Author Archive

Playful Maths

April 16, 2018

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.

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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.

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Some BCME9 Thoughts

April 7, 2018

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

February 11, 2018

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…

June 2, 2017

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.

Critic

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!)

Fool

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.

More Definitions

October 21, 2010

Following on from the variety of things we mean when we use the word “maths”, I’ve been thinking a bit of what we mean when we say someone is “good at maths”.

Of course, some of this is tied up with the previous question; if I describe someone as being “good at maths”, I presumably mean they show some aptitude in whatever set of skills I have identified as being “maths”. But I think there are also lots of other issues in play here. It depends who is identifying someone as being good at maths, and it depends where this identification happens. I’m sure there are whys and hows and whens too, but I’ll stick to those for now.

The who is interesting, because as teachers, we are often asked to make judgements about whether pupils are good at maths, and our judgements can have important consequences for a child’s maths education. In schools where setting is the norm, teachers have to make a decision about whether children are “good at maths” or not, and decide which group to put them in. This is often done through testing, so “good at maths” becomes a shorthand for “good at passing maths tests”. Children might assess how good they are at maths based on a teacher’s judgement of them. So children who are not in “Set 1” may consider that they aren’t very good at maths, and might consider that the children who are in “Set 1” are. But in the top sets I taught, there were several pupils who thought they shouldn’t be in Set 1, and that they weren’t as good at maths as others in the group, so based on children’s own assessment of who is good at maths, there are only a handful in each school! Perhaps in schools or classrooms with a culture of rewarding and recognising mathematical process and thinking, many more children identify as being “good at maths”.

I’ve already touched on where a little bit – within a school where setting is the norm, there may be lots of children who don’t consider themselves good at maths. More widely, in a society with lots of “mathsphobia”, or where being mathematical is seen as being a bit nerdy, people might shy away from maths and not recognise their own abilities to think mathematically. In many circles, I am considered by others as being “good at maths” – I have a degree in maths, I successfully taught maths in secondary schools for a while, and a large part of my job involves writing maths problems. And yet, when I’m with a group of friends who went on to do PhDs and study an area of maths intensively and in great depth, I don’t consider myself to be good at maths, because it seems that they know so much more (and more difficult) maths than me!

I think it’s vitally important that we look out for opportunities to recognise good mathematical thinking with praise, and to challenge people’s assumption that they can’t do maths. Competitive testing and setting seem to reinforce lots of children’s feeling that they are not (and can never become) good at maths. If we expand our notion of what it means to be good at maths to include generalising, modelling a problem mathematically, expressing mathematical ideas in conversation, looking for different methods, posing good questions, making connections… rather than just being good at passing maths tests, then all of a sudden, we have lots more young people willing to consider themselves good at maths.

A great way to spend a day

July 14, 2010

Yesterday I attended a day of mathematical fun to celebrate the retirement of Jenny Piggott, who stepped down as NRICH director earlier this year. In the morning, we worked on problems which had been suggested by people who had worked with Jenny over the course of her career. These were varied – we talked about Pythagorean triples, methods of buttoning up one’s shirt, and what happens when you drop elephants into a lake! Some of these problems will no doubt be developed into tasks on the NRICH site in the next few months.

The Pythagorean triples task reminded me of something else I’ve tackled recently, as I ended up with the same set of equations to solve as I had done when working on another problem. Unfortunately I can’t find such a problem in my notes, and I can’t remember what it was I was working on – maybe it was all a very vivid dream! We were seeking triples (sets of three whole numbers that could be sides of a right-angled triangle, so satisfying Pythagoras’s theorem a^2+b^2=c^2) where two of the numbers differed by 1. Having worked on Pythagorean Triple problems before, I quickly found examples where the longer two sides differed by 1. It wasn’t immediately obvious whether it was possible for the perpendicular sides to differ by 1 though – I won’t post my further musings on this just yet in case people want to investigate it for themselves.

In the afternoon, we did some lovely mathematical origami. Alas, no camera with me today but my model ended up very like this. Read more about making them here. Then while tea was served, Bubblz entertained both young and old with mathematical bubbles which took me back to vague memories of an undergraduate course on calculus of variations that I attended ten years ago! It’s days like this that enthuse and energise us as maths educators to remember why we love our subject so much, and help us to pass on our passion to those we work with.

Magic Vs

July 6, 2010

It’s been a busy couple of weeks for me, so I’m going to try to round it off by blogging some of the things I’ve been doing.

Firstly, I was asked to participate in a day aimed at Gifted and Talented Year 7 pupils. Last year, I worked on the problem Odds and Evens for a similar session, which was great fun, but I didn’t feel we had time to get the most out of the task, so this year I chose to work with them on Magic Vs instead. The idea is to arrange the numbers 1-5 in a V shape so that both arms of the V add up to the same “Magic total”. I challenged the pupils to find every way they possibly could to do this, and then to come up with a convincing explanation of how they knew they’d found them all. Along the way, we had some interesting conversations about when two Vs count as the same and when they are different. I think it’s valuable to allow these discussions to emerge from the class, and for them to decide on whether two Vs are the same, rather than me deciding for them.

After they’d justified their answers and had a go with the same activity with the numbers 2-6, I asked them to suggest lines of enquiry that they might explore next. I was amazed by their inventiveness – some wanted to come up with general rules for magic Vs with consecutive numbers, some explored Vs made with all odd or all even numbers, some looked at Vs with more than 5 numbers in, and some investigated Magic Ws or Magic Xs.

We finished the session by giving pupils the chance to share anything interesting they had discovered, and the explanations they’d used to convince themselves of what they’d found. Where appropriate, I introduced a little bit of algebra to help their proofs along, but for the most part their reasoning did not rely on algebra for justification.

This is the first time I’ve used Magic Vs, but I can see that it will become a firm favourite in my repertoire of rich tasks. Even though I ran the session four times with four groups, it felt fresh and exciting each time because each new group of pupils came up with their own justifications and their own ideas to explore next. The only maths knowledge they needed in order to begin the task was an ability to add numbers, but the level of mathematical thinking they got out of it was higher than I could have hoped!

Words from the past

June 24, 2010

Last night I was reading through some things I wrote a few years back when I stumbled across this sentence:

I wish there was more time in the curriculum to focus on proof, which is the cornerstone of mathematics but tends to get buried between attainment targets and numeracy strategies.

It brought back to me memories of the frustration I felt as I started out in teaching, that so many of the things I wanted to try I felt that I couldn’t because of the pressure to deliver exam results that met targets.  These words were written towards the end of an academic year, after exams, about the joy and delight I felt after a successful lesson on investigation and proof, having recaptured the enthusiasm that drew me into education in the first place, but my longing was for maths teaching to be like this all the time.

Since joining the NRICH team, I have been privileged to meet teachers, departments and schools who are striving to make every lesson like this, and I hope I have contributed in some way by working on developing support materials for teachers wanting to use our resources. The best resource of all in creating confident learners of mathematics with a firm grasp of the importance of mathematical proof is to have confident teachers with the freedom to express their own enthusiasm for maths in the classroom.

On another note, I had a piece of excellent news from a former pupil yesterday; he has achieved a First Class Honours degree in Mathematics. He plans to train to become a maths teacher – I shall be keeping a close eye on him to make sure he uses NRICH resources as often as possible!

Mathemagic

June 3, 2010

I know many mathematicians who enjoy various forms of mathematical recreation – sudoku, mathematical art, solving really hard differential equations for fun… and lots who dabble in mathematical “magic”. There is something that entrances young and old alike when a card trick is performed with verve and showmanship, but there is even more satisfaction in seeing a mathematically based card trick and figuring out how it is done.

I’m preparing for a session working with some Year 10 students in a few weeks. It’s only a short session – 45 minutes or so – and most of my favourite maths tasks take rather longer to get into than that. So a colleague suggested I did some mathematical card magic with them, and suggested the problem The Amazing Card Trick. His idea is that he’ll come in, do the trick, and then leave us to puzzle out how he did the trick – something he’s done with great success working with groups of children in the past.

I wanted to make sure I’d cemented the trick in my mind before trying it out in public with actual youngsters, so after dinner with friends (and a few glasses of wine) at the weekend, I requested a pack of cards and tried it out. My friends loved it, and had the trick figured out (complete with algebra to explain how it worked) in just a few minutes, so I’m hoping that 28 year olds are at least twice as quick as 14 year olds, and the Year 10s need a little longer to puzzle it out. Have a read of the trick as it’s explained on the NRICH site, have a go at it and see if you can figure out how it can be done. And then why not share with me your own favourite pieces of mathematical magic, just in case the Year 10s are smarter than I thought and I have 35 minutes and nothing more than a pack of cards with which to entertain them!

The language of maths

May 21, 2010

I’ve been thinking a lot recently on the language we use when we are teaching mathematics. This was partly prompted by a thread I saw on the TES maths forum discussing the use of the phrase “smaller than” to describe relationships between negative numbers. The discussion became quite heated as people argued their point, but one point that cropped up which resonated with me is that the most important people in a discussion like this are the learners – as teachers we must equip the children in our care to function in the real world and have a good grasp of mathematics.

For me, clear unambiguous language and definition is one of the most important things a mathematician does. If we all mean something different, we can never agree on our mathematics. In all mathematical communication it’s important to understand what everyone else understands by the terms we use. A well-documented problem, researched by many finer minds than me, is that we often use words that have a meaning in English as well as in mathematics. One example of this is the word “multiply” – in English usage this word has connotations of growth, which goes some way to explaining the common misconception multiplication always makes the answer bigger.

On to the less than/smaller than debate. I risk revealing my vast ignorance about primary mathematics and the way my primary colleagues work, but I would suspect that when children first meet the symbol < it is in the context of ordering positive whole numbers. If we are thinking about comparing measured quantities in the real world, “smaller than” is a very natural phrase to use. Alison is 1.70m tall, Sam is 1.87m tall. 1.70<1.87 so Alison is smaller than Sam.

However, I am uncomfortable about using the phrase “smaller than” when comparing negative quantities. Maybe my unease is unnecessary, but I worry that it may be ambiguous to some children in a way that “less than” wouldn’t be. “Less than” can clearly be defined to indicate that a number is positioned further to the left (or down if we’re working vertically) on a number line. I would rather not define “smaller than” in the same way, because to me, “small” is for talking about real world, positive things. I can imagine contrived examples about one person’s debt being smaller than another, and I often speak of behaviour near an asymptote at x=0 by talking about x being small and positive, or small and negative.

People will disagree with me, I’m sure. But my own personal opinion is that once children’s understanding of the concept of < is secure enough for them to be considering ordering negative numbers, we should be insisting on precise, clear unambiguous language and reading the symbol as “less than”.