# 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?

# Preoccupations with naive interpretations of reality

Constructivism begins with an emphasis on the constructed world of the knower and the relationship of that world to reality. As von Glaswerfeld observes: It is necessary to keep in mind the most fundamental trait of constructivist epistemology, that is, that the world which is constructed…makes no claim whatsoever about ‘truth’ in the sense of correspondence with an ontological reality…

Among philosophers, Kant usually is recognized as the main person to introduce the notion that the mind does not passively reflect experience but rather actively creates meanings by attributing patterns and regularities that cannot be perceived directly. So, two centuries after Kant’s death, radical constructivists’ preoccupations with naive interpretations of reality seem quaint — especially in a field like mathematics where the notion that mathematics is about truth was abandoned hundreds of years ago — following the discovery of non-Euclidean geometries. In fact, since the discovery of Godel’s Theorem, the Axiom of Choice, and other related issues in mathematics, mathematicians have recognized that they can not even guarantee the internal consistency of any system that is more complex than the integers.

From Beyond Constructivism, Lesh and Doerr. I don’t think they’re right about the field of mathematics having discarded truth in light of non-Euclidean geometry. (Quine and Putnam would certainly disagree. Frege wrote after the discovery of non-Euclidean geometry, and was an unabashed realist. Godel too. And so on, many others.)

The critique of radical constructivism is sharp. It applies well to the writings of Constance Kamii:

Piaget’s theory provides the most convincing scientific explanation of how children acquire number concepts. It states, in essence, that logico-mathematical knowledge, including number and arithmetic, is constructed (created) by each child from within, in interaction with the environment. In other words, logico-mathematical knowledge is not acquired directly from the environment by internalization.

From Young Children Reinvent Arithmetic. This sort of talk either ignores philosophy or rests on a really ungenerous interpretation of Locke, Hume and other empiricists.

More from Lesh and Doerr:

As constructivists surely would agree, the key issue is not whether a theory is true or false, but rather whether it is useful. So, one implication of this policy is that, for researchers whose goals are to test and revise or refine theories, a philosophy that is accepted by nearly everyone is not useful. In other words, constructivism itself ceases to be useful to theory developers precisely when virtually every potentially competing theory claims to adopt this philosophy.

In sum, radical constructivism (and Lesh and Doerr) claim that realism in mathematics is dead. It aint. But to the extent that there is necessarily a gap between reality and our thought, that mismatch is agreed upon by basically everyone and has been since Kant pointed it out, so there’s no real need to shout it from the rooftops today. And since everyone basically does agree on that, radical constructivism is not a particularly useful framework for math educators or researchers.

# A methodology, I guess?

We start by wanting to know what hints or feedback we might give a student who is stuck on a fairly typical area problem.

(Or, we want to know when a student’s struggle is likely to be productive and we should just smile and listen instead of yammering away.)

Right away, we run into a serious problem. It’s frankly impossible to talk about this while ignoring the details of the child, the course and the classroom. Kids don’t universally enter school with a wealth of experience that would help them dive into this problem. (This is a sense in which this problem is quite different from addition and subtraction word problems.) We need to know, what relevant experiences will this imagined kid have had before working on this problem? That question, is essentially, a curricular one. What comes before this problem?

There’s no easy way to answer this, because there is no agreed on curricular sequence. We can imagine that students have experience finding the area of shapes by chopping up and rearranging shapes, but there’s no guarantee of this. (Kids have some everyday experiences with chopping up areas, but maybe not the kind that would help them with this.) We can imagine kids have thought about area formulas for different shapes and how they’re similar or different, but who knows if a class studied that.

If we can’t know the experiences that students are likely to draw on, then we’re unlikely to be able to plan out helpful feedback or hints. This speaks against the possibility of pursuing a hint-gathering project in all its generality. Boo.

We’re left with some options, both of which are pretty promising:

1. Focus on particular curricular sequences. Make those curricular assumptions explicit. (e.g. My kids have experience chopping up and rearranging shapes. They also have experience with finding the area of right triangles and rectangles and some triangles. They know triangle area formula.) Craft feedback and hints that are particular to the course and the curriculum.
2. Start with problems at the beginning of the development (with problems that students are very likely to be able to use their everyday experience to solve) rather than the problems at the end (the problems we eventually want our students to be able to solve). (e.g. What strategies do kids tend to use with Ramp Steepness Comparison problems? vs. What strategies do kids tend to use for right triangle trigonometry?) Outline student thinking, in general, for problems that children’s common experiences are highly likely to equip them to solve. These problems can’t just be accessible, they also need to be promising, in the sense that they have to yield mathematical thinking that can be directly applicable to problems that we eventually want our students to solve. (The strategies you develop from getting better at comparing the steepness of ramps can be directly applicable to finding missing sides of right triangles using trigonometry.)

The first option seems promising on a personal level. After all, I know my kids and my course and my classroom. I’ll be able to directly observe my students’ thinking, so I can articulate a bunch of learning trajectories that are consistent with my curriculum, and then adjust these in response to what I actually see in my students. I think this would make me a better teacher for my students, as I could also plan for feedback and hints in a way that is sensitive to my students’ likely thinking. (As a bonus, any teachers or departments that are working together could work on developing frameworks in this way.)

The big downside to that first option is that the work is unlikely to be helpful for teachers who don’t share my curricular assumptions. That’s a bit of a downer.

The second option seems better suited for coordinating a broader conversation around student thinking. Since we’re starting with problems that students can almost certainly solve without instruction (using everyday or very common school experiences and knowledge) we can all understand each other when we talk about how students will think about these problems. We can then show the way that students can develop in their thinking about these very accessible problems. To be useful, we’d have to show how these strategies are useful for the problems we eventually want our students to solve.

When it comes to area, what might this project look like?

• I use the CME Project’s Geometry text in my classes. I could look at the way units are sequenced there, along with what I suspect (based on experience) that my students will know from their previous courses. I can use this to detail the thinking that my students are likely to use on this problem. I can use my guess at the thinking that students will use to plan for feedback/hints in advance.
• The other thing I could do is identify a problem that most students are almost certainly able to work on productively using everyday knowledge (or strategies that they almost certainly would have from earlier coursework), but that can also lead to being able to find the area of parallelograms and other shapes. Then I could use student work and experience to outline how thinking might develop as students get a lot of experience with these sorts of problems. I’d want to draw a line from early strategies to advanced strategies, and show how these strategies can be applied to finding the area of a parallelogram.
• These projects could go together. If I look at the CME Geometry text, I might find problems that students could work on using everyday strategies. If I find problems that kids could make sense of using earlier experience, I might want to rewrite parts of my curriculum around them.

Long-winded, but I think that’s the plan.

# 5 Research Results on Problem Solving

Does the problem-solving literature have anything of value to guide instructional decision making? The answer is that, although the literature on problem-solving instruction presents ambiguous messages, five results stand out:

1. Students must solve many problems in order to improve their problem solving ability.

2. Problem solving ability develops slowly over a prolonged period of time.

3. Students must believe that their teacher thinks problem solving is important in order for them to benefit from instruction.

4. Most students benefit greatly from systematically planned problem-solving instruction.

5. Teaching students about problem-solving strategies and heuristics and phases of problem solving (e.g. Polya’s, 1945/1973, four-stage problem-solving model) does little to improve students’ ability to solve general mathematics problems.

Source: “From Problem Solving to Modeling,” Frank Lester and Paul Kehle, in Beyond Constructivism.