(First draft, obvi.)

(**Update: **By the way, I didn’t pull this out of thin air. I used the Math Forum’s EnCoMPASS software and some of their student work along with some of my own to develop this trajectory.)

Short version:

- Don’t see the pattern
- See the pattern’s growth recursively
- Calculate recursively
- Take shortcuts that involve multiplication
- Take shortcuts that involve multiplication, then tinker

- See the pattern relationally — i.e. relate the pattern’s dimensions to its step number
- Calculate by multiplying and systematically tinkering
- Calculate via reference to the dimensions of the shape
- Calculate via reference to a linear function

- See the pattern purely arithmetically

**Don’t See The Pattern**

- Kids with very, very little experience with growth patterns might struggle at first to see how this pattern is growing. Unlike other patterns that tend to be trickier, the growth here goes in only one direction, and there are relatively strong visual cues that preserve the overall shape while it is growing. Go young enough, though, and you’ll find kids who struggle to see the growth here. These kids will see that the shape is growing, but will have trouble being precise about how. They might be taking the object in all at once, they might not yet have the categories necessary for analyzing the growth of this shape. (e.g. they lack the ability to talk or think about the ends vs. the middle).

**See The Pattern Recursively**

- Kids who are able to be precise about the growth here — i.e. most kids — might describe the growth of this shape recursively. They’ll say things like “it goes up by 2 each time.” Kids who can
*only*see this pattern recursively will have the resources to determine the number of triangles in the (say) 60th step, but only through a laborious counting. Some students out there will attempt this, though. Depending on their comfort with all this counting by twos and how systematic they are, these kids might make arithmetic mistakes. (Organization matters for doing such a big counting accurately, and the structure imposed via an organization on the counting would likely be the beginnings of some sort of shortcut.) - Students who, in essence, see the pattern recursively can start developing resources for a relational view by taking shortcuts in the repeated additions that they are recursively performing. Some students will do this crudely:
*ah, we’re growing by 2 each time, I’m looking for the 60th step, there will be 120 triangles in that step.*Another way of stepping in this direction is to break apart the step number they seek, or build up to it.*I know there are 10 triangles in the 4th day. This means that there will be 20 triangles in the 8th day. I already know that there are 6 in the 2nd day, making 26 in total.*How do students come to use these shortcuts? They either are cued by the context or they are making connections to what they know about repeated addition and multiplication. (I think it’s mostly the “repeated addition” thing.) These students will get wrong answers to questions that students using a recursive perspective answer correctly, but they’re far closer to the sophisticated approach then those using repeated addition of the growth rate. - Some students learn to use a “tinkering” approach to develop a general method for calculating any step while still operating with a recursive view of the pattern. How? They know, using the reasoning mentioned above, that they’ll need to multiply by 2 to find a “jump ahead” step. They also know from experience that just multiplying by 2 won’t cut it all on its own. So they multiply by 2, and then adjust.
*I realized that I had to multiply by 2, and then I added 2 because that makes the pattern work in the first three cases.*This means that a student who develops a general equation to the pattern might not actually see the pattern as linear relationally — they might still be operating with recursive resources. (Though such kids are well-positioned to understand the linear function approach, I’d bet.)

**See The Pattern Relationally**

- Students can learn to see this pattern relationally. This means they relate the step number to the dimensions of the worm at each given step. Where does this perspective come from? With time, it becomes automatic — a way of seeing — but at first it rests on the recursive perspective, almost an inductive perspective.
*If the number of triangles is 2 in step 1 and 4 in step 2, and we always add 2 it’s just always going to be twice as many as the step.*For other (fewer?) students this perspective might be developed by noticing regularities in the calculations they perform:*All my tinkerings always have me adding 2…because there are always 2 at the ends!*Is the recursive perspective necessary for seeing the pattern relationally? I wonder if we could test this by showing just**one**image to students who are experts at these types of problems, and ask them to describe the shape. Will they connect aspects of the image to the step number? - Seeing the pattern relationally is consistent with seeing it recursively. Many students with little experience with linear functions and equations will have a relational perspective that is built atop their recursive perspective. These students will use their knowledge of the relationship of the pattern to systematically and purposely tinker. This stands in contrast to the flaily tinkering described above. These students might say,
*You are adding 2 each time so you multiply by 2, but then there are the two at the ends so that means you are always going to have 2 more that that.*This could explain how a relational perspective coexists with a recursive description of the pattern. - With experience, students will no longer refer to the shape’s growth recursively. Some students will continue referring to the shape of an element in the pattern when predicting how many triangles it contains. Maybe they would say,
*There are 2 on each end and then there will be 60 pairs of triangles because there’s a pair for each step.*An interesting subtlety that this might explain is why some students would describe their thinking as 2 + BLA instead of BLA + 2. - With experience, though, many students will leave the shape behind entirely in their calculations. They’ll look for the growth rate, immediately see the multiplicative relationship, and then add 2 on. (Students with lots of experience seem to rarely read the “constant” first. Their eyes immediately head towards the growth rate.) This model can either be implicit or explicit. Implicitly, these students will calculate a step using the relationship as in,
*I did (2×4) + 2 to find the 4th step.*Otherwise, they might start by making their model explicit:*First I tried to figure out an equation, and I found 2n+2*. Students with lots of experience with these sorts of problems likely know that, eventually, they will need a general equation, so they might as well figure it out first. (It’s more reliable, they know, anyway.) Importantly, though, students who are very young and who have very little experience with algebra can express this linear relationship explicitly when prompted to do so — “Write a set of instructions…” or “How do you find the BLA step?” Whether they make an algebraic equation explicit or not, for these linear patterns I think that these students all calculate a “jump ahead” step in the same way. The difference is in their ability to express that calculation in algebraic terms without prompting. (I could be very wrong about this?)

**See the pattern purely arithmetically**

- Some students — mostly students who have seen a ton of these problems, I think — come into the habit of ignoring the shape and reducing each step to just a number. These students would perhaps not see the linear relationship from the shape itself, but would instead determine the number of triangles in each step via counting, and then look for a linear relationship between the step and the number of triangles directly. These students either have consolidated knowledge about how to model a linear relationship with a “constant”, or they are adept at tinkering to adjust for that constant. These students might produce equations that model the pattern in a way that seems to go against the “natural” way of seeing the shapes:
*I saw this as 2(n+1).*Of course, it’s fun to find a way of seeing that fits this equation, and it’s nearly always possible to do so (I think). Still, these students might not be thinking about the shape at all.

That was a huge barf. I’m not sure if there’s value in all this analysis, or the above was just systematically detailing a bunch of obvious things.

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I think there’s value in it. It reminds me of CGI’s detailing of direct modeling —> counting —> derived facts. I don’t know if you intended these to be sequential in the same way, though.

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How sequential is the direct modeling –> counting –> derived facts progression? I suppose I’d say that it’s reliably sequential, but not strictly sequential, and I intend the same thing here.

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Have you had a chance to do this with students yet? Thoughts on feedback that you would give to students in each category? I have some ideas on this, but find it really hard to script in advance because I’m so not used to planning like that. Is there value in preparing that type of feedback in advance or is it sufficient to be aware of the different types of responses likely to happen and be able to respond in real time? You already know about my thoughts on teacher-student versus student-student interaction on these kinds of problems…

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Meaning, have I had a chance to plan for feedback in advance? And, in particular, have I had a chance to plan for specific responses in advance?

If that’s what you mean, yeah, this is a sort of planning that I do often, if not regularly. I was exposed to this sort of planning in Stein and Smith’s 5 Practices, the Thinking Through a Lesson Protocol, and (especially) the Shell Center lesson materials.

I do agree that being aware of the type of responses is more important than crafting individual responses in advance. But drafting potential responses is all about thinking about where the learning is headed and what the learning will consist of, and not really the exact wording. For me, anyway.

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