Pythagorus's theorem doesn't work.

Is a circle that's not a perfect circle still a circle?

The difference between models and reality is a complete muddle in school. In class, I try to get students thinking about models vs. reality by working up a confusion about what we mean by 'a circle' and then about how we can use something if we don't know what it is or whether it even exists.

I draw a figure on the board and ask the kids to tell me what it is. So far, they've always said it's a circle. Look at my circle, the first one, below. It has problems, so I redraw it, but still it's not a 'perfect circle' we all agree, so I improvise a compass and try to construct a 'more-perfect' circle, but usually give up half-way through. What about one drawn by a computer, is that a perfect circle?

How 'perfect' of a circle can we get?

What is the definition of a circle? What did you learn in geometry? A standard textbook definition, a 'formal' definition, of a circle is typically something like

those points in a plane which are equidistant from a given point called the center (wikipedia)

They remember some version of this, albeit pretty dimly. Do any of my pictures match the definition?

Be honest.

After thinking about it a while, in class we realize that it is impossible to draw such a thing, as all paper has an irregular surface; all pencil lines have a thickness and therefore 'points' on the outer edge of the line are farther away then points on the inner edge. Even computer screens have their own granularity, their pixel size. Anyway you try, no physical shape matches the definition.

There are no circles.

Have you been lied to all these years?

That's what I ask my kids. How would you answer? Aristotle, thousands of years ago, coined the term 'equi-vocal', or same-sound for words that really have different meanings depending on the context. Science and math education is full of equivocal words. These are one of the hardest things to overcome, I think, and create a near hopeless muddle.

Mathematics and the world are separate. Their terms are equivocal. There are no circles, but we use them all the time. This is the essence of modeling. We look at a messy shape but think of an idea that has a formal meaning. Then we pretend.

A formal meaning? I borrow the term from computer science and linguistics. The main idea is that a formal meaning is a set of rules. The rules are ideas, not experiences. Lines on a piece of paper do not obey these rules. Two thousand years ago, Euclid took the messy idea of 'circle' and made it into a rule, as above. Four hundred years ago, Descartes took Euclid's idea and turned it into an equation. Soon after, Newton used these.

There's the thing, and there's the ideal thing. Which are we talking about?

When the kids say 'perfect circle' they mean the formal idea, the model, yet 'circle' is a fuzzy experience to them, hard to define, but they can point to it. If you asked them to measure the diameter of a 'circle,' they would eventually pretend it was a 'perfect circle' and measure across it and come up with a single number to represent its size. If you asked them to find the perimeter, they would again pretend it was a 'perfect circle' and use a formula. If you ask them to measure the perimeter, they are typically confused, and return to the formula.

What's fun is to ask them to measure the perimeter and calculate it, that is predict it using a formula, and compare the two. This is where we can begin to unmuddle the muddle.