Making Your Own Sense

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

Tag: life

  • Sleeping through Miss Marple

    My wife and I like to watch mystery shows together like Poirot, Midsomer Murders and Miss Marple. Unfortunately I have a slight problem: when watching television in a comfortable position, I tend to drift in and out of sleep, no matter how interesting the show might be. This can be quite disasterous for mystery shows, especially ones with major unexpected plot twists.

    Just yesterday we were watching an episode of Marple called The Pale Horse and I woke up from a doze at the scene where everyone was gathered in the dining room to reveal the killer. And I had not the slightest clue what was going on.

    Later I went back to see the bits I had missed and it turned out I had missed a total of about three minutes of viewing in small snippets, but these were precisely the key moments I needed to be able to follow that final revelation.

    This morning it occurs to me that some students I have helped in the MLC have been in a similar position with their maths courses. A particularly common example is in Maths 1A too with the word “span”. It is mentioned in passing in one of the early Algebra lectures, with the discussion lasting for a total of about a minute, and it doesn’t seem related to the content of the course at the time. But then later it becomes one of the most important ideas and is talked about as if they already know what it is. If they “slept through” the first mention, they’d be most confused! It happens in our own bridging course too, with the idea of a “unit vector”. It’s mentioned on precisely two pages in our course materials, and is very easy to miss. Students almost always completely ignore the word in their assignment and then struggle to get what we are asking them to do. (This is one of the reasons we hope to rewrite the resources in future.)

    As teachers, we need to remember that a maths course is not like a murder mystery. In a murder mystery it’s the fact that working out the case hinges on small details that makes it mysterious and fun. But maths courses don’t hinge upon small details, they hinge upon big ideas. We need to make sure that anything pivotal is mentioned several times and discussed deeply so that even if their attention wanders for a minute once, they can still pick it up again and still follow the story.

  • 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.”

  • The advent calendar function

    In a previous post I discussed how we need ways to think about functions that are not curves on an x-y-plane. Well I have a seasonally-appropriate one for you: the Advent Calendar.

    The advent calendar I have in mind is the kind where there is a little cardboard door for each day in December up to Christmas, and behind each door is a little chocolate. (Yes I know it might just have a picture, but seriously, the chocolate ones are better, right?)

    An advent calendar. Each window has a picture of a cat on it. A hand is holding one of the windows open, showing an empty space where a chocolate used to be.

    Most of the time when we’re not drawing graphs, we talk about a function as a sort of machine, which takes some sort of input and produces some sort of output. This is a very dynamic view of function which I like very much. I imagine putting an object (usually a number) in a funnel at the top and the machine churns and whirs and gurgles until a new object (usually another number) shoots out of a chute at the bottom.

    But a function doesn’t necessarily have this time element. The set-theory definition of function is simply a correspondence between one set and another, so that every object in a domain has associated with it one object in a codomain. (The domain is the set of things we usually call “inputs” and the codomain is the set of things from which we choose the “outputs”.) In this sense the “output” is there all the time whether we calculate it or not.

    This is where the advent calendar comes in. The chocolate is there whether you open the door or not. Opening the door to see for yourself what shape the chocolate is corresponds to what we do when we calculate the value of the function. But fundamentally all the numbers in the domain have a value for the function before you calculate it, just like all the unopened doors have a chocolate.

    I find this particular picture works well for vector calculus, where every point in a plane or in space has a function value, which may be a number (in the case of a “scalar field”) or a vector (in the case of a “vector field”). In the vector field case, the vector you find doesn’t really interact with the ordinary points, and indeed the vector “output” at one point doesn’t really interact with the ones at the other points. It’s almost as if at every point there is a little room where the vector lives all by itself. All we need is a little door to let us into this little room…

  • Jamie Oliver’s teaching lesson

    This is a guest post by MLC lecturer Nicholas Crouch

    So it has to be said that I do like what Jamie Oliver does. I have always liked watching his shows and some of the messages that he aims to get across to the community are ones that I believe in. However when it comes to following his recipes and repeating what he does on television, well I figured it was like any other cooking show: the steps are there, but it required being a competent cook to begin with before you would get anything edible from doing what he said to do.

    With this in mind when I saw his show on how to make pasta, I was keen to give it a try as I have always loved the idea of making my own pasta. In fact I had given it a shot in the past. My pasta took a very long time and really turned out rather ordinary (and for 3 hours of cooking there was about 5 mins of eating).

    This time, however I listened carefully to what Jamie told me, created myself a mental list and believed that he was not going to lead me astray (and when I listened carefully, he did say pretty much don’t bother trying unless you have a pasta machine, which might be a lot of the problem with my first attempt). I followed his instructions and in very little time at all I had beautiful pasta which those who tried it all commented on in a very positive way. So why was this attempt so much more successful than the first?

    Well in thinking about that question I felt there could be some lessons that all who aim to pass on their knowledge could learn from. Jamie’s shows could be considered like a lecture, where he can talk about a topic, but there is no opportunity to give feedback on what the student has done. What did he do that was worthy of note?

    Firstly he was clear about what exactly to do. Most cooking shows are, but often get distracted by other less important details. He told me just what to do.

    Next he sign-posted. “If this is the situation you find yourself in at this point, this is your problem, and this is the solution.” Or “Do this until these conditions are met”. For example, I think he mentioned that if your pasta is too dry you can add more water to make it softer (but not too much). However, do not over complicate it with all possible cases, just the important ones. He also gave some explanation as to why we would do something in that way.

    Lastly he inspired me to believe that I could do exactly what he had done and achieve a similar result.

    So what are the lessons about teaching?

    Firstly, be clear, don’t overcomplicate something. You probably know a lot more than is required about your topic. It can be very difficult to cast your mind back to a time when you didn’t know that topic extremely well. However this is what you need to do to distil the essences of what you are trying to teach.

    Secondly, signpost (OK I should know the proper pedagogical term for this but I don’t). By this I mean, talk about the reasons for making the decisions that you have, not just the steps that need to be taken! For a lot of what I do, this comes down to trying to state the connections between what is being asked and what I am doing. Even to the point of listing my options and then saying why I would chose this option.

    As for inspiration, well that can be hard to give people advice on. I personally sound excited when talking about certain topics, and it is infectious. Other things that work are giving your topic some historical significance, or talk about how it is used.

    The last part is keeping the goal in mind. In Jamie’s case, we want to eat what we are working on. In the case of teaching we want the student to have a greater understanding. So look back at what you have done. (I know about this term – the cognitive closure) But this also can be a time where we talk to students about how they could re-order the concepts for themselves so that they don’t just have the linear connections in their head. The way the material has been presented to them, we want them to do more than that with it. So perhaps connect the dots for them, even mind map it for them if you have to!

    So next time you are in front of a class, sound excited and have pasta!

    If you are the student in the class, create your mental list and have confidence in your instructions!

    Thanks for your time

    Nicholas Crouch

  • 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!

  • Elsa’s Freedom

    Disney’s Frozen came out on DVD last week and my family and I watched it on Saturday. It’s a very good movie with an excellent theme about the real nature of true love which is not usually seen in a “princess movie”. There are also two different stories about freedom, which is pretty common in a princess story (consider Rapunzel in Tangled and Jasmine in Aladdin). It’s this I want to talk about today.

    In case you don’t know the story of the film it starts something like this: Princess Elsa is born with the ability to create snow and ice, and while playing with her younger sister Anna there is an accident and Anna is blasted in the eyes with ice. The trolls are able to cure her but warn that Elsa will need to learn to control her power. The King and Queen take this to mean she needs to try not to let her power out at all and so Elsa spends much of her time locked in her bedroom, trying not to feel. Well this all comes to a head at Elsa’s coronation where it all gets too much for her and she lets it out spectacularly and runs off into the mountains. Cue the Oscar-award-winning song Let it go.

    The song is all about how, now that everyone knows about her powers, Elsa is free to let it go and really see what she is capable of. This is where I want to draw my parallel to maths.

    I have been a mathematician from a very young age – my mother says at the age of five I was playing number and letter games in my head. But it wasn’t too long before I realised that a love of maths attracted resentment and even hostility in others. I soon learned that the only way to enjoy maths was in my head, all by myself. I had to work hard to turn off my natural instinct to see the mathematical in things when I was with others for fear of getting the wrong sort of attention.

    In my first year at university, the people I spent time with expressed resentment at being forced to learn useless and boring maths, so even then I felt I had to hide. It wasn’t until my second year at university that I found others who shared a similar joy in maths and was fully free to be myself with them and “let it go”.

    Strangely, I had to go into hiding again as a maths teacher in a high school. Despite the fact that my job was to teach maths, I was looked upon as strange by the other teachers for my love of it. And worse, the many and burdensome responsibilities of day-to-day work in a school meant that I did not have time to play with the maths I was teaching and I struggled having to hold it all in. Coming back to University was like Elsa escaping to the mountains – I felt a great relief that I could play freely.

    Of course, this story is a little sad because there are still throngs of people who don’t share my love of maths and I shouldn’t have to feel trapped when I am with them. Well I’m happy to say I learned a long time ago that it’s not worth worrying about what others think of me, and I am actually free to be myself no matter who I am with. As long as I’m not getting in the way of their joy, I should be able to express mine. (Incidentally this allows me to publicly express my love of other strange things like childrens fiction and singing.)

    The key point I want to make is that it was being part of a community where it was ok to love maths that helped me realise this, and I found that community here at University. There must be scores of students with a similar experience to mine – those who have a love they’ve had to hide and who discover the freedom to express it here at University (and not just maths either). This is much better than Elsa’s so-called freedom, where she still had to be alone to express herself.

    I dare say, however, there are many other students who still don’t find that freedom here because for whatever reason they don’t connect with the right community. Like me in first year, they still feel like they have to hide their love. This is one of the reasons we do public puzzles and sculpture, and talk excitedly in the Drop-In Centre about maths beyond the curriculum. Who knows who might find the freedom to let it go?

  • Life Impact

    The University’s current slogan is “Seek Light”, but the one we used to have before that was “Life Impact”. I have decided that at least for myself I would like to keep the old one, because recent events have shown me its true meaning.

    My mother-in-law Merle died almost two weeks ago, and last week was the funeral and memorial service. She was not just my mother-in-law, but my second mother, and life will never be the same without her. The main reason I feel this way is because my life was already never the same because of her. And not just for me: more than three hundred people came to her memorial service and Dad has been receiving twenty to thirty cards in the mail every day since the news was announced. Clearly, Mum made a huge impact on hundreds of people during her life.

    How did she do this? She wasn’t a well-known public figure famous for her musical skill, or for curing a disease, or  for changing history. She was just an ordinary woman – a wife, a mother and a nurse.

    I think the impact Mum made was at least in part due to the fact that she believed in the power of little things. She believed in doing what she promised to do, in asking people how they were and actually listening, in writing thank-you notes, in telling shop assistants they were doing a good job,  in baking cupcakes just because, in offering to do the dishes, in opening doors for people, in learning her patients names, and in simply giving a smile. I am sure that it was all these little things that made such a huge impact on so many people.

    The University used to post stories about how people’s work at the Uni has had “life impact” because of the big changes it made to the world, but Mum has shown me that the greatest impact you will ever have is on the people around you every day, and that this impact will happen through all the little things you do and say.

    I feel humbled and sometimes overwhelmed to think that I am in a position in the University that puts so many people around me every day who I will impact in this way. I only hope I can continue the legacy she has left, and remember the power of little things. I would feel truly blessed if I had even the tiniest fraction of the positive life impact Mum did.

  • Why I’m not a “lecturer”

    Every so often a student asks me why I am not a “lecturer”. Often it happens after I’ve helped them understand something from their course, or (as it did this week) after I’ve given a revision seminar on some topic from their course.

    Now I do realise that they are giving me a compliment by saying this. They are saying in their roundabout way that I did a good job of my teaching and therefore deserve to be a “real” lecturer. And I’m flattered.

    But even so, I do like to answer the question they actually asked, and it’s high time I did it publicly so everyone knows.

    The first and most banal answer to the question is that, actually, I am a lecturer – “lecturer” is my official title. I don’t need to be a “real” lecturer in the sense of giving lectures to large classes in particular courses because what I do already makes me a real lecturer – it’s a perfectly legitimate activity for an academic (though not everyone in the University agrees with this).

    The second and truest answer is that it’s much more fun teaching in the Maths Learning Centre than it is lecturing! I really love my job and I wouldn’t trade it for anything else!

    As a lecturer in the Maths Learning Centre I get to do things I have always loved doing as legitimate parts of my job. I get to do papier-mache and play-dough and puzzles, and I can be publicly excited about maths on a daily basis. And I can be publicly excited about the learning of maths on a daily basis too and discuss with others who are interested about how people learn maths and how to help them learn maths.

    And even more than these, I get to spend most of my time talking to actual students one-on-one! I get to be there at the moment they learn and see the spark in their eyes as they realise how it all fits together. I get to help them realise they actually can do it themselves, and sometimes see their tears of joy as the walls of their bad experiences with maths begin to fall down. I get to see people progress from terrified at the sight of a mathematical symbol to being able to construct a proof all by themselves to passing on their knowledge to other students.

    And that’s why I’m not a “lecturer”. How could I possibly trade all of that in to “lecture”?

  • Wrapping up integrals

    I love wrapping presents. I’d like to say it’s because of the warm glow I have inside from giving a gift to someone else – and that feeling is certainly there to an extent – but I’m sorry to say the main reason is because I like the process of wrapping presents itself.

    A rectangular box wrapped in red, gold and green Christmas wrapping

    I like putting the present on the paper and making a judgement of how much paper to cut; I like using the scissors like a knife to cut a clean edge; I like folding the edge of the paper so that it looks nice and clean when you fold it over the present; I particularly like the part where you do the fold-in-the-sides-then-fold-up bit on the sides; and most of all I like the part where it’s all finished and your present is neatly encased in a piece of paper just the right shape with all the bits folded in neatly.

    Yes, I know I’m weird.

    But I reckon I’m not that weird. My daughter at 10 years old, still likes reciting the alphabet, though she learned to do this 6 years ago. My other daughter at 5 years old, will write her name over and over and over and over, seemingly getting pleasure out of the simple act. A musician will sometimes play a song they know well, for the sheer pleasure it, and almost any person will go up to a piano and play chopsticks. Many people I know like the experience of making scrambled eggs, no matter how many times they have done it before.

    It seems that all people derive some pleasure in doing things well that they know how to do well, even though they have done it before. There is something about the repetition that gives you a sense of pleasure. Perhaps your brain likes to have the electrical signals pass down the well-worn paths where it’s not so much effort. Perhaps the experience helps you remember the buzz when you learned it for the first time.

    I think perhaps the second reason is pretty accurate because I see myself doing it all the time in my work as well: guessing eigenvalues, calculating integrals, adding fractions and drawing conics. I love them all. I jump at the chance to do them with students in the MLC because I love doing them, no matter how many times I’ve done them before. And every time I do them, I remember with pleasure the first time I figured out how to do them myself.

    But whatever the reason, I do get pleasure from doing the integral of ex cosh x or (cos x)2 or 1/(x2 – 1) – integrals I have done a hundred times – and it coming out to the answer it ought to. It’s the same pleasure I get from wrapping a present.

    Sometimes you just enjoy doing something you know how to do.

  • Birth stories in the MLC

    One of my favourite memories of the Drop-In Centre happened not too long after I started here. One of our regular visitors happened to be pregnant at the time, and as always happens when parents are in the presence of a pregnant woman, it wasn’t long before we began swapping birth stories. And not just ones from our own experience, but also the ones related to us by other parents before the births of earlier children. I won’t relate any of these birth stories here, because I don’t want to freak you out (like the way we freaked out those poor 18-year-old male students studying at the same table as us during this conversation).

    I remember thinking later how strange it is that parents naturally want to tell birth stories to pregnant mothers. And the pregnant mothers listen to these stories with interest, no matter how gruesome and frightening the story is. Why is that?

    The reason came to me during the last couple of weeks of summer semester. It came to me because large numbers of students asked me what their maths courses in second year would be like. They soaked up all the information I was willing to give even if it frightened them a bit that there would be statistics, or more proofs, or they would have to remember this infinite sum stuff. And why? Because the fear of the unknown was worse than the fear of the known.

    This is why pregnant mothers listen to birth stories: so that when their own birth experience comes, they know what might or might not happen and how to deal with it if it does – in other words, to reduce the fear of the unknown. They even listen to the stories when they have had a child already, knowing that one birth is not enough to know about all possible births that could be.

    And in a flash I realised that one of our longstanding policies in the MLC was wrong. In the past we had a general policy of not allowing the second years to stay. Of course we’d been as nice about it as we could be – telling them that the first-years need us more, that not all of the staff are experienced in their particular maths, and that they should make their lecturer work for them – but to all intents and purposes we had turned them away.

    But I realised there was a certain advantage to having them in the room with the first-years: they can tell stories from first-hand experience about what the future will be like for the first-years! If they weren’t there, it’s just us, and they only get our stories, and our stories become less relevant as time goes on, like the stories my great auntie told us about having babies in her hallway in Wallaroo 50 years ago. (Ok, I had to tell at least part of one actual birth story.)

    So we have a new policy: we’ll still tell the second years they should talk to their own lecturer, and that first-years have a greater need, but we won’t turn them away. Because they can help the first-years with their fear of the unknown.