Making Your Own Sense

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

Tag: books

  • More wisdom from the Dodecahedron

    After a long hiatus, I am dusting off my blog and I’ve moved it here to a new home. While I was going through the process of transferring everything here, I re-read the very first post I ever wrote, called Wisdom from the Dodecahedron. And I also found my drawing of the Dodecahedron that the original banner on the very old blog site was based on.

    A net for a regular dodecahedron, with twelve pentagons joined together in two flower-like arrangements. Each pentagon has a face drawn on it, each showing a different emotion, including happy, sad, surprised, suspicious, angry, confused, wistful, and ashamed. Some edges have tabs saying "glue under" and some edges have hands and feet coming off them. There is also a little beret with tabs to help it stand when it is cut out.

    The Dodecahedron is a character from The Phantom Tollbooth by Norton Juster, and he lives in Digitopolis, the city of numbers. Many people prize mathematics for its cool logic and want to hold it up as emotionless as if that somehow makes it better. But here is the first person we meet in Digitopolis and he has on display twelve emotions!

    And that appeals to me a lot, because I find that maths is full of emotion. Frustration, curiosity, surprise, satisfaction, pride, sadness, companionship, wonder, silliness, joy — they’re all there, sometimes in quick succession. Talk to any mathematician about their work and those emotional words are guaranteed to spill out. We’re human, and humans feel emotions, and humans do maths in a human way, which is an emotional way.

    The Dodecahedron reminds me to feel my emotions as I do maths, and to make space for others to do the same.

  • Quadric Cameo

    As I said recently, quadrics hold a special place in my heart and I get excited every time the topic comes around in Maths 1B. Quadrics have so many cool things you can say about them, and are such a great opportunity to talk about the deep connection between algebra and geometry. I personally could teach an entire 12 week course on nothing else. But paradoxically, this is also why I feel a sense of frustration every time the topic comes around in Maths 1B.

    You see, quadrics have such a short appearance in Maths 1B – just long enough to be an application of orthogonal diagonalisation, but not long enough to get anything close to understanding them. To have even a slight feeling that you understand quadrics, you actually need to spend a bit of time near them investigating how they behave, but there’s barely long enough to learn half their names (indeed, I’m told that this semester they’re not going to be expected to know their names for the exam at all).

    Some people might say this isn’t a problem because the students can just learn the bits they’re told and leave it at that. But the students have been told so many times that it’s better to understand things than just do them, and have failed in the past when they took a surface approach to maths, so they try to understand things if they can. Plus, many of them can see that quadrics might actually be quite interesting and they want to understand.

    Normally I would rejoice that students had a desire to understand something I love, but in this case, when quadrics are such a tiny bit of their course, it stops them putting energy into other, more mark-worthy bits of the course. Like the Dufflepuds in the Voyage of the Dawn Treader movie, I would prefer that either we did them the justice they deserve or left them out entirely.

  • Moses loved numbers

    Many traditions hold that Moses wrote the first five books of the Bible. If we assume this is true, then there is one thing I think is clear about Moses, based on the things he wrote: he loved numbers. I’m pretty sure he was a mathematician at heart, or at the very least an accountant, because his books are littered with numbers which are not entirely necessary to get his overall point across.

    Just look at this passage from Genesis (NIV):

    When Adam had lived 130 years, he had a son and he named him Seth. Afer Seth was born, Adam lived 800 years and had other sons and daughters. Altogether Adam lived 930 years, and then he died.

    When Seth had lived 105 years, he became the father of Enosh. And after he became the father of Enosh, Seth lived 807 years and had other sons and daughters. Altogether Enosh lived 912 years, and then he died.

    When Enosh had lived 90 years, he became the father of Kenan. And after he became the father of Kenan, Enosh lived 815 years and had other sons and daughters. Altogether, Enosh lived 905 years, and then he died.

    When Kenan had lived 70 years, he became the father of Mahalalel. And after he became the father of Mahalalel, Kenan lived 840 years and had other sons and daughters. Altogether, Kenan lived 910 years, and then he died.

    When Mahalalel had lived 65 years…

    And this one:

    He spent the night there, and from what he had with him he selected a gift for his brother Esau: two hundred female goats and twenty male goats, two hundred ewes and twenty rams, thirty female camels with their young, forty cows and ten bulls, and twenty female donkeys and ten male donkeys.

    The emphasis on numbers is striking.

    Now I’m pretty sure Moses didn’t mean to place such an emphasis on numbers in his writing. Presumably his main aim was to let his readers know about the history of Israel, and the nature of God and his relationship with humankind in general and Israel in particular. But still, the numbers are there. Why?

    I argue that the reason the numbers are there is because Moses himself loved numbers. I think he couldn’t help the numbers appearing in his writing because he wasn’t even aware he was doing it. He liked numbers, so he thought about them a lot, and so they just turned up in his head when he was writing his books.

    And if it can happen to Moses, then it can happen to anyone. I know myself that I can’t help references to childrens literature turning up in my lectures, and I can’t help maths turning up in my everyday conversation, just because I love those things. And I can’t help turning every discussion about maths into a discussion about problem-solving, because I think about the process of problem-solving a lot and it just happens.

    But the danger is when the things we are interested in distract from the message we want to get across. For example, what if a teacher absolutely loved sport to such an extent that every example in class was about sport, and some of the students who disliked sport were turned off because of the association? And what if the thing a teacher most loved in the solution to a problem was the fancy trick? Then when they presented the solution they couldn’t help getting excited about the trick and it would seem to their students that fancy tricks were what problem-solving was all about.

    But what can these teachers do, since they can’t help the things they love coming through in their communication? Well I think they can simply be aware of it. Then at least they can make sure that even though the things they love are there, the overall message isn’t obscured by them. (Of course the ultimate would be to love the thing you are trying to teach!) 

  • Bathelling in assignments

    The Deeper Meaning of Liff by Douglas Adams and John Lloyd  defines the word bathel like this:

    bathel (vb.) To pretend to have read the book under discussion when in fact you’ve only seen the TV series or movie.

    I do not like to bathel, and in fact it is one of my life’s ambitions to find and read the books on which the TV series and movies I have seen – especially those I saw as a child. This ambition has inspired me to read Tom’s midnight garden, The Children of Green Knowe, Anne of Green Gables, The Hundred and One Dalmations, Babe (aka The Sheep Pig), Archer’s Goon, Jumanji, Dot and the Kangaroo, Peter Pan, The Wizard of Oz, The Last Unicorn, Halfway Across the Galaxy and Turn Left, Finders Keepers and I’m sure several others I can’t think of right now.

    I was talking about the word bathel at the AUMS barbecue yesterday, and Nicholas called me to remind me that I had lost track of time and what I should be doing was helping students in the MLC Drop-In room. So off I went to help people with their t-tests, conics and subspaces. And it occurred to me while doing this that a small number of the students I was talking to were attempting to bathel about their coursework.

    These few students were attempting to use information they’d been told in their lectures to talk knowledgeably about a problem, without having tried to organise and connect the ideas first. They hadn’t sat down with their notes and some problems and tried to grapple with how these ideas can be applied. They had only seen the movie and not taken the time to read the book.

    (I should say at most students I talk to have a very positive attitude and do try to think through their course content deeply, using the MLC to help them learn to do this thinking!)

    A movie presents the ideas in a book most pertinent to the film-makers’ intepretation of the overall theme. And it does so in a small window of time without any pauses or breaks for thought. On the other hand, when you read a book, you can savour a particular page for quite some time, and flick back and forth as you read to check something you might have missed. And you can think about what the book means to you in the gaps between reading sessions.

    In the same way, the lecturer presents the ideas of a particular area of learning most pertinent to the overall themes of the course. And you don’t get the chance in the lecture to think through what it means to you and how these ideas are connected. To really understand you need to sit and savour it like you do when reading a book.

    I’d like to hope that I can encourage students to take the time to savour it, but if not, I’d at least like to teach them that bathelling is not the best way to go. Lecturers are pretty good at spotting people bathelling on assignments!

  • The shoemaker and Dobby

    Do you know the story of the Shoemaker and the Elves? Well, I’ve known it since I was very young. It’s a Brothers Grimm, and it goes something like this:

    A poor shoemaker is down on his luck and can’t make enough to feed himself and his wife. All he has left is enough leather for one pair of shoes and he works late into the night preparing the leather but falls asleep at the workbench. In the morning the shoes are all made with such fine and perfect workmanship that they are snaffled up quickly by the next person to pass the shop window.

    The shoemaker of course buys some more leather and gets it ready and tries the trick again. And again all the leather is sewn into wonderfully well-made shoes. Soon he and his wife are very well off.

    Eventually they decide to ask the question of how this is happening, and they hide themselves so they can see who is making the shoes. As it turns out, it’s a team of little elves, who are all completely naked.

    The shoemaker and his wife feel sorry for the little elves who have helped them so much and decide to make clothes for them, which they leave out the next night. The elves are so delighted with their clothes that they declare they don’t need to work all night anymore and dance away into the night.

    Now those of you who have read Harry Potter may recall a character called Dobby – a house elf, who had to remain in servitude until such time as his master presented him with clothes…

    Just a moment! Doesn’t that sound familiar? Of course it does. It’s right out of the Shoemaker and the Elves!

    For no reason that I can see, I suddenly came to this realisation this weekend. Jo Rowling rose again in my estimation as being a very clever woman. And I sank just a little in my estimation because I knew this story from when I was very young – why on earth did I not see this connection earlier?

    Still, it’s not worth kicking myself over it – this sort of thing happens all the time with learning maths. Students say to me all the time: “I just realised these things were connected! I never knew I didn’t understand how this worked until I suddenly understood how it REALLY worked!”

    It’s nice for the feeling to happen to me for a change

  • Books in the 22nd Century

    I’ve just read a book called “Written for Children” by John Rowe Townsend. It was published in 1974 and gives the history of writing for children (in English) up to that time. It was very interesting reading. What I’d like to comment on here is the final chapter, where the author talks about the future of books (p333 onwards):

    The question that arises next is whether changes in the book world might be overtaken by technological developments which would make the book itself, or at any rate reading for pleasure, obsolete. … Myself, I have an instinctive faith in the ability of the book to keep going. It is a tough old bird, after all. People thought that the cinema and radio and television would kill it, but they have not done so yet. Perhaps it is not too wildly optimistic to hope that in the twenty-first century, when all the modern miracles and some we have not yet dreamed of have come to pass, a child will still be found here and there, lying face down on the hearthrug or whatever may be then have replaced the hearthrug, light years away from his surroundings, lost in the pages of a book.

    It makes me happy to know that Mr Townsend’s vision did in fact come to pass and that children can still be found lost in a book even here in the twenty-first century. And it gives me hope that in the same way that the book was not killed by cinema or radio or television, that it will also survive the internet and the ipad.

    And finally it makes me think of a parallel situation in mathematics. I have heard people say that the computer is forever changing the way mathematics is done. This is definitely true, but I don’t believe that the “old ways” will die. I believe that there is a certain joy that comes from doing something yourself, from scratching out a problem yourself on paper, from playing with symbols and pictures, from visualising things in your own mind, from dreaming about new ideas – a joy that is absent when the computer does things for you. So I hope that even in the 22nd century you’ll still see people sitting down with a pencil and paper scribbling as they try joyfully to solve a problem all on their own.

  • Pushing your own “Dawn Treader”

    I went to see the new movie version of The Voyage of the Dawn Treader on my birthday and I was sorely disappointed. I liked The Lion the Witch and the Wardrobe and I was a little disappointed with Prince Caspian – if the pattern continues I wonder what depths of disappointment I might sink to if they ever make The Silver Chair.

    But my disappointment itself is quite another matter – the point of this post is why I’m disappointed: I’m disappointed because they removed most of the wonder and innocence of the book. They decided their version would be more exciting or interesting or mysterious, but in the end they just made it less than it was before. Many of the scenes of the book with the most meaning and feeling for me were cheapened by the film-maker’s agenda. And as to some of my well-loved characters – like the Dufflepuds – the film-makers thought that merely including them somewhere was the important thing, but I think it would have been better not to include them at all rather than the half-hearted cameo they got.

    *sigh*

    But what has this rant got to do with Maths or Learning or living in the Maths Learning Centre? Well here’s my thought – how often do we, with our students, do the same thing to the subjects we teach?

    In order to serve some agenda – say the usefulness of a particular bit of maths to some obscure application – we remove the wonder and innocence of our subject. The joy of discovery and learning, and the meaninfulness that comes from an encounter with something rich and wonderful are lost in the agenda we must keep.

    And we keep some things in our courses in order to appease the people who say it must be there, but it would be better to leave it out entirely than leave behind the shameful passing wave they have – like “informal induction” in the current South Australian year 12 maths curriculum.

    But what are we to do?

    If we have the choice to change our curriculum we are like the writers of the film – we have a responsibility to capture the wonder of the original and not push our own agenda.

    If we must teach the curriculum we have, then we are like the actors in the film who have to do the best we can with the script they’re given, even if it is only a hollow copy of the original story. We can put as much emotion and feeling as we can into our delivery.

    And if we are neither and like me aren’t directly involved in the curriculum at all, what then? Well the film did remind me of the book, and made me think of all those things I did love about it (even if they weren’t in the film), and it will probably inspire many who have never read the book to read it now. In the same way, maybe I can work with those students who are involved. I can tell them about the story I remember that is similar but so much better than the version they have seen; I can encourage students to have their own moments of wonder; and I can encourage them to investigate the half-hearted cameos further and gain a true love for the characters they missed out on the first time.

  • Wisdom from the Dodecahedron

    The Dodecahedron is a character from the book The Phantom Tollbooth by Norton Juster. He lives in the city of Digitopolis at the base of the Mountains of Ignorance. Here is his description from the book (page 145)

    He was constructed (for that’s really the only way to describe him) of a large assortment of lines and angles connected togehter into one solid many-sided shape – somewhat like a cube that’s had all its corners cut off and then had all its corners cut off again. Each of the edges was neatly labelled with a small letter, and each of the angles with a large one. He wore a handsome beret on top, and peering intently from one of his several surfaces was a very serious face. Perhaps if you look at the picture you’ll know what I mean.

    Now we can learn a lot from what the Dodecahedron says. Look at this exchange (page 148):

    “I’m not very good at problems,” admitted Milo.
    “What a shame,” sighed the Dodecahedron. “They’re so very useful. Why, did you know that if a beaver two feet long with a tail a foot and a half long can build a dam twelve feet high and six feet wide in two days, all you would need to build the Kariba Dam is a beaher sixty-eight feet long with a fifty-one foot tail?”
    “Where would you find a beaver as big as that?” grumbled the Humbug as his pencil point snapped.”I’m sure I don’t know,” he replied, “but if you did, you’d certainly know what to do with him.”
    “That’s absurd,” objected Milo, whose head was spinning from all the numbers and questions.
    “That may be true,” he acknoledged, “but it’s completely accurate, and as long as the answer is right, who cares if the question is wrong? If you want sense, you’ll have to make it yourself.”

    To me, this encapsulates a lot about how mathematicians think: to a mathematician, it’s the problem that’s the interesting thing, not the usefulness. In fact, we even define usefulness differently – note how the Dodecahedron uses his beaver example to show the usefulness of problems. Clearly this is not the same as the Humbug’s definition of usefulness. I rather suspect that to the Dodecahedron, it’s useful because it highlights how maths can solve problems – whether the answer is realistic or useful is a side issue.

    Still, Milo clearly doesn’t get it, but I’m not sure it’s the maths itself that’s the problem – it’s the mathematician: Milo and the Dodecahedron think differently about what is useful. Maybe as teachers, we should help the students understand mathematicians a little more, as opposed to just understanding mathematics.