It was a sea of dots.

Eduardo’s work for 47 + 27

Over and over again, for both computational and story problems, third grade student Eduardo drew dots.

Eduardo’s work, question from Cognition-Based Assessment & Teaching of Addition and Subtraction, by Michael T. Battista (Heinemann, 2012)

It was fascinating to watch him — meticulous, methodical — draw dots and dots and dots. It was perhaps more fascinating to me that he never even counted the first set. Although he drew the 47 (for 47 + 27), in painstaking detail, he started at the second set of dots, the 27, and counted on: 48, 49, 50… But why draw all those unnecessary dots?

“Wow, Eduardo! Sometimes I lose track when I have to count so many things, but you managed to stay focused,” I gushed.

“Slow and steady,” he responded, with all the panache of the Tortoise who bested the Hare.

This is not a strategy I would expect from the “typical” third grader. Most of the student’s in Eduardo’s class add numbers using partial sums, or the traditional algorithm, or larger jumps on a number line. These strategies rely on number relationships and knowledge of place value. Eduardo? He’s counting on — a strategy we are working on in kindergarten right now.

Meet Eduardo

Eduardo is struggling in math class. I could write a long list detailing the signs that he is struggling in class, but, perhaps most poignant for me is that Eduardo recently told his third grade teacher, Caitlin, that he feels anxious about math. So Caitlin I have been collaborating to learn more about him. 

We want to know:

  • What are Eduardo’s current (independent) strategies for addition and subtraction?
  • Why is he struggling in class? Are the tasks inaccessible to him to him because of content — or something else?

From there, we want to figure out:

  • How can we build off his current strategies and understandings in class?
  • How can we help him push some “unfinished learning”? (e.g. continue to develop his addition and subtraction strategies even as he and the rest of the class move onto fractions, etc.)

So far, I’ve observed Eduardo in Caitlin’s class, and conducted 3 sessions of clinical interviews with him (Michael T. Battista’s Addition and Subtraction CBA, and the GloSS from the New Zealand Numeracy Project).

Eduardo is puzzling to us. He is generally slow to process new information. Eduardo will stare at a blank page. It’s unclear if he’s continuing to think, or avoiding work, like so many students do. Caitlin spends a lot of time working one-on-one with him over the course of the math block, and sometimes in a spare moment before morning meeting, as well. She usually pulls him to the side to watch him complete an exit ticket after a lesson, so that she can learn more about his thinking, and determine how to help him access tomorrow’s math lesson.

Eduardo and the Dots

So we return to Eduardo and the dots. Is he struggling in math class because he doesn’t recognize number relationships? Is he struggling in math class because his only strategy for addition is counting on or back, and this crumbles to bits when presented with larger numbers?

267 + 189.

It involves three digit numbers, and regrouping multiple times. I expected Eduardo would attempt to draw dots again, the only strategy he had shown me in 5 consecutive problems. I was wrong.

“I don’t need to draw dots. Since it’s written like this, I can do it a different way,” he told me.

Oh. So a problem’s format can really throw him off. All of the other problems were couched in stories or presented horizontally.

“2 + 1 = 3. Then 6 + 8… that’s, um, 14… huh.” He paused, and allowed his hunched shoulders to roll forward. “I’m not sure what to do, since 6 and 8 are 14.”

He wasn’t exactly asking for help. I watched his head sink down towards his lap. He balled himself up inside his large winter jacket, which he insisted now wearing despite the comfortable temperatures in his classroom. I thought I’d lost him.

Before starting the interview, I placed some strategic math manipulatives — counters, base 10 blocks, unified cubes — next to us. “Feel free to use them,” I had said. He hadn’t.

Since Eduardo looked ‘stuck,’ I decided to reintroduce the manipulatives. “Would any of these tools help you?” I asked, trying to maintain a neutral tone so as not to push him towards a particular representation.

Eduardo’s hands emerged from the sleeves of his jacket, reaching for the base 10 blocks. I wasn’t surprised; Caitlin told me that she had been using the base 10 blocks a lot in her work with Eduardo.

Slow and steady, Eduardo modeled both 267 and 189, and then joined the groups, trading in groups of ten 10s for hundreds, and ten 1s for tens. He worked at an even pace, never once looking up or allowing himself to be distracted by the fourth grade class walking past us in the hallway.

Eduardo whispered to himself: “300…. 400… 440… 456. Done.”

“Please write that here, on the page. After, we’ll figure out how to represent what you just did and the thoughts you just had on the paper.”

356, Eduardo wrote, instead of 456. I had given too many directions at once. He’d lost the number in his head.

Is he struggling in math class because he struggles with short term memory? Is he struggling in math class because there are too many interruptions to his thoughts? What can he keep in his head once his thinking has been interrupted?

Eduardo and I worked on representing the Base 10 blocks on paper. He seemed pleased with how much faster it was than drawing 267 dots. We tried one more problem: 284 + 257. He went straight to drawing, rather than picking up the blocks.

Eduardo’s work for 284 + 257

I was surprised how quickly this seemed to come to Eduardo, who had been stuck on dots and counting on just minutes earlier.

Is he struggling in math class because he does not always generalize strategies? Is he struggling in math class because even comfortable strategies don’t come to him without some cuing? How does he access previous learning?

Thinking Slowly and Deeply

In our next session, I used the New Zealand Numeracy Project’s GloSS (Global Strategy Stage) assessment. It is a clinical interview that allows the practitioner to observe strategies across three domains: (1) addition & subtraction, (2) multiplication & division, and (3) proportions, fractions & ratios.

For 9 + 7, he used his fingers to count on.

The next task involved a 5 x 6 array of cups. How many cups in all? Eduardo quickly responded with 30. “Well, I just go 5, 10, 15, 20, 25, 30. I know my 5s,” he said with a hint of pride.

When does he count on and when can he use larger groupings? Is he struggling in math class because he doesn’t see groups, including equal size groups, unless they’re the 5s and 10s he’s memorized?

Eduardo continued to surprise me as we spiraled up and up with the GloSS, hitting questions intended for more advanced strategies.

Eduardo drew 57 in base 10 block format: 5 sticks, 7 dots. He then erased the equivalent of 25: 2 sticks, 5 dots. 32 left.

(from the New Zealand Numeracy Project’s GloSS Assessment)

So he can generalize more about the base 10 model! Caitlin said that she has done subtraction using base 10 blocks, but only using the manipulatives, not the drawings. He must be reactivating some learning from the other day. What cued him? Is it merely my presence?

Eduardo drew 24 with base 10 blocks here, too: 2 long sticks, and 4 dots. Oh, no! He’s overgeneralizing!

Eduardo continued to surprise me, though. First he drew the four dots on the right, clustered into groups of 2. “those 4 make 2.” Then he got stuck. “Do… do the 10s count as ten pegs? I’ll draw them.” And he unpacked the tens.

Eduardo’s work on the peg problem (from the GloSS: Interview 1, Task 7)

He started to erase the dots, two at a time. “1, 2, 3, 4…” he whispered. “12. It’s 12.”

So he can make sense of stories involving the four basic operations! This one didn’t have any key words to signal division, and he did not indicate that he thought of this as division. He’s just making sense of it.

He continued to make sense of problems drawing either grouped dots (e.g. for multiplication and division) or using base 10 blocks (for addition and subtraction). He hit another (temporary) wall with a fraction problem:

Eduardo drew 12 in base 10 format: 1 long stick and 2 dots.

“What does this sentence mean?” He asked, pointing to the second line.

“The pizzas are cut into slices, and each slice is 1/4 of the whole pizza. Each slice is 1/4 in size.

“So those don’t help,” he murmured, pointing to his base 10 drawings. He then proceeded to write out the fractions.

Eduardo’s work on the pizza problem, GloSS Interview 1, Task 8

His class has been working on understanding fractions greater than 1, and building them from unit fractions! He’s activating the learning! Why doesn’t he do this independently in class? Is he struggling in math class because of an emotional block?

For the first time in our sessions, Eduardo looked at me expectedly. “I don’t know what to do next.”

“Well…” I paused. What hint to give? What question will help push his learning? “Did they eat a whole pizza?”

“Yes. 4/4.”

“So how many pizzas did they eat?”

He carefully tapped each number in the sequence, saying “1” aloud when he hit 4/4 ,”2″ when he hit 8/4, and “3” when he hit 12/4. “Three pizzas.”

Honestly, as much as I like to think I believed in Eduardo before we started this interview, I didn’t expect this from the kid that had produced the sea of dots. With each question during the interview, he showed me understandings that forced me to shift my thinking — maybe rewrite the entire narrative of what I know of him a student?

Presuming Competence

One of my favorite parts of using interviews written by someone else is that I end up giving tasks, or asking questions, that I wouldn’t have thought to ask. It opens my mind. It opens my understanding of the student as a mathematical thinker. It reinforces the idea that we should presume competence — look at everything Eduardo can do! All the learning he can leverage! I had no idea!

Eduardo continued to think deeply about the questions. For a joining problem, with the start unknown (a story problem that matches __ + 26 = 86) he drew 26 in base 10 block form, and then figured out how much more to add to get to 86. This demonstrates an understanding of the commutative property of addition.

To figure out a story problem that matched 6 x 8 (8 packs of 6 cans of soda), Eduardo drew 8 groups of 6 dots, and, again, counted by 1.

Can he skip count by numbers other than 5 or 10? This reminded me of the beginning of our interview, watching him laboriously count on his fingers for 9 + 7, and then skip count quickly to calculate 5 x 6.

“I wonder if there are faster ways to figure out how many in all…” I said, engaging in a ‘thinkaloud.’ “I remember how you skip counted with the cups.”

“Oh, I guess someone else could skip count by 6s.”

Heartbreaking. I don’t want him to think that math is for “other” people! “I bet you could, too!” I added, perhaps too cheerfully for him to stomach at that hour in the morning.

“6… 14? 25?”

He was right. He didn’t know how to count by 6s. I briefly considered modeling using his fingers to help him keep track, since that’s such a comfortable strategy already, but I decided to move on.

What We Know About Eduardo Now

My thinking, about both Eduardo as a thinker and Eduardo as a student, evolved quickly over our time together, and perhaps didn’t reach any real conclusions. I’m left to wonder:

  • How can we help him leverage his comfort with 10s (and sometimes 5s) to add and subtract more efficiently? How can we leverage this to skip counting by other groups?
  • Eduardo likes to make sense of problems using visuals, particularly dots, groups of dots, and base 10 images. How can we help him connect these representations to numbers?
  • He likes to erase things as he works. Why? What does this say about him as a student, and his relationship with math?
  • He processes slowly but deeply. Is something interrupting his thinking process in the classroom? What? How can we minimize the distraction?
  • He is able to access some recent learning (e.g. things that he worked on, in and out of the class, in recent days), but does not independently access learning that is from more than a day or two ago. How can we help him build more mathematical connections? Is there something in his cognitive profile that makes this more challenging? How can we learn more about strategies that may have greater success

What does it mean to be struggling in math class?

I prefer to say that Eduardo is struggling in math class, rather than struggling with math. He showed me time and time again in the interviews that, even if he is not able to complete a task independently, he thinks deeply about it. Maybe staring at a blank page really is related to processing time, and not avoidance.

Caitlin and I will meet again soon to talk more about what this means for Eduardo. We would love to build in some time for him to work more on (and process) addition and subtraction, connecting to numerals, but we are struggling with when to do it. He does not initiate work on his own, and most tasks take several times longer for him to complete than one of his classmates.

How can we play to his strenghts more in math class? How do we balance playing to his strengths with the strengths of the other 23 students in his class… among other constraints his classroom teacher is saddled with?

Does learning more always lead to more questions than answers?

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Some of my favorite published clinical interviews: