At every point in the real plane, there are a real-plane’s-worth of complex points attached. The complex points attached to the real point (p,q) are all of the form (p+si,q+ti). That is, the real parts of the two coordinates are (p, q). I imagine them as a plane, with x-axis showing what imaginary part we’ve added to the x-coordinate, and y-axis showing what imaginary part we’ve added to the y-coordinate. The real point itself is in the centre of this plane. I call this plane of complex points the **iplane at (p,q). **Every complex point is in one of these iplanes. The complex point (3+i,5-2i) is in the iplane attached to the real point (3,5); the complex point (2i, -6+7i) is in the iplane attached to the real point (0,-6); and the complex points (i,0) and (-i,0) are both in the iplane attached to the real point (0,0).

I imagine the iplanes as transparent sheets attached at each real point, that are usually rolled up like an umbrella, but can be flattened out to sit on top of the ordinary real plane when you need to. And yes I imagine them as transparent red or pink, because that’s the colour cellophane I had in my original model.

Last time, I did a thorough investigation into the points on a complex line, stepping through the points on a real line, a line with real slope but non-real intercept, and a line with non-real slope. But I’ve always been a little unsatisfied with my treatment of it. I really wanted to have a unified approach to all three types of lines, since they are all just complex lines. I was sure there was a way of dealing with them that could allow me to just take a complex line with a standard equation and know what it should look like.

Well there is a way and I’ve done it. I am very proud of it, and not least of the things making me proud is a GeoGebra page that allows you to choose any complex line and then see what points are on the line in the iplane attached to a moving point, but with all kinds of complex lines together in one GeoGebra tool, rather than separate like last time. But we’ll get to that soon…

First, the calculations.

Consider a line L with equation (a+αi)x+(b+βi)y=(c+γi), for real numbers a, α, b, β, c, γ with a, α, b, β not all zero. This is the most general form of a complex line. Given a real point (p,q), I am going to find all the complex points in the iplane attached at (p,q) that are on the line L. I will suppose that the point (p+si,q+ti) is on L and try to find the imaginary parts of its coordinates (s and t).

Since (p+si, q+ti) is on the line L, it satisfies the equation, and so:

(a+αi)(p+si)+(b+βi)(q+ti)=(c+γi)

ap + asi + αpi + αsi² + bq + bti + βqi + βti² = c+γi

ap – αs + bq – βt + (as+αp+bt+βq)i = c+γi

Equating real and imaginary parts, I get two equations:

1: ap – αs + bq – βt = c

2: as+αp+bt+βq = γ

Which I can rearrange into:

1: αs + βt = ap + bq – c

2: as + bt = γ – αp – βq

Whatever points (p+si,q+ti) inside the iplane at (p,q) are on the line L, they must have their s and t’s satisfy these two equations.

But look closer at them! These two equations are the equations of lines! So whatever points in the iplane lie on both of these lines are the points of the original complex line L in this iplane. Two lines meet in exactly one point, which means there is exactly one point in the iplane on the complex line L!

They are not just any lines either: the coefficients are chosen from the real and imaginary parts of the original line equation. Equation 1 has s and t coefficients which are the imaginary parts of the original line’s coefficients, while Equation 2 has s and t coefficients which are the real parts of the original line’s coefficients. That means these two lines inside the iplane are parallel to the “real part” and “imaginary part” of the original line equation, with equations ax+by=c and αx+βy=γ respectively.

And those two equations appear again in the equations inside the iplane at (p,q). The constant term in Equation 1 is ap+bq-c, which you would get if you substituted the point (p,q) into the real part of the original equation. The constant term in Equation 2 is γ-αp-βq, which you would get if you substituted (p,q) into the imaginary part of the original equation. If (p,q) satisfies both of these equations, then Equation 1 and 2 both have constant 0, and so have s=0, t=0 as a solution. But if s=0 and t=0, that means the point in the iplane is (p,q), which is a *real point*! That is, there is one real point on the complex line L with equation (a+αi)x+(b+βi)y=(c+γi), and this one real point is the intersection of the two real lines with equations ax+by=c and αx+βy=γ. I think that is *so cool*!

Of course, I have glossed over a few technical details. What if the two lines with equations αs+βt=ap+bq-c and as+bt=γ-αp-βq are parallel? Then there would be no intersection and so no points of the line L at all in this iplane. When would this happen? This would happen when the coefficients α and β were the same multiple of a and b, so α=ka and β=kb for some k. And then the original equation for L would have been

(a+kai)x + (b+kbi)y = c+γi

a(1+ki)x+b(1+ki)y=c+γi

ax+by=(c+γi)/(1+ki)

So the line L can be rewritten to have real coefficients for x and y, which means it has *real slope! *So a line with real slope will have many iplanes with no points of the line at all. Which ones will have actual points? Well, two parallel lines will share points (indeed *all *their points) if they are actually the same line after all. This would happen when as well as α=ka and β=kb, we also have the constant terms being related in the same way:

ap + bq – c = k(γ – αp – βq)

ap + bq – c = kγ – kαp – kβq

ap + bq – c = kγ – kkap – kkbq

ap+k²ap + bq + k²bq = kγ+c

ap(1+k²) + bq(1+k²) = kγ+c

ap + bq = (kγ+c)/(1+k²)

This is the equation of a straight line parallel to the one with equation ax+by=c, which is the real part of the original line L’s equation! So either an iplane will have no points of L at all, or it will have a *whole line *of points of L, whenever the point (p,q) is on a specific line parallel to the real part of the original line.

If the line L itself was a real line all along, then this could only happen if (c+γi)/(1+ki) was real, so γ would have to be kc as well, and the original line equation becomes ax+by=c. There will be points in the iplane at (p,q) when ap+bq = (kγ+c)/(1+k²) = (kkc+c)/(1+k²) = (k²+1)c/(1+k²) = c. This is the actual real line again! So a real line has points only in those iplanes attached to the real line itself, and in those iplanes, the points form a line parallel to the real line.

There is one final technical detail. What if both a and b are zero, or both α and β are zero? If both α and β are zero, then the two equations inside the iplane become

1: 0 = ap + bq – c

2: as+bt=γ.

If ap+bq-c is not zero, then Equation 1 has no solutions and there will be no points of the original line L in this iplane; if ap+bq-c *is *zero, then Equation 1 is 0=0, which is always true, and so any point satisfying Equation 2 is a point in this iplane. But wait, the points satisfying ap+bq-c=0 are exactly the points on the real line with equation ax+by=c. And since α and β are zero the original equation was ax+by=c+γi anyway. So this matches with what we found before: a line of real slope has points only in the iplanes attached to a real line parallel to the real part of the original line. In this case, not just parallel but equal to the real part of the original line!

The same thing happens when both a and b are zero. Then the two equations inside the iplane become

1: αs + βt = -c

2: 0 = γ – αp – βq.

And now there are no points of L in the iplane unless αp+βq=γ , which is still the equation of a real line. Indeed, the original equation is (αi)x+(βi)y=c+γi, which is the same as αx+βy=-ci+γ, so the line with equation αp+βq=γ is actually the real part of one version of the original line’s equation.

So after all this, we find that the complex line L with equation (a+αi)x+(b+βi)y=(c+γi) behaves like this in the iplanes:

- The points (p+si,q+ti) of the complex line L in the iplane attached at (p,q) satisfy the two equations αs+βt=ap+bq-c and as+bt=γ-αp-βq.
- If the complex line L’s equation cannot be turned into one with real coefficients for both x and y, then every iplane has one point of the L, at the intersection of two lines inside the iplane. There is exactly one real point on the line L, which is the intersection of the real lines with equations ax+by=c and αx+βy=γ.
- If the equation
*can*be turned into one with real coefficients for both x and y, then there are no points of the line L in most iplanes. The only iplanes with points on L are those attached to a real line parallel to the real (or imaginary) part of the original line equation. - If the equation can be turned into one with real coefficients for x and y and real constant term, then L is a real line and the only iplanes with points on L are attached to the points on L itself. The points of L in each iplane form a line appearing to lie on top of L itself.

You can investigate all of this in this GeoGebra applet, which allows you to choose coefficients for the equation of L and move the point (p,q) to see the point(s) of L in the iplane attached there.

Thanks for being here for my mathematical scribblings. I worry that it might not make any more sense than the old version, but I feel a deeper sense of satisfaction having tried it both ways (and having one GeoGebra applet with all the types of lines at once).

]]>— David Butler (@DavidKButlerUoA) March 26, 2020

The story of how this object came into the MLC is the reason it is so special to me.

It began with a Christmas countdown calendar that my wife’s brother and his wife gave to us one year. It has two blocks that are able to spell out how many days there are until Christmas. Every year in our house we hang it up on the 1st of December and take turns changing it each day.

A couple of days before Christmas in 2012, I was thinking about how very soon it would be 365 days until Christmas. I knew that even though we only use our countdown calendar for the numbers from 25 to 00, it is possible for it to count down from 32, and I wondered if you could make a set of blocks that counted all the way down from 365.

I asked this question in the online One Hundred Factorial discussion area that we had at the time. Over a couple of weeks of very slow discussion, we discovered you can only count to at most 87 with three blocks if you must have all three on display, and so you need at least four and (possibly even more) blocks to be able to count all the way from 365 to 0.

At this point someone suggested that since we already had the ability to spell 1 to 31 with two blocks, maybe we could use three extra blocks to spell the month. This was an intriguing suggestion. If you’d like to think about it yourself for a bit, then I suggest you look away now, because there are about to be spoilers…

If you look at the first three letters of all twelve months, there are 19 letters: JANFEBMRPYULGSOCTVD. This would seem to say it’s not possible to spell all the three-letter months with three blocks because three cubes only have 18 faces.

However, the number blocks that inspired all of this are only able to produce all the numbers they do by allowing the 6 to also represent a 9 when you turn it upside-down. So we thought maybe there are letters in the list that could be other letters if you turn them the other way. The only candidates we could see were U and C if we drew them in exactly the right way.

Unfortunately, even with the C and U as the same letter, it still wasn’t possible. Since we have to spell both JAN and JUN, that means that A and U have to be on the same block. But since we also have to spell AUG, that means that A and U have to be on different blocks. They can’t be on both the same block and different blocks, which means it’s not possible to do it with three blocks. (This right here is one of my favourite proofs by contradiction of all time.)

Or is it really impossible after all? We couldn’t see any letters that could be turned into each other, but we were using *capital* letters. I suggested that perhaps we could use small letters instead. Then the u could be turned upside-down to be an n, which means it might be possible for a and u to be both on the same block and different blocks, if the u from the other block was actually an upside-down n. We still need to have both u and n for this of course, so unfortunately that’s still 19 letters to fit onto 18 faces. The letters p and d to the rescue! They can be rotated into each other, and they don’t have to both appear in the same month abbreviation!

Now you know why I used lower case letters for my date blocks: because it’s not possible with upper case letters!

Of course, you still have to figure out exactly how to put the letters onto the blocks, and that I was able to do on the 14th of January 2013. This is a photo I took at the time.

But I couldn’t stop there. The days of the week also have three-letter abbreviations, and so I made a set of three blocks for those too. These were much quicker to figure out because there are only seven days after all!

This left the day number, which I knew already could be done with two blocks, but it seemed weird to have only two blocks for that when the weekday and the month had three each. I decided that since I wanted to use three blocks, I might have the leeway to have blank faces so I could spell the one-digit numbers without a leading 0, and so that there weren’t redundant faces, I decided to try to include the “st”, “nd”, “rd”, “th” suffixes too. I did in fact figure out how to do this, though I am not at all sure it’s the most efficient solution.

And so after a day of avoiding other work by playing with cubes, I came up with a solution on the 15th of January 2020. I chose to display the 1st of January so that I could make it clear there were blank faces and more than just “th”.

I put these paper blocks on display in the MLC next to the sign-in sheet box (yes we had a paper sign-in system in 2013), and I changed them every day through Summer School. When first semester arrived with a crop of new students, I decided it was high time for a sturdier more permanent-looking set. This set is made of cardboard, which was painted in dark paint, pasted with printed letters, then painted over with PVA glue as a lacquer.

This would be the end of the story except for one extraordinary epilogue. One of our regular students had observed me changing the blocks every day, and had noticed that while I could stack them on top of each other to make the date visible, they were a little precarious. So he took measurements and made a case for them that they could fit into, to be properly on display. He presented it to me on 22nd April 2012. The most impressive part is the handle at the top, which is an MLC and Writing Centre branded pencil. The MLC and Writing Centre shared a room when we moved into the Hub Central building in 2011 and even though they moved into their own room just around the corner in early 2012, we used these pencils for some years later.

So that’s the story of the MLC date blocks. They are special to me because they grew out of a wondering, that became a puzzle at One Hundred Factorial, that inspired a new puzzle following our MO of “the goal is not the goal”, and the solution has some really lovely reasoning and problem-solving. They remind me of all this wonderful mathematical thought, but even more than this, they are also connection to a student who loves the MLC environment so much he would build something for us.

Not many things hold so many special memories, and a calendar seems an appropriate sort of thing to do so.

]]>**The very beginning**

Once upon a time I was a PhD student in the School of Mathematical Sciences at the University of Adelaide. Sometime during the third year of my PhD program (2007), I was asked to give a talk to the first year undergraduate students as part of an evening event where the goal was to hopefully convince them to keep studying maths at a higher level next year. I titled my talk “How to Tell If You Are a Mathematician”. I don’t remember any of the things I spoke about, except for one thing. Before I started talking, I put a puzzle up on the document camera. I did not mention the puzzle in any way or look at the screen at all. I just did my little talk as if it wasn’t there. But right at the end of my talk I said this:

The final and truest way to tell that you are a mathematician is that you haven’t been listening to any of what I just said, and instead have been trying to solve this puzzle.

Cue guilty looks and nervous laughter from all of the academic staff in the audience, which successfully proved my point. Anyway it worked. Several students came up to me to talk about the puzzle, and I was able to direct them to lecturers who could talk to them about their study options. Yay for puzzles, right?!

This was the puzzle I used so neatly to make my point about the mathematician’s mind:

The number 100! (pronounced “one hundred factorial”) is the number you get when you multiply all the whole numbers from 1 to 100.

That is, 100! = 1×2×3×…×99×100.

When this number is calculated and written out in full, how many zeros are on the end?

I don’t remember where I got the puzzle from, but it is a pretty famous one that’s been around for some time. I actually hadn’t even thought through a solution at the time either. I just knew that it mentioned a concept that had been in the first year lectures recently.

**The puzzle sessions begin**

The other thing that happened that night was that a group of students and staff stood at the blackboard in the School of Maths tea room to nut out a solution to the 100! puzzle. I can’t even remember if we finished it or not, but we did decide that we should get together regularly to solve puzzles together, and a weekly puzzle session was born. At the first session, we started with the 100! problem again, and an extension of it, which is to find out what the last digit is before all those zeros start. Then as the weeks went on, we would do puzzles that I would find and bring to the sessions.

When I finished my PhD in mid-2008 and took up the job in the Maths Learning Centre, I took my little puzzle session with me, and was able to invite more students to come along, and it slowly morphed into a student event more than a staff event, which really pleased me. In fact, a regular at these puzzle sessions for years was that first student who had come up to me after my talk at the first-year event, and he eventually became one of my tutors at the MLC.

**The name of the event**

Over the years the puzzle session has had many names. We started out calling ourselves “People with Problems”, and then simply “Puzzle Club”. For a while it was called “The Hmm… Sessions” after the sound we made very often while thinking about puzzles. Indeed, there is a reference to the Hmm Sessions inside this very blog. But in 2012 after the website where I was hosting our online discussion was decommissioned, I decided it was time to change the name. I was also starting to think about moving the sessions out of the MLC itself and into a public space, and to match with this move I wanted a new name. I thought long and hard, and decided to name it after the first puzzle we ever did, the puzzle that first inspired staff and students to talk and think about maths together, the puzzle that helped students decide they really were mathematicians after all.

**The legacy**

So the regular puzzle session of the MLC became One Hundred Factorial at the end of 2012, and here we are in 2020 still going, so that now it’s been One Hundred Factorial longer than it’s been any other name. It’s been my testing-ground for new puzzles and games and teaching ideas, a place where I have made friends and welcomed people from around the country and the world. And it has become a glowing island of mathematical play in the middle of the stressful university life, and indeed the middle of a stressful life generally. In recent weeks it is a glowing island of community in a world of pandemic-induced isolation.

One Hundred Factorial reminds us that there is always something joyful to think about if you are looking for it, and that it’s okay to pause and ignore your responsibilities for a while to think about it, and that doing this with people is a source of *shared *joy. I hope the puzzle and the event can keep reminding us of that for a long time yet.

In case you haven’t heard of BODMAS/BEDMAS/PEMDAS/GEMS/GEMA, then you should know they are various acronyms designed to help students remember the order of operations that mathematics users around the world agree upon so that we can write our mathematical expressions simply and unambiguously. For example, the most common here in Australia is BEDMAS, which stands for “Brackets, Exponents, Division, Multiplication, Addition, Subtraction”. The idea being that things in brackets are calculated first, followed by the rest. There are some technical details there, where Division and Multiplication are not strictly in that order, and instead ought to be done in whatever order they come, with a similar rule for Addition and Subtraction. This is one of the reasons I don’t like these acronyms, because if people only remember the list, then they tend to think that addition is *always *before subtraction. Indeed, I observed someone doing this within the last month.

The acronym GEMA is supposed to get around this by not explicitly mentioning the Division or Subtraction, and by using Grouping symbols to cover all grouping-type things. (If any Australians are wondering what the P in PEMDAS stands for, it’s “parenthesis”. People in the USA actually do use this fancy-sounding word and in their language they think that “brackets” means only the square ones.)

But this still makes me uncomfortable. The first way it makes me uncomfortable is that all of these acronyms seem to imply that brackets (or parentheses) are operations. But they aren’t. I mean, some types of brackets are operations, sure, but at the life stage when you usually learn this order, they’re not. Even if you do know about something like the absolute value, when you do calculate |-4+8| you *don’t* do the absolute value first do you? You do the *thing inside *the absolute value brackets first and it would be just wrong to do the absolute value to each piece first.

For your ordinary brackets this idea of focusing your attention on the thing inside is their whole purpose. In general brackets aren’t operations but just a way of holding things together so you do the actual operations at the proper time. It really bothers me to have them in a list of operations at all. I do understand they need to be mentioned somewhere, but still it bothers me.

The second thing that makes me uncomfortable is naming the concept by the acronym you use to remember it. I can’t count the number of people who I have heard say “should I use BEDMAS?” when they ask how to evaluate or read an expression. The concept is called “the Order of Operations” or “Operation Precedence” people! Get it right!

Anyway, this makes me less uncomfortable than it used to. It feels more like a mild transient little itch than a raging angry rash. I mostly made my peace with it by imagining they are saying “should I use BEDMAS to remember the right order?” which is perfectly fine. While there is reasoning behind why we do it in the order we do, it is more-or-less an aesthetic choice that could have been made a different way in an alternate reality, and aesthetic or arbitrary choices often need supports to help remember them. Also I realised that people need names for things, including mnemonic tools, and one of the nice things about acronyms is that they are already a name. It’s natural for the acronym, which has a ready-made name, to end up being the name for the thing itself. It doesn’t mean the itch isn’t there though!

So how do I reconcile all of this when I teach people about the order of operations? I use the Operation Tower.

It’s a visual representation of the order of operations, that keeps the same-level things at the same level, and carefully separates the brackets and other grouping symbols out from the operations themselves to make the point that they are different things. The innovation in the last couple of days is giving this thing a name, so that students can talk about it both to themselves and to others (thank you to a friend online for pushing me to do so!)

Here is how I usually introduce it: I start at the bottom, drawing the + and – first, saying that they are the most basic operations. Then I draw the × and ÷ above that, saying that they have to be done before + and – nearby. Then I draw the ^ and √ above that, saying they have to be done before any of the lower ones nearby. Finally I draw the box on the side with the (), [], ___, saying they are designed to hold things together in order to *override the usual hierarchy*. I also point out that the horizontal bar is usually seen as part of the root symbol, which is why it holds things in.

(Note I have been careful to say “nearby” in that previous paragraph. It’s not actually true that all the multiplications have to be done before all of the additions. Only the ones that are near to additions. Indeed, the whole point of the brackets is to put new boundaries on what “nearby” means!)

I also use the operation tower to help students remember some other rather nice properties of operations. Notice how the operations in each box distribute across the operations in the box below. For example √(4*9) = √4*√9 and (2+10)*4 = 2*4 + 10*4 (though you have to be a bit careful with division). Also notice how the higher things inside a log turn into the lower things outside the log. That makes the Operation Tower a reusable tool for multiple different things, which I rather like.

But the thing I like the most is how students respond to it. It really seems to make sense to them to organise the operations spatially as they learn for the first time that there is an order mathematicians prefer to work in, and they seem to understand that the brackets are a different sort of thing and appreciate having them listed separately. For those who are like me and have serious trouble remembering the correct order of letters, it’s much easier to process than an acronym, too.

I hope you like using the Operation Tower yourself.

PS: I have actually written about this before, and you can track the change in how I have used it over time if you read the previous blog posts about it: The Reorder of Operations (2015) and Holding it Together (2016),

]]>Players:

- This game is designed for two players, or two teams.

Setting up:

- Each player/team has two grids with the letters A to J, one labelled MINE and one labelled THEIRS, like the picture below.
- Each player/team writes the ten digits 0 to 9, each in a separate box in the MINE grid, keeping the grid where the other player/team can’t see it. (I actually have a printable version with the rules on them that can be turned into battleships-style game stands here.)

The goal:

- Each number has been disguised as a letter. You need to find out which number each of the other player/team’s letters is, by finding out what calculations with the letters produce.

Your turn:

- On your turn, ask for the result of a calculation involving exactly two
*different*letters and one of the operations +, -, ×, ÷. Some examples are A – B or C÷J or H+D or E×G. - The other player/team answers your question truthfully, either telling you which of their letters is the result, or telling you the result is not a letter. To be clear: players
*do not ever say a number*in response to a question. They only ever say a letter or “not a letter”. - You can write notes to help you figure out what you know from the information you have so far.
- Now it is the other player/team’s turn.

Ending the game:

- Once during the game, instead of asking a calculation, you can say you are ready to guess. Then you say what number you think goes with every letter. The other player/team tells you if you are right or wrong.
- If you are right for all letters, you win! If you are wrong for any letters, you lose and the other player/team wins! Either way the game is over.

This game was inspired by a puzzle I wrote called “The Number Dress-Up Party“. In that puzzle, all of the numbers are at a dress-up party and you have to find the identity of just a few of them by asking them to perform operations. I was reminded somehow of the puzzle and I was wondering about modifications of it. One thing I was wondering was if I had only a few numbers at the party, how long it would take to find out what all of them were, and so this game was born. Later that same day I played it across Twitter with a friend, on either side of a whiteboard at One Hundred Factorial, and with my daughter at home. I was then *completely obsessed. *

I learned a lot from these three early games.

In the first game with Benjamin, I was struck by how quickly the logic got complicated, and then how quickly it all cascaded into finding everything when I finally got a few different numbers. It was interesting thinking about what it meant to get the response “not a letter”. I loved finding ways to keep track of the information I knew so far, and making sure I was using all the information I had so far. The presence of the 0 really made for an interesting ride on Benjamin’s side.

I’ll go first!

A-B— David Butler (@DavidKButlerUoA) September 17, 2019

When I played at One Hundred Factorial, we played in teams on either side of a whiteboard and it was way more awesome! Firstly, it was heaps more fun to play in teams — talking through the logic so far and finding ways to represent it so the rest of the team understood was a really pleasurable experience. I loved hearing other people’s thoughts about what we knew so far and what we should do next. Not to mention having people notice when we had made a mistake in our logic. Secondly, having a huge space to write all of our reasoning was really nice. It was fascinating to see the other team’s approach, which was to have a big grid of which letter could be which number and slowly cross off the possibilities. I had never even considered doing something like that!

Digit Disguises was SO MUCH FUN today at #100factorial! Thank you for playing @zithral @JeremyInSTEM @EmilyLHughes pic.twitter.com/ueVuumJSD3

— David Butler (@DavidKButlerUoA) September 18, 2019

When I played with my daughter C (she’s just turned 11 years old), it was a whole different experience. C had no trouble answering my questions using the letter/number correspondence. In fact, she really enjoyed that part. (Indeed, when I offered to play with C the next day, she was happy just to be the keeper of which letter was what number and let me play Mastermind style.)

I found it rather fascinating that even though they haven’t really done algebra at school, it was perfectly natural for her to refer to these numbers by their letter disguises and to write stuff down on the page in terms of the letters. That is, she didn’t need to be taught about using letters for numbers to really get into the idea of the game. I think that is pretty awesome, actually!

What C was having trouble with was making sense of the information she was getting. She immediately realised that getting something like A-B = “not a letter” meant that A was smaller than B, but she didn’t really know what to do with the information that B-A=C. A little discussion helped her realise that this meant B had to be A+C and both A and C had to be smaller than B. But then it seemed like a huge task to find any letters!

It took a few tries to come up with a representation that was helpful, and a bar model really worked to make sense of it, and even allowed us to pull out information such as I=2F. It was still a little difficult for her to understand how knowing I=2F and I/F=C meant she could know that C=2.

I was thinking that maybe all ten digits was too much for someone her age, but then after finding the numbers 2 and 1, the cascade of finding all the other numbers really gave her a feeling of both power over it, so maybe ten digits was fine..

— David Butler (@DavidKButlerUoA) September 18, 2019

So I reckon that for kids her age, definitely playing in teams is a good idea, or having the whole class be a team against a mastermind, or even the very first time with the teacher against the class as the mastermind with the teacher describing their thought process, to get an idea of the strategies involved. I’m also imagining a version much more like the original dress-up party puzzle where a collection of *students* are the numbers and they know which student is which number and the rest of the class have to figure out which student is which number by asking them to combine with operations.

On the topic of “mastermind”, my friend Alex has created a little python script that will allow you to play the game against it, mastermind-style. If you prefer to play in teams but you only have a few players, then you can play with all of you on one team against the computer!

The other thing that is so very awesome about Digit Disguises is how many mathematical wonderings flow out of it so very easily. On top of the usual things that come up during the game, such as what getting an answer of “not a letter” tells you about the numbers involved, you can go a long way wondering about the game as a whole and what happens if you change it.

I have wondered all of these things, but only investigated some of them. I won’t ruin the answers for you for the ones I have thought about already.

- How many questions do you need to finish the game? Is there an algorithm that will finish in the smallest number of questions possible?
- What if the goal was to correctly identify just one number? How quickly can you find the first number?
- Which number is the easiest to find? Which number is the hardest?
- Can you identify all the numbers using only one operation, such as only using subtraction? What is the smallest number of questions to finish the game if you’re only using each operation?
- What if the operations were mod 10? You wouldn’t be able to use subtraction to tell you greater or less than in this case, but would it still be possible to find everything?
- What if you had 0 to some other number, like 0 to 25 or 0 to 5. How long would the game be then? Would your strategy be any different?
- What if you didn’t have 0 or 1? What if you had some negative numbers? What if you had some completely other collection of numbers?
- Is it even possible to find all the numbers if you have a different collection? Are there collections of numbers any size you like where you can’t find any of them? Are there collections of numbers where you can find some but not others?
- For the previous question, if there are only a small number of numbers (like two or three), what are
*all*the sets of numbers that can all be identified?

I hope you like my game of Digit Disguises. I think it’s *AWESOME! *If you do play it yourself or with your students in a classroom, or you have thoughts about the answers to my wondering questions, or have anything you are wondering about yourself, please do let me know.

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The problem with maths in context is that the contexts themselves require understanding of their own in order for the maths to make sense. This is nowhere more true than in statistics, where you have to use your understanding of whether you expect the relationship to exist, what direction you expect it to be, and whether you think this is a good or bad thing. The classic one in my head is an old first year statistics assignment where they used linear regression to investigate the relationship between manatee deaths and powerboat registrations in each month in some southern coastal American city. You have to know what a manatee is, what it means to register a powerboat, and why those things might possibly be connected in order for the statistical analysis you’re asked to do to make sense, not least because at least one part of the question will ask you to interpret what it means. When helping students read published articles recently, I’ve had to find out what’s been done to the participants in the study, how things have been measures, what kind of measurements those are, why they’ve been measured that way, and all sorts of little details to decide how to interpret the numbers and graphs that are presented.

Even ordinary everyday word problems are a minefield. Across two recent assignments, some financial maths students had to cope with album sales for AC/DC, flooding of the land a factory is built on including insurance, bull and bear markets, machines in a mining operation, committees with various named positions, road testing electric cars, contraband being smuggled in shipping containers. This is a *lot *of context that has to be made sense of before you can get a handle on the maths, and there is nothing in the question itself to tell you what any of this context means if you don’t already know. Even if you are already familiar with the context, you actually have to suspend some of your understanding in order to do the maths problem, because it’s much simpler than the actual situation any of the questions are talking about.

All of this interpreting is exhausting stuff! It just tires you out if you have to even a moderate amount all at once. You just feel like you don’t have any more energy to deal with any more today. That feeling there is **context fatigue**. Yesterday the first year maths students were doing related rates and every question was a new context with little nuances created by the context that had to be dealt with. Those poor students were exhausted after just one problem, letalone three or four.

As teachers, we need to realise that as the people writing the assignment questions, or at least people who have dealt with them before, we are much more aware of the details and nuances of the context than the students are, so we don’t have to work so hard to make sense of them. Not only that but we’re usually simply more experienced in both life and language than most of our students so it’s easier for us. Imagine the context fatigue you would get reading ten research papers in an unfamiliar area in one day (I feel this in real life regularly). That’s the sort of context fatigue your students have just from your assignment questions. Cut them a little slack, and make sure there is adequate time to process the context with appropriate rest time between context-interpretation. Also it wouldn’t be the worst thing to explicitly teach them strategies for making sense of context, such as ignoring the goal, and finding out about what some of the words mean. Strategies can make the work less intimidating, especially in the face of knowing how tiring it is already!

PS: If you’re in charge of tutors in a drop-in support centre, especially one that deals with statistics, please be kind. Context fatigue is real and tends to wear us down some days!

]]>But there is an important thing I almost never talk about, which is that sometimes listening is actually *awful*. I can think of not many more debilitating curses to lay upon people than to wish them the ability to listen.

Because listening is *exhausting*.

Because I am listening to students, I know an explanation doesn’t work, so I have to come up with new ones, usually on the fly. Because I am listening to student thinking, I come across new ways to think about all sorts of things that I had never considered before, which I then have to process. Because I am listening, I am faced with people’s feelings and stories, which I have to process emotionally. Because I am listening, I can easily become fascinated with new ideas and problems which take up my mind. Because I am listening, I hear things that need changing in teaching methods or university systems, and either try to work to change them or worry that I can’t. In short, because I am listening I am constantly processing information and emotions both in the moment and later on. It’s exhausting.

I don’t always cope well with it. In person with students I can just deal with who is in front of me and it’s ok, though there are times I need a break and just walk away for a few minutes. Unfortunately when I’m apart from the students, I can’t leave my brain behind and I carry with me the swirling thoughts in my head all day long caused by the listening to students. One way I have to cope with this is to talk with people about those thoughts, in person or on Twitter. But I actually can’t talk about *all* of them, so I have to choose one thing to think about and ignore everything else. There are times I have to say to students or my tutoring staff that actually no I can’t think about that right now, which is really really hard. And there are times I can manage to do an activity like origami or folding or watching tv to turn off my brain for a while to give it some rest. Still the call to listen is back again soon enough.

This isn’t a whinge session to get sympathy, it’s a warning. Be warned that if you choose to listen, you too will have to find ways to ignore some things, to find moments of brain-calm, and to find ways to process the thoughts you do choose to entertain.

Was my aim to scare you off? Certainly not! I wouldn’t ever give up listening and sacrifice the pleasure and learning I get from it, or the benefit it has for students. The blessing far outweighs the curse.

Just be prepared, ok?

]]>There is a lot that staff can do to engage students in the university community and in their learning, and a lot of these things have to do with the staff being engaged with the students. One way that any staff member can show their own level of engagement with the students is to learn the students’ names.

Names are important. Your name is a part of your identity, and not just because it is what you call yourself. Your name may tie you to the culture or the land of your ancestors, or it may speak of your special connection to those you love. You may prefer to be called by a different name than your official one because your chosen name is more meaningful to you. What all of these have in common is that your name is an important part of your identity.

For myself, my name is David, and I don’t like to be called Dave. I grew up in a community with several Davids and other people were called Dave, so being David kept my identity separate to theirs. Yet many people give me no choice and call me Dave without asking for my permission, despite me introducing myself as David. I find it intensely rude that someone would choose to call me by a different name than the one I introduce myself. On top of this, I am a twin, which means as a child I was forever being called by the wrong name entirely. We are not identical twins, and yet this still happened, because we were introduced as PaulandDavid, without an attempt to give us a separate identity. The fact that I was called Paul, or “one of the twins”, meant that I had no identity of my own separate to my brother. Being called David means that I have an identity of my own and this is important to me.

For many students, these and worse are their daily lives. Imagine a student who no-one at university knows their name. They have no identity at university, can feel very alone and can quickly disengage. Yet according to “The First Year Experience in Australian Universities” by Baik, Naylor and Arkoudis, only 60% of first year students are confident that a member of staff knows their name.

Not having your name known at all is one thing, but being called by the wrong name can be worse. An international student has to deal constantly with being different to other students, and in the community at large has to deal with a lot of everyday racism. To have your name declared “difficult to pronounce”, or to have it declared as not possible to remember, is just another one of these everyday racist events. The person doing so may not be meaning to be racist, but it adds up to the students’ feeling of not belonging, to their feeling that they themselves are not worth remembering. Similar to me and my twin brother (only worse), they may have the feeling that others believe all international students are the same, so why remember them separately. In “Teachers, please learn our names!: racial microagression and the K-12 classroom” by Rita Kholi and Daniel G Solórzano, there are many examples of the hurt that such treatment of student names can have.

So what can we do to learn our students’ names? Members of the Community of Practice suggested several strategies.

One idea is to spend time talking to them and ask them what they would like to be called. You can’t learn their names unless you find out what they are! Be visible in your effort to pronounce it correctly, be adamant that you want to call them by the name they ask to be called. If you get multiple chances to talk to them one-on-one, ask their name again if you can’t remember and try to use it as you talk to them.

Another idea is to print out photos of your students and to practice remembering their names. If you don’t have access to their photos, then it should not be hard to find someone nearby who can. (Though of course it would be excellent if there were a simple system whereby anyone teaching a class — including sessional staff — could get photos of their students!) Even if you can’t get their photos, simply working your way down the roll and remembering how to *pronounce* those names, or what the students’ actual preferred names are, is good exercise. The students are likely to appreciate the effort you put in here, even if they can’t know how much time you did put in!

You may have your own ideas on how we can make sure we know students’ names. I’d encourage you to share them in the comments, along with any stories of how it made a difference to student engagement.

I would like to work in a university where 100% of the students are confident that someone knows their name. We have hundreds (possibly thousands) of staff in contact with students on a regular basis. If each of us only learns a tutorial-worth of names, then we can surely meet that goal easily!

]]>This is lovely, but one problem is those students who on the face of it don’t *want *to play. The majority of students I work with in the MLC are not studying maths not for maths’s sake, but because it is a required part of their wider degree. That is, I am helping students who are studying maths for engineering, or calculations for nursing or statistics for psychology. A lot of these students just want to be told what to do and get the maths over and done and don’t like “wasting” time playing with the ideas.

Or so I thought. I have realised recently that actually they *do *like playing with the ideas. I just couldn’t see that this was what they were asking for.

One of the questions I like to use to play with maths is “what if?” To ask it requires the asker to notice a feature they think might be different, and wonder what would happen if it was, and so investigate how this feature interacts with the other features of the situation. This investigation of how ideas interact is exactly what mathematics is, to me. I am very very used to doing it at extracurricular activities like One Hundred Factorial, and it’s easy to do with students who are doing very well with their maths and have the breathing space to wonder about this stuff because they’ve finished their work. For students working on an assignment they are really struggling with that they wonder about the usefulness of for their degree and which is due in the next 24 hours, it’s not so easy.

Only maybe it’s a little easier than I thought, because I started to notice that the students were already asking questions about the connections between things.

A very common question students ask around exam time is “What would you do if the question was like *this …*?” as they suggest a change in a small detail that makes it more similar to the past exam question they have at hand. I used to get really annoyed at this sort of question, but then I realised that in order to ask this question they had already noticed that the two problems were similar in most respects and different in this one. Sure, they may be motivated by a belief that success in maths is about a big list of slightly different problems and remembering ways to deal with each, but on the other hand they have noticed a relationship and are trying to exploit it, This is a mathematical kind of thought, to look for similarities and differences and relationships, and we can hang on tight to it and actually learn something!

Another kind of question students ask is “Why is this here in this course?”. I used to get annoyed at the whinging tone of voice here, but then I realised that a student is begging for a connection between this and the rest of what they are studying, and connections is precisely what understanding maths is all about. I can respond to it by saying that yes I am also confused about the curriculum writer’s logic for including it, but then we can search out connections together to see if we can find them.

A third kind of question is the one where the student looks at the lecture notes or solutions and asks “How did they know to do this?”. Sure, it’s usually motivated by wanting to be able to successfully do it in their own exam in restricted time, but on the other hand it does recognise that there ought to be reasoning involved in that decision, as opposed to just guessing. This expectation of reasoning is the beginning of believing that they themselves could reason it out too.

My second-last kind of question is “Why is this wrong?” as the student points to the big red cross on their MapleTA problem on the screen. Sure they just want to make it all better so they can submit the damn thing. But they are also recognising that there must be a reason why it’s wrong, and so are looking for meaning. These students are often ripe for the experience of looking at the information and how it’s related to what they’ve entered, thus doing exploration. They are also usually ready to try various different ways of entering the result, or different strategies of getting a right answer to see where the edges of the idea are.

The final kind of question I want to mention isn’t even a question, it’s an exclamation of “It doesn’t make *sense!*” Even this is telling me that the student thinks it ought to make sense. They are crying out to do something to make sense of it. Often they describe being almost there and needing something to push them over into understanding. This student is ready to explore the edges of the understanding they do have to see where the nonsensical stuff can fit, or where their ideas need some tweaking to fit together better.

It is very helpful to me as a one-on-one or small group teacher, or even a lecturer with a big room of students in question time, to be able to see the questions struggling students are asking as cries for sense-making exploration. It doesn’t matter that they are struggling or don’t understand things. Indeed, being in a situation of not knowing is exactly what we are doing for the fast students when we give them extension activities, so why is the everyday maths the slower students are not understanding any different? In my experience students like being treated as if they are behaving like mathematicians when they have these struggling sorts of questions. And it was so much easier to treat them that way after I realised they *are *behaving like mathematicians when they have these kinds of questions, even in some small way.

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One reason I wanted to do this was because too often we give the impression that maths is all so slick and easy, when really it is actually messy and hard. This issue is exacerbated by the nature of puzzles, which often have clever simple solutions, and when people share only that, we give this impression that maths is done by some sort of divine inspiration of the clever trick. I wanted my responses to problems to show the honest process of muddling through things, and getting them wrong, and only seeing the clever idea after I’ve done it the unclever way, and even if a clever way did occur to me, what it was that made me think of it.

Over time, this has grown into me doing epically long tweet threads where I live-tweet my whole solution process, including all the extra bits I ask myself that are sideways from or beyond the original goal of the problem. I have to say I really really love doing this. I love puzzles and learning new things, and it’s nice to do it and count it as part of my job. I also get to have a record of the whole process for posterity to look back on. Even in the moment, it’s very interesting to have to force myself to really slow down and be aware of my own thought processes enough to tweet them as I go. In my daily work I have to help other people to think, so the sort of self-awareness that this livetweeting of my thought process brings is very useful to me, not to mention fascinating.

In the beginning, I had some trouble with people seeing my rhetorical questions as cries for help and jumping in to try to tell me what to do. I also had people tell me off for not presenting the most elegant solution, saying “Wouldn’t this be easier?” Every so often I still get that happening, but it’s rarer now that I have started to become known for this sort of thing. I am extremely grateful that a few people seem to really love it when I do this, especially John Rowe, who by rights could be really annoyed at me filling up his notifications with hundreds of tweets with a high frequency of “hmm”.

I plan to continue to do this, and so if you want to keep an eye out for it, then I’ll happily let you come watch. From this point forward I will use the hashtag #trymathslive when I first begin one of these live problem-solving sessions, so I can find them later and you can see when I’m about to do it. (I had a lot of trouble looking for these tweet threads, until I remembered that a word I use often when I am doing this is “try” and I searched for my own handle and “try”. In fact, that’s the inspiration for the hashtag I plan to use.)

Note that I am actually happy for people to join in with me — it doesn’t have to be a solitary activity. I would prefer it if you also didn’t know the solution to the problem at the point when you joined in, but were also being authentic in your sharing of your actual half-formed thoughts. I definitely don’t want people telling me their pre-existing answers or trying to push me in certain directions. If you feel like you must “help” then, asking me why I did something, or what things I am noticing about some aspect of the problem or my own solution are useful ways to help me push my own thinking along whatever path it is going rather than making my thinking conform to someone else’s. I will attempt to do the same for you.

To finish off, here are a LOT of these live trying maths sessions. (If you click on the tweet, you’ll go to the thread without having to log into twitter. You’ll have to scroll up a tweet or two to get to the original problem and scroll down, often a long long way, to find what happened.) I hope you enjoy reading them as much as I enjoyed doing them.

4th July 2019

Do you mind if I give this a go?

— David Butler (@DavidKButlerUoA) July 4, 2019

28th June 2019

Let’s go!

— David Butler (@DavidKButlerUoA) June 28, 2019

23rd June 2019

Ooh! Surprising!

— David Butler (@DavidKButlerUoA) June 23, 2019

12th June 2019

I am going to live-tweet my solution process for this. Look away if you don’t want to know… https://t.co/av1yPL98R8

— David Butler (@DavidKButlerUoA) June 11, 2019

May 28 2019

OK let’s try it!

(1+9^(-4^(7-6)))^(3^2… hmm. That stack of powers I’m a bit confused about. Is it 3^(2^5) or (3^2)^5? I may have to come back to that later.— David Butler (@DavidKButlerUoA) May 27, 2019

May 17 2019

I wonder: at what height is the volume of a cone above that height equal to the volume below? What about the surface area? Are there any cones where it’s the same height?

— David Butler (@DavidKButlerUoA) May 17, 2019

17th April 2019

Ooh! I’ve done this before but I can’t remember. So let me try again now…

— David Butler (@DavidKButlerUoA) April 17, 2019

20th March 2019

The 72 makes it more accurate but when I do it for a class I tend to use 70 as well and stress the fact it's an approximation but very useful without having to use logs

— Gavin Scales (@ScalesGavin) March 19, 2019

24th February 2019

I don’t know how to solve it in my head yet (or at all) — and I don’t want any hints — but I do notice I can put a circle here: pic.twitter.com/QTjRqlKln1

— David Butler (@DavidKButlerUoA) February 24, 2019

21st February 2019

Ooh! Let me try!

— David Butler (@DavidKButlerUoA) February 21, 2019

28th November 2018

Hmm. If assuming that top edge is divided exactly in half. I hope that’s ok. I’ll figure out it it’s necessary later.

— David Butler (@DavidKButlerUoA) November 29, 2018

5th November 2018

Ok. Not sure where to start but I do see a halved triangle up the top there. pic.twitter.com/a0bPMVkR2R

— David Butler (@DavidKButlerUoA) November 5, 2018

14th October 2018

All right. I will live tweet my process. Be aware I will go to bed very shortly, so there will be a several-hour gap.

— David Butler (@DavidKButlerUoA) October 14, 2018

4th October 2018

Ok, I don’t know how to do this already so here are some live thoughts about it…

— David Butler (@DavidKButlerUoA) October 3, 2018

11th March 2018

My first thought is: how would I even start thinking about that?!

(This is not a request for help, just being honest about my thought process.)— David Butler (@DavidKButlerUoA) March 11, 2018

21st Janurary 2018

Oooh!! Fun!

— David Butler (@DavidKButlerUoA) January 20, 2018

5th December 2017

]]>Ooh! Fun!

— David Butler (@DavidKButlerUoA) December 14, 2017