Restless Teachers

My favorite educators are restless.

They have that itch. Nothing is ever completely satisfying. If you chat with them after a lesson, they’ll tell you three things they would change.

This can translate into a beautiful urgency with students. Restless educators are curious about student thinking, and then just as curious about how to nudge students deeper into the mathematics.

They Got It!

“Everyone got it!” I heard the teacher exclaim, as we surveyed the mess of student work on the table. “That’s great!”

“It is!” I affirmed. We’ve been working, across the district, to increase access to grade level content for all learners. Everyone arrived at the correct answer? Yes! That’s something to celebrate! That’s…

It can’t have been more than a second or two before the other teacher started to rake the papers into a pile, armed with a ready paperclip. Neat. Tidy. Done.

I could feel my restless eye twitching. Yes, that’s great that everyone had the correct answer, but that wasn’t the end. The show wasn’t over. It wasn’t paperclip time.

“Can we take a look at the student work for another minute?” I asked.

“Oh, sure,” the teacher replied. She swept her hand over the first few papers. “Look at this. Students had so many different ways to solve the problem. So many strategies! Beautiful, right?”

I nodded. She knows my Achilles’ heel: diverse collections of student work samples. So lovely…

I scanned the work again. Some students had solved the problem using multiplication, and others had used addition, and still others had drawn dots or tally marks. They were all correct! And that’s great… as a starting point. (Restless twitch.)

I celebrate that every student had an entry point, and found success. But… drawing dozens of dots to solve a problem that amounts to 8×6? I would have expected a 9 year old to have a more efficient strategy. There must be a way to honor the beauty that we see while also figuring out how to nudge the student up the mathematical progression. I also wonder about the teacher. Are all strategies equally good? (Why does that question sit strangely in my stomach?)

Teachers Notes

My district is fortunate to use Investigations as our core curriculum, K-5. In the back of each teacher’s guide are gorgeously crafted Teacher’s Notes — diving into algebraic connections throughout the unit, sharing ways in which students might engage with the ideas during classroom discourse, and detailing what student work might look like. These notes read like a conversation with the authors, and, without exaggeration, I would never turn down a conversation with the TERC authors. The credits page is a veritable who’s who of elementary math educators.

Unfortunately, teachers are tired. My colleagues and I walk on average 1.7 inches shorter than we did before the pandemic hit. We still love students and love teaching, but the air in the hallways is thicker.

These beautiful Teachers Notes are somehow incompatible with how most teachers are doing their jobs right now. But I don’t think we need to give up on them…

Bringing the Teachers Notes to Life

So how do we help teachers engage with these PD opportunities that are baked into the curriculum? Over the summer, I worked on a PD project with another math specialist from my district (Tara Washburn, my unofficial mentor). We designed slide decks for individual units that outlined the story of the unit, listed the mathematical benchmarks, and dug into an assessment. The goal is to orient teachers to the mathematics of the unit while remaining firmly anchored on student thinking.

Here’s one slide from the section about the story of a unit. The colored arrows indicate that this is a main math idea that continues across multiple investigations within the unit.

Overview of Grade 3, Unit 1

After discussing the story and the mathematical benchmarks, Tara and I decided teachers should do some math together.

Investigations Grade 3, Unit 1, Session 4.6

And here’s the framework we developed for looking at the student work.

What does this student know? What are our next steps? What might we say to or do with this student? How might this change our plans for future lessons?

We ended up changing the first question — what does this student know? Perhaps it’s all semantics, but I think it’s hard to determine precisely what a student knows perdurably. Assessment is so contextualized. It’s in the moment.

Instead, we showed student work, and asked teachers to compare it to the benchmark.

What do you notice about this student work?
What about the work samples for ‘meeting the benchmarks’ indicates that these students are, in fact, meeting the benchmark? 

For work samples that are ‘partially’ or ‘not’ meeting the benchmarks:
What about the work samples indicate that these students are partially meeting the benchmarks?
What would you say to or do with these students next?

The Insect Problem

So let’s do the math.

Insects have 6 legs. How many legs are on 3 insects?
How many legs are on 6 insects?

How do you anticipate students would solve this?

This problem comes from the end of unit 1 in grade 3, which is the first unit about multiplication and division in the K-5 continuum. Students are beginning to think about equal groups. They are building off their knowledge of addition and how many in all to develop a multiplicative lens.

The first thing I noticed in the problem is the relationship between the numbers. After figuring out how many legs are on 3 insects, we can double the number of legs to determine the number on 6 insects. I wondered it students would make that connection, and how they might double the number.

What Does It Mean to Meet the Benchmark?

The Investigations Teachers Guide helpfully maps out mathematical benchmarks for students. The Insect Problem aligns with three of the benchmarks for this particular unit.

  • Benchmark 1: Demonstrate an understanding of multiplication and division as involving equal groups. 
  • Benchmark 2: Solve multiplication and related division problems by using skip counting or known multiplication facts.
  • Benchmark 3: Interpret and use multiplication and division notation.

With those in mind, Tara and I examined the work samples that are “meeting the benchmark.”

Tara and I included facilitators notes, including questions to ask, in our slide decks. We want to make these usable for other math specialists and teachers across our district.

First, here’s what we noticed:

  • All of these students included a multiplication equation.
  • Some students used only multiplication (e.g. Adam).
  • Some students recognized that doubles relationship between the total number of legs for 3 insects and the total number of legs for 6 insects. This relationship was sometimes expressed as addition (Kim’s 18 + 18 = 36) and sometimes expressed as multiplication (Gina’s 18×2).
  • Some (but not all) students identified the label for the unit: legs.

We then compared what we noticed to the benchmarks. We could see that every student showed some understanding of equal groups, by preserving the 6 legs per insect either in skip counting or multiplication and addition. When students did use addition, it was in combination with a multiplicative strategy, as in Gina’s work to discover that half of the legs on the insects (the more familiar 3×3) yields 9 legs, and then she doubled that using addition (9+9) to arrive at 18 total legs on 3 insects. Every student used multiplication notation.

What Does It Mean to Partially Meet the Benchmark?

Next, Tara and I examined the two student work samples that were identified as partially meeting these benchmarks.

Here’s what we noticed:

  • Benjamin used repeated addition to solve both problems.
  • Benjamin formed additive groups (6 + 6 = 18, 12 + 12 = 24) and showed these on his work.
  • Kelley used multiplication notation, although it is not true that 6 x 2 = 18.
  • Kelley didn’t solve the entire problem.

When we compared this to the three benchmarks, we realized something critical: Benjamin correctly solved the problem, but he did not use any multiplicative structures, and that means he’s only partially meeting the benchmark.

Is That Bad?

My first grader came home from school the other day, and announced, “isn’t it wonderful that I always get 100% on my Friday spelling tests?” This was clearly her cue for me to break out the trumpets and confetti — as if I have any energy left at the end of the day to clean up confetti! I smiled at her, and thought for a moment before responding: “I love to see how proud you are of your work!” That’s circuitous Mom Double Talk. I think my daughter and I both know that she knew how to spell those words before they were assigned to her. So, while I’m happy that she’s being careful with her work and enjoying her success, I think I will hold off on the parade for now.

And when a student does not make the grade? Is that bad? At least implicitly?

Emphatically not. In the case of Benjamin’s work, there is so much to build on here. He has entered third grade with a great understanding of additive properties, and he interpreted the “3 insects, each with 6 legs” scenario to mean equal groups, and then he added those groups.

It’s all about how we build.

What Comes Next?

The next step is to nudge Benjamin towards multiplication. How can we help him connect these different ways of thinking about the problem — as addition and as multiplication? How can we help him look for ways to combine groups faster, so that he can scale quantities.

Two work samples that did not “meet expectations” for the end of unit benchmark. These students drew the insects, did not use as many numbers, and did not include any equations that match the context.

Even the student work samples that our Investigations teaching guide identified as not meeting expectations were not bad! They give us a different starting point. I’d love to see how Ines (below) determined the total number of legs for three insects. Did she count by 1s? Did she skip count? How did the picture support her thinking?

With more understanding of what Ines was thinking, we can determine what we might do with or say to her next. Sometimes, there are small nudges we can make. Sometimes, we add in entire lessons and learning experiences. This is the art of teaching.

Are all strategies equally good? All strategies give us a starting point, and that’s good.

In the Classroom

Earlier in September, I observed third graders in another classroom writing their own equal groups situations.

Student: Here are 6 paks of markers. Each pack has 11 markers. There are 66 in all.

It’s beautiful! She created her own equal groups situation. I love how she decomposed the packs of 11 markers into 10s and 1s, and then worked diligently to combine them.

“I wonder if there’s another way you could have figured out how to add up all of those numbers.”

Silence.

“It looks like you added the tens first,” I mused. “And then the 1s. Is there another way you could have added them up even faster?”

“Six ones… that’s just six!” she replied.

“Yes! Because 6 x 1 is 6.” I affirmed.

Her eyes lit up. I’d made a connection that made her feel fancy. Multiplication is, after all, new at the beginning of third grade, and for many students this feels like Big Time Adult Stuff. They’ve made it.

“Let’s look for more opportunities to use our brand new multiplication skills,” I suggested, cheerily. “I bet you’ll find a lot of them.”

When we know the direction we want to go, we can nudge students there.

Removing the Paper Clip

And so, though tired and weary, I remain restless.

I’ve seen the Paperclip Teacher — who is, by all accounts, a wonderful teacher! — increasingly restless, too. It helps to know what we’re looking for. It helps to realize that right answers are not always the end of the journey.

Restless teachers are also prone to overwork. (Hahahaha ask me how I know.) Particularly in our burnout culture, we want to preserve everyone’s humanity. Even restless teachers are not indefatigable. Sometimes, it’s okay to realize there’s more work to do, and to say no.

But isn’t it also delightful when we feel that little spark, that curiosity, that restless twitch, that drives us to continue with the work.

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