Category Archives: Questions from middle school students

Questions from middle schoolers V: Where do mathematicians come from?

Were mathematicians chosen by god?

I could get into trouble here, so let me refocus on a related, profound question that mathematicians struggle with…

Do we “discover” or “invent” mathematics? That is, does mathematics exist in the universe, like the Sun does, and we discover it? Or is it something that never would have existed without us, like automobiles, and we invent it?

If you think it’s discovered, you are called a “Platonist,” after the Greek philosopher. If you think it’s invented, you might be a “formalist”. Either way, it’s a lovely debate to have with yourself. Careful…it might keep you awake at night.

Questions from middle schoolers IV: The hotel problem

What happened to that dollar in the hotel problem?

Are you asking, “What hotel problem?Click here to read it.

Nothing. Each guy paid $9. So $27 altogether. The hotel kept $25, the bellman kept $2.

The key is that the bellman’s $2 is part of the $27 total. So it makes no sense to add $27 + $2.

I remember my best friend in fifth grade, Matt, posing this problem to me as a “hole in mathematics”. He was a wonderful story teller and it made the puzzle very compelling. It’s a lovely one. But it’s not a hole in mathematics.

Questions from middle schoolers III: Doing math all day

Do you get bored having to do math all day?

What I am really interested in is how people’s minds work mathematically. That includes my own, so I enjoy working new problems and trying to see new connections. But for me the really interesting work involves understanding someone else’s way of thinking.

I have learned a lot of math that way. It has even resulted in my developing a new algorithm for adding fractions: the common numerator algorithm.

Questions from middle schoolers II: New numbers

Are mathematicians looking for any new numbers?

They are always looking for the next biggest prime number. We know that there are infinitely many of them, and we know some really big ones. Every so often, it makes headlines when a supercomputer finds another, bigger one. The goal isn’t to prove that there is another one; we know that already. The goal really is to test the computational power of computers, since all the undiscovered primes have many, many digits (in 2008, mathematicians found one that has 13 million digits).

Instead of new numbers, mathematicians are always on the lookout for new categories of numbers. Whole numbers, rational numbers, irrational numbers, imaginary numbers, transcendental numbers, etc. These are all categories of numbers that mathematicians discovered (created?) in the process of their work. Surely there are more to come…

Questions from middle schoolers I: Pi

I travel a lot to work with middle school math teachers. In one classroom recently, I was oversold as a visiting mathematician. Visiting I was. Mathematician I am not. I do think about mathematics for a living, but I do not create original mathematics.

Nonetheless, students in the class generated a number of questions they wanted to ask a mathematician. So this series is for Ms. Otto’s eighth-grade students at KMS…

Who discovered Pi?

Can you believe that an entire book has been written about this subject? It’s called A History of Pi by Petr Beckmann.

Here’s the very short version…

In order to discover pi, people needed to understand ratios. That is, they needed to not just be able to count objects, they needed to be able to see relationships between quantities. That took a long time, until about 2000 B.C.

At that time, there were two major societies doing serious mathematics-the Babylonians and the Egyptians. Each of these societies appears to have noticed that there is a relationship between the diameter of a circle and its circumference, and both societies had decent estimates for pi (although neither called it that).

The next several thousand years involved people getting better estimates of the value of pi. There are lots of interesting mathematical advances that came about as people tried to do this.

In the 1700’s, at least two mathematicians demonstrated that pi was irrational (Lambert and Legendre). In 1882, Lindemann proved that pi is not just irrational; it has an even more profound property-it is transcendental. This means that it is not the square root (nor cube root, nor…) of any rational number. It is an even more special number than the square root of 2.

So who discovered pi? That’s a bit like asking who built the Empire State Building. Pi is an achievement of human intellect more than of a single person.