# Category Archives: Reflection

A few months back, I got an email from a local suburban elementary school. They had been given a bit of money to “give all of our fourth graders a unique math experience,” and they were seeking advice.

My first thought was, “Send them all to New York to visit the Museum of Math!” but this was off by a couple orders of magnitude.

As the conversation continued, it became clear that they weren’t seeking advice so much as someone to make it happen. So I said yes.

I am spending three Thursday mornings, and one afternoon, with these fourth graders. Today was day 1.

The theme of the residency is scale. We are playing with small versions of big things and big versions of small things.

A few favorite moments from today:

### Horses

When asked to share a big version of a small thing, one girl said “Horses”. I pressed her to state her meaning. “If you had a map with stables on it, the horses in those stables would be really small, then when you went to the stables, the actual horses would be really big.”

Ladies and gentlemen, I give you the big math idea of inverse!

I thought the horses on the map are small versions of the big real-life horses. But she was very clear that her experience was small horses on the map, then see the big ones. The small-to-big relationship isn’t just the opposite of the big-to-small one; it is its own relationship. These two relationships are inverses—each existing on its own, but with a special connection to each other.

### Which One Doesn’t Belong?

I cooked up a little Which One Doesn’t Belong? set in preparation for our work.

Which One Doesn’t Belong? never disappoints. (Student/home version and Teacher Guide coming this summer from Stenhouse, by the way!)

We noticed all the things I had hoped for, and more. And then afterwards a girl came up to me to make her case that we weren’t being totally precise about our description of the upper-right image. If—as we claimed—the shape in the upper right is composed of four of the upper-left triangles, then the big triangle wasn’t exactly the same size as the one in the lower left because the triangles have outlines which are not infinitely thin.

### Composing triangles

I brought in many small laser cut triangles of these seven types:

I gave them time to play with these triangles. One student said she knew what we were going to do with them. So I asked her what that was, and she replied that we were going to see which ones could fit together to make other ones. This was not the plan, but was behavior I was eager to encourage.

She asserted that the pink and the black make the red.

This was a detour worth five minutes, so we took it. Arguments were presented pro and con. The major pro argument was based on the close enough principle. Con arguments were of two flavors: (1) put the red underneath and you’ll see some red peeking out from underneath, and (2) the long side on the pink plus black shape is not straight, while it is on the red one.

### Composing similar triangles

The main question I wanted to get to—remember that our focus is scale—was Which of the triangles in our set will do what the upper-right shape in our Which One Doesn’t Belong? set does? Which of our triangles can you make into a larger version?

All triangles do this. But these fourth-graders don’t know that. And because they don’t know that, they got to feel a little thrill of success when they found one that did.

And of course they produced some evidence that the relationship we’re investigating is a challenging one.

This is what we had on the document camera at the end of one of three sessions this morning.

HOLD THE PHONE! LET’S LOOK AT ONE OF THESE CLOSE UP!

Do you see? All the others use four triangles to make the bigger version, and this one can too. But this can scale up to make a bigger version that uses only two of the original!

Of course there is a part of my math-major brain that knows this about isosceles right triangles, but it’s a wonderful wonderful thing to have pop up unexpectedly in the middle of fourth-grade math play.

Overall, a delightful morning of math. We got to only a small fraction of what I’ve got chambered so we’ll pick up where we left off next week. I’m hoping I can get them to build one of these.

Either way, I am thankful for the opportunity to play math with this group of kids. They are creative, enthusiastic, curious, and delightful. Their teachers have been very welcoming and open to the intellectual chaos I began to unleash today.

I chose a set of triangles that would have interesting variety and some discoverable properties.

Purple: 3-4-5

Pink: Isosceles obtuse

White: Isosceles right

Red: 30-60-90

Light blue: One-eighth of a regular octagon

Black: Equilateral

Dark blue: One-fifth of a regular pentagon

I also made some yellow obtuse scalene triangles, but they are missing so they didn’t make the trip. Within these classes, these triangles are all congruent. Each class has at least one side that is one inch long.

## Project Pentagon

Pentagons are taking over my life.

You may have heard the announcement this summer that mathematicians found a new tiling pentagon. Previously, there were 14 known classes of convex pentagons that tile the plane. Now there are 15. Maybe that’s all there is; maybe there is another class, or even infinitely many classes, remaining. No one knows.

My Normandale colleague Kevin Lee brought some samples of this new pentagon to Math On-A-Stick this summer, mere days after the announcement. This led to discussing the nature of sameness of the pentagons with my father, which led to further reading, and so on…

I am now drawing an example of each of pentagon type using Geometer’s Sketchpad and Adobe Illustrator, cutting them out of wood on a laser cutter, and then figuring out how they go together. No phase of this project is simple.

I consider a pentagon “solved” if I have at least once figured out how it tiles.

I have successfully drawn and cut pentagons 1 through 11. I have solved all of these but number 9.

The project is making me think a lot about learning.

For example, tonight I was working on pentagon number 8. I solved it.

These sets of four can continue to go together in a way I see and can describe.

But that’s not the only way to view the solution. Maybe someone else solves it using sets of three.

This is the exact same arrangement—the same solution—organized differently. The threes are meaningful here, whereas the fours were meaningful in the first solution. Which is better? Which is right?

Another solution uses sixes.

With that set of six pentagons, you can tessellate by translation only. The three pentagons at lower right are the beginning of the next set of six. Each of these has the same orientation as its corresponding pentagon above it. Does that make it a better solution?

I’m thinking a lot these days about the kinds of questions I’ve posed here. I’m trying to sort out my answers to a larger question:

What is (or should be) the relationship between informal outside-of-school math, and school math?

I have given a couple versions of a talk that asks four basic questions about people’s mathematical activity that occurs outside of school:

• Is this math?
• Is it school math?
• Do we value it?
• Why or why not?

I invite you to join me on this journey.

I’ll keep you posted on the pentagon project.

## Kindergarten questions

I am spending a bit of time a couple days a week in kindergarten this year. It was part of the now-changed sabbatical plan, but important to me to follow through on.

Today was my first day. It was awesome.

The young ones are working on patterning. AB patterns, AAB patterns, ABB patterns and ABC patterns. I’ll leave the curricular questions for when I know more. Today I’ll take these activities at face value, which is to say: this is the mathematics these children were working on today.

The children were instructed to use square tiles to make an ABC pattern. If you haven’t spent time studying curricular approaches to patterning in early elementary, this means that they were to use the tiles to make something such as this:

Color is the only variable attribute of the tiles the children were using.

Girl: Yes.

Me: How do you know?

Girl: [blank look; long pause]

Me: Show me how you know it’s an ABC pattern.

She carefully points to each tile, saying one letter per tile in the following way.

She pauses.

Girl: And you need another C.

Learning is messy—beautifully messy.

I left today with two big questions on my mind; each relating to this exchange, and to others not documented here.

1. What would these children have done if asked—prior to instruction—to make a pattern with their tiles?
2. How does this kind of patterning work interact with learning to count?

I hope my readers will see that these are not questions I expect to be answered in the comments. I hope you will see that these are big and important questions worthy of wondering about for days, weeks, and beyond. I hope you’ll join me in wondering about these questions, and the consequences of potential answers to them.

I argued a while back that learning is having new questions to ask. I hope you’ll join me on my learning journey.

## The Twin Cities Shapes Tour

I recently put out a call for K—2 classrooms in which I could talk shapes with students. As a result each of the next several Mondays (Presidents’ Day excluded), I will be in a different early elementary classroom somewhere in the Minneapolis/St Paul metro area.

Last week I was at two schools: Dowling in Minneapolis and Echo Park in Burnville. I talked with one kindergarten class, three first grade classes and four second grade classes. I have learned a lot.

In particular…

Young children find composing and decomposing shapes to be much more compelling than adults tend to. They nearly all saw the bottom-right figure here as being a square and four circles. Adults can see that, of course, but we are more likely to think “not a polygon”.

On that note, I am now quite certain that we spend way too much time having young children sort polygons from non-polygons. That bottom-right shape has many more interesting properties than that of not being a polygon.

For example, a class of second graders on Friday were variously split on the number of “corners” that shape has. Is it 0, 4 or 8? Second graders can understand each other’s arguments for and against these possibilities.

These arguments can lead to the reason that mathematicians use vertex instead of corner. “What exactly is a vertex?” is a much richer and meatier mathematical question than “How many vertices does this shape have?” But if that latter question only comes up with respect to convex polygons, then it is unproblematic and not interesting for very long.

So imagine for just a moment that the lower-right figure has 8 vertices (and it wouldn’t be too difficult, I now believe, to get a classroom full of second graders to agree to this perspective, whether it agrees with the textbook definition of vertex or not).

Now kids can work on stating exactly what makes a vertex.

And what makes a vertex is going to be awfully close to what makes a point of non-differentiability (large point at apex of figure below).

I’m telling you: in twenty minutes with second graders, we can get very close to investigating things that are challenging for calculus students to describe. My point is that second graders are ready to do some real mathematics, and that sorting polygons from non-polygons is not the road to it.

Other things I found interesting:

• When kids give us something close to the answer we expect, it is easy to fool ourselves into thinking they understand. Example: on the page below, one boy said about the lower left shape that “if you tip your head, it’s a square.” A couple minutes later, it occurred to me that there might be more to the story. I asked whether the shape is a square when your head isn’t tipped, or whether it only becomes a square when you tip your head. He confirmed that it’s the latter.

• Another second grade class was unanimous that the one in the lower right doesn’t belong because it’s not a square. When I asked “is the lower left now a square, or does it only become a square when you tip it?” the class was evenly split. This was surprising to both me and the classroom teacher.

• Diamondness is entirely dependent on orientation in the mind of a K—2 student.

• The 1:1 correspondence of sides of sides to vertices in polygons is not at all obvious to young children. I sort of knew this but saw it come up again and again in our work.

• A first grader said that the spirals below didn’t belong with all the other shapes we had seen that day because “you can’t color them in”.

Even the unshaded ones that had come before could have been colored in, you see. These spirals you cannot color in even if you try. What a brilliant and intuitive way into talking about closed figures—those that can be colored in.

## Standard algorithms unteach place value

I found a page full of computations sitting around the house this evening. Naturally, I picked it up and gave it a look.

Griffin (10 years old, 5th grade) had been doing some multiplication in class today. Somehow his scratch paper ended up on our couch.

Here is one thing I saw.

Me: I see you were multiplying 37 by 22 here.

Griffin (10 years old): Yeah. But I got it wrong so I did it again with the lattice.

Me: How did you know you got it wrong?

G: I put it in the answer box and it was wrong.

It turns out they were doing some online exercises. There is an electronic scratchpad, which he found awkward to use with a mouse (duh), plus his teacher wanted to be able to see their work, so was encouraging paper and pencil work anyway.

I was really hoping he would say that 37 times 22 has to be a lot bigger than 202. Alas he did not.

Anyway, back to the conversation.

Me: OK. Now 37 times 2 isn’t 101. But let’s imagine that’s right for now. We’ll come back to that.

G: Wait. That’s supposed to be 37 times 2? I though you just multiplied that by that, and that by that.

He indicated 7 times 2, and then 3 times the same 2 as he spoke.

Me: Yes. But when you do that, you’ll get the same thing as 37 times 2.

A brief moment of silence hung between us.

Me: What is 37 times 2?

G: Well….74.

Let us pause to reflect here.

This boy can think about numbers. He got 37 times 2 faster in his head than I would have with pencil and paper. But when he uses the standard algorithm that all goes out the window in favor of the steps.

THE STEPS WIN, PEOPLE!

The steps trump thinking. The steps trump number sense.

The steps triumph over all.

Back to the conversation.

Me: Yes. 74. Good. I like that you thought that out. Let’s go back to imagining that 101 is right for a moment. Then the next thing you did was multiply 37 by this 2, right?

I gestured to the 2 in the tens place.

G: Yes.

Me: But that’s not really a 2.

G: Oh. Yeah.

Me: That’s a 20. Two tens.

G: Yeah.

Me: So it would be 101 tens.

G: Yeah.

I know this reads like I was dragging him through the line of reasoning, but I assure you that this is ground he knows well. I leading him along a well known path that he didn’t realize he was on, not dragging him trailing behind me through new territory. We had other things to discuss. Bedtime was approaching. We needed to move on.

Me: Now. We both know that 37 times 2 isn’t 101. Let’s look at how that goes. You multiplied 7 by 2, right?

G: Yup. That’s 14.

Me: So you write the 4 and carry the 1.

G: That’s what I did.

Me: mmmm?

G: Oh. I wrote the one

Me: and carried the 4. Yeah. If you had done it the other way around, you’d have the 4 there [indicating the units place], and then 3 times 2 plus 1.

G: Seven.

Me: Yeah. So there’s your 74.

This place value error was consistent in his work on this page.

Let me be clear: this error will be easy to fix. I have no fears that my boy will be unable to multiply in his adolescence or adult life. Indeed, once he knew that he had wrong answers (because the computer told him so), he went back to his favorite algorithm—the lattice—and got correct answers.

But I want to point out…I need to point out that this is exactly the outcome you should expect when you go about teaching standard algorithms.

If you wonder why your kids (whether your offspring, your students, or both) are not thinking about the math they are doing, it is because the algorithms we (you) teach them are designed so that people do not have to think. That is why they are efficient.

If you want kids who get right answers without thinking, then go ahead and keep focusing on those steps. Griffin gets right answer with the lattice algorithm, and I have every confidence that I can train him to get right answers with the standard algorithm too.

But we should not kid ourselves that we are teaching mathematical thinking along the way. Griffin turned off part of his brain (the part that gets 37 times 2 quickly) in order to follow a set of steps that didn’t make sense to him.

And we shouldn’t kid ourselves that this is only an issue in the elementary grades when kids are learning arithmetic.

Algebra. The quadratic formula is an algorithm. Every algebra student memorizes it. How it relates to inverses, though? Totally obfuscated. See, we don’t have kids find inverses of quadratics because those inverses are not functions; they are relations. If we did have kids find inverses of quadratics, they could think about the relationship between the quadratic formula:

$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

and the formula for the inverse relation of the general form of a quadratic:

$y=\frac{-b \pm \sqrt{b^2-4ac+4ax}}{2a}$

Calculus. So many formulas (algorithms) that force students not to think about the underlying relationships. If we wanted students to really think about rates of change (which are what Calculus is really about), we might have them develop a theory of secant lines and finite differences before we do limits and tangent lines. We might have Calculus students do tasks such as Sweet Tooth from Mathalicious (free throughout October!). There, students think about marginal enjoyment and total enjoyment.

On and on.

This is pervasive in mathematics teaching.

The results are mistaken for the content.

So we teach kids to get results. And we inadvertently teach them not to use what they know about the content—not to look for new things to know. Not to question or wonder or connect.

I’m telling you, though, that it doesn’t have to be this way.

Consider the case of Talking Math with Your Kids. There we have reports from around the country of parents and children talking about the ideas of mathematics, not the procedures.

Consider the case of Kristin (@MathMinds on Twitter), a fifth grade teacher, and her student “Billy”. Billy made an unusual claim about even and odd numbers. She followed up, she shared, we discussed on Twitter. Pretty soon, teachers around the country were engaged in thinking about whether Billy would call 3.0 even or odd.

But standard algorithms don’t teach any of that. They teach children to get answers. They teach children not to think.

I have read about it. I have thought about it. And tonight I saw it in my very own home.