Birthday, again.

Last year, I blogged about turning 29 and never being perfect again. It seems I was somewhat unenthusiastic about this year’s birthday, but it’s not such a boring birthday after all: tomorrow my age in years will be the smallest number that’s the sum of three distinct primes.

There are other special properties of 30 of course (after all, every number is interesting) – it is a pyramidal number (1^2+2^2+3^2+4^2) and apparently it’s the largest number such that every smaller number coprime to it is prime.

But the best way(*) to celebrate my 30th birthday will be by making a dodecahedron out of post-it notes. I’ll need 30 post-it notes for the 30 edges of course!

(*) Don’t panic. I’m not so much of a geek that I’ll forget to celebrate with a couple of beers and some good friends too. And maybe some Mobius strips.

The F Word

Last month, Rob Eastaway wrote this piece about Maths and Fun. It really resonated with me, because I often come across this notion that we ought to be making maths fun. In fact, one popular perception of the NRICH project is that we exist to try to make maths fun.

This came to mind today as I re-read some of the feedback I got after working with some Year 8 students recently. The project was an extended one, and at the start I explained to the students and their teachers that we would be trying some challenging maths problems and learning new techniques for solving problems, as well as developing ideas like working systematically, conjecturing and proving.  Some of the feedback included exhortations to “Make it more fun” or “More games”. Thankfully, there was also feedback with comments like “Being challenged made me think more”, and some students appreciated that although they sometimes found the maths hard work, they enjoyed the satisfaction they got from solving a difficult problem.

I wrote a while ago about the use of rewards in mathematics classrooms. It all seems to be part of the same issue to me – whether trivialising maths as ‘fun’ or make a physical reward the motivation, we take away the potential for them to derive joy from solving really tough problems and rob them of the discovery that maths can be intriguing and engaging (words I would much rather use than ‘fun’).

Gender, Maths and Car insurance

The portions of the internet that I frequent have come alive over the past few days with heated discussion about the European Court of Justice ruling on insurance and gender. I posted something about it on Facebook the other day and got a stream of comments, mostly defending the status quo and saying “Of COURSE girls should have cheaper car insurance than boys!”

What has this got to do with maths education, the subject of this blog, I hear you cry? Well I was thinking about the many studies into differences between boys and girls’ experiences of learning maths, and various measures that are taken to correct the so-called “gender gap” whenever one group is outperforming another. People seem to accept as a given that boys and girls aged 16 should achieve broadly similar results in GCSE maths. When someone says “Well maybe one gender is naturally predisposed to be better at maths than the other”, they are (in my opinion rightly) jumped upon from a great height. Any systems in place in schools which seem to favour one group over another are challenged, and where one gender is slipping behind, initiatives are put in place to challenge this.

So why is it ok to say that boys are naturally predisposed to be more dangerous drivers than girls? And why is there no outcry to close the gender gap in car insurance prices? Why are we not putting initiatives into place to raise our boys to be safer drivers, so they too can benefit from cheaper car insurance?

Teacher support materials on NRICH

This month we republished an old NRICH favourite which we dusted off and repackaged to reflect the way we’re thinking about task design for the secondary site at the moment. If you’ve never met the problem Tilted Squares before, I urge you to take a look and have a play, as it’s one of my favourites.

We’ve also done something slightly different with the Teachers’ Notes to this problem. Charlie and I were recently working on Tilted Squares with a group from a school in London, and we arranged to have the session filmed. The notes include extracts from the video footage – the clips are quite long, I’m afraid. Having watched the sessions back (which is always going to be nerve-wracking – I don’t think I’ve ever met a teacher who enjoys watching video footage of their teaching!) we have our own ideas about what was successful and what we maybe should have done differently, but I am very keen to know what others see in them. So if you have time to watch some of the footage and then leave a comment here, I’d be very grateful. More generally though, is it useful to include video in our Teachers’ Notes? Do you have suggestions of other changes we could make to our teacher support materials?

Maths on the underground

As I was on my way to catch the train home this evening, I received a text from a friend with the following question:

Two flagpoles, heights a and b, are separated by a horizontal distance d. The top of each flagpole is joined by a wire to the bottom of the other pole. At what height do the wires cross?

I solved it on the metropolitan line with a few minutes’ thinking and scribbling time. Let me know how you get on, and feel free to share your methods in the comments.

Its own reward…

I had an interesting thought the other day. It was about motivating students by offering them rewards. Someone suggested to me that I could enthuse some students I was working with by bringing some sweets for the team who finished first, and my knee-jerk response to this was that I didn’t want to do that.

I went away and analysed my knee-jerk response, and came up with a few thoughts as to why I was so reluctant to offer rewards. Firstly, there have been many times when I’ve been given something to do in competition with others, and the nature of the competition has caused me to want to opt out. After all, if there’s a risk of coming second, of not winning, then at least by opting out I can say that I CHOSE not to compete. I think I am far from alone in this, so I prefer not to introduce competition into the maths activities I offer to students.

Secondly, I want the students I work with to think positively about maths. I don’t want them to work hard on a problem because they’ll get a mars bar if they do, I want them to work hard on a problem because it engages and interests them, and they are curious about the result. I want them to work on the maths for its own sake, not because they want to please me and win a prize.

Thirdly I’m aware of studies showing that although offering rewards can increase motivation in the short term, in the longer term groups who are not offered rewards catch up and then overtake the rewards groups.

Students often ask the question “Why do I have to do this, miss?” It has always seemed to me that an answer about the benefits to the student’s mathematical understanding and progress in the subject is a more honest and better answer than “Because if you do, I will reward you with a mars bar/A grade/not giving you a detention!”

Comfort zone

This morning I did something that took me quite a long way out of my comfort zone, and I’m very glad I did it.

We have a study school here in Cambridge at the moment, forty Year 10 students from London who are spending four days with us becoming better problem solvers. As part of this, I agreed to model solving a problem I hadn’t seen before, in front of the students. My colleague Charlie had found a suitable problem he didn’t think I was familar with:

Find all positive integers m, n with n odd such that 1/m + 4/n = 1/12

My first reaction was panic. I didn’t immediately see a complete route to a solution so I was very worried that I wouldn’t be able to solve it. Eventually though, I made a start and tried some things out, worked out some conditions that m and n had to satisfy, and found one solution. But the question had asked for all solutions, so I knew I needed to work more generally.

As I continued to work on the problem, people in the room started to help me out. The Year 10 students made a couple of suggestions, as did the student ambassadors sitting at the back of the room. Of most help to me was Charlie who knew one possible route to a solution and made suggestions that he thought might help me to get there.

Eventually, I realised that I had found an equation the solutions to which would give me all possible values for m and n. At this point, I was very happy to stop but the students wanted me to carry on and find at least one solution using my method so I continued, and once I’d shown them that my method worked I went and hid in the corner to recover my wits while Charlie drew out the key points of what I had done.

Charlie talked very little about the maths I’d done, but rather identified ways in which I’d used my problem-solving toolkit to help me. Having read Polya’s “How to Solve It” quite recently, the idea of problem-solving heuristics was on my mind so the narrative I gave while thinking out loud about the problem had phrases such as “This has helped me in the past so I’ll try it here” and “This isn’t going to work, what else can I try?”

The best part of the morning came quite a bit later. The students were working on the problem solving booklet we’ve put together for them, and one was working on Hidden Dimensions. He wrote down some algebraic expressions, and then used exactly the same technique I’d used on the problem earlier to solve his problem. Making that connection for himself, without any prompting, is exactly the sort of thing we were hoping this week would foster in these students!

International perspectives…

The part-time Masters course I’m following is called “International Perspectives on Mathematics Education”, and so a lot of our reading and discussions focus on the way maths is taught throughout the world. One of the course requirements is for all students to present on an aspect of maths education that interests them, drawing on their own experience, and to lead the group in discussion on that topic.

Last night was my turn, and I chose the topic “Grouping by Ability”. I started by outlining my own experiences as a learner – I was educated in Lincolnshire where there are still grammar schools, but my dad chose not to enter me for the 11+ and instead I went to the rural comprehensive school just up the road. The grammar schools tended to cream off the top, so it wasn’t a true comprehensive. Thinking back, in Year 10 and 11 we were in mixed ability groups for our option subjects and it was only English, Maths and Science where we were setted. The top Maths set ended up with a wide variety of grades anyway; only a small handful (four or five of us maybe?) were entered for the Higher paper at GCSE and the rest of Set 1 sat the Intermediate paper. In explaining all of this background to the rest of the students on the Masters course, I trotted out a little mantra of mine that I used to remind myself of regularly, whether teaching a “mixed ability” or “setted” group:

Every classroom is a mixed-ability classroom

The top set I found myself in as a learner was a mixed-ability classroom. My teacher had to work very hard to make sure there was an appropriate level of challenge for pupils like me who were very quick on the uptake and did lots of maths outside the classroom, while making sure there was appropriate support for people who took longer to come to grips with certain new concepts. She encouraged us to talk to each other and seek help within the classroom community rather than all lining up at her desk to ask her for help. These are the sort of strategies that teachers adopt through necessity when faced with learners working at a wide range of levels, but I think they are good teaching practices whatever the spread of achievement within a class. Of course my underlying philosophy of maths teaching is one where learners explore, conjecture and develop new understanding through collaboration with each other rather than one where the teacher imparts knowledge from the front, so this model of mixed-ability teaching doesn’t seem so alien to me as it might to others. The idea of having a set where everyone is “working at level 6″ so the lesson is pitched at some pupil who is somewhere in the middle of whatever level 6 means, and those who are thinking above that level just have to slow down a bit, and those working below that level will have to follow on as best as they can – that seems like laziness to me. If we accept that we’re going to have to differentiate within whatever class we are faced with, why should the range over which we need to differentiate make that job any harder?

The most interesting part of my student-led seminar was the discussion which followed. Most of the cohort were educated overseas, and when I took them through my own experiences not just as a learner but also as a teacher in schools where setting was the norm, they were quite shocked. Unfortunately, the half-hour we had for me to present and for us to discuss wasn’t enough to get into the intricacies of other countries’ education systems, but I need to have a lot more conversation with my peers from overseas about how exams are structured and how lessons are taught to cater for a wide range of pupils. I have been so embedded in a system where grouping by ability is the norm that I find it very hard to imagine how it can be otherwise.

Were you educated in the UK? In a setted or mixed ability classroom? Or in an area with grammar schools? How did it affect your experiences of learning maths? Or perhaps you were educated somewhere where mixed ability teaching is the norm, and my experience is as alien to you as yours is to me – tell me about it in the comments!

More Definitions

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.

Definitions

Although my focus in this post is Key Stage 3 and 4 maths in UK secondary schools, some of what I have to say may well apply to other phases and other places too.

I realise I have been remiss. Some time ago, I started this blog claiming it would contain thoughts about maths, but I have never actually defined what I mean by that! I think I’m far from alone in this – throughout the blogosphere it’s very common to talk about well-used terms and concepts without ever unpacking exactly what is being talked about. So I thought I’d devote a short entry to what I mean when I use the word “maths”.

There are two categories of activity that “maths” as I use the term falls into. Firstly, there is what some people might call functional maths – making sense of number, graphical data, statistics, money, measurement… all the skills that young people need to master in order to function in society when they leave school (or the skills that young people need to master in order to access mathematics in higher education or within their career, for that matter!) Secondly, I use the word “maths” when I’m talking about an activity which involves such skills as working systematically, looking for and explaining patterns, generalising, and proving those generalisations.

I think both of these types of activity are important – maths shouldn’t be an either/or thing; maths classrooms can involve both sorts of “maths”, and in fact one particular maths lesson might be targetting skills from both sets. I think denying children either would be wrong, and saying “these children only need functional maths because they are never going to be professional mathematicians” perpetuates a mis-understanding within society of what maths is, and what mathematicians do. We should make sure that all children have the chance to explore, conjecture, and prove, and to know that these ideas are at the heart of what mathematics is all about. So I think when I think of the word “maths”, it’s the second meaning that jumps to my mind first.

When you use the word “maths”, what do you mean by it? Have you ever come across anyone who thinks about and defines “maths” in a completely different way from you?