## Getting math problems wrong for cognitive science reasons

I have three Algebra I classes this semester. Here is a problem I gave them on a test:

$3 - 3 \times 3 - 3$

Out of 105 students, only one got it right.

Some students did order of operations wrong (3-3 first, getting 0 as an answer) but a good number were saying 3-9 is positive 6.

Informal checking later led to many students (but less than the test) getting this wrong

$3 - 3 \times 3$

and even less getting this wrong

$7 - 3 \times 8$

Two things seem to be at work here:

1.) When the problem is difficult to do “in one’s head” and requires some pause, the integer rules are properly applied. However, when it the answer is “obvious” the associativity of the brain triggers the instinctual answer, bypassing any rules (9-3 = 6, so 3-9=6).

2.) The $3 - 3 \times 3 - 3$ seems to cause extra-super confusion in that there’s an added indirection. Many students who knew the order of operations quite well were answering zero because the associative part of the brain put an override (3-3 instantly shouting out “zero”) and their rationally thinking got blasted. If they managed to resist their mental instinct and get past that successfully they had even less resistance to prevent jumping to answer of 6 in the 3-9 statement.

I suspect even some strong advanced students would get the first problem wrong; it’s almost like a trick question even though there’s none of the traditional hallmarks of one.

### 19 Responses

1. Interesting. I wonder how often the format of a question (rather than the content) is what causes difficulties for students?

Perhaps there is research that has already been done in this area?

• Yes, there’s research in this area and it turns out that these kind of difficulties can manifest themselves in similarly-structured algebra problems:

I’m guessing that most people will hit the paywall, but the abstract is worth reading. I believe the group consistency/individual inconsistency is precisely what Jason found as he kept the structure the same but made the work less “obvious.”

2. This seems to me to be thoroughly bad teaching, since the given expression has five different interpretations depending on how it is bracketed. You appear to be assuming that a particular convention is to be applied to all such cases, say that operations are to be done from left to right. You may know which convention you intend, but if the expression came from some other source, how would you know?

• Order of operations always has multiplication to come before addition and subtraction, just like in coordinate point (4,3) 4 is the x and 3 is the y. Unlike the goofy Facebook problem that was floating around a few months ago, there’s no alternate convention in this case (unless you’re meaning something other than infix notation, like HP calculators with postfix, but that looks entirely different).

• Math is not interpreting, is applying rules.
Whether you know or you don’t know the rules makes the difference between a correct answer and a bad answer.

• This seems to me to be thoroughly bad commenting, since the given comment has many different interpretations depending on whether you read the words left to right or right to left, or whether you interpret it as being in proper English, English through a substitution cipher that just happens to come out looking like proper English, badly misspelled Italian, or terrible, terrible phonetic Japanese. You appear to be assuming that there is no way we could tell from context which language or order of words you could possiby mean in your comment, when, in fact, given that I am replying to you, that does not seem to be true. But how could you know, since you don’t know what language this reply is written in either?

3. Just as another data point, I tried this with my 5th period of 7th grade algebra students today.

1 out of 20. Most common incorrect answers were 0 or -3, just like yours (though a fair number of those came from the 3 – (9-3) variation)

4. I often give this (kind of) explanation for the order of operations to my students:
You have 10 $(€ here) and pay for 3 sodas, 2$ each. How much do they cost? How much does the waiter return? Can you figure out the one-line solution? (that’s not exactly the question, just for quick)
After the right result for the former question, it’s worth-seeing the absurd 10-3·2 =7·2=14 \$, where I guess you cheat the waiter.
And I don’t have a remedy for this math-disease.

5. The order of operations seems predicated on the intuitive appeal of the reduction of planar spaces to scalars? Perhaps part of the difficulty is that our brains are designed to think in terms of three dimensional objects, not scalars.. or at least not in the sort of refined-and-labeled magnitudes that scalars represent. (Trying to think a bit like Gerd Gigerenzer. I’ve always thought geometry… Euclid.. should come before algebra.)

• The order of operations is more likely predicated on a nested inner to outer sequence, like scripting code. Everything in brackets is in more than anything out of them. Operations within terms – exponents (including roots), multiplication and division – are in more than operations between terms – addition and subtraction. And, within terms, exponents are in more than multiplication and division. When solving algebraic equations and expressions, we reverse this nesting order to isolate the variables.

I wonder, in addition to all the explanations above, if students aren’t also distracted by the symmetry of the original question. The two congruent subtractions seem to dominate the expression, making the multiplication barely noticeable. One pays more attention to the wheels and less attention to the axle, even though the axle does the work.

• The symmetry is indeed tempting; those 3-3s look like they are just asking to be cancelled.

• I stumbled onto something similar with a problem that I use as the first problem on my first assignment (“Review of Prerequisite Knowledge”) in both College Algebra and Calculus I:
10[7-4(7-5)]
The siren call of symmetry tempts even college calculus students. We work this in groups on the first day of class, and after letting them get started, I always announce to the class that the most common wrong answer to #1 is 60. There are always audible groans.

Even though I never considered that as a stumbling block when I first wrote the problem, I keep it now *because* of that stumbling block. It leads to some good discussions.

6. Did anyone answer -3? I.e. 3-(9-3).

• Yes, but based on their work it seemed to be they were doing 9-3-3 but then answered negative just because they sensed something was wrong and were confused (as opposed to doing 3-(9-3)).

7. “The symmetry is indeed tempting; those 3-3s look like they are just asking to be cancelled.”

But they are indeed asking to be cancelled, leaving $-3 \times 3$. Is there an easier way to do the problem?

• Heh, I suppose. But they would generally do straight order of operations: 3 x 3 = 9, 3-9-3 is -9.

8. The 6th graders in the classroom im working in just learned about order of operations. I had a good laugh when I read that even the students whom understand order of operations struggled. I know my students everyday struggle with acting before the think. im considering offering this problem for a challenge to see how the my students do.

9. […] Getting math problems wrong for cognitive science reasons […]

10. […] 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. Like this:LikeBe the first to like this […]