Making Your Own Sense

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

Tag: mathematicians

  • 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.

  • The unexpected fear of statistics

    Statistics is the cause of a lot of fear. There are thousands of students studying psychology, sociology, economics, biology, medicine, animal science and education who thought they would be free of mathematics and suddenly discover they have to deal with statistics. In the case of psychology it is absolutely everywhere: both in whole courses about statistics, but also embedded in almost every other course they do. For most of these students, their fear of statistics carries over from their existing fear of mathematics, and so as sad as it is that they are afraid, it’s not wholly unexpected.

    What is unexpected is the the thousands of students who have done the highest levels of mathematics at school, and are doing the most mathematical disciplines like physics, engineering and mathematics itself, and yet somehow have a deep aversion to statistics when it appears in their degree. Indeed, my own staff at the Maths Learning Centre often express a fear at having to help people with statistics. As I often say “Mathematicians are afraid of statistics in a similar way to how other people are afraid of mathematics.”

    But why? What is it that causes this fear of statistics in those with lots of mathematical experience? I have some ideas…

    One of the biggest reasons is that statistics isn’t 100% maths. A large part of statistics is whatever discipline the statistics is being used for today. That discipline dictates the kinds of data that can be collected, how it is recorded, and most importantly how it is interpreted after the statistics is done. In a generalist statistics course this will change moment-to-moment as each new assignment question brings up a new context. In Question 1 it’s ecology, in Question 2 it’s quality control in food production, and in Question 3 it’s engineering. Each new question requires you to think about a new context and understand various subtleties about what the context means. For a professional statistician, this is often what they say is the most exciting thing about their job. They absolutely love that they get to “play in everyone else’s backyard”. They love that the same tools can be used for a large variety of different problems. However, for many a maths student, this annoys them at best and terrifies them at worst. They didn’t sign up to learn ecology/food science/engineering; they signed up for maths. They prefer to work with the numbers and word problems have made them worried from a young age. Think of how terrifying it would be to suddenly be forced to do a course where every single problem is a word problem! And spare a thought for how hard it is to do the statistics when you don’t have a clue what the context means. As I said, each new context has subtleties that impact a lot on how the statistics is applied and interpreted, and not everyone will have the general knowledge (or indeed language skills) to understand those subtleties without considerable effort. And now imagine having to go through that effort for every single assignment question. It’s exhausting!

    I’m not really sure what to do about this particular problem, other than making sure you give students space and time to talk about the contexts. Don’t treat them like idiots for not understanding contexts they have never experienced before, and definitely don’t think they don’t understand the statistics just because they don’t understand the context. Allow them the grace to have to ask what a manatee is and to ask why manatee deaths would be expected to have anything to do with powerboat registrations. I can imagine assignment questions having links to further information so they can find out more, or a quick whole-class discussion about contexts when you hand out assignments – possibly something like a numberless word problem. They might go a long way to alleviate context fatigue.

    The second reason is that statistics involves making decisions, the biggest of which is deciding what statistical procedure to do and which bits of your situation go with which parts of the procedure. With so many to choose from, and no consistent naming system for the various procedures even inside the one discipline, this is a hugely daunting task for the beginner. It all just seems like a big cloud of random stuff and the students often can’t see what it is that distinguishes between the procedures and what information is being used to decide one over the other. This is only compounded by the fact that part of the decision is made based on information that comes from the context the statistics is being used for today, which was already a problem. It’s further compounded by the fact that many who succeed in mathematics at school have done so by having a list of problem types and how to solve each one, and are not actually used to making decisions at all.

    I think a good dose of actually analysing that decision process and comparing situations that produced different decisions would go a long way to helping this, rather than leaving the decision to chance. Indeed, I’ve written before about how important it is to give students practice at the act of making the decision.

    The final reason I can think of right now is that doing statistics requires making a computer do what you want. This is a completely separate skill from understanding the context, understanding the maths and deciding what stats to do, and has a whole host of its own frustrations, not least of which is just getting access to the computer program itself or figuring out how to install it! And yet it is the gatekeeper of producing actual statistical results. Learning how to communicate with the statistics program is just one more language process that has to happen to succeed in statistics, on top of the decision-making and context-interpreting language processes I already mentioned! Added to this, for the mathematically experienced, they have spent a lot of time learning how to assert their independence from technology and rely on their own reasoning. To not be able to do something themselves and be forced to get a machine to do it for them leaves a bad taste in their mouth.

    Again I’m not too sure how to do something about this problem. Certainly you can make sure there is a lot of support available for getting the program to work, and for asking help specifically with the program. At university you could elevate it to the regular lecture time rather than leave it to practical classes that students may avoid. (Yes I know if they struggle with computer stuff they should go to computer classes, but humans are nothing if not illogical when emotions like guilt and fear are involved.)

    Now that I have written this all down, it occurs to me that these problems are a lot about language, and so this issue may be related to your high-maths-experience students avoiding language in much the same way that other students avoid maths. Perhaps the main thing we can do for them is help them process the fact that it will be about language and support them in their language, and perhaps help them realise that they have a lot more language skills than they thought they had. (Those of us who teach students at earlier stages in their lives might do well to help them realise that maths is all about language anyway!) On top of that, we can have some compassion on them because learning statistics actually is hard work.

    Megan 15 May 2018:

    I personally love statistics but I certainly know plenty of maths learners and educators who don’t. I think statistics is often mathematically simple (mostly plug and chug often with technology) but conceptually challenging. For example, even a simple p-test (covered in Year 12) is not entirely intuitive (Wouldn’t we want a large probability to reject H_0?…We reject for a large test statistic…). As a result, students who usually do well in maths are suddenly struggling not because they can’t do the maths but because they need more time to understand the context and interpret what the numbers actually mean as you say. Maybe more discussion of what is actually happening when we perform different tests could help.

    David Butler 25 May 2018:

    Thanks for the comment Megan. Yes I agree that people might struggle because it’s not the maths that’s the problem! I reckon more discussion of how the context relates to the maths might help them make the connection.

  • Book Reading: Becoming the Math(s) Teacher You Wish You’d Had

    This post is about Tracy Zager’s most excellent book, Becoming the Math Teacher You Wish You’d Had.

    (You can read this blog post and all other Book Reading posts in PDF form here. )

    I actually finished reading it back in January, and I live-tweeted my reading as I went.  The process culminated with this tweet:

    A tweet from David Butler @DavidKButlerUoA 25 Jan 2017. Text: I've just finished reading your #becomingmath book @TracyZager. This is the bit I liked. Video: Me flipping through all the pages in the entire book. https://twitter.com/DavidKButlerUoA/status/824183955012415495
    https://twitter.com/DavidKButlerUoA/status/824183955012415495 

    That’s what I thought about it at the time, but I haven’t sat down to organise my thoughts on it. Until now.

    I was first drawn to the book based entirely on its contents page. Check this out:

    • Chapter 1: Breaking the Cycle
    • Chapter 2: What Do Mathematicians Do?
    • Chapter 3: Mathematicians Take Risks
    • Chapter 4: Mathematicians Make Mistakes
    • Chapter 5: Mathematicians Are Precise
    • Chapter 6: Mathematicians Rise to a Challenge
    • Chapter 7: Mathematicians Ask Questions
    • Chapter 8: Mathematicians Connect Ideas
    • Chapter 9: Mathematicias Use Intuition
    • Chapter 10: Mathematicians Reason
    • Chapter 11: Mathematicians Prove
    • Chapter 12: Mathematicians Work Together and Alone
    • Chapter 13: “Favourable Conditions” for All Maths Students

    Is this not awesome? Here was a list articulating things about maths that I know are important and yet that I’ve struggled to articulate all my life as a mathematician and maths educator. Many of them cut straight to the heart of the difference between how I experience mathematics and how it usually is experienced in a classroom.

    • “Mathematicians use intuition” you say? Well, yes. Yes we do.
      But many a maths classroom is about following rules and avoiding the need for intuition.
    • “Mathematicians work together” you say? Well, yes. Yes we do.
    • But so many students think maths is only a solitary activity.
    • “Mathematicians make mistakes” you say? Well, yes. Yes we do.
      But mistakes are feared and avoided in most maths classes.
    • “Mathemaicians connect ideas” you say? Well, yes. Yes we do.
      But so many maths curriculums are just so many piles of disconnected procedures, even here at my own university.

    The contents page promised a book about the most important aspects of mathematical work and thinking, and a hope that it would give ways to bring these into the experiences of students in all maths classrooms.

    And the hope was made real.

    Each chapter starts out comparing how mathematicians talk about what they do and what students’ experience of it is. Then it moves on to detailed examples of the aspect of maths thinking in action in real classrooms, as well as strategies to encourage it both in your students and in yourself as a teacher.

    I didn’t expect to see this last point about encouraging these attitudes and thinking in yourself as a teacher. Yet it is the most compelling feature of the book for me. Indeed, I don’t think the book would have had nearly the impact it had on me (or the impact I see it having on others) without this constant message that to help your students experience maths differently, then you yourself need to experience it differently too. More than this, Tracy doesn’t just make this need clear, but actively and compassionately empowers us to seek out ways to fill it.

    Somewhere inside you is a child who used to play with numbers, patterns and shapes. Reconnecting with your inner mathematician will improve your teaching and benefit your students, and it will also benefit you.
    – Tracy Zager, Becoming the Math Teacher You Wish You’d Had, p39

    These comments were left on the original blog post:

    Tracy 10 May 2017:

    Misting up over here, David. You’ve cut right to the heart of it. Thank you so much. I feel incredibly lucky to learn with you.

    David Butler 10 May 2017:

    And I am lucky to have been able to sit and talk with you by reading your book. I really felt like you were there with me, encouraging me to be more.

    Susan Jones 10 May 2017:

    I’m still reading it. Your post makes me realize my privilege in not majoring in math 🙂 I never did lose the “play with the numbers” thing.

    David Butler 10 May 2017:

    Sadly, you don’t have to major in maths to lose the play-with-numbers thing. A good dose of standard high school maths teaching can safely banish that tendency, as Tracy described in the first chapter of the book!

    For me, my maths university degree is what actually freed me to play. I took Discrete Maths II in second year of university and it felt like all we were doing was playing with these ideas, and it was play encouraged by the lecturer.

  • Money and me

    In the online resources for Becoming the Math Teacher You Wish You’d Had, Tracy Zager provides information about the benefits of writing a “math autobiography”. I really have tried to do this, but I am having a lot of trouble organising my thoughs and memories. However, I reckon I can track some of my memories about one particular application of maths: money.

    As a child, I hated money. My mother says as a young child I would actually cry if someone gave me money as a gift. She thinks it might have been related to having to make a decision about what to spend it on, which was way too big a responsibility for little me to handle.

    This sounds right, based on a very specific memory to do with making decisions. I don’t know how old I was, but I wanted to buy some lollies at the local deli. I talked to my older brother and sister about the process and had a well-formed plan. I could go up to the counter and ask the man for 5c worth of lollies, he would give me lollies and I could go home with them. I walked to the deli with my 5c and went in the door. I reached up to the counter and placed my coin on top and asked “Can I have 5c of lollies?” The man asked me “What would you like…” and proceeded to describe about five options. I couldn’t cope. He was just supposed to give me a selection of lollies and I could have them and go! I wasn’t supposed to have to make all these decisions! I left my money on the counter and ran all the way home. It turns out that using money meant making a whole lot of decisions on the fly and I did not like having to have that sort of pressure.

    At some later time, I remember plucking up the courage to try and buy lollies again. I went to the school canteen and put my 20c coin on the counter and asked for 10c of lollies. The canteen lady looked at me and said, “You’ve given me too much.” I said, “But that’s all I have.” After a pause, she reiterated, “You asked for 10c and you’ve given me 20c.” I looked at her, not knowing what to say. I was certain that if you gave the lady more than you needed, they could just give you whatever was leftover. I don’t think I knew the word “change” to describe this, but I certainly knew how the process worked. Still, I wasn’t sure how to explain the concept to the canteen lady on the fly. She stared at me. Acutely aware of the line of people behind me, I took my coin and left. Looking back on this as an adult, I wonder if she simply assumed I didn’t know what the value of the coin I gave her was. Clearly it never occurred to her that I didn’t actually want to spend my whole 20c. I learned that sometimes people treat you like an idiot when you use money.

    Later again – I think it was during Year 4 – the teacher had set us an assignment asking us what we would do to spend one million dollars. I couldn’t do it. I had done absolutely nothing on the assignment right up until the very last minute, because I simply couldn’t face it. My mother sat me down to talk to me about it and tears streamed down my face as we tried to figure out what the problem was. I described all the examples the teacher had given, which were about cars or houses or things that people might want for themselves, but I didn’t want any of those things. Cars didn’t interest me, I liked the house I lived in, and I simply couldn’t imagine wanting to spend that huge amount of money entirely on myself. When my mother realised what the root of the problem was, she encouraged me to think about how I might spend the money on someone other than me. How could you use the money to make life better for someone else? In the end, I wrote my assignment about using a million dollars to start an animal shelter in order to look after lost animals. Not until now do I realise this assignment was actually about the size of the number one million. For me at the time, it was about money and spending, which were emotional topics and not maths at all.

    This aversion to money-related maths never really went away. In High School, I found topics on compound interest intensely uninteresting. I could “do” them, and I understood their application to my future life, but I never cared about them. When teaching financial maths to high school students, the explanations of which numbers in the graphics calculator had to be positive or negative were never natural to me. Even now I can’t process explanations of probability which define probability as “how much you’d be willing to pay” for something. How much I’m willing to pay for something is such a complex issue, which requires me deciding if I actually want the thing and is dependent on how much money I actually have to spend and my emotional state, not to mention the horror of having to manage the interpersonal minefield of actually negotiating a price.

    I don’t have any particular goal in relating all of this and I wonder if it will mean anything to anyone else. For me, it has helped me realise just how much of my like or dislike of particular applications of maths has to do with emotional and interpersonal things. It makes me aware that there will be internal battles inside my students that affect how they respond to maths and its applications that I can’t see or even they can’t see.

    These comments were left on the original blog post:

    Paula Krieg 19 April 2017

    One thing that’s really powerful about this story,David, is how a person can be struggling so intensely with a problem has little to do with the problem that is presented. About the canteen lady, it’s easy to think that she was being a jerk to little David, but the fact is, she really may not have known how to make change. Hard to know if in that case she was the one that was struggling with problems that were hard to pinpoint. A good reminder to ask questions of the person who is befuddled. but what a great string of stories. thanks.

    David Butler 19 April 2017

    Thanks Paula. It was a tough thing to write.
    That may possibly have been true about the canteen lady. Still, 6- or 7-year-old David could hardly have helped her understand at the time!
    I guess a lesson for 37-year-old David is indeed to ask and listen to people.

    Nicola Petty 2 May 2017

    Kia ora David
    An interesting story, and almost the opposite of many people, like my father, who felt he could only do maths if it involve money.

    I have an aversion to stars and anything much to do with space. The size of it all makes me feel uncomfortable. I know for many people, space is really interesting, but I would rather not think about it. I don’t know why I feel like this, and I have a hard time seeing why other people are so interested in space. We just don’t know what will make people uncomfortable!

    David Butler 4 May 2017

    That’s really interesting Nic. Another fine example of how we can never really predict what might be going on inside our students’ hearts.

  • 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?

  • Individual Ahas

    At the Hmm… Sessions in November, something cool happened when a couple of the students were showing the rest of us the solution to a puzzle.

    (For those who don’t know, The Hmm… Sessions are a regular gathering that I run where staff and students solve puzzles together in a group.)

    (Update: Later this year, the Hmm Sessions were renamed “One Hundred Factorial” after the first puzzle we ever did.)

    The puzzle was this one from the AustMS Gazette November 2010:

    Three boxes are on the table. One has red balls, one has blue balls, and one has balls of both colours. Three labels are made for the boxes, but they are misplaced so none of the boxes are labelled correctly. How many balls would you need to retrieve from the boxes in order to determine the correct labelling?

    Well, as I said, a couple of the students were showing us their solution to this puzzle at the big whiteboard. As they were doing this, each person watching cried, “Oh! That’s cool!” in turn. Each of us came independently to a sudden realisation of how the solution worked – we had “seen” the solution before it was actually presented.

    I think this is very cool because it says to me two things about how mathematicians work:

    One: Mathematicians don’t really listen fully to other mathematicians. Our minds are always racing ahead making connections and figuring things out on our own. Perhaps this is because we get the most thrill from seeing things ourselves rather than being told.

    Two: Mathematicians come to understanding at their own pace, which may be fast or slow. At the Hmm… sessions, each of us had our Aha momentat a different time.

    Apart from these two, the other thing that really impressed me was that none of us looked down on anyone else for taking longer but merely rejoiced when someone did see it. I couldn’t hope for a better atmosphere in the Hmm… Sessions than that!

  • 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.