Month: August 2016

Total 7 Posts

Testify

a/k/a Oh Come On, A Pokémon 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 Pokémon Go into the future of education, posts which no one will admit to having written in three months, once Pokémon 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 – a testimonial – where by “this” I mean this blog, these tasks, and my career in math education to date.

I don’t care about Pokémon Go. I don’t care about multimedia. I don’t care about the sociological importance of a question.

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 Pokémon 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 – intrinsically interesting or uninteresting. Every last thing – pure math, applied math, your favorite movie, everything – 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.

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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?

Related: How Do You Turn Something Interesting Into Something Challenging?

[Download the goods.]

A Very Valuable Conjecture

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Here is a very valuable conjecture:

The spelling of every whole number shares at least one letter with the spelling of the next whole number.

Which is to say that:

  • “one” and “two” both share an “o”
  • “two” and “three” both share a “t”
  • etc.

Could that possibly be true for every whole number?

If I were starting a course on geometry or a unit on proof or an activity on deductive logic, I would introduce this conjecture very early in the process. Let me explain what I find so very valuable about this conjecture.

Deduction is hard. It’s an abstract mental act that adults find difficult. (See: the van Hiele’s and their levels.) Too often we rush students to that abstract act, rushing them past the lower van Hiele levels, and we ask them to argue deductively about objects that, to them, are also abstract.

I suspect that, to many students, those proof prompts read something like this:

Given that the base bangles are twice the tonnage of the circumwhoozle and the diagonalized matrox is invertible, prove that all altimeters cross the equation at Quito, Ecuador.

The word “prove” is weird. And, unfortunately, so is every other word in the sentence.

So I cherish opportunities to help students argue deductively with concrete objects, which is what we’re working with here, with the spelling of whole numbers. This conjecture also gives students several different angles on the proof act.

You can ask students to find a counterexample, for example, a useful strategy when first interrogating a conjecture.

Once students have tried several different numbers they may satisfy themselves that the conjecture is true. This is one of the naive proof schemes Harel & Sowder observed in the students they studied. When this proof scheme surfaces in conjectures about geometric shapes, it’s challenging to summon up one new shape after another to challenge the student’s proof by example. It’s trivial, by comparison, to summon up one new number after another and ask the student to check her hypothesis again.

At a certain point in this process, likely after you give several numbers in the millions, your students may transform in two ways:

  1. They’ll get tired of trying example after example. “Proof by examples means you have to try all the examples,” you can say, giving you both a moment to reflect on the need for a more rigorous proof scheme, like deductive reasoning.
  2. They’ll notice that every number in the millions shares an “n” with every other number in the millions. And same for the billions. And same for the trillions. And … same for the hundreds. And so on.

And suddenly we’re on our way to a proof by exhaustion, which is much more rigorous than a proof by example. Nice.

This conjecture also leaves ample room for you and your students to pose follow-up conjectures. Like, “Does it work for all integers, or just whole numbers?”

I saw the conjecture and saw its value immediately. This is a very valuable kind of conjecture, I thought. But I don’t have many of them. Do you have another you can trade?

[via Futility Closet.]

BTW. You’re worse off in at least one way now than before you knew the conjecture was true. Now, when you ask your students, “Could that possibly be true?” you’re going to have to pretend.

Featured Comments:

Max:

What I like about game strategies is you go from “what seems to work,” to “will this always work,” to “here’s why this does/doesn’t work” pretty seamlessly.

Paul Hartzer:

Is it true for any other language with an alphabet? It fails for German (5/6 using umlauts, 7/8 otherwise), French (2/3), and Spanish (7/8).

Michael Serra:

What letter(s) of the alphabet do(es) not appear in the spelling of the first 999 whole numbers? Prove it.

Lynn CP:

List all the factors of every number from 1-100. What do you notice? Which numbers have an even number of factors? An odd number? What do you notice about numbers with an odd number of factors? Can you prove which numbers beyond 100 will have an odd number of factors?