Tag Archives: nature of mathematics

Your Daily Wu: Research

Wu on making claims about teaching and learning that are based on even a shred of research evidence

What i think he’s saying

Despite strong research evidence to the contrary, and despite the well-known historical trajectory of the development of the ideas that now comprise elementary school mathematics (including operations on both fractions and integers), the human mind is unfailingly logical.

Because all mathematical ideas can (post facto) be put into a logico-developmental framework, they must be. Anything else lacks rigor and misrepresents the nature of the field of mathematics.

These truths are self-evident. By definition, they do not require substantiation.


Questions from middle schoolers IX: Length

Can you have a negative length?

Preliminary confidential to the KMS middle schoolers who asked this question: I am continually impressed by your active minds; it was a delight to meet you and be your teacher for a day recently. And you should be very grateful for the teacher you get to have every day. Ms. O cares deeply about you, about keeping your minds active and about your success. Thank her for that now and again in 10 years when you realize the impact she has had on your lives.

Back to the question: Technically, no. “Length” is always positive. As is “Area”. As is “Speed”.

But if you are willing to expand your mind and imagine a world with negative lengths, you don’t cause any problems. But you will have to be ready to consider negative areas. And negative speeds.

In Physics, they deal with this by the difference between speed and velocity. Velocity can be negative, as it includes direction. Forward velocity is positive; backward velocity is negative. Speed is the absolute value of velocity. Whether my velocity is 10 mph or -10 mph; whether I am running forward or backward, my speed is 10 mph (positive).

And this requires a negative version of distance. In Physics, this is displacement. Displacements can be positive or negative. Distance is the absolute value of displacement.

In Calculus, it can be useful to imagine a world in which area (and length) can be negative. It’s not usually encouraged, but it causes no problems and in fact can make some concepts more intuitive.

How can negative areas and lengths be intuitive? Well, you’re an eighth grader and you thought of them-shouldn’t a college student be able to also?

Questions from middle schoolers VI: Irrational numbers

How do they know irrational numbers never repeat?

What a lovely question. When you are told that “irrational numbers have decimal representations that never repeat,” it’s a good instinct to say, “That’s just because you haven’t looked hard enough.”

There are only 10 digits, right? So they must repeat eventually. There are only so many ways to arrange 10 digits. So it makes sense that they don’t repeat very often, but they must repeat eventually, right?


And the way we know they don’t repeat isn’t what you expect. You expect that mathematicians have looked, but not found patterns in the digits of irrational numbers, and that from this they conclude that the digits don’t repeat. You are correct in being suspicious of this argument.

But it’s not the one mathematicians use.

Instead, we know that all repeating (or terminating) decimals are rational. And we know that rational numbers have other properties (like that they can be simplified-or reduced-to a fraction with whole number numerator and denominator). And then we know that a number such as pi or the square root of 2 cannot be reduced to a fraction with whole number numerator and denominator. Therefore pi and the square root of 2 must have decimals that don’t repeat (or terminate…because if they did, they would be rational, which would mean they could be reduced which they cannot).

This is common in mathematics-knowing that something is true in a roundabout way.

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.