Tag: learning culture

Running out of puzzles
Because people know I run the One Hundred Factorial puzzle sessions, they often ask me if I have a repository of puzzles they can use for their classroom, enrichment program, maths club, or their own enjoyment.

Sometimes I feel embarrassed because I don’t actually have a big repository of puzzles. Surely since I am a person known for promoting problem-solving and puzzles, I should have such a thing. At the very least I should have a record of the puzzles we did do. But I don’t.

It turns out my scatterbrained tendency to forget record-keeping is not the main thing that caused this lack of puzzle repository, but only in the last few weeks did I realise what the main cause actually was. It’s that I don’t feel the need for lots of puzzles. A person recently asked for my advice on where to find puzzles and told me the reason was they were worried their maths club would tear through them and so have nothing to do. Only when they gave this reason did I realise I don’t worry about this at One Hundred Factorial. But why?

Firstly, puzzles are not the main food at One Hundred Factorial. I usually have exactly five activities available: a logic puzzle (eg sudoku), a word/geometry puzzle, an art activity/construction toy, a game, and the Numbers Game. If people get to the end of the puzzles, there is always other stuff to do instead.

Secondly, and much more importantly, the whole vibe of One Hundred Factorial means that puzzles do not end. I have carefully cultivated a culture encapsulated in the mantra:

The goal is not the goal.
The end is not the end.

What “the goal is not the goal” means is that the stated goal of a puzzle or problem is not the actual goal. The “goal” might be to find the area of a shape, or the probability of some event, or count how many of something there is, or whatever. They are not the goal. The real goals are to learn something, or understand someone’s thinking, or make something beautiful, or find a connection to something else.

What “the end is not the end” means is that even if you do get to the stated goal of a problem, it doesn’t mean the thinking stops. You can ask if there’s another way, or what the problem would be like if you changed this aspect, or look for a connection to something else, or build something cool out of the answer or process. The truth is there is no end.

The mantra of “the goal is not the goal, the end is not the end” means that we can get by at One Hundred Factorial with just one puzzle. In fact, we can get by with no new puzzle at all. Maybe someone was at the previous session and we want to continue with the non-end of last week’s puzzle. Or someone saw a random thing during the week that inspired their thinking and turn up ready to include others in their thinking or find out what thinking it might inspire in others. Or someone pulls out a puzzle that’s been done before and wants to find out how other people might think about it.

As far as I can see it, my approach to cultivating a “goal is not the goal, end is not the end” culture had three aspects:
1. Constantly ask goal-free, non-end questions like “what are you thinking?”, “is there another way?”, “what would happen if?”, “what can we make?”, “what is this connected to?”.
2. Notice when other people ask those sorts of questions and run with it. I found that once I became attuned to them, I noticed people asked them a lot more often than I realised.
3. Provide open-ended things other than just puzzles, like construction toys or art activities. There is nothing like an activity with no goal to foster a more goal free attitude. Even just puzzles with more than one solution foster a more open-ended attitude.
So that’s how I don’t run out of puzzles: I don’t only use puzzles, and when I do, we go further or in different directions than the puzzle says to.
2 February, 2022
My first Maths Teacher Circle

Last week I participated in my first Maths Teacher Circle . I just want to do a quick blog post here to record for posterity that I did it and it was excellent. I choose to take the practical approach of just relating what happened.

I had been interested in somehow going to one since I heard about them a while ago, and then the founder of the Aussie Maths Teacher Circles, Michaela Epstein , contacted me through Twitter back in November to ask if I might like to facilitate an activity at an online session in 2021, and of course I said yes. She invited me to a session about mathematical games, and I was so excited to share some of the games I have invented with some interested teachers.

Of course, the closer it got, the more nervous I got. When I heard there would be 40 or so teachers ranging all through primary to secondary to post-school teachers, I was rather intimidated! But Michaela and Alex  assured me I would be ok and that what I had planned would work. And they also put up with my scatterbrained discussion of random maths stuff whenever I met with them too. So, feeling a little reassured, but still nervouscited (as Pinkie Pie would say), I dove right in feet first last Wednesday morning.

To start off with, Michaela invited past Maths Teacher Circles participant Samantha  to set the scene by sharing what she has gotten out of Maths Teacher Circles in the past. This was a nice way to begin by grounding it in a real teacher’s experience. Then Michaela shared the goals of Maths Teacher Circles, which were exploring maths, strengthening classroom practice, and bringing maths enthusiasts together. I was so glad I had come to a place that resonated with all the things I love. It really matched with the goals of One Hundred Factorial, which is probably why Michaela invited me to present in the first place. This was all a really smart way to begin, because it set the tone for the rest of the session. Even when the housekeeping notes about breakout rooms and whiteboards and chat windows came, it was clear that these were there to support the overall vibe.

Then we had a very quick chat in breakout rooms with a couple of people. We were supposed to talk about Noughts and Crosses too, but we only just made it through the introductions! But honestly I was happy to just have met a couple of friendly faces to help reduce the nervous part of the nervouscieted.

By this time, so much had happened already, yet it had only been a few minutes. And now it was my turn. Michaela introduced me and I was now responsible for the journey of these 45-ish hopeful people. I put up the rules for Which Number Where, and asked everyone to quietly have a read, then ask any questions they might have. People had some very useful questions in the chat and out loud, and I felt we were ready to try it live. I asked for volunteers and described how to play the game Mastermind-style, with one player being the Secret Keeper and the other players asking questions. After a couple more questions, we were ready to break into groups to play.

Michaela put people into groups of fourish, and I popped into about half of them to have a chat. I asked people how they were going and played with them for a bit, seeding a different kind of question than the ones they had been asking so far. I found everyone to be gracious and thoughtful and engaged. Such a thrill to meet such wonderful people and play maths with them. These moments when I was in a small group with people were my favourite parts of the session.

I brought everyone together into the big group to discuss how the game went. I started by asking people if they had a favourite question that was asked. And then people shared any thoughts they had at all about how to use this in a classroom.

Suddenly it seemed my time had run out, so I quickly showed everyone my other two games Digit Disguises and Number Neighbourhoods, and encouraged them to go back to their breakout rooms to keep playing Which Number Where or to try a new game instead. I stayed out in the main room where Michaela made sure I was ready to do a wrap-up when people returned. I very much appreciated being able to think in advance about that part!

One question Michaela asked was why I chose the game I did. I said I chose Which Number Where because it’s about logic, and not any particular maths topic per se. As someone said earlier, it’s about locations rather than numbers per se, which means it’s really about the yes-and-no questions, and about logical arguments and joining information together, and those are skills you use everywhere in maths, which is why I like it so much. Plus I just love to hear how people think and this game gives me a chance to do that.

And then it was time for me to participate in someone else’s activity. Toby  and James  shared the Multiple Mysteries game and some problem-solving/proving prompts to go with it. I got to play the game with some lovely other people and join in with the play. It really was a lovely thing to just play around with something that someone else shared that they were excited about. I am very grateful to Toby and James for providing such a great game to play and think about, and to the members of my little breakout room who I had such fun with.

After this, it turned out that Michaela had read the time wrong and had cut short my activity the first time! So I got to have a few more minutes! I decided to share Digit Disguises properly, and instead of using breakout rooms, to play a game as a whole room with me as the Secret Keeper. Some brave souls shouted out questions and I wrote the questions and responses on a Word document on the screen. After a few questions, I decided that I would stop people and ask them what they can figure out from the information we have so far. This part was just wonderful. People had multiple different ways of gleaning new information about the numbers and their letter disguises from what we already knew, and quite a few of the participants expressed a satisfying amount of delight at these fascinating new possibilities. It was extremely gratifying to have people so excited about something that I am excited about (and egotistically, satisfying that people liked something I had invented).

At this point, my laptop ran out of battery power and I had to scramble to find the power cord. By the time I came back, things were starting to wrap up, with participants filling out a Padlet with their thoughts. And then it was over. It felt like almost no time at all had passed, which is a good sign that I’ve been deeply engaged.

After all the other participants left, Michaela, Alex, Toby, James and I had a debrief, which was some lovely discussion about how it went and how cool it was to work mathematically with people rather than just present them with stuff, and just some nice discussion about teaching and learning maths with some lovely people. And after that, I couldn’t help but keep working on  one of the investigations that Toby and James set me off on, because that’s how I roll and is the sign of a good maths problem.

So that was my first experience of a Maths Teacher Circle. For me, the best part was the chance to think and play together with other teachers. The environment was so safe to just play and talk, and this was very carefully set up by Michaela in the first place, by discussing what was important and how to keep it safe. Being told explicitly that we were allowed to adjust the activities to match the level of the group made us free to play in our own way. And really, everyone was just so gracious and excited and, well, lovely. I am so grateful to have been a part of it.

21 January, 2021
The Seven Sticks and what mathematics is

This week I provided games and puzzles at a welcome lunch for new students in the Mathematical Sciences degree programs. I had big logic puzzles and maths toys and also a list of some of my eight most favourite puzzles on tables with paper tablecloths to write on.

One of the puzzles is the Seven Sticks puzzle, which I invented:

Seven Sticks
I have seven sticks, all different lengths, all a whole number of centimetres long. I can tell the longest one is less than 30 cm long, because it’s shorter than a piece of paper, but other than that I have no way of measuring them.
Whenever I pick three sticks from the pile, I find that I can’t ever make a triangle with them.
How long is the shortest stick?

I sat down at one of the tables and I could see the students there were working on the Seven Sticks, so I asked them to explain what they had done. (WARNING! I will need to talk about their approach to the problem, so SPOILER ALERT!)

They told me about their reasoning concerning lengths that don’t form triangles, and what that will mean if you put the sticks in a list in order of size. They had used this reasoning to write out a couple of lists of sticks on the tablecloth which showed that a certain length of shortest stick was possible but that a longer shorter stick wasn’t possible. And so they knew how long the shortest stick actually was.

Only they said to me they hadn’t done it right.

I was surprised. “What?” I said. “All of your reasoning was correct and completely logical and you explained it all very clearly. Why don’t you think you’ve done it right?”

Their response was that what they had done wasn’t maths. They pointed across the table to what some other students were doing, which had all sorts of scribbling with calculations and algebra, and said, “See? That’s maths there and we didn’t do any of that.”

I told them that actually what they did was exactly what maths is – reasoning things out using the information you have and being able to be sure of your method and your answer. Just because there’s no symbols, it doesn’t mean it’s not maths.

Only later did I realise the implication of what had happened: these students are coming into a maths degree, which means they have done the highest level of maths at school, and they perceived that something was only maths if it had symbols in it. A plain ordinary logical argument wasn’t maths to them. And a scribbling of symbols was better than a logical argument, even if it didn’t actually produce a solution.

This made me really sad.

I wonder what sort of experiences they had that led them to this belief. I wonder what sort of experiences they missed out on that led them to this belief. I wonder how many other students have the same belief and I will never know. I wonder how I can help all the new students to see more in maths than calculations and symbol manipulation, and allow themselves to be proud of their work when they do it another way.

19 February, 2019
A public health approach to improving teaching and learning

Making a big difference to student learning is a tricky business. Here at my university, there are a certain number of (wonderful) teaching staff who are champions of innovation, always making big changes to the way they do things and jumping onto any innovation as soon as it comes around. Yet the students not in those classes don’t see much benefit from it. Indeed, those staff who are not champions of innovation may do nothing for fear of having to adopt all at once All The Things they see the champions doing. A student who seeks regular support for their learning may make spectacular gains, but there are literally thousands of other students who don’t seek such support on a regular basis, and thousands of students who don’t really need spectacular gains but just a little bit extra. I have started to think that perhaps the best way to make a big difference is to find some way of encouraging a large number of small differences.

This is essentially the way Public Health works. In Public Health you are concerned with whole population health initiatives, which are often of necessity a large number of small differences. For example, you may not cure the flu, but you might encourage 20% more people to wash their hands and so prevent the spread of infection and stop so many people getting the flu in the first place.

Imagine the benefit that might happen, not if a few lecturers rub out their courses and start again with flipped learning, but just if every lecturer simply labelled everything in Canvas/Blackboard so the students could easily find stuff. Imagine the benefit, not if a few course coordinators completely changed their tutorials to be about group discussion, but if every classroom tutor asked one “what if” question in every tutorial. These are not big things to change, but if a lot more people did them, I think the overall effect would be far-reaching. And they might seem like something you could actually do, as opposed to the big changes that are the usual fare of innovation.

Personally I am trying to do more Public Health approaches to student support too. Instead of just visiting lectures to tell the students how to seek one-on-one support, I’m visiting with a five-minute message about interpreting assignment questions, or choosing to put in more explanatory working, or what a standard deviation is. If I can reach even half of a lecture of 500 students with one of those little messages, then I have made a big difference by making a lot of small differences.

Unfortunately, Public Health doesn’t make for spectacular stories. Giving one person brain surgery to save their life after a horrific traffic accident is a spectacular story. On the other hand, lowering the speed limit in urban areas in order to make horrific accidents less likely is not a spectacular story, but it can be argued that it saves a whole lot more lives. I only hope I can convince the Powers That Be that my Public Health approaches to learning and teaching improvement are worthwhile, if not spectacular.

16 May, 2018
Book Reading: Becoming the Math(s) Teacher You Wish You’d Had
This post is about Tracy Zager’s most excellent book, Becoming the Math Teacher You Wish You’d Had.

(You can read this blog post and all other Book Reading posts in PDF form here. )

I actually finished reading it back in January, and I live-tweeted my reading as I went. The process culminated with this tweet:

https://twitter.com/DavidKButlerUoA/status/824183955012415495

That’s what I thought about it at the time, but I haven’t sat down to organise my thoughts on it. Until now.

I was first drawn to the book based entirely on its contents page. Check this out:
- Chapter 1: Breaking the Cycle
- Chapter 2: What Do Mathematicians Do?
- Chapter 3: Mathematicians Take Risks
- Chapter 4: Mathematicians Make Mistakes
- Chapter 5: Mathematicians Are Precise
- Chapter 6: Mathematicians Rise to a Challenge
- Chapter 7: Mathematicians Ask Questions
- Chapter 8: Mathematicians Connect Ideas
- Chapter 9: Mathematicias Use Intuition
- Chapter 10: Mathematicians Reason
- Chapter 11: Mathematicians Prove
- Chapter 12: Mathematicians Work Together and Alone
- Chapter 13: “Favourable Conditions” for All Maths Students
Is this not awesome? Here was a list articulating things about maths that I know are important and yet that I’ve struggled to articulate all my life as a mathematician and maths educator. Many of them cut straight to the heart of the difference between how I experience mathematics and how it usually is experienced in a classroom.
- “Mathematicians use intuition” you say? Well, yes. Yes we do.
  But many a maths classroom is about following rules and avoiding the need for intuition.
- “Mathematicians work together” you say? Well, yes. Yes we do.
- But so many students think maths is only a solitary activity.
- “Mathematicians make mistakes” you say? Well, yes. Yes we do.
  But mistakes are feared and avoided in most maths classes.
- “Mathemaicians connect ideas” you say? Well, yes. Yes we do.
  But so many maths curriculums are just so many piles of disconnected procedures, even here at my own university.
The contents page promised a book about the most important aspects of mathematical work and thinking, and a hope that it would give ways to bring these into the experiences of students in all maths classrooms.

And the hope was made real.

Each chapter starts out comparing how mathematicians talk about what they do and what students’ experience of it is. Then it moves on to detailed examples of the aspect of maths thinking in action in real classrooms, as well as strategies to encourage it both in your students and in yourself as a teacher.

I didn’t expect to see this last point about encouraging these attitudes and thinking in yourself as a teacher. Yet it is the most compelling feature of the book for me. Indeed, I don’t think the book would have had nearly the impact it had on me (or the impact I see it having on others) without this constant message that to help your students experience maths differently, then you yourself need to experience it differently too. More than this, Tracy doesn’t just make this need clear, but actively and compassionately empowers us to seek out ways to fill it.

Somewhere inside you is a child who used to play with numbers, patterns and shapes. Reconnecting with your inner mathematician will improve your teaching and benefit your students, and it will also benefit you.
– Tracy Zager, Becoming the Math Teacher You Wish You’d Had, p39

These comments were left on the original blog post:

Tracy 10 May 2017:

Misting up over here, David. You’ve cut right to the heart of it. Thank you so much. I feel incredibly lucky to learn with you.

David Butler 10 May 2017:

And I am lucky to have been able to sit and talk with you by reading your book. I really felt like you were there with me, encouraging me to be more.

Susan Jones 10 May 2017:

I’m still reading it. Your post makes me realize my privilege in not majoring in math 🙂 I never did lose the “play with the numbers” thing.

David Butler 10 May 2017:

Sadly, you don’t have to major in maths to lose the play-with-numbers thing. A good dose of standard high school maths teaching can safely banish that tendency, as Tracy described in the first chapter of the book!

For me, my maths university degree is what actually freed me to play. I took Discrete Maths II in second year of university and it felt like all we were doing was playing with these ideas, and it was play encouraged by the lecturer.
10 May, 2017
Really working together

Yesterday, I had one of those experiences in the MLC that makes me love my job.

The Maths 1B students were working on a linear algebra proof today, and as I came up to one of the tables, Fred (name changed) was explaining the beginning of his proof to the rest of the table. When I arrived at the table, he was leaning over two of the other students to point at a section of the lecture notes. He noticed I was standing there and said, “But David can probably do this better than me.” I responded, “Not necessarily. You keep going,” and I sat down in his chair.

Fred continued to explain, and I think he did a very good job. I was very pleased that he kept flicking through the lecture notes to point out different theorems, though I thought it was interesting that at no point did he write anything down.

Then one of the other students said, “so is that the end?” And Fred said that no, this was just the beginning, there was still more after that, and I could see in his eyes he was having a sinking feeling as he tried to think of how to move on to the next bit.

So I asked him if he could pass me a whiteboard marker. I stood up to the wall and said, “I just want to write down where we’re up to.” I asked the students he had been talking with to tell me what they’d done so far, and I transcribed it to the wall, asking them to explain why each line worked. And then we got up to the end of what they had already done.

“So what now?” I asked. There was a short silence, and then Fred piped up with a comment about what we needed to know next. I asked why that was important to know, and this started a discussion of what goal we were heading for.

And here is where the really great stuff happened. The students at the table offered suggestions of things to try, looked up definitions and theorems in their notes, helped each other refine their maths language, asked each other questions when they weren’t sure of things, welcomed new students into the discussion when they wandered over to listen, discussed how to make the proof their own when they wrote it to hand in, and basically really worked together to construct the proof. It was a pleasure to be a part of it.

It’s this sort of thing that makes my job such a joy – seeing students learning and supporting each other to succeed. On a day containing many other parts of my job that are much less joyful, it was something I really needed to see.

This comment was left on the original blog post:

Steven 8 February 2016:

Indeed, I also did enjoy reading your post regarding on how crucial and effective to have a group discussion i.e. working together. I’ll be sitting UMAT this year and I hope i can find someone/group as well to discuss on some UMAT questions and produce interesting results like the above.

5 February, 2016
Moses loved numbers

Many traditions hold that Moses wrote the first five books of the Bible. If we assume this is true, then there is one thing I think is clear about Moses, based on the things he wrote: he loved numbers. I’m pretty sure he was a mathematician at heart, or at the very least an accountant, because his books are littered with numbers which are not entirely necessary to get his overall point across.

Just look at this passage from Genesis (NIV):

When Adam had lived 130 years, he had a son and he named him Seth. Afer Seth was born, Adam lived 800 years and had other sons and daughters. Altogether Adam lived 930 years, and then he died.

When Seth had lived 105 years, he became the father of Enosh. And after he became the father of Enosh, Seth lived 807 years and had other sons and daughters. Altogether Enosh lived 912 years, and then he died.

When Enosh had lived 90 years, he became the father of Kenan. And after he became the father of Kenan, Enosh lived 815 years and had other sons and daughters. Altogether, Enosh lived 905 years, and then he died.

When Kenan had lived 70 years, he became the father of Mahalalel. And after he became the father of Mahalalel, Kenan lived 840 years and had other sons and daughters. Altogether, Kenan lived 910 years, and then he died.

When Mahalalel had lived 65 years…

And this one:

He spent the night there, and from what he had with him he selected a gift for his brother Esau: two hundred female goats and twenty male goats, two hundred ewes and twenty rams, thirty female camels with their young, forty cows and ten bulls, and twenty female donkeys and ten male donkeys.

The emphasis on numbers is striking.

Now I’m pretty sure Moses didn’t mean to place such an emphasis on numbers in his writing. Presumably his main aim was to let his readers know about the history of Israel, and the nature of God and his relationship with humankind in general and Israel in particular. But still, the numbers are there. Why?

I argue that the reason the numbers are there is because Moses himself loved numbers. I think he couldn’t help the numbers appearing in his writing because he wasn’t even aware he was doing it. He liked numbers, so he thought about them a lot, and so they just turned up in his head when he was writing his books.

And if it can happen to Moses, then it can happen to anyone. I know myself that I can’t help references to childrens literature turning up in my lectures, and I can’t help maths turning up in my everyday conversation, just because I love those things. And I can’t help turning every discussion about maths into a discussion about problem-solving, because I think about the process of problem-solving a lot and it just happens.

But the danger is when the things we are interested in distract from the message we want to get across. For example, what if a teacher absolutely loved sport to such an extent that every example in class was about sport, and some of the students who disliked sport were turned off because of the association? And what if the thing a teacher most loved in the solution to a problem was the fancy trick? Then when they presented the solution they couldn’t help getting excited about the trick and it would seem to their students that fancy tricks were what problem-solving was all about.

But what can these teachers do, since they can’t help the things they love coming through in their communication? Well I think they can simply be aware of it. Then at least they can make sure that even though the things they love are there, the overall message isn’t obscured by them. (Of course the ultimate would be to love the thing you are trying to teach!)

21 May, 2013