Visual Patterns – Who Needs Them?

(An excerpt from this essay.)

Visual patterns – who needs them?  After all, very little in the world comes in the form of a neat little sequence of growing Tetris pieces. (A growing doodle, perhaps.  Windows of a rising building. Towers of children’s blocks. Apples, being laid out for display.)

Far more common in school than visual patterns are patterns that show themselves through numbers, graphs, or tables.  The L-Shape pattern that appears above could easily been presented in any of these three other forms. These other forms are more common, flexible and useful. Why bother with all this picture-pattern stuff?

I see three types of thinking about visual patterns: recursive, relational and functional thinking. Relational thinking – that connecting of the step and a dimension of the picture – is not available when the pattern is presented numerically, or in a table or a graph. Relational thinking is this perspective that is only useful for visual patterns. It’s what makes visual patterns different.

(Don’t graphs allow for special, graphical ways of finding a step in a pattern? Graphical patterns are different, too.)

In a sense, visual patterns are easier for students than other representations of patterns. I see this most often when my students work with non-linear visual patterns. Recursive and functional thinking often doesn’t occur to them. Relational thinking, on the other hand, eventually occurs to many of my young students, and they’ll use this to make sense of patterns that would otherwise be inaccessible.

Relational thinking is great, but it’s not broadly useful. The most powerful perspective on a pattern is functional thinking, the holy grail of many a high school course. It’s the sort of thinking that helps an expert quickly look at a pattern and make careful predictions about any step in the sequence. Many students don’t get there, though. The journey from recursive to functional thinking can be rocky. It’s hard for a lot of kids.

Relational thinking can only really be applied when the pattern is presented in a visual form. It’s certainly beautiful, but it’s not broadly useful all on its own. To the extent that relational thinking isn’t just beautiful, but also useful, it’s because relational thinking can help students gain this hard-to-obtain functional perspective. The important question, then, is how do students develop a functional perspective out of a relational one?

(For more, read here.)

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Essay: On Visual Patterns and Feedback

can you find a pattern in every direction?

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.

From Thinking to Goals, From Thinking and Goals to Feedback and Hints

How many cubes are in the 10th step of this pattern? The 43rd step of this pattern? The nth step?

I’m interested in how a bit of theory about student thinking (proposed here) can help answer pressing teaching questions I’d have about this problem (listed here).

What is this problem good for? What math would students be able to learn from working on this problem?

Students aren’t going to learn about Shakespeare from working on this problem, because Shakespeare is unlikely to come up while working on this visual pattern. So, what thinking and ideas are likely to come up during this problem?

Our systematization of student thinking on visual patterns gives us (at least) a very good start, I think.

Roughly, there are three sorts of ways that students might think about this pattern:

1. Recursively, seeing its growth in terms of the previous stages
2. Relationally, connecting the step number to aspects of the shape at each stage
3. Other students might reduce this pattern into numbers (essentially ignoring the shape) and then see the growth in recursive or relational terms (except now the relationship is an arithmetic one) [Example, here]

There’s also the business with expressing the generalized rule, no matter what it is or how it’s arrived at.

4. Students will either express a rule for the nth step in terms of algebra or in terms of instructions or steps for a calculation.

This, then, is roughly the math that my students will learn from working on, discussing and revising their work on this problem:

1. They can learn to see this pattern (and others like it) recursively
2. They can learn to see this pattern (and others like it) relationally (as in Kathryn’s lovely dialogue)
3. They can learn to see this pattern (and others like it) only in terms of numbers and their growth or relationships
4. They can learn how to express generalized rules in terms of algebra (or using some other sort of language)

That’s the learning I’m identifying that can happen from working on this problem, though I don’t claim to be exhaustive (more on that below). What potential learning does this analysis not include?

• Contra Mimi, I don’t see think factorization and expansion, or other algebraic manipulation skills are likely to be improved while working on these problems. That’s not something that students are thinking about when they think about these problems.
• Contra Tina, I don’t think that these problems will help students recognize patterns more globally, unless those patterns can be seen recursively or by relating the step number to aspects of the shape.

There are a lot of caveats that I need to add to the above. First, teachers are endlessly creative and I don’t doubt that Mimi and Tina’s classes worked on these skills. My point is that because algebraic manipulation and other sorts of pattern-reasoning don’t show up in most student thinking about finding “jump ahead” steps in visual patterns, some other prompt for thinking about those ideas has to come-in.

For example, Fawn does this. She tries to teach equivalence of expressions (i.e. factorization and expansion and like terms) via discussion after students have worked on these problems. So during discussion she prompts students to think about the equivalence of different people’s expressions, and that draws attention to thinking about algebraic manipulation. To my mind, this is a new call for thinking, one that is connected to the visual pattern but also substantively different from it.

Megan shares another great example of algebraic manipulation coming up from visual patterns. My only point is that kids won’t get better at algebraic manipulation from trying to find the 43rd or nth step.

Of course, there are lots of good reasons to do something in class besides helping kids learn something. Dan uses visual pattern problems to help foster a culture and expectation that students will work and figure out puzzles on their own. While I worry a bit about how expansive that culture and those expectations are (could a divide develop between opener problems and everything else that happens in class?) I think that this is certainly a legitimate goal that isn’t excluded by anything above.

One last word: I simplified the story and left out important parts. In the previous post I dug into some of the details about how kids use relational or recursive models for calculating a given step in the pattern. We can get into much more detail. A goal for a class or a student might be to relate the step number plus/minus some amount to a given step’s shape. A goal might be to calculate using multiplication (rather than addition) when using a recursive model. The more careful our analysis of how kids think about these questions, the sharper our goals can be.

What should I do if my students are stuck on this problem?

Part of what excites me about all this is that this analysis gives me a substantive way to work on the feedback and hints that I give. If we know how students think about a problem, we might get a sense of what ways of thinking a particular student could aspire to, and then we can think about hints and feedback strategically.

In particular, we can anticipate seeing various bits of the thinking that we outlined. We can identify productive next steps for students who are stuck, since we know where they are headed. Essentially, we can prepare a repertoire of hints or feedback to offer. Of course, we might adjust these hints/feedback on the fly given the specifics of context, but we can do significant preparation. (I’m not saying anything here that isn’t said better by Smith here.)

As illustrated by oodles of student work (again, as in Kathyrn’s post), students often see a pattern’s growth recursively and struggle to adopt a relational model. These students will likely express stuckness, since the “what’s the 43rd step?” question is laborious using the recursive model. (I also think it’s likely that a student with the recursive approach correctly infers that her methods are insufficient when she reads the call for the 43rd step, because why would a teacher ask you to do something simple and repetitive? That subtext is part of why a kid will report being stuck, even though they could add 2 43 times.)

What are questions, suggestions, hints, feedback, etc. that could help a student move from a recursive model to a relational one?

• In the “Growing Worms” post I identified some middle-steps within recursive thinking that kids take on their way to a relational approach. This suggests that if a student is able to see the pattern recursively, we might benefit by nudging them towards multiplication.
  • “I’m hearing you say you could add 2 all these times, but that doesn’t feel right. I agree. What shortcuts can we come up with?”
• What if a student uses multiplication, but fails to adjust or tinker? As a result, they’d end up using false proportional reasoning and likely have a false answer. This student likely wouldn’t feel stuck and wouldn’t ask for a hint. How can you help this student reach the “multiplies and tinkers” approach to calculating? (After all, that approach is right on the horizon for this student.)
  • “I notice that your answer is different than Dylan’s. Could you two chat about that?”
  • “One way that we can check a technique is by using it on really low numbers in a pattern. Can you show that your trick either does or doesn’t work for the low step numbers?”

And so on.

—

The big concern (raised on twitter in conversation with Lani Horn) is that classrooms, students and teaching situations are too different from each other for these sorts of feedback/hint suggestions to be useful.

I gain confidence from the big-girl and big-boy projects that, I think, this little one is aspiring to: Cognitively Guided Instruction [Fennema, Carpenter, Franke, Levi, Jacobs, & Empson (1996)] and the Math Assessment Resource Service (http://map.mathshell.org/). These projects — especially CGI — make generalizations about student thinking that travel across classrooms. The MARS project offers suggestions for feedback that are specific to the task, but presumably grounded in their a learning progression that a kid might take.

(On the other hand, those are big-boy and big-girl projects done by lots of people over a very long time. An important question is how much a teacher, or even a few teachers working together, can reasonably expect to accomplish in this area.)

Ultimately the proof is in the pudding and I think I need to start working whatever this all this might turn into.

Next steps:

• Go through a lot of student work on visual patterns. Make sure that we’re capturing the thinking that is going on.
• Look at non-linear patterns. How is student thinking similar? How is it different?
• What is something that this could turn into, to support teachers who are thinking of using visual patterns in their classes? An article? Maybe additions to visualpatterns.org? A pdf guide, ala MARS’ lessons? A poster?
• On the horizon: how would this generalize to function-finding, more generally? I know students look at tables of inputs/outputs in roughly similar ways — recursive, then relational once they have a model to relate outputs to inputs. Could all this be about a larger class of problems? And what sorts of problems are on the horizon after that? (Maybe this is as general as it’s worth going?)

How Theory Could Help Me Teach Visual Patterns

How does the analysis I gave of the worm pattern (here) apply to this pattern?

Wait, I was going to answer this question in a post, but then I realized that I need to take a step back. How do I judge the success of an analysis of student thinking? What would it mean for the analysis to apply to this pattern? What should a “theory” do here?

Ultimately, I want a theory that helps me answer teaching questions. (I don’t need a theory that speaks to psychologists or math ed researchers, or explains phenomena more broadly.) So, what are the teaching questions that I might have to answer about this problem?

• What is this problem good for? With an abundance of problems on the internet and in books, it becomes necessary to have some way of choosing the ones that are best for my kids. Since I’m in the business of helping my kids learn more math, and since I often am looking for problems that fit into a particular sequence of ideas, I really need to know what math a problem can help my children learn. (Citing a standard is unhelpful here. A standard tells me nothing about where a problem falls in the development of learning that standard.) Further, there’s a whole website of visual patterns. How do I know which ones to choose for my class?
• What should I do if my students are stuck on this problem? Should I let them struggle? Should I have a “mathematical chant” that I repeat over and over (e.g. “Pick another step to find”)? Should I show them how the pattern grows?
• What should we talk about in the beginning/end of class? Mathematical conversations need to be purposeful — otherwise they sprawl and frustrate my students. What could we talk about?
• Will my students find this easy? Hard? I don’t want to walk into class and be surprised by the difficulty, since that will just screw with everything else that I planned.

In turn, I think these questions boil down to just two concerns:

1. What are the mathematical goals I could have for a lesson that uses this problem? [This would guide my selection of the problem, and tell me what we can talk about. It also gives me a gauge on the difficulty of the problem.]
2. What sorts of feedback/hints would be helpful to give in/after class, if somebody needs them? [What if my kids are stuck?]

Any theory about how kids think about these problems, then, needs to answer to the need for mathematical goals and feedback.

Am I missing anything? Err, of course I’m missing something. Any thoughts on what I’m missing, or where I went wrong here?

Thinking You Might See With the “Worm” Pattern

(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.

Visual Patterns: Pre-Recursive Thinking

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.

Structure of Solving Visual Pattern Problems, III

My 3rd and 4th Graders worked on this visual pattern today, trying to find the 10th, 34th and nth steps. Some stray observations, none of them Earth-shattering:

• With time, most of my students were able to understand this pattern recursively, i.e. they understood and were able to articulate how it grows from the 9th to the 10th step, the (n-1)th to the nth step.
• Many of them articulated a description of the nth step in terms of the bottom row (n+1 long), the height of the leftmost column (n high) and the change from row-to-row (2 less than the bottom row, then 1 less each successive row.)
• This description — the rows and columns in terms of the step — is precisely the sort of description that would have helped for a linear growth pattern. Since the growth is constant, this sort of description turns the laborious addition problem into a multiplication problem.
• When my 4th graders were discussing this pattern in whole-group at the beginning of class, one student noticed that the pattern looks like a pyramid. This is precisely the way of seeing this pattern that is helpful for this sort of visual pattern.
• It seems to me that the strategy that students could eventually develop would be to see a rectangle in this pattern, and patterns like it. This would be a new way of seeing, and could help with lots of non-linear patterns. For example…

…and so on.

In short, the problem space of visual pattern puzzles has a structure, and we can probably specify it.

Structure of Solving Visual Pattern Problems, II

Let’s give this a first shot.

Following Lesh, let’s think about what needs to happen with a constant rate of change visual pattern in terms of the modeling cycle.

• Description: Articulate a “way of seeing”/model that allows one to describe the growth through the given steps.
• Prediction: Turn that “way of seeing” into a mode of calculation
• Verification and Modification: See if the model works. Tinker.

There is important thinking, variation in strategy, and levels of sophistication at each of these stages. (Another yiddish word that we lost the chance to incorporate into the broader culture is chochmah. There is a chochma involved in each of these components.)

Description:

One way or another, the point is to identify what’s changing in each step. Here are some levels of this thinking, roughly inspired by CGI.

1. Imagining the action – At an early level of experience with these problems, it is important for students to imagine being able to “add on” the new elements of the step at each stage. Young students can be given manipulatives to act this out. This sort of thinking is, by definition, recursive.

2. Relational model – Students begin to see growth that relates the step number to the shape of the pattern elements.

There is a ladder of sophistication that students show in their representations of a relational model. This is where, for many students on their way to algebraic models, the magic happens. At an early stage, students will fail to draw a picture that exposes their way of seeing. They might be thinking about the given stages of the pattern with an “acting it out” model and not making the jump into a relational perspective.

Other students will draw out fully specified sketches of the “jump ahead” step.

Other students make good use of a “block” model that doesn’t draw out each individual object, but instead exposes the groupings that they see in the pattern.

Even students with fairly sophisticated representations are not sensitive to under- and over-counting in their models, counting in full groups of objects that “naturally” fall together in rows, columns, lines or other familiar shapes.

With time, though, students become sensitive to compensating or overcounting and develop ways of adjusting their models.

3. Algebraic relation – Students, with experience, will begin to selectively drop their visual models when determining “jump ahead” steps. They’ll sometimes very quickly abstract an algebraic model.

At times, students will attempt this and result in error.

There’s more to do, of course, since we haven’t talked about prediction or verification. But let’s identify in a nice list all the models above.

• Acting it out
• Relational Model – Detailed drawing of the step
• Relational Model – Schematic drawing of the step
• Relational Model – Unsensitive to over- or under-compensating
• Relational Model – Sensitive to over- or under-compensating
• Arithmetic Model 
• Algebraic Model

I’ll end this drafty post by trying to draw out some of the feedback/hints/goals that each of these models suggests.

Goal come quickly: use of each of these models would serve as a perfectly sensible mathematical goal for a group of students that hasn’t attained them or their more sophisticated descendants. If my class has never worked with patterns before, maybe my goal is for them to solve visual patterns using, at least, an acting it out model. If I’m working with students that use relational models pretty well, maybe my goal for the day is for them to be able to use arithmetic models of the pattern.

Feedback and hints can help with this, and our articulation of the models can help us come up with sensible hints and feedback.

• Acting it out

“Can you act this out with some bricks?”

“How would you build this?”

“Can you imagine building this next step with your hands?”

“Which blocks were added on?”

“If you’re not sure how a pattern is growing, imagine yourself building each step. Maybe you’ll realize something new.”

• Relational Model – Detailed drawing of the step

“Show me a picture of this step.”

“Why does it make sense that this row is 5 long and 4 up?”

“What instructions could you write someone for drawing out this step?”

“Drawing a picture sometimes helps us get clearer about how we’re seeing the pattern.”

• Relational Model – Schematic drawing of the step

“Do you have to draw each box?”

“Why does it make sense that you counted 10 in this row?”

“If you’ve got a big step, it can take a long time and be hard to count all those pieces. That’s when you might try making a drawing that doesn’t count each little thing.”

• Relational Model – Sensitive to over- or under-compensating

This gets sort of close to verification and that’s a whole other set of thinking tools. I want to focus just on the model, though, and getting to this way of seeing a pattern.

“Why doesn’t it make sense to count this square twice?”

“When we’re working with patterns, we sometimes have to worry about counting things too many times. Can you find the part of your diagram that you counted twice?”

• Arithmetic Model 

“What about the 89th step? Or the 103rd? Could you figure those out too?”

“Why does it make sense that this turned out to be 10+10 – 1?”

“If you feel that you understand a pattern really well, it’s often helpful to look for ways of calculating that only use numbers and not pictures, because that gets us closer to finding a rule that always works. Can you try writing this in terms of only numbers?” (I feel somewhat weird about this hint, but I think that it would be entirely appropriate, depending on our mathematical goals for the student.)

• Algebraic Model

“What about the millionth step?”

“Can you write some instructions that anybody could use to calculate the number of squares in a step?”

“When you’re looking for an algebraic rule for a pattern, it’s often helpful to try calculating a few “jump ahead” steps, because we can often just tinker our arithmetic and end up with a rule. Try to find a rule that always work using this strategy.”

—

Like I said, first draft. I feel this is promising, but has issues that will become easier to notice on a reread.

The Structure of Thinking About Visual Patterns

I’m planning a lesson for 3rd Grade right now. We’ve been working on making representations of patterns using graphs and models, and as part of that work I’ve been asking them to find “jump-ahead” steps, e.g. the 43rd or 21st step or whatever.

Their work on Friday was great to see. In particular, there was a controversy over the number of total squares in the 21st step. Some students drew attempts at accurate diagrams that they miscounted or misdrew, or that somehow didn’t capture the structure of the pattern. Other students reasoned without trying to draw the 21st step, and they often made misgeneralizations.

The most accurate students tended to do a particularly sophisticated thing: they drew schematic diagrams of the highest step that accurately captured the structure of these patterns.

Coming in to today, I’m thinking about making explicit two important features that distinguish the most accurate and efficient work on this problem in my 3rd Grade class from the least.

• Seeing The Structure: The strongest work articulated a productive way of seeing the pattern and then applied that to a higher step. That productive way of seeing it involves thinking about how the number of WHATEVER is related to the step, e.g. “There are 2 columns in the first step, then there are 3, then 4” or “There are two rows on top of each other and at first the grey are just 1 long, then 2 long, etc.”
• Drawing a Schematic, Rather than Accurate, Picture: The strongest work also used a schematic representation of the jump-ahead step rather than trying to draw an accurate picture and count it.

If I knew what productive thinking looked like for “jump-ahead” work with visual patterns over a wide-span of grades, I could have a bank of mathematical goals, feedback, and hints to give students. Further, I could tightly integrate goals, feedback and hints and really focus my instruction in productive ways.

I’m about to head into class. I wonder what other productive mathematical moves I can find in their work today. I also think that explicitly articulating these important features of the strongest work will help close the gap in the student work. I’m OK modeling it, though I think I could also draw their attention to these features by analyzing some excellent student work.

Maybe articulating the structure of student thinking about visual patterns would be a helpful first mini-project to take on for me?