I think this is a really cool piece of student work. Some background: it came from a 3rd Grader, and this was the second (or third?) day in our visual patterns lesson sequence. We launched with a discussion of the (linear) pattern that we had studied the day before (this one), and I made sure that we discussed two different choices that were made by kids when representing the “jump ahead” step: 1) Drawing a super-duper careful diagram of the jump-ahead step with lots of squares and 2) making a “block diagram” that doesn’t include every little square but still shows how many squares there would be.

This kid (“Toni”) calls me over because she’s staring at a blank page. Toni is absolutely stuck on the pattern at the top. No progress. No thinking going on. I come over with the agenda to figure out why and to see if I can restart her thinking.

“There is no pattern,” she tells me.

“Interesting. Tell me more.”

I wish this was fresher in my mind, but I don’t remember exactly what she said. One way or another, she having trouble seeing the constant growth in this pattern, and the reason was because this pattern here grows on both ends. Looking at the scratch work in her pattern, I have a clue as to how she was seeing it, and why she got stuck. There are faint lines, running horizontally across the bottom of those Ls. I believe she was seeing these shapes as made of this horizontal line and then this tower that grows. She could see the line growing, and she could see the tower growing, but she had no framework for seeing how they were growing in concert.

And that’s why she called me over. This is new territory for me — I very rarely see kids who can’t see a pattern recursively.

I try a few things, fumble here and there, looking for a foothold. (“It’s getting bigger?” “How is it getting bigger?” “What do you notice?” etc.)

Here are the two questions that ended up helping:

- “Imagine that you had the second picture, and you were trying to make the third picture. Where would you add on the bricks?”
- “Do you see the second picture in the third picture? Can you show where?”

These two questions got her thinking recursively. She drew the second “L” in, and she drew a line between the two added squares to indicate that they were new. She looked at the fourth picture, was puzzled for a second, had an “aha” moment and then found the new bricks in the fourth picture.

So far, her work is interesting to me because it shows how a student who doesn’t see a recursive pattern can get started. But the rest of her work is fascinating too. Everything that follows shows how students don’t just think at any neat developmental stage. Toni’s work shows an attempt to reach for shortcuts and generalizations, right after getting comfy with seeing this pattern recursively.

Here’s what I’m noticing:

- Toni uses proportional reasoning to derive the number of bricks in the 10th step of the pattern. Now, this turns out not to be quite right, but it’s something quite more sophisticated than recursive reasoning. It’s on the path towards algebra.
- Then, for the 43rd step she uses a “block diagram” to correctly analyze the number of bricks that would be present. This is also well on the path to a full generalization.
- When it comes time to state a rule, though, we’re back at a recursive pattern — “add 2 on the ends.” Notice that she says “on the ends,” because
*of course*it’s on the ends and not just “add 2 to each step.” She has a distinct, hard-won way of seeing this pattern’s growth.

Potential take-aways:

- The path towards a full, easy recursive perspective on these patterns can involve a stage when only
*some*kinds of growth can be seen as constant. Linear growth in one dimension is easier to see than multiple linear growths. - Asking students to find the previous pictures in the next picture can help, and I think it’s part of how experts know how to “see” these patterns.
- I’ve seen this in a few places: students will use less powerful techniques when trying to find lower stages of the pattern. They have more powerful techniques, but they don’t always use until the problem demands it. Eventually, though, some students become so comfortable that they often just find the nth step first, then apply that rule to various n.

Have you read Making Sense of Algebra? There’s that whole section in Chapter 5 (I think) about visual pattern problems. I like how they have the students draw what they see in the pattern, and then draw several more iterations, and they analyze the kids’ drawings to see how they’re seeing the patterns. You can tell that some of the kids don’t see the pattern and so might get stuck.

I also heard Jo Boaler talk about when she does visual patterns, she spends a long time having each kid share how they see the growth pattern. I think that might help give kids some ways of seeing that they might not have realized they were missing — look at the ends, look for the previous shape in the new shape, look at adding rows/columns, look at adding to existing columns, etc. It would be neat to think about if there were different ways of seeing that we could be on the lookout for in student work. Did you see in Kristin Gray’s class there were students looking at a patter like this who saw both 2n – 1 and 2(n-1/2) and could see that in the pattern by cutting the “overlap” block in half?

I also LOVE how clearly Toni’s work demonstrates that just because there are stages of conceptual development doesn’t mean any individual student lives in any one of them at once, even within a problem!

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From “Making Sense of Algebra,” Chapter 4:

More more:

That’s a lot all at once, and I’m not sure that I get it yet. Seems like a really interesting framework.

I’m inclined to think that this doesn’t get the progression quite right. True beginners, I think, don’t typically look just at the numbers for a relationship. I think that looking just at the numbers is a habit that kids develop from lots of work with constant-growth patterns, since “just the numbers” often allows them to quickly identify the relationship, an nth step and an equation.

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