Building Mental Representations

“Sabrina is fascinating. She will look at a problem for three minutes, not appearing to be thinking about it at all, and then come up with the correct answer,” her classroom teacher told me. “And I have no idea how she figured it out.”

How do we understand the thinking that is in someone else’s head?

I interviewed Sabrina about her strategies for addition and subtraction using an interview designed by Michael T. Battista, who also wrote the interview that I used with Ali. It’s the same exact interview protocol that I used with Eduardo.

In a box, there are 35 red apples and 27 green apples. How many apples are in the box?

Battista, Michael T. Cognition-Based Assessment & Teaching of Addition and Subtraction. Portsmouth, NH: Heinemann, 2012.

Sabrina responded with:

I took a 3 from the 35, which left a 2 and a 30.
27 + 3 = 30, and then I have 30 + 30 = 60 and 2 more makes 62.

She was right. 36 + 27 = 62.

“Huh! So first I heard you say that you would take a 3 from the 35. Why take 3 away? Why not 4? Why not 5?” I cued.

“Because of the 27,” she responded prosaically.

“Interesting. It looks like you’re trying to add up to the next group of ten. So if you want to get from 27 to 30, you need 3 more.”

“Yes.” Sabrina looked almost disinterested at the conversation. She didn’t feel like I was unraveling her thinking.

So we moved on. She did it again for 267 + 189. We waited, suspended in silence for at elast a minute or two, before Sabrina spoke:

Okay, so take a 1 from the 267, and you have 266, and… no, wait, take 11 from the 267. And then you have 256 and 200 and that makes 456.

I couldn’t tell if she was using a compensation strategy here, or one of incremental jumps. I decided that the only way to tease this out might be to draw it using open number lines.

As I drew, I attempted to narrate for her, to make explicit the connections between my visual representation and her strategy.

So start with 189. I want to add 267.

So first I add the 1 to get to a nice, benchmark number of 190… oh, wait, if I just add 11 I can get up to 200!

Then there is 256 left to jump. 200 plus a jump of 256… I land at 456.

“Does that match your thinking?” I asked her, perhaps with too much confidence.

“Uh… I don’t think so. I don’t really understand what you’re doing. Jumps?”

I attempted to keep a neutral expression, but I am sure I flinched. I pride myself on understanding student thinking. However, in this case, I had projected something onto Sabrina’s thinking: that she was thinking about addition as adding onto another quantity, bit by bit. Maybe she wasn’t. Maybe she was thinking about this in a much more compensatory way.

I decided I would model her thinking again using Base 10 blocks. Instead of using the real blocks, I decided to use the virtual version from Math Learning Center. Their app (available as both as web app and for iPad & chrome) makes it much easier to show regrouping. Circle 10 unit blocks, hit the regroup button, and… voilá! A ten block!

So this is 189. Right?

Sabrina nodded.

Now this is 189 and 267.

Sabrina nodded again. She was still with me.

Then we add 1 from the 267 to the 189. Now it’s 190.

But it doesn’t look exactly like 190, because I see 100 + 80 + 10. So let’s regroup it to make those 10 ones a single ten block.

There. So now we have 190 and 266. Let’s take a ten from the 266 to make the 190 an even 200.

And now we have 200 and 256, which is 456.

“I guess…” Sabrina shrugged. “I get what you did. But it wasn’t the same.”

“Were there parts that felt the same? Which ones felt the same?”

“You regrouped like I did.”

“Tell me more about regrouping.”

“I started with the 1s because it could end up equalling more than 10, and then I’d have an extra 10.”

Sabrina’s explanation was in her own language, and felt so clear. She owned this idea.

“So what felt different about the visual I showed you?” I asked Sabrina.

“Well, it just… it’s not how I pictured it in my head. No blocks. Your way is slow and you have to change things.”

So in her head, 189 + 1 instantly became 190, but my visual showed a more laborious regrouping process — a limitation of using these blocks. Her way felt more natural. 190 didn’t need to turn into a different version of 190, it just was.

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We were only halfway through the interview, and already I was thinking deeply about building up mental representations. Is internalizing these visuals helpful? When and why? And, if so, how do we help students internalize some of the visuals? Or how do we help them connect their natural strategies to some of the visuals? I think these connections can add shades of meaning, that push thinking deeper.

I watched Sabrina use a similar strategy for 306 – 128. I was impressed that she was able to keep these numbers in her head.

306 minus 6 is 300.
Then I have to take away 2 more to make the 8. 300 – 2 is like 299, 298.
Then 298 minus the 20 is 278.
Then 278 minus the 100 is 178.

I didn’t think she was regrouping here. It seemed like she was doing ‘take away,’ like incremental jumps. She split up the 8 in 128 to make it easier to bridge across the benchmark of 300. That doesn’t feel like something she would do with the base 10 blocks. This felt more like the open number line.

Sabrina clearly demonstrated an understanding of place value, and applied this understanding to decompose and recompose numbers.

Sabrina used the “same” strategy for both addition and subtraction, moving fluidly with ideas about taking away, adding to, and perhaps even compensation (taking from one of the quantities to add to the other quantity). It didn’t seem to bother her that she needed to think of addition and subtraction in different ways. These were all ideas that she had ownership over, and these ideas seemed to join at the intersection of getting to the next group of 10.

So do we need to build up more visuals that Sabrina can use mentally? I think they will help her, but I also think that the visual representations are just that — representations of ideas. They are symbolic of what Sabrina is already doing.

This brings me back to the original dilemma. Sabrina’s third grade teacher says that Sabrina’s approach feels almost “magical,” because she rarely describes it and never shows it. How can we help Sabrina show her thinking? Will the visuals support her? 

How might visual representations support Sabrina as she builds understanding of multiplication and division? 

Conducting clinical interviews with students always leaves me wanting to reread books like Children’s Mathematics: Cognitively Guided Instruction, by Thomas Carpenter, Elizabeth Fennema, Megan Franke, Linda Levi, and Susan Empson. Although I think I understand how Sabrina is adding and subtracting, that’s just the tip of the iceberg.