Unlike toys, you should always try to break your mathematics.

An example: at some point, we get used to the idea of powers as being a shorthand for repeated multiplication. What is 3^{4}? It’s three multiplied by itself four times, 3x3x3x3, which comes out to 81. No problem. But things don’t get interesting until we try to break it.

How do we break exponents? It’s no trouble to find 3^{4}, or 3^{99} (if you have the time). But what is 3^{-1}?

A few weeks ago, a teacher and a parent (independently) told me that they’d been approached by students with this very question. Kids trying to break mathematics! The questions is, does the mathematics, being challenged, grow to handle the attack, or does it simply break? That’s up to the kid. I didn’t hear what happened with the teacher, but the parent told me that she started to tell her child the answer, and he stopped her, wanting to figure it out herself. “Is it even something you can figure out?” she thought. “Isn’t it just a definition you learn in math class?”

A little while later, her son came back with the following argument: if you multiply 3^{-1}x3^{2}, you get 3^{1} (recall your rules of how to multiply the same number raised to different powers). That means 3^{-1} times 9 is 3, which means that 3^{-1} must equal 1/3. And indeed, he was right. The parent was astonished, and thrilled. The child was now ready to extend the definition of exponents to any negative number.

In fact, it’s possible to work out all the rules of exponents for yourself, and if you keep trying to break them, you can figure out what 3^{1/2} must be too. It is easy to forget that *definitions come from people trying to break mathematics*, and unless you try to break the math too, it’s hard to see what people were thinking when they came up with the definitions and rules we have today.

So don’t be gentle. Bend it, and see if it will break.

## Comments 11

Lovely! I’m going to use this next time I work on exponents: “Can we break this?”

What a fantastic story!! I’ll have to incorporate that into Camp Ling for this summer, thanks!

Great post. This reminded me of my posts on Problems vs Exercises as well as Polya and Algebra. It’s amazing what can be done by “breaking” and fixing by connecting to what we already know.

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I love discovery and the Socratic process. I appreciate the fact that the student was confident enough to tell his parent that he didn’t want the answer and was able to leave and figure it out (aka Break the Math).

If only more individuals were comfortable enough Breaking the Math academic scores would soar and our world would be a better place.

Glad to follow your blog via Beth’s Bookshelf. I look forward to more stories such as these and sharing them with our families.

Dan, thanks for sharing.

I love the story of the boy who didn’t want to be TOLD what 3^-1 equals, but wanted to figure it out – just brilliant!

If only we could get all students to want math to be learned in this way 🙂

Excellent , I will use it next time when I teach exponents.

By the way , I am Sanjay Gulati from India.

Have a look at my blog http://mathematicsbhilai.blogspot.com/

Please link it to your blog.

Sanjay Gulati

I have used the term “experiment” and “get your hands dirty” but never “break it.”

This is fabulous!

And Peter – they DO all want to learn this way. We just suck that passion and desire out of them. They want to please us more than they want to break math. It takes the strong willed kid to say, “no, let me figure it out.” Good for him!

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Thanks, everyone, for the enthusiasm.

I find myself saying this more and more–every time there’s a student (or teacher!) conjecture about some problem. Does it work in every case, or can we find the example that breaks it? It’s a great way to push towards that ever-important goal of students owning their mathematics.

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