It’s hard not to let our own bias creep into teaching situations. For one, our decision making processes thrive on context: what thinking has this student demonstrated about this topic? How can we nudge them forward? This colors what questions we ask, what mathematical tools and representations we use, and whose thinking we amplify within the classroom. These are all generally good things! But we have to be careful to check our bias — for a million reasons. Today, I’m going to write about how we look at student work.
Amalia
Amalia’s kindergarten teacher referred her to me for intervention. “She’s really struggling. She can barely count or recognize numbers,” I was told. It was October.
Amalia is now in fourth grade. In the years that have passed, her teachers have usually expressed some concern. She just doesn’t seem to be getting it. Her strategies are inefficient. She’s quiet. She never shares her thinking. I don’t know how she got this. Her test score last year was a few points shy of passing.
When we launched the latest multiplication division unit in grade 4, I asked the classroom teacher to help me gather lots of formative data. Multiplication and division are two of the focus areas for the grade, and we want to take action now. (We wouldn’t want to get to the end of the year and realize, oops, a bunch of kids in this class can’t really multiply multi-digit numbers.) The classroom teacher and I decided to incorporate regular exit tickets/cool downs, as well as some larger benchmark assessments.
November 15: Drawing an Array for 18×7
November 15. At that point, we had finished the first of three investigations in the unit, and most of the students were successfully splitting a factor to determine the products of larger numbers, e.g. 
Amalia wrote 
Beneath it, she drew a detailed array. The dots and dashes in each ‘box’ made me think that she may have counted all of them, one by one.
Below it, she wrote:
She decomposed one factor! But then it all seemed to go terribly wrong. Did Amalia understand how the array could have helped her? How did she arrive at 36 and 58? The classroom teacher and I placed her work in a pile alongside students we wanted to check in with the following day. What was she thinking?
We never figured it out. The classroom teacher and I agreed that we should look closely at her exit tickets for the next few days so that we can make a plan for support. We did not want a fourth grader attempt to count out 126 boxes in an array by ones.
November 16: Introducing Division
On November 16, we started the second of three investigations in the unit, which focuses on division. The first problem asked students to determine whether 13 tables that each sit 4 people will be enough to seat 56 people? Amalia said no, and then offered some multiplication. This shows some understanding of multiplication and division as inverse operations. I wasn’t certain how she got it, and she anxiously shrugged off my line of questioning when she handed it in. I wasn’t looking to scare her, only to learn more about her thinking so that I could leverage it with the class, so we continued on.
November 29: Quiz, 104 ÷ 8
On November 29, we assessed students more formally. The first problem was to write a story about 104 ÷ 8. Every student except Amalia wrote a story problem, like “There were 104 chocolates, and they are being put in boxes of 8.” Amalia actually wrote out a paragraph detailing some of her ideas around the distributive property.
Amalia wrote:
104 ÷ 8 can be broken up in parts to make solving the actual problem easier. Like, if someone didn’t know what 20 x 4=, They could break it up. Like, 10 x 4 = 40. So the answer to the problem would be 80 if you add 40 + 40 = 80. You could also brea apart 104 ÷ 8 into parts.
While this response wasn’t what we had intended, it was interesting to see her breaking things apart. There’s something going on with halving for her. This made sense given what we had been working on in class. Here’s a record from a number talk that followed 
Amalia had actually solved 
At a first glance, all I saw was addition. In fourth grade, we push students to record thinking as multiplication and division. Addition does not meet fourth grade benchmarks here (blog post: Restless Educators). We sigh and move on, but… wait! Enshrouded in this string of 2nd grade addition problems is some rich multiplicative thinking. She’s doubling, and doubling, and doubling again. Her record is what she needed to solve the problem, and not what she needed to justify her thinking. (blog post: Look! I Showed my work!)
Honestly, neither the classroom teacher nor I had anticipated she would do anything like this. Only two weeks earlier, to solve 
December 8: What.
Amalia demonstrated increasing accuracy with multiplication and division problems in class. She often did not want to share her thinking aloud, but there was a stronger record on paper. Here she is solving 17 x 20 by decomposing the less friendly factor of 17 into 10 and 7.
(The class was almost equally split between students who solved this problem as 
However, it was Amalia’s work on another problem that stopped us in our tracks.
“Wait… 46?” The classroom teacher looked at me. I pulled my chair closer to examine the paper.
Two groups of 23.
Two groups of 23… two times! Four groups!
Two groups of 23 + Four groups of 23 = 6 groups of 23
I went over to Amalia to have her explain her thinking. Was this actually what she had done?
She had.
(I posted it to twitter with a simple message: “Today, in grade 4… 👀“)
Looking & Listening
Amalia’s work on 23 x 6 left me a bit tongue tied, but it also wasn’t a complete shock. The classroom teacher and I have been working to build up the formative assessment practices within the classroom, so that we can closely monitor students’ progression of thinking over the course of a week, a unit, a year. We try to listen interpretively and generatively, not only in an evaluative way. (blog post: How We Listen: Astrid’s Thinking) Because of this, I had already seen glimpses of Amalia’s multiplicative thinking, and also her work on doubling.
From Amalia’s work on December 8th, we could see how she deeply understands the commutative property as well as the distributive property. She is thinking hard about groups. She moves fluidly between multiplying groups (by doubling, and doubling again) and adding groups together (2 groups of 23 plus 2 groups of 2 groups of 23, or 
If I had assigned Amalia the role of “struggling student,” I might not have given some of her more sophisticated work, like the multiplication masked in addition, a second look. The way that we look at student work is framed in the way that we think about that student.
Those strictly stratified roles — the low students, the high students — are shorthand for context, but they inadvertently remove all of it. Thinking of Amalia just as a “struggling student” — and, to be sure, there are days that she struggles in math class! — means that we expect less, and may miss important thinking that they share. This isn’t universally true, but it’s true enough that we need to assess ourselves for bias.
It’s always worth digging into the student work, and asking questions.
Jenna Laib // Embrace the Challenge 
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