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Daniel Finkel's Broken Calculator

Susan Notes:

I wish that members of the New York Times editorial board would read this blog--and take it to heart: Einstein called play the highest form of research; happily, play is also the best way to win anyone over to math. Play is where love begins.

Other entities that should read it: Achieve, Inc., Advanced Placement, ALEC, Lamar Alexander, Alliance for Excellent Education, Jonathan Alter, American Enterprise Institute, American Federation of Teachers, America's Choice, America's Promise, Amplify, Annenberg Challenge, ASCD, Aspen Institute.

And that's just the A's. I have about 400 more on my list.

By Gary Antonick

Our puzzle this week was suggested by Daniel Finkel of Math For Love, the Seattle-based team committed to sharing a love of math through seminars, festivals and camps.

Broken Calculator

Meredith had been warned about playing catch with the calculator, but the moment it slipped through her fingers and banged on the tile floor, she knew it was too late.

"Meredith!" cried her teacher. "What have I told you about throwing the calculator?"

The teacher picked up the calculator and pressed a button. Nothing happened. Then another. Nothing. She pressed a third, and the calculator registered a 7. She hit it again, and it went to 14. Picking through every key, Meredith and her teacher realized only three keys did anything, and what they did was strange.

Button A added 7.

Button B added 5.

Button C took the square root.

"Meredith," said her teacher, "you've certainly broken this calculator. I used to be able to type in any number I wanted, and now I can't even make it display a 2."

"Yes, you can!" said Meredith. And with that, she picked up the calculator (which displayed 0 to start) tapped on the buttons, and turned it around to show her teacher a 2 on the screen.

How did Meredith make 2?

Bonus: What is the fewest number of buttons Meredith would have to press to make 2?

Super Bonus: What are the positive whole numbers that the calculator can and cannot display with only the three working buttons?

That's it for our puzzle this week. Dr. Finkel concludes with a few thoughts on the role of play in developing an authentic relationship with mathematics.

Daniel Finkel on Play in Math

No matter whom we work with, our initial goal is for them to have an authentic, mathematical experience; that is the first step to helping anyone -- teachers, students, parents, kids, you name it -- develop a love for math. The question then becomes, what does authentic mathematics look like, and how can you get people to experience it, especially if they've been turned off from the subject?

The answer is play. Play is the engine of learning, and our minds are wired to play with mathematics. When people are able to play with mathematics, they learn more profoundly and expertly than they would otherwise. The enemy of play is fear, and insistence that others do things your way, the "right" way. If you share mathematics as something you love, a gift you and everyone else gets to play with, and have the discipline to reserve judgment, what you end up seeing is that people of all ages will quickly take up the invitation to play with you, and soon they are doing math with a level of creativity, interest and rigor that you might have thought would be impossible.

All we need to do to open people up to mathematics then, is give them permission, and the right circumstances, to take ownership of their mathematical experience and begin playing. There are critical elements to provide:

  • A worthy mathematical question or problem, with a low barrier to entry and a high ceiling,

  • A safe atmosphere, free of judgment,

  • Time, and whatever encouragement and support is necessary to coax people into playing.

  • I think the Broken Calculator problem qualifies as the right kind of mathematical question, potentially, though working with teachers or students I usually change the question to: What numbers are possible to get on the calculator and what numbers are impossible? That way, people can mess around and get positive feedback. You made 12 on the calculator? Great! That tells us something new -- what else is possible? That's what I mean by a low barrier to entry. But the problem is designed to resist easy answers, too (unless you've worked on this kind of question before), so it is unlikely that anyone without training would be able to say, "Here's the answer. Now what?"

    I recently demoed a pared-down variation of the Broken Calculator problem in a third grade classroom (without square roots) as part of a teacher professional development partnership. I was worried that I'd have to sell the problem to the students, but the opposite was true: The moment the kids saw that they were in control of how they approached the question, they couldn't get enough. Kids know how to play.

    In the end, play really is the magic ingredient. Einstein called play the highest form of research; happily, play is also the best way to win anyone over to math. Play is where love begins.

    — Gary Antonick and Daniel Finkel
    New York Times blog, 3/23/15



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