If you had to pick just one Mathematical Practice to focus on in your classroom, I would strongly suggest picking Practice 3, “Construct viable arguments and critique the reasoning of others.” Yes, even more than Practice 1, “Make Sense of Problems and Persevere in Solving Them.” Why? Because Practice 3 is the most fun. Making sense and persevering are hard work. So is constructing viable arguments. But critiquing the reasoning of others? That’s just plain fun!

In all seriousness, productively critiquing the reasoning of others truly is hard work. It’s even harder to do in a way that is accountable and doesn’t leave others feeling put down or attacked. But having productive arguments in math class can be a very engaging activity, and one that helps students get into other practices, such as making sense of problems (so they can successfully convince others of their ideas), using appropriate tools strategically (as they defend their decision to solve a problem in a certain way), and modeling with mathematics (as they argue over the best way to simplify a messy quantitative situation).

Does the idea of starting with Practice 3 feel scary to you? It does to me. How would I open up my classroom to debate and argument from the get-go? What if my students are very novice at sense-making and persevering? What if they don’t know how to critique without being mean? What if someone says something mathematically incorrect and no one knows and they learn something wrong? These are all valid concerns! Maybe some of these stories will help us think about them …

**Peanut Butter Jelly Time!**

It was the very first week of school in a 9th grade class of students who were specifically identified as needing a double-dose of math every day. Their teacher was working on establishing classroom norms, especially around communication. She decided to use the classic activity, “Making a Peanut Butter and Jelly Sandwich,” in which students write directions for making a PB&J sandwich and then an alien tries to follow those directions. Hilarity and messiness ensue!

I was watching and videotaping the lesson and noticed something really profound happening. The students started out with completely reasonable steps for another normal person to follow: open the jars, spread equal amounts of peanut butter and jelly on the bread, put them together, eat. If you’ve ever seen a PB&J before, and if you’ve ever used a knife before, you could totally follow those directions.

But then the teacher started to follow the directions under the persona of Bob the Alien. She did crazy things like smush the full (open) jars of peanut butter and jelly directly onto the unwrapped loaf of bread. She used her hands to smear peanut butter all over the unwrapped loaf. She put enormous globs of peanut butter and jelly on the bread and then put the slices together, bread side in and gooey side out. The students started calling out

“Bob doesn’t have any common sense!”

“Bob is crazy!”

“We’re going to have to be so specific!”

The teacher agreed, “Bob is an alien after all. He doesn’t know anything about peanut butter and jelly sandwiches.” So the students got back to work. Orally, their directions got much more precise and specific as they thought about what to change. In writing, they didn’t get nearly as much on paper as they said out loud (pretty typical!), but even their written directions improved.

What I took away from that is that students really do have a sense of audience and purpose when they argue. If they give you an incomplete argument, they aren’t necessarily being lazy or sloppy. They’re giving the (minimal) amount of information they think you’ll need to understand their ideas. With feedback about how their audience really understands their thinking, even math-phobic 9th graders see the value in revising and re-explaining their ideas.

Having an audience, someone to test their ideas against and gauge the reaction, really helped the 9th graders rise to a level of specificity and clarity they hadn’t reached before. The “critique” of their arguments wasn’t someone saying, “that’s not clear; fix it.” It was someone acting out what they thought the students meant — albeit from a very silly perspective! Students can practice the art of constructing arguments, getting feedback, and revising by writing for an audience and then having their audience read their work and say, “Here’s what I think you’re saying…. Is that what you meant?” It’s simple, motivating, not too threatening, and helps students communicate more effectively!

**Lessons from the Little Ones**

And how do students learn to be helpful audiences? This year I get the joy of spending time in Kindergarten and 1st grade math lessons, as students participate in routines like Number Talks and independent Math Centers for the first time. It’s fascinating to watch what the little ones need to learn, and what they already know how to do… and I see the same needs and knowledge in their 5th – 8th grade schoolmates upstairs!

In a Kindergarten Number Talk, the students are practicing some really important habits by talking about them all the time! Before we do a Number Talk, we discuss what it looks like and sounds like when you listen to a friend. We talk about how we stay quiet, track them with our eyes, keep our feet still, and use hand signals to show when we have a “brain match” or a question. Then we do a Number Talk, which involves the teacher putting up a visual prompt and calling on several students to tell what they see or how they think about a certain question about the visual. The Kindergarten students aren’t really sure yet how to talk about what their brains do, and so their teacher really has to work to ask questions to help them reveal their explaining.

But it’s working! The students are learning to say things like,

“I saw 2 red beads and 3 blue so I counted 3, 4, 5 and I knew there were 5.”

“I counted 1, 2, 3, 4, 5.”

“I just saw 5 and I knew it was 5.”

“I know 2 + 3 is 5.”

The students are also learning to recognize if they thought the same thing as another student. And slowly, slowly they are learning to put their “That’s not right!” reflexes into a question. So when a student says they saw 4 beads when others saw 5, instead of shouting out, the students show they have a question. Their teacher has given them some question ideas like “Could you check that again?” or “Are you sure it’s 4?” and the students are learning to use those questions instead of calling out, laughing at a friend, or saying “NO!” when they hear something they don’t agree with.

**Some Ideas to Try**

If you want to explore Math Practice 3 with your students, here are some activities you might try:

- Take a look at the Teacher Packet from a recent or upcoming PoW. Copy some of the novice and apprentice student solutions from the packet and share them with your students. Ask your students, “What do you notice about this work?” and “What’s one question you could ask this student to help them revise their thinking?”
- Invite students to share their written work on a PoW with the class. Make copies of the work from your volunteers and give students 2 colors of post-it notes. Put classmates in small groups to read the work and write “I notice…” positive statements on one color of post-it note, and “I wonder…” curiosity statements on the other color. Collect up all the feedback and give it to your brave volunteers.
- Try some Math Talks with your student. PoW user and Powerful Ideas columnist Fawn Nguyen has a great blog including a description of Math Talks that is applicable to students of many ages.
- Start some arguments! There’s a list of mathematical argument starters that ranges from upper elementary to upper grades of high school just getting started here: http://matharguments180.blogspot.com/

Enjoy!

Max

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