Yesterday we held the first of this year’s Maths Experience days. We invite students in Years 10 and 11 from different schools onto campus for an intensive one-day program. Students find out about mathematical research, talk to professionals who use mathematics in their careers in some way, and participate in hands-on mathematics workshops. Importantly, they also meet and connect with other students who enjoy mathematics.

One of the activities I chose for this year was Conway’s Rational Tangles. I’ve previously written detailed notes about running the activity with pre- and in-service teachers. For the Maths Experience, apart from the inherent fun of ‘playing’ with ropes, I wanted students to have a collaborative and authentic problem-solving experience. I introduced the activity as one that mirrors mathematical research — full of questions, puzzling moments, uncertainty, frustration and hopefully also joy. I emphasised that we might not solve the problem, but that the experience of working mathematically was our goal, which includes making wild conjectures and having out-of-the-box ideas!

In this post I want to highlight one addition I made to the activity described in my earlier blog post — the inclusion of the ‘Notice and Wonder’ prompt^{1}.

I started the session by showing students the short video below, edited from one I found on Youtube by Tom Hildebrand. Specifically, I turned off the sound, cut out the whiteboard, and sped it up significantly. Then I asked the two magic questions: ‘What do you notice? What do you wonder?’ Take 70 seconds to watch the video, and see what you think.

Here is what they **noticed**.

**Group A**

- They are trying to untangle the ropes.
- One person hangs on to one end of the rope for the whole time.
- They rotate 90 degrees clockwise.
- There is a plastic bag.
- Twist involves exactly two people and occurs in exactly the same position.
- They untangle using exactly the same types of moves they used to tangle.

**Group B**

- Four people holding two ropes.
- Same person holding the same end for the whole activity.
- When rotating, one person moves clockwise. (Later refined to each person moves one position clockwise.)
- The twist movement always involves the two people on the right. The same position goes under each time.
- There was some pattern they kept repeating.
- They did some moves to get a knot. Then they did some more moves and there was no knot.
- There was a bag.
- There were four rotations before the bag appeared and eight rotations after.
- Sometimes there is a different number of twists after a rotate.
- A twist after a rotate goes ‘perpendicular’. (Not sure what that means!)

And here is what they **wondered**.

**Group A**

- What’s the deal with the plastic bag?
- What’s the deal with the teacher?
- How did they decide when to stop tangling and start untangling?
- How tangled was the rope?
- What did the teacher and the student pass to each other? (Scissors.)
- How did they work out how to untangle? (I explicitly prompted this question — although I’m sure they were all thinking it.)

**Group B**

- How did they know how to untangle the ropes? Was it from memory?
- What is the point of rotate? It doesn’t seem to change the rope.
- Does the bag have something to do with the tangling?
- Is it a proper knot? Or just a tangle?
- What is the teacher doing?

There was more conversation that I didn’t manage to capture. (Next time I’ll record it!) Group A spent around 10 minutes on Noticing and Wondering. Group B spent 15-20 minutes. We then largely ran the session as I’ve detailed in the earlier blog post.

What effect do I think Notice and Wonder had? I noticed that students were keen to try the problem for themselves. They made sense of the situation, became intrigued and engaged, and then made the problem ‘theirs’. As a group, students saw that others had interesting ideas. They added on to each other’s thinking. I suspect that it also smoothed the way for working together more intensely once we broke into smaller groups where students didn’t necessarily know one another. It also became more natural for them to Notice and Wonder as the session progressed. All in all, it’s a great modification to a thoroughly engaging activity.

[1] I trialled this with teachers at the MASA conference in April.

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