Making Your Own Sense

Reflections on maths, learning and the Maths Learning Centre, by David K Butler

Category: Reflections

Reflections on learning and teaching and research and life.

  • Dagwood Dogs at the Gawler show

    I went to the Gawler Show with my family the weekend before last, and it was a wonderful day. We had camel and pony rides, patted the animals, looked at all the stalls, bought some toys, got given balloons and generally had a most excellent day.

    And as we left, we decided to indulge in some show food. One of the food vans was selling what they claimed to be “The best Dagwood Dogs in the land”. And you know what? They were! If hadn’t already left the show and walked halfway down the street to our car when we had finished, we would have bought another one.

    But even as I ate this faboulous Dagwood Dog, I wondered, “Sure it’s good, but why is it so much better than any other one I’ve ever had?” And soon I had quite a list:

    • It had just the right level of salt. Most Dagwood Dogs are way too salty, but this one was just right.
    • The batter wasn’t greasy. Instead it was fluffy and light.
    • The batter had corn and peas mixed into it! I have never seen this before but I’m amazed no-one has ever thought of it.
    • The flavour was so good I wanted to keep eating it even after the bit with the tomato sauce was gone.
    • The stick they used had a wider bit at the bottom so you could properly hold onto it.

    Later that day, it occurred to me that I very naturally evaluated my Dagwood Dog. It was so easy for me to make the decision of whether it was good or not, and to come up with a list of reasons why it was good.

    So why is it so hard to do this when it comes to teaching and learning? When I have a particularly good class, do I stop to think about why it was good so I can achieve it again? When my students fill out a SELT for my seminar, they quickly decide if it was good or bad, but do they give me a list of things that made it good or bad, so I can do better next time?

    Yet, it was such a natural thing to do this for my Dagwood Dog. I reckon we could all start using our natural food-evaluation instincts on our teaching and learning, and then perhaps we could claim we have “The Best Teaching in the Land”.

  • 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

  • Rule collision

    The same experience has happened to me several times in the Maths Drop-In Centre recently – with different students from different courses – and it was such a strong pattern I need to talk about it.

    The students are doing some algebra involving negative powers on the tops of fractions.  Something like this:

    \[\frac{1-x^{-2}}{1+x^{-2}}\]

    Now they remember this rule (probably from school) which says that a negative power belongs on the bottom of a fraction but as a positive power. And so they do one of these:

    \[\frac{1-x^{2}}{1+x^{2}}\]

    or

    \[\frac{1+x^{2}}{1-x^{2}}\]

    Both of these are, of course, TOTALLY WRONG. But the students have a hard time being convinced of this fact.

    The problem is, that that rule only works if everything involved in your fraction is multiplication and division. It doesn’t interact with the plus and minus that are trapped there on top and bottom of the fraction. And why doesn’t it interact with the plus and minus? Because the rule is based on the definition of what a negative power means. This is what it means:

    \[x^{-2}=\frac{1}{x^2}\]

    What this means is that multiplying by a negative power is the same as dividing by the matching positive power. And this gets to the heart of the issue: adding a negative power is not at all anything to do with multiplying it, so the nice “switch to the bottom, make positive” rule just isn’t going to work, because you have to do the addition first.

    The rules for negative powers are colliding with the rules for addition, and for fractions, with unpredictable results! If only the students had been encouraged more to work from the original definition rather than it being all about remembering a rule. Then maybe the results wouldn’t be quite so unpredictable! If only the students had attempted a few things like this in the past in a situation where someone could notice it and talk to them about it! Then maybe they would have found this glaring gap in their understanding of algebra!

    PS: If you’re wondering how to go about simplifying that fraction, then you have to first deal with the negative power using its original definition – which means it will become a positive power on the bottom of its very own little fraction. Like this:

    \[\begin{aligned}\frac{1-x^{-2}}{1+x^{-2}} &= \frac{1-\frac{1}{x^2}}{1+\frac{1}{x^2}} \\ &= \frac{\left(1-\frac{1}{x^2}\right)\times x^2}{\left(1+\frac{1}{x^2}\right)\times x^2} \\ &= \frac{x^2 -1}{x^2+1} \end{aligned}\]

  • Rapunzel’s Epiphany

    We bought Disney Studio’s newest film “Tangled” on the weekend and I have to say it’s one of my favourite movies ever. It’s certainly Disney’s best movie since “Beauty and the Beast”, and I dearly loved “Beauty and the Beast”. I should warn you now that in order to say what I want to say I’m going to have to reveal a bit of the plot, so let this count as your spoiler alert.

    OK. So Rapunzel grows up in her tower thinking that the old lady is her mother and not knowing who she really is. During the film she escapes and goes to the town where there are a lot of sun-shaped motifs. She brings one home to the tower with her on a piece of cloth to remember her experience.

    We see her lying on her bed staring at the ceiling, which she has completely filled with painted pictures during her life in the tower. She looks at the sun-shape and notices something remarkable about her painting: the sun-shape from the cloth is there in her paintings, and not just once, but over and over and over, and the repeating pattern sparks a memory of seeing the shape when she was a baby. The music swells as she realises who she really is. In short, the cloth and the paintings spark an epiphany.

    But it occured to me that she would never have had this epiphany without two important factors. Firstly, she had to bring the sun-shape home with her on the cloth. Secondly, and more importantly, if the sun-shape had not been in her paintings so many times, she may not have noticed the connection.

    And here’s where it relates to learning maths:

    We want the ideas we show our students to connect together so that the students realise the true nature of things and the realisation changes them. In short, we want them to have “learing epiphanies”.

    I’ve seen it happen for students when learning about subspaces in first-year maths. There are a lot of ideas but they are all highly connected, and sometimes while they are trying to solve a particularly difficult problem they suddenly realise that they’ve been seeing the same pattern over and over and that it all just makes sense.

    I want this experience to happen for all my students.

    But is it possible to set up these learning epiphanies in advance? It could be argued that epiphanies are highly personal and can’t be engineered. But I think perhaps we can make them more likely by putting certain things in place…

    Firstly, the connections between the ideas have to be there all along, just like the sun-shapes in Rapunzel’s paintings. If they weren’t already there, the realisation wouldn’t have been so powerful. We need to make sure that there are patterns in what we do and say from the very beginning.

    Secondly, the connections have to be there many times – so many times that once they have been noticed you wonder why you didn’t notice it before. It gives a huge sense of sureness to the realisation that you have, so you don’t just discount it as your imagination. So in our examples and explanations, we need to repeat and repeat the same pattern over and over and over.

    Finally, there needs to be an event to start it off, something to help you notice that first connection. Just like Rapunzel’s cloth – she needed the shape to be marked out simply so she could notice it in her own work. So we need to stop and point out the pattern in what we’ve said every so often, and get the students to do activities that hold the patterns up close to each other so they can notice.

    I think keeping these things in mind as we choose what examples to show our students, and choose how to present them, and choose what activities to get them to do, may just make it possible to help them have an epiphany like Rapunzel’s.

    This comment was left on the original blog post:

    “Humane Pain” 26 March 2013
    I agree (with both your opinion of *Tangled* and with the learning strategy), but wanted to add an additional benefit besides helping students to grasp the concept: epiphanies also add the element of excitement, that “aha!” or “Eureka!” emotion that is such a rush, such a natural high, that they want to study and learn more and more, in essence, the epiphany becomes a vehicle for motivating the students as well as grasping concepts, and this makes them lifelong learners. It makes maths *fun*.

  • Discounting your problem-solving

    As I was leaving the other day, a student said that she would come to see us the next day to ask some questions about her assignment. She said she had tried to do as much of it herself as she could, and had only done 70% of it.

    The “only” made me start – she had done most of it herself but that wasn’t good enough because she still had to ask for help. And somehow in the hurry of the moment, this came out of my mouth: I said, “And how much did you do on your own last time?”

    It was her turn to start – “Oh!” she said, “A lot less I suppose.” And then I had to keep walking or risk missing my train. But as I walked the incident ran around my mind: it’s amazing how many people can discount the evidence of their own problem-solving ability, simply because they still need help.

    I’ve seen it before, but I’ve never seen a way to fight against it. I’ve always tried to tell them that they can do it, and point out the bits they did do, but it always seemed to wash over them without leaving an impact. I’ve been focussing on a single moment of problem-solving.

    What I’ve learned from my thirty second conversation is that perhaps I should help people focus on more than just today. Instead, maybe I can help them look at their journey so far and focus on the improvement.

    But my student has given me even more: she’s given us something concrete to focus on: the amount they have done on their own. This is so much more tangible than “problem-solving ability” because it’s plain numerical data. The student can compare this time to last time and feel success as long as they’ve done that little bit more on their own.

  • Frayed research

    Phew! I submitted our article for the MERGA conference last week and now I feel like I’ve come out of hibernation: I’m standing blinking in the sunlight wondering what happened to everything I was doing before I started work on the article. (One of those things was this blog, which is why I’ve been quieter than usual lately.)

    One thing that caused me to descend deeper into research-hibernation was when I stopped to check the word count after getting halfway through what I wanted to say, only to discover that I was already 1500 words over the limit. I had to sacrifice a lot of what I had planned to say, and was left feeling like my research had a lot of loose ends flapping about everywhere.

    This is not the feeling I get from maths research. I’ve published very short articles in maths journals before, but felt no qualms about them at all because they were all tied up. I don’t mean I had finished everything there could be to do. No, I mean that there was a proper result – something about which you could really say, “This is it. This is true. This is why it’s true. And that’s all I need to say.” It’s neat.

    Education research is not neat. I’m always left with the feeling that you haven’t said anything. It seems more like, “This is sort of it. This is what might possibly be considered reasonable. This is why I think I might be in some way justified by holding the belief that this might possibly be reasonable. A lot more could be said but I have to stop now.” See? Not neat.

    My experience researching maths has left me with the feeling that things ought to be neat, and I stress myself out trying to tie up the loose ends in my education research. What I’m learning to realise that loose ends are the way things are in education research and saying why you think something is possibly reasonable is actually enough.

  • Only one chance

    We’ve been running Drop-In Centre tutor training recently, and as part of the training we discussed the statistics on how students use the Centre. The focus of this post is the following graph:

    A column graph titled "Number of visits per student 2010 year". The x-axis is labelled "Number of visits", and the y-axis is labelled "Percentage of students". The highest column is the first one for 1 visit, reaching to almost 50%.

    The graph describes how many students visited the Centre various numbers of times across 2010.

    There are many things you could notice about this graph, but one thing you might notice is that almost half of the students who use the Centre only visit once.

    We all discussed the possible reasons for this, ranging from those students who only come to be reassured that they’ll be fine with their maths this year, to those who came for one specific assignment in a course like Geology, to those who didn’t like the way we helped them and never came back.

    But one trainee Nick made a very important point which is the one I wanted to share with you:

    Although there are many reasons why it may be true, the shocking truth is that for most students, we only get one chance to help them. Therefore, the way we treat the students and the words we use with them are important every single time because that time we talk to them may be the only time.

    Suddenly our responsibility seems so much greater…

  • Charlotte’s Sudoku

    The other night I was doing a Sudoku, and my two-year-old daughter Charlotte decided she wanted to help, as she always does at any time when I have a pen and paper she could steal.

    So she bent over the paper concentrating very hard and a while later she threw her hands in the air and declared in her loudest voice, “I did it!!!”.

    And sure enough, she had. Every single box was filled with a letter-like squiggle.

    And isn’t this the most basic rule of Sudoku, when you really get down to it? Sure, the symbols in the boxes are supposed to be the digits from 1 to 9 and you’re supposed to have one of each in every column, row and box, but really if there’s not one symbol in every box, you just haven’t finished have you?

    I was extremely impressed that she was able to figure out that rule just by watching me for a few minutes!

    And, as usual, it made me think of my students. Often they latch on to one idea from their lectures and do their maths problems following that one idea, even though there are quite a few important details that make it actually work. But should I really be upset? They’ve picked out the most important idea and that in itself is impressive. Next time I’ll try my best to let them know how impressive it is before telling them how to be better.

  • The Fairyland Clickety-Clock

    This post is again inspired by the television show The Fairies (you have been warned!).

    In Fairyland, there is a peculiar procedure you have to go through in order to find out the time. The following sequence is played out several times across different Fairies episodes:

    RHAPSODY: I wonder what time it is.
     
    HARMONY: Well there’s only one way to find out the time in Fairyland…
     
    CUE MUSIC.
     
    ALL SING (with appropriate actions):
    Fairyland Clickety-Clock we need to know the time.
    Please show yourself in Fairyland and let us hear you chime.
    Tick tock Clickety-Clock,
    Tick tock Clickety-Clock,
    Tick tock Clickety-Clock,
    Chime!
     
    ENTER CLICKETY-CLOCK to tell everyone the time.

    You can understand why they lead such a carefree existence in Fairyland: if you had to do this every time you needed the time, you wouldn’t find it easy to stick to a timetable!

    Anyway, while watching the little sequence above, I was reminded of the Gramm-Schmidt Process, which the students doing Maths 1B in Summer Semester have been studying recently. It’s pretty much the only surefire way to create an orthonormal basis for a vector subspace. (If any of the words “orthonormal”, “basis”, “vector” and “subspace” are not familiar to you, then I wouldn’t worry about it too much – the point is that the Gramm-Schmidt Process is the only way to do a particular task.)

    The problem with the Process is that it can take quite a lot of calculation, especially if you’re working with long vectors. This of course inspires the students to ask if there is an easier way, or at least a short-cut sometimes. It usually takes some convincing for the students to believe that in fact this is the only way and they’re going to have to get used to how long it takes. Sometimes there really is only one way to do something, and it does take a while.

    RHAPSODY: I wonder what an orthonormal basis would be for this subspace.
     
    HARMONY: Well there’s only one way to find an orthonormal basis in Fairyland…

  • Very Unique

    I often hear that the phrase “very unique” is not a correct thing to say. The explanation is that the word unique means “there is nothing else like it” and as such is already an absolute. So there’s no grades of unique: something is either unique or it’s not – there is nothing else like it or there is someting like it. This is a good explanation, so I agree we shouldn’t use the phrase “very unique”.

    But here’s the big problem: what are we supposed to use instead? What if you want to say something is unique, but add some strength to your statement? It’s all well and good to tell me what I can’t say, but perhaps you could also tell me what I can say?

    And, as always, it makes me think of my own teaching. How often do I tell students not to do something, with all the excellent reasons why it’s a bad idea, only to leave out the important part of telling them what to do instead?

    As a student, this has been done to me a lot when writing proofs. The lecturer could often tell me that what I have written is not correct or understandable, but usually did not think to (or know how to) tell me how to find a correct or more understandable way to write it. I hope I have the presence of mind not to leave students with this frustration – because it is damned frustrating!

    PS: I think a good alternative might be “truly unique”.