Take a moment  before reading on. How many squares would be in the 7th step of this pattern? In the 43rd? In the nth?

Take another moment: what mistakes would you expect to see?

From looking closely at student work with other visual pattern problems, you’d expect kids to think about the change of this pattern in two different ways.

1. Thinking about the pattern change recursively – Students would think about the pattern as adding four squares on to the previous image at the corners.
2. Thinking about the pattern change relationally – i.e. by relating the step number to some part of each picture (e.g. number of squares in diagonals,  number of sets of four squares on the corners, etc.)

The relational goggles are more powerful and useful. It helps us calculate any step of the pattern efficiently. It can be generalized to linear functions. Further, most students have an easy time seeing this pattern’s recursive growth. The real learning that can happen with this pattern, for most students, happens in the move from a recursive to a relational perspective.

With that in mind, I want to share some mistakes that my students made on this pattern. I’ve organized the mistakes into two categories, and I’m curious if you’ll see them the way I do.

Category 1:

Category 2:

The way I see it, all the mistakes that I placed in Category 1 show strong evidence of seeing the pattern’s change relationally. Both of the students in Category 2 show a recursive perspective. In fact, the students in Category 2 don’t even make any mistakes!

What feedback do you think the students in Category 1 should get? What about the students in Category 2?

If all you care about is whether a student’s answer is right or wrong, then all the students in Category 1 will get some sort of nudge towards the right answer, while the students in Category 2 will be praised for their correct answers and maybe encouraged to keep on going.

But the students who are able to relate the step number to part of each picture are actually in pretty great shape. Yeah, they made some mistakes, but most of those mistakes are “off by 1” or “sloppy errors,” the sorts of mistakes that are almost always the result of paying attention to something besides the calculation or step number. (In this case, attention is being sucked up by the need to focus on the structure of the pattern at each step, a way of thinking that is brain-consuming when it’s new.)

On the other hand, the second group of students are getting right answers using a limited perspective. Ultimately, we’d like to help them see a relational perspective. Even though they have the right answers, they’re struggling here.

It’s not news that kids who get the wrong answer might be thinking in more sophisticated ways than students who got some question correct. What is news, I think, is that we ought to be as explicit as possible to ourselves about how those students are thinking with more sophistication. That’s the sort of thinking that can help us be strategic about the sort of feedback that we can give.

What feedback should Category 1 get? I’m inclined to use a very light touch with these students. They’re working within a powerful framework — they’ll likely be able to tease out where they went wrong. Even though they are using a strong perspective to analyze the problem, I still think it’s worthwhile to ask them to correct the calculations. First, because even though getting a correct answer isn’t all that matters, it also matters to students and to me. I want to show that I value correctness. Second, because seeing what doesn’t need to change in their answer is ultimately good for learning. I see this as a chance to adopt that relational view on the pattern again (“Oh wait how did I do this…Oh yeah!”).

Here are some comments I’d give Category 1 kids:

• I love the way you brought the step number into your calculation.
• Can you revisit this? Something’s wrong, but I’m not sure what.
• Your rule here is excellent. Can you check these answers again?

Some teachers will be tempted to encourage Category 2 students to continue their work, even if it’s within a recursive perspective. They might agree that the goal is ultimately for these students to adopt a relational perspective, but they’re willing to bet that kids will come to a “realization” while working recursively all on their own. Or, teachers want to affirm these students’ good thinking, so they are reluctant to offer them another way of thinking. They’re willing to defer the relational view to some other time, and maybe the kid will just pick up the relational view during a class discussion or by talking with a classmate.

Those are all legitimate moves, depending on the kid and the classroom and the course. But what if it’s important — for the kid, classroom, course — to help these students move from a recursive to a relational perspective? What feedback could they get then?

For these students, we want to offer them a new way of thinking. Here’s what I might say:

• Lovely work so far. Can you see where the step number appears in each diagram, and use that to find the 43rd step?
• I see the 4th diagram as made up of 3s. Can you see it as made up of 4s? Try to use that to find the 43rd step.
• Nice job noticing the growth pattern. Can you find a solution to the 43rd step that doesn’t involve adding 2 forty-three times?
• Can you show that there’s a counter-example to the “multiply the step number by 4” rule?

Any other ideas, people?

I’ve squawked a bunch about feedback. I’ve likewise done my share of squawking about student mistakes. I’m realizing now just how much that squawking has been missing out on by failing to get specific about student thinking. This isn’t the familiar complaint (familiar to me, at least) that by focusing on mistakes we only see students for their errors. Or maybe this is that “deficit model” complaint, but I had always interpreted as saying something about what we value in our students, and now I’m seeing how only thinking about mistakes really gives you nothing to latch the errors on to. It’s really limiting.

The flipside of this realization is that to really get at mistakes, feedback, hints or next instructional steps, we need to map out the terrain of student thinking. And there’s no way to do that without looking at sets of student work, rather than some single kid’s  thinking. And there’s no way to do that without getting messy with the details of particular mathematical topics.

This is as true in my teaching as it is for my work here or anywhere else. My best feedback comes when it’s purposefully guided by some sort of explicit story about how student thinking develops for this type of problem. This is probably something I first really learned how to do with multiplication in 4th Grade, and it’s heavily influenced by the way I read the work of the Cognitively Guided Instruction team.

This post is a long, long way of saying that while I’d still love it if you send in individual mistakes that tickle your fancy in any way, I would LOVE it if you could send me a class set of really anything that your students have done, and especially if it’s from a geometry unit or a geometry class. I would be eternally grateful for your class scans: michael@mathmistakes.org. (I’m really good at quickly anonymizing student work.)

Next post: more on why class sets are the best.

Previously: http://mathmistakes.org/visual-patterns/

• Just a curiosity: For the students who have solved this recursively, what do you think would happen if you then gave them the same three figures — but removed the central square from each? Could they answer the corresponding battery of questions? If so, would they still answer it recursively (i.e., the previous number +4)?

My guess is that a fair number of them would be able to see an answer of 4n. For those students who now answer in this way: When you then ask about adding back the 1 central square, could they make the connection that an explicit formula is given by 4n+1?

(Such an approach — though a bit heavy-handed with its scaffolding — is reminiscent of examining lines of the form y = mx in order to understand better lines of the form y = mx + b.)

• Michael Pershan

Maya, this is a cool idea! I hope if you have the chance to try this out you will and report back?

I will say, though, that lots and lots and lots of students would see the “4n” pattern recursively. When I was looking through student work on the “Growing Worms” pattern (here) I saw repeated addition many times. I’ve seen this sort of thinking with 4th and 9th graders. Of course, your students might be able to see a 4n relationally more easily than my students.

Another complication: you can use multiplication and still see the pattern recursively. Lots of students, as a mid-step between repeated addition and a relational view, will use multiplication as a shortcut for their repeated addition calculation. At first glance, this might seem equivalent to a relational view, but it isn’t. First, because it’s a substantively different mental representation of the pattern (shortcut for repeatedly adding 4 vs. relating number of 4s to the step number). Second, because it’s harder to generalize the “shortcut” move. Third, because the “shortcut” approach often leads to errors with patterns such as these, where students use false proportional reasoning. (In fact, you can see this in one of the Category 2 pictures.)

I’d even say that students who are able to figure out 4n+1 still might not be relating the step number to any feature of the diagram. Another approach students have is to still, essentially, see the pattern’s change as recursive. Then they know that they’ll end up with 4n, either via a calculation shortcut or experience with these sorts of problems, and then they’ll tinker around until they get the right “y-intercept.” This thinking — though valuable — still isn’t a relational view.

All that said — maybe this line of thinking would lead to relational connections with our students. I’m not sure. If I’m trying to lead kids to a relational view, though, I tend to see value in making that relational view explicit, and in order to do that I think we need to explicitly talk about the way the step number relates to the diagram.

Otherwise, we might just be helping students be more effective within a recursive perspective. Which I’m not poo-pooing. That’s a valid goal for various classrooms, kids, courses for lots of reasons. It’s sufficient for finding linear functions given a table of data. It’s generalizable. For a lot of kids being more effective within a recursive framework is a step on the road to being able to see these things relationally.

• Looking back at the worm task: You can see what my analogous comment might be about removing the central square in the current post, namely, removing the worm “ends” and seeing if students can find a pattern for the “middles.” As I read along in the earlier post, it seems that some of the students did this (implicitly, at least). As you write, “They’ll look for the growth rate, immediately see the multiplicative relationship, and then add 2 on.”

The language in the current post provides an interesting viewpoint, i.e., contrasting recursive and relational thinking. In my own estimation, what I believe you are observing (at least in the given example and earlier worm task) is the lack of transition between additive and multiplicative thinking. For more on these terms, one place to look is the following (along with its references):

Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1-5. Journal for Research in Mathematics Education, 41-51.

As a side-note, one task that I think is helpful for identifying the struggle with such a transition is the “squareness” one: Present students with a number of differently oriented rectangles with integral sides, and ask them to make up a definition for “squareness.” That is, given an NxM rectangle, how might you assign a number to describe its squareness? How does this allow you to compare two different rectangles and decide which one is “squarer” (or if they are equally square)? What happens when your definition is applied to an actual square? [All of these can fit in nicely with mathematical modeling SMPs.)

Perhaps you are familiar with this task. If not, then my experience suggests that students (and teachers) fall into two camps: Additive and multiplicative.

An example of an additive definition (perhaps after some refinement) would be something like: I look at |N – M|; the smaller it is, the more square the rectangle is, and it gives 0 for squares. Alternatively, some students think about comparing, say, a 3×4 rectangle and a 7×5 rectangle, each represented as arrays of unit squares on the initial prompt, by counting how many unit squares need to be added in order to turn the rectangle into a square. For a 3×4, you need 4 more; for a 7×5, you need 14 more. Since 4 < 14, we have that the former rectangle is "squarer" than the latter. (What's non-ideal about this approach?) Again, with a square: the number of unit squares to be added is 0.

The multiplicative answers are usually some variation of Length/Width, and an ensuing discussion (I would say a precise answer would be: squareness(LengthxWidth rectangle) = min{Length, Width}/max{Length, Width}, which is <= 1 with equality only for squares, and allows you to compare any two rectangles by using the bigger number to indicate a "squarer" rectangle).

Anyway: Googling ["squareness" tasks] should lead you to some materials, which you might modify for your own use. My hunch based on just the two blog-posts (cautiously remembering that "the plural of anecdote is not data"…) is that much of what is attributed to a split between recursive vs. relational thinking can be rephrased/reframed/reseen as additive vs. multiplicative thinking.

MQ

• Michael Pershan

Maya — I shared your hunch about additive/multiplicative reasoning, though I haven’t read that paper of Kamii’s yet, so I’ll take a look.

I certainly agree that additive/multiplicative reasoning is a huge theme in the sort of mathematical thinking that we observe. But when I was trying to sort out whether that lens applies to the recursive/relational distinction, I came up short.

What I’m stuck on is that students seem to be able to think multiplicatively within a recursive perspective, or additively within a relational framework.

Multiplicative/Recursive: “This pattern grows 4 every time, so instead of adding 2 forty-three times I’m going to multiply 4 by 43 and add that on to the start.”

Additive/Relational: “This pattern adds one square in each of the four diagonals for each step in the pattern, so the 43rd step is going to have four lines of 43. I’ll add 43 four times and that’ll tell me the 43rd step.”

What do you say, Maya? Does this possibility cause trouble for our additive/multiplicative theory, or is this still an instance of the additive/multiplication distinction?

Another possibility: maybe the recursive/relational perspective isn’t really what matters, and maybe all we care about is the additive/multiplicative relationship. Maybe!

• I think the Multiplicative/Recursive example you have given seems just fine as far as approaches go. My first thought about an issue with Multiplicative/Recursive reasoning would be around a pattern that grows exponentially, and for which students were giving the next term as e.g. 2 times the previous (rather than deriving a formula like 2^n).

The Additive/Relational thinking seems pretty okay to me, too, though I would test to see if the student adding 43 four times is a result of the # of emanating strands being small. Novel strategies might be devised for computing 43×4: for some, adding it up four times might be easiest; for others, perhaps doubling it twice would be easiest; for yet others, carrying out the multiplication might be ideal. But the 43 arises initially (as I understand) because you do not want the solution to be brute forced. So if you had a central point with 9 strands emanating from it, each of which lengthened by one at each step, I’d be concerned by someone who thinks about step 43 by literally adding up 43 nine times (rather than computing 43×9 using a multiplicative strategy).

Still: Your articulation of these additional approaches gives more strategies to discuss with students!

So here is what I can say for sure(ish): Both perspectives can be fruitful, and considering their manifestations within one another is a great way to delve deeper into student thinking. I wonder, if you look over student work (for this problem or another), how often the four categories we have arrived at [mul/rel, mul/rec, add/rel, add/rec] appear in clear ways. It may be that some of the categories are non-ideal for this particular problem; in such a case, are there other problems (e.g. ones around exponential reasoning) for which the richer classification scheme is more helpful?

I realize this does not address your call for student work (especially) in geometry, but I expect these ideas will rear their heads again.

RE: Maybe!
In fact, I am not very familiar with the math-ed blogging world, but I do keep a weblog entitled “Throw Out The Maybe.” YMMV.
https://mathwater.wordpress.com/

• Owen Thomas

great title (and subtitle), there “mathwater”. prove everything. hold fast to what is true.

• Anna Blinstein

Honestly, instead of having feedback come from me in this case, I would group students who found the rule explicitly together and those who found the rule recursively together first to compare their approaches within each framework. I bet that the kids in Category 1 who made errors applying their rule would figure this out when confronted with another kid’s work that was different. Then, I would have groups share the two ways and have a class discussion about the connections and differences between the explicit and recursive rules and why we might prefer one or the other.

In general, my experience has been that kids learn more from feedback that’s along the lines of, “Your work is different from that other kid’s work. Can you guys discuss this and figure out what’s going on?” than they do from direct feedback from the teacher that their method or answer is incorrect. I think that it provides more cognitive dissonance (which motivates learning) than simply being told that there’s an error in their work. Similarly for the second group, I think that there is more motivation to think about alternative approaches if it’s a comparison between peers rather than an instruction to look at the pattern in another way. You would perhaps also want the kids who didn’t do it recursively to be thinking about that way as well so there actually is value to having everyone understand both ways even if you ultimately want them working more in the land of explicit rules.