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.

Formula Sheet – A Toolbox or Takeout Menu?

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During the IB Exams, students get a set of equations and constants to use. Part of the motivation behind them is to reduce the amount of memorization required. There’s no sense in students memorizing Planck’s constant or the Law of Cosines in a context that emphasizes application of these ideas.

That said, I’ve heard variations on the following from different students just in the past three days:

  • I thought I was right, then I looked at the formula sheet, and realized I was wrong. (She was right the first time.)
  • I didn’t study it because I knew it was on the formula sheet.
  • I don’t know what formula to use.

If you read my blog, you know that I don’t test formula memorization for all sorts of reasons. You get it. I get it. It has a place, but that place isn’t one I want to be spending my time.

You might also know that I’ve experimented with different versions of resources available to students during a test. I’ve done open note-card, open A4 sheet, open A5 sheet, open computer/closed network, open computer/open network, open notebook, and open people (i.e. a group test) formats.

I believe that the act of students creating their own formula sheets is more effective than handing one to them. The process of seeing how a formula is applied in different contexts and deciding what needs to be remembered is valuable on its own. Identifying that one problem is similar to another for reasons of physics shows understanding. I want to make opportunities for that to happen. Reducing the size of the resource requires students to prioritize. These are all high level skills.

The difficulty is that students see formulas directly as a pathway from problem to solution. Most problems worth solving don’t fit with that level of simplicity. Formula sheets give you the factual information, and rely on the user to know how to connect that information to a problem. The student thinks that the answer is staring at them in the face, and they just have to pick the right one. As teachers, we want students to identify information they need, then look at the reference to get it.

This is part of the reason I like standards based grading, as it justifies assessing students through conversation. A student asks me for a specific piece of information. If it’s how to calculate something, I’ll tell them if the related learning standard is about applying a concept, not calculating a quantity. If their request directly asks for the answer to the question, I don’t tell them. If they ask for a hint, I give them enough to get them moving, and adjust their proficiency level for the related standard according to the amount of help I give them.

In the long run, however, students need to know how to use the resources available to them. This is one of those big picture skills everyone talks about. Students need to know how to use Google to effectively find what they are looking for. They need to know that typing the text of a question into Yahoo Answers is not going to get them the answer they are looking for. I do know that if a student directly says “I can’t remember a formula for [ ]”, and I give them an equation sheet, they can usually find it. If they use the formula sheet as step one, they are not likely to complete the problem on their own. Having the sheet there in front of them makes it far too easy to start a problem that way. Would having students tally the number of times they looked at their sheet be enough of a feedback mechanism to keep this in check?

I don’t know what the answer is right now.

How do you help students treat a formula sheet more like a tool box, and less like a restaurant take-out menu?