Category: ontheroad

Total 9 Posts

Pick A Point

Here’s my favorite moment from a workshop in Spokane last week:

It’s about the quickest and most concise illustration I can offer of Guershon Harel’s necessity principle. The moment of need is brief, but really hard to miss. It sounds a lot like laughter.

2014 Feb 19. Christine Lenghaus adapts the interaction for naming angles:

I drew a large triangle and then lots of various sized ones inside it and asked the students to pick an acute angle. I asked a student to describe the one they were thinking about and then another student to come up and mark it! This lead to discussion on how best to label so that we both agree on which angle we were talking about. Gold!

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Joel Patterson:

There’s an easy way to do this in Geogebra.
Open up blank Geogebra file, viewing only the Geometry window (no Algebra window).

Click the point tool and make a bunch of points like in Dan’s video.

Then there is a small button AA with one A in black and the other in grey. This button shows and hides labels for all points.

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Hans Freudenthal:

Youngsters need not repeat the history of mankind but they should not be expected either to start at the very point where the preceding generation stopped.

2017 Oct 16. Here are the slides.

Negative Times A Negative

Check you out, Twitter!

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:

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I am not sure why these need to be either/or or why they rise to organizing principles.

Just based on the history of mathematics, some parts of it are very practical and driven by the real world. Other parts are more abstract and were discovered and elaborated long before anybody found a practical use for them or a connection to the rest of mathematics.

Rebecca Phillips:

The “because you need this” and “because it’s possible” are going to tap different kids, and shouldn’t I be trying to inspire them all?

Here is my response.

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Scott Farrand:

Many of the examples you shared did not apparently address multiplication. For example, some of them gave an explanation for why the negative of a negative number is positive, which is not quite the same thing as explaining why a negative number times a negative number is positive.


On the present question, about Dan’s two alternatives for organizing a math class, I prefer the second. Creating perplexity and curiosity in students requires that they have some comfort and understanding that leads to a little intuition or projection that can appear to be contradicted by something, hence the surprise. If a student simply gets used to applying a formula that they don’t understand, then it is difficult to surprise them about a result related to that formula.

2014 Mar 10. James Key contributes a valuable entry to our project.

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.


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Kate Nowak:

Corny as it sounds, don’t give up. The first and second and tenth attempt at -whatever it is that’s a very different approach in your class – a 3Act, a project, a whatever it is — is probably going to either fall flat or fail spectacularly. The kids might get mad and weirdly uncooperative. Things might happen that you didn’t anticipate and don’t have the skills to handle. You aren’t going to get good at planning them until you get some experience planning them. You’re going to suck at this for a while. [..] You need to keep stretching the rubber band over and over until it loosens up and doesn’t snap back all the way.

Great Moments In Mathematical Invention

I was in Australia this last week, working with some teachers at MYSA on You Pour, I Choose. It’s a task that asks which of two glasses has more soda and involves, among other skills, a fairly straightforward application of volume.


A teacher in the workshop called me over. “I’m not a math teacher,” she told me, and then pointed to the person next to her who had calculated the formula for volume of a cylinder.


“But that seems like more work than you need to do,” she said. “We don’t care about the exact amount. We care which one has more. With both glasses, we multiply by pi and square the radius. So all you really need to do is multiply the radius by the height for both glasses and compare the result. That’ll tell you which one has more.”


This was a rather stunning suggestion, made all the more impressive by the fact that this woman doesn’t immerse herself in numbers and variables for a living like the rest of us.

I have two questions:

  • Is she right? She’s certainly right in this case. Both the volume formula and her shortcut indicate the left glass has more soda.
  • As her teacher, what do you do next?

I’ll update this post tomorrow.

2013 May 29. I knew just telling her, “That’s wrong.” would be unsatisfying because, explicitly, she said, “Am I wrong?” but, implicitly, she was saying, “If I’m wrong, then make me believe it.” I knew the current problem wasn’t helping me out because her shortcut actually worked.

I knew this woman had dropped me off deep in the woods of “constructing and critiquing arguments” but I didn’t know yet what I was going to do about it.

“Whoa,” I said. “Does that work? If that works that’s going to save us a lot of time going forward. Let me bring your idea to the group and see what everybody thinks.”

In the meantime I stewed over a counterexample. It took me more than a minute to think of one because a) I was kind of adrenalized by the whole exchange, and b) I don’t do this on a daily basis anymore so my counterexample-finding muscle has become doughy and underused.

I posed her shortcut to the group and said, “What do you think? Does this work?” I gave them time to think and debate about it. Someone came back and said, “No, it doesn’t work. Imagine two cylinders with different heights and a radius of one.”

Awesome, right? This particular counterexample doesn’t disprove the rule. The square of one is also one so her rule works here also.

Eventually someone suggested two examples where the product of the radius and height were the same but where the radius and the height were different in each cylinder. The shortcut says they should have the volume. The formula for volume says they’re different.

Final note: students are often asked to prove conjectures that are either a) totally obvious (“the sum of two even numbers is even” in high school) or b) totally abstract (“prove the slopes of two perpendicular lines are negative reciprocals” in middle school). It’s rare to find a conjecture that is both easily understood by the class and not obviously correct or incorrect. I’m filing this one away.

Two PD Opportunities

One, I’ll be chatting with three-act mavens Chris Robinson and Andrew Stadel in the Global Math Department Wednesday 10/10 8:00PM Central Time. Here’s the agenda.

Two, I’ll be offering two sessions in San Francisco on Monday 10/14 for Integrate|Ed along with a pile of other really great educators. Details here. Tickets are ordinarily $275 but if you type “vendor sponsorship” in the coupon code blank you get it for $75, which is kind of an insane discount.

2012 Oct 11. Here’s the recorded version of the Global Math Department discussion.