I’m really fascinated by Craig Barton’s idea of problems that have the same surface features, but whose deep structure is different. He has started a website to collect them, and has started taking submissions.

Here is Craig’s explanation of what the thought behind these problems is:

What I needed instead were a new set of problems – ones where the surfaces were similar, but the deep structures were very different. By exposing students to problems like that, I would ensure that they learned to recognise not just the similarity between problems, but also the

differencesbetween them.

I love this idea. Here’s an example of one of the “SSDD” (=”same surface, different deep structure”) activity that Craig created:

There are now lots of these types of 4-sets of these problems on Craig’s website. As I scrolled through some of them, I found myself with questions. Here are some of them:

- What makes for a good SSDD activity?
- Is it important that the four pictures resemble each other precisely?
- What sort of thinking does a student have to do with similar “surfaces” that they wouldn’t equally have to do with four unrelated problems?

I set out to make a SSDD activity myself to mess around with some of these ideas. Here is what I came up with, intended for my geometry students:

Along the way, I tried to ask myself “would this work just as well with four separate diagrams?” For a lot of what I tried, it did.

The thing is that four separate, unrelated problems call on students to think about deep structure just as much as four different questions about the same diagram, I think. When I thought about reasons to keep the surfaces similar, I came up with two possibilities.

First, students often get confused between two different prompts that often come with the same diagram. This has been happening all week with my kids and arcs. They learned to find arc *degree **measures *first, and they often don’t realize that a question is asking them to find arc *length. *For that reason, I tried to include a problem that asked for arc degree measures and another one calling for arc length.

Second, an important idea in geometry is that the same diagram might have different assumptions associated with it. We want to reason about what can be guaranteed by the *information *we have at hand; this version of proof isn’t about observing what happens to be true of a given diagram. So I think it’s helpful to show students that different problems can use the same diagram but represent two different sets of information, depending on what else is given. For that reason I tried to contrast two cases, one where the diagram is known to be symmetric and one where we lack any such info.

I think that’s my takeaway for now about these SSDD problems. There isn’t always a tremendous difference for the student between problems that look different (and are different) and problems that look the same (and are different). In fact, I think part of what’s fun about problem solving practice is playing around with a variety of problems that look (and are) different — the variety can provide a sort of buzz.

SSDD problems do seem like a helpful tool to use when there are important contrasts to make between things that look awfully similar at first. I think my best practice resources already incorporate some of these, but Craig’s identification of this as an activity type is very helpful to me. I’m adding it to my mental bucket of practice formats.

Interesting. I think this is great for working on SMP see and make use of structure.

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There is some interesting expert/novice research related to this specifically in the field of physics. Many physics problems, on the surface, look very similar (think block sliding down a ramp) but depending on the specific conditions involved might require different approaches. Experts tended to categorize problems around big ideas (Newton’s Laws vs. Energy Conservation vs. Conservation of Momentum, etc.) while “competent” beginners focused on the equations they knew and knew how to manipulate.

I like this SSDD approach as a nice way to help students begin to focus on the big ideas (i.e. “different deep structure”) rather than all the surface level features. Since the surface level features are the same, you’ve got to focus on the deeper structure to distinguish the problems. Categorizing based on deep structures will only be helpful though if those categories then evoke strategies or tools that are helpful in solving the problem, so the deep structures need to be meaningful themselves.

This is also very reminiscent of Dienes’s principles of mathematical and perceptual variability. Mathematical variability is all the ways that the numbers or problem specifics themselves can change without changing the underlying idea (i.e. graph y = 2x + 1, y = 3x + 1, y= 1/2 x + 1, y = -x + 1 etc.) and perceptual variability is all the ways we can represent an idea without changing the underlying principle (e.g. y = 2x + 1, a graph of the line y = 2x + 1, a story about starting with 1 comic book and getting 2 comic books each day, a picture of a steep hill, an appropriate visual pattern, etc.) To help students gain deeper understanding of ideas we want to provide them with mathematical and perceptual variability. This seems like it adds another “variability” to the mix–concept variability.

I originally saw the physics expert/novice study in How People Learn, but here is the original citation:

Chi, M.T.H., P.J. Feltovich, and R. Glaser (1981) Categorization and representation of physics problems by experts and novices. Cognitive Science 5:121-152.

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Thank you so much for bringing this type of problem to our attention. It is fascinating how little my students want to READ the directions. I also teach Geometry and getting students to list what they know vs. what they THINK they know can be perplexing. Some kids just get it, and I so want to understand how they know what they know. I like how a set of problems like this can help the Ss realize they need to think about what they know and what they think the picture is saying without enough information.

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