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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.

Posted in Being a good teacher, Isn't maths cool? | Tagged ,
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This blog post is about the Solving Problems poster that has been on the MLC wall for more than ten years in one form or another. The most current version of it in handout form is this:

You can download this handout in PDF form here.

I’ve been meaning to blog about it for some time, but I never got around to it. However, I’ve shared it online a few times, so I think it’s time I did get around to it.

How it started

Nearly 20 years ago, when I was training as a high school teacher, I read Polya’s “How to Solve It” and thought it was very cool. Polya divided the problem-solving process into four helpful stages: “Understand the Problem”, “Formulate a Plan”, “Carry Out the Plan” and “Look Back”, and then crammed the space under those headings full of advice for problem-solving, with all sorts of helpful questions you could ask yourself to help your problem-solving along. I translated it into handouts for my students and thought I was doing a great job.

Over time, I realised it was all a bit too much information. A list of a hundred questions is just too much information to sift through to find the advice you need right now today while you’re in the middle of a problem. Not just this, but many of the questions on Polya’s list only apply to a particular niche problem type, and sometimes they are nonsensical for some kinds of problems. I needed something with just a few bits of advice that could be used to encourage problem-solvers in many situations.

Also, I always felt rather uncomfortable with Polya’s second stage, which he calls “Formulate a plan”. It took me a while to figure out why I felt uncomfortable with it, but eventually I realised that my problem was twofold.

Firstly, it is extremely rare to sit down and map out a plan for solving a maths problem. This sort of well-formed plan will only happen in the situation where you have seen this sort of problem before and you know exactly the steps that will need to be taken to get to the solution. And I would not really call that problem-solving!

Secondly, when you look at Polya’s advice under the “formulate a plan” heading, you find that much of it will require you to actually get going on attempting to solve the problem in order to decide if the action will be part of the plan at all. His titles imply that you neatly devise a plan and then neatly carry out the plan and tada you have solved the problem. But his text under those titles implies a more messy process of tentatively proposing an action, then giving it a go, then going back to see what you can try now.

So after thinking about Polya’s original heuristic scheme, I decided that what I wanted was to do three things:

  • Change the titles so that they better reflected how the problem-solving process works.
  • Choose only a few questions and advice to keep it manageable.
  • Choose advice and questions that would spark action in as many situations as possible.

Of course, I didn’t actually do this until many years later, after I became the coordinator of the Maths Learning Centre.

Something for/from the MLC

When I became the coordinator of the MLC back in 2008, I completely redesigned the learning environment. I rearranged the tables so we could fit more students, removed 80% of the textbooks so there wasn’t so much decision paralysis, got nice containers to put the stationery in on each table so it was easier to use, and I put a few displays on the walls. I put up the NZQRC posters, the Greek alphabet, and some quotes from the Phantom Tollbooth. I also removed everything off a very busy pinboard, but I didn’t put anything back on it. It just sat there looking very grey for seven months while I decided on what it was that I thought would have the most impact.

I realised that what I needed was something that could give the students and the staff some inspiration during problem-solving. So I set about trying to achieve the three things I wanted to do to Polya’s “How to Solve It” to make it more practically useable.

First, I changed “Formulate a plan” to “Decide what to do”, and “Carry out the plan” to “Do it”. This was much closer to how my experience of problem-solving works: I don’t make a plan, I just choose something to do and then do it, and then maybe come back and decide what to do next. To reflect that back and forth, I put arrows both forward and backwards between those steps.

After this, I thought of just three questions or pieces of advice to go under each heading, instead of the big list of questions Polya had. It was a tough thing to decide, since Polya had so many questions in his book you could ask yourself, and I had some others I had used myself that weren’t in his book. I sat down with my colleague Nicholas and we brainstormed all the things we could say. We took from our experience of being with students during the problem-solving process and thought about the places they got stuck or needed encouragement most. As my experience has grown, and as I have used it with students, I have changed what is written in all of the boxes. Looking back at the original version now, I cringe to see some very vague things like “Keep trying!” and “Start with what you know.”, but to be fair to past me and Nicholas, even these were better than nothing at all.

Finally, I added two extra stages to the problem-solving process: “Prepare yourself” and “Reward yourself”. These were not in my original plan of things to change that I made when I was a school teacher, but I had grown a lot since then, and my experience in the MLC has shown me that there were important emotional and cognitive aspects to problem-solving that Polya’s original book didn’t really cover.

Preparing yourself has two meanings. One is to be prepared by having the resources you need like paper or play dough or textbooks or calculators. The other is to be prepared mentally and emotionally, by realising that even if it’s hard, you do have the skills to finish or at least you are capable of learning them as you go. I had seen too many students decide it was impossible for them to succeed before even starting and so see any sign of struggle as confirmation of that impossibility. They needed reminding that they could do it or that they could learn to do it.

Rewarding yourself is there because I had seen so many students successfully solve problems and then next time still not believe they were capable of doing it. They never stopped to notice that they actually did solve the problems and so their experience never changed their view of their abilities — I could see their growth but they couldn’t. I wanted something on the wall I could point to and say that they had to stop and take note, or they hadn’t finished the problem-solving process yet. The sorriest cases were (and are) all the people who would completely discount their problem-solving if they had any help, and I wanted something there to remind them that they were still the one that did it, not me, and they could still count it as a success.

Even after all the changes over the years to the Solving Problems chart, I am still most proud of these two additions.

And so I was ready with the design of my poster in eary 2009, and I printed the headings and the prompts on coloured paper and arranged them on the pinboard. (This photo is from 2010.)

The poster has remained a part of the MLC learning environment since that time in 2009 when I first put it up. I have taken it with me to the later MLC locations, and referred to it hundreds of times when I have been helping students. And I have edited it many times as I have learned more about how students learn and how they respond to being given advice during problem solving, so it is very personal to me, and everything there has a reason to be there.

Reasons behind the current version

Over the years I have changed every one of the questions and advice on the problem-solving poster, as I read more research about how people learn problem-solving, and reflected more about how I myself go about problem-solving, and observed more about what things students find most helpful to be told or asked as they do problem-solving. So everything on the latest version has a lot of thought behind it. This is what the final section of the blog post is about.

(I just want to pause for a moment to say thank you to Nicholas Crouch,  Cass Lowry and Tierney Kennedy for looking over the latest version. I really appreciate it.)

PREPARE YOURSELF

You can do it! As I said already, this is here to highlight that part of success in problem-solving is to be both physically and mentally prepared. You have to have all the stuff you need and you have to be emotionally ready to keep trying even when it seems like you are failing.

UNDERSTAND THE PROBLEM

Write or draw — it helps you think. This began its life as “Just start writing!” under the  DO IT heading, but I realised it needed to be more specific and also include a reason why you wanted to do it. Many students actually believe that they have to do all their thinking in their head and we wanted to make it clear that drawing and writing are actually ways to support their thinking, not cop-outs for people who can’t. Later, I also realised that I didn’t want students to wait until late in the piece to start writing or drawing, but to start right at the beginning, so I moved it up to UNDERSTAND THE PROBLEM.

Make sense of all the words and symbols. When the poster was first made, this said “What do the words mean?” in both the sense of what are they trying to tell you and what does the technical terminology mean. We later added symbols too because there is so much meaning hidden in the symbols in maths problems, and so much the students need to learn about how the symbols behave before the problem becomes understandable! This month I changed it so it was an action, and I very much liked that the action told them to make it make sense, because I believe that maths should make sense. (Thank you to Cass Lowry for helping to make me brave to move away from questions and towards actions.)

Look for other related information. This one used to be a question too, “Can  you find other related information?”, but long ago we changed it so that students couldn’t just answer “no” and had to actually go looking. This one is deliberately vague, because it could be a worked example, or a relevant theorem, or more information about the rules of manipulating the symbols, or information about the context of the problem, or any number of other things that might be different in different contexts. But the message is the same: outside information is often needed to understand a problem.

DECIDE WHAT TO DO

I already mentioned that I changed this from Polya’s original “Formulate a plan” because you don’t actually formulate a plan most of the time, but instead just decide what to do, and often quickly. The advice here is for what to do when you don’t immediately see what to do.

Understand the goal. This used to be “What do you have to do?” and then later “What is the goal?” and when I decided to remove all the questions and make them directions, I decided the best verb was understand. The word goal is important because it focuses specifically on the idea that the problem has a specific outcome it is asking for, and it sets it up in the students’ mind as a location to get to. I like the word understand here too, because it refers back to understanding the problem, and says you are now understanding something else to figure out what to do. This mention of the goal used to be up in UNDERSTAND THE PROBLEM, but I moved it down because I know that actually the goal rarely makes any sense at all without all the surrounding context of the problem, and also starting with the goal tends to produce means-ends analysis, which is known to often be a big waste of energy. Putting it under DECIDE WHAT TO DO sends the message that you actually don’t need to think about the goal until you are ready to decide what to do.

Look at other problems for inspiration. This started life as “Look at already-solved problems for inspiration”. The “already-solved” was a direct reference to Polya’s work, but I later decided to make it “other problems”, because they don’t have to be actually already solved to be helpful, let  alone already solved by the student themselves. Plus, the other problems could be generalisations or specialisations of the current problem you’ve made up yourself, as opposed to worked solutions to similar problems (which are of course helpful). The “inspiration” was a deliberate choice because the point is not to copy the other problem, but to take inspiration from it for how you might proceed yourself. Too many students just try to copy the other problem rather than learn from it!

Choose a smaller part to try. The advice in this spot used to be to break the problem into smaller steps, but this almost never happens unless it’s a very familiar problem you already know how to solve. If it’s an unfamiliar problem, then you don’t know what steps there will be, and usually you just try to do something. It could be just to write or solve an equation, or it could be to ignore one aspect of the problem — Polya has a lot of advice on this front, actually. This new version covers all of it. The fact that it’s so general allows you the freedom to pick something on the fly. As you look back later, you can figure out more specific strategies.

(Thank you to those who replied to this tweet, and so made it clearer that I really did need to change this particular piece of advice.)

DO IT

Polya’s original book has very little in this step, because his premise is that you have made a plan and now you’re carrying it out. But that’s not my experience of problem-solving. The original version of this step had vague motivational phrases like “Keep trying!” I’ve slowly moved away from that and towards more specific advice to help students keep focussed and decide when to go back a step.

Focus on one part at a time. So many students that we observe get themselves tied up in knots trying to think about every aspect of a problem all at once, and we wanted to remind them that it was ok to ignore everything else for a bit and just deal with one thing at a time.

Regularly check with the goal. This used to be “keep the goal in mind”, because we wanted students to not forget what they were trying to achieve, which can happen in complicated problems. But of course, you can’t simultaneously focus on one part and be thinking about the goal! Your head would explode with the cognitive load of that! The compromise is to ignore the goal to focus on the bit you are doing right now, and then stop working to check in with the goal every so often. (Thanks also to Alex who summarised Ollie Lovell‘s book on cognitive load theory where this strategy is also mentioned.)

If it isn’t working, try something else. This one is here because many students will valiantly commit to a course of action in the face of all evidence that it is not working. You have to admire their persistence, but sometimes you do want to tell them to just go back a step and try a different approach. It’s not giving up, it’s being strategic. (Of course there are some circumstances when they have to retry something they gave up on earlier, but you can’t have it all.) I specifically have the “if it isn’t working” because you can’t just tell people to try something else if it is working!

LOOK BACK

The LOOK BACK step has always been my favourite bit of Polya’s original heurstic process. I loved that he explicitly included a stage where you evaluated what happened with the problem-solving. The three pieces of advice here have all been here in one form or another from the beginning. They are quite different from Polya’s advice and are based on our experience with solving problems in a school or university context.

Be clear that you reached the goal. The reason we put it here is seeing student work where they wrote it all up but didn’t actually answer the question being asked! For example, they give the inverse of a matrix instead of its determinant. Or they find the value of x where the maximum of a function happens but not the actual maximum. Or just that they stopped at some point halfway and honestly forgot to finish it. The latest change I made was to make this not a question, and also add the word “clear”. The students need to be clear to themselves that they have reached the goal — which can involve them checking the result like Polya suggests under LOOK BACK — but they also need to make it clear to whoever is looking at their work that they reached the goal. It’s amazing how useful a neat summary sentence at the end is to a marker!

Rewrite so other people can understand. The purpose of this is to make sure students consider that their work is actually almost always going to be read by someone else, especially in a school or university context, where problems are often for the purposes of assessment. But also in a work context you will always have to tell someone, whether in a published article or a presentation. It used to be “Could someone else understand your work?” but people can blithely answer “yes” to this without really thinking about it. So this month I changed it to a direct command to rewrite it. To direct people to rewrite implies that some rewriting will always need to be done, especially if it’s been a complicated process to get to the solution!

Find something you can learn. The purpose of this is to remind students that the actual point is to learn something. In Polya’s original list, he focuses explicitly on using the result or the method for another problem, and that is certainly a thing you can learn. I do like how that completes the circle by making this problem you just solved one of the other problems you use for inspiration in the future. But this is not the only thing you can learn from doing a problem. You may learn a key maths concept, or learn something about how maths is applied, or learn something about maths language, or learn something about yourself. All of these things are perfectly good things to learn from a problem, and I don’t want to dictate something in particular. What I do want to dictate is that you learn something, and that’s why years ago I changed it from “Is there something you can learn?” where the answer could be “no”, to a direct command to find something to learn.

REWARD YOURSELF

You did it! As I said above, I added this when I first created the poster in order to remind students that they need to notice when they solve a problem. Too many students don’t learn the most important thing they could learn which is that they are capable. They do any number of problems and exercises and still believe they aren’t capable of solving problems. I really want them to stop and notice and revel in this feeling. And it’s also a non-negotiable part of the process even when they get help. I have pointed to this step so many times to say to students that they still need to reward themselves for what they did, even if all it was was to stop and find something they can learn when someone solves the problem for them.

Conclusion

So that’s my Solving Problems poster and handout. Thank you for sticking with me to hear the story of how it was created, and the reasons why it is the way it is. It’s something very special to me and I hope it might be useful for you too.

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This blog post is about a new variation on the classic Quarter the Cross problem, which I call Quarter the Cross: Connect the Dots.

Background

Here is the original Quarter the Cross problem:

To catch you up, here is everything I’ve written about Quarter the Cross up until now:

Even without reading those posts, you can probably infer that I really love Quarter the Cross. And you’d be right. I love how you have to think a bit hard to get any solution, but once you get started there is so much freedom to be creative.

But sometimes, you feel like you don’t want quite so much room for creativity. You want some more constraints so you don’t feel awash in the entire universe of possibilities, most of which you can’t even think of. Alternatively, you might enjoy the creativity but you are running a bit dry and need some more constraints to push you to try new things. Finally, Quarter the Cross might seem all a bit familiar to you, and you still want to play, but you need something to make it new. This new version of Quarter the Cross provides a solution to all of these problems.

A new constraint: connect the dots

Here is the new version of the puzzle, to use when you feel the need for an extra constraint.

(A downloadable Word document with this cross made of 3cm squares and the instructions is here.)

This Connect the Dots version is an easy way to turn the original Quarter the Cross into a new challenge. One bonus feature is that someone else can put the dots in for you, making it more of a surprise. If you would like a computer to choose for you, I made this Desmos graph. Note that you could choose a different number of dots than four (and the Desmos graph allows you to do so) but I find it’s about the right number to make the challenge easy to set up and not too annoying to do.

I personally very much enjoyed this challenge. It forced me to think in new ways, because I couldn’t just put the shapes I would normally use wherever I wanted. I had to do a lot more thinking about how pieces added up to a quarter because I had to stretch them out to meet each other. I also had to let go of an attachment to symmetry. (Though I now realise it could have been an extra extra challenge to make the solution symmetrical in some way as well as connect the dots!)

I’ll finish with some tweets with solutions to Quarter the Cross: Connect the Dots challenges. If you want to try the challenge yourself before seeing others’ solutions, please look away now! Either way, I hope you enjoy this variation on a classic.

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This blog post is about a metaphor I use when I think about the order of operations: the idea that the various operations are stickier than the others, holding the numbers around them together more or less strongly.
The idea begins with the fundamental idea in arithmetic, that maths working proceeds by replacing something with something […]

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I have had many people say to me over the years, “But algebra is easy: just tell them to do the same thing to both sides!” This is wrong in several ways, not least of which is the word “easy”. The particular way it’s wrong that I want to talk about today is the idea […]

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I think asking students questions is an important part of my job of helping students succeed. Good questions can help me see where they are in their journey so I can choose how to guide them to the next step, or can help to make clear the skills they already have that will help them […]

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I conscripted the game Numbers and Letters seven years ago to help promote the Maths Learning Centre and the Writing Centre at university events like O’Week and Open Day. Ever since then, it has always bothered me how free and easy participation in the Letters game is, while the Numbers game is much less so. […]

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This blog post is about a game I invented in February 2020, the third in a suite of Battleships-style games. (The previous two are Which Number Where and Digit Disguises.)
NUMBER NEIGHBOURHOODS: A game of analytic deduction
Players:

This game is for two players, or two teams.

Setting up:

Each player/team choose six different numbers between 0 and 10 (not including 0 […]

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Introduction
In 2016 I created the iplane idea, which allows you to locate the complex points on a real graph.
In case you haven’t heard of it or you need reminding. The idea is that at every real point (p,q) of the real plane, there is a planes-worth of complex points attached, all of which have coordinates […]

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Once upon a time, I met a His Royal Highness the Duke of Kent.
The story of how that happened was pretty cool from my perspective, but every so often I wonder about it from his perspective. The Duke is the patron of the Royal Institution of Australia, and was in Australia just as they were […]

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