Stacking cups and bunny rabbits

Last weekend, it was Big MathsJam, and among the mathematical delights I sampled, there was a talk by Christian Lawson-Perfect about stacking cups. You can see a really quick version of his talk on the Aperiodical website. Before reading on, you might like to think about the problem a bit…

When am I ever going to need to know this, Miss?

Last week, I posted an innocuous-looking question on Twitter:

Maths teachers! Which topics in the curriculum do you find are most likely to get a response “when am I ever going to need to know this?”

— Alison Kiddle (@ajk_44) October 25, 2018

And a fascinating conversation followed, with loads of suggestions, as well as a meta-discussion about why this question gets asked. This seemed like a good opportunity to dust off my blog again, to gather together the ideas and add a few thoughts of my own.

Firstly then, the most straightforward answers were from teachers who actually did what I asked, and told me curriculum topics! This is particularly useful, because both in my work for NRICH and also my wider engagement with maths outreach, it’s good to reflect on what teenagers perceive as relevant and worth learning. Many people mentioned geometry, and particularly circle theorems. Other topics mentioned included trigonometry, surds, algebra, quadratic equations, Pythagoras’s Theorem, inequalities …

The general theme that emerged, which was picked up on by a couple of commentators, was that anything which was difficult and where it was not immediately obvious why it might be important prompted the “When am I gonna need to know this, Miss?” cry. (Or “Sir”, or a gender-neutral term of your choice.) Some people connected it with student attitude, others with level of confidence. Distracting Teacher by making them justify the choice of lesson material is a technique as old as the hills for a student who is unwilling or unable to engage with the topic at hand.

Another interesting type of response is illustrated nicely by Colin’s post:

Analysing the themes of Measure For Measure. Being able to identify a minor third. Knowledge of mediaeval agriculture systems. Names of parts of a cell. The départements of Ile-de-France and their numbers. Or is it just maths that’s supposed to have everyday use?

— Colin Beveridge (@icecolbeveridge) October 25, 2018

As I have never had the good fortune to teach anything other than Maths, I can only assume that teachers of other subjects DO hear the same plaintive cry. Certainly I know my head is still full of half-remembered facts from humanities lessons that I rarely use in my day-to-day life unless I’m solving a general knowledge crossword or entering a pub quiz.

This brings us nicely on to the question of why we teach, why we choose to include certain topics into the curriculum. At least some of the role of education is to turn out well-rounded young people with a decent grasp of the world around them and their place in it. As far as the maths curriculum is concerned, I want students to have a sense of what mathematics is all about, the way in which different mathematical topics link together, the way in which mathematical knowledge and proof is different from the way things are done in other academic subjects, and yet the way in which it fits into and supports other disciplines, particularly science, technology and engineering.

This comment sums up one of the real challenges that faces us as maths teachers:

The problem isn’t the content per se in my opinion but the schizophrenic nature of the GCSE. It is both the entry to further study for some who question all this boring numeracy and others who are ending their mathematics here who question the need for all the abstract algebra.

— theperfectlanguage (@theperfectlang1) October 26, 2018

At least some of the “When am I ever going to need to know it” content is there because some students WILL need to know it someday. And there’s no way of telling at age 11, or 14, or even 16, which students are which. Sure, you get people like me who knew from upper primary school that I probably wanted to do a maths degree someday, but you also get the kid who isn’t bothered about maths until they suddenly find the spark, the part that brings it alive for them, because of their hobby or their dream job or something that requires them to use trigonometry, or statistical analysis, or circle theorems…

I don’t really have any answers for teachers who want a ready reckoner of places where maths is used in the Real World so they can give a glib response every time this question comes up. What I would recommend is for all of us to encourage engagement with maths for its own sake, the beauty of number patterns, the elegance of geometrical reasoning, the stories of people who find maths a joyful experience, so that even if students aren’t ready yet to feel that joy for themselves, they are aware of it as a possibility. When am I ever going to need to know this miss? Well maybe you won’t NEED it, but you might WANT to know it, you might ENJOY learning it. And you might surprise yourself and find that you are actually better at it than you thought.

A weights puzzle, and reflections on how I solve problems

At the weekend, I went to a lovely party, and on finding out I was a mathematician, one of the other attendees shared a weight puzzle that I think I have come across before but never tried to solve. I have been thinking about it on and off since then, and I promised to the other party-goers that I’d share my solution once I got there, so here we go!

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.

Playful Maths

In my BCME thoughts post, I kind of promised that I might maybe do a bit more blogging, and one of the topics that I did quite a bit of musing on at BCME was playfulness and its importance in understanding mathematics. So I guess I should flesh out some of those thoughts a bit…

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.

Some BCME9 Thoughts

I spent Tuesday-Friday at the University of Warwick for the 9th BCME conference. I reckon there’s at least twenty blog posts worth of material from the conversations, thoughts, problems and games of the last few days, the most important of which is that I really need to get into the habit of writing more about maths and maths ed. It has taken me a long time to realise that I have worthwhile things to add to the conversation, and that my community appreciates my contributions even if not everyone agrees with me!

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:

1. The importance of play and playfulness in mathematics education at all ages
2. The tensions that are introduced by high stakes assessment
3. Public perception of what mathematics is and what mathematics teaching should achieve
4. The role of computers and calculators
5. How to engage in respectful dialogue and find common ground when talking about emotive topics within maths ed (and how to pick your battles)
6. Diversity and representation
7. 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!

 

Coming out

I have been thinking for a little while about whether to write this post, and what to write in it. This post is about Autism, Asperger Syndrome, and a personal journey. There are risks to writing this down publicly, but I think the benefits outweigh the risks, so here goes.

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.

A STEP too far…

This morning I performed an experiment. Around a quarter of my time at work this year has been spent working on materials for the STEP Support Programme, and I thought it would be useful to put myself through the experience of doing a timed test. I chose last year’s STEP II paper, printed out a copy of the formula book, got a fresh pad of paper, some pens and a pencil, and set the timer on my phone for three hours.

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.

A decent sketch makes all the difference. This wasn’t.

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

1. 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.
2. 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.
3. 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!)

Further up the page, I failed to integrate correctly. Then I criticised myself.

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.

That’s easy!

There are various words I would like to ban, or rather, as I am generally quite liberal in my views and don’t tend to go in for banning, words I would caution against using carelessly. I may blog more about such words in the future (Ha! Who am I kidding? When did I last blog?) but for today let’s talk about the word… easy.

“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 x²+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.

Thinking about Dotty Grids

There’s loads of stuff about dotty grids on NRICH at the moment. I’m trying to get my head round Pinterest, and figure out what content can be shared there, so why not check out my dotty grids pinboard? I’m sure I’ll add more to it once I figure out what can go there!

Meeting maths

At NRICH meetings, we like to devote a little bit of time to working on some maths together. Today I’d like to share the problem that m’colleague Mike presented.
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!