Making Your Own Sense

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

Tag: mathematics

  • 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)!

  • Likeable primes

    There is a Twitter account that tweets the prime numbers once an hour in sequence. (The handle is @_primes_ .) Since before I joined Twitter, it’s been working its way through the six-digit primes and some of them are very nice. A lot of other people think they’re nice too, based on the fact that they are given likes and retweets. But what is it that motivates people to do this? What is it that makes a prime likeable? Well, that’s what this post is about.

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

  • 65536

    I have a whole suite of maths t-shirts that I made myself. One of them simply has the number 65536 on it. It’s been getting a bit of attention over the past couple of weeks, so I thought I might write about it.

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

  • Childhood memories

    Two books I’ve read recently have encouraged me to investigate my memories from childhood. In Tracy Zager’s “Becoming the Math Teacher You Wish You’d Had“, she urged me to think about my maths autobiography to see what influenced my current feelings about maths. In Stuart Brown’s “Play“, he urged me to think about my play history to see what influenced my current feelings and tendencies about play. In the spirit of those two, here are some of my earliest memories about maths and play.

    In primary school, I have very few memories of actually being in a maths class, and all of them are negative. I’ve related two of them already in this blog. One was my memory of doing a maths assignment about one million dollars, where the financial aspect distressed me to tears. Another was my memory of my Year 6 teacher attempting to teach us averages using cricket.

    The only other maths class memory is of a test I did in Year 3. I had been sick with asthma for a couple of weeks and came back to school on the day of a test. I dutifully did the test and actually got almost full marks. The only thing I got wrong was the meaning of the word “net” in the phrase “net weight” as you might see listed on a packet of food. I distinctly remember it being a multiple choice question and ruling out two of the answers as ridiculous, but basically having to guess between the other two. I was angry because how could I possibly know that? Everything else was just logic and so I could figure it out for myself, but you can’t figure out the meaning of a word without more context. Eight-year-old me was an astute little person.

    Across my primary school career, I do remember a strong feeling of pleasure and fascination associated with construction toys. I remember absolutely loving the MAB blocks, in particular the moment when I replaced ten units with a long, and ten longs with a flat and ten flats with a block. Interestingly, my memory is only of the blocks themselves and I can’t pinpoint a year level or a teacher that goes with this. I also remember loving playing with polydrons and attribute tiles, but again the memory is just about the fascination of playing with them, and not about any particular maths class. In fact, thinking carefully about what is around me in these memories, I seem to be in a hall or a library, rather than in a classroom.

    Outside of school, I remember playing a game in each new playground, where I would try to do every part of the play equipment exactly once without crossing my path. Would I have to interpret the slide as both a sliding down and a climbing up in order to do it? Would I end up trapped on the top, or could I finish on the ground where I started?

    At home, we’d build elaborate maze-like cubby houses out of spare mattresses and sheets (we lived in a house where visitors often stayed over). I remember planning these out with my brother with explicit conversations of how we would fit more rooms and pathways into the space of our shared room. I also remember spending hours making designs with a ruler and compass. Or by folding paper several times and cutting out holes then unfolding and sticking on a contrasting colour.

    It seems that for me, geometrical play holds the strongest positive mathematical memories from my primary school years.

    Indeed, my very first memory of primary school is about geometrical play. It’s the moment I walked into my kindergarten classroom for the first time. We walked into a carpeted play area, and the desks and blackboard were some distance away at the other end of the classroom. Here in the play area was a bookcase filled with big thick brown blocks. Some of them were on the floor being made into a car track by some other children. I remember immediately wondering about how the various straight and curved pieces might fit together. I have some vague memories of tying various combinations on other days in kindergarten.

    Earlier than this, one of my only memories of Happy Days Pre-School was getting out the giant foam blocks from the store room under the building and playing with them on the grass.

    It’s funny that so many of my positive mathematical memories are geometrical when now I also have such a love of the structure and behaviour of numbers. Maybe that came later, though my mother says as a very young child I was always “playing number and letter games in my head”. I myself can’t remember doing that, but my mother is a very astute person and I am not about to doubt her observations.

    My earliest memory of any kind is of a cool hard flat greenness. My mother says this is probably a memory of the back verandah at the house we lived in before I was two years old – it had a green-painted concrete floor. I wonder if other people’s earliest memories are about feelings of space and colour. If so, maybe it means we’re all geometers from birth. Or maybe it’s just me.

    What is clear is that it’s hardly surprising that I ended up doing a PhD in finite geometry even though the original undergraduate degree I enrolled in was mathematical physics. I think the fundamental pull towards that geometrical play was calling me all along, considering how strongly I gravitated towards it in primary school despite the rest of maths not being so inspiring.

    If you’re reading this, I don’t know what you might learn from my story. But for myself I realise I am right where I belong.

    This comment was left on the original blog post:

    V Lakshmi 27 September 2017:

    Nice article! Infact, childhood memories have something to learn and plays an important role in future they are like the learning stages check this peace very interesting http://www.publicdebate.in/childhood-happiest-part-life-agree/ 

  • Actually, I am a maths person

    I am a mathematician and a maths teacher. Therefore it is an occupational hazard that any random person who finds out what my job is will respond with “I’m not a maths person.” The most frustrating people are my own students who I am trying to tell that my actual job is to help them learn maths. I used to tell them that there was no such thing as a “maths person”, but I have recently come to the conclusion that this is a lie. There is definitely such a thing as a maths person because I am a maths person.

    Let me explain.

    I used to think that the phrase “maths person” meant “a person who naturally finds maths easy and without working can do all the maths”. I’m pretty sure a lot of people do mean this when they say they are not a maths person, as if I’m going to force them to knuckle down and learn complex differential geometry at any moment.

    But it occurs to me that a more literal interpretation of the phrase “maths person” would be “a person who is maths”. That is, a person for whom maths is part of their identity. And in that case, there is absolutely no denying that actually, yes, I am a maths person.

    Maths is a huge part of my identity as a person. I have a favourite fraction (3/8), and a favourite fraction fact (1/3 + 1/6 = 1/2). I love the classification of quadrilaterals. I can’t help but see shapes in a building, or try to tell if a friend’s age is a prime on their birthday. I actively seek out puzzles to try. For goodness’ sake I wear home-made maths t-shirts to work every day!

    Of course, maths is not the whole of my identity. I am a Christian, a husband and a father. I love to read children’s books aloud, and to write stories, and to draw and to sing. I design board games for fun. It’s just that maths is a big part of who I am. I simply would not be me without my love of maths.

    So when I hear a person who says they’re not a maths person, maybe they mean that maths is not a part of who they are. Which is perfectly acceptable, to be honest. Maths doesn’t have to be an overtly obvious part of everyone’s personality!

    Still, I suspect a lot of people actually see not liking maths as a part of who they are. I wish they maybe allowed themselves to have a tiny corner of themselves to be a maths person. Maybe a maths little toe, perhaps. If only so that they can incorporate approaching maths into their study of, say, nursing or economics or teaching. What frightens me most is how difficulut it is to help people when they don’t see something as part of their identity. I know I can be gentle and calm and patient and encouraging, but I still worry how much of a difference I can really make.

    I am also afraid that they might look at me – clearly a maths person – and be intimidated by that part of my personality. Yet I can’t stop being who I am. I can only hope that my playful approach to it might alleviate some of that identity threat. Maybe seeing it as play will allow them to do it without seeing it as a change to their identity?

    That descended a long way into despair in only a couple of paragraphs, I’m sorry. But once I noticed that there was such a thing as a maths person, it really did create this spiral of doubt. I’d love to hear some words of wisdom from the people out there, so please do leave a comment or join in the conversation on Twitter.

    These comments were left on the original blog post:

    John Rowe 8 June 2017:

    Such a nice post, David, I really enjoyed reading this. I also see myself as a maths person but have great difficulty in describing it in that way for perpetuating a fixed mindset towards learning maths. One thing I still hear some people say, which I resent, is having a “maths brain”, which I think is much different to being a maths person. I do think, and worry, that when people see me as a maths person, some think it’s because I have a “maths brain”. Not sure if that made much sense… haha

    Great post – it resonated with me significantly.

    David Butler 9 June 2017:

    Thanks for the reply John. I agree completely about the idea of a “maths brain” being unhelpful. In fact, there’s a blog post on my list waiting to be written all about that. It looks like I’d better do that one next.

    Telanna 8 June 2017:

    I wonder if by the time we become adults with jobs and families of our own, our identities feel like they all the pieces of them fit together like a puzzle. And after all this painful teenager/young adult self-exploration it feels comfortable. Changing something that you alreasy settled upon might not be as fast and would require multiple experiences.
    It took me about 2 years of hanging around #MTBoS and actively seeking engaging mathematical experiences to turn from “definitely not a math person” to “becoming a math person”. But I had intention; what if someone doesn’t?

    I think you are right about playful and enjoyable experiences that can nudge people to looking at math differently. My workshop with most engagement this year was about manipulatives when even self-identified “not math people” had fun with hands on puzzles. I remember one of the comments, “Think I’ll get some wine and continue on the weekend.” I think the math/play/art events that you organize at MLC are great way to invite people in. Maybe “math person” will never become a big part of their identity. But then maybe “not a math person” will stop being a part of it. Like, I am not a mountain biker, but I do enjoy taking my bike to the mountains on the nice summer days and stick to the easier trails with beautiful views.

    David Butler 9 June 2017

    Thanks Lana. Even if I can only help people who are seeking, I think I can take heart that the seekers can be helped! The people who do visit me in the MLC are at least seeking to not be unmaths people, and you’ve given me hope that I can help them on that journey.

    John Golden 8 June 2017:

    I also self-identify… I guess the majority of those who read this will be, though it would be great to hear from others. Maybe we can share on FB where we intersect with a more general audience?

    I tend to think of this as the result of a kind of abuse. Not to minimize other forms of abuse, but convincing someone through repeated messaging that they lack a capacity which they really do have (in my belief) is really cruel. That it is done with often the best intentions of a teacher is deadly irony.

    My usual response is to ask what they do or enjoy and then share some of how that is like math to me, and if they were taught in a way that emphasized connections, they’d see that they are doing maths already.

    David Butler 9 June 2017:

    Thanks John. The cowering people do when they hear I teach maths certainly is consistent with a response to abuse. I wonder even more about how I can be a little positive experience on the day I meet them, rather than reinforce their abuse. Asking them about what they enjoy sounds like an interesting approach, and actually I have had some success with that sort of discussion too. That is, helping people realise that my ability with maths uses all the same skills as their ability with, say, poetry. There’s a blog post upcoming about that.

    Gregory Taylor 9 June 2017:

    There’s sort of an interesting distinction there, “doing” maths versus “being” maths. Being a teacher myself, and more to the point having personified something like 50 graphs into people, I can hardly deny having it as a part of myself too. On the flip side though? I am pretty terrible at finance.

    Like, I can calculate a tip… but budgeting, income tax, even knowing my own income, I’d much rather go to the dentist. I guess what I’m saying here is, “maths” is a huge umbrella. People can dislike part of it, even while accepting that other pieces are an integral (ha ha) part of themselves or other people. Trouble is most don’t get past the “dislike part of it” stage, assuming everything under the umbrella is the same, and hence being intimidated. Well, there’s a random thought, any way.

    David Butler 9 June 2017:

    Thanks Greg. That’s a really interesting point. Would I claim I’m not a fruit person because I don’t like all fruit? Or does being a [insert thing here] person mean you have to like all of it all the time? I certainly don’t enjoy all maths all the time, having a similar aversion to things financial as you do. Thank you for the thought.

    Mike 9 June 2017:

    G’Day from the USA,

    I am not a maths person. I say that from the experience of never having found maths to be an easy subject throughout my academic career. I like to joke that I was okay in maths class until letters made their appearance.

    This is not to say that I don’t, at this stage in my life, appreciate the application and use of maths in my life and the world around me. I am profoundly fascinated by the scientific facts that humanity has and continues to uncover thanks in large part to maths. I’m also very fond of using maths and logic to my own advantage in my personal and professional life.

    After reading your blog post, you definitely sound like a maths person; which I would define as someone who is fascinated with and enjoys thinking about and working with mathematics separate from its applications. I don’t share that fascination. Advanced maths to me remain a bit of a mystery. I can grasp the concepts that the maths operate on, or understand what the maths are trying to prove, but it is the how of maths that eludes me.

    To use an analogy, for a non-maths person it’s like being a traveler in a foreign land. It’s fascinating and exciting, but I don’t fully understand it. I can’t speak the language, I don’t understand the culture, I’m not used to the social and physical environment.

    A maths person is like a native of that land. They have an understanding and a feel for the culture. The language comes naturally to them. They can navigate the land of maths with confidence, if not ease. I may, through time and effort, come to understand maths to a level where I am more comfortable, and can get by okay, but I don’t feel as though I can ever assimilate to the point where I will have the same experience as a native of mathsland.

    My educational journey took me on a much different path. I am a lawyer by trade and education, and a philosopher at heart. That’s the land I feel home in. I enjoy thinking about and working through logical dilemmas, moral questions, and the why of human nature and human existence. Like you, I find myself pondering such things after a conversation, or while reading a book or news article, or even while washing the dishes. It’s endlessly fascinating and a significant part of my personal identity.

    I think this is true for everyone. We all have something that truly fascinates us, and for some those things come more naturally than others. For you it’s maths, for me it’s law and philosophy, for others it’s music, or poetry, or science, etc. Like you, I like to share my interest with anyone who has the patience to listen. It’s important that the “natives” share their interests with the “non-natives.” It makes us all better as people, and deepens our shared knowledge as a species.

    Thanks for sharing your perspective.

    David Butler 9 June 2017:

    Thank you so much Mike for sharing your thoughts on this! It’s a really interesting perspective to me.

    The comparison to a native of a country is making me think of immigrants. I would like people who have come to live here in Australia to see themselves as Australians, even if they weren’t born here. How can I, as a native, make them feel welcome? Even more, I’d like people who are only visiting to maybe not see themselves as Australian people, but maybe at least see themselves as “Australian people people” – people who like being around Australians even if they don’t fully understand them. It sounds to me like you’re happy to be a “maths-person person” and for others to be “philosophy-people people”.

    On that note, I would have to say that while I certainly wouldn’t consider myself a philosophy person, I definitely wouldn’t say I’m a non-philosophy person. That particular handle seems to me to be an unhelpful way to see yourself. From what you say about your appreciation of maths’s place in your life, I wouldn’t consider you a non-maths person at all! Somehow I think we need a middle-of-the-road word that doesn’t sound like it excludes all maths.

    Michael Way 9 June 2017:

    I am a maths person. I remember a co worker ( also a maths person) once say ”we as mathematician (teachers) like to count in our moments of idleness.” Yeah I find my self counting between light changes at an intersection, time between TV commercials, etc. That was one of he first moments I recall calling my self a mathematician and not just a teacher of math and not feeling afraid to say it.

    David Butler 9 June 2017:

    What a simple and lovely idea “counting in moments of idleness”. For me it’s drawing figures in my head. Thank you for sharing.

    Sally 10 June 2017:

    David’s original pat and the ideas of everyone here have made me stop and wonder: Am I a maths person?
    I enjoy thinking about maths, wondering about maths, playing with maths, teaching maths. Is this enough?
    My background is in philosophy which is lucky: thinking about thinking let’s you try on many hats. Sometimes I like wearing my mathematician hat and thinking about maths. Other times I wear my scientist hat and plan experiments; or my artist hat and create new things. There are many hats I wear, but always for me, the attraction is in the thinking that underlies each discipline.
    Perhaps the most important thing about the labels we give ourselves – “maths person” “not a maths person” “becoming a maths person” – lies in the activity of making the distinctions. We discuss what each one means to us and to others; we make distinctions and give examples in our quest to convey our feelings and desires, the things that give us joy (and the things that don’t). This conversation is so important because the same label can mean different things to different people and then lead to all sorts of misunderstandings!
    Thanks David for a post that helps us explore our own definitions and how they interact. Am I a maths person in the way you are? Sometimes? Maybe…. I don’t know yet – but I loved the opportunity to think this through some more!

  • Finding an inverse function

    There is a procedure that people use and teach students to use for finding the inverse of a function. My problem with it is that it doesn’t make any sense, in two ways.

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

  • Holding the other parts constant

    It seems like ages ago – but it was only yesterday – that I wrote about differentiating functions with the variable in both the base and the power. Back there, I had learned that the derivative of a function like f(x)g(x) is the sum of the derivative when you pretend f(x) is constant and the derivative when you pretend g(x) is constant.

    Since then I have realised that this idea actually dictates ALL of the differentiation rules where two functions are combined through an arithmetic operation! It’s everywhere!

    You can read the two blog posts in this series in PDF form here. 

    The titles of the two blog posts in the series are:

    • Differentiating exponentials: two wrongs make a right
    • Holding the other parts constant: it’s everywhere!

  • (Holding it together)

    Last week, I helped quite a few students from International Financial Institutions and Markets with their annuity calculations, which involve quite detailed stuff. One of the more important problems was about how the calculator interprets what they type into it, which is really in essence about the order of operations.

    You can read the rest of this blog post, and the other posts in the series across the years, in PDF form here. 

    The titles of the five blog posts are:

    • The reorder of operations
    • (Holding it together)
    • The Operation Tower
    • Replacing
    • Sticky operations

  • One reason I’ll still use pi

    Every so often, someone brings up the thing with tau (τ) versus pi (π) as the fundamental circle constant. In general I find the discussion wearisome because it usually focuses on telling people they are stupid or wrong for choosing to use one constant or the other. There are more productive uses of your time, I think.

    But for a while I have wanted to add just this one thought to the conversation and now is as good a time as any.

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

  • Where the complex points are

    When you first learn complex numbers, you find out that they give you ways to solve equations that were previously unsolvable. The classic example is the equation equation \(x^2 + 1 = 0\), which if you’re only using real numbers has no solutions, but with complex numbers has the solutions \(x=i\) and \(x=-i\).

    As someone who likes to imagine the physical reality of everything, this has always caused me considerable difficulties. The equation \(x^2 + 1 = 0\) can be thought of as the equation that tells you where the parabola with equation \(y = x^2 + 1\) meets the x-axis.

    Only the parabola with equation \(y = x^2 +1\) doesn’t meet the x-axis. If our complex number solutions are to be believed, then it meets the x-axis in the points \((i,0)\) and \((-i,0)\), but I certainly can’t see those points on my graph. Where are they?

    Presumably there are a whole host of points with complex coordinates, which are points where various things meet that don’t look like they meet. These points must be somewhere, and they must be some place that is somehow related to the graphs I see in the real plane. But where is this place?

    Well, about a week ago, I finally found the place where the complex points are!

    You can read the rest of this blog post, and all seven of the blog posts in the series, in PDF form here. 

    The titles of the seven posts in the series are:

    • Where the complex points are
    • Where the complex points are on a line
    • Where the complex points are on a parabola
    • Where the complex points are on the graph of a function
    • Where the idea came from for where the complex points are
    • Where the complex points are on a complex line (again)
    • Where the complex points are on a real circle

    UPDATE: There was a later blog post in 2016 where I slightly modified the idea from i-planes to i-arrows, and a later blog post in 2024 further investigating the line joining two complex points using i-arrows.