My school recently adopted TERC’s Investigations in Number, Time, and Space. The cornerstone of this curriculum is listening to, centering, amplifying, and leveraging student ideas. Many of the embedded assessments ask students to explore just one or two problems, with plenty of black space for communicating their thinking. (As you may know, I’m a big advocate of #TeamBlankspace.)
|Guiding Principles of TERC’s Investigations|
1. Students have mathematical ideas.
2. Teachers are engaged in ongoing learning about mathematics content, pedagogy, and student learning.
3. Teachers collaborate with the students and curriculum materials to create the curriculum as enacted in the classroom.
In my work with teachers and teacher teams this year, the underlying theme is ‘assessment as learning about student thinking.’
- How can we learn about this student’s thinking?
- What are our next steps?
What might we say to or do with this student?
- How might this change our plans for future lessons?
Here’s how that played out today in first grade.
The assessment activity from the curriculum was to “draw 20 circles.”
That’s it. It seemed so simple. Worth, what, 2 minutes of class time? 3? We’ve seen students counting in counting collections, and surely many students could complete this task quickly. After talking it over, the classroom teacher and I decided that it might still be worthwhile, especially as we are wrapping up the investigation on counting and moving onto addition as ‘combining.’
We printed out the Assessment Checklist about counting, which urged us to think about how the student ‘knows how many,’ and whether or not they double-check their work.
We launched the math block with a choral counting exercise. Students counted aloud together, and then searched for patterns in the record of their count.
We talked about how we’ve been working on counting carefully and recording our thinking, and how curious the classroom teacher and I were to see how they did this on their own.
So students returned to their desks to draw 20 circles.
Students that finished early would continue their work with counting collections, allowing us additional opportunities to observe students at work on quantity.
How Do We Know How Many?
Mira drew circles, and recorded numerals inside to keep track.
At first glance, I thought that Elan had drawn his circles from left to right, and counted from 1 to 20. But looks closely at how his circles align into columns. Elan counted by 2s: “2, 4, 6, 8, 10, 12, 14, 16, 18…” He indicated each pair with two fingers positioned vertically. Then, when he got to the bottom row, he rotated his fingers horizontally: “aaaaaand 20!”
Jesse had also counted by 2s. He drew rings around his horizontal pairs.
Lila enjoyed using ten frames during our counting collections work last week, so I wasn’t surprised to see that she’d drawn them here, too. I asked her to tell me about her work.
“I made two ten frames!”
I glanced down at the assessment checklist, and was reminded to inquire about how students determined how many. Did she double check?
“I drew my 10 frames carefully, and two tens makes 20, so… I didn’t even need to count!”
Omar started drawing a ten frame, and then erased it. “I just want to show my groups,” he later told me. He made 5, followed by another 5, and then 10.
“Five and five makes ten, and then another ten makes 20.”
“Why did you choose these groupings?”
“Fives and tens make it really easy to count.”
“How did you know that this group down here was 5?”
Omar knit his brow. “It looks like a 5.” I pressed to see whether he knew it perceptually or had quickly calculated 3+2, but his skeptical expression told me it was just a familiar visual pattern. (He was able to subitize it.)
Lexi had organized her circles into a neat 4 by 5 array. She labeled the dimension. I loved how she dealt with labeling the shared dot in the lower right corner — how this dot counted towards both the length of ‘5’ (five in each row) and the width of ‘4’ (four rows). She recorded a multiplication equation.
Across the room, another student — Caiden — told me he had drawn a 4 by 5 array, but he had really drawn an arrangement of 5 by 5 because he had not wanted to count a dot “twice,” like Lexi had. He later corrected his work by erasing a few circles.
More and More Questions
Bella’s work flows like a river. The classroom teacher sat with Bella for a while, listening and observing. She encourage her to label her circles with numbers to check, and asked her about how she kept track of her thinking. Bella circled groups of 5, but did not necessarily use them in her counting. She had originally written “12” to represent twenty, but the classroom teacher supported her in using tools around the room to check how to write it. She eventually wrote “02.”
Alma is an active participant during class discussions. She shared her thinking about place value during the choral counting warmup, and confidently links equations to her work on paper. I did not get a chance to speak with her about her work, but I’m so curious to learn more. It looked like she started with two groups of 5, and then drew on 10 more…? But her lines don’t show the same focus on groups of 5 or 10, and her equation doesn’t match.
Using What We Learned to Plan for Future Learning
The classroom teacher and I learned a lot about student thinking — how they count, how they organize their thinking mentally and also on paper, and how they consider using groups — as we listened and observed students!
The next investigation is all about thinking about addition as ‘combining’ sets, so it was encouraging to see a lot of thinking that we can leverage. In particular, we could use Omar’s work as the focus of a conversation about determining ‘how many’ using addition.
This first grade teacher fosters a warm and inclusive culture within the classroom. It has always been clear to me that she values student thinking, and with this simple action — taking the time to listen to students talk about their 20 circles — she made this clear to students, too.
I posted a few of these work samples on twitter. I was thrilled to get a quick response from the Investigations twitter account — which I think is Megan Murray, but maybe don’t quote me on that — affirming that this small task does, in fact, show some beautiful things. Kids expect to solve problems in ways that make sense to them!
That’s another statement that seems so simple, but really… it’s foundational.