Category: Reflections

Reflections on learning and teaching and research and life.

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.

24 May, 2018
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.

16 May, 2018
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.

15 May, 2018
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

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)!

26 February, 2018
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!

3 January, 2018
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.

3 October, 2017
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/

17 June, 2017
The Arts student’s maths brain

Yesterday I talked about one of the common responses to people finding out I am a mathematician/maths teacher, that of saying, “I’m not a maths person.” The other common response I get is, “I don’t have a maths brain.” (John Rowe mentioned this in his comment on the previous post.)

This is how I reacted last time someone said this to me:

https://twitter.com/DavidKButlerUoA/status/871637804916195329

It may not have been the best response, but I stand by the sentiment. I strongly believe there is no such thing as a maths brain. Or at least, that all brains are maths brains. I believe all human brains are wired in such a way to be able to learn and do maths, not least because I observe babies engaging in mathematical thought long before they can talk, so that capacity is there in all of us from the beginning. But more than this, I believe that the skills that I use to be good at maths are the same skills that other people use to do other things that they wouldn’t call maths.

I have one specific story to tell about how I helped an Arts student to believe that maybe she did have a maths brain after all.

Earlier this semester (a few months ago now), several student services were invited to an orientation event for Arts students, to make sure they knew about what was available for them. So I went along with a Writing Centre staff member to do our usual joint activity of Numbers and Letters.

A student came along to see what we were doing and happily engaged with the Letters game. She then glanced over at the Numbers game and I asked if she’d like to join in. With a rather green look on her face she said, “I don’t have a maths brain!”

I said, “I’m not sure I believe there’s such a thing as a maths brain.” Then I asked her what she was studying, and she revealed it was mainly poetry. “That’s really cool!” I said. “I reckon the skills you use to analyse and create poetry are tha same ones I use to do maths. Did you want to try a different sort of activity?” She graciously agreed and so I wrote this haiku on the board:

Word points, letter lines:
Fur ute you oft try fey roe.
My geometry

“What do you notice?” I asked.

“It really is a haiku. And there’s a lot of really interesting words in that middle line.”

“Yeah I know right? I particularly like the concept of fey roe. What else do you notice about the words in that middle line?”

“Well, they’ve all got three letters…”

Following this was a most wonderful conversation about the letters they start with and end with, and which letters appear and how many times, scribbling notes on the board. This all culminated in the beautiful moment where the student realised the symmetrical nature of the words and the letters here and made an “oh!” of satisfaction.

“That was cool,” she said, after declaring she had to go. “Maybe I have a maths brain after all.”

This was one of the greatest moments of my entire teaching career, right then.

9 June, 2017
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!

8 June, 2017
Money and me

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

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

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

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

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

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

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

These comments were left on the original blog post:

Paula Krieg 19 April 2017

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

David Butler 19 April 2017

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

Nicola Petty 2 May 2017

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

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

David Butler 4 May 2017

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

14 March, 2017