A few days ago, I registered at bettermarks, a German startup co-founded by successful entrepreneur Arndt Kwiatkowski. I played around a little pretending I was in 7th class and looked at what they’ve built. This is certainly a very interesting project, and I hope that it will help to improve math lessons in Germany – but that is a long way to go.
Anyway, I don’t want to rant about bettermarks. On the contrary, I think bettermarks is great. But I’m worried about the content of math classes. The content that is specified by politicians. 7th-grade students in Germany seem to constantly deal with units in math. I guess the correct English term is “physical units”. Physical units because units are a concern of physics and, more importantly:
Mathematics is not concerned with units.
That’s the whole point of mathematics: You make things simple so you can handle them. In mathematics, there is no unit of area and unit of length.
Mathematicians are happy if something can be measured at all, a property that not too many mathematical entities have.
Take fractals for example: Suppose I drew a Sierpinski
Triangle on a piece of paper, how much space do I need to do that?
Obviously, I will need some space if I wanted to actually draw it (physical reality), otherwise I couldn’t see it. Mathematically speaking, however, it has Lesbegue measure 0 and thus, vanishing area. Not square kilometers, apples or Newton, just a number.
How about a Hilbert Curve? I can draw an approximation on some finite area. Now I can refine it, thereby increasing its length, but I don’t need to extend my drawing. Eventually, the length will grow to infinity – how can this be? In the context of soccer fields and living rooms, this doesn’t make sense – it’s purely theoretical. It’s mathematics.
I agree, it is important to teach children about the typical size of a living room, a soccer field or the distance between some well-known cities. But that shouldn’t be done in mathematics: Mathematics should be concerned with abstract thinking only. Many people believe abstract thinking makes classes less appealing to children, but honestly: Do they really think that “Martha owns a set of Miley Cyrus stickers of cardinality 8″ type of exercises make math more interesting, let alone be taken seriously by children?