My regular gig is teaching 8th grade math, but on the side I work twice a week with a small group of 7th graders who struggle with math, supporting their regular curriculum. Stumbled across a fascinating misconception from one student today.

She was solving a problem about vertical angles, where one angle was equal to 7x, and the other was 4x + 18. She set up an equation, and copied the two expressions straight from her page:

7x = (4x + 18)

Then, she called me over and told me she was stuck.

She was completely, 100% convinced that, because the right side of the equation was in parentheses, she couldn’t solve it. She wasn’t articulating very much about it — she clearly didn’t understand that the parentheses in the problem were superfluous, and was reaching back to a well-learned instinct about order of operations. But I was fascinated by the fact that such a simple change in the way a problem was presented stopped her cold.

Later that period, another student was working on a problem about circle graphs. He had been rolling through similar questions without a problem. This one presented a circle graph where 32 people corresponded to 40% of the graph, and asked how many total people were represented by the graph.

This student told me that he couldn’t solve it because 40 doesn’t go into 100 (he wanted to solve it by setting up a proportion, and multiplying top and bottom by a whole number to make 100). We talked it through, and got him to figure out that 20% of the total would be 16 people. Here I was trying to get him to see that 20% goes into 100, or that 8 people would be 10%, and go from there to the answer — but he did an end around that redeemed his problem solving for the moment. “Wait — if I have 40%, and now I know 20%, then I can figure out the whole thing — 2 40%s and one 20%!” Definitely an impressive stroke of reasoning given where he’d been a minute ago.

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howardat58In each of these examples it seems that the student sees problem solving as “How to figure out which technique is the right one for this problem”, and accident or guesswork play a big role. Real problem solving should start with ” What is this problem about? ” and proceed to “Can I guess the solution (the answer in most cases)?” and then to any attempt to get a result by any means. Only then does it make sense to go for the formal approach. So, problem solving is not just about those big problems that they can argue over and spend an hour on, it is an attitude of mind, which mathematicians, engineers and a few others have arrived at. (here I go again!)