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.
- 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.