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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.

Another foray into Facebook Maths

March 4, 2010

After the success of my last problem-posed-on-facebook, I decided it might be fun to post some more maths for my friends, so on Tuesday afternoon I updated my status:

Alison would love to share some maths with you this afternoon – do you want number patterns or shape?

Well the bad news is that the first couple of responses were very negative – despite the friendliness of my language, and the offer to share some maths rather than impose it on people, led them to post things such as “eeek”, and a sentiment that they wished to run away and hide, with the follow-up comment “Maths is seriously Not Fun”.  This is from people I respect greatly and who I know to be intelligent, articulate, witty individuals, and I dread to think what their experience of maths at school must have been for them to feel such distaste for my favourite subject.

Luckily for my self-esteem, another old friend came along and requested a shape problem, so I posted the following:

You’ll need a ruler, something to write with, and something to write on. Got them? Great, let’s get started!

Draw a triangle. Any triangle you like, as long as it has three straight sides I don’t really mind. Now find the middle of each side, and mark it with a dot. If you wanted to be really clever, you could find the midpoint using a straight-edge and compasses the way the Greeks did, but if you’d rather just measure with the ruler (or even fold your triangle) I won’t mind.

Once you’ve found all three mid-points, join them together: this should split your original triangle into four smaller triangles. Do you notice anything interesting about the smaller triangles?

What happens if you try joining the midpoints of each side of a quadrilateral (four-sided shape)?

And sure enough, a few people rose to the challenge, and started sharing their ideas. One commented several times, first explaining that he’d found it split the original triangle into four that weren’t necessarily the same size or shape, and then coming back to say he always ended up with four congruent triangles, all similar to the original.

Another commenter used vectors to prove the conjecture about joining the midpoints giving congruent triangles with sides half the length of the original, and went on to think about quadrilaterals. His first comment suggested that repeating the process might get you to a smaller version of the original quadrilateral, but he sent me a delighted message later on to say he’d experimented and found he always ended up with parallelograms, and that he’d proved it using his result for triangles.

Someone else saw the triangle problem in a very visual way – he could convince himself very easily what would happen with an equilateral triangle, and then just imagined all other triangles as being transformed, or viewed from a different angle. Some of my facebook friends are easily as imaginitive and happy to explore as the classes I’ve worked with in schools.

There have been plenty of comments along the lines of “Keep them coming” to the maths I’ve posted, which heartens me, but I know that my friends on facebook are not representative of the population as a whole – many are people I studied maths with, or who share an interest in technology and geekery.  I’m glad that each time I do this, some people I wouldn’t necessarily expect to respond are willing to dive in and have a go, and share their insights which are every bit as valuable as the fully and carefully constructed mathematical proofs that others send, but the real challenge will be to engage some of the doubters and maths-phobes, and convince them that actually, Maths is Seriously Fun!


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