Making Your Own Sense

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

Author: davidkeithbutler

  • The Human Galton Board

    Last week we were booked in to do Human Markov Chains with several groups of school students, but it turned out there would be a lot fewer of them than we expected, and I didn’t think Human Markov Chains would work very well with under 20 students. I still dearly wanted to do a moving maths activity, and I still wanted it to be about probability, but I wasn’t sure what to do. Then, on the morning of the day the students were coming, I had an inspiration and quickly knocked together the Human Galton Board.

    You can read the rest of this blog post in PDF form here. 

  • Human Markov Chains

    This blog post is about a moving maths activity that I have wanted to do for years and finally got an opportunity to do this year in 2018. It’s a model of a concept called a “Markov Chain” using human movement.

    You can read the rest of this blog post in PDF form here. 

  • Leaving the most important teaching to chance

    Something is bothering me about teaching at university: we are leaving the most important teaching to chance.

    In most tutorials, there is an opportunity to try out things with a tutor there to talk to about it, or deep discussion of course content, or at the very least worked examples of using the ideas in practice with a higher chance of asking questions. In a lot of ways the tutorial is the place where the majority of the classroom learning actually happens in a university course. Indeed, students often say that tutorials are the most important part of their learning at university and will go to them even if they don’t go to lectures. I talked to a student just the other day who was still catching up watching the lectures online from two months ago, and yet has been able to do his assignments because he has been attending the tutorials.

    So, if the tutorial is the most important class for student learning, then you would think that the tutorial would be the class where you put the most effort into making sure it was as good as you could achieve. Yet in so many disciplines in so many universities, the tutorials are given to their current postgrad students to teach, with minimal or no training. (Not to say the postgrad students can’t be great teachers, just to say they don’t have much teaching experience yet.)  By not carefully considering our tutorials and training the tutors, it’s like we’re leaving the most important teaching to chance.

    Even more than this, most initiatives to improve teaching at university focus on the lecturers. We give support for designing what happens in lectures and online, but somehow we don’t provide any time or resources for training the hundreds of tutors running the thousands of tutorials. Again, we spend all this effort improving lectures, but leave the most important teaching to itself.

    This really surprises me, and I really wish there was something I could do about it. What I wish for is a funnelling of funding into designing effective teaching in tutorials, and even more importantly, funnelling funding into training tutors in effective teaching in tutorials. I think this might have a huge impact on learning at university.

  • Mr Johnson’s Rainbow

    I love reading and writing, and the way that people use words to express ideas fascinates me. So it is no surprise that when I was in Year 12, I studied the highest level of English available. My English teacher was called Mr Johnson and I hated him. (It wasn’t really, Mr Johnson – I’ve changed his name to write this.) The reason I hated him is expressed in this poem I wrote at the time:

    MR JOHNSON’S RAINBOW

    The afternoon sky was fretted
    With cotton shades of blue
    And the rainbow came, inspiring us all
    And on some old scrap paper
    My thoughts and feelings grew
    Some lines of verse upon the page did fall

    And then I took the poem
    A work all of my own
    And to my English classmates did I show
    That my poem was Quite Good
    To me it was made known
    Because my fellow classmates told me so

    But the teacher, oh my teacher
    Said: I know it is Quite Good
    But it is not what I would call My Way
    Your verses on the rainbow
    Are not the way I would
    Ever say the things that I would say

    For I, yes I my student,
    Am like the poet Donne
    As you are like the other poet Keats
    You like to write, like he,
    On emotion by the ton
    Where I do so much higher mental feats

    He went on by relating
    All the things that he would write
    And prattle from his open mouth did flow
    He said: the rainbow is
    All the colours making light
    So there must always be thingy, you know

    And I was quite inspired
    By this brilliant oratory
    And thought:
    Why don’t you write your own bloody poem if mine isn’t good enough for you?

    The same theme appeared in all of his feedback about all of my creative writing: he disapproved of the content I chose to write about, often saying that it wasn’t clever enough. He never gave me feedback on my expression of those ideas – no discussion of flow or characterisation or word choice or metaphor – only ever that the ideas themselves were not to his taste. One notable example was when we were asked to write a short story about a Far Side comic involving butterflies from the wrong side of the meadow, and so I wrote about the flowers sending rogue butterflies to attack the flowers on the other side of the meadow. I was marked down because I didn’t instead do something more clever, like write about some completely other thing only tangentially related to the theme of the comic. As you can imagine, I did not choose to study creative writing at university, and to this day I still have quite a fear of sharing my writing.

    Thinking about how this applies to my maths teaching, I wonder how often we tell students in maths “but it is not what I would call my way”. For example, those times when a student does a perfectly wonderful and correct solution to a problem, but then we tell them it has to be done this way instead. Or those times when we discount their excitement of maths applying to something that interests them to tell them they should be interested in the beauty of the maths itself. Alternatively those times when we get annoyed at the student who wants to understand the ideas behind a method and tell them to just do it and not worry about that. When a student asks me to check their work, do I critique their execution or do I criticise their ideas? What about all those times when I ask the class for what they notice/wonder and then wait until I get the one I was really hoping for? I am worried about students choosing to stop studying maths because we always judge them on their ideas.

    As usual, I don’t know what to do about this other than just be aware of it. Just yesterday when this was on my mind I was careful to say to a student how awesome I thought their Quarter the Cross solution idea was, before talking to them about how they might be more precise in their execution so other people could also be sure it was a quarter. I only hope I can have it on my mind a bit more often as I work with students on the everyday stuff in the MLC.

  • A public health approach to improving teaching and learning

    Making a big difference to student learning is a tricky business. Here at my university, there are a certain number of (wonderful) teaching staff who are champions of innovation, always making big changes to the way they do things and jumping onto any innovation as soon as it comes around. Yet the students not in those classes don’t see much benefit from it. Indeed, those staff who are not champions of innovation may do nothing for fear of having to adopt all at once All The Things they see the champions doing. A student who seeks regular support for their learning may make spectacular gains, but there are literally thousands of other students who don’t seek such support on a regular basis, and thousands of students who don’t really need spectacular gains but just a little bit extra. I have started to think that perhaps the best way to make a big difference is to find some way of encouraging a large number of small differences.

    This is essentially the way Public Health works. In Public Health you are concerned with whole population health initiatives, which are often of necessity a large number of small differences. For example, you may not cure the flu, but you might encourage 20% more people to wash their hands and so prevent the spread of infection and stop so many people getting the flu in the first place.

    Imagine the benefit that might happen, not if a few lecturers rub out their courses and start again with flipped learning, but just if every lecturer simply labelled everything in Canvas/Blackboard so the students could easily find stuff. Imagine the benefit, not if a few course coordinators completely changed their tutorials to be about group discussion, but if every classroom tutor asked one “what if” question in every tutorial. These are not big things to change, but if a lot more people did them, I think the overall effect would be far-reaching. And they might seem like something you could actually do, as opposed to the big changes that are the usual fare of innovation.

    Personally I am trying to do more Public Health approaches to student support too. Instead of just visiting lectures to tell the students how to seek one-on-one support, I’m visiting with a five-minute message about interpreting assignment questions, or choosing to put in more explanatory working, or what a standard deviation is. If I can reach even half of a lecture of 500 students with one of those little messages, then I have made a big difference by making a lot of small differences.

    Unfortunately, Public Health doesn’t make for spectacular stories. Giving one person brain surgery to save their life after a horrific traffic accident is a spectacular story. On the other hand, lowering the speed limit in urban areas in order to make horrific accidents less likely is not a spectacular story, but it can be argued that it saves a whole lot more lives. I only hope I can convince the Powers That Be that my Public Health approaches to learning and teaching improvement are worthwhile, if not spectacular.

  • 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.

  • Stop hating on cis(θ)

    I met with some lovely Electrical and Electronic Engineering lecturers yesterday about their various courses and how I can help their students with the maths involved. And of course complex numbers came up, because they do come up in electronics. (I have not the slightest clue how they come up, but I am aware that they do.)

    I asked them what notation they used for complex numbers, and they told me about using j instead of i, and then about polar form using either reᶿʲ and r∠θ. I told them that the students will have seen the notation r cis(θ) at school for those and I was met with righteous indignation. What a horrible notation! How could anyone think that was a good idea!

    It was rather a surprise considering their wonderful openness and friendliness up to this point. But this is not the first time I have been met with this sort of response to cis from more-or-less nice people. Many a mathematician has expressed to me their disdain for this little notation, and I have to say I am bamboozled. Why do people hate it so very much?

    I actually rather like cis. I like how cuddly-looking it is with all those curvy letters. I like how wonderfully easy it is to write simply with no special symbols and with no powers that have to be typeset at a higher level than the text. I like how it sounds like you’re singling out one of your many sisters when you say it aloud.

    I think cis(θ) is friendlier than eⁱᶿ because it doesn’t require you to suddenly believe that imaginary powers of real numbers are a thing and believe they ought to be this unusual combination of cos and sin. I think it’s also friendlier than ∠θ because we’re not co-opting a symbol we saw in geometry for a new and strange usage. (Not to mention the fierce pointiness of ∠ compared to the cuddly roundness of cis.)

    Finally, the thing I like the most is how cis(θ) is a function that takes real numbers and produces complex numbers, and those numbers are on the unit circle. For me It’s such a nice thing right from the start to recognise that there are such things as functions that produce complex-number outputs, and that they can have rules to manipulate them just like ordinary functions. Indeed, writing it with a multiple-letter name like all their other familiar functions forges this lovely connection. I also  love that multiplying a complex number by cis(θ) rotates it around the origin θ anticlockwise, because of course it does since cis(θ) is a position on a circle. (If anything, I think it wouldn’t hurt it if we renamed it “cir” to highlight its connection to a circle rather than highlighting its connection to cos and i and sin, even if that means losing its familial pronunciation and a little roundness.)

    Here’s a little geogebra applet to play with cis as a function from the real numbers to the complex numbers: https://ggbm.at/MWnm5TJA 

    Two graphs side-by-side. The left-hand one is labelled INPUT and shows just an x-axis with a point marked A. The right-hand one is labelled OUTPUT and shows an real and imaginary axis and a point marked cis(A) somewhere in the first quadrant.

    I know it hides the thing about complex powers of real numbers, but I think it’s important to hide that for a bit, lest the eⁱᶿ feels too much like a fancy trick. Plus I think it’s good to be familiar with this friendly little function for a while before discovering that it also solves a problem of complex powers. Indeed, I would love it if students were familiar with this friendly little function even before introducing the concept of polar form.

    So there are the reasons why I actually rather like cis. But there is one more reason I think mathematicians and professional maths users shouldn’t hate on cis: because it means hating on the students themselves. Our students put a lot of effort into understanding cis, and to loudly announce how much you hate it is telling them that they wasted all that effort and that their way of doing things is less sophisticated and babyish. Not the best way to start your relationship with them as their university teacher. I’d prefer it if people started off with acknowledging the coolness cis and then pointed out that the other notations are more convenient for doing what they want to do, or allow them to write formulas in more magical-looking ways, or allow them to do some cool stuff with powers. Then maybe we might not get them offside early in the piece.

    So please, stop hating on cis(θ)!

  • Remainders remain a puzzle

    My first post of 2018 is a record of some rambling thoughts about remainders. I may or may not come to a final moral here, so consider yourself warned.

    What has prompted these ramblings today was reading this excellent post by Kristin Gray  about her own thoughts on division and remainders. In that post, I saw the following: 

    7÷2 = 3R1

    For some reason, this bothered me. For some reason it’s always bothered me. Today I think I realised what the problem was: In my head “7÷2” is a number, and “=” indicates that two things are equal, but 7÷2 can’t be equal to 3R1 because 3R1 is not a number. It is only today that I realised that 3R1 isn’t a number.

    How do I know 3R1 isn’t a number? Well firstly it’s two numbers. One is a number of groups and the other is a number of objects. I don’t even know how big the groups are – it could be 3 groups of 2 and one left over,  it could be 3 groups of 7 and one left over, or 3 groups of 200 and one left over. I can hear people saying that actually all plain numbers could mean any number of different units, and a 7 could be 7 cm or 7 ducks or 7 groups of 200. But the 1 here is definitely 1 of something, while the 3 is some unknown size of groups of that same something. That seems like a totally different kind of unit issue than with a plain ordinary number.

    Also, if it really is a number, then I should be able to place it on a number line, but where does it go? The 3 I certainly know where it goes, but what about the 1. Where does that live? It lives in a completely different land to where the 3 lives, and I can’t really put it on the number line until I know how big the groups are that the 3 represents.

    It occurs to me that this is a good way of transitioning to a fraction sort of idea. The fact that the 1 is small relative to groups of size 200 and large relative to groups of size 2, and needing to encode this relative size would lead nicely to a need to write this as 7÷2=3+1/2. What an interesting idea.

    My other really big issue is that the “=” sign in this context doesn’t work the way an “=” sign works. If 601÷200 = 3R1 and 7÷2=3R1 then usually the properties of “=” would mean that 601÷200 = 7÷2. But they aren’t equal. I suppose they both produce 3 groups with 1 left over, but that 1 is very different in size relative to the group in each situation! So they’re not really equal are they? Actually, this idea is going back to the same idea I had with the number line, where you need to encode the relative sizes.

    My final problem is that if it really is a number, then surely you’d be able to do operations on it. But I don’t really know how you’d do that. You’d expect that if the 3R1 came from 7÷2, then 2×(3R1) would produce 7, but if it came from 601÷200, then what would 2×(3R1) even mean? I’ve been trying to figure it out, but to no avail.

    It might be possible to do addition and subtraction, if you knew the groups were the same size. In that case 3R1 + 5R3 would be 3 groups and 1 plus 5 groups and 3, so it should be 8 groups and 4. So 3R1+5R3 should be 8R4. It seems you add the two numbers separately, which is actually super interesting. I’m still a bit worried about what would happen if the groups were size 4, say, because then 8R4 is actually the same as just 9. So now it seems like they are a lot more like numbers than I originally thought. This seems like a very interesting thing to investigate.

    As you can see, I’m rather puzzled by remainders and where they stand numerically. I get that the idea of dividing a collection into groups requires us to have a concept of remainder. I just feel weird writing it in this way because these symbolic representations feel like they ought to make numerical and algebraic sense, but here they don’t.

    In Kristin’s post, she floated the idea of the equation being not 7÷2=3R1 but instead 7=3×2+1. This second equation I feel completely comfortable with. It is 100% clear what the numbers are doing and the “=” really is acting as an equality here. I still wonder if there’s a more helpful way of representing the division-producing-a-remainder thing though.

    And maybe that’s another issue I have with it, that this statement “7÷2=3R1” is about doing an operation and producing a result, as opposed to declaring a relationship, which is what I have come to believe the “=” sign is for. By using something that is not like a normal number and just encodes a description of an answer, are we reinforcing that “=” means “here is the answer”? I don’t know what to do with this question yet, or if it even really matters.

    So there you go. I’ve rambled through a whole lot of thoughts and worries about remainders. I don’t have any conclusions or morals or recommendations. But it’s certainly helped me to try to write it all down. I hope it helped you to read it. I’d love to hear your own thoughts on it.

    This comment was left on the original blog post:

    Deborah Peart 4 January 2018:

    You make a great point! Students have enough confusion around equality and the equal sign. Truthfully it should be expressed
    7/2=(3×2)+1
    Interesting thought!

  • Home in One Piece: a game of strategy using play dough

    This post is about a game I invented called Home in One Piece.

    An image of the Home in One Piece board. At the top is a grey zone, at the bottom are white and grey circles alternating. The board has wierdly shaped balls and tendrils of pink and purple play dough over it. You can also see three dice with blob shapes on the faces.

  • Three hours in the MLC Drop-In Centre

    Last week, I had one of those days in the MLC Drop-In Centre where I was hyper-aware of what I was doing as I was talking with students and by the end I was overwhelmed by the sheer volume of things I had thought about. I decided that today I might attempt to process (or at least list) some of it for posterity.

    A Real Analysis student was worried about a specific part of his proof where he wanted to show that a function was increasing. The function was a kind of step function with more and more frequent steps as it approached x=1. It was blindingly obvious from our sketch of the graph that it was increasing, but every time we tried to come up with a rigorous argument we just ended up saying something equivalent to “it is because it is”. I remember thinking about whether it was just obvious enough to just say it without proof, but shared the student’s desire to make the nice clean argument. I don’t remember now if he ended up with a neat argument or not, but I do remember finishing with the moral that obvious things are often the hardest things to prove.

    Before we even got to this frustrating little bit of the proof, I was reading through what he had done already, and I noticed that he had written “x is < 1” and commented that this wasn’t a grammatical sentence because the “<” contains an “is” already – it’s “is less than”, which makes his sentence “x is is less than”. This led to quite a discussion about the grammar of “<“, which is complicated by the fact that it can be read aloud in multiple ways depending on context. We investigated some other sentences containing < or > he had seen written before to see how it was pronounced there. In my head I was reminding myself to pronounce written maths aloud more often when I’m with students so they can learn the correspondence between speech and writing in maths.

    I was called over to help a Quantitative Methods in Education student with her research assignment because she had mentioned she was struggling with statistics (and I’m the default stats support person). Eventually I helped her figure out that her problem wasn’t statistics at all, but rather that she didn’t understand the wider context of the research, or what the data actually represented, or what the actual goal was. Mainly this came about by me not understanding the context, data and goal and kept asking her questions about those things in order to try to figure out where the statistics fit into what she was trying to do. We talked about how she might go about gaining the understanding she needed in order to come up with the specific-enough questions about variables that statistics would be able to help her to answer. I could see her heart sinking, so I reassured her I would still be here next week to talk about the statistics should she need that support later, but also from our discussion that she would probably be okay with that part when it came.

    I returned to the table I came from to discover even more statistics, this time a statistics theory course in second year of the maths degree. This student was struggling to come up with a linear regression proof which was pitched in the assignment using linear algebra. For this small part of the proof, he had to show that the columns of some matrix were linearly independent. I did what I always do with linear algebra proofs and helped him remember various definitions and facts relating to the vocabulary words in the problem, then we chose a representation of linearly independent from the list we had made. He commented that I go about proofs in a very different way to him, and I said I was merely following the problem-solving advice I had already put up in that big poster on the wall, see? In the end we finished with the moral that coming up with connections between things is a great way to solve problems and to study, even if it means ignoring the goal for a moment.

    I saw a student on the other side of the room that I had talked to months ago and I went to see how she was doing. When I saw her last, she was questioning her decision to study Statistical Practice I and indeed her whole decision to come to university at all, while simultaneously worrying about letting down her son who was the one who had urged her to come to university. Now she revealed she was systematically working her way through the course content so far, making sure she understood everything before classes returned and the last set of new content arrived. She asked if that seemed like a good plan and I said I was so pleased to see her making plans for her study and persevering with her course. I suggested she do some work soon to connect together the ideas in the different topics, since it’s the connections between ideas that will create understanding. I recommended a mind-mapping idea I had seen on Twitter recently (thanks Lisa).

    At the next table I came to, some students and one of my staff were having a discussion about function notation, in order to help a student in our bridging course. They were talking about how unfortunate it is that something like f(x) could be interpreted as multiplication in some contexts and function action in another. I agreed that it was unfortunate, but it was just the way maths language has come to work and it was too late to change it now. I related this to how the word “tear” is interpreted differently depending on context, which seemed to help everyone accept the fact, but not necessarily be happy about it! My staff member brought up how it’s made more complicated by the fact that we then write “y=f(x)” or even “y(x)=…”. I flippantly said that this was making a connection between graphs and functions, which is a whole separate discussion. Of course everyone wanted to hear more about this and suddenly I was giving a little seminar on the fly about function notation and graphs and the connection between them. I was keenly aware of everyone watching me, and of the choices I had to make with my words, especially when I made some mistakes along the way. Here’s more-or-less what I said [with some of my thoughts in square brackets]:

    There are many ways to think about functions, but one of them is that a function takes numbers and for each one it produces another number. [I thought about saying it doesn’t have to be numbers, but decided that was just going to distract from the discussion today.] It could be a formula that tells you how to get this other number, or there could just be a big list that says which ones produce which ones. For example, there’s a function which takes 1 and gives you 4, takes 2 and gives you 5, takes 3 and gives you 7, takes 5 and gives you 8. It’s the one that adds 3 to every number. The brackets notation is supposed to highlight that there’s a starting number and the function is a thing that acts on it to produce a result. So we could write something like “addthree(2)=5”. [I’m not sure what motivated me to write this. I’m sure I’d seen someone talk about this on Twitter somewhere, but anyway it seemed right at the time.] Sometimes we want to talk about the function as a thing in its own right so we give it a letter-name like “f” (for function) so that we can talk about it more easily. Or we only have some information about it but don’t know what it really is, so we can give it a letter name until we know its proper name. [I was reminded at this point about a Rudyard Kipling story featuring the origin of armadillos but chose not to mention it. I also thought here that I could have done another example earlier of a function acting like lookupthelist(2), but we’d moved further than that and I didn’t want to go back.]

    And what about the connection to the graph? Well you could draw a nice number line and write down what result each number produces. [I started drawing pictures at this point.] You could visualise this by drawing a line of a length to match the result attached to each number on the line. [I accidentally drew the lines representing the input numbers when I drew my picture, and had to go back and change them later. No-one seemed to mind that much.] You could have a ruler to measure how long each of these is when you needed it. Or you could attach the ruler to the edge of the page and line up the drawn lines with it. And now suddenly you’re locating the end of each drawn line by numbers on two axes. This is coincidentally just like the coordinate grid which is something else we’ve learned about before but not directly connected to functions. In the coordinate grid, the y refers to this second number and the x refers to this first number and you can describe shapes by how the x and y are related to each other. It doesn’t have to be y = formula in x, but that is a useful way to do it if you created your shape from a function.

    I think it was at about this point that I asked the students how they felt about this. My staff member said he quite liked the “addthree(x)” thing, which he’d never seen before. The bridging course student said that really helped her too, as well as realising that there were two different things happening at once when you write y=f(x). I decided to move on at this point.

    The next student was studying Maths 1B and was doing a deceptively simple MapleTA problem which he was stuck on. It listed an open interval – I think his was (-4,5) – and asked him to give a quadratic function with no maximum on this interval, a quadratic function with neither a maximum nor minimum on this interval, and a cubic function with both a maximum and a minimum on this interval. He had answers entered for the first two questions, which MapleTA had marked correct, but he wanted to know why. He revealed a little later he had gotten these answers off a friend and that now he wanted to know how to get them himself. I’m always glad when students do this, because it’s the first step to them really understanding themselves. With the first question, we discussed what it meant to be a maximum, and how that definition played out on an open interval. Once we had done this, I decided that I should take my own advice and make the question more playful, more exploratory. So I asked him what he would need to put in in order to make the answer wrong. He was really intrigued and started thinking through what the function would have to look like in order to have a maximum. After a while, he hit upon the idea that the leading coefficient could be negative, and this is all it would take. This was followed by several attempts to test his theory by putting in bigger and bigger constant terms in order to see if they really didn’t make a difference. They didn’t, and he was most impressed with himself.

    In the second part, we talked about moving the function around until the minimum wasn’t part of the domain. I went to get a plastic sleeve and drew a parabola on it in whiteboard marker, then asked him where he could move it to get what he wanted. At first he rotated it. I congratulated him for thinking creatively but did point out that it’s not the graph of a function like it was before. After that he only moved it up and down. Finally I gently moved it a tiny bit to the left, and he caught the idea that he could move it sideways, which he did until the turning point was outside the domain. The issue then came with how to change the function to do this, and we discussed why his friend’s solution was able to achieve it. Again he went playful and started putting in really big numbers and numbers really close to the boundaries to confirm that yes really he could move it anywhere outside that domain.

    Now we moved on to the third part and we discussed how cubics look and tried to figure out how to make the two turning points higher or lower than the end points. He wanted to use derivatives, so I let him do that, and it was quite a long journey, though he learned a lot about derivatives and specifying function formulas along the way.

    At the end, he asked if he had done all of this the correct way. I replied that it was definitely a correct way. He took this to mean there was a better way, and I said there was possibly a faster way. This meant a further long discussion about specifying zeros of functions and talking about whether we needed them to be zeros or if the function could be higher up or lower down. When we had finished this he asked me to explain the steps of this method again so he could write them down. I said that he didn’t need to remember the method because he’s never going to see this question again. The point is not to remember methods of solving all questions, but to learn something, and look how much stuff you learned today!

    Even though it was very close to my home time, I decided to talk to one more student. This one was doing a course specifically designed for the “advanced” maths degree students. He had been set a problem in Bayesian statistical theory. This is not familiar to me, so he spent a lot of time explaining what was going on to me. There were a whole lot of things in his working that I thought could be solved by simply referring to facts he had learned in previous courses, and I asked if he had done those courses. He revealed that they were in fact prerequisites for being allowed to do this course, at which point I said it was okay to use results from prerequisite courses. The biggest problem at the end was that he had assumed he had to integrate the normal distribution density function by hand, because the lecturer had gone to the trouble of giving its formula in the assignment. I assured him that it was in fact impossible to find areas under that distribution by hand, and he was allowed to use a table or technology. Sometimes you just have to tell a student what’s impossible so they can do it another way.

    So that was my day. I had three hours in the MLC and talked through so many ideas about learning and studying and maths. I had to make so many decisions about my words and actions and teaching tools. And I was left with a lot to think about. I hope you’ve found it interesting too.