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

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

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

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

]]>MR JOHNSON’S RAINBOW

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 fallAnd 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 soBut 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 sayFor 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 featsHe 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 knowAnd 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.

]]>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.

]]>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.

]]>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 re^{θj} 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 e^{iθ} 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 e^{iθ} 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)!

]]>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, or 3 groups 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.

]]>Home in One Piece is a game for two players that is played with play-dough. Each player has eight blobs of play-dough and the goal is to join the play-dough together and move it across the board, so that finally you win by having all your play-dough together in one blob in the home zone (hence the name Home in One Piece). The original version was a pure strategy game, but then I remembered that I prefer games with a chance element, so I created three special dice to control how many of your blobs you are allowed to manipulate during your turn.

To make the game you need the following components:

- A game board.

This is the size of four sheets of paper, laminated so the play-dough doesn’t stick. Download a printable template with instructions here. - Three special dice.

The dice have faces that are either blank or show one blob, each with a different distribution. The distribution is described in the rule sheet below. You can also download a printable template with instructions for making your own dice here. - Two blobs of play-dough in different colours (about 100g of play-dough each).

There is a recipe in the rule sheet below. - The rules.

Download the rule sheet here, which also includes a description of the dice and a recipe for play-dough.

Here is a YouTube video of a game in action, so you can have a better idea about how these rules work in action.

There are so many things I love about this game, if I do say so myself. I love the physical and three-dimensional nature of it, with the manipulation of the play-dough in space. I love the contradictory feeling of freedom and constraint of being able to make the play-dough whatever shape I want, yet having to hold it down in one spot. I love the fact that the movement condition is continuous, whereas in almost all other games it is discrete. I love sitting on the same side of the board as my opponent, so that we are both looking at the same game and reaching past each other to play. I love the fact that I’ve arranged the dice so it’s possible to roll 0. (If you’ve played my other games you might notice that particular fascination of mine.)

Mathematically I love thinking about what it means to be “one blob” and about the interesting topological shapes that happen when people join and rejoin blobs. I am particularly in love with my dice, which have been arranged carefully to give the right probabilities for the different numbers with only one dot per face. (That was a whole wonderful investigation, I can tell you!) I love secret problem zero of dividing your blob into eight equal pieces.

But most of all I love watching people play it. I love the shapes they make, and the realisations they have about what is possible and not possible — I love the walls and lakes and bridges people build, especially bridges over bridges over bridges. I love the cries of despair when the play-dough breaks and when someone rolls zero. I love how lax people are about their play-dough touching at the beginning and how unforgiving they are about it at the end. I love watching people wander over, fascinated at what the hell is going on with the play-dough here.

I hope you enjoy the game as much as I do. Please let me know how it goes!

To finish off, some tweets of the game in action (note this is the first version of the game, which had the rules printed on the game-board.)

Home in One Piece drawing a crowd at #tmc17 pic.twitter.com/CQs2Em0nl7

— David Butler (@DavidKButlerUoA) July 30, 2017

]]>Another game in progress #100factorial pic.twitter.com/w1ADRPUHob

— David Butler (@DavidKButlerUoA) November 23, 2017

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.

]]>I was planning to do this investigation over Summer, and at the time I got Lewis and Tobin to help me get the data on the prime tweets over the previous few months. Tobin did some analysis of his own, but I haven’t looked closely at it because I wanted to have the fun of doing it myself. Not until now have I had the chance to actually do it. So here goes! What you see here is a record of my thoughts and investigations as I did them.

I have data on primes tweeted between the 11th September 2016 to 26th January 2017, a total of 3217 tweets. I thik there’s a gap of time in there with no data, but for our purposes it should be enough. You can get the raw data here: likeable-primes-data.csv

First up I’ll just have a look at how many likes each prime has gotten and see what we might see.

Oh my! Well one prime in particular is way more likeable than all of the others, and there’s a couple more there that are quite a lot higher, but not nearly in the same league. Let’s have a look at the top ten and see what they are.

Well would you look at that top one?! It’s the first several digits of pi. So it seems that the *most* likeable thing about a prime is *being pi*. It seems you have to be right on the dot though – being close just isn’t all that likeable as this list of primes starting in 3141 shows:

I could try to search to see if being the digits of other special numbers is likeable, but it seems the digits of other special numbers just aren’t prime. Pi itself was last prime at 31, which doesn’t stand out as pi-ish, and it’s not going to be prime again for quite some time. Phi and e aren’t going to be prime until seven digits, and the square root of 2 won’t be prime until after 50 digits, so it’s going to be a while before I can check the effect of this on prime likeability. (Check out pi-prime, e-prime, phi-prime and this wolfram alpha search.)

I wonder what it is that made those other highly likeable primes have so much love? I see in the top ten list a whole lot of primes with lots of the same digit, so it seems repeated digits is something highly likeable.

I might come back to that later because I also see 300007 and 299993, which are the first primes before and after 300000. Maybe there’s also something in primes being close to milestones. I’m not sure if there’s any other good milestones in this range of primes to test this theory. Let me go searching for the prime tweets near to other milestones…

Aha! Just a couple of weeks ago we reached the 400000 milestone and there was a spike in likes before and after. There were also spikes when 200000 was passed in 2015 and when 100000 was passed in 2014. So yes it does seem that primes before or after milestones are liked more. It’s interesting that 199999 got so many more likes than 200003. I’m wondering if it’s to do with the repeated digits thing that I mentioned earlier.

So what about these repeated digits then? It seems primes with a lot of repeated digits get more likes. There’s a lot of factors there that might be at play – is repeated digits in a row more likeable than separated? How many repeated digits do you need to get more likes? So many questions!

Well first I’ll count how many of each digit there are in every prime, and I’ll find the maximum number of repeated digits. I’m not looking at repeated digits in a row right now. I’m not sure how to do that yet. Let’s look at the relationship between highest number of repeated digits and likeability.

Oops! That 314159 is making it hard to see what’s going on here! Those other two really big ones could make it hard to see too. I could remove those top three, but I don’t want to lose the fact that they are there. What I’ll do is replace them with something just above the next one down, like 150, 155 and 160. Let’see if I can get a better look at what’s going on down the bottom there after that.

This is much better — there’s definitely something going on there. Primes with five repeated digits are certainly more likeable than primes with less, and four repeated digits definitely seems to increase the chances of likeability.

My attention is drawn to those few extra-likeable ones leaping out of the clump of primes with digits repeated three times. I want to look closer at those.

Some of those are rather nice, but there’s nothing I can see that they share which makes them particularly likeable. Let me widen the search to include the next few most likeable.

Ah! I see most of these have their triple digits all in a row as opposed to separated. A lot of them also have a double-digit too. Other than that, I can see ones with an alternating pattern. Those are going to take a bit of learning for me to figure out how to find them…

Phew! That was some hard work! And unfortunately it doesn’t really tell me anything that much different than the number of repeated digits ignoring the number in a row.

Let’s look at them together: in the graph red is “in a row” and blue is “repeated at all”. If a prime already has repeated digits then having several in a row might make it a little more likeable, but I don’t know if it was worth all the effort to figure out how to get R to count them. The graph is pretty though.

I’m not sure I want to figure out how to search for an alternating pattern. Someone I’m sure will say “just use regular expressions” and I would say in response that I don’t know how to use regular expressions and I’m not sure I care to learn right now. Plus I suspect it probably doesn’t add much more to the likeability compared to just having repeated digits in any order.

Well that pretty much exhausts the things I noticed looking at the most likeable primes. The first thing anyone has suggested when I have mentioned I was doing this was that perhaps time when it was tweeted has an effect, so I might as well take a look. I think the primes with above 50 likes can be attributed mainly to repeated digits and piness, so I’ll just adjust those like I did before. These graphs show the likes on days of the week or time of day, with the mean marked by a red dot.

There’s variation between days/times certainly, but I don’t see that it makes all that much difference to the number of likes compared to the total amount of variation. Actually, my stats-ear is saying I should back that up with some tests. These ANOVAs say that the day of week or time of day don’t really make much of a difference compared to the rest of the variation.

I suppose I could look at the grander sweep of time to see if there’s anything interesting going on there.

Well first I notice a gap in October. Now that I see it I think I do remember Tobin saying something about some missing data. I also notice that in October there aren’t many primes with a lot of likes. I’m not sure what caused that. I do know that most of those highly-liked primes are repeated digits, so what if I colour by the maximum number of repeated digits to see how that relates?

Wow! It seems that almost all of those primes above the river are ones with 4 or 5 repeated digits, and there just happen not to be many of them in October. I can see a few orange dots in the mix there and I do wonder why those ones aren’t very likeable. Interestingly, there’s a lot of orange dots down the bottom there in January. But there’s also a lot of yellow there which means three repeated digits. Maybe in comparison to the general repetition of digits at the time, they just didn’t seem as special as they did back in November when there had been a repeated-digits drought. (Upon closer investigation, that January period is when we were passing through the 330000’s so we were guaranteed to get double 3’s for a while.)

I also notice that there is an upward bend in the river around 314159 in early November and the milestone of 300000 in late September. I think perhaps the twitter account generates higher levels of attention around an important event, which means that likes are more likely. This might explain the high number of likes for 301333, even though it only has three repeated digits a row: it’s the first prime after 300000 with three nonzero repeated digits in a row, so it got more attention because of the afterglow of the milestone.

The final thing to consider is if there is anything that makes a prime specially *un*likeable. Let’s have a look at the bottom twenty or so.

I can’t see anything in particular that sets these ones apart, which I is the point I suppose! I do feel sorry for poor 324619, which has the dubious honour of being the least-liked prime in this timeframe. (And as a result, it’s no longer the least-liked prime in that timeframe. )

324619

— Prime Numbers (@_primes_) December 15, 2016

I suppose it’s time to sum up. What have I found out here?

The following things seem to make a prime number more likeable:

- Being pi
- Being close to a milestone
- Having a lot of repeated digits, especially if not near other primes with repeated digits
- Being near pi or a milestone

Well. That was fun!

]]>This thin little book is about how words have power to help children learn about reading, writing, learning, themselves and their place in the world. The majority of the book is a list of sentences spoken by teachers followed by an analysis of what those words mean for children’s learning. The focus is mostly on helping children learn to read and write successfully, but don’t let the “children” or the “read and write” fool you — I have so many thoughts swirling in my head about how this might possibly apply to my own teaching, and indeed my life.

Unfortunately, “swirling” is the appropriate word for my thoughts right now. The fact that the book is structured around analysing specific utterances by teachers made it all very concrete, but on the other hand it is making really hard for me to process the information coherently. At the moment it’s just a big cloud of things to think more about, a lot of which overlaps. I’m finding it hard to tease things apart to find something I can apply first, or a way for me to consistently apply it so it’s useful for my students. I’ve decided the best thing to do is to write this post so I can attempt to process it all.

The chapter titles might be a good place to start. Here they are:

- The Language of Influence in Teaching
- Noticing and Naming
- Identity
- Agency and Becoming Strategic
- Flexibility and Transfer (or Generalizing)
- Knowing
- An Evolutionary, Democratic Learning Community
- Who Do You Think You’re Talking To?

Even just listing those titles is helping me focus a bit more. While I was reading it, it might have helped me to keep a bookmark in the chapter heading so I could look back and remind myself what the big idea of the chapter was. Instead I found that I got a bit bogged down in some of the details as I went along and lost the focus. Now that I can look back from a higher vantage point, I reckon I might be able to pull out some bigger ideas…

Chapter 1 is about how much our language has power to *create *reality, in particular the reality of the listener’s identity. If I were to hold on to just one thing from the whole book then maybe this message would be it: I can make the world different for another person by choosing the words I use.

Chapter 2 is about how in order to learn and know what you have learned, you need to notice things. You need to notice how things are similar or different, how they are related or not. And then, things need to be named, so that it is possible to talk about them. This is remarkably similar to the Notice and Wonder idea from the Math Forum people, and to Chris Danielson’s way of getting to geometry ideas via Which One Doesn’t Belong. But here, Peter goes deeper than this. He suggests that you can notice and name not just content, but also your processes as you work as a group, your thoughts about yourself as a learner, the things you have learned so far, and your behaviour. It is a fascinating idea to me that you can apply the same noticing and naming to mental and social processes as you can to the properties of quadrilaterals. Something to hold onto from this chapter is that my words can draw attention to features worth noticing, and the act of noticing itself.

Chapter 3 is specifically about identity. Peter talks about how we construct a narrative with ourselves as one of the characters and the words we use to tell this story shape the sort of person we see ourselves as. We as teachers can make a difference to identity by the words we choose. Something that struck me most strongly was using words that don’t give people a choice to opt out of the identity. For example, the question “What problems did you have?” assumes that there must have been problems, and asking someone what choices they made assumes they made a choice. This is what I want to hold onto from this chapter, that I can give someone courage to be a writer or mathematician by using words that put them into that character.

Chapter 4 is about agency, and in a way is an extension of the previous chapter on identity. The identity in question here is that of a person who has power over their own choices. This chapter spoke to me most strongly as a maths teacher, since maths is a subject where so many students feel they have no choice and that choice isn’t even a thing that people ought to have (as evidenced by the constant request to tell them what to do). Peter advocates talking to students as if they did make a choice, and analysing the choices they could have made. This is one of the biggest ideas in the whole book to me, and I want most to hold onto this one as I go forwards.

Chapter 5 is about transfer, that holy grail of teaching where students are able to apply what they learn in one area to another. Peter pulls together the agency and the noticing/naming from the earlier chapters as the main mechanism for this. More explicitly, the questions listed here focus on noticing explicit connections between things and also exploring the “what if” questions. He ends with a comment about the importance of play, which of course resonates strongly with me. The thing I want to hold from this chapter is the focus on connections, over and above answers.

Chapter 6 is about knowing, and in particular about who holds knowledge and who decides when we know something. In many teacher-student interactions, the assumption is that it’s the teacher who knows and the teacher who decides what is true and when we are correct. Yet really one day when they leave our care, our learners will need to know how to be sure of things for themselves. The thing I want to hold onto here is that I can give my students the power over knowledge. This is especially important in maths, which is set up so that you actually can be sure of things through your own arguments, rather than having to rely on the authority of others.

Chapter 7, while it has a very long title, is really about how our words can help people learn to work together. Peter has a lot of examples where teacher words encourage learners to consider the feelings and ideas of others, and to choose shared goals. He reuses the noticing and naming power of words to help learners notice their own group processes, and the identity-forming power of words to help learners put on the mantle of people who care about others. The thing I can hold onto from this chapter is that words can make group social and cognitive processes explicit in a way that makes them learnable.

Chapter 8 is about the interplay between your beliefs and your words. As a teacher, if you believe your students are not capable of learning something, your words (and your silences) will reflect this. However — and this is the big thing I want to hold on to here — if you choose to change your words, then some of your beliefs might follow. I see this in using SQWIGLES with myself and my staff where choosing to ask open-ended questions changes the ways that students respond to you and therefore ways that you respond to them. Your beliefs about what students have to say can change through this change in your words.

I think I’ve achieved my goal in writing about this book, in that I have a much clearer idea about how I want to respond to it in my work. I have clarified how much of an impact my words can have on learners’ realities, which I knew, but not to the level of specific detail I did before. In particular I think I want to hold on most strongly to the idea that I can help learners to see themselves as having choice and capable of making that choice, changing both their view of mathematics and of their place in it.

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