My Unscientific Case Study on Helpful Explanations

I’ve been fascinated by the discussion on Dan Meyer’s blog about explanations and their role in a math class. This was prompted by this article that makes assertions about the usefulness of these explanations to indicating understanding. The question of what merits the label of explanation and how that relates to ‘showing work’ is an important one, and has been hashed around by the commenters on Dan’s blog. I decided to pitch a question to students that asked them to explain and nudge them in a discussion to get meaning out of their responses.

Here’s the question, which is from the Amsco Integrated Algebra textbook on page 115:
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I took pictures of their responses and then put them up two at a time in front of the class. I didn’t pair them up deliberately, which might have been more interesting. After putting them up, I asked students to first share their observations about what made them different. There wasn’t much of a responses, but I wrote what was shared underneath. I also asked for each pair to vote on which of the two was more helpful to understanding the answers. Here are the results:

Pair A:
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3 voted for the one on the left, 11 voted for the one on the right.

The one student that spoke up said that the one on the left makes more sense because the one on the right merely shows the pattern. I didn’t get more out of this student in terms of explanation, and other students weren’t stepping up to share.

Pair B:
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10 voted for the one on the left, 4 voted for the one on the right.

The left example is the sort of diagram that I think I’ve seen in those Facebook posts knocking Common Core. I’ve never shown them this kind of diagram though – this was 100% from the student who, knowing this student’s history, has never stepped into a CCSS classroom in the United States to be taught this explicitly. This student decided to make this diagram because she felt it best showed her understanding of the problem. On the right is a set of arithmetic problems that show precisely the same thing, and the students preferred it, but weren’t willing to share why.

Pair C:
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In a move that surely would appease the writers of the article Dan referenced in his post, 2 students voted for the left one, and 12 voted for the right.

I’m not sure what these results mean aside from the comments I’ve already shared. I think it would be easy for Garelick and Beals to point to the preferences of my students as evidence that supports their argument. I think the role of showing answers in this context are different from one of testing, which is one complication of this result. The other is that my question on answers being ‘helpful’ might be dramatically different from asking which answers best show ‘understanding’.

Certainly a more carefully designed experiment might tease out more. This might be the sort of task to use Desmos Activity Builder or PearDeck to give students a chance to share their thoughts in a less public setting.