# Thinking Through “Falling Leaves”

How will kids think about this? As part of the Math Forum’s EnCoMPASS project (http://mathforum.org/encompass/) I sorted through and categorized about two hundred student responses to this problem. Here’s the thinking that I saw, with sample responses that seemed typical of the things that I saw.

Now that I’ve systematized student thinking, I want to use this to analyze the possible uses of this problem for learning math, and the feedback and hints that can help kids meet those goals.

Goals:

My first instinct is to say that any sort of thinking that you can use to solve a problem might be a helpful goal for a classroom. In this case, though, there’s little to be gained by encouraging students to see this as a linear pattern!

Instead, I’d say the two ways that students think this through with an exponential lens would make good goals:

• To recognize an exponential pattern and use any method (e.g. recursive or explicit) for calculating a given step in the pattern.
• To recognize explicit functions to describe exponential patterns.

Seeing this pattern as linear at first could be productive for the first goal, as the contrast between the linear and exponential model could make their differences seem even more stark. (See “Teaching to the Negative Space”)

This sort of analysis leads me to a different conclusion than some of the teachers who also considered this question. Other teachers and I are on the same page, almost. Some samples:

• I’d want them to make sense, organize info, realize it’s non-linear.
• Goal: for students to compare and contrast this with linear sequence.
• (I’ll just add one goal) Make sense of the story.
• I think I used this problem with my 7th graders. Wanted them to be able to to explain in word number relationship in pattern.
• Goal: organize the information in a clear manner, look for and identify patterns
• Identify pattern, represent algebraically & graphically; identify function.
• Goal: organize the sequence/data, make conjectures, discuss a limit to # of leaves on tree
• Goal: Connect the pattern to geometric sequences and exponential functions

I don’t think “make sense” is a particularly helpful goal for students, because they’re already trying to make sense of problems, and the way they make sense of problems is through specific acquisition of new models, concepts, paradigms, representations. So I don’t buy the existence of some universal “sense-making” skill that kids can practice, and therefore I want to know how kids are going to make sense of this problem. That probably means recognizing this as a “doubling” pattern.

Likewise, I’m not a huge fan of “explaining in words,” “organizing in a clear manner,” “identify patterns” or “make conjectures” as a goal of the problem. These are all things that people do through specific models, paradigms, etc. and if a kid can’t already do them for this problem, they’ll need these new sense-making, pattern-identifying, organizing lenses.

If I’m being too hard on anyone, it’s on “organizing in a clear manner.” This seems like a broadly useful way of working on patterns, and it’s fairly useful no matter what the pattern is. I didn’t include that in my description of student thinking above. I wonder how much “make a table” helps before you know what patterns to anticipate. Maybe I should’ve included “make a table” as a goal above. I’m not sure.

Hints/Feedback

I’ve tried to identify thinking that could happen with this problem, and then I tried to use that to identify the goals we might have when using this problem. (Roughly, the goals are for all students to use some of the thinking that can happen with this problem.)

We’ve made the decision to intervene, either through feedback, hints, lecture, whole-group discussion or something else. How can we do so in a way that helps kids learn something?

There are two situations that I think we need to worry about.

Helping Kids See an Exponential Pattern Rather Than a Linear One

The trick here, as Max puts it well, there’s no sense in which the exponential model emerges out of the linear one. It’s sort of a dead end. Sure, we’d like students to check their model against cases, but even knowing to do that comes from an awareness that the model might not be linear. This is something that is learned through experience with non-linear models, and so “check your work!” might not even be an accessible habit to these kids.

In this case, the teacher can help learning by drawing attention to the wrongness of the linear model. Some teachers will do this with questions that put kids in the right zipcode for noticing their own mistake, others will be more direct. The choice is partly a matter of style, values, knowledge of the particular kids and the teaching situation. I think it’s important to see that all of these choices help learning in the same way: by drawing attention to the insufficiency of the linear model.

• Can you check your work with Sam’s group? I think you’ll have an interesting chat with them.
• Can you show me how you got 3 for the 3rd minute?
• Can we act out how the leaves are falling in the first few minutes?
• Another group got 4 leaves falling during the 3rd minute. Could you prove them wrong?
• Can you revisit the 3rd minute? I think there’s something there you haven’t noticed yet.
• The 3rd minute is sneaky, actually. Could you try it again!
• Can you show that the 3rd minute is not 3?

These go across the “question to request” axis and the “vague to specific” axis, but they all aim to draw attention to the 3rd minute. Because one way or another, you need to see that this is not linear if you want to learn something new and cool here.

Helping Kids Calculate a Step In This Pattern Explicitly Rather than Recursively

How does a kid move from successively doubling to explicitly using exponents?

One pathway towards conceptualization in math is the “process to object” pathway. (And maybe it’s a really really important pathway. See Sfard.) This is where we take some process and gradually see it less as something do be done or followed (like a law or rule of a Soccer) and more like something that can have properties and do stuff itself (like a person or an Oak tree).

Maybe this “process to object” is a helpful lens through which to see the movement from successively doubling entries in a table to coming up with an explicit formula for any step in the pattern. After all, the exponential notation isn’t strictly necessary for describing the nth step. You could say (as a few students did) that the nth step involves 2*2*2*2…*2 n-times. So there’s a way of describing the nth step that doesn’t strictly depend on notational knowledge that kids are lacking, yet most kids don’t go for it. The conceptual gap (between seeing the growth as a process versus an object) might explain way.

If this is true, then we’d want to ask questions and make requests of students’ thinking that help them see doubling as an object. I think this means that we’d want kids to notice the properties of this growth, and this means that we want students to compare this growth with other sorts of growth.

I think a particularly helpful way to make these comparisons is to follow-up this activity with an activity with some other sort of growth and then explicitly ask students to compare the growth. The other growth might be linear, quadratic, or even exponential with another growth rate (tripling, or halving or something). Feedback/Questioning/Hints on the given problem alone might not be the best way to support the move from recursive to explicit thinking about this pattern.

That said, if we are giving hints and feedback (for whatever reason) we can talk about the feedback and hints that might be helpful. They’d draw attention to the properties of the growth in this problem:

• I notice that you can rewrite the first minute as 2*1 and the second as 2*2. Can you keep on doing this? Can you explain what you find?
• How would this table look different if instead of doubling each minute it tripled?
• Can you show how this table would look different if instead of doubling at each minute it added 2 at each minute?
• Can you prove that this table only ever uses powers of 2?

There are other questions we could ask. A common suggestion from other teachers, I bet, would be “How many 2’s are you multiplying in step 3? Step 4?” because we teachers love asking our students no do some inductive reasoning. I’m not such a fan of that question, because I feel like that doesn’t leave much for students to think about, besides noticing the pattern. Maybe it’s just a style thing, but I would replace that with my first bullet-pointed question. Maybe that’s not an important difference. Really not sure. (Thoughts?)

Questions I Have

There are things that I’m not sure about. For example, some students (3 out of 222, to be exact) used a sort of proportional reasoning to find the answer to this problem. That reasoning goes, like, “Oh so I figured out at 12 minutes the answer is 78 and so the 24th minute is just double 78.” That proportional reasoning is false, but I’m not sure exactly why to say it’s false. At first I’m inclined to say that this proportional reasoning is just an instance of using a linear lens on the pattern (and using a false linear model for that matter), but then one kid used exponential doubling to get to the 12th step and then used proportional reasoning to get the 24th.

Should we think of that kid as quickly switching from an exponential to a linear lens on the growth? Or should we think of proportional thinking as a sort of patch that can be a component of any way of seeing the pattern?

I’m inclined to say that while there are no hard and fast rules for how thinking matches up, it’s most helpful to see proportional reasoning as something that tends to mean that you’re seeing the pattern linearly and recursively. But dunno if that’s really a true or useful generalization.

Another question I have is how much “make a table” is its own way of thinking about this question, or how useful “make a table” is. In particular, would a lesson that had its goal to help students “make a table” be really helpful? Or is the “be organized” instinct one that comes more or less naturally once you see the growth in an organized way? Or do they come together? Basically, I’m struggling to bring together my instinctive dislike for all-purpose problem solving moves with the fact that making a table is actually pretty useful for a large class of pattern problems.

I’m also struggling to figure out the right terminology for what I called “input-to-output” reasoning above. This is lousy terminology, since a recursive perspective also has inputs and outputs. What else could this sort of thinking be called? “Functional reasoning” sounds wrong, since recursive thinking is still functional. “Explicit reasoning” (ala “explicit formula”) doesn’t sound right to me, since that sounds like you’re just making your reasoning explicit. And “Closed form reasoning” doesn’t really describe anything unless you know what “closed form” means, and honestly I find that language weird so I’m guessing others do too? I wish I had a better way to talk about the kind of reasoning that generalizes a pattern into a closed-form formula (be it in algebraic language or not).

Finally, I’m generally concerned that I’ve gotten the analysis of student thinking wrong. Wrong, in this case, would be not useful, or unable to productively guide teaching. It feels right to me, but maybe I’m on the wrong track?