An Investigation of the Learning of Addition and Subtraction, Carpenter and Moser, 1979. Their three year study ran from 1978 to 1981, forming the foundation of CGI.

What are the major structural differences that distinguish different “find the area” problems?


Thinking about Problems vs. Thinking about Topics

Over at Lani’s place, Max talks about expanding CGI beyond CGI.

“Another grand challenge could be to extend the kind of work that went into CGI [Cognitively Guided Instruction] to high school Algebra, Geometry, Probability, Statistics, and Calculus concepts, so that teachers teaching, say, right-triangle trig, or similarity proofs, or solving quadratic equations, or standard deviation, had a map of learning progressions in that space and ideas about kids’ informal intuitions and misconceptions, multiple representations that connected to various aspects of a fully operational mathematical understanding, and then linkages among those multiple representations.”

Max talks about detailing how kids learn topics. But I’m not sure that this is exactly how I read CGI, though I can see it through that lens. The question is, do we seek learning progressions mapping how students think about problems or how students think about concepts? Meaning, do we want knowledge about how kids think about addition or about how kids think about addition problems?

I don’t think that this distinction really causes issues at the elementary level — to know addition is to know how to solve addition problems, and we all agree on what addition problems are — I wonder if it’ll be harder to nail down on the secondary level.

I was working on something I noticed in a trigonometry unit I recently ran with 9th Graders. Following CME Geometry, I see a road into trigonometry concepts that begins with comparing the steepness of different right triangles.
Here’s the different sort of thinking that I saw students doing in “comparing steepness” problems during the unit:
  • Comparing steepness visually, or by measuring the angles of the ramps
  • Comparing steepness by comparing the sides of the triangles
  • Comparing steepness by looking at both the height and width of the triangles, but not distinguishing between ratio of sides, difference of sides, etc.
  • Comparing steepness using the height/width ratio of each triangle
  • Comparing steepness using the ratio of any two sides of the right triangles

Here are some samples of student thinking to prove that I’m not totally pulling this out of my elbow.




I think it would be wrong to say that this is a way that my students thought about trigonometry. It’s certainly how my students thought about problems that are deeply related to trigonometry. It’s definitely how my students thought about “steepness comparison” problems.

Where does this leave us, if we’re interested in providing something of value to other trigonometry teachers?

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.

A Small Accomplishment

This was an exciting moment in my day. I was stuck on this problem for weeks, and yesterday I decided it was time to check out the solutions. Reading solutions was very satisfying, but there was something else that struck me today.

The problem I was stuck on was this:

After reading the first solution I found (it involved auxiliary lines and jumps of intuition that struck me as prophetic) I read a second solution that hit me hard.

Let E be the intersection of AC and BD. For triangle AED, angle(AEB) = x + 60. For triangle EBC, angle(ECB) = angle(AEB) – angle(EBC) = x + 60 – (x + 30) = 30.
So angle(ADB) = 2.angle(ACB), points A, B and C belong to a same circle with center D, AD = DC, triangle ADC is isosceles and x = 10.

In my own work, I had figured out that angle BCA is 30 degrees, but this is new for me. Since that angle is 30, and it connects up with angle BDA, you can see BDA as a central angle and BCA as an inscribed angle. Do D is the center of this added circle, and that means DC is a radius and that means x is 10.


This feels so new to me. I immediately felt the urge to find more problems that I could tackle with it. I started looking back on other geometry problems I had worked on in recent weeks and months. I soon spotted GoGeometry Problem #3.

If D is the center of a circle, DC and AD would be radii, and then x would be the inscribed angle that goes with 45, and therefore be 22.5. That is a cool new way to see this problem! Is it powerful?

Now, to be sure, there’s more work to do in Problem #3. In particular, you have to worry about whether B actually lies on the edge of the circle. If it does lie on the edge of the circle, then a theorem that I teach my kids says that it has to be 90 degrees, so the work becomes trying to show that angle B is a right angle. And if you knew that angle B is a right angle, well, you’d be done for reasons that have nothing to do with this circle perspective.

But the circle perspective tells me something else very quickly. It tells me that, ultimately, we’re dealing with two isosceles triangles glued together along BD, since BD has to be a radius. This is a new “way of seeing,” and it’s one that is going to help me see possible ways forward in other problems.

What I’m most excited about, though, it the promise of this perspective for helping me make sense of when and how to add auxiliary lines. Consider Go Geometry Problem #1:

I looked at someone’s solution. They added auxiliary lines. It was mysterious to me. Then, I looked at this scenario again from the circle perspective, and things opened up quickly.


Oh, wow. So that’s how you could know to extend into a right triangle!

This month, I’m reading Making Sense of AlgebraLesh & Zawojewski (2007), I’m working on these geometry problems (and a few complex numbers problems), I’m looking carefully at visual pattern puzzles and I think I’m starting to see one way that these disparate sources come together: we need an expanded notion of what students learn in math.

These “ways of seeing” aren’t any more central to mathematical learning than content knowledge, mathematical practices, strategies or metacognition, but they are as crucial to learning as any these things. Likely, for any given topic or sub-domain of math (e.g. linear equations, auxiliary line geometry, visual patterns) these aspects of mathematical knowledge grow together in a way that makes causality hard to distinguish. (Is my “circle perspective” a development of my habit of looking for new perspectives? Am I using finding a new way to use the solve a simpler problem strategy? Am I learning a theorem?) This tangling doesn’t bother me very much, but maybe it should.

We talk about pedagogical content knowledge. Maybe I’m just arriving back there. Truly being aware of all aspects of a specific content area that are especially important and challenging is the sort of sensitivity to content that teachers need to be armed with. Sometimes those sticking points will be ways of seeing (e.g. visual patterns, auxiliary lines). Other times they’ll be self-conscious use of a strategy (e.g. adding decimals, solving 1-variable equations). Maybe other times the sticking point for kids will tend to be something dispositional, like a lack of perseverance (e.g. debugging software).

So, maybe, that’s the project. To focus in on narrow areas of content. To smoke out the different ways that kids think about these, and to make explicit the whatever that makes this specific area of learning challenging. To then share ideas for moving kids past those stuck-points so they have more great ideas.

Maybe there are no shortcuts. Maybe the only way to talk about what feedback is best to give, or how to craft learning goals or to give good hints is to deeply ground this sort of talk in specific content. This would be a big change in how I do 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.

Completely Stuck, Looking at a Solution

Completely stuck on this problem, so I decide to look at a solution.


I look at an answer. I find this:


I find the art of adding auxiliary lines entire mysterious. There are a few moves I have, there are a few things that I notice help pretty often (adding altitudes to diagrams; finding right triangles; adding lines that create congruent angles and possibly congruent/similar triangles) but I have no sense of when to use which move. My scrap work on each of these problems involves lots of random line-drawing. It’s entirely guess-work, random for me.

I don’t know this area of math very well. Am I having fun?

The short answer is, yes, I am.

Do I want a teacher to show up and make this mess more systematic?

This is a tougher question. I’m having fun with this, I’m in no rush to get better at this. It is clear to me at this stage that if I wanted to get better at these problems in a jiffy I should spend a lot more time trying to make sense of the answers that other people have written than I currently am. It’s unclear to me how much time spent working on the problem was beneficial for my learning.

I was also hit hard by this paragraph from Max’s post:

I also believe that students’ beliefs about things like who does math, what it means to do math, how math is learned, etc. are so important that pedagogy has to take them into account — even if that means sacrificing clarity or efficiency to support the belief that math can make sense to students, and good math ideas can come from anyone.

I have no idea how these experiences are shaping my perceptions of myself in relation to math. I think my mathematical personality is pretty stable — stuff doesn’t come so easily to me, but I know if I work at it eventually I can get it — but I’m not at all confident that I can see myself clearly.

I’m also working on these problems by myself. I feel no social pressure to “keep up” with anyone else. So little is at stake with my work here. I think that we often underestimate the feeling of bla that comes from seeing everyone around you engage with math while feeling lost. In such situations, I think arguably Priority #1 for fostering a productive mathematical disposition is getting the kid doing the same mathematical task as everyone else. (This is why I get a bit queasy when people recommend giving kids a special new task as a form of feedback.)

Another sort of stray observation: it’s damn near impossible to generalize from one problem. If I want to get any better, I need to go back and try to find some connections between the problems that I’ve seen so far. This is really tough math work — trying to cull some generalizable principles from different-seeming problems.

Some questions, to wrap this up:

  • If I keep at these geometry problems, will I eventually start to see some connections and patterns and get auxiliary lines down? Probably, right?
  • I wonder if I can get better at sensing and responding to when my students are feeling the way I felt — like I had gotten everything that I could out of the problem.

Dylan: “Hints can promote learning in five ways”

Michael cites Kluger & DeNisi (1998), and argues that hints can cause learning in two ways:

  • Locus of Attention – Feedback can draw the student’s attention towards something that matters for their learning.
  • Motivation – You can ask a student to do something that they otherwise wouldn’t do.

This is interesting, and I think I buy it, but I think we can make this more specific and actionable. Here’s my proposal, which is really just breaking down the two ideas above:

Hints can promote learning in five ways:

  • Redirect attention to features of the problem (What is it saying?)
  • Redirect attention to student knowledge (What do I know?)
  • Redirect attention to student cognition (How am I approaching it?)
  • Promote students’ beliefs in their mathematical efficacy (I think I can solve it)
  • Provide missing information (I know what I need to solve it)

This is from Dylan, and it’s such a great start. One piece of work that we need to do is get clearer on what we mean by “promote learning.” I think we probably need to make a basic distinction here too, and I’ll start by distinguishing two different goals that I have with hints.

  • Promote learning, in the moment – It’s during problem solving, and I see an opportunity for the student to learn. (e.g. this)
  • Preparation for future learning – We’re going to have a follow-up activity, or a whole-group conversation, and in order to participate in that opportunity to learn the student will have to notice certain problem features.

A lot to think about with Dylan’s post. I’m going to stew on it for a while.

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.