This post contains recordings from a maths club session in which I invited some students to re-construct this animation of van Shooten's theorem. I chose this animation because I found it surprising, intriguing and simple (SIS). I would invite you to watch it before reading on.

Having watched the animation, I asked students to describe it step by step, in as much detail as possible -

Insisting that no-one draw or point seemed to lead students to,

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Initially, students had great difficulty describing what was happening, trying to describe the whole animation in one go. This example was typical:

Having watched the animation, I asked students to describe it step by step, in as much detail as possible -

*without drawing or pointing*- that is, only with words. This is following experiences at IMP16, and upon reading this article by John Mason.Insisting that no-one draw or point seemed to lead students to,

*'contact the deep desire to illustrate, to extend their mental screens...'*as predicted in the article. This may result in these students being more likely to*want to,*or see the point of, drawing a diagram in the future.---

Initially, students had great difficulty describing what was happening, trying to describe the whole animation in one go. This example was typical:

We had to slow down.

I asked:

*"Forget about trying to describe the whole thing, what is the first thing we see?"*to which a student replied,

*"There is a triangle inside a circle."*A good start. What type of triangle is it? I had assumed it was equilateral, but we were only convinced following this moment of inspiration:

One of the students suggested labelling the vertices of the triangle (ABC from bottom-left) and the point (S). This took a while (and some encouragement) to catch on, but was essential to forming a shared language:

With labelling, descriptions became more fluent, although students used some non-normative words (such as translation instead of rotation). I made a choice not to 'correct' this language, as students seemed to understand what

*they*meant by it.

I also attempted to remain as impartial as possible so as not to influence the discussion unduly; unfortunately this sometimes meant that useful observations didn't get taken up, floating off into the ether.

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Here is a description of the movement of the triangle ASB, followed by the first vocalisation that a new equilateral triangle, ADS, has been formed. Note also the growing sophistication in the description:

As the session progressed, I asked if we could now tell the story of the animation. To my surprise and delight, one of the students who had said very little throughout asked: "Can I start off?"...

This student adds the detail that A stays fixed during the 'translation' of triangle ABS. Although I sense that the absence of the word rotation might be inhibiting description, I allow it to continue; I am more interested in this student expressing himself, and how the other students are naturally contributing.

I ended the session by asking if anyone can now state van Shooten's theorem:

The session ended here. I asked students if they could explain why the equilateral triangle ADS 'appears' for next week's session.

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I found this session so interesting. A huge amount of emotional energy was invested. A number of times, I noticed students (and teachers) sitting with eyes closed, heads on desks. I was staggered at the levels of concentration, for well over an hour, as they struggled to find the language to re-consruct this 1-minute animation.

The increase in sophistication of description over the hour was remarkable. It convinced me that we (maths teachers) might want to spend more time working with mental images.

Discussing the session with the other teachers in the room, we wondered if the difficulties the children had in describing at the start was indicative of a loss of mental imagery due to modern technologies; it was noticeable that the reaction of a number of students at the end was to take a photo of the final image, even though I felt they could almost certainly recreate it in their heads at that point.