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.

]]>I found out about TMC last year, when Tracy Zager mentioned me in her keynote at TMC16, effectively yanking me right into the thick of it. I could see that this was one of the things that cemented together the people of the MTBoS and I really wanted to feel first hand what it was like.

And somehow I managed to do it. I applied for a Learning and Teaching Development Grant from my University, available to me because I am part of the Adelaide Education Academy, and submitted a couple of proposals for sessions at TMC. Everything was accepted and I was able to go. Some of the process of organising the money and the travel was a bit arduous, with more red tape than I was expecting in order to make it happen, not to mention the close to 70 hours of travel time involved and the longest time I have ever spent away from my wife since we were married. But it really was totally worth it to have this (possibly once in a lifetime) experience.

I really do want to reflect on why I think it was worthwhile, but I’m having trouble doing that right now. So at the moment, I will simply describe what I actually did at TMC, diary style.

**Tuesday**

I left home early Tuesday morning and spent about 35 hours in taxis and airports and aeroplanes to arrive in Atlanta airport on Tuesday night (there’s some mind-boggling timezone maths in there if you want to figure it out). I consider this journey to be part of TMC because I spent a large amount of the time crocheting corals to be used in the triplet of morning sessions I would be running with with Megan Schmidt, called Mathematical Yarns. I also made a timetable and chose what sessions and activities I would go to across the five days I would be in Atlanta.

Tuesday night I was launched right into TMC when I shared a taxi to the hotel with Annie and Greta, with some pretty intense discussion of finite geometry along the way (not sure what the taxi driver thought of us!). When I checked into the hotel I didn’t even make it to my room for several hours because there was a crowd of people hanging out in the lobby/bar and I stayed to meet and talk with them (and to give out Tim Tams and Mint Slices).

**Wednesday**

On Wednesday, I went out on a tour of some of the sights of Atlanta with a small group of other TMC attendees: Megan, Henri and Andria. Megan, Henri and I went to the Civil Rights Museum and learned about the history of the struggle to gain rights for black people and immigrants, including the work of Dr Martin Luther King. This was extremely powerful. We wandered around the Olympic Park and ate in the CNN building food court. Megan and I went to the World of Coke, which was way more fascinating than I had imagined it would be. The innovations in marketing were quite amazing and mildly scary. Also worth noting this pro tip: thongs are not appropriate footwear for the sticky floor of the tasting room! Lastly Megan, Henri, Andria and I went to the Atlanta Aquarium, which was also way more fascinating than I imagined it would be, which is *really* saying something because I had pretty high standard of imagined fascination to meet! Whale sharks! Belugas! Sea turtles! Sea otters! Dolphins! Sea lions! Between and among all this, we discussed maths and teaching and how our lives intersect with that. Oh, and I also saw a bumblebee for the first time.

On Wednesday night, I went to the registration evening in one of the hotel ballrooms, and spent the night talking and playing games with even more people, and giving out Tim Tams to the crowd. Also my roommate Andrew arrived, though I didn’t see much of him until the next day!

**Thursday**

On Thursday, the program of sessions began at the School which was hosting TMC. We started bright and early with the newbies session. I was late to this (because I had met Megan to do some last planning and smoothie breakfast along with Stephen), so all I heard about was how to use Twitter. I have a suspicion that before this there was advice about surviving your first TMC, which would have been nice, but it’s my own fault for being late. Then we had an opening session with the official welcome and those of us giving morning sessions each doing a one minute pitch to the crowd about what we were going to do. The one Megan and I were doing was called “Mathematical Yarns” and our pitch was literal because I threw corals into the crowd (though not far because corals are not known for their aerodynamic properties). Then off we went to actually give the first of our three sessions. How our morning sessions went deserves a whole blog post of its own, which I’ll do later. Spoiler: it was wonderful!

At lunch time, food trucks came to the School and we lined up in the sun to get something to eat together in the Dining Hall. Of course I also pulled out a game to play with people. After lunch we went into “My Favourites”. This is where anyone can go on the list to present for 5 to 10 mintues on something they like about a resource, a teaching strategy or anything at all really. Today I was particularly impressed by Sam‘s description of the “math joy bell” to make mathematical joy audible in his classroom.

Then we had the first keynote, which was by Grace Chen. It was about how teaching is political because it intersects with stories about who gets to do things or have things or be things. She was immensely brave and honest telling us about her own story and the stories of her parents and grandparents. I was partiularly struck by her comments on the power of listening to someone’s story how they tell it, rather than how others tell it to us.

The keynote was followed by two afternoon sessions. I went to Max and Malke‘s one about bodyscale maths learning and made some cool things out of rolled-up newspaper and sticky tape. I got talking to people after this session and so was a little late to Megan’s one on the patterns you can find in number spirals, though it was still great to be part of some investigations with the people there, like Christopher, who co-opted several of us into a collaborative effort to investigate quadratics.

At the end of the day there was supposed to be speed dating, but I just wasn’t sure I could cope with meeting any more people that day. In hindsight maybe I should have done it after all, based on the glowing reports I got from others about it. Oh well. It allowed me to have some more quiet conversations. Plus Christopher gave me a signed copy of Which One Doesn’t Belong!

On Thursday night, we all went to a restaurant for the newcomer’s dinner. I had lovely conversations with some great people. The hashtag controversy hikacked the discussion towards the end, but still it turned out ok. After dinner I went back to the hotel and spent most of the night playing games and singing. Later, I got back to my room and prepared for my My Favourites session tomorrow and had a longer chat with Andrew than I’m sure he really wanted before finally falling asleep much earlier in the morning than I had planned!

**Friday**

Early this morning, Andrew and I made a failed attempt to go to get bread (and breakfast) which made us late for My Favourites. One consolation was that I got to see two live squirrels as we drove along, which I have never seen before. We weren’t too late for me to do my My Favourites, which was all about SQWIGLES. It was nice to share something to the whole of TMC that was important to me, even though it was a lot less practiced than I had hoped. (YouTube Video here, if you’re interested.) I was mildly surprised by how much people were interested in it. Of course I very much tried to listen to the rest of the My Favourites, but I was coming down off the thrill/terror of doing my own.

After this it was the second of our three Mathematical Yarns sessions. It was an island of calm and interesting discussion in the wild tumultuous sea that is the rest of TMC.

Lunch today was brought in by the school and so we all sat in the Dining Hall together to eat. There was a true community atmosphere to this that that I really loved. I used my time well in the lunch line by teaching Taylor how to crochet as we walked along, with a couple of other listeners on either side.

After lunch it was another My Favourites. I was particularly impressed with Pam‘s ideas for encouraging students and teachers using each of the five fingers with a meaning, especially the pinkie promise of I will be here with you. Graham Fletcher gave a keynote session about various ideas he learned from the MTBoS, including some interesting estimation challenges. The strongest idea I took from it was that everyone sees things differently – both different students and different teachers – and we need to realise that different is still smart and work together to learn.

After the keynote I did my afternoon session on One Hundred Factorial. I was deliberately late this time because I set up a One Hundred Factorial play space in the Dining Hall in order to bring my participants back to it when I’d finished presenting. I actually left this play space set up for the rest of the weekend and it was very gratifying to see it being used on and off throughout the next several days, even by people who hadn’t been at my session. But I’ll talk about that more in its own blog post.

And then there was one more afternoon session to go. I wandered around aimlessly and found my way into Bob and Scott‘s session about engaging with reluctant colleagues. I think this was something I really needed to be in, even though I had originally planned to go to a different session. (Still I was bummed to find I’d missed Kent’s session on base 8.) The biggest message I took from it was to start with your colleagues’ strengths and ask them to help you as a way of opening discussion.

I had the unexpected pleasure of having a group of people invite me out for dinner because they wanted to talk to me more about One Hundred Factorial. I don’t remember if we actually did talk about it in the end, but it was a wonderful time of fellowship and maths discussion nonetheless. Thanks to Jill, Kent, Jasmine, Ethan and Taylor I will always laugh when the mode is 1.

After dinner it was the second annual Trivia Night. I didn’t have a group pre-organised, so I just wandered in and a table of ladies flagged me down and said I had to join their group. It was loud loud fun. I particularly liked the round of books with numbers in the title. (Though again, the mode was 1.) When trivia was over, I wandered out into the bar/lobby to discover Malke and Max doing some Sierpinski Sponge and some pattern-machine/accordion music, which was fun to be part of. Later I pulled out Home in One Piece and Justin was unreasonably amazed by it. So much so that if I don’t blog about it eventually, I’m sure he’ll kill me. We had some pretty deep discussion along with Taylor until I finally had to call it and stagger back to bed.

**Saturday**

Andrew and I were up bright and early again and this time we really did manage to get to Whole Foods, though not without missing a little of My Favourites. I did get to see Bob show us how-old.net, which will attempt to guess your age from a photo and therefore provides a nice set of data to analyse.

Then we had our final Mathematical Yarns session and set up a gallery for everyone else at TMC to peruse our work. I was so grateful to Megan and all our participants for a really lovely time. (But as I said I’ll blog about that separately.)

At lunch time, I picked a table and slowly made tray after tray of Fairy Bread, which I gave to everyone in the whole room. People were sufficiently impressed with Australia’s favourite children’s party food. I did manage to squeeze in a moment for a quick sandwich, which I have to say was a welcome change to all the aeroplane and takeaway food I’d eaten so far!

After lunch it was My Favourites again. Joey began with a description of his Play With Your Math project, which reminded me a lot of One Hundred Factorial. I particularly liked the focus on making the puzzles very accessible with clear design and few words. I may have gotten nerdsniped by one of his puzzles and not paid too much attention to the other My Favourites sessions after Joey.

The final keynote came next, which was given by Carl Oliver and basically asked us to be brave and vulnerable and to just push send on our blog posts and tweets. There was also some fascinating analysis of when various people first used the hashtag #MTBoS. This made me reflect on what it was that brought me into the #MTBoS and I went trawling my old twitter to see how I had thought about it at the time. It was most interesting to reflect on my persistence with the #MTBoS despite varying levels of engagement at the beginning.

And then, there was one last afternoon session to go to, and I decided to go to Jonathan‘s one on Calculus for Algebra teachers. While I know a lot about calculus, I was interested in the sorts of things the others might like to know and how they responded to the explanations. I was so impressed with Jonthan’s gentle and respectful approach where he listened to the participants’ needs and responded to them gently and respectfully. I got into some quite deep discussions with Nik, which distracted the other participants, which I am very sorry for, everyone!

In the final afternoon timeslot, there were “Flex” sessions, which allowed people to do impromptu sessions on things that came up across the week. Malke and I decided to do one on Bodyscale Prime Climb because we had been itching to play it all week and hadn’t had the chance. To our surprise, eight other people arrived to join in too. We all had a great time walking on the numbers and noticing all sorts of patterns and relationships between the numbers. At the end of the session, I gave Malke my set of giant Prime Climb cards, which was completely worth it just for the expression on her face!

After the end of this session, I was invited out to tea again with a bigger crowd. After they waited for me to talk with my family because I was missing them very very much, we had a lovely conversation about all sorts of things, not least of which was the dangerousness of Australian spiders.

Finally, I made it back to the hotel for Games Night and played Home in One Piece and Flamingoes and Hedgehogs for many hours, plus had some wonderful talk with Max about the process of designing games, until the hotel staff chucked us out of the ballroom. Somewhere in the middle there, John asked me and a few other people (Jasmine, Edmund, Jim) to help make a video for his online Calculus students. We went up to Jim’s room and had a very pleasant conversation about derivatives and integration and problem-solving. It was an honour and a blast to be a part of this.

**Sunday**

I got up bright and early and packed up all my stuff so that we could check out of the hotel before the final sessions. I got there nice and early to pack up the coral display and the One Hundred Factorial stuff only to discover people using them, so I just left them there until the last minute! Anyway, I had to go practice for the traditional TMC song, which I was roped into by Julie when she discovered I could sing on Wednesday night.

The final My Favourites was even more wonderful than the rest because so many people did one. I was particularly impressed with Glenn‘s about the power of small changes in the words you use, especially the words you use to name people, like “learner” versus “student”, and something other than “guys”.

After this, we did the TMC song, which was a bit of sustained ridiculousness celebrating the things that happened across the week, written by Sean, Julie, David and several others I shamefully never learned the name of! I really didn’t realise just how much of a public figure I had been at TMC until I saw just how many of the photos I or one of my games was in.

And suddenly the offical stuff was over! Lisa announced the date and location of TMC18 (Cleveland, July 21 2018) with the flair of a reality singing competition presenter, and that was it. There were a lot of hugs and “will you be coming next year”s. I got some lovely thank you cards from people, which made me cry completely unexpectedly. I gave away a lot of my stuff to people as souvenirs, including the giant Galaxy to Taylor, the giant Jigoku to Anna and the Dragonistics cards to Bill (which I originally got from Nic Petty) . By this stage I had arranged to wear all 12 of my home-made maths t-shirts, plus the TMC17 one Glenn brought me.

Yet it wasn’t quite over yet. Taylor and Emily invited me out to lunch and we had a most interesting discussion about the nature of MTBoS and TMC and welcomingness and joining in.

And then it really was over. I did try to connect to people who were still in the hotel or airport, but always contrived to miss them. Finally, I was on the first of three planes home and 33 hours later was relieved to run into my beautiful and longsuffering wife’s arms.

**Conclusion**

So that was what I did at TMC17. Looking back I crammed weeks of stuff into a few short days, and as I said to Megan, it was unreasonably awesome. It is going to take me months to process all of it, so keep an eye out for the various reflections to come. For now, thank you to everyone for welcoming me and making me part of your family for this week and beyond.

]]>65536 is my favourite power of 2. More specifically, it’s 2^{16}, which means you can make it by starting at 1 and multiplying by 2 sixteen times. Even better…

But this cool stack of powers is not why it’s my favourite powet of 2. It’s my favourite power of two because of its connection to two very cool ideas in maths.

Firstly, 65536 is the last known power of 2 for which the next number is prime. It’s known that if a number one more than a power of 2 is a prime then it must be 2^(2^n)+1 for some n. The first five are all prime

- 2^(2^0)) +1 = 3
- 2^(2^1)) +1 = 5
- 2^(2^2)) +1 = 17
- 2^(2^3)) +1 = 257
- 2^(2^4)) +1 = 65537

Fermat apparently conjectured in 1650 that they were all prime, which is why numbers of the form 2^(2^n)+1 are called “Fermat numbers” and if they’re prime they’re called “Fermat primes”. But so far, no more Fermat primes have been found. That is, every bigger number of the form 2^(2^n)+1 that we can calculate has been found to not be prime after all. Yet we haven’t been able to prove that there are definitely no more of them.

Isn’t that amazing? In close to 400 years we haven’t been able to find any more Fermat primes, but neither have we been convinced beyond a doubt that there aren’t any. I think it’s awesome that in maths there are things so simple that at the moment are unknown.

Secondly, 65536 is the only known power of 2 with no powers of two in order among its digits. Every other power of two where we have a list of most its digits, you can cross out some of the digits and have a power of two left behind. But we just don’t know if somewhere out there there’s a really big power of 2 that again lacks any smaller powers of two among its digits. Even stronger than this, every other power of two we’ve calculated has a 1, 2, 4, or 8 among its digits, but again we don’t know if somewhere out there in the distance there might be one that lacks these four digits.

What makes this surprising is that there is a perfectly good pattern to the final digits of the powers of 2. The last digit goes in the pattern 2, 4, 8, 6, 2, 4, 8, 6, … and the last two digits go in the pattern 04, 08, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 04, … You’d think that the repeating pattern of the final digits might make it easy to tell what digits were in a power of 2, but it’s not nearly so easy.

What’s even more surprising is that the same concept for prime numbers is completely solved. If you cross out some of the digits of a prime number you might have a prime number left behind. For example, the prime number 16649 leaves behind the prime number 19 when you cross out the 664. So which prime numbers have no prime numbers among their digits? Well there’s exactly 26 of them and they are 2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049. That’s it. All of them. Every other prime number has at least one of these in order among its digits.

This is from a paper in 2000 from Jeffrey Shallit: “Minimal primes,” J. Recreational Math., 30:2 (1999–2000) 113–117. He talks about it here. Within a set of numbers, he calls the “minimial set” those ones with none of the others in the set in order among their digits. He references another author’s theorem which says that given any set of numbers, the minimal set within it must be finite.

Isn’t it amazing that the prime numbers with all their apparent randomness have allowed us to find their minimal set, but the powers of two with their obvious regularity haven’t?

So that’s why 65536 is my favourite power of 2. It represents to me some cool ideas, and more than that, it reminds me that maths is far from all done in the distant past, it’s got unanswered questions alive right now.

]]>Prime Climb is a wonderful game by Dan Finkel (aka @MathforLove), which you can find out more about here. The board is a path made of the numbers from 0 to 101, coloured by an ingenous and beautiful system. Each player has two pawns which they move around the board by applying numerical operations to the number the pawn is sitting on. If you finish on a red prime you get an action card to use now or later. If you finish on another pawn, they go back to start.

The first time I played it at work back in December 2016, we had so many players we decided to play in teams of two. I have to say I enjoyed playing in teams so much more than playing for myself. There was someone to talk to about how we would move our pieces and I have to say the talking about what was happening in the game was the most fun part for me.

Enjoying the introduction to #PrimeClimb at #100Factorial puzzle group. Thanks @DavidKButlerUoA @reSolveMBI pic.twitter.com/99xojgffnH

— Matt Skoss (@matt_skoss) December 14, 2016

**The idea of bodyscale Prime Climb**

Playing in pairs gave me this most fabulous idea. Each team had two players and each team had two pawns. What if the players *were *the pawns? What if we could actually *walk *on the board? That would totally take this game from wonderful to wonderfully *awesome! *

Unfortunately, the rigours of life and work meant that the idea had to go on the backburner for five months. Moreover, I had a couple of issues to work out with the walk-on version: how would I deal with the fact that I need two 10-sided dice, and what do I do about the deck of action cards you draw when you land on a prime? It occurred to me I could use spinners instead of dice, but try as I might I couldn’t find any that would allow me to make the spinner myself.

But then a couple of things came together that made the dream possible. First, I was in the Reject Shop and found travel Twister for $2 each, which meant I finally had a cheap spinner I could use instead of dice. Also, one of my staff/students played the game at a games night and reported his frustration that you couldn’t move the other players more often. This gave me an idea of an easy way to replace the cards by simply spinning one spinner and allowing you to apply that to the other players. Finally, I was home sick with a chest infection with plenty of time to individually design all the cards and easy access to a laminator. And with that, Bodyscale Prime Climb was born! I was itching to try it out at the next available One Hundred Factorial session.

**Playing bodyscale Prime Climb for the first time**

And it really was wonderfully awesome when we did play. We laid out all the cards in a back-and-forth line on the floor.

Last #100factorial of the semester: #bodyscale Prime Climb and skyscrapers. Super fun! https://t.co/M0serlYoWe pic.twitter.com/Ug1vNdJ27V

— David Butler (@DavidKButlerUoA) May 31, 2017

From the outset we had people walking up interested in what was happening and trying to figure out the colouring scheme on the cards. It was one of the best levels of engagement I’ve seen at One Hundred Factorial this year.

We played with three teams of two using the same rules as for ordinary Prime Climb, except for what happens when someone lands on a prime. In that case, the team spins one spinner, and then applies that number with a + or a – to any player on the board (whether in their team or not). We also ignored the usual rule for rolling doubles (which is to actually count it as four of that number) to counteract the extra freedom expected from our new land-on-a-prime rule. Here’s an action shot:

Body scale Prime Climb at #100Factorial thanks to @DavidKButlerUoA. cc @mathinyourfeet @MathforLove pic.twitter.com/MPFyHch97q

— Amie Albrecht (@nomad_penguin) May 31, 2017

I thoroughly enjoyed the game because of how it felt and because of the talk it generated.

The feeling while playing it was very different from the hand-scale Prime Climb. There was something totally engaging about standing on the board. You could really *feel *how far you had to move to get where you want to go, and you had to look all around you for numbers you might be able to get to. Bumping people back to start felt so much more intense when you actually forced a person to walk all the way back to zero.

The talk between players was also very different from the hand-scale Prime Climb. I already mentioned how I loved the talk that happened when we played in pairs before, but this was at a whole different level. It’s impossible to hide your discussion when your partner is standing three metres away from you! This meant that all our talk was much more public so that everyone could follow what was happening, even the bystanders trying to figure out what we were doing. Also, because the board was so big and pointing was therefore so inaccurate, we had to be a lot more explicit about our language to each other. On the other hand, having the pawns being different people meant it was easier to talk about where they went. There was a lot of talk like “If we add 2 to me and multiply you by 3, then…”.

**Some thoughts**

One lesson we learned along the way was that we really shouldn’t have tried to get both players to 101 but instead have the shorter goal of just getting one player to 101. This would have made for a much quicker game and allowed us to play more than one in the time we had, and to more quickly get our bystanders into the game. I’d also like to get some coloured hats or sashes or something to wear so that it’s easier for observers to tell who is on what team.

I’m not 100% sure that our land-on-prime rule was the best replacement for the cards. It would be kind of cool if you could save them up to use on a later turn, but my memory’s just not that good and I don’t want anything that requires us to pass out or hold onto cards. One alternative that just occurred to me is that if you land on a prime, you can add or subtract one of the prime factors of the number where the other player is standing to any pawn on the board. That would make it very highly strategic!

The spinners worked really well. To start with, I held the spinners and did it for everyone. Later when we had an extra person, they became the spinner like in a game of Twister. It worked really well to have this extra person because they could freely move around to help people think and explain what we were doing to observers. I’ll have to appoint an official spinner next time I play!

I do think that maybe I could make some dice instead of spinners using 12-sided dice. I have no shortage of 12-sided objects to use as dice right now! My idea was to have 1-10 on each dice, but then an extra 2, 5 on one and an extra 3, 7 on the other. I think it might work really well. Still, I will miss the spinners, which had a certain charm about them.

**The moveable board**

One thing I didn’t really expect was the swirling vortex of ideas that were created by having the board made up of cards. The moment we started playing I started thinking about all sorts of ways that I could rearrange the structure of the board and thereby change the way the game felt.

The first thing I want to try is setting it up with one long straight line. Then there would be a real feeling of *distance* when you multiply or divide. Indeed, you could guess where double or triple your number is by doubling or tripling the distance between you and the 0 card. I also want to try it in the traditional 100’s chart with each row going left-to-right to match up how movement across that chart feels. Adding 10 would be a particularly pleasant experience I’d wager.

I also wonder how the game might work if we arranged the cards not in numerical order. Then to add or subtract, we’d have to actually calculate where to go rather than just step it out. I’m not sure I want to lay them out completely randomly, because it would be very hard to find the number you wanted. What if you laid it out like the 100’s chart but started a new line every time you got to a red prime? What if you had all the composite numbers in order in one row and the primes in order in another? What if you made a big Venn diagram of the multiples of 2, 3, 5, 7? I want to feel the feeling of what it’s like to adding or multiplying in these situations and move from one collection of cards to another. Not to mention the very interesting task of simply arranging the cards themselves!

On that note, I actually reckon I might make a small version of the Prime Climb cards to use in the original game. How cool would it be to sit and sort the coloured numbers to see what sorts of patterns you could find before actually playing the game? (EDIT Aug 2017: Amie Albrecht has done just this with her class of pre-service teachers. Check it out!)

**Stay tuned!**

So stay tuned! At the July One Hundred Factorial session I want to try all the variations we have the energy to try and see how they turn out. And then a week later I’ll be at Twitter Math Camp and I’ll be trying it again there. I’m going to have so much to write about so check in in a couple of months to see how I went ok?

**The resources**

If you want to make your own bodyscale Prime Climb board (or indeed a set of Prime Climb number cards) then you can access a PDF of the cards and the spinner template at the link below. (I don’t recommend printing it at home because of the density of the ink! Best to use your workplace’s laser photocopier/printer.)

If you do play it then please let me know! I’d love to see how it turned out for you.

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

I fear I may have been just a bit rude when I almost yelled "There is no such thing as a maths brain!"

— David Butler (@DavidKButlerUoA) June 5, 2017

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:

Ute fur 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.

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