This post is about my response to TMC16.

For the uninitiated, TMC is short for Twitter Math Camp. This is a conference designed by teachers for teachers with teacher speakers, organised through the collective efforts of the Math Twitter Blog-o-Sphere (MTBoS) — a group of people who blog and tweet about their experiences teaching math(s). That description is not the best description of the MTBoS, but I’ll get to that later.

For those who were actually at TMC16, you may be thinking, “But David, you weren’t at TMC16!” And I wasn’t, not physically at least. But then Tracy Zager mentioned me in her keynote, and Megan Schmidt told me that she did, and suddenly I was drawn into a most interesting world I had never observed before.

Over the next week, I followed the #TMC16 hashtag as people live-tweeted sessions, live-recorded them and posted them on YouTube. I participated via Twitter in the activities they did during sessions, and in discussion in the days afterwards. I read any number of blogs about people’s experiences attending.

Now I have so many thoughts spiralling through my head, and I need to write about them to process them. I also need to put TMC16 to bed because semester starts again today and I have so many jobs to do, some of which I didn’t do during the mid-year break because I was virtually attending TMC. So here goes: my TMC16 reflections.

The conference structure you wish you had

The conference itself sounds like the best setup for a conference I have ever known: The morning sessions were all connected across the three days to allow for focussed development of ideas. There was one keynote speaker a day. There are afternoon sessions based on accepted proposals, with one more timeslot on the last day for people who weren’t accepted to do sessions of their own (called “Flex” sessions) [Note: see Lisa’s comment below for what Flex sessions are actually designed for.]. Anyone at the conference could sign up to do “Favourites” talks of up to 15 mins which were done at whole-conference sessions at various points during each day — literally anyone from the newcomer to the seasoned veteran. Plus there was a full 90 minutes for lunch!

I have never experienced a conference with such a diverse set of ways to learn and interact: three-day workshops, short sessions, long talks, short talks, long lunch. I have also never experienced a conference with such an explicit focus on encouraging community interaction. At what other conference has a newcomer ever been encouraged to actually do a 10-minute talk to the whole conference, and cheered on when they do so?

I have been to several conferences in the past and felt like an outsider. I’ve felt like others there were concerned with meeting up with their once-a-year friends more than welcoming newcomers. I’ve felt like somehow you have to be better than you are to be allowed to participate or have a voice. Watching the Favourites talks of people, many of whom were newcomers, some of whom revealed quite personal stories, made me wish for something better. Reading people’s thoughts about how welcoming everyone was to their conversations made me wish for something better.

Keynote speakers who are honest, encourage you think and then do something about it

There were three keynote speakers across the three days, chosen from among those in the MTBoS: Dylan Kane, Jose Vilson, and Tracy Zager. Each of them talked with humour, passion, compassion and honesty. They brought their perspectives on what teaching can be about, what the MTBoS can be about, and then had calls to action to encourage the listeners to do something about what they had heard. This is a brilliant concept and really brings home the idea that the conference wasn’t just a love-in and a place to passively receive information, but is supposed to empower others to change.

What I particularly appreciated was the three speakers’ honesty. Each of them told their own journeys and struggles with teaching and participating in this community. It’s a different type of thing from keynote speakers I’ve seen before, who are called in as experts and always seem to have all the answers. These three on the other hand clearly admitted their own shortcomings and how they still have a lot to learn. It was so refreshing.

A paradox: more than resources, but was it ever just resources?

Dylan in his keynote mentioned how the MTBoS needs to become more than resources. He noted how when he was a new teacher, he was a magpie bringing in shiny new resources and activities into his classroom from the MTBoS and “throwing them at the wall to see what would stick”. It was a long time before he realised that what he needed was ideas that underpin these to make them work in his own context. He encouraged the MTBoS to be more than just a collection of resources people have made.

Tracy in her keynote mentioned how she as an early school teacher was intimidated by the number of high-school teachers in the MTBoS discussing on Twitter, and how she overcame this to become part of these conversations. The point is that she seemed to see the MTBoS as a conversation to be joined.

I find this contrast most interesting. I’ve been watching the MTBoS for about a year, and I never had the impression of it as a list of resources at all! Only now am I starting to find these resources that Dylan mentioned. Perhaps this is because it was introduced to me as a Twitter discussion hashtag first. I saw it as a place where people asked questions and in response got various answers and mostly thoughtful discussion. That’s what I was seeing on Twitter. When I followed links to people’s blogs, I mostly saw discussion of how they had tried to implement things, or thoughtful pieces about what teaching is and could be. Very rarely did I see a blog post simply listing resources people had made, and if I did I never found them that inspiring, so I didn’t go back there.

In light of Dylan’s talk, I realised I had somehow selected my experience of the MTBoS to be about the discussion and not the resources. Another reason for this self-selection may be because I am not a classroom teacher any more and so it’s not the resources that I think I need. I can’t use resources in my teaching most of the time because of the teaching situation I am in, so I gravitated to the discussion. That discussion fed my thinking about teaching, rather than some other more physical need.

I have to say that I agree with Dylan that there needs to be more about the thinking when people do post resources. As a new teacher in a school where I was it for maths and science for the whole school, I really needed that thought process to make sure I actually went through a thought process, rather than “throwing to see what sticks”. As a teacher in university looking at resources that are at the school level, I need the thinking to help me sort out if and how I could apply this stuff to my own teaching situation. I suspect Tracy would agree that the thinking would allow an elementary teacher to relate things talked about by a high-school teacher to their situation, too.

The really interesting thing is that TMC itself shifts the focus from resources to discussion. I can’t count the number of TMC recaps that have mentioned the fact that going to TMC and interacting with people in person made them feel more part of the community and therefore more able to engage with discussion. I can’t count the number that said that it was not the resources and activity ideas they appreciated but the chance to thrash out their thinking about teaching. I think there’s a lesson there about what the MTBoS is really about.

Post-TMC malaise

One problem with watching but not attending TMC (and possibly the same for attendees too) is that it only heightens my sense of disconnectedness at home. My teaching situation is quite rare, and it can be hard to feel connected. Also, at my university at least, not everyone is willing to have discussions about teaching at all, and sometimes only willing to do so if there is a rigorous research backing first, rather than the refreshing can-do attitude I see in the members of the MTBoS. Even when I do have them, it’s hard to feel like it’s a valid use of my time when there’s the huge pile of mostly administrative jobs that have to be done. I don’t know if there’s anything I can do about this.


Thanks to Megan and Tracy for inviting me in to the TMC experience. Despite the malaise, I count it as a most rewarding experience. Oh, and I gather I am supposed to choose one TMC thing that I plan to work on for the next year. Well, I choose to be more active in welcoming people to the teaching discussion. I want the teachers around me to know that I really care about their thoughts and want to learn from them. I choose to ask others for their thoughts on teaching and to really listen.

So there’s my TMC16 reflections, from someone who didn’t actually go.



Posted in Being a good teacher, Education research reading

Once upon a time, I did a PhD in projective geometry. It was all about objects called quadrals (a word I made up) – ovals, ovoids, conics, quadrics and their cones – and the lines associated with them – tangents, secants, external lines, generator lines. During the first two years, I did talks about my PhD research, which I could not resist calling “David Butler and the Philosopher’s Cone” and “David Butler and the Chamber of Secants”.

At that time, my use of JK Rowling’s titles had to stop because there was no suitable mathematical thing to insert into the third title. It’s been ten long years since “David Butler and the Chamber of Secants”, and finally I have found something to use. Hence, welcome to…

David Butler and the Prisoner of Alhazen

Chapter 1: In which David finds out about lunes for the first time

As you may have noticed, I have quite a love affair with the Quarter the Cross problem, having blogged about it here and here. This blog post is not about Quarter the Cross, but the story did start there.

At the December holiday One Hundred Factorial session, we did Quarter the Cross together, and these were some of the solutions we came up with:


You may notice some rather interesting solutions involving curved shapes. Well, one of the students (Adam) created the first of these, and of course we all asked him how he was doing it. He told us the shapes were called “lunes” and showed us how to draw them: you draw semicircles outside the short sides of the triangle, and then one big semicircle inside the longest side, which has to go through the point because semicircles on hypotenuses always pass through right angles. Then he declared that the area of the two lunes was the same as the area of the triangle. (That is, in the following triangle, the striped area is the same as the pink area.)


Of course we needed a proof of this, and we duly worked together to construct one. Here’s a pretty version of the proof we came up with:


Later that day, I tweeted about this, and someone asked what this theorem was called. I didn’t know, so I went looking online for more about these things. My first problem was that I was searching for “loons” instead of “lunes”. Afterwards, it made perfect sense that they were called “lunes”, since “lune” is the French word for “moon”. But when I was searching, I had only heard the word spoken aloud and not seen it written down, so I couldn’t know that it was actually a French word could I? After what seemed like hours, I finally did find it. I was not disappointed.

These things have the coolest name of any maths object I know, worthy of any Harry Potter title: the Lunes of Alhazen! They are named after an Arab mathematician/scientist/philosopher who lived about a thousand years ago, and was once so famous that if you simply said “The Physicist” then people knew you were talking about Alhazen. (Funny how most of us educated in Western countries have never heard of him.)

Chapter 2: In which David is imprisoned by Alhazen’s lunes in two different ways

But the story didn’t stop there. As with most very cool maths things, it sort of takes over a corner of your mind, holding you prisoner, as it were. This Alhazen and his lunes held me and others prisoner in various different ways.

One way I was held prisoner was that I found myself trying to see lunes everywhere. Every time I saw a right-angled triangle in a design, I’d wonder about the two lunes you could make from it. MLC staffer Damien confessed this same obsession to me not long after the One Hundred Factorial session, except that he was actually carrying a pair of compasses around to do the actual construction. In Quarter the Cross itself, I had quite an ambition to find a single lune that had an area of a quarter of the cross. I did find it, and here it is:


(Technically, this lune is the Lune of Hippocrates. You can make two of them as lunes of Alhazen  on the short sides of an isosceles right-angled triangle, but one big one can be drawn on the hypotenuse too, to give the same area.)

The other way the Lunes of Alhazen held me prisoner began when I realised that our earlier proof didn’t rely on the fact that the shapes I had drawn were semicircles. I could use any shape as long as they were all similar to each other, because the areas of similar shapes are always proportional to the squares of matching lengths. So I started thinking about the concept of the “Things of Alhazen” – objects constructed in this way from a right-angled triangle. Here are a few:


The convex hexagons of Alhazen


The connected triangles of Alhazen


The strange shapes of Alhazen

And a few more, after I realised they could go inside the triangle too.


The dolphins of Alhazen


The loons of Alhazen

And a few more, after I realised the big shape didn’t technically have to go through the right-angle.


The big convex hexagon of Alhazen


The almost lune of Alhazen

(Note they’re all polygons because I was doing them in Geogebra and it was the easiest way to colour things in the way I wanted. I suppose I could have done it in Geogebra and copied the picture and bucket-filled, or drawn them in some other program. But this is what I did.)

Chapter 3: In which Tracy is also a prisoner of Alhazen

But I wasn’t the only one who was imprisoned by these lunes of Alhazen. This week it was “Twitter Math Camp”: a conference where those in the MTBoS come together to share their reflections and learning, organised by teachers for teachers.  At TMC, Tracy Zager did a keynote talk where she talked about the benefits that happen when people at all different levels of education come together to learn about teaching maths from each other. (You can see the part of the video directly related to this story here, and see the slides or give comments to Tracy at her blog here.)

Tracy told the story of Quarter the Cross, and how it leaped up and down the levels of education and back and forth around the world. When she originally saw the work me and my students did, she was instantly caught by the curved shapes, and wanted to know more. Upon hearing the name of them, this was her response:

She was instantly captured by the wonder and the history and the awe, just as I was. But the imprisonment is deeper than this.

That tweet was seven months ago, and yet here is Tracy in a keynote talk at a conference, remembering this as a key moment in her journey as a maths educator. In a way, Alhazen has captured a part of her memory. She will forever think back to this time and have it all wrapped up in these lunes. And so will I.

We are prisoners of Alhazen in the best possible sense.

Posted in Isn't maths cool?, One Hundred Factorial, Thoughts about maths thinking
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A few months ago, I learned a new word: “mansplaining”. You may have heard it before, but I never had until this year.

The general idea is that very often, a man will explain something to a woman in a way that seems to be based on the assumption that the woman is incapable of understanding the concept themselves and requires the man to rescue them from their misunderstanding. Often it is very explicitly patronising or condescending. This is a mansplanation.

In recent weeks, I have seen people I greatly respect being treated this way in the online space, and they have called out the man in question by telling him that he was mansplaining. Quite often, he has responded with quite a bit of vitriol, claiming that the word “mansplaining” is in itself sexist and they were just “trying to help”. This very vitriol is of course really not supporting the man’s case, and tends to show that his assumptions actually are that the woman did need to be rescued from her ignorant state. You can see some classic examples of this sort of assumption in Fawn Nguyen’s excellent blog post “Baklava and Euler”.

I had formed the idea that mansplaining was really just assholesmanplaining, and it didn’t have all that much to do with your general everyday respectful man.

But then something happened that hit me in the guts. Megan Schmidt started a conversation on Twitter about notation, and it had a flurry of responses, all from men, one of whom was me. She tweeted separately that “the mansplaining game is strong right now”. I was not consciously responding from an assumption that Megan needed to be rescued from confusion, and yet the conversation was called mansplaining. Clearly Megan’s use of the word didn’t fit with my understanding that only assholes mansplain.

It was time to get to the bottom of this, so I asked Megan to help me understand what she meant and how she felt about it. I have to thank her a hundred times for the thoughtful and gracious responses that she gave. I hope I will do justice to what you taught me, Megan!

I learned that there are times when offering an explanation at all is actually mansplaining. Not because the explainer is an asshole, or because they meant to be condescending or sexist, but because the explainer is unwittingly playing to a wider cultural assumption that the woman needs an explanation at all.

When a woman expresses frustration or anger or worry at something, a man’s common response is to offer an explanation to clear up confusion. Do you see the disconnect there? The man is rescuing the woman from confusion, but the woman wasn’t expressing confusion. She didn’t need an explanation – she didn’t need to be “rescued”. It’s most likely that she actually does understand the nuances of the concepts involved. Indeed, she would usually have to understand in order to have the emotional response she is having.

An unfortunate part of it is that the majority of men in this situation, especially in a professional setting, actually do realise that the woman does have the same or greater experience and training. It’s just that they are culturally conditioned to offer explanations in response to frustration. Indeed, it seems to be that men in professional settings are expected to engage in more “academic” conversations than “emotional” ones. Yet by doing so, we are still mansplaining.

The problem is that it opens the door for assholemansplanations, which are sure to follow. Even worse, it is adding to the hundreds of tiny  sexist events that occur for a woman every day. And it reinforces the very cultural norm that produces those daily tiny sexist events. It’s important to give the experience a loaded name like mansplaining to make sure that those of us who do care have our attention drawn to these problems.

But how, as a man, can I fight back? Well, I can certainly call out others when they are mansplaining. Assholemen need to hear it from other men to have a chance of hearing the message — they’ll never listen to a woman. Ordinary men need to know about the damage they do unintentionally.

And what about my own daily actions? All I can think of is to be more aware. I can listen to the actual words people are saying and notice the emotional part of what they say. I can choose to respond by asking for more information first, rather than launching into an unwanted and unnecessary explanation. It takes a lot of energy to watch your own words and actions, and sometimes I will slip (sorry in advance) but with practice I’ll get better at it. And then one day maybe I’ll find I never offer a mansplanation again.

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The story so far
I promised Tina on Twitter that I would write about how the line at infinity relates to conics, and I’ve been doing it in the last two blog posts.
First, I talked about what the line at infinity is. We noticed that a set of parallel lines all share a slope, so we […]

Posted in Isn't maths cool? | Tagged ,

The story so far
I made a promise to someone on Twitter to talk about how conics relate to the line at infinity, but when I came to do that, I realised it’s a can of worms that will take a few blog posts to untangle. Last time, I talked about how I construct and think […]

Posted in Isn't maths cool? | Tagged ,

Why I’m doing this

I foolishly said this on Twitter about a month ago:
@DavidKButlerUoA @MathguyArt on the line at infinity? Tell me about those!
— Tina Cardone (@crstn85) June 14, 2016

At the time I declared this was a bit of a can of worms and I promised to write something and post it later. Well, here it […]

Posted in Isn't maths cool? | Tagged ,

And there you have it. All my thoughts about the Day of Maths I had in a Year 7 classroom.

The Best! Day! Ever!
Quarter the Cross
The Zero Zeros
Spotless Dice
Mathematical Art Appreciation
Hotel Infinity

Thank you to everyone who has followed my journey all the way here. One thing I find about teaching is that it often leaves me […]

Posted in Being a good teacher, Thoughts about maths thinking | Tagged

This post is the sixth in a series about a Maths Day I did in a Year 7 class last week. I’ve talked about my feelings about the day overall, and  all but one of the activities we did: Quarter the Cross, the Zero Zeros, Spotless Dice, and looking at my maths art. In the […]

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This is the fifth in a series of posts coming from my Maths Day in a Year 7 classroom. So far I’ve talked about the following:

how awesome the day was
how I implemented Quarter the Cross
the unexpected things that happened with Zero Zeros
how I implemented Spotless Dice

You may be thinking by this stage, “David, how much […]

Posted in Isn't maths cool?, Other MLC stuff, Thoughts about maths thinking | Tagged ,
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This is the fourth in a series of blog posts about a Maths Day I did in a Year 7 classroom. So far I’ve discussed

how awesome the day was
how I implemented Quarter the Cross
the unexpected things that happened with Zero Zeros.

In this post, I’ll be talking about the activity we did in the recess-to-lunch period. […]

Posted in Being a good teacher, Thoughts about maths thinking | Tagged