# Lessons from teaching gifted elementary school students (Part 5a)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received (though I probably changed the exact wording somewhat):

Exponentiation is to multiplication as multiplication is to addition. In other words,

$x^y = x \cdot x \cdot x \dots \cdot x$,

$x \cdot y = x + x + x \dots + x$,

where the operation is repeated $y$ times.

So, multiplication is to addition as addition is to what?

My kneejerk answer was that there was no answer… while exponents can be thought of as repeated multiplication and multiplication can be thought of as repeated addition, addition can’t be thought of as some other thing being repeated.

Which then naturally led to my student’s next question, which I was dreading:

Can you prove that?

This led to another kneejerk reaction, but I kept this one quiet: “Aw, nuts.”

I suggested that $x + y$ can be thought of as starting with $x$ and then adding $1$ repeatedly $y$ times, but my bright student wouldn’t hear of this. After all, in the repeated renderings of $x^y$ and $x \cdot y$, there’s no notion of starting with a number and then doing something with a different number $y$ times.

So I had to put my thinking cap on, and I’m embarrassed to say that it took me a good five minutes before I came up with a logically correct answer that, in my opinion, could be understand by the bright young student who asked the question.

I’ll reveal that answer in tomorrow’s post. In the meantime, I’ll leave a thought bubble if you’d like to think about it on your own.

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### 2 Comments

1. #### howardat58

/  November 24, 2015

Since 2 to the power 1 comes out of the definition as 2 times itself one time, which is actually a meaningless statement I prefer to see 2 squared as 1x2x2, 2 cibed as 1x2x2x2,
and so 2 is then 1×2, and with almost zero need for hand waving 2 to the power zero is 1.
This now solves your problem:
“I suggested that x + y can be thought of as starting with x and then adding 1 repeatedly y times, but my bright student wouldn’t hear of this. After all, in the repeated renderings of x^y and x \cdot y, there’s no notion of starting with a number and then doing something with a different number y times.”

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