First of all then, the problem. You have 12 objects which look identical. 11 are the same weight; one is different, but we don’t know if it’s heavier or lighter. You have a set of balancing scales, and you can do up to three weighings. How do you determine the odd weight out and determine whether it’s heavier or lighter?

You might want to have a think about the problem for yourself before reading on.

To start with, I thought about similar, simpler problems I’ve solved before. The difference about this variant was that we didn’t know whether the odd weight out was heavier or lighter, and after playing in my mind with a few possibilities this was the point where I kept getting stuck. In my mind, I was picturing a balance with some balls on one side and some on the other, tipped up, and having no idea whether it was because one on the left was too heavy, or one on the right was too light!

In my last post I talked about playful maths, and one of my strategies for the weights problem was to get a feel for it by playing around. Could I solve the problem if I knew one weight was heavier? Could I solve it if I had fewer weights? This phase of the problem solving was about building up instincts and understanding about different situations. We had been chatting a bit about the problem and different people at the party had suggested various starting strategies (6 against 6? 4 against 4? 3 against 3? What happens if I start with each of these, where do I get stuck?) Eventually I realised I needed some thinking time on my own (and other people were less invested in solving the problem than I was, so conversation moved on), so I went up to the loo and used it as an excuse to think on my own for a bit. I made a bit of progress, worked a few things out that weren’t going to work, got frustrated because it felt like the answer was close, and then parked the problem, rejoined the party, and talked about Greek Tragedy, Educational Policy in the late 20th and early 21st Century, and the benefits of halloumi as a protein source.

That night as I was trying to fall asleep, I thought about the problem some more. Without pencil and paper, I started thinking about how I would record ideas if I did have something to write on. I began to come up with possible notation, and fell asleep thinking of how the right notation would help me to play with different possibilities.

The following day, I was travelling so didn’t really have time and headspace to think about the problem, but I did think a bit about how I would write a simulator which randomly created a heavier-or-lighter object and allowed you to trial different weighings, in order to work out a strategy. I planned to try writing it the following day, but as it happened, I did some more thinking about the problem as I was trying to fall asleep that night, and managed to solve it! So instead of writing a weights puzzle game, I am writing up my solution instead.

So, notation. I will call the objects 1,2,3…,11,12. I will append in {} after each one L/N/H to say if it is possible for it to be lighter, normal weight, or heavier as a result of each weighing.

Stage One: weigh 1,2,3,4 against 5,6,7,8.

Two cases; they balance, or one side goes down.

**Case one: they balance.**

We have 1,2,3,4,5,6,7,8{N}, 9,10,11,12{L/N/H}.

Second weighing: 1,2,3{N} against 9,10,11{L/N/H}.

Three possiblilities: a) they balance b) 1,2,3 goes down c) 1,2,3 goes up.

a) We now know 12 is the odd weight out. We use our third weighing against any of the others to determine if it’s lighter or heavier.

b) As 1,2,3 are normal weight, we know 9, 10 or 11 are lighter. We use our third weighing to test 9 against 10, if they balance 11 is the lighter weight and if they don’t, whichever is lighter out of 9 and 10 is the lighter weight.

c) as b) above, except 9,10 or 11 is heavier.

**Case two: one side goes down.**

Let’s assume it’s 1,2,3,4 (the argument follows in a similar way if it’s 5,6,7,8).

This means we have 1,2,3,4{N/H} 5,6,7,8{L/N} with 9,10,11,12{N}.

For the second weighing, we are going to weigh 1,2 and 5 against 3,4 and 6.

Three possibilities: a) they balance b) 1,2,5 goes down c) 1,2,5 goes up.

a) We now know that either 7 or 8 is the odd weight out, and it’s lighter, so we can weigh 7 against 8 to determine.

b) We now know that either 1 or 2 is heavier, or 6 is lighter. We weigh 1 against 2; if one of them goes down we know it’s heavier, and if they balance we know 6 is lighter.

c) We now know that either 5 is lighter, or 3 or 4 is heavier. We weigh 3 against 4; if one of them goes down we know it’s heavier, and if they balance we know 5 is lighter.

So there we have it; three weighings is enough to determine which is the odd weight out! I am tempted to still make the simulator when I have time so that I can play some more and start determining a general strategy for finding the odd weight in n…

]]>

Firstly, some moments from BCME that got me thinking about playfulness. I attended an excellent session by Helen Williams and Mike Ollerton about using Cuisenaire rods at all ages. All good practice with Cuisenaire seems to be rooted in giving learners lots of time to play and become very very familiar with the rods, the way they feel, the way they fit together, and the colours, before starting to write things down more formally. I went to the second of their two sessions, and those who’d attended both sessions had a whole wealth of shared experience to draw on from having had “playtime” in the first session. And in videos where children work with Cuisenaire, it is a joy to watch the almost instinctive way in which they reach for the piece they want.

Talking of joy, I grabbed a lovely snap of John Mason after he’d completed the snake puzzle challenge from one of the stands at coffee time. He had been challenged to turn the snake into the ball-shape. I was familiar with the puzzle as I’d played with one before, and I can’t help thinking John would have been less frustrated if he’d had five minutes of being playful with the snake first before trying to make a particular given object! But he showed perseverance and resilience, and enjoyed a feeling of success at the end, so that’s OK.

Becky and I did quite a lot of playing with Pattern Blocks at the BCME workshop. Becky has written up some of the experience on her own blog, but I would like to add a little – we created some cards as a prototype of a game that can be played with squares, rhombuses, hexagons and equilateral triangles. While trying to formalise the rules of our game, lots of people joined us at different times and wanted to join in, make suggestions, and talk about the maths classroom implications. The context of a game and the associated playfulness can direct attention and awareness to particular attributes of a situation; by focussing on shapes meeting at a point, our game provided a natural way to think about angles and tesselations. (I really hope we can find the time to write our game up properly at some point.)

Ruth Bull led a session about geometry and paper folding. Many of the ideas were ones I’d seen before, but what was really valuable for me was seeing them woven in together, with time allowed for me to fold, play, think, and reflect on old ideas in new ways. In fact, I used one of the ideas, folding a hexagon, with a workshop group of my own last week – a story for another blog post, perhaps.

Now, thinking more generally about playfulness and why these moments at BCME were important. When I talk with students and teachers about problem solving, I talk about understanding the problem. Some mathematicians talk about getting stuck in, getting their hands dirty, digging into a problem. For me, my mindset when I am first working on some new mathematics feels very similar to my mindest when I am trying out a new craft, or exploring a new place. I am excited, a little apprehensive perhaps, thinking about all the possibilities ahead, ready to make a start. Perhaps I take a left turn over a bridge, to see where it goes! Or prepare some materials, and try to join them together. If I make a mistake, it doesn’t matter – there are no wrong answers at this point. If the bridge leads nowhere, I can turn back. If I cut out something and it doesn’t fit, I can cut again. I am being playful, and I learn a lot from the early explorations. Then I can draw on that experience of being playful later on. When I get down to work – trying to find a route, or make a particular item, or solve a particular problem, I can draw on my explorations and make plans based on the experiences I had. I know where the dead ends are, what will work and what probably won’t.

Sometimes, having a problem to solve can be a really good motivation for learning. Without knowing the problem, there can be too much open space, too little direction. But there has to be room to play too. If the space is closed right down and the paths are prescribed, some of the joy goes out of learning, and the opportunity to make connections is lost.

]]>Here are some ideas that I will try to find the time to blog about and flesh out a bit over the next few days:

- The importance of play and playfulness in mathematics education at all ages
- The tensions that are introduced by high stakes assessment
- Public perception of what mathematics is and what mathematics teaching should achieve
- The role of computers and calculators
- How to engage in respectful dialogue and find common ground when talking about emotive topics within maths ed (and how to pick your battles)
- Diversity and representation
- How to run a quiz without controversy (or The Time I Lost a Tie Break but am Not Bitter, honest)…

For the most part, the people I talked to at BCME are on the same page as me, trying to improve maths understanding, trying to instill a love of and enjoyment for mathematics, trying to change the notion that maths is for a select few (and that you need grey hair and a white beard to do it…) and I hope by blogging and continuing to tweet I can be part of the ongoing conversations that started last week. Let’s make the world a better place, one maths classroom at a time!

]]>

Towards the end of 2017, after a year on a waiting list, I had an assessment and received a diagnosis of Asperger Syndrome. In some senses this marks the end of one journey and the beginning of another. So let’s start with the journey that ended with diagnosis.

I don’t know where to start the story. Birth? Childhood? We discussed my schooling in the assessment, and my dad accompanied me to tell the psychologist about my early life. (I also took all my school reports to the assessment, but that’s a whole other blog post!) Long story short, any peculiarities in my childhood were probably masked by the fact that I was very highly achieving at school, so I was seen as being unusual. Studying maths at Cambridge, I really wasn’t that much of an outlier – even though I did not suspect I was on the autism spectrum, I was aware that I fitted in very well with fellow mathmos and their geek culture, pedantry and precision.

I then started my teaching career. How I became a teacher is another good story, but again, one for another day. The structure and routine of teaching fitted me very well, and my Asperger Syndrome continued unnoticed and undetected. I was me, quirky, mad about maths, a bit eccentric, but no-one ever suggested autism. I taught kids with AS diagnosis, and I was patient and kind and understanding of their needs, and angry when they didn’t get the support they needed. My little brother was making his way through secondary school with his Asperger diagnosis, occasionally struggling, but excelling at all things mathematical and logical. (He followed me to Cambridge, did maths, and is now a software developer.) I read books about autism, I attended training to better help the children I taught who were on the spectrum, and I never applied any of it to myself.

After five and a half years in the classroom, the opportunity to work for NRICH presented itself. I was terrified about leaving the safety of the classroom and stepping into the unknown, but I did it, and started to carve a niche for myself. I have been at NRICH for 9 years now, and I think I’ve been pretty successful.

Throughout my adult life though, there was a bit of a black cloud looming. In 2010, my GP diagnosed depression, and prescribed me antidepressants. I also sought counselling, and had various types of talking therapy on and off for the next half dozen years. One of those counsellors, when I was talking about Aspergers Syndrome in my extended family, asked as a throwaway question in the way counsellors do, whether I thought I was on the spectrum. At the time, I dismissed it, but then in recent years I kept coming back to it.

I read the experiences of late-diagnosed women, and it resonated. I read about mathematicians with autism, and their experiences sounded familiar. I started writing things down. I did the “Autism Quotient” quiz, and consistently scored in the 40s (People on the autism spectrum usually score in the 30s or higher). After following autism advocates on Twitter and reading about the diagnostic process, I went to my GP, and said I thought the depression and anxiety were symptoms of undiagnosed autism. He agreed it was a possibility worth investigating, and referred me.

When I told people I thought I might be autistic, there were a few common responses: “of course you’re not, autistic people can’t communicate and you communicate very well!” Or “even if you were, why does it matter? You’re doing fine!” Or “my cousin’s son is autistic and you’re not like him.” But other people, particularly autistic people, were supportive and encouraging. Check it out, they said. Yes, that sounds familiar, they said. You are not alone in feeling like that, they said. I am grateful for friends both autistic and not, who supported me and encouraged me to search for an answer.

Then in December I spent three exhausting hours talking through every aspect of my life, looking at all the ways in which I am different, odd, peculiar, dysfunctional, a misfit. And at the end, the psychologist said yes, I have Asperger Syndrome. I hide it very well; I have excellent coping strategies, but ultimately I have spent 36 years trying to make sense of a world designed for neurotypical people without ever realising I wasn’t one of them.

Diagnosis was a huge relief. Many things suddenly came into focus, I have begun looking back over things that have happened and been able to make sense of them through this new lens. My depression was a result of exhaustion at masking my natural ways of being and responding in order to try to fit in. My anxiety was a fear of the disruption of routine and the security of the known. My low self esteem was from a sense of not being very good at being neurotypical. Just those simple words, “you have Asperger Syndrome” made me feel amazing, because I don’t have to look at my failings as a “normal person” but rather celebrate my successes as an autistic one!

People weren’t sure how to respond when I shared the news. Those I love most dearly were quick to remind me that nothing has really changed, I am still unique, quirky, wonderful, expressive Alison, I just have a new label that better explains me to others. I think “congratulations” is a nice response, although those who took the time to ask “how do you feel about it?” are my favourites. Those who say things like “are you sure? You don’t seem very autistic to me…” can… well, I think maybe those are words that are inappropriate for this blog.

And why have I chosen to post this rather personal narrative on what is usually a maths/pedagogy blog? Well from now on, my journey through life includes the knowledge that I see the world differently from most. When I work with students and there are aspiring young mathematicians who are autistic, I am a role model for them in a slightly different way than before. I am fascinated at how many people on the spectrum seem to find a niche in mathematics and I would love to explore that further. But most of all, I am not ashamed of my diagnosis, and I want people to know, this is what an Actually Autistic person looks like. I am still me, but now you know a little bit more about who I really am.

]]>Wow, what an experience. I have learned some valuable lessons that will certainly inform the responses I write on the STEP Support forum, and the messages I share with students when talking about STEP at events. And the messages I share with teachers, for that matter.

Having spent lots of time this year working through questions, I was pretty confident that even under time pressure I’d be able to produce some pretty good mathematics. And in places, I did! I came up with sensible ideas, did sketches, wrote stuff down. But I also did some disastrous mathematics, and exhibited some of the worst exam technique you could imagine. (I also took a fifteen minute break in the middle to check email, get a glass of water and go for a wee. I promise I didn’t cheat or think about the questions during that break though.)

Having looked at the mark scheme afterwards I think I probably scraped enough marks for a comfortable Grade 3… disappointing.

Here’s where I think I went wrong, and what I have learned:

- I’ve been doing a lot of STEP III questions lately, so I think I overcomplicated things in a lot of places because I had forgotten that STEP II is on a narrower and simpler syllabus. I expected it to be harder than it actually was, thought to myself “It can’t be that easy…” so I did lots of unnecessary algebra. And once I was in that mindset, I couldn’t see the wood for the trees.
- Having got myself into lots of messy algebra, I found that I made lots of mistakes. This is the difference between me at 18 and me at 36 – at 18 my full time job was preparing for A Levels and STEP, and as I was doing Maths, Further Maths and Physics, I was spending a significant proportion of every day performing integration, differentiation, curve sketching, algebraic manipulation, trigonometry… Let’s face it, I’m rusty! I only spend a few hours a week working on STEP level maths these days, so it’s unsurprising that these skills are no longer fresh.
- Question choice. The bit that I did right was in reading through the whole paper before I started – I checked the timer and I think I took about 8 minutes circling things, annotating the paper, and thinking about what I might do. Then I did something daft and picked a Mechanics question, to prove that I can do mechanics now. And got stuck. And panicked. And spent too long. When I looked through with the mark scheme, I reckon I would only have got around 10 or 11 marks for what I did, but I spent more than a quarter of my time on it. As it was, I only attempted 4 questions, and two of those were little more than fragments.

But hang on – isn’t this exactly WHY we tell students to do a timed test before their exams? (Or preferably more than one!)

Doing STEP questions with no time pressure, with the ability to look things up, to go away and think about it, to concentrate on one topic at a time, is a million miles away from actually sitting the exam. This exercise of trying a paper under near-exam conditions helped me to reflect on ALL the skills students need for STEP. Because as well as the problem solving mentality, the good ideas, the willingness to try things out, you also need fluency, timekeeping, common sense, self-discipline… I think my work this year has developed the first set of skills with regards to STEP, but it was never intended to address the second set. Perhaps I was too unkind to myself calling me “FOOL” but the sense of frustration that I have lost the ability to integrate accurately under pressure and concentrate on STEP questions for hours without a break overwhelmed me. And perhaps this is the final lesson to take from my experiment – preparing for an examination like STEP *is* overwhelming. It’s not just about developing the fluency, practising lots of questions, managing time effectively; it’s also about being kind to yourself, remembering that you are only human, and acknowledging that it’s just an exam. Once it’s over with, there will be music, dancing, flowers, love, and other things that really matter.

“Let’s start with something easy!” “I know it looks hard, but don’t worry, it’s easy!” “If you can do x you’ll be able to do y because it’s much easier!” Familiar? These sort of phrases trip off the tongue, particularly if you are an educator who wants to make your learners feel safe. They are all messages designed to make the listener less anxious, and more capable. They are intended to empower! Unfortunately, I know from personal experience that such messages can be the opposite of empowering.

You see, “easy” is not a property of a task or a concept. It is a relationship between the task or concept and a person. There is no such thing as an easy question, because it depends on whom you are asking. (Don’t even get me started on political interviews in which someone is badgered to answer a “very easy question, yes or no” where actually a more nuanced answer is necessary and neither “yes” nor “no” is a satisfactory answer).

In some cases, it is glaringly obvious that “easy” is not a straightforward absolute concept. For example, if I were to ask an A Level Further Maths student to find the values of x such that x^{2}+5x+6=0, I would hope they would agree with me that it is an easy question. If I asked my 11 year old niece, she would find it very hard. If I asked my friend’s toddler, he would find it impossible to even understand the question.

I can see two problems that may arise when using the word “easy”. Firstly, using the word glibly without knowing your audience. This can happen when teaching or presenting to a group you do not know well, or a group where you make assumptions based on their prior knowledge, achievement and experience. You start off with an icebreaker, something everyone will be able to handle, and you introduce it as such. Then you find out that you’ve massively misjudged the situation, and people are stuck on your easy task! Or, perhaps worse, everyone *does* find it easy, except some poor soul who is then left behind (or hides the fact they don’t understand and just feels utterly rotten). This can be mitigated against by using “Low threshold, high ceiling” activities where literally everyone can get started and you can assess what “easy” means in the context of the group in front of you. And if you introduce the task with “here’s a thing” rather than “here’s a lovely easy thing”, so using neutral language, you’re not setting people up for failure if they don’t get it straight away. The flip side of this is that if you introduce something that many people might find difficult, but with neutral language, you’re not in danger of setting up a self-fulfilling prophesy. I remember teaching the technique of completing the square to a group who were not expected to tackle such questions because they were in one of the lower sets. I didn’t tell them it was a “hard” topic until we’d finished. Their response? “But that was EASY, miss!” It wasn’t often I heard that class say THAT!

The second problem is more subtle. This can happen when you know someone well, and make assumptions about what they will find easy from what you already know they can do. The problem with this is that there isn’t a nice linear spectrum from easy to hard with everything in the same order for everyone. This one has bitten me in both directions. It has taken me decades to understand that just because I find some things very easy that other people find hard, it doesn’t mean I won’t find hard the things they find easy! For example, I am pretty good at solving STEP maths questions, and I am terrible at recognising faces or noticing when people have changed their appearance. There have been times when people have made me feel awful by saying things like “but you can do x, of course you must be able to do y!” I am pretty sure I have also made other people feel rotten by assuming that they would find something trivial based on my knowledge about other things they could do. (Sorry! I really will try harder in future not to do this! If you catch me doing it, call me out please.)

In general, I think as educators we should use the word “easy” with caution. There are better and clearer ways to express the meanings we are trying to capture, and if we allow learners to make up their own mind whether something is easy or hard, and listen to what they have to say, perhaps they will become resilient and resourceful, rather than feeling rotten.

]]>Mike provided dotty paper and circular objects to draw round, and invited us to investigate the convex hull of the points contained within circles of our choice. (The convex hull is what you get when you join together the points nearest the edge of the circle without being allowed any concave bits – imagine the circle is a rubber band on a pegboard and when you let go of it it springs round the outermost points.)

Mike threw a few possible questions at us, and then let us get on with it. This is my favourite type of maths investigation; very open ended, and no compulsion to work on something that someone else finds interesting, at the expense of exploring my own avenues. I started by centring my circles on a grid point, and exploring possible shapes. After a very short while, I got distracted and wanted to know whether I could create something in GeoGebra to help me. Others were busily discussing symmetry, whether shapes with different numbers of sides were possible, what happened as circles got larger, and much more. We even talked about practical applications of the mathematical ideas, approximating circles on a square grid such as pixels on a computer screen.

Alas, the meeting was over all too soon (not something you’ll hear me say very often!) and I had to get back to other things, but I saved my GeoGebra file to explore a bit more when I have the time. And if this starting point provokes any interesting questions for you, do let me know in the comments!

]]>I picked this puzzle up in a charity shop somewhere down south while waiting for my brother to have a job interview this summer. I recognised the puzzle because my other little brother had it as a child and I used to spend hours playing with it after he’d gone to bed, when I was babysitting.

The object of the game is to use the Knight’s move from chess to swap all the blue marbles with the… well… I think they’re brownish, or maybe pink? Anyway, the marbles of the other colour. According to the box, 50-55 moves is average and 45 is excellent. I seem to remember I used to be able to solve it reliably, efficiently and quickly, but having played around with it again I have forgotten all the little tricks and skills I had as a teenager.

If I get any spare time in the next twenty years or so, then implementing a computer version of this puzzle would be an interesting programming challenge. It’s probably already been done, but it’s the sort of thing I can imagine rather enjoying having a go at for myself. Meanwhile, when I get bored of using my Tower of Hanoi as a stress-reliever that lets my mind wander, I have the Knights problem to occupy my hands too.

]]>I found this in one of the charity shops in Ely, near Cambridge, some time ago. It looks hand made, and cost a couple of quid.

For those unfamiliar with the old problem, this is a Tower of Hanoi puzzle. The object is to transfer all the rings from one peg to another. You can only pick up one ring at a time, and you can never place a ring on top of a smaller one.

This shows the puzzle after a few moves have been made. (How many?) Altogether, there are nine rings. I did move all the rings successfully but not all in one go. This was a great find, because I’d been familiar with the Hanoi problem for many years, but actually having a purpose-built puzzle to play with it ‘hands on’ refamiliarised me with the task. If I was introducing the problem to kids, I’d want them to have something to manipulate. When we were little, we used to do it with the brass weights that went with the kitchen scales, as they were little discs of different sizes that stacked.

Have you met the Tower of Hanoi before? Have you used it in a classroom or masterclass situation? Have you ever found anything cool, mathematical and geeky in a charity shop?

]]>