Math Talk by Necessity: a 4-year-old’s story


My 4-year-old is chatty. And while he will occasionally talk to himself, he’d much rather talk to you. And to him nothing is more frustrating than not being understood.

So, he’s realized that he needs detail. Words like “lots” or “just a little” or “soon” are easily misunderstood. Like when Dad says, “Hang on, dinner will be ready soon.” He’d prefer to have a better understanding of whether you mean 30-seconds soon or 10-minutes soon.

The math teachers among us see this as adding units to your answer. We ask our students, “What was the final speed of the car?” We don’t expect them to say, “fast.” We’d prefer something like “54 m/s”. So would my 4-year-old. “Soon” is too loose a word.

Except, most of the typical units don’t mean anything to him because he is still developing his understanding of what “a mile” is. First, that it’s a unit of distance. Second, how far is one of those. These two things are essential to making meaning out of a statement like, “Well, the store is 6 miles from here.” Without those, statements like “This box is heavy. It’s, like, 6 miles!” are not uncommon around my house right now.

But yesterday, something new happened. Together, he and I made Alton Brown’s Hot Cocoa Mix . To go from ingredients to a drink, you have four units of measure. (cups, teaspoons, tablespoons, and ounces)

So, midway through as we were chatting about this and that… together making the distinction between teaspoons and tablespoons… why two cups of one ingredient looks so much different than two teaspoons of another… stuff like that. Then, we came to the part where we needed to scoop the mix into the mug and I handed him a “scooper” and he says, “I need two… two… of THESE, but I don’t know what it’s NAME is? Dad… what’s this one’s name?”

Did you catch that? There was a step forward in that.

He knew the number wasn’t enough. “Put two in there” is really easily misunderstood. That would be a problem. Two of what?

He also knew that it totally mattered what “name” he gave the scooper. If he called it “a cup”, he risked calling it the wrong thing. Two cups of mix makes a totally different amount of hot cocoa that two tablespoons. And both could be called “a scoop”. (Two scoops from the coffee can is different that the two scoops of raisins on the cereal box.)

Finally, he figured out that this thing had a name. That I knew it. That he needed to know it and that it wouldn’t work to make it up. If he is going to communicate this amount to others, he needed to know the ACTUAL name of this scooper. Not his preferred name.

There you have it. Math talk by necessity. Lends credence to the notion that if you put students in rich enough environments, you won’t have to mandate good math talk. They simply won’t be able to communicate effectively without it.

Now, we just have to deal with the fact that when Chef Alton says “two tablespoons”, Dad tries to measure accurately while the 4-year-old would prefer that mean “the maximum possible amount of hot cocoa mix that the tablespoon will transfer to the mug.”

Why I love this picture…

I want to tell you why I love this picture.


I took this picture this morning in Lansing, MI during some wonderful small group math talk. There is one device, an iPad, with an Osmo setup attached to it.

So, here’s why I love this picture.

There’s tech and…

… manipulatives and whiteboard markers and collaboration. Tech fits among the variety of tools available. It’s not the best tool unless it best supports the learning. And sometimes other tools work better. And in this case, the students were being led into learning with all the different tools.

The activity is built around the social nature of learning. 

The kids are clearly sharing their answers with each other and the teacher… there is a constant back-and-forth, sharing ideas and discussing them. They were seeing each other ideas, but…

Their strategies aren’t all the same. 

One girl is using an array. One girl is using groups of three. One girl wasn’t quite sure what to do (so it was a good thing she could see the other two girls’ work.)

The teacher’s hands are off. 

The students are doing the reaching, arranging, manipulating. Remember, the one that does the work will be the one that does the learning.

That’s why I love this picture. it captures so many wonderful things about the right kind of teaching and learning.

Excellent Classroom Action – Art and Geometry in the Elementary Classroom

I’ve written about the connections in math and art before. The visual nature of Geometry lends itself quite nicely to this. I found that the right pairing could bring in the engagement of the visual arts while maintaining fidelity to the content.

Exhibit A: Sarah Laurens, 5th grade teacher at North Elementary in Lansing. Mrs. Laurens reached out to me excitedly while I was in her building to see a math activity that she was leading her students through. They involved quilts hand sewn by Sarah’s grandma.

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The activity went like this: Students were in groups of threes and fours gathered around one of grandma’s quilts. Each quilt was made of a series of geometric shapes. Sketch the primary “unit” shape of each quilt and identify each of the polygons that are contained within it. On the surface, it is a fairly simple activity, but listening to the students talk to each other.

[Students looking at the black and blue quilt above left]

Student 1: “Those are just a bunch of hexagons.”

Student 2: “Hexagon’s, no… no… those are octagons”

Student 1: “Yeah, yeah… same thing.”

Student 2: “They’re not the same, one’s got six sides and one’s got eight.”

Student 1: “Well… wait… one… two… three… four… five… six… six sides! See I told you!”

[Students looking at the purple and while quilt on the above left]

Student 1: “That’s an octagon with kites around the outside.”

Me: “Are they kites?”

Student 2: “They look like kites.”

Me: “They sure do. How many sides do they have?”

Student 1: “One… t, th, f… oh! five…. They’re pentagons!

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Ms. Laurens and her students were comfortably saying and hearing words like “regular”, “tessellation”, and using definitions to make sense of what these shapes are, and using the definitions to settle disagreements (the foundations of proof…)

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Interesting images like these were leading to some interesting conversations as well. I’m thinking of  a conversation I heard between two students who were trying to make sense of the shapes they were in the picture directly above to the right. They were trying to to determine if the purple section in the middle was one big shape or two smaller shapes put back-to-back.

Then after some discussion, they realized that their answer would be the same either way. (Quadrilateral was their choice for the shape name. The word “trapezoid” was getting thrown around, but the students were having to be prompted for it).

I very much enjoyed getting to see these fifth graders exploring. Ms. Laurens was excited, the students were engaged (and this was clearly not the first time they were expected to be a self-directed and collaborative).

I’m just bummed that my schedule forced me out the door before I got to see Ms. Laurens’ closure of the activity. The students were wrapping up their discussions as I had to head out the door.

The Role of Feedback

I had to learn this lesson the hard way.

In the fall of 2012, we had to part ways with our Geometry textbook. We replaced it piece-by-piece by what became our Geometry Course It was definitely a net positive for everyone involved, but as you might expect, it didn’t come without some growing pains.

As it turns out, in order write a decent course, it helps to know a little something about how young people learn. When I built in the activities that made up the course, I tried my best to pay attention to what the students would be doing. What was my target content and how was I giving the students a valuable experience with that content that would help reduce as many barriers as I could to the learning. I tried to pay attention to foundation knowledge, reviewing old content and building in new understanding in reasonable chunks with practice.

What would the handouts look like? What should the wording be? What sequence will I ask the questions? How many practice problems should they have? What type of product should they produce?

I thought of all of these things. The first year, there were lots of positive signs, but I still noticed students struggling in areas that I wasn’t expecting. The students were getting better at verbalizing their thoughts both in writing and speaking. They were using the vocab fairly well. But they struggled in ways that after a while, got pretty predictable. It started to seem like the more extensive the paper-and-pencil process, the more likely the students were to struggle, and not just clumsy-style, like in ways that seemed to suggest that they didn’t really know how to complete the process.

What I never thought of: How am I going to build in opportunities for the students to receive feedback on the products they were producing?

As the year progressed, the seemed to get better and better at discussing the math and using the words, sometimes with some wonderful results, but that was because the whole time they were talking, I was walking around talking with them. Interacting, correcting verbiage, modeling effective uses of the essential vocab and connecting it back to the problem. I was responding to their curious looks and asking them to clarify their confusion. These were my favorite times in math class. So I did them as often as I could.

No wonder they showed growth in this way.

But what was I doing once they completed a handout like this? That first year, a whole lot of nothing. We moved on. Or we would discuss the broad topics and I would pose a problem like this with a mind to formatively assess their progress.

But I wouldn’t simply show them the answers. I forgot about the feedback. And it got in the way.

A few quotes from Dr. John Hattie on this topic:

“In short, receiving appropriate feedback is incredibly empowering. Why? Because it enables the individual to move forwards, to plot, plan, adjust, rethink, and thus exercise self-regulation in realistic and balanced ways.”

“Imagine a group of students who are about to embark on a series of lesson, and during these early experiences we pause and show them the various way they will be successful at the end of the lessons, or tell them how they will know when they have been successful in these lessons. The is a relatively cheap investment, takes little time, but of course, it provides teachers with a challenge – that of working alongside the students to maximize the number who reach the success criteria.”

“When we interview students on what they understand by feedback and why it is important to them, on theme emerges almost universally: they want to know how to improve their work so that they can do better next time.”

“The feedback you offer your students provides the tools they need to be able to perceive the immediate path ahead, and so decide that it is really worth effort.”

Second year we determined that we were going to have an answer key (and if possible two or three) so that the feedback could start by them checking their answers. I instructed them to ask anytime they had questions (even if the question was “mine looks different than yours, but I still think it’s right.” I actually rather like that question.) Dr. Hattie is indeed right that that was a cheap investment. My goal was for every student to get to compare his/her solutions to the correct ones. It slowed progress a bit through a lesson the first time a bit. But I had to do significantly less reteaching, so I got that time back.

When the students were left to practice and then draw their own conclusions about their own comfort level of their progress, I had forgotten one rather important understanding that both myself and the students were largely aware of. They mostly lacked the expertise to properly determine whether or not they should feel comfortable with the work they just completed.

But given the appropriate feedback, they can see where they were successful, were their focus needs to be for future success and how much room their is between the two.

Quotes taken from Chapter 8 of Visible Learning and the Science of How We Learn by John Hattie and Gregory Yates. Routledge Publishing – 2014

Vocab: The Foundations of Math Talk

I want to talk about vocabulary for a minute.

Specifically, I want to describe one way that can support students struggling to learn vocabulary that is necessary for effective math talk. Often times, struggling math students are willing math talkers, but the math talk is filled with pronouns (that thingy right there, you know what I mean?), hand gestures, and rough sketches.

This can make communication a bit of a chore, especially if the student is talking to another student who is also still in the beginning stages of developing understanding in that topic.

So, here’s a way that you can change the conversation a bit.

Consider this handout combined with this Google Form.

In short, the students would take a few minutes exploring different formal definitions for the vocab words you are exploring. There are always differences if you go to different sources, so by forcing them to explore a variety of interpretations, you can really help the students see that this vocabulary is about describing an idea, not memorizing a specific wording. This is important, at least to me.

Pair the students up (or group them in 3’s) and have them consider the big ideas and develop their “own-words” definitions for each vocab word. Then they plug them into the Google Form.

At that point, you’ve got a nice collection of what your class currently has taken away from their time. (And by “time”, I mean about 20 minutes from start to submission.)

As you wander around listening to the conversations, you’ll likely notice a few definitions that are coming together more slowly than others. This is inevitable.

At that point, I’d bring in Wordle. Wordle is a free tool that build word clouds. Word clouds can have a nice visual appeal and change the way a block of text looks by changing the physical size of a word based on its frequency in the text. For confusing definitions, this tool has the potential to help the students hone in on the big ideas. For my students, it was often “circle.” Most kindergartners could draw a decent circle (or at least a shape that you would guess quite quickly was supposed to be a circle.) But, that visual understanding is often as far as students get.

Want to see how well your students understand what a circle really is? Ask them why this is a lousy attempt to draw a circle. See what they say.

bad free hand circle

If your students are struggling with an idea, though, go into the Google Form responses, copy all of their definitions and paste them into a Wordle.

My third hour Geometry last year, produced this word cloud.


This isn’t going to solve all of your problems by any means, but with a visual like this, your students will have the opportunity to see that the major idea of this concept of circle revolve around the notion of a center point, a distance, and points in a curved shape. Those are major steps in the right direction when we are moving students away from a purely visual understanding of what a circle is and helping them understand the geometric properties of a circle.

Remember, the goal of learning vocabulary is to facilitate student understanding of the content. Memorizing a definition only goes so far. We have to continue to develop the means to push students to internalize the bigger ideas within the vocab so that those words then become foundations to build more sophisticated ideas.

If you feel like you’d like a tutorial of how to use Wordle, I went ahead and made one.