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!

Featured Comment

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.

Featured Tweets

@ddmeyer @JustinAion @TracyZager awesome. Basic premise to lesson design: What happens if we *don't* have definition/concept/skill X?

— Casey Rutherford (@rutherfordcasey) August 29, 2014

@fawnpnguyen @JustinAion @ddmeyer @TracyZager Yes, this. Naming points wasn't boring for folks that came up with it, I expect. We made it so

— Casey Rutherford (@rutherfordcasey) August 29, 2014

Featured Academic

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.

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:

@ddmeyer I think of it as subtracting debt repeatedly, or reducing the money loss until you get a gain.

— Chris Adams (@MrAdamsProblems) June 17, 2013

@ddmeyer In a movie scene. If a car in the scene is moving backwards, playing the scene in reverse cause the car to go forward

— Chris Adams (@MrAdamsProblems) June 17, 2013

@ddmeyer Would this example work (See attached picture of an example)? Just a thought. pic.twitter.com/xZSLSkguaG

— Gary A. Petko (@GaryPetko) June 17, 2013

@ddmeyer If a negative charge is moved towards a negative electric potential, the electric potential energy increases positively by Vq

— Doug Smith (@bcphysics) June 17, 2013

@dcox21 @ddmeyer My student: "when you love love it's love, if you hate love it's hate but if you hate hate its love." #realworldexample

— Eric Benzel (@mrbenzel) June 17, 2013

@ddmeyer taking away a penalty in football moves the team forward.

— Jeffery Baugus (@baugusj) June 18, 2013

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:

@PaiMath @ddmeyer Somewhat. From what I know neg #s were invented for debt. Once they exist, we might ask what happens when we multiply?

— Bryan Meyer (@doingmath) June 17, 2013

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:

• Samuel Otten’s Cornered by the Real World.

Featured Pushback

Brian:

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.

Featured Comment

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.

Also:

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 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

• No blogger tore through a teacher’s conviction that teaching would be great if only we had different students harder than Kilian Betlach, whose nom de blog was “Teaching My Ass Off.” If you only came to the blogosphere in the last four or five years, after Betlach moved on from blogging to school administration, you should carve out some time this summer to dig into his entire catalog. Here is a sample I quoted on this blog, as well as his own highlight reel.
• If you work with teachers, you’ll appreciate Grant Wiggins’ recent post where he describes some useful interventions for situations like mine.

Featured Comment

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.

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, 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.

I was at urinal during a break in my Grand Forks session. “I’ll give you this,” the guy said next to me. “You walk your talk.”

Two things:

1. The etiquette on urinal interaction must be a little more relaxed in North Dakota than in California.
2. You get the subtext right? “I’m not buying any of this stuff, obviously, but you put on a good show.”

I knew exactly what he meant.

I’ve facilitated enough PD to not feel new at it. I’ve taken enough coursework in PD at Stanford to feel like I get some of the theory behind teaching adults about teaching children. Whenever I’m planning a session or a talk, though, I don’t lean on the theory or my experience half as hard as I do on the fear that I’ll be working with a teacher who’s exactly like me, and he’ll hate me. Which is to say, rather, that I’ll hate me.

My urinal buddy helped me understand that whenever I blog or facilitate PD or give a talk or drive in traffic or cook a meal or talk to my friends, subconsciously, I’m always wondering, “Would I hate me?” It’s a coin flip, really, whether that’s evidence of personal integrity or flagrant self-absorption.

Give yourself one photo or one minute of video to tell a mathematical story so perplexing that all of your students will want to know the ending, without you saying a word or lifting a finger.

I received e-mails all the way through the night and into the morning before the second day’s session. I loaded each entry into a slidedeck and reeled them off to the group over 25 minutes.

At the same time, I had the participants working in a Google Form. For each entry, they’d write up a) the name of its designer, b) the question it provoked, c) the perplexity of that question (ie. how bad they wanted to know the answer).

The yield on that investment of 25 minutes was incredible. We spent the next hour mining and interpreting data and drawing conclusions about effective curriculum design with digital media. It was some of the productive professional development I’ve ever been a part of.

I posed several rounds of questions for table discussion. The first round began after they had just submitted all of their questions:

1. Was the exercise fun? Was the exercise useful?
2. What will be the most common question for your entry?
3. How curious will people be about it?
4. Which ones were you most curious about?

Then I sent out a link to the Google Form results and posed the second round:

1. How effective was your entry at provoking a common question?
2. How could you have made your entry more perplexing?
3. Whose entry was the “best”?
4. What do we mean by “best,” anyway?

While they batted those questions around, I dug into the spreadsheet and found the median response for each entry’s perplexity rating. Six of them tied for the highest median rating:

• Cindy
• Dan
• Cathy
• Sandy
• Nancy
• Jessica

We reviewed those six, briefly, and then I posed a third round of questions:

1. What’s special about these entries?
2. If you had a favorite that didn’t make the cut, what isn’t captured by this measurement (the median perplexity rating)?
3. Dan and Nancy both feature leaky faucets filling up a container. In what ways are those two entries different?

Selected Answers To Those Questions

• The unanimous consensus was that the exercise was fun. Fun isn’t necessarily a prerequisite for an effective PD exercise, but man does it ever help.
• One participant said the #anyqs exercise was useful, mainly, for “training my eyes.” She elaborated that after just one pass at #anyqs the day before, she was already more alert to the applications of math in her life outside the math classroom.
• The issue of subjectivity has been one of the most fascinating conversations about #anyqs online, and so it was in Grand Forks also. Are the questions of math teachers about this image a useful proxy for the questions of students? Will a student from North Dakota have a different question about a video of a wheat thresher than a student from California? One participant noted that the high school and middle school teachers in the workshop asked questions that were linked closely to their content areas? (Strong data mining, right?) The sum of my thinking to date? Yes, the process is subjective. In spite of its subjectivity, field testing my curriculum with teachers has improved it immensely for students. Perplexity can transcend our demographic differences.
• I forgot to mention this in Grand Forks but the easiest, best way to make your video-based problems more perplexing is to use a tripod. Or to simply put your camera down on something sturdy. The reason being is that it’s so much harder to gauge so many different measurements (speed, for instance, or height) when the camera is wobbling back and forth and up and down, throwing your subject around in the frame.

General Remarks

• I have designed a lot of different constructs to explain to myself (and others) perplexing curriculum design. None has been as effective as mathematical storytelling. I’m particularly chagrined to think back on all the times I’ve broken a problem down into the four tasks of “verbalization, visualization, abstraction, and decomposition.” That construct resonates with my grad school peers, but it’s terrible vocabulary for teacher professional development. (ie. “Okay, so where do you find the decomposition of this task. How would you help the student abstract the problem space?” etc. Gross.) I’ve never heard table groups reference “decomposition” in one of my design activities. The language of storytelling, in contrast, was a constant feature of their conversations.
• One participant: “We need a website for sharing these.” Yes.
• Another participant: “Kids should bring in their own photos and videos.” Maybe.
• This is the dy/dan drinking game: every time I put my readers to work to make me smarter or more effective in my studies or at my job, drink. I was legally unsafe to drive after receiving hundreds of pages of student work for my Michael Benson experiment. I was black-out drunk after using the work of @salmathguy, @reimerpaul, @eduz8, @techsavvyed, @fnoschese, and @wpeacock202, for fodder in my #anyqs workshop. Y’all should be so lucky to have readers like y’all.
• There was a horrible moment in the early morning of the last day when I planned to hand each participant 28 strips of paper (one for each person’s #anyqs entry) on which she’d write her question and the name of the designer. Once we finished, I figured we’d trade the slips back to each other, a process that would probably take forty-five minutes on its own, right? So take a shot for Google Forms as long as you have the bottle out.

Your Homework

• How are the two leaky faucet videos [Dan, Nancy] different? Which one is better? Define “better.”
• One participant submitted this video of Carl Lewis’ 1984 Olympics long jump. He was perplexed by the parabolic motion of Lewis’ jump. Instead, nearly all of his colleagues (and yours truly) wondered how fast Lewis was running at lift-off. Given infinite resources, tools, the ability to travel anywhere in time and space, how would you capture Lewis’ long jump in a way that highlights the perplexity of his parabolic motion?

And Now A Word From Our Sponsor

If any of this seems interesting to you, let me recommend my Perplexity Session, which I’ll be hosting in Mountain View on 9/10/11.

Here’s John Scammell with a celebrity endorsement:

As someone who was fortunate enough to see one of the early incarnations of this workshop, I can tell you that it is incredibly valuable professional learning. Dan is a skilled facilitator, and more importantly a great teacher. I highly recommend it.

In addition to the material I facilitated on instructional design, the staff at Colchester High School wanted to work on their implementation of standards-based grading. Happily, they had already agreed on the fundamentals:

1. We should assess students on what they know now, as opposed to what they knew when we first assessed them.
2. Assessment should be atomized to the point that it empowers teachers and students in their remediation.

This left me all the creative, interesting parts. We talked about reporting methods for keeping students apprised of their progress, both individually and as a class. We talked about the effect of SBG on retention. Then we picked a concept and had pairs come up with a score of 1, 2, and 3.

We debated productively about marginal scores – when a 2 turns into a 3, specifically – and concluded that, in a system this forgiving, we’d rather underestimate a student (who could return to improve her score whenever, wherever) than overestimate her.

We discussed, afterwards, how to construct valid, manageable assessments. I gave them four test questions, each of which, in its own way, invalidated what it claimed to measure or was unmanageable at scale. I’ll leave them here. Feel free to kick them around in the comments.

Alex:

The trouble with the two-step equation problem is that it’s also an intimidating decimal arithmetic question. If a student fails it, you don’t know which skill needs work.

Erick:

The issue with the Law of Sines / Cosines problem is that you do not have to use the Law of Sines / Cosines to solve it. A student can get those right WITHOUT using the Law of Sines / Cosines, especially the 30-60-90.

Also, the concept is too broad. If a student has a 2/4 on “Law of Sines / Cosines,” how do you know which one to remediate?

“Quadrilaterals” is also too broad a concept. If a student has a 3/4 on “Quadrilaterals,” do you know what the student knows about quadrilaterals? Which ones she understands and doesn’t?

We decided “Linear Pairs of Angles” is too small a concept. If every concept were this granular, we’d have several hundred concepts to manage by semester’s end.

I take these speaking jobs for three reasons.

1. To maintain the tuna-casserole lifestyle to which I have become accustomed, even though I’m only bringing in the part-time research assistant money these days.
2. To compel me to find better structures, metaphors, visuals, and exercises for communicating good curriculum design.
3. For the helpful feedback and criticism the attendees offer.

These groups of grownups are my classroom for the foreseeable future. It’d be a waste of a blog if I didn’t share what I learned last weekend.

1. Mathematical notation isn’t a prerequisite for mathematical exploration. Mathematical notation can even deter mathematical exploration. When the textbook asks a student to “find the area of the annulus” in part (a) of the problem, there are at least two possible points of failure. One, the student doesn’t know what an “annulus” is. (Hand goes in the air.) Two, the student knows the term “annulus” but can’t connect it to its area formula. (Hand goes in the air.) ¶ That’s the outcome of teaching the formula, notation, and vocabulary first: the sense that math is something to be remembered or forgotten but not created. ¶ Meanwhile, let’s not kid ourselves. The area of an annulus isn’t difficult to derive. Let the student subtract the small circle from the big circle. Then mention, “by the way, this shape which you now feel like you own, mathematists call it an ‘annulus.’ Tuck that away.” ¶ Similarly, if I give you this pattern, I know you can draw the next three pictures in the sequence. That’ll get old so I’ll ask you to describe the pattern in words. You’ll write out, “you add two tiles to the last picture every time to get the next picture.” I’ll show you how much easier it is to write out the recursive formula A_n+1 = A_n + 2. ¶ I’ll ask you to tell me how many tiles I’ll find on the 100^th picture. You’ll get tired of adding two every time, and we’ll develop the explicit formula A = 2n + 3, which makes that task so much easier. ¶ Terms like “explicit” and “recursive” and “annulus” can do one of two things to the exact same student: make the kid feel like a moron or make the kid feel like the master of the universe.
2. “Talk to someone who actually makes ticket rolls. What kind of math does he have to do to make the thing,” said Russ Campbell, a community college adjunct instructor at least twice my age. Great idea, Russ. Speaking of which, pursuant to some harebrained WCYDWT idea, I spent twenty minutes on the phone with my local and state Departments of Transportation last week and it was almost too much fun to handle, peppering questions at engineers who were all too delighted that anyone gave a damn about how they calculated recommended speeds for curved roads. More of this.
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