## Negative Times A Negative

In my workshop this week in Monterey, CA, a math teacher named Paul came up and said, "I ask everybody the same question: what is a numeric illustration of the fact that a negative number multiplied by a negative number is a positive number?"

I put his question out to Twitter and more than one hundred responses came in over the next few hours. You can click through to my tweet and see many of them. I've pulled out a sample here:

I appreciated a lot of these illustrations (as did Paul, though he pointed out that many of them aren't numeric) but my heart belongs to Bryan Meyer's response:

See, there are these things called negative numbers. Our students understand that they're useful descriptions. They understand how to add and subtract them. (Perhaps with a metaphor like going into more or less debt.)

We know how to add and subtract positive numbers, sure, but we can also multiply and divide them. Is the same true of negative numbers? What would multiplying and dividing negative numbers look like? What are your theories?

In this post we have two very different organizing principles for a math class:

1. Students will commit to difficult math work if we can cite some job that uses that math or some moment where it occurs in the world outside the math classroom.
2. Students will commit to difficult math work if we can put our students in a position to experience what's curious and perplexing about it.

There's some overlap, sure, but not a lot. Over a year, those organizing principles create very different classrooms. Over a career, those organizing principles create very different teachers. Let's talk about those differences in the comments.

Always Related:

## [Makeover] Bedroom Area

That's from Connected Math.

What I Did

• Simplify the prompt. It's already pretty spare, but I'm going to get rid of the information for a minute.
• Add intuition. A choice between two items like this lends itself really well to a guess. But when the information is already included, students will start to calculate right away.
• Justify the constraints. One set of dimensions is given in feet and the other in meters. Why? Is that just a contrivance for the sake of a math problem?

That's everything. If I'm teaching this material tomorrow, I don't have time to whip up a video or a photo.

So I'll tell students, "Two students drew pictures of their rooms. Which is bigger or are they both the same?" (Good catch from Chris Lusto on Twitter: "'How much greater?' omits the possibility they have equal area. Why do that?")

I'll ask them to write down their guesses, then share with a neighbor. Then we'll take a quick poll.

A student may ask if the two drawings are at the same scale, which would make for a nice, quick discussion ("Good eye. That's really important to know. They are.) but it isn't an essential moment.

I'll say, "Okay. I'm going to give you the width and height of the rectangles and we'll find out who guessed correctly. But first I have some bad news. Rodney is from the United States and Emile is from France. Do you know why that's bad news?"

Here's what I expect to be pretty interesting as students work with the fact that there are 3.28 feet in a meter:

• I imagine most students will convert the meters to feet. But some may run the other way. Do they arrive at the same conclusion?
• I imagine most students will convert the linear dimensions and then multiply. But will other students multiply the linear dimensions and then convert the area? Will they arrive at the same conclusion?
• If they convert in their last step, will they multiply by the conversion factor twice as they should? (ie. 2.5m • 3.5m • 3.28 f/m • 3.28 f/m) or just once? If no one makes that error, I'm for sure going to throw it out there that "a student from another class got a different answer." Then they'll construct an argument for or against.

What You Did

Good ideas from the blogs:

• Andrew Shauver made Rodney and Emile brother and sister and brought in some realia with a floor plan.
• Caren Hickman makes over a different task, one about food, with the goal of using real data and giving students some choice over the constraints of the problem.

Call for Submissions

If you have a textbook task you'd like us to consider, you can feel free to e-mail it. Include the name of the textbook it came from. Or, if you have a blog, post your own makeover and send me a link. I'll feature it in my own weekly installment. I'm at dan@mrmeyer.com.

## The Unengageables

Halfway through my curriculum design workshops, I ask teachers to share their "secret skepticisms." These are the sort of objections to new ideas that often take the form, "That would never work in my class because …. " They share them anonymously in a Google Form before lunch.

The secret skepticisms came back in Phoenix two weeks ago and these four were easy to group together:

This process assumes every student wants to learn or has the motivation to learn.

How do I get students to buy-in when they struggle with any problem solving skills at all?

What if my kids don’t know enough math to be engaged?

This approach is very compelling but this lesson will have additional challenges with students who could care less about getting involved. It is difficult getting any engagement by students who have little interest.

These responses were troubling. They seemed to emerge simultaneously from a deficit model of student thinking (ie. students lack engagement in the things we think they should be engaged in) and a fixed model of student intelligence (ie. these students are unengageable and that's just the way it is).

Neither idea is true, of course.

What is true is that after years and years of being asked questions every day, students may find it odd to be asked to pose their own. After years and years of associating "math class" with a narrow range of skills like computation, memorization, solution, they may find it odd when you try to expand that range to include estimation, abstraction, argumentation, criticism, formulation, or modeling. After years and years of acclimating themselves to their math teacher's low expectations for their learning, they may find your high expectations odd.

They may even resist you. They signed their "didactic contract" years and years ago. They signed it. Their math teachers signed it. The agreement says that the teacher comes into class, tells them what they're going to learn, and shows them three examples of it. In return, the students take what their teacher showed them and reproduce it twenty times before leaving class. Then they go home with an assignment to reproduce it twenty more times.

Then here you come, Ms. I-Just-Got-Back-From-A-Workshop, and you want to change the agreement? Yeah, you'll hear from their attorney.

"But it's tough to start something this new in April," a participant said.

That's true. For similar reasons, it's tough to start something new in a student's ninth year of school. That doesn't mean we don't try. Thousands of teachers successfully change their practice mid-year and mid-career. Luckily, there are also steps we can take to acclimate our students gradually to new ways of learning math.

Here are three of them:

• Model curiosity. I asked some kind of miscellaneous question on every opener. The questions weren't mathematical. (eg. How much does an average American wedding cost? What's the highest recorded temperature in Alaska?) I pulled them from different published books of miscellaneous facts and figures. This cost me very little classroom time and bought me quite a lot. It benefited my classroom management but it also built general, all-purpose curiosity into our classroom routine. That helps enormously when it comes to mathematical modeling where we're telling students that we welcome their curiosity.
• Ask the question, "What questions do you have?" Show any image or video from the top ten of 101questions. At the longest, this will take you one minute. Then ask them to write down the first question that comes to their mind. Take another minute to poll the crowd for their responses. (I model one polling procedure in this video.) This will also help your students to become more inquisitive and it will demonstrate that you prize their inquisitiveness.
• Make estimation part of your daily routine. Modeling takes place on a cycle that runs from the very concrete to the very abstract and back again. Typically, we drop students halfway into the cycle with all kinds of abstract representations (formulas, line drawings, graphs) already given. Give your students more experience with concrete aspects of modeling like estimation by taking an image or video from Andrew Stadel's Estimation 180 project and showing it to your students at the end of class. Ask them to write down a guess. Poll their guesses. Find out who has the highest guess and the lowest guess. Then show the answer.

Your students will come to understand you prize curiosity in general and their curiosity in particular. They'll understand that mathematics comprises more than the intellectual hard tack and gruel they've been served for years. At that point, you can help walk them through activities involving estimation, abstraction, argumentation, criticism, formulation, modeling, and more, aware that each of your students can be engaged in challenging mathematics, that none of them is unengageable.

Related

## Makeover Monday: Introduction

Here is a "high-leverage teaching practice," according to Deborah Ball:

Teachers appraise and modify curriculum materials to determine their appropriateness for helping particular students work towards specific learning goals. This involves considering students' needs and assessing what questions and ideas particular materials will raise and the ways in which they are likely to challenge students. Teachers choose and modify materials accordingly, sometimes deciding to use parts of a text or activity and not others, for example, or to combine material from more than one source.

So every Monday this summer, I'll post a problem from a textbook and start a conversation about how we could modify it. The details of that makeover may take the form of a loose sketch or something more formal. In either case, I'm going to be explicit about the goal of the makeover.

Fawn Nguyen, who's been on an absolute tear lately, illustrated this process recently. She took this task:

And then she showed how she implemented it with her students. Her goal wasn't something formless along the lines of, "Well this sucks and I want to make it more engaging." In the title of her post, she says explicitly she wanted students to have some personal, creative input on the constraints of the problem. So she had her students start by drawing their own golf course. She set a high bar for the rest of us.

You should play along. You can feel free to e-mail me a textbook task you'd like us to consider. Include the name of the textbook it came from. Or, if you have a blog, post your own makeover and send me a link. I'll feature it in my own weekly installment. I'm at dan@mrmeyer.com.

## The Do You Know Blue Student Prizewinner

Rebecca Christainsen had the highest score of any student on our Do You Know Blue machine learning activity. Yesterday was her last day of school at Terman Middle School in Palo Alto, CA, so Evan Weinberg, Dave Major, and I sent her math class a pizza party in her honor.

Because we're keeping the activity available for you and your students to use as they study inequalities, we aren't going to go into much depth on all the different rules contestants used. But I asked Rebecca how she came to her final, game-winning rule, and she told all:

My teacher first showed me the website, and I decided to try it out. My first attempt scored me only around 18%, but since hardly anyone had tried it out yet, I was ranked 33rd. After that, I was encouraged to try more equations, and suddenly thought of all the different types of equations that I could use, and moved to squared terms. One of the first equations that I came up with was b2>r2+g2. I simply used trial and error to come up with new equations, and I recorded each equation that I used and the percentage. I combined different equations together, and a few different combinations even had the same percentage.

Nobody beat that.

Extra Credit: How many of the Standards of Mathematical Practice does Rebecca evoke in that quote?

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