# Inventing a knot theory for eight year olds (I)

Last year I experimented with teaching seven and eight year olds about monoidal categories using a very concrete model of cardboard rectangles and bits of string. The experiment was a success, with children becoming able to represent formally a model or to construct a model from a formal specification. They also developed an understanding of some key concepts, such as coherence. The physical manipulation of cardboard and strings was fun and relating it to notation and equations was surprising to the children and, I think, interesting.

This year I am reprising the experiment, with some changes. The first thing I wasn’t too happy with last year was the overhead of constructing models of cardboard and string. Too many things can go wrong, from poking holes too close to the margins so that they tear to making the wires too long or too short. Not to mention that some children did not know how to tie a knot! The motivation of building structures of cardboard and strings was also unclear, I didn’t even have a name for these structures. So this year we are only looking at knots. They are a simpler to build physical structure since all you need is the strings — no more cardboard and hole punchers. They are also something children already know. And if they don’t know how to tie them it’s OK, because that’s what we are learning. The second change I am making is slowing down the pace and making the process more exploratory. I will not try to teach them anything in particular but we will aim to develop together a term algebra for knots.

The first meeting of our club started with a general discussion about what is mathematics. Third-graders are taught a very narrow view of the subject, which they tend to identify with arithmetic. “Tricky problems” was another common view of what mathematics is about. When I asked whether a triangle is something that mathematics deals with the kids started to question their own understanding of what it is really about.

I proposed that mathematics is the general study of patterns and this was immediately accepted. Then we talked a while about the need in mathematics to represent concepts and I introduced the concept of “notation”, illustrated with a potted history of numerals, notations for the numbers. We spent some time on hieroglyphic numerals, a topic which turned out to be very enjoyable in its own right. I wanted children to discover the inconveniences of this system compared with our Indo-Arabic system and they indeed came to that conclusion quite soon.

The next step was to introduce knots. The fact that mathematics would have something to say about knots struck many children as implausible but even the most skeptical changed their tune immediately when reminded that “knots are some kind of patterns we make with strings” and “maths is about patterns”. This abrupt change of mind and wholehearted acceptance of a previously rejected thesis makes working with children a great pleasure.

We started by making knots. The first bit of technical terminology we established was calling anything we make with strings “a knot”, including braids or bows or even a plain bit of string. This was deemed acceptable. Making knots and experimenting with various ways of tying them was of course a lot of fun.

This is also a knot!

The second step was representing knots by drawing them. The challenge was to draw a half-knot:

The children under-estimated the difficulty of the task, producing something like this:

This drawing of course makes line crossing ambiguous. Which line is above and which line is below?  I suggested leaving small gaps to indicate line crossings:

The principle was readily understood but the execution was problematic. Drawing the knot correctly proved a real challenge, with more than half the children failing to draw it correctly.

The upshot of all this struggle was a consensus that we really need a better way to write down knots, a “notation” (knotation?). The groundwork has been laid for creating a term algebra for knots next week.