We bought Disney Studio’s newest film “Tangled” on the weekend and I have to say it’s one of my favourite movies ever. It’s certainly Disney’s best movie since “Beauty and the Beast”, and I dearly loved “Beauty and the Beast”. I should warn you now that in order to say what I want to say I’m going to have to reveal a bit of the plot, so let this count as your spoiler alert.
OK. So Rapunzel grows up in her tower thinking that the old lady is her mother and not knowing who she really is. During the film she escapes and goes to the town where there are a lot of sun-shaped motifs. She brings one home to the tower with her on a piece of cloth to remember her experience.
We see her lying on her bed staring at the ceiling, which she has completely filled with painted pictures during her life in the tower. She looks at the sun-shape and notices something remarkable about her painting: the sun-shape from the cloth is there in her paintings, and not just once, but over and over and over, and the repeating pattern sparks a memory of seeing the shape when she was a baby. The music swells as she realises who she really is. In short, the cloth and the paintings spark an epiphany.
But it occured to me that she would never have had this epiphany without two important factors. Firstly, she had to bring the sun-shape home with her on the cloth. Secondly, and more importantly, if the sun-shape had not been in her paintings so many times, she may not have noticed the connection.
And here’s where it relates to learning maths:
We want the ideas we show our students to connect together so that the students realise the true nature of things and the realisation changes them. In short, we want them to have “learing epiphanies”.
I’ve seen it happen for students when learning about subspaces in first-year maths. There are a lot of ideas but they are all highly connected, and sometimes while they are trying to solve a particularly difficult problem they suddenly realise that they’ve been seeing the same pattern over and over and that it all just makes sense.
I want this experience to happen for all my students.
But is it possible to set up these learning epiphanies in advance? It could be argued that epiphanies are highly personal and can’t be engineered. But I think perhaps we can make them more likely by putting certain things in place…
Firstly, the connections between the ideas have to be there all along, just like the sun-shapes in Rapunzel’s paintings. If they weren’t already there, the realisation wouldn’t have been so powerful. We need to make sure that there are patterns in what we do and say from the very beginning.
Secondly, the connections have to be there many times — so many times that once they have been noticed you wonder why you didn’t notice it before. It gives a huge sense of sureness to the realisation that you have, so you don’t just discount it as your imagination. So in our examples and explanations, we need to repeat and repeat the same pattern over and over and over.
Finally, there needs to be an event to start it off, something to help you notice that first connection. Just like Rapunzel’s cloth — she needed the shape to be marked out simply so she could notice it in her own work. So we need to stop and point out the pattern in what we’ve said every so often, and get the students to do activities that hold the patterns up close to each other so they can notice.
I think keeping these things in mind as we choose what examples to show our students, and choose how to present them, and choose what activities to get them to do, may just make it possible to help them have an epiphany like Rapunzel’s.