Who Is Doing the Math?

When I spend time in a school visiting math classrooms, I'm inevitably asked questions like: "How are we doing?" "If we stay the course, will our students show growth this year?" In other words, many people want to know leading, classroom-level indicators of student learning in mathematics. Concretely, what do classrooms look like where students are learning math? As we gear up for another new school year, I've been thinking about these questions again.

I think where I've landed is in the power of a single question that may sound kind of silly: are all students doing math every day? I've come to this admittedly obvious-sounding conclusion for three reasons. First, it's easy to get caught up focusing on teacher moves (and I think this was especially true during remote instruction, when students weren't as visible), and then naturally paying less attention to how students respond. For example, when coaching it's natural to focus a lot on developing good questions and deploying them at the right time; when observing instruction the coach may, with very good intentions, focus on the content of the teacher's questions at the expense of what students are doing. This can then lead to false positives; the teacher is doing "the right moves" that the coach has suggested but we haven't paid enough attention to how students are responding. So much of coaching and supervision is focused on the work of adults.

Second, it's important to look critically at what constitutes "doing math" in the first place. In reading class, I think it's relatively clear when students are reading. (Though I wouldn't be surprised if there's plenty of nuance I'm not aware of.) But in math there is often less consensus, both because we can get distracted by "shiny objects" ("Look, they're using manipulatives!") and because we have culturally-imposed beliefs about what it means to do math ("That's not how I was taught!"). And of course, "school mathematics" is mostly not how mathematicians conceptualize or do mathematics (see for example Wu's conception of "TSM" or "textbook school mathematics"). It's also often (especially at the secondary level) not the mathematics that the vast majority of adults do in their daily lives.

And third because of the key qualifiers "all" and "every." Even if we can focus on what students are doing, and even if we can reach consensus on what it means to do math, are all students engaged? Or just some? By the end of class, what do they produce? And critically, do we really believe that all students can do math? So our singular focus becomes noticing who is doing math, and how:

• Who is thinking about math?

• Who is talking about math?

• Who is doing written math work, or writing about math?

Is it mostly the teacher doing these things? Or the students? Some students? Or most, or all? Let's look, for example, at teacher questioning. A lot can go into choosing or crafting the right question, but I've found that how the question is posed matters just as much and can influence who is doing math; in particular, who is thinking about math. Here are some characteristics of questioning I've noticed:

Characteristics of questioning that does encourage student thinking	Characteristics of questioning that does not encourage student thinking
Questions are posed to the entire class, without naming a specific student to respond.	Questions are posed with a specific student named to respond.
Once a question is posed, students are given plenty of time to think independently about the answers to questions.	Questions are posed without time for students to think independently.
After wait time, students are selected by the teacher to respond.	After wait time, students volunteer to respond.
After a student responds, the teacher does not immediately reveal whether the response was correct or not.	After a student responds, the teacher immediately reveals whether the response was correct or not.

Consistently using questions as described on the left develops a culture of thinking in the classroom; when students consistently experience questioning like on the right, they learn they do not have to think in class. Here's how that righthand flavor can typically look:

Teacher: "Marc, is 4 the solution to 20 = 5x?"

Marc: "Yes."

Teacher: "That's right."

This is sometimes known as "IRE" questioning, which stands for "Initiate, Respond, Evaluate." The conditions are set to ensure that only one student needs to think about this question. Here's a different vignette that shows questioning of the lefthand flavor.

Teacher: "Is 4 the solution to 20 = 5x? Take a moment to think about that question."

...

Teacher: "OK, is 4 the solution to 20 = 5x...Tiffany?"

Tiffany: "Yes."

Teacher: "OK, why do you think 'yes'?"

Tiffany: "Umm...because...20 divided by 4 is 5?"

Teacher: "Gerald, what do you think? Is 4 the solution to 20 = 5x?"

Gerald: "No."

Teacher: "OK, why not?"

Gerald: "Because if you substitute 4 for x you get 54."

Teacher: "OK interesting. Tiffany says yes, Gerald says no, turn and talk to your neighbor and decide for yourselves. Is 4 the solution to 20 = 5x?"

An expert teacher could easily continue in this manner for some time, further building uncertainty and therefore discussion, about this one question. I think when students regularly experience this kind of facilitation, they learn to think about each question because: (a) they know the teacher may call on them, even if they call on someone else first; (b) time is allocated for them to think; (c) the teacher doesn't give the answer away immediately, so they have to figure it out for themselves. Seemingly small moves like these can make a big difference in how much math students end up doing; it's also important to start the school year by exposing students to these routines, so they come to expect them.