**If we want students to harness the power of visual representations — to own them — students need to understand why that representation is important. **

Visual representations can be elegant metaphors for the underlying mathematics. They can help learners develop an understanding of the mathematical concepts and procedures. They can reveal hidden nuances of the mathematics at hand. They can function as a tool, assisting in problem solving.

However, visual representations are really only useful if the learner can understand then use them. It can take time to process and build an appreciation for a representation. In the image you see above, I was working with kindergarten students to build up an appreciation for the **ten frame**.

### What is it about ten frames?

Ten Frames are among the most important representation we introduce to our primary classes. Ten Frames (and Five Frames) support using benchmarks of five and ten to compose and decompose numbers. If a ten frame’s top row is full, and there is only one on the bottom, that’s 5 and 1 more: 6. When kindergartners first start learning about how teen numbers are like a “ten with some extra,” ten frames provide a powerful visual. One full ten frame and six more: 16. One full ten frame and eight more: 18.

When first graders are learning about addition and subtraction, we use ten frames to model bridging across 10. For example, to show 8 + 5, we might show 8 dots in one ten frame, and then decompose the 5 into 2 (to fill up the first ten frame) and 3 (in the second ten frame). That shows how 8 + 5 = 10 + 3, which is 13.

### Setting the stage for debate

I work with a group of kindergarten students on Thursday and Friday afternoons. They have been identified as being “in need of additional support,” typically because they get lost in the weeds of counting through the teen numbers, or because they do not consistently demonstrate an understanding of one-to-one correspondence.

These students count quantities — whether it’s a loose collection, or dot images fixed on a paper — by ones, sometimes laboriously. We have had conversations about what collections and images are the trickiest to count. From there, we had conversations about strategies for keeping track of what we are counting, even when the collection/image is ‘tricky.’

That had me thinking about ten frames. Quantities are often easier to count when they are within this structure. Everything is aligned, and there are also perceptional benchmarks — 5 in the top row, 10 in the full ten frame, etc. I want these students in my “intervention” group to begin to identity known quantities, so this felt like a natural direction.

### Introducing: Tiny Polka Dot

Tiny Polka Dot is a deck of cards with perhaps unlimited application. (They were designed by Dan Finkel/@MathForLove). It’s available for purchase on Amazon or as print & play files on the website.

The cards come in different mathematical flavors.

###### (image from TinyPolkaDot.com)

The **purple** cards show the numerals 0-10.

The **blue** cards show the numbers 0-10 on ten frames.

The **teal** cards show the numbers 0-10 using familiar dot patterns, like the ones on dice.

The **green** cards show the numbers 0-10 using differently sized dots, arranged randomly. (I call these Dinosaur Spot cards.)

The **orange** cards show the numbers 0-10 on a circle. (I call these the Atomic Cards.)

The **red** cards show the numbers 0-10 as a series of doubles and doubles + 1 representations. Even numbers are symmetrical — three blue dots, and three green dots reflected across a diagonal line. Odd numbers as asymmetrical — three blue dots, and four green dots.

### Priming the students for debate

I took out the **blue** (ten frames) and **teal** (dice dots) decks of cards. “Today, we are going to see how fast we can find different numbers,” I explained to the two girls present. (It was convenient that I had only two kids in group. This year’s flu is *nasty*. Get your flu shot, everyone!)

“Oh! Dice!” Emma exclaimed.

From each deck, I placed the cards for the quantities 0, 1, 2, 3, 4, 5, 6, 9, and 10. Emma traced her finger along the edge of one card, while Ava looked at me expectantly.

With an arched eyebrow and a dramatic flair, I offered the following: “see how fast you can find… the number…”

They both leaned forward.

“1!”

Emma quickly grabbed the teal 1. A second passed, and Ava grabbed the 1 in the ten frame. We would try this a few more times to build up some ideas before starting our debate.

“Great. Now… see how fast you can find… the number… 4!”

Ava swooped up the 4 in the teal dice pattern, while Emma found the 4 in the ten frame. “1, 2, 3, 4… oh, it didn’t make it to 5. It’s 4!” She affirmed.

In general, I noticed that the girls seemed to claim the dot pattern cards faster, until we tried to find 9 and 10. This makes sense. The dot patterns for 1-6 are very familiar. Ava told me that she and her parents like to play lots of games at home. These are representations that she demonstrates ownership over. We had some quick discussion, and then moved onto our featured task: finding the quantity card when the choices are the blue ten frames cards or the randomly placed, randomly sized dinosaur spot cards. This would incite a real desire for the structure of the ten frame.

### Is it easier to find the number shown on the ten frames or with dinosaur spots? Why?

I had been reading a bit about Chris Luzniak’s Debate Math structure. (His book, *Up for Debate*, was published by Stenhouse in 2019.) The day before planning this K lesson, I read a blog post from Janaki Nagarjan (@janaki_aleena) about using this structure in her third grade class. I was not ready to go all in with the structure just yet, but I loved this idea about creating a need for talk. What’s the easiest/trickiest/coolest/etc. I thought about asking what the coolest way to determine the number of dots on the 10 dinosaur spot card, but that didn’t seem to get at my big goal of revealing the beauty of the ten frame structure.

**Is it easier to find the number when it’s shown on a ten frame or modeled with dinosaur spots? Why? **

We tried a few rounds. *Find four!* Ava pulled up the dinosaur spot card faster.

“It’s easier to find the dinosaur spot card, because it looks like two and two. 2 + 2 = 4.”

*Find zero!* Emma pulled up the dinosaur spot card, again.

“It’s easier to find the dinosaur spot card. The other one has too much stuff on it.” (The zero in the ten frame deck has an empty ten frame, which is more visual information to process.) There isn’t much debate when everyone agrees.

*Find ten! *The girls fought over the ten frame card, before Emma begrudgingly attempted to find the ten amongst the dinosaur spot cards. She first pulled up the six card and counted it. “1, 2, 3, 4, 5, 6… nope.” Then the nine card. Finally the ten. I was curious to see what they would say now.

“The ten frame is so easy! It’s full!”

“It was really hard to find the ten frame in the dinosaur spots.”

“Why do you think that is?” I asked.

“I don’t know what ten spots looks like. First I got nine.”

“No, first you got six,” Ava corrected Emma. Kindergartners!

“Oh, yeah, first I got six. Then nine. I had to count so much!” lamented Emma.

“I didn’t have to count. The ten frame is just ten,” Ava trumpeted.

I suggested we try another one. *Find nine!* Emma reached for the full ten frame, then reconsidered and grabbed the one that was missing a single dot. Disappointed, Ava was left to count all of the dots on the 9 card. Even though she had just watched her friend count it, she couldn’t remember which one it was.

“The ten frames was so easy!” Emma gushed.

“I wish I had saw it first!”

“It was like a ten but it’s missing one.” Honestly, I was surprised. I hadn’t seen Emma using structures like this to determine quantities before. She always counts by ones in our work together. I wondered if this might be a fact that she memorized from class, or if it represented something deeper.

“Yeah, that’s nine. It’s so close to 10. But on the dinosaur spots it just looks like a lot.”

“All of these cards look like *a lot*,” Emma added, pulling out the cards for 5, 6, 9 and 10. “There are so many dots.”

“*Soooo* many! 1, 2, 3, 4… so many!” (I think I laughed here.)

“I’m going to put the ten frames all closer to me,” said Emma, creating a winning strategy.

“Nooooo!”

Yes, it is easier for two children to carry on a conversation without adult intervention, rather than a whole classroom full. Still, I was impressed with the potency of the debate-like question. These girls had a lot to say, even when they agreed. Some of this boils down to the culture we have created in our group, and the culture was created in part through the use of tasks like this: ones that are designed to get right at the heart of the mathematics, and inspire conversation.

I look forward to my next session with the K crew, when we can continue to explore and debate the potency of the ten frame. I want these kids to own it.

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