The category of cardboard rectangles and bits of wool string (I)

I am conducting an educational experiment: teaching children in year 2 and 3 (7-8 years old) about categorical concepts, more precisely about monoidal categories. Monoidal categories are an appealing formalism because of their connections with diagrams [1]. So we just discover together a bunch of stuff about diagrams.

The first lesson started with a general conversation on what Mathematics is and, unsurprisingly, children were surprised to learn that it is not all about arithmetic. The idea of mathematics without numbers seemed strangely exciting at least to some of them — a boy clenched his fist and muttered ‘yes!’ to himself when I announced that we will do the mathematics of shapes. Because this is highly experimental only 10 children were invited to participate, among the more gifted in their year. I divided them up into teams and assigned them tasks. The first task was to cut rectangles out of cardboard, punch holes in the four corners and label both the rectangles and the corners: these were going to be the morphisms of the category. Preparing 8 such rectangles proved to be unexpectedly challenging and took quite a lot of time. The children in each team first spent quite some time deciding on how they will cooperate. Two common models were parallel (everyone does the same thing independently) and pipelined (each one does a specialised task). There was also the unexpected difficulty of operating a hole puncher so that you punch just one hole rather than a pair of holes, with at least one team being unable to figure out how to insert the cardboard into the puncher to achieve this. The same team struggled with 5 or 6 ruined “morphisms” because the hole was too close to the edge.

As is usually done in categories of diagrams, identities and other structural morphisms are represented by lines. In our case we were going to use wool strings, tied to the holes. The hole patterns are, of course, the type of the morphism, i.e. the objects.  The next task was just to get familiar with the mechanics of tying bits of string to little holes in cardboard rectangles. This was a skill essential in building more complex morphisms out of simpler morphisms and compositions. Just like before, the level of dexterity of the children was unexpectedly low. For example several did not know how to tie a double knot and they had to learn that first. In the end, however, all teams mastered the task.

The final task was for each team to construct a composite morphism, which was specified in a relational style as a list of connections between node labels: 1c to 2a, 1d to 2b, 2c to 1a, 2d to 1b. With the cardboard morphisms at the ready and the string-tying skills mastered, this task was completed relatively painlessly and error free. Each team displayed their creation.

In the conclusion of the lesson, with all the diagrams displayed, I asked the children whether these were the same diagrams or different diagrams. Intuitively, because they knew they built the diagrams using the same instructions, they all answered that these were the same diagrams. I challenged their answer, objecting that the cardboard was different colours and sizes and the strings were different colours and lengths, to which they replied that these don’t matter. Indeed, what we wanted to matter is just the what-is-connected-to-what-ness of the diagram.

To summarise, in this first lesson we covered the basic skills of constructing morphisms in the the category of cardboard rectangles and bits of wool string: making cardboard morphisms and composing by tying the bits of string, using a relational-style specification. The children also started to develop the basic intuitions which will lead up towards a topological notion of equality of diagrams.

Figure. Two equal morphisms in the category of cardboard rectangles and bits of wool string.

About Dan Ghica

Reader in Semantics of Programming Languages // University of Birmingham // https://twitter.com/danghica // https://www.facebook.com/dan.ghica
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2 Responses to The category of cardboard rectangles and bits of wool string (I)

  1. Dan Piponi says:

    You can bring the arithmetic back in with rational tangles: http://fawnnguyen.com/2013/03/12/20130312.aspx

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