Author Archive

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”.

BCME7

April 22, 2010

I spent a few days in Manchester at the start of the month for the British Congress of Mathematics Education and I really wanted to note down a few thoughts while it was fresh in my mind. Well, I’ve failed miserably at that, because it seems to be a distant memory now, but I’m going to dig through my notes anyway and record some of it here for posterity.

Obviously the conference sessions were a very important part of the week. I attended a variety of things, from the thought-provoking to the creative to the just plain fun things. Highlights included some mathematical origami, some reasoning with Tri-squares, and entertaining sessions from James Grime on recreational maths and Matt Parker (aka Stand Up Maths) on everything from cube-rooting large numbers to knitting Klein Bottle Hats. I went to a session about Statistics, something which I’ve not always had the most positive relationship with, but had the chance to think about some really inspiring ideas to capture the attention of young people studying the subject. I also had the chance to see my NRICH colleague Steve Hewson sharing some of his calculus ideas, and he reciprocated by coming along to my session on proof and being very well-behaved.

Hearing some of the great minds in the maths education community sharing their ideas was very inspiring for me, particularly as I am planning to apply to do a Master’s Degree in Maths Education in the autumn – I am going to have to get used to reading about these ideas and thinking hard about how they impact on my practice. But possibly the most useful part of BCME was meeting maths teachers and talking about what’s going on in their classrooms, what they are trying to do, and how we at NRICH can support them in their work.

The free Rubik’s Cube and Casio Calculator were pretty nifty too though 🙂

Birthday

March 31, 2010

As of tomorrow, my age will no longer be a perfect number. A perfect number is the sum of its proper divisors (the numbers that go into it, not including itself). 28 is divisible by 1,2,4,7, and 14, and 1+2+4+7+14=28. The last time my age was a perfect number, I was too young to appreciate it really (1+2+3=6) and I doubt I’ll live to see my next perfect birthday, which would be 496.

It’s been a good few years. Last year, I was a cube (27=3*3*3) something which won’t happen again until I’m nearly retired (64=4*4*4). The year before that, I was one more than a square and one less than a cube. The year before that I was square (no nasty comments about me always being square please!) 24 was exciting because it had lots of factors, and of course 23 was the last time I was prime. Tomorrow I’ll be in my prime again!

Next year’s birthday I’m struggling to see any reason to look forward to, but the year after that will be very exciting as my age in binary will be 11111, and I will be a “teenager” for the last time in hex, being 1F years old. When we were young, we sometimes had binary candles on our birthday cakes, with candles lit or unlit to represent 1s and 0s. In two years, it’ll be the last birthday I can do with just 5 candles.

Finally, some maths – when I was a teacher, more often than not each class would have a couple of pupils in it who shared a birthday. Should we be surprised by this? If you’ve never come across it before, the Birthday Problem is an interesting bit of probability theory, with a nice graph to show the probability that there will be at least two people who share a birthday for increasing sizes of group. How many people would you need to be 100% sure that there would be at least two with the same birthday?

Homework sucks

March 16, 2010

I usually listen to Simon Mayo’s Radio 2 Drivetime show on my way home from work. One of the features is Homework Sucks, where a listener explains a piece of homework they’re having problems with, and then other listeners contact the show to offer help.

Since the feature started in the new year, I’ve only heard a couple of maths-related questions. The more interesting of these was the following question: how many ways can you arrange three eggs into a six-egg box? The respondant is a regular poster on the TES maths forum, and you can read the thread here.

Some teachers have objected to a radio feature called Homework Sucks, but it’s a sentiment I find myself agreeing with. As a learner, homework didn’t have a great impact on me – at primary school we were expected to read every night (which for me was a joy rather than a chore), learn a few spellings, and learn our tables. Secondary school brought with it the structure of a homework timetable, and tasks which were intended to take an hour or so each evening to complete, although more often than not I finished them off on the bus or in the library at break or lunchtime. It was only really at sixth form that the workload associated with four A levels dictated that I had to spend lots of time on independent study.

As a teacher, I had to fit in with the homework policy of the schools where I taught, which sometimes meant that I had to set a homework task even if there was nothing appropriate which fitted in with what I had been teaching. I would much rather have had the freedom to set homework when I felt it would support the learning that had been going on, and the freedom to give my pupils a night off when there was nothing I wanted to set. I know that a homework timetable avoids the congestion of nineteen subjects all wanting you to spend an hour on a task on the same night, but it should not constrain teachers into feeling they have to set busy work in order to be able to tick a box on their markbook.

I could rant all day about the pro-homework arguments about instilling discipline and work ethic, preparing children for the long days and overtime they will put in as adults in corporate careers, giving children something purposeful to do with their evenings… I could rant about the unfairness of expecting a child living in a cramped untidy flat with several younger siblings to produce the same piece of work as an only child living in a mansion with a spacious study with an anglepoise lamp… but instead, I will just say that in a perfect world, I would want to encourage learners to discuss their maths with their family and friends between one lesson and another, and maybe do the odd practice-paper in the run up to their GCSEs and A levels, but otherwise, I don’t really mind what they get up to in the evenings.