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:
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?”.
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.
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.
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.
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.
These comments were left on the original blog post:
Tierney Kennedy 19 November 2020
Great post! I love the description of how your thinking has changed over time. I’ll be sharing this with the teachers I work with.
David Butler 20 November 2020
Thanks Tierney. This means a lot to me.
Ian Thomson 10 December 2020
Thank you so much David! This gives a wonderful overall view of the process. It is very practical and addresses some potential stumbling blocks in an insightful way. As always you are respectful and understanding of the learner.
David Butler 13 December 2020
Thank you so much Ian
Shane Dye 1 March 2021
Great poster and a great post. Reading through the way you developed this, I am even more convinced this will be useful to our students. And, you have given me some great ideas for making use of it. I would love to be able to share this with our students both on the wall in our Maths Room and as an electronic version through our Learning Management System. Are you happy for it to be shared in these ways?
David Butler 4 March 2021
Absolutely ok with you sharing it with your students in both places. So glad it might help your students.
Quarter the Cross: Colouring — in which I draw All. The. Lines. and so create a colouring template for Quarter the Cross
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 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.
I made these solutions in May 2018 when I first was trialling the Connect the Dots challenge to see how it felt.
I made these two at the same time as the ones above. They are two different solutions for the same set of four dots.
These solutions were made by people at a One Hundred Factorial session not long after I thought of the Connect the Dots challenge.
And one more that I made in 2020 after creating the Desmos graph that randomly chooses dots.
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 else you know it’s equal to. When working on an expression involving numbers and operations, to proceed further, you need to find a piece of the expression you can replace. The order of operations describes the rules for when you are allowed to replace right now a part of an expression involving numbers and an operation. (I usually draw the order of operations using the Operation Tower.)
For example, look at the following expression:
6÷ 3 + 7 – 2 – 5 + 8× 2³ + 6× (8+2)
There are several pieces of this you can replace with their result safely without breaking the rules of the order of operations. You can replace the 6÷ 3 with 2, or the 7 – 2 with 5, or the 2³ with 8, or the 8+2 with 10.
When people describe the order of operations, we usually talk about it saying which operations can be done first, saying that some operations should be done earlier than others, and you do them left to right if there’s no preference, and brackets override the order. That’s how I usually describe it.
But this doesn’t quite match with what I describe above. For a start, it’s actually correct to do the subtraction before the power even though the power is supposed to be done earlier than subtraction. I usually tell students that the order of operations just tells you which operations go before others when they’re nearby to each other. So that subtraction is too far away from the power to really make a difference if I do it first or second.
But then there’s weird problem with subtractions near to additions and each other. I can’t replace the 2 – 5 in the above expression because that would stuff up the 7 – 2. But there is nothing in my description of “nearby” that would tell you there is this preference. You have to fall back on the “left to right” idea from the traditional order of operations again.
For some reason I really don’t like this. The left-to-right idea seems completely against all the effort we take to teach students to be flexible with their arithmetic, where we encourage them to pull numbers apart and put them together in new ways to make the arithmetic easier to do and to understand. Plus it’s not how I actually do it. When I read an expression, I scan across the whole thing first, and then I pick a spot to work on that feels right to me, deciding that it’s ok to work on that bit because it doesn’t cause problems with the things around it. I make my decision of which operation to do locally, not globally.
And there is one final thing: when we port all of this order of operations stuff up to algebra, we start to think about terms, which are pieces of expressions joined together with multiplication and exponents. We learn to see those things as one blob, separated by the minus and plus signs.
So I reckon that the order of operations in my head is actually not about which operation should be done first, but which operations do a better job of sticking numbers together.
In the above expression, the smaller expression 6÷ 3 is stuck together more strongly than the expression 3 + 7. It’s safe to pull out the 6÷ 3 and replace it with a 2, but not safe to pull out the 3+7. So ÷ is stickier than +. I imagine pulling at the 3+7 to try to replace it with a 10, but the 3 breaks off when I do that because the “6÷” is stuck more strongly than the + 7.
Also, in the above expression, I notice that I think it’s safe to pull out the 7 – 2 to replace it, but it’s not safe to pull out the 2 – 5. It must be that – is stickier on its right-hand side than it is on its left-hand side. If I see a ” – 2 –” then I know the whatever number is before the 2 is stuck more strongly to the 2 than whatever number is after the 2.
So it seems that the operations have different levels of stickiness, and the ones that aren’t commutative stick stronger to one side than the other.
But there is one thing left to think about: grouping symbols like brackets and the horizontal bar. They’re not operations, but they do play a part in holding expressions together. Indeed, for brackets, holding things together is their entire purpose. In fact, the stuff between the brackets, or under the bar in a square root, all that stuff is wrapped up together into one number and can be treated as one number for the purposes of moving stuff around in algebra. So maybe the grouping symbols work more like a box you put things into, with operations outside stuck to the box itself rather than to the stuff inside it.
This box idea is actually quite useful, because it helps me to sort out problems with how I think about the distributive law. A lot of students will see something like 3(x+2)² and try to do (3x+6)² . What they did there was replace 3(x+2) with 3x+6. But the brackets are holding the x+2 in a box, so that it might as well be 3B². In that case, the exponent sticks to the box much more strongly than the 3, so you can’t pull out the 3B to replace it with something else, though you can pull out the B² to replace it with something else.
So those are my fairly rambling thoughts that explain why I rather like this way of describing the order of operations. I’ve used the stickiness metaphor before, but not until now have I thought about it as deeply as this, and I actually like it even more than I used to, now that I’ve connected it to replacing.
So, to sum up, here’s how I think about the metaphor:
To do maths working, you pull out an expression to replace it with something it’s equal to.
Operations are glue that stick numbers together, preventing you from pulling them out if they’re stuck too well.
Some operations are stickier than others: × and ÷ are stickier than + and -; ^ and √ are sticker than × and ÷.
Some operations are stickier on one side than the other: – and ÷ are stickier on their right-hand side.
If you see …number[operation]number…, you can only pull that expression out to replace it if the operations on the other sides of the numbers equal or less sticky than the one between.
Grouping symbols are boxes that hold expressions inside them.
You can always replace the stuff inside a box without affecting the stuff outside the box.
And so in my head, the Operations Tower has become a Tower of Stickiness:
PS: I had this thought that I need to add here, lest I forget it. I reckon that it would be much easier to help people understand the idea of an order of operations that allows for flexibility if you did just addition and subtraction first, without any other operations. Knowing that 7+5 – 3 – 2 has a few choices for how it can be done would be a great advantage before you introduce multiplication and division into the mix. Plus, a string of additions and subtractions could easily represent a story of adding and taking and so your learners would know what the correct answer is supposed to be! Anyway, that’s a thought to wrestle with another day.
NOTE: My thinking about this evolved over the years. Earlier blog posts on related ideas are here:
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 that doing the same thing to both sides is somehow the only move in algebra, because it’s not even the most important or the most common move.
I think the most important and most common move in algebra is this:
Replace something with something else you know it’s equal to.
This rule isn’t even an algebra rule, it’s a rule you’ve used in plain arithmetic. Look at this working:
(3+4)×5-6 = 7×5-6 = 35-6 = 29
All of those steps were replacing something with something else I know it’s equal to. First, I replaced the (4+3) with 7, and then I replaced 7×5 with 35. Finally I replaced 35-6 with 29. The reason I was able to write the “=” signs there was because I knew each expression was produced by replacing something with another thing it’s equal to. Of course they are the same. If you have ever written your working in the way I did there, then you were using the replacement move.
Indeed, this move is the heart of how mathematicians write calculations. We always move from one step to another by replacing something with something it’s equal to. Knowing that this is what we do, we read other people’s working by comparing each expression in a chain to the next to see what it is that has been replaced.
It’s interesting that nobody has ever told me explicitly that this is how to read maths working that’s written with a chain of equal signs. I just somehow figured it out. I am sure quite a few of my students don’t actually know this strategy.
This idea that anything can be replaced with something it’s equal to is for me the major thrust of all the algebra laws and identities.
For example, the distributive law―a(b+c) = ab+ac―doesn’t tell me how to “expand brackets”. Instead it tells me any time I see a(b+c), I’m allowed to replace it with ab+ac, and every time I see ab+ac, I’m allowed to replace it with a(b+c). The same goes for all the algebra laws, even though some of them seem very complicated.
For me, completing the square illustrates this very well. For example:
The first move was to replace 3x² + 24x with 3(x² + 8x) by the distributive law. Then I replaced 0 with +16-16. Then I replaced x² + 8x + 16 with (x+4)². Then I replaced 3((x+4)² -16) with 3(x+4)² – 48. Then I replaced -48 + 7 with -41.
The second pair of moves are usually extremely surprising to students, and even I have to stop and look closely when I read already-written-down completing the square working. But they make sense when I compare the line of working to the one above and see which parts are the same, so that I can deduce that the parts that change must have been replaced because they are equal.
Incidentally, this is why I am such a stickler for keeping things lined up when you do maths working if you can. Changing the order makes it hard to deduce what’s been replaced from one line to the next. Of course, you can change orders, but I prefer to do that as its own move so you can see when it happened.
Finally, the replacing move is actually used when solving equations too, even though it’s usually hidden. Look at this working that I did in a chemistry lab with a student once.
ρ = m/V ρ × V = m/V × V ρ × V = m
The move from line 1 to line 2 was multiplying both parts of the equation by V, but the move from line 2 to line 3 was replacing m/V × V with m because you know they are equal.
The student who I did this with was ok with the multiplying both parts by the same thing, but they were not ok with the replacing. They complained that I hadn’t done the same thing to both sides. And this was when I realised that there was another move in algebra and it’s much more fundamental than doing the same thing to both parts of an equation. And we need to tell students explicitly about the existence of this move.
So there you have it. Algebra is not all about doing the same thing to both sides, it’s very very often about replacing something with something else it’s equal to. Keep an eye out for it next time you read or do any maths working, and maybe explicitly remind your students every so often when it happens.
Actually just a bit of an epilogue: the replacing move is really rather fun to do “in reverse” as it were. Usually we do it in arithmetic by replacing an expression with a single number, but there’s nothing stopping you replacing a single number with an expression and so finding some rather complicated expressions for familiar numbers. This first one is based on the fact that if 1 = 2-1, then any 1 can be replaced with 2-1 at any time.
Replacing 1:
Replacing 2:
Replacing 3:
Replacing 4:
Replacing 5:
Replacing 6:
Replacing 7:
Replacing 8:
Replacing 9:
May you enjoy your replacing too.
NOTE: My thinking about this evolved over the years. Earlier blog posts on related ideas are here:
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 figure things out for themselves. But there is a class of questions that shuts all of this down immediately. Here are some examples:
“Did you go to the lecture?”
“Have you started yet?”
“How many of the exercises have you done?”
These questions all have answers that are morally Right or Wrong. The answers a student gives make the student out to be a Good Student or a Bad Student. And if a student has the Wrong Answer, they will feel ashamed.
I know many people who believe it is very important to send students the message that they should go to lectures, start assignments straight away, and do all the exercises. While these are all things students could do to help themselves, they’re not the most important thing to focus on when they are here seeking support from me. They can’t change any of those things right now, so all a question like those does is make them feel ashamed. And, as Turnaround for Children CEO Pam Canto says in this blog post , “shame is toxic to positive outcomes”.
Shame is the feeling that you are a bad person, that there is something wrong with you. Guilt is a bad feeling about your actions, which is unpleasant, but may make you want to change those actions in the future. Shame is the next level, where you feel you have been exposed as the horrible person you really are. A person who feels shame won’t try to change their actions, they’ll just try to avoid situations that expose them, which will just make the problem worse. I don’t want this to happen to my students, and I certainly don’t want them to think that seeking support from me will expose them to shame, or they will decide not to seek help.
Once upon a time, I realised that I was causing a student shame, and I decided that I would give myself a new principle.
Never ask a question that has a morally wrong answer.
This is one of the rules I use to evaluate if my question is useful and choose a better alternative.
For example, I could ask “Did you go to the lecture?”, but there is definitely an answer to this question that is morally wrong and having to give that answer will cause shame. Do I really want to know if they went to the lecture? How will that help? Maybe what I really want to know is what the lecturer has to say about the topic, since that might be useful. In that case, I could ask “What did the lecturer have to say about this?” The student doesn’t have to reveal their attendance status to answer this question, thus avoiding the shame. Even better would be to avoid the awkward moment where they have to reveal they don’t know, and say, “It would be useful to know what the lecturer says about this. Can you tell me what they said, or tell me where we might go looking for that?”
For my second shame-inducing question of “Have you started yet?”, the first simple fix is to remove the “yet”. That implies they should have started already. The second fix is to think about why I want to know this? Maybe I want to know what they’ve done already so we can build on it. In that case I could just ask “What have you done so far?”, since that’s directly asking for the information I want. But there is still an implication that they should have done something, so causing shame if they have to reveal they’ve done nothing. So instead I could ask “What are you thinking about this problem?” or maybe “How do you feel about this problem?”. These let me get into their head and heart and I can help them move on from there. I might be able to ask them about what they’ve done so far later, or it might not even be important because they’ll tell me what they need to help themselves.
This second example highlights another principle, which is to ask open ended questions, preferably about student thoughts and feelings. This makes it much easier to ask questions without morally wrong answers, because there are no specific predetermined answers in particular! (Asking open-ended questions is actually one of the factors in SQWIGLES, the guide for action I give to myself and my staff at the MLC.)
So, I urge you, think about whether the questions you ask have a morally wrong answer, and if so, try a more open-ended question that is less likely to cause the shame that is so toxic to success.
These comments were left on the original blog post:
Todd Feitelson 12 September 2020:
Thanks for this, David. I appreciate as always the attention you give to the details of your interactions with your students. It can be excruciating to have to consider every word you speak, but it’s really important. It’s a big source of stress and anxiety for me. But, it’s critical.
I do wonder about the difference between a direct question (“Were you at the lecture?”) and a less-direct question (“Can you tell me what the lecturer said?”) As you stated, you want to avoid that awkward moment when the student might have to admit not being there, but it feels way more awkward (almost like a trap) to ask the indirect question. For me, setting up the situation and relationship where the direct question can be seen as just a way to gather the information without judgement is the trick. My goal can be to get that all out on the table and move forward, with the student seeing me (I hope) as an ally.
I’m dealing with younger students, but there is also some value in helping students see that going to the lecture is a valuable tool in learning. It’s water under the bridge once they’re asking for help, but they are asking for help, so it seems important to help them learn for the next time. In a shame-free way — that’s the challenge.
(Somewhat irrelevantly, it reminds me of what my English teacher wife wants to say when kids come and ask her how to improve their vocabulary scores on standardized tests — “Have read a lot since you were ten.” Not so helpful, so she remains positive and forward-looking, and avoids the shaming.)
Thanks for writing!
David Butler 12 September 2020:
Very helpful thoughts Todd. A decent relationship where you can ask questions to keep them accountable is definitely something that is desirable. For me, this is usually the first time I’ve met them so I don’t have that luxury. I will say that when I have used “It would be good to know what the lecturer said…” and we’ve looked it up, then they often say later that going to the lecture or at least consulting the notes is the thing they learned today.
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. People seem determined to only put answers up on the board if they are right, and even rub off answers that don’t produce the target after all. And it’s a real hard sell to get people to try for an alternative solution too. Once a solution is found, people stop. I have always been worried that the game sends the wrong message entirely.
This Open Day I had a remarkable idea: instead of stating in the rules that the goal is to achieve the target, and trying to encourage people to take a different approach, what if I just changed the stated goal! I don’t know why I didn’t think of it before, to be honest!
So this is my new version of the Numbers game instructions:
NUMBERS
Help to make calculations that produce as many numbers near the target as possible, each calculation using some or all of these two big and four small numbers, and any combination of +, -, ×, ÷, and brackets. (Numbers can only be used in each calculation as many times as they are in the list.)
For posterity, I choose the numbers randomly from pop-sticks that I painted:
the small numbers are chosen from a set with two each of of the whole numbers 1 to 10
the large numbers are chosen from a set with one each of 25, 40, 50, 60, 75, 100, 120, 125.
the target is chosen from a set with two 0’s and three each of the digits 1 to 9.
Since coming up with this on Open Day, I’ve put a daily Numbers game on the board in the MLC Drop-In Centre, and it’s been a delight to have students and staff join in and add their solutions.
After explaining to students that the goal is now to get as many numbers near the target as we can get, all of a sudden they start just saying things to write on the board. Mostly we don’t even get the target until quite late in the piece, because people are excited that they can modify what’s there to get other numbers. Even if the target is produced early, there is still a desire to fill in the numbers on either side, and then the whole set of 10 that contains the target.
That modification of existing answers is my favourite part. I used to work it in before when I would put a wrong answer up and fiddle with it to get closer to the target. But now it just happens naturally because the goal is just to make as many different answers as possible.
Here are four games from the last couple of weeks:
As a testament to how absorbing this new version is, I came back to work on Monday last week, after having worked from home on Friday and a student arrived early to see if there was a game yet, as well as to show me all the ones they came up with when I wasn’t there on Friday.
I am so very pleased that this new version of the game works so well, and a little ashamed that I didn’t think of it earlier. As much as my catch cry is “the goal is not the goal, the end is not the end”, really sometimes you have to explicitly change the goal.
PS: Every day, people have come to look at the board when I am starting the game, and they say “Ohh, this is like Countdown!” My response is always that yes it is like Countdown, but it’s BETTER because of three major differences:
There is no clock, so it’s not stressful.
We are working together, not competing, so it’s friendlier.
The goal is to get all different answers instead of just one, so we get to keep playing.
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 and 10), each number with at most one decimal place. For example, you might choose 0.8, 3.2, 5.6, 5.9, 6.0, 8.7.
Without the other player/team seeing, write the numbers in increasing order in the MY SET template. For example, the numbers above would end up looking like this: (I have a printable version of the game with the rules and templates to fill in, which can be turned into battleships-style stands.)
This notation represents a set on the number line made of three separate open intervals. Colour in the intervals on the MY SET number line. For example, your completed MY SET notation and coloured intervals might look like this:
Set up so that you can write and draw on the THEIR SET template and number line without the other player/team seeing the MY SET template and number line. One possibility is to use the printable template to create battleships-style stands. Another possibility is to play on either side of a freestanding whiteboard, though you would have to draw your own set and number line.
The goal:
The goal is to guess what the other player/team’s three intervals are. In other words, you must guess exactly their six numbers.
Your turn:
On your turn, you suggest up to five sets of numbers like “all the numbers less than D away from C”.
After each suggestion, the other player/team gives one of three responses:
“All of those numbers are inside my set” — or “inside” for short.
“All of those numbers are outside my set” — or “outside” for short.
“Some of those numbers are inside my set and some are outside” — or “both” for short
Note: the responses do not include the endpoints of any of the intervals. If the whole interval except the endpoints is inside their set then the answer is “inside”, and if the whole interval except the endpoints is outside their set, then the answer is “outside”. There are examples of suggestions and how to answer them in the next section.
The suggestions made on the same turn must all have the same centre C, and the distance D must be smaller each time. For example, you could make the following suggestions on the one turn:
“All the numbers less than 2 away from 6.5”
“All the numbers less than 1 away from 6.5”
“All the numbers less than 0.5 away from 6.5”
“All the numbers less than 0.2 away from 6.5”
“All the numbers less than 0.1 away from 6.5”
You don’t have to use all five suggestions. If you want to ask about a different centre or a wider distance, you have to wait until your next turn.
You can write any notes in any way you like at any time in the game. It is a good idea to keep a record of the suggestions you have made so far.
Ending the game:
Once during the game, on your own turn and instead of making any suggestions, you can say you are ready to guess. Then you say the six endpoints of the intervals.
If you are right for all six numbers, you win! If you are wrong for at least one number, you lose and the other player/team wins! Either way the game is over.
Examples of answering suggestions:
These examples show visually the situations where you would say “inside”, “outside” and “both” in response to the other player’s/team’s suggestion. In the examples, your set is (0.8, 3.2)U(5.6, 5.9)U(6.0,8.7), which is drawn in blue in the diagrams. The other player’s suggestion is drawn in yellow slightly higher on the diagram, with the centre marked as a red vertical line.
They say “all the numbers less than 0.3 away from 0.3”. Your response: “Outside”.
They say “all the numbers less than 0.6 away from 5”. Your response: “Outside”. (Notice how the answer is still “Outside”, even though the other player’s suggestion shares an endpoint with your set.)
They say “all the numbers less than 0.5 away from 2.4”. Your response: “Inside”.
They say “all the numbers less than 1 away from 7”. Your response: “Inside”. (Notice how the answer is still “Inside”, even though the other player’s suggestion shares an endpoint with your set.)
They say “all the numbers less than 1.5 away from 3”. Your response: “Both”.
They say “all the numbers less than 0.3 away from 6.1”. Your response: “Both”.
Introductory play:
When you are first learning to play, you may want to agree to each have only one interval in your set for your first game, then increase the number of intervals to two and then three in later games.
Thoughts about Number Neighbourhoods
In this section I’ll describe some thoughts about the game I’ve had through playing it. Note there are some spoilers, so I recommend playing the game yourself before reading this section!
I’ve played several games of Number Neighbourhoods across the months since February, and I’ve thoroughly enjoyed them all. Even now, after getting quite a bit of experience, the game feels much harder than Digit Disguises, but I don’t think that’s a bad thing. I think one of the things that makes it feel harder is that the information you collect doesn’t join together as easily as it does in Digit Disguises, which tends to produce a cascade of figuring stuff out towards the end of the game. But there are still some really cool moments of logic that happen in most games.
Here are the whiteboard of notes from a recent game, so you can see the logic that goes into it. The left-hand set is me and a student Kristin. The right-hand set is another pair of students.
Interestingly, though Kristin and I were certain of our set, we got it “wrong” because our opponents thought that the answer should be “both” when an interval was inside the set but included an end-point. I’ve actually updated the rules here in the blog to explicitly point out that case, so it’s less likely to happen in the future!
One thing that’s really interesting about the game is the amount of concentration it takes to even answer the question by another player! The act of seeing whether an interval specified as centre and radius is inside an interval specified with two endpoints is surprisingly difficult, no matter how many times you do it. But that’s really cool, because it’s definitely a skill that Real Analysis students need to think about!
My absolute favourite moment is when someone picks a boundary point as their centre, in which case the answer is going to be “both” all the way down. If someone playing the game with me watched closely, they could probably guess they had a boundary point by my excitement and the sudden increase in the speed at which I answer.
I also like how every new player suddenly realises in the first two turns that once they get an answer of “inside” or “outside” they can stop that turn, because going further in won’t change the answer. I did actually toy with putting that stopping condition in the rules, but decided against it because it is such a nice little moment of “I figured out something cool” I can give to new players. (This is one of the spoilers I warned you about earlier.)
I haven’t really come up with any decent strategy for winning the game yet, or for choosing a “most hard to guess” original set. I have thought about it a bit, but I don’t know if I really want a perfect strategy, so that I don’t feel tempted to use it and ruin the game for new players!
For now I am enjoying the experience of playing, and especially playing in teams and discussing the logic and strategies with my team-mates, which is my favourite part of all my battleships-style games.
Along the way in the game above, since it was sometimes days between turns and I had the time, I made a Desmos thing for recording guesses and answers, and you might be interested to see that too. This graph is Lyron’s guesses of my set.
The story of Number Neighbourhoods
Finally, as a bit of an epilogue, I thought you might like to hear how I went about designing the game and the choices that I made along the way.
The game Digit Disguises, which I invented last year, was inspired by a puzzle which I created specially to give a similar feel to the maths in a course like Abstract Algebra. In Algebra you investigate the properties of numbers (and other things) by the way they interact with each other, and so a game where you find numbers based on their interactions with each other matches the vibe of such a course. But there are two broad flavours of pure maths that tend to feel different. A course like Algebra is of one flavour, but a course like Real Analysis is a different flavour. So ever since I created Digit Disguises with Algebra in mind, I thought it would be cool to make a similar game that had Real Analysis in mind.
In Real Analysis, you define a set as open if at every point inside it, you can find an interval centred at that point that fits completely inside the set; and you define a set as closed if at every point outside it, you can find an interval centred at that point that fits completely outside the set; and a point is a boundary point if no matter how small you make an interval centred there, it always has some of it inside and some of it outside the set. So given any set, there are three kinds of points and you can decide between them by looking at intervals centred there. And as long as you know in advance if boundary points are in or out, you should be able to find the entire set in this way. This seemed like an idea you could build a game around!
Riffing off Digit Disguises, which uses the digits 0 to 9, I decided the set you were looking for should be between 0 and 10. I also wanted it to feel small because Real Analysis is about things being close to each other, so I went for decimal places. I did briefly toy with the idea of making it numbers from 1 to 99, since that would be equivalent and not require operations with decimals. But that felt so huge and also put a focus on integers, whereas Real Analysis is about the continuously full number line in all its detail and the tiny distances between things. Plus I decided that actually, I wanted a chance to practice my decimal calculations. So I stuck with 0.1 to 9.9. Plus as I said before, it did feel like it was a proper partner to Digit Disguises then.
And so the idea for the game was born. I wanted the idea of boundary points to happen several times during a game, so I decided that the goal set should be three intervals, in order to have six boundary points to consider. And I also liked the idea of something being outside the set even though it was between pieces of the set.
I thought about the concept in Real Analysis of finding an interval small enough to fit inside (or outside) the set, and decided I wanted that idea to be in the game too, so I came up with being allowed to make several guesses on the same turn as long as you make your intervals go inwards. After playing it against myself a couple of times, I decided I should restrict the number of guesses to five so each turn didn’t go too long.
I briefly considered changing it so that you specified the endpoints of the interval you guess on each turn, but it lost the cool |x – b| < epsilon feel, so I abandoned that idea.
The game was almost complete, and just needed a name. Neighbourhood is a terminology they use in Real Analysis to describe an interval centred at a point, and we were using numbers, and I used alliteration already for the titles of the other games, so Number Neighbourhoods seemed like the most natural choice.
I played the game against myself a couple of times, and conscripted a student in the MLC to play with me too. Then I played with some students at a games night. These helped me to define what examples I needed to include in the rules. And that’s how I created the game of Number Neighbourhoods.
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 \((p+si, q+ti)\). The collection of all the points with real part \((p,q)\) I called the iplane at \((p,q)\), and I imagined the iplane as a transparent sheet attached at the point \((p,q)\) which I could flatten out to see its points when I needed to.
Ever since I had this idea, I have wondered on and off about the complex points on a circle. It’s time to write about what I’ve found.
In this blog post, I will investigate the complex points on a real circle centred at the the origin. (In later blog posts, I’ll think about “unreal” circles, but you may need to wait a while for those.)
Investigating complex points on a real circle
I’ll consider the circle with equation \(x^2+y^2= r^2\), where \(r\) is a real number with \(r>0\), and I’ll try to find the complex points on this circle in the iplane attached at the real point \((p,q)\).
The points in this iplane have coordinates \((p+si,q+ti)\) for real numbers \(s\) and \(t\). If such a point is on the circle I’m considering, then it would have to satisfy the equation. So I’ll sub it in:
These equations together describe the \((s,t)\) coordinates of the complex points in the iplane at \((p,q)\) that are on the circle. Let me investigate them separately.
Equation 1 has real solutions for \(s\) and \(t\) — and they have to be real because \(s\) and \(t\) are real numbers — when
Now \(\sqrt{p^2+q^2}\) is the distance of the point \((p,q)\) from the origin, so there are solutions for \(s\) and \(t\) in Equation 1 when the point \((p,q)\) is at least \(r\) from the origin. That is, when \((p,q)\) is on our OUTSIDE the circle!
Just to be absolutely sure, let me just check.
When \((p,q)\) is inside the real circle, then \(p^2 + q^2 < r^2\). So \(p^2 + q^2 – r^2\) is negative, which means that \(s^2+ t^2 = p^2+ q^2- r^2\) has no solutions.
So there are no points of the circle in the iplane at \((p,q)\) if \((p,q)\) is inside the real circle.
When \((p,q)\) is on the real circle, then \(p^2 + q^2 = r^2\). So \(p^2 + q^2 – r^2 =0\), which means that \(s^2+ t^2 =0\), And that means \(s=0\) and \(t=0\). This also satisfies Equation 2 since \(p\times 0 + q \times 0 =0\). So the only pair \((s,t)\) that satisfies both equations is \((0,0)\), which corresponds to the centre of the iplane at \((p,q)\).
That is, when \((p,q)\) is on the real circle, then the only point of the circle in the iplane at \((p,q)\) is \((p,q)\) itself.
Ok, so I can now get back to the iplanes at the points outside the circle.
Let \((p,q)\) be a point outside the circle, which means \(p^2+ q^2 > r^2\) and so \(p^2 + q^2 – r^2\) is positive. Therefore \(s^2 + t^2 = p^2 + q^2 – r^2\) is the equation of a circle (inside the iplane at \((p,q)\)), centred at \((p,q)\) and with radius \(\sqrt{p^2+q^2-r^2}\).
That radius \(\sqrt{p^2+q^2-r^2}\) ought to be a short side in a right-angled triangle with hypotenuse \(\sqrt{p^2+q^2}\) and other short side \(r\).
But I know those distances! The distance \(\sqrt{p^2+q^2}\) is how far \((p,q)\) is from the origin, and the distance \(r\) is the radius of the real circle. If the distance \(\sqrt{p^2+q^2-r^2}\) meets a radius in a right angle, then that means it’s a tangent to the circle! So that means I am looking for the distance along a tangent drawn from the point \((p,q)\) to the real circle! Of course, this construction is happening inside the iplane, as it is lying flat on top of the real plane so I can see the real circle through it.
The points I am looking for are on a circle of radius \(\sqrt{p^2+q^2-r^2}\) drawn inside the iplane centred at \((p,q)\). But this is only based on Equation 1. The complex points on the original circle in the iplane also satisfy Equation 2:
So the vector \((s,t)\) is perpendicular to the vector \((p,q)\). The vector \((s,t)\) is the vector from the centre of the iplane to the point \((s,t)\) inside the iplane, while the vector \((p,q)\) is the vector from the origin to \((p,q)\). So the line \(ps+qt=0\) inside the iplane passes through the centre of the iplane at \((p,q)\) and is perpendicular to the line from the origin to \((p,q)\). Which means the points I am looking for can be constructed like this.
To find the points of the circle with equation \(x^2 + y^2= r^2\) in the iplane at a point \((p,q)\) outside the circle:
Lay the iplane at \((p,q)\) flat over the real plane.
Draw a tangent from \((p,q)\) to the real circle.
Find where this tangent meets the circle, then draw a circle inside the iplane through this point, centred at \((p,q)\).
Draw a line perpendicular to the vector \((p,q)\) inside the iplane at \((p,q)\).
Find where this line inside the iplane meets the circle inside the iplane.
These two points are the points of the original circle inside the iplane.
This GeoGebra graph allows you to see the complex points on a real circle in the iplane at a moving point \((p,q)\). You can turn on and off the construction lines used to find the points.
I feel like I need to do one final thing: to find the actual coordinates of those points, which means formally solving those equations.
\[\begin{aligned} ps &= -qt\\ s &= \frac{p}{q} t\end{aligned}\]
So if \(t =p\sqrt{1-\frac{r^2}{p^2+q^2}}\), then \(s = -q \sqrt{1-\frac{r^2}{p^2+q^2}}\) and if \(t =-p\sqrt{1-\frac{r^2}{p^2+q^2}}\), then \(s = q \sqrt{1-\frac{r^2}{p^2+q^2}}\).
Technically, this argument doesn’t work if \(p=0\), but then a similar argument using \(q\) instead will produce the same result. (And it doesn’t matter if they’re both zero because I already know there are no solutions then anyway.)
So, the points of the circle with equation \(x^2+y^2=r^2\) in the iplane at \((p,q)\) are \[\left(p-q p\sqrt{1-\frac{r^2}{p^2+q^2}},q+p\sqrt{1-\frac{r^2}{p^2+q^2}}\right)\] and \[\left(p+q p\sqrt{1-\frac{r^2}{p^2+q^2}},q-p\sqrt{1-\frac{r^2}{p^2+q^2}}\right)\]
(Note how these are the same point when \(\sqrt{p^2+q^2}=r\) and they are undefined when \(\sqrt{p^2+q^2}<r\). )
A new representation
The fact that the two points on the circle in the iplane at (p,q) are both on a circle centred at (p,q) tells me they are the same distance inside the iplane from the centre.
Thinking all the way back to my physical model, the iplanes can be folded up like umbrellas, like this picture but in reverse:
If I do that, then every point inside them would end up on a vertical line, at a height equal to the distance away from the centre point. That seems to me like a very nice representation! In this case it’s particuarly convenient because both points are the same distance from the centre of the iplane, and so the same height.
For our circle \(x^2+y^2=r^2\), the points on the circle in the iplane at \((p,q)\) are a distance \(\sqrt{p^2+q^2-r^2}\) from the centre of the iplane, so my “folded umbrella” representation will have a point at height \(z =\sqrt{p^2+q^2-r^2}\) at each point \((p,q)\) and it will come out like this:
You can see an interactive version of this graph which you can drag around to look at from different perspectives here: interactive folded umbrella iplane graph.
One thing I love about this graph is the sense that some complex points are closer to being real than others. In the graph, you can see there are no points of the circle (real or otherwise) inside the circle, and as you move further away from the real circle, the complex points get further and further from being real. The points on the circle far away from the real circle are very far from being real, which appeals to me.
(Of course, if I had two such graphs, there would be no guarantee that the places where they met would actually indicate shared complex points, because every height can be created by anything in the iplane on a circle of that radius. But still useful for thinking about one graph.)
Where next
I am going to have to investigate the folded umbrella representation of the complex points on an object more at some point. I really want to see how it looks for the complex lines and parabolas I already investigated four years ago. But first I would like to investigate what happens for an unreal circle. What does the circle look like when it has \(x^2+y^2=R\) for some negative number \(R\). In a sense, that would be a circle with negative area. And what if the area was any complex number? But those are other stories and shall be told at another time.
