Making Your Own Sense

Reflections on maths, learning and maths learning support, by David K Butler

Tag: play

  • Pure play

    The other day I did a workshop with students from Advanced Mathematical Economics III, which is more or less a pure maths course for economics students. It covers such things as mathematical logic, analysis and topology – all a bit intimidating for students who started out the degree with almost no mathematical background!

    We had just spent an hour looking at relations: the definition as a subset of the set of pairs, the common usage as statements in natural language, various ways to visualise them, examples and non-examples of various properties relations might have, and proofs involving those properties. After all this point, one of the students said, with some surprise, “It really feels like it’s just playing around here.”

    The student was right, of course, and I told him so! Pure maths is play. You have some ideas and you fiddle around with them to see how they fit together, how they’re similar or different, what other things are really the same things in disguise. Most of the time there’s no particular goal in mind, but even when there is, you often get distracted by something cool and end up somewhere totally different. This is exactly what play is. Watch any child in a sandpit and you’ll see the same behaviour.

    It suddenly occurred to me that I have always seem maths as play, and have liked it the least when it looked like work. More than this, the maths courses I have had the most academic success in were the ones where I allowed myself to see it as play. These were the courses where I took the concepts I was learning and pulled them apart and put them together in new ways, where I tried to do things that may or may not be possible just to give it a go, and where I drew lots of pictures in vivid colour just because. And importantly, where I didn’t question what the point of any of it was but just ran headlong into whatever crazy idea the lecturer presented next to see what happened with it.

    So perhaps the best way to approach a pure maths course is to see it as play! Surrender to “just playing around” and see where it goes. That’s the advice I gave to these students, and I’m hoping it helps them to have the freedom to learn.

  • Jack Frost’s Centre

    On the weekend I watched the film “Rise of the Guardians” by Dreamworks Pictures, and it is a very enjoyable film. In it, Jack Frost is enlisted by the Man in the Moon to join the Guardians of Childhood—who already have Santa Claus, the Easter Bunny, the Sandman and the Tooth Fairy in their ranks—and together they fight the evil Pitch Black, who is the Bogeyman.

    There is a scene in the film, where Santa Claus (called “North” by the other Guardians) gives some advice to the new recruit Jack. He tells Jack that he needs to find his centre – that part of him that guides how he will guard the children of the world against fear.

    Spoiler alert! Jack’s centre is fun. His great epiphany is when he finds himself saying the phrase: “Don’t be afraid, we’re just going to have a bit of fun instead.” He realises that for hundreds of years he has been warding off the darkness of winter by helping children have fun in the snow. Because they are having fun, they forget to be afraid.

    And then it was Monday. And not just any Monday either! It was the first day of O’Week – one of the busiest days in the year for the MLC. From 8am it was printing handouts and setting up the drop-in-centre and visiting preliminary lectures and playing Numbers and Letters in the Hub Central welcome centre.

    This year we had PASS leaders helping us out in the welcome centre and I must say they were a great bunch of people who leaped into the role of seeking out students with cheerful fervour. Georgia in particular was super enthusiastic about getting people involved in the games, and indeed playing the games herself.

    Interestingly, Georgia started the day saying she wasn’t all that good at calculations, and stood back letting others do it. But I persevered with her, encouraging her to have a go, asking for any ideas, writing down her slightly wrong solutions so we could work together to tweak them to make them work. By the end of the day she was our greatest champion for the numbers game, itching to rub it out and start a new one every time we drew the last stroke of the previous solution.

    That night when thinking about this MLC success story, I suddently made the connection to “Rise of the Guardians”. Georgia had forgotten to be afraid because she was having fun. She had forgotten to be afraid of calculations, forgotten to be afraid of failing, forgotten to be afraid of looking stupid in public – all because what we were doing was playing a game.

    Of course this is precisely why we have a mathematical art and play program at the MLC, but I had never so clearly realised the power that it actually had to change someone’s perspective.

    And I never realised how much of the time that Jack Frost’s centre is mine too. Any number of times a day, I will do something to distract a student from their fear so that they can get on with learning the maths they need to learn: wear a maths-themed t-shirt, pull out the play dough, draw their diagram in texta or crayon, ask about their lecturer’s lovable quirks, or just simply yell out “That’s so COOL!”. In all of this I am saying to the student, “Don’t be afraid of the maths, let’s just have a bit of fun instead.”

  • Too much time on his hands

    On the train a while ago I overhead some people talking about Heston (the celebrity chef). Apparently he had been doing a series on giant food. It involves him trying to figure out the physics and logistics of trying to produce food on a giant scale – for example, a three-metre tall soft-serve ice-cream cone.

    After describing all the care and effort Heston took to produce this giant ice-cream, the first person declared, “He’s very clever.” Her friend’s response was, “He has too much time on his hands.”

    Clearly this person could easily do a better job than Heston if she wanted to, but she chooses not to because she has so many more important things to do with her time. Apparently Heston’s not clever, he’s just idle.

    I had a strong desire to lean over and ask her if JK Rowling had too much time on her hands, or if Stephen Spielberg had too much time on his hands, or Michaelangelo had too much time on his hands. If you think about it, what they did was more or less for their own enjoyment too and wasn’t “important” either.

    Of course, it wasn’t Heston I was really indignant about. The statement brought up several unpleasant memories when people have said this to my face when they have seen me making models of fractals, crocheting hyperbolic coral, drawing digits of pi on the pavement or solving puzzles, or even just doing maths in my own time. They seemed to feel that they needed to make those things seem trivial.

    Perhaps they felt cheated that they don’t spend more time doing things they actually enjoy. Perhaps they felt like I was making them look stupid and they needed to make me feel bad for it. Or perhaps they are just grumps who are unable to share in others’ fun.

    Now that my indignation has faded a little, I feel sorry for them. I remember what it was like to be in a situation where I felt it was somehow wrong to choose to do things I enjoyed, and it wasn’t a pleasant place to be. It can colour your view of the world and frankly it does make it difficult to enjoy other people’s fun.

    Still, it’s no real excuse for making people feel bad about things they have spent a lot of time acheiving. Sure, they may have a lot of time on their hands, but at least they are using it well!

  • Kindy is awesome

    My younger daughter started kindy last week, and I got to actually be there for the beginning of her first day. It was one of those moments only a parent can understand as I realised with both excitement and sadness that my little baby was not a baby any more.

    But this is a maths learning blog, so as much as the above point really is all that needs to be said, I will make just one more: kindy is awesome!

    Of course I already knew it was awesome – my wife is a kindy teacher and director and anything she is involved with is definitely awesome – but the true awesomeness of it was brought home to me once more as I stood there in my daughter’s own kindy.

    I looked around, and everywhere I looked was something specifically designed for learning, and learning was actually happening there.

    The activities chosen allowed each child to choose to learn in their own way: Some activities were quiet and some were loud; some required social interaction and some were individual; some involved running, some hand-eye coordination, and some sitting still; some needed deep thought and some creativity – and every activity was encouraging learning.

    The free-form structure allowed each child to choose what to learn at their own pace: children decided what to do as the whim took them, and didn’t need to wait for anyone else to tell them it was ok, and didn’t need to wait for everyone to be ready. They just started playing, and therefore just started learning straight away.

    The staff were in the thick of it helping learning to happen: they moved from one place to the next, talking to the children and turning ordinary moments into teaching moments. It was clear that at some point they would move to the “group time” areas and help the students draw together the ideas they had encountered in their play. And finally, they were also constantly setting up and re-setting up the environment so that new children could keep learning there.

    I was in total awe of the whole thing, and it made me wish again that we could make uni more like kindy – a place where the staff choose learning environments and materials that students could interact with in their own way and still learn, but where staff are always ready to help make learning happen.

    I have been accused in the past of making the Maths Learning Centre too kindy-like. But in my mind, I could never make it too much like kindy.

    Because kindy is awesome.

  • The Road to Royalty

    Last week, I met His Royal Highness Edward the Duke of Kent. I’d like to tell you the story of how this came about.

    His Royal Higness was in Adelaide because he is the patron of the Royal Institution of Australia and was presenting an award to a scientist there. But it just so happened that the Hyperbolic Crochet Coral Reef exhibition was still on display at the Royal Institution during his visit, so he dropped in for half an hour to see it.

    “How nice for him,” you might say, “but how did you get to be there while he dropped in?”

    Good question! Well, less than a week before, the curator of the exhibition Julie had emailed me to ask if I would like to attend afternoon tea with His Royal Highness in the Reef exhibition. She said that I was one of the very few men who was involved in crocheting the reef, and moreover she would greatly appreciate the knowledge I would bring about the mathematical aspects of the exhibition. After making sure the students in the Drop-In Centre would be looked after, I said yes – and that’s how I got to be there.

    “But David,” you might continue, “how did you get involved in crocheting the reef in the first place?”

    Another good question! Well, let’s see… another RiAUS staff member Cobi who I happened to know dropped in to my office in February asking me if I knew anything about hyperbolic geometry, and also if I happened to have any 3D models that might be used as part of an art exhibition. And of course I knew about hyperbolic geometry, and I had a couple of ready-made models of hyperbolic quadrics rigt there.

    That was the start of a year-long involvement in the project. During the course of the year I wrote a poster for the exhibiton to explain the geometry, learned to crochet, crocheted about forty corals, and ran three crochet coral workshops. I was probably the most involved of the very few men who were involved.

    “Wow! You really committed yourself to this, David,” you are probably saying. “But how did you happen to know Cobi?”

    Ah! Yet another good question! Well Cobi had once worked as a tutor for a course called Research Methods in Media at the University. The very first tute was supposed to be refreshing the students’ memories of statistics, and she had come to me as the coordinator of the Maths Learning Service for some help running that tute. I jumped at the chance and we had a great time getting the students to draw graphical representations of data on the windows. It’s one of my first memories of working at the Maths Learning Service.

    “Cool!” I’m sure you are saying. “But why did she think you would know about geometry and have 3D models?”

    You are full of the great questions today! Well if you know me at all you’d know that I can’t help being excited about maths in general and geometry in particular, and also that I also have a collection of models of geometries that I pull out at every opportunity. So it shouldn’t really be a surprise that I did talk excitedly to Cobi about geometry, and show her the models when we met, even though my task was to help teach statistics.

    “But,” you must certainly say, “that doesn’t explain why you happened to have the models of quadrics there.”

    That is an excellent point. Well, I’ve always been a model-maker (I remember making things even in primary school). So it was a natural thing for me to try to make models of the quadrics I was studying when I came back to Uni to do my PhD. Towards the end of my first year of PhD I spent a few weeks making paper-mache models of quadrics, constructing the underlying structure with carboard and string. I have fond memories of sitting in the School of Maths tea room with my hands covered in maper-mache glue and paper and cardboard all over the table.

    Later, towards the end of my PhD, I ended up on a team designing interactive activities for open-day and the string quadrics seemed like a reasonable thing to get passers-by to engage in. When I went to the Maths Learning Service, I took the string models with me.

    Now you probably have more questions at this stage, but our conversation has been going on for some time now, and I think I’d better make some sort of point, don’t you?

    If you trace the story back, you’ll find that there are two reasons I ended up meeting His Royal Highness. The first is that I never shyed away from doing things that would be considered play – things like paper-mache and crochet. Other people were too busy or too embarassed to do this sort of thing, but that never stopped me. The second is that I was always willing to share my love of maths – with Cobi, with Cobi’s students, with the passers-by on Open Day, with the visitors to the reef, and finally with His Royal Highness himself.

    If you want to learn anything from this, then learn what I did: never shy from playing, and never give up sharing the things you love with others. You really never know what good may come of it if you do.

  • The Pied Mathematician of Hamelin

    Have you ever been in a situation and felt like you were reliving a scene from a book or movie? Well it happened to me the other day when I went to visit my daughter’s school. I felt exactly like I was the piper in the Pied Piper of Hamelin, because an ever-growing crowd of children followed me across the oval as I walked in.

    And why were they following me? Well I had brought a Stage 4 model of the Sierpinski Sponge with me for my daughter’s Show and Tell. She had come with me to university the day before to help me make Stage 6, and we thought the other kids would be interested – and my goodness they were!

    At this point I should probably tell you more about the Sierpinksi Sponge Project…

    The Sierpinski Sponge is a fractal constructed in the shape of a triangular pyramid. It has a giant hole in the centre, and each of the corners around this hole is a copy of the whole thing. This means that each corner is a pyramid with a big hole in the middle, and each corner of those pyramids is a pyramid with a hole in the middle, and each … and you go on like this forever until you have an object with a great number of holes (infinitely many in fact) – hence the name “sponge”.

    Of course, you can’t make a real Sierpinksi Sponge because it goes inwards forever; you can make a decent model though. What you do is you get four small pyramids and you join them together at the corners to get a bigger one with a hole in the middle. Then you take four of these bigger pyramids and you join them together to get a bigger one with a hole in the middle. And so on. The individual small pyramids are called Stage 0, then when you join four together you get to Stage 1, and then Stage 2, and so on. We made a Stage 6, which contained 4096 individual pyramids.

    (To see video of the Stage 6 Sponge and us making it, check out the YouTube videos: http://youtu.be/A7YbmITSck8  , http://youtu.be/W0uLbhRR-Hw  .)

    So back to the story… on the day after we made Stage 6, I walked through the school yard with a Stage 4 Seirpinski Sponge, and all the kids crowded around to see this remarkable thing and ask questions. And then in the classroom, the kids just couldn’t keep quiet with the questions and fought over who would be first to hold the smaller models. Their little eyes lit up as they imagined standing inside Stage 6, and imagined the awesomeness of Stage 7, Stage 8 and Stage 100.

    But it did make me think: We made our Stage 6 in a public place in the Uni where hundreds of people walk past a day. Several people looked at it as they walked past, and a few came close to touch and ask questions, and a handful of those actually stopped to help make it bigger. Before construction day, I had spent every train journey sticking pyramids together, usually reaching Stage 4 by the time I got to my destination. On one particular day I finished a Stage 5 on the train and carried the metre-wide pyramid it through the crowds. Not one person stopped to look or ask about it. Yet the schoolchildren couldn’t keep themselves away.

    When did all the adults lose their sense of wonder? Because even the the Stage 4 Sponge is truly wonderful to me (and to hordes of children too).

    What would it take for adults to crowd around like the children in the Pied Piper of Hamelin?