Last summer I wrote an essay about how feedback and the math that visual pattern problems can help students learn.
Looking back, I don’t think this essay ever worked entirely, as a piece of writing.As my initial excitement about the piece soured, I never got around to giving it the big edit that it needed. Still, there are some good ideas in there that it helped me to figure out.
Here’s the essay: On Visual Patterns and Feedback
Here’s an excerpt:
I knew what I wanted to help Toni see. She was looking for a pattern in the growth, but she was having trouble getting specific about it. I wanted to ask a question that would draw Toni’s attention to helpful features of the pattern’s growth and help her get specific about precisely how this shape is changing.
This would involve a bit of guessing on my part, though, since I didn’t really know what question would work!
My first question was a promising dud: “Can you see the previous step in the following step?”
To which Toni responded, “no.”
I tried again, this time directing her attention more directly: “Do you see the second picture in the third? Imagine that you were building the third picture from the second. Where would you put the extra bricks?”
Bingo. She grabbed her pencil and started sketching.
Why did that question work? I think it’s because it encouraged Toni to see the static picture on the page as a changing thing. Toni had lots of experience playing with blocks and adding on parts to existing doodles. By asking her to think of one picture in the next, I helped direct her thinking to this analogy, and she was able to see the pattern’s growth in a useful way that related to things she had lots of experience with.
Like I said, an interesting failure. Enjoy! Let me know if you find parts of this useful.
Teaching With Problems 
1) On pages 2 and 3, you have a wonderful juxtaposition of a visual pattern and the number pattern 28, 14, 24. That was a big moment for me. I think i had a student all last year who I never got to understand patterns becasue she didn’t see how figure 3 grew from figure 2. She was also incredibly uncommunicative and disengaged, but thinking back, this may have been the source of her struggle.
What do you think about creating Figure 1, 2, and 3 using actual blocks on a document camera or something? It might help kids see how one shape builds from the previous one. I wonder what the downside of that is – framing all visual patterns as an extension from the previous figure. Maybe it limits kids’ ability to see the pattern in ways that don’t stem from the way it grows from the previous figure.
2) I have one kid who loves to find the common difference, multiply if by 40, and then add in Figure 3 to get figure 43. Where would you categorize that strategy?
3) “Relational thinking is great, but it’s not broadly useful” – Here is my first real disagreement. I think it’s broadly useful in comprehending each function family, and therefore can be used in all sorts of useful situations. If I give students the sequence of numbers 3, 10, 29, 66, it’s going to be hard for most kids to write a function for that sequence. But if I show them a nxnxn cube with 2 blocks stacked on top, many more kids could write a function. They wouldn’t have to find the first, second, and third difference to determine that the function is cubic. They would see the cubic nature of the function from the cube.
To me, relational thinking is what connects linear, quadratic, and cubic functions to 1, 2, and 3-dimensional objects. It helps me see exponential growth such as as a fractal in its 4th iteration, such as the picture below.
Having written two paragraphs about relational thinking, I am now wondering whether I have misinterpreted your definition of it, or whether the definition is still sort of inchoate and standing in for anything that isn’t recursive or purely functional.
4) “If kids stop seeing things recursively the project would collapse” Why? And do you think kids lose their recursive tools? Certainly not with linear growth. With quadratic growth, I can see it. But isn’t recursive thinking less useful with nonlinear growth? Or, it’s still useful but can be easily found from a table of values after the fact. I recognize that the slope of the function is represented by the first difference, etc. I wonder whether it’s best to loop back to recursion once you’re in formal-function-table-of-values land, rather than within a visual representation. Hmm.
5) Ooh, one big category of student thinking is improper applied proportionality. Finding figure 5 and multiply by 10 to get figure 50. How do we deal with that? It comes up a ton for my students.
6) I never thought of giving kids the rule for Fig n and asking them to draw a pattern. Seems fun, and the sort of thing you could post in the hallway to impress administrators. There could be a lot of cool colorful visual representations of 3n+5
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