Tag Archives: elementary math

Adventures in fourth grade

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:


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:

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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.

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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?

Spoiler Alert!

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.

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


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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.

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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.

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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.

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).

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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.

2• 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.


Skills practice [#NCTMDenver]

I attended E. Paul Goldenberg’s session on Thursday of NCTM in Denver. It was not at all, as advertised, in keeping with the proof strand. But that does not matter.

What matters is this. Goldenberg shared the video below. The whole video is worth your time, but I have queued it up to the 2-minute mark, where a beautiful classroom sequence unfolds (give yourself about 5 minutes for it).

My eyes tear up watching this sequence. I am neither kidding nor exaggerating. It gives me hope for quality classroom instruction in elementary mathematics.

Be sure to notice the transition to a new task at the 4-minute mark, and how the teacher deals with the struggle that occurs at the 6-minute mark.

Also please look in the kids’ eyes. Watch their body language and their waving hands. Watch them think.

Kids are practicing facts in this classroom. The teacher is providing instruction. Contrast with this.

[NOTE: As of 5/2/2013, the video referred to seems to have been removed from YouTube. My apologies. Go search YouTube for “EDI math” and you’ll find plenty of examples that are essentially equivalent to the one I refer to below.]

You can flip this latter instructional sequence because it involves telling and choral response.

You cannot flip the first instructional activity because it involves  adapting instruction in response to student ideas, and it involves students justifying their thinking to the teacher and to each other.

You can’t flip that.

[NOTE: I have edited some of the comments below in order to focus on the practices that were exemplified in the videos (one of which is now private), rather than on the teachers in them. See my post on norms a while back. My apologies to anyone who feels their words have been altered in ways that do not convey their original meaning.]