heh, not blaming them or anything. It’s not so much making the error as making the error only on that problem and not any of the others which is what I was puzzling over. There were 7 problems of a similar type.

  4. I agree that it’s the lack of co-efficients that likely messed them up. Interesting…

  5. Quick update: I had a one-on-one session with a student where I asked about this problem explicitly, and as Brent guessed he did in fact have mental interference from the Pythagorean Theorem.

  6. […] and helpful. If you haven’t already, you should find your way to his blog and see his ongoing investigation into the cognitive science behind student’s wrong answers. Also this week, Sam Shah talks about a question on his most recent test that works wonders at […]

  7. I come across this mistake among my students all the time. I wonder if the students are “primed” to add the coefficients if there are already some present in front of the variable, but if there are none, then they put the “number” of a’s or b’s wher eever – like in the exponents. I know that sometimes I tell my students things like: “How many x’s are there?” And I would use this phrase for both multiplication and addition. I’m sure I don’t help the cause.

    Also, I wanted to let you know that this post has been featured on the Math Teachers at Play Blog Carnival for November: http://www.nucleuslearning.com/content/math-teachers-play-november-blog-carnival.

  8. I feel like x and y are often used in the context of problem 14, while other letters are often used for rules of exponents problems. I don’t have evidence for this. The suggestion about the Pythagorean Theorem may be relevant, too.

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