I wrote the other day about the five practices in an article titled “Orchestrating Discussions“. Also, I made the bold claim that nothing else has influenced my teaching in the last two years as much as this article.

The following are some consequences of the article…

## 1.

I refuse to teach in this room.

I cannot get to the students along the wall, so I cannot **monitor** them. I have absolutely no idea what their strategies are because I cannot be physically close to them.

## 2.

I work to be explicit with myself about the strategies I **anticipate** students will use. For example, I asked my College Algebra students to do this computation mentally on the first day of class last week: 45×12.

I had three strategies in mind. I list them below in order of my expected frequency (most common to least):

- 45×10+45×2
- 45x2x6
- Visualize the standard algorithm, including carrying.

Turns out far more than half of my students did 3. A bunch did 1, few did 2. One student visualized the partial products algorithm. And one student added 45+45=90, then 90+90=180, etc. She kept track of the number of 45s on her fingers until she got to 12 of them.

The fact that I committed to an order greatly impacted how I was able to interpret what my students did.

## 3.

I force myself to **sequence** the solutions and strategies we discuss. Here’s my note card from class today. The task was, *How many tens are in 268? What possible answers can you imagine to this question?* I made physical notes of who had which answers, mental notes of what was behind these answers (**monitoring**) and sequenced them. I called on these students in this order and helped the class to make **connections** among the strategies.

All in all, a much better lesson than I would have taught without explicitly thinking about these five practices.

## 4.

I make these practices explicit in my professional development work. I show how I use Post-Its (or scrawled note cards) to monitor, select and sequence. I give teachers time to anticipate student strategies, and then to add to their lists after we have worked a problem together.

I am a better teacher when these practices infuse my classes.

Thank you for posting this. These last two posts have been incredibly helpful. The book has gone on my first to buy list once the three month drought of paychecks ends.

In the article-version of the book, Smith et al offer an anecdote from “Mr. Crane’s” class. I love this device. They set up what seems to be a prescription for good teaching only to twist it around into a proscription against lousy teaching. This is exactly what’s needed for teachers like me who think they’re already great. Here’s the turn:

“The upshot of the [end-of-activity] discussion appeared to be ‘the more ways of solving the problem the better,’ but, in fact, Mr. Crane only held each student accountable for knowing one way to solve the problem.”

But I don’t think I understand, either from their article, Smith’s talk in Florida this summer, or your summary here, how you hold each student accountable for knowing every way to solve the problem. Can you help me out with that?

From Dan:

“But I don’t think I understand, either from their article, Smith’s talk in Florida this summer, or your summary here, how you hold each student accountable for knowing every way to solve the problem. Can you help me out with that?”

I agree with Chris that accountability for making connections and tracing the logic of another student strategy is more the focus than being accountable for being able to use all of the strategies. Using re-engagement lessons has been a way that I have accomplished this in the past. The idea is that students will take a look at completed strategies from work they have previously engaged in with a group. The group analyzes the work and unpacks the logic so that each student can explain and justify the strategy.

I typically followed up such a session by having student discuss and debate the strategies for efficiency or breadth of application. Often I would challenge groups or individuals to apply one of the unfamiliar strategies to a novel problem.

Re-engagement lessons are a great way to respond to ‘worked problem’ camp that utilized that practice in a way that still quotes student work and remains student-centered.

Danwrites:Good question. It’s challenging stuff.

First, I don’t think the standard of holding

each student accountable for knowing every way to solve the problemis a realistic one. Holding students accountable to make connections among a variety of strategies is achievable, though.A quick example and I’ll try to find some ways to flesh out others in more detail soon…

In College Algebra the other night, I worked with my students’ work on the Prius problem. They were all over the place in their thinking. Some wrote equations with miles as the dependent variable, others with dollars as the dependent variable. Some made graphs, some made tables. A few were still stuck doing arithmetic computations.

They had spent a bit of time working outside of class, so I opened class by having them share their work so far with a neighbor and writing down their progress so far on a blank sheet of white paper. I collected the white papers, took a deep breath, turned on the document camera and proceeded to do what I could to help them to connect and understand the various ways of looking at the problem.

For us, the fundamental question turned out to be interpreting meaning in the various representations.

What do these numbers mean? What is the independent variable? The dependent variable? How is this thinking like yours or different from yours?These kinds of questions hold students accountable to make connections in the sense that if I ask the questions and give the class time to discuss with partners, they are responsible for doing that. And they are held accountable in the sense that I may call on them to share what they are thinking about the work we are looking at.

That’s a different sort of dynamic from your Act 3,

Dan. You’re thinking about Act 3 as the place wherethe answergets revealed. And this is important. But I think Peggy Smith and her colleagues are arguing that rich mathematical discussions from which students learn mathematics require careful, public analysis of various ways of thinking.How say you?

Christopher:

“That’s a different sort of dynamic from your Act 3, Dan. You’re thinking about Act 3 as the place where the answer gets revealed. And this is important. But I think Peggy Smith and her colleagues are arguing that rich mathematical discussions from which students learn mathematics require careful, public analysis of various ways of thinking.”

All of my problems feature dramatic catharsis in act three. That catharsis isn’t always the answer’s revelation, though. In many cases there isn’t an answer to reveal, in which case the catharsis comes from seeing several students demonstrate work that all leads to the same (or similar) answers. With some groups (teachers, namely) that’s all the catharsis they need. They don’t care about the reveal.

I’m an ace with the reveal. I’m much less competent facilitating productive class discussion.

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