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.