Coherence, Round 2: The Power of Connecting Instructional Routines

For a million reasons, I love, love instructional routines*. They engage students, active thinking and discourse, and, with the structures in place, students are given the freedom to focus on the MATH instead of having to remember new directions.  (The website linked above, TEDD.org, offers some amazing resources and ideas about instructional routines like number talks/strings, choral counting, and contemplate then calculate.)

In one of my recent posts, I focused on the importance of coherence in telling a story, and using this story to construct understanding. The architecture of that story may draw out different ideas, or emphasize different connections. It’s important for this coherence to be deliberate over the course of the year, a unit, and a lesson. (This design principle is fundamental to the authoring of the new Illustrative Mathematics middle school curriculum.)

I am using this Illustrative Mathematics middle school curriculum with a group of 7th grade students this year. We have only met for a few days, so today marked our third lesson in the first unit of the year: Scale Drawings. Day 1 had us searching for precise language to describe scale copies. Day 2 had us applying some new terms (corresponding parts, scale factor) to describe some important attributes of scale copies. Now, I was all set to launch into day 3: exploring the multiplicative nature of scale.

The launch for lesson 1.3 in 7th grade is a variation on a number talk, called “more or less.” Instead of evaluating the expression given, students are asked to determine whether the product will be more or less than a given value, and to justify their thinking. Each problem was revealed one at a time.

from Illustrative Mathematics, grade 7, unit 1, lesson 3 (“Making Scale Copies”)

The students offered some nice thinking about partial products for the first problem. A few had to puzzle things out, and revise their thinking as they spoke. “Well, I know that 8 groups of 25 is 200, and then half a group of 25 is 12.5, so that’s more than 205,” Caleb explained.

We spent more time zeroing in on that second problem.

The students offered a few different explanations, mostly centered on the second factor.

“So I started with 9.93. I knew that if I multiplied by 1, it would stay the same,” Lily began. “But we aren’t multiplying it by 1. We’re multiplying it by less than that, so the number has to get smaller.” Isabelle and Matthew emphatically signaled “me, too” with extended thumbs and pinkies.

“Please repeat that last part,” I prompted Lily.

“What, about it getting smaller?”

“Yes, but the whole idea.”

“We’re multiplying by less than that, so the number has to get smaller?”

“Less than what?” I pushed for her to clarify.

“We’re multiplying by less than 1, so the number has to get smaller?” Elijah started to twirl his pencil absentmindedly. Sergei looked towards the window. This seemed to be a very comfortable and familiar idea to the group.

“Hmm. So yesterday we were talking about scale copies, and, specifically how to determine whether they increase or decrease in size. I wonder how that connects to our number talk, and what Lily just said…”

There was a pause, then Caleb lit up. “Oh! We can tell a scale copy will be smaller if the scale factor is less than 1, just like with the number talk.” I saw students nodding in agreement.

This then launched into a slight mathematical tangent, with students musing about whether a scale factor could be negative. We had to table the discussion in order to advance the lesson’s key focus — how scaling is a multiplicative process — but the connection had been illuminated. Elijah and Sergei, and all the others, looked reinvigorated with purpose.

I’m pretty certain this brilliant warm-up comes from Kristin Gray, aka @mathminds. I know Kristin would not want me to call her a genius, but, seriously, how elegant is it that a single warm-up allowed me to offer some beginning of year work around

• reviewing decimal operations
• estimating to determine reasonableness
• justifying thinking/constructing and critiquing arguments,
  AND THEN
• enhanced our understanding of the investigation we were about to launch?!

It was quite the 10 minutes! I’ll be honest: I feel really good at my job when I’m able to capitalize on these moments. It gives me a temporary ego boost. (Naturally, this is often tempered by something else falling flat a minute later. Sigh. Teaching.)

This kind of work takes careful, deliberate, and measured planning. It takes having your mind centered both on the big ideas at the heart of your lesson, while also being trained on important concepts for the grade level at large. It takes someone with a coherent understanding of mathematics as a landscape and a continuum, and not just an isolated part like a lesson.

I tried to mirror this in my own planning of a whole class demo lesson in third grade, to be delivered minutes after this 7th grade lesson. I met with the third grade teacher last week, alongside another math specialist and our ECS (“enrichment and challenge support”) specialist, to discuss differentiation in her classroom. I then offered to model a high leverage instructional routine — I chose “choral counting” — and then an open middle challenge. (She’s a first year teacher, technically, and I want her to build up some sustainable ways to differentiate for the wide range of readiness levels in her third grade class.)

Knowing that the third grade teacher was about to launch into a unit on addition and subtraction, I designed an open middle-ish task to explore the concept of place value.

I knew I wanted to launch this lesson with a choral count that would allow us to talk about some regrouping, so I chose skip counting by 20 from 164.

The students spent some time discussing what they noticed from the column of “84s” to the column of “04s,” but they were more interested in some other patterns. So it goes. It meant that there were some nice connections to be made while they puzzled through the “sum to 500” challenge, but it didn’t feel as wonderfully and magically coherent as the 7th grade lesson. It’s not easy.

Coherence within a lesson relies on me, as the planner, knowing what mathematical ideas and concepts are baked into the very heart of the lesson. I can then design — or implement — tasks that spotlight these ideas from a variety of perspectives, and help students construct their own understanding.

In looking at the 7th grade students’ homework from the previous night, many of them had listed the scale factor of a shrinking copy as “÷3” instead of “•1/3.” In the exit ticket, called a “cool down” in IM, none of them made this mistake again. All of them were thinking about how a scale factor between 0 and 1 will decrease the size of the copy. Maybe they would have arrived at this solid understanding without that connection to the “more or less” routine, but I know I saw metaphoric lightbulbs going off. I have to believe it helped facilitate this.