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## Can We Model Generosity With Mathematics?

Which of these bushfire relief donors was the most generous? What’s your ranking? What information matters here? What would your students say?

Some teachers quickly identified a connection to ratios.

Meanwhile, Lee Melvin Peralta critiqued ratios as too limited to fully model generosity.

I tend to side with George Box here, who wrote:

All models are wrong, but some are useful.

Anyone who thinks that proportional functions fully describe runners in a race, or that linear functions fully describe the height of a stack of cups, or that quadratic functions fully describe the height of objects under gravity, or that ratios fully describe generosity is, of course, kidding themselves.

But those models are all useful. Ratios are a useful way to think about generosity.

Emily Atkin originally stirred this question up for me in her fantastic climate change newsletter:

Chevronâ€™s donation is paltry, however, given its earnings and relative contribution to the climate crisis. Not only is Chevron the second-largest historical emitter of all the 90 companies, it also earned about \$15 billion in 2018. So a \$1 million donation amounts to about .00667 of its yearly earnings. To the average American, that donation would amount to about \$3.96.

So Atkin is evaluating generosity as ratio of net worth/earnings to donation size. But then she also considers the donor’s contribution to climate change.

The model is complex and grows more complex!

One teacher wanted to add fame and notoriety to our model, something Chris Hemsworth donates that Mariam might not. (Maybe she’s a TikTok teen, though. We can only speculate.) I talked with someone who lives in Australia about this question, and she said Hemsworth is less generous than someone from another country donating the same amount because of his identity as an Australian citizen. Robyn V wondered how to evaluate time donations, and even the donation of one’s life.

So ratios aren’t a perfect model for generosity, but they do offer us an important insight that, under some circumstances, someone who donates \$75 is more generous than someone who donates one million dollars, which one teacher noted is a quantity that is really hard for students to fathom!

One teacher preparation program asked the question:

If the ambiguity of the original question strikes you as anything other than a feature, then please don’t risk the conversation.

If you go into the conversation presupposing a model for generosity rather than admitting to yourself in advance that all the models are broken, you’re likely to diminish students who suggest variables you had already excluded.

Okay, yes, um, ‘whether or not someone lives in Australia.’ Okay, that’s one idea, but can I get some other ideas, please? Perhaps ones more related to the math we’ve been studying?

All of these models are complex. All of them are certainly broken. And all of them offer you the opportunity to celebrate and build on your students’ curiosity and contextual knowledge, an experience that is all too rare for students in math class.

BTW:Â Shout out to Christelle Rocha for her observation that individual generosity is no way to solve the climate crisis.

## The Teaching Muscle I Want to Strengthen in 2018.

It’s the muscle that connects my capacity for noticing the world to my capacity for creating mathematical experiences for children. (I should also take some time in 2018 to learn how muscles work.)

By way of illustration, this was my favorite tweet of 2017.

Right there you have an image created by Brittany Wright, a chef, and shared with the 200,000 people who follow her on Instagram. Loads of people before Ilona had noticed it, but she connected that noticing to her capacity for creating mathematical experiences for children. She surveyed her Twitter followers, asking them to name their favorite banana, receiving over one thousand responses. Then on her blog she posed all kinds of avenues for her students’ investigation â€“Â distributions, probability, survey design, factor analysis, etc.

That skill â€“Â taking an interesting thing and turning it into a challenging thing â€“Â is one of teaching’s “unnatural acts.” Who does that? Not civilians. Teachers do. And I want to get awesome at it.

But Ilona ran a marathon and I want to run some wind sprints. I need quick exercises for strengthening that muscle. So here are my exercises for 2018:

I’m going to pause when I notice mathematical structures in the world. Like flying out of the United terminal in San Antonio at last year’s NCTM where I (and I’m sure a bunch of other math teachers) noticed this “Suitcase Circle.”

Then I’ll capture my question in a picture or a video. Kind of like the one above, except pictures like that one exist in abundance online.

Civilians capture scenes in order to preserve as much information as possible. That’s natural. But I’ll excerpt the scene, removing some information in order to provoke curiosity. Perhaps this photo, which makes me wonder, “How many suitcases are there?”

In order to gauge the curiosity potential of the image, I’ll share the media I captured with my community. Maybe with my question attached, like Ilona did. Maybe without a question so I can see the interesting questions other people wonder. You may find my photos on Twitter. You may find them at my pet website, 101questions.

I want to get to a place where that muscle is so strong that I’m hyper-observant of math in the world around me, and turning those observations into curious mathematical experiences for children is like a reflexive twitch.

(Plus, that muscle will be more fun to strengthen in 2018 than literally any other muscle in my body.)

BTW. Check out the 3-Act Task I created for the Suitcase Circle. It includes the following reveal, which I’m pretty proud of.

BTW. The suitcase circle later turned into Complete the Arch, a Desmos activity, which has some really nice math going on.

[Suitcase Circle photo by Scott Ball]

Featured Comment

I would just add that we shouldn’t forget that the classroom is a world within a world for us to notice, and that while many great, unforgettable tasks are based on interesting phenomena that we’ve observed or collected outside of school, on a day-to-day basis, high-impact tasks are probably more likely to be rooted in our observations and interactions with our students (in fact, even the banana tweet and post were sparked by a conversation with a student who was eating what was, to me, an exceptionally green banana). They tend not to be as flashy, but can have just as much impact because they’re tailored to the kids, norms, relationships, and histories in our classrooms.

## My Winter Break in Recreational Mathematics

Chase Orton asked all of us, “What is your professional New Year’s resolution?

I said that I wanted to stay skilled as a math teacher. As much as I’d like to pretend I’ve still got it in spite of my years outside of the classroom, I know there aren’t any shortcuts here: I need to do more math and I need to do more teaching. I have plans for both halves of that goal.

In support of “doing more math,” I’ll periodically post about my recreational mathematics. Please a) critique my work, and b) shoot me any interesting mathematics you’re working on.

When Should You Bet Your Coffee?

Ken Templeton sent me an image from his local coffee shop.

Should you bet your free coffee or not? Under what circumstances?

This question offers such a ticklish application of the Intermediate Value Theorem:

If the bowl only has one other “Free Coffee” card in it, you’d want to bet your own card on the possibility of a year of free coffee. But if the bowl had one million cards in it, you’d want to hold onto your card. So somewhere in between one and one million, there is a number of cards where your decision switches. How do you figure out that number? (PS. I realize the IVT doesn’t hold for discrete functions like this one. Definitions offer us a lot of insight when we stretch them, though.)

I asked some of my fellow New Year’s Eve partygoers this question and one person offered a concise and intuitive explanation for her number, a number I personally had to calculate using algebraic manipulation. Someone else then did his best to translate some logistical and psychological considerations into mathematics. (eg. “Even if I win it all, I won’t likely go get a drink every day. Plus I’m risk averse.”) It was such an interesting conversation. Plus what great friends right?

How Many Bottles of Coca-Cola Are in That Pool?

When I watched this video, I had to wonder, “How many bottles of Coca Cola did they have to buy to fill that pool?”

I tweeted the video’s creator and asked him for the dimensions of the pool.

12 feet across by 30 inches high,” he responded.

Even though the frame of the pool is a dodecagon, the pool lining itself seems roughly cylindrical. So I calculated the volume of the pool and performed some unit conversions to figure out an estimate of the number of 2-liter bottles of Coca Cola he and his collaborators would have to buy.

How Do You Solve Zukei Puzzles?

Many thanks to Sarah Carter who collected all of these Japanese logic puzzles into one handout.

Carter describes the puzzles as useful for vocabulary practice, but I found myself doing a lot of other interesting work too. For instance, justification. The rhombus was a challenging puzzle for me, and this answer was tempting.

So it’s important for me to know the definition of a rhombus â€“ every side congruent â€“ but also to be able to argue from that definition.

And the challenge that tickled my brain most was pushing myself away from an unsystematic visual search towards a systematic process, and then to write that process down in a way a computer might understand.

For instance, with squares, I’d say:

Computer: pick one of the points. Then pick any other point. Take the distance between those two points and check if you find another point when you venture that distance out on a perpendicular line. If so, see if you can complete the square that matches those three points. If you can’t, move to the next pair.

Commenter-friends:

• What are your professional resolutions for 2017?
• What recreational mathematics are you working on lately?
• If any of you enterprising programmers want to make a Zukei puzzle solver, I’d love to see it.

2017 Jan 2. Ask and ye shall receive! I have Zukei solvers from Matthew Fahrenbacher, Jed, and Dan Anderson. They’re all rather different, each with its own set of strengths and weaknesses.

2017 Jan 2. Shaun Carter is another contender.

## #BottleFlipping & the Lessons You Throw Back

2016 Oct 7. I was wrong about everything below. After admitting defeat to #bottleflipping, my commenters rescued the lesson.

I’m sorry. I went looking for a lesson and couldn’t find it.

Relevant background information:

Last spring, 18-year-old Mike Senatore, in a display of infinite swagger, flipped a bottle and landed it perfectly on its end. In front of his whole school. In one try.

That thirty-second video has six million views at the time of this writing. Bottle flipping now has the sort of cultural ubiquity that can drive even the most stoic teacher a little bit insane.

Some of my favorite math educators suggested that we turn those water bottles into a math lesson instead of confiscating them.

I was game. Coming up with a math task about bottle flipping should be easy, right? Watch:

Marta flipped x2 + 6x + 8 bottles in x + 4 minutes. At what rate is she flipping bottles?

Obviously unsatisfactory, right? But what would satisfy you. Try to define it. Denis Sheeran sees relevance in the bottle flipping but “relevance” is a term that’s really hard to define and even harder to design lessons around. If you turn your back on relevance for a second, it’ll turn into pseudocontext.

For me, at the end of this hypothetical lesson, I want students to feel more powerful, able to complete some task more efficiently or more accurately.

Ideally, that task would be bottle flipping. Ideally, students who had studied the math of bottle flipping would dazzle their friends who hadn’t. I don’t think that’s going to happen here.

But what if the task wasn’t bottle flipping (where math won’t help) rather predicting the outcome of bottle flipping (where math might). You can see this same approach in Will It Hit the Hoop?

The quadratic formula grants you no extra power when you’re in mid-air with the basketball. But when you’re trying to predict whether or not a ball will go in, that’s where math gives you power.

Act One

So in the same vein as that basketball task, here are four bottle flips from yours truly. At least one lands. At least one doesn’t. Each flip cuts off early and invites students to predict how will it land?

Act Two

Okay. Here’s a coordinate plane on top of each flip.

If you’ve been around this blog for even a day, you know what’s coming up: we’re going to show which flips landed and which flips didn’t. Ideally, the math students learn in the second act will enable them to make more confident and more accurate predictions than they made in the first act.

But what is that math?

I asked that question of Jason Merrill, one of the many smart people I work with at Desmos. I won’t quote his full response, but I’ll say that it included phrases like “cycloid type thing” and “contact angle parameter space,” none of which fit neatly in any K-12 scope and sequence that I know. He was nice enough to create this simulator, which has been well-received online, though even the simulator had to be simplified. It illustrates baton flipping, not bottle flipping

Act Three

Here is the result of those bottle flips. For good measure, here’s a bottle flip from the perspective of the bottle.

I’m obviously lost.

Here’s a link to the entire multimedia package. Have at it. If you have a great idea for how we can resurrect this, let me know. I’m game to do some video editing on your behalf.

But when it comes to bottle flipping, if “math” is the answer, I’m not sure what the question is. Please help me out. What is the lesson plan? How will students experience math as power, rather than punishment.

Sure, it’s probably a bad idea to destroy the bottles. But it’s possible we shouldn’t turn them into a math lesson either. Maybe bottle flipping is the kind of silly fun that should stay silly.

2016 Oct 7. Okay: I was wrong about #bottleflipping. A bunch of commenters came up with a great idea.

I see a couple students playing the game during some down time and my immediate reaction is, â€œThereâ€™s gotta be some great math in there!â€ One of the boys who was playing sees my eyes light up. He looks at me in fear and says, â€œMrs. Raskin. Please. I know what youâ€™re thinking. Please donâ€™t mathify our game. Let us just have this one thing we donâ€™t have to math.â€

Mr K:

I suspect I should put as much effort into making this teachable as I would for dabbing.

Meaghan found a nice angle in on bottle flipping, along with several other commenters:

It would be neat if you could spend a, for example, physics class period talking about experimental design (for fill ratio questions or probability questions) and collecting the data, and then troop right over to math class with your data to figure out how to interpret it.

Paul Jorgens has the data:

It started with an argument in class last week with the optimal amount of water in the bottle. Should it be 1/4 filled? 1/3? Just below 1/2? I told the group that we could use our extra period to try to answer the question. We met and designed an experiment. Thought about problems like skill of tosser, variation in bottles, etc. We started with 32 bottles filled to varying levels. During class over 20 minutes 32 students flipped bottles 4,220 times.

## Testify

a/k/a Oh Come On, A PokÃ©mon Go #3Act, Are You Kidding Me With This?

Karim Ani, the founder of Mathalicious, hassles me because I design problems about water tanks while Mathalicious tackles issues of greater sociological importance. Traditionalists like Barry Garelick see my 3-Act Math project as superficial multimedia whizbangery and wonder why we don’t just stick with thirty spiraled practice problems every night when that’s worked pretty well for the world so far. Basically everybody I follow on Twitter cast a disapproving eye at posts trying to turn PokÃ©mon Go into the future of education, posts which no one will admit to having written in three months, once PokÃ©mon Go has fallen farther out of the public eye than Angry Birds.

So this 3-Act math task is bound to disappoint everybody above. It’s a trivial question about a piece of pop culture ephemera wrapped up in multimedia whizbangery.

But I had to testify. That’s what this has always been â€“Â a testimonial â€“ where by “this” I mean this blog, these tasks, and my career in math education to date.

I care about math’s power to puzzle a person and then help that person unpuzzle herself. I want my work always to testify to that power.

So when I read this article about how people were tricking their smartphones into thinking they were walking (for the sake of achievements in PokÃ©mon Go), I was puzzled. I was curious about other objects that spin, and then about ceiling fans, and then I wondered how long a ceiling fan would have to spin before it had “walked” a necessary number of kilometers. I couldn’t resist the question.

That doesn’t mean you’ll find the question irresistible, or that I think you should. But I feel an enormous burden to testify to my curiosity. That isn’t simple.

“Math is fun,” argues mathematics professor Robert Craigen. “It takes effort to make it otherwise.” But nothing is actually like that â€“Â intrinsically interesting or uninteresting. Every last thing â€“Â pure math, applied math, your favorite movie, everything â€“ requires humans like ourselves to testify on its behalf.

In one kind of testimonial, I’d stand in front of a class and read the article word-for-word. Then I’d work out all of this math in front of students on the board. I would circle the answer and step back.

But everything I’ve read and experienced has taught me that this would be a lousy testimonial. My curiosity wouldn’t become anybody else’s.

Meanwhile, multimedia allows me to develop a question with students as I experienced it, to postpone helpful tools, information, and resources until they’re necessary, and to show the resolution of that question as it exists in the world itself.

I don’t care about the multimedia. I care about the testimonial. Curiosity is my project. Multimedia lets me testify on its behalf.

So why are you here? What is your project? I care much less about the specifics of your project than I care how you testify on its behalf.

I care about Talking Points much less than Elizabeth Statmore. I care about math mistakes much less than Michael Pershan. I care about elementary math education much less than Tracy Zager and Joe Schwartz. I care about equity much less than Danny Brown and identity much less than Ilana Horn. I care about pure mathematics much less than Sam Shah and Gordi Hamilton. I care about sociological importance much less than Mathalicious. I care about applications of math to art and creativity much less than Anna Weltman.

But I love how each one of them testifies on behalf of their project. When any of them takes the stand to testify, I’m locked in. They make their project my own.

Again:

Why are you here? What is your project? How do you testify on its behalf?