Conventional Wisdom

November 10, 2011 by

There are some truths in mathematics that are true because they are true because they are true. For example, if I have a right-angled triangle in the plane, the square on the hypotenuse has to be equal to the sum of the squares on the other two sides. Other truths are true in a different way. It’s true to say that this:

is a square, but it’s not true in the same way that the statement of Pythagoras’s Theorem above was true. And of course, saying that the statement is Pythagoras’s Theorem is true, but not in the same way that the Theorem itself is true.

Confused yet? Great!
The notion of different types of truth has been around for a very long time, and has been recast using many different descriptions. I’m rather fond of Dave Hewitt’s designations “arbitrary (socially agreed names and conventions) and necessary (properties and relationships)”, so calling that red shape a square is arbitrary, because I could call it anything I liked, I just choose to follow the social convention to call it a square, but Pythagoras’s theorem is necessarily true, it is a property of right-angled triangles, it is a relationship I can derive for myself.

These necessary truths are vital to mathematics, in fact, perhaps in some sense they are mathematics. And in teaching mathematics, the NRICH philosophy draws on lots of great thinkers in maths education and comes to the conclusion that these truths are something that children should have the opportunity to explore and discover for themselves. They will never be able to discover that a square is called a square without some external influence (an adult, older child or dictionary telling them that a regular quadrilateral has the name “square”).

But despite my strong feeling that it is the necessary truths that are core to mathematics, I also think that educating children into the conventions of mathematics is important. Part of being a mathematician is being able to speak a common language with other mathematicians. This means knowing the definitions, being fluent in the notation, understanding the conventions.

I ranted a bit on Facebook earlier about the questions like “20 + 20 x 0 + 1” that have been popping up, with a popularity contest where people vote on what the answer should be. An idea that came out of that discussion was that one of the reasons that people don’t remember BIDMAS, BODMAS, PEMDAS or whatever it’s called in their local language is that they don’t see a need for such a convention. For me, the link between arithmetic and algebra means that order of operations is firmly embedded – if I was evaluating 2 + 3n, of course I would do 2 + (3 x n), so if I’m doing 2 + 3 x 4 I think of it in the same way – in my mind, the “three times four” is grouped together. Given the widespread lack of awareness that mathematicians have a convention for order of operations though, I think if I have need to write a calculation down for others I will use extra brackets just to be on the safe side!

Enrichment

October 21, 2011 by

This week’s MEd session was on the subject of Enrichment. Our pre-session task invited us to first consider the following questions:

  • What is ‘enrichment’?
  • What roles could ‘enrichment’ play?
  • Who should be ‘enriched’?
  • Where and when should ‘enrichment’ take place?

After writing our answers to these questions, there were three pre-session readings. The idea was to reflect on our first feelings to see if they changed based on the readings.

Here are my thoughts from before the session:

Enrichment is that which enriches… it is something that gives students a richer experience of learning mathematics than they otherwise would have. For me, a rich experience would be one that involved making the connections between diverse areas of mathematics, and being exposed to the big ideas of mathematics. It would be a learning experience that offered young people the chance to work like mathematicians – exploring, making conjectures, justifying, proving.

Enrichment could happen on different levels and in different ways. An otherwise dry and dusty textbook lesson could be enriched slightly by passing reference to the connections with other areas of mathematics. A particular task cannot necessarily be considered an enrichment task – it is the way that the task is presented that makes it enriching. Enrichment is relative to the child, the teacher, the cultural norms of mathematics education in that time and space.

I firmly and strongly believe that every child should be entitled to an enriching learning experience in mathematics. Chances to explore, make connections and work like a mathematician should be a part of normal maths lessons. This is not to say that extra-curricular enrichment activities are unnecessary; on the contrary, young people who have maths brought alive for them within the classroom may well get more out of enrichment opportunities outside the classroom, who knows? I am very glad that maths masterclasses take place, and it would be great if there were chances for every child who got excited by maths to participate in such events.

The discussion on Wednesday was fascinating, as different people shared their views on enrichment. Our Masters group are quite a diverse bunch, with lots of different educational backgrounds, so this all fed into the discussion. I don’t think my views have changed very much from the above; I don’t think I will ever stop believing that we should aim to give everyone a good experience of learning maths and a good understanding of what maths actually is – so much more than just arithmetic and accountancy.

Maths and…

September 14, 2011 by

If you’ve visited the NRICH website in the last year or so, you’ll probably have noticed that we’re doing a lot more cross-curricular stuff lately. This began with stemNRICH, an effort to examine the M in STEM and provide good quality mathematical resources with a scientific and/or technological focus. Such was the success of stemNRICH at key stage 5 that it is now being rolled out into key stages 3 and 4.

In addition, we’ve had monthly themes on both art and sport recently. So I thought I’d blog a little bit about some of these cross-curricular themes, and the issues and opportunities they raise. Today I’m going to talk about sport.

I admit, when the decision was taken to publish maths and sport resources my heart sank a little. Although I love spending cold Saturdays in December standing on a terrace watching football, and will watch almost any televised sport that isn’t golf, I tend to keep the sporting part of my life very separate from the maths part of my life. When I was at school, there was virtually no overlap between the sporty kids and the maths geeks, and I was definitely one of the kids who was picked last for everything because I was too busy calculating square roots in my head to notice if a ball was travelling towards me. To me, there was a massive gulf between sport and maths, and until I was forced to think about it, I couldn’t see how that gulf could be overcome.

Luckily, I am part of a wider team with lots of vision and ideas. The Maths and Sport website is beginning to grow, and as we add to it, we are getting new ideas from each other’s work. I think my favourite of the problems I’ve been involved with creating is Charting Success. Teaching about graphs and data representations is always going to be easier if we use data that has some sort of an impact, and by choosing several different sports we hope the majority of students will be able to engage with at least one of the representations. It actually started out as a problem just about data representation and a chance conversation about where I was going to find suitable graphs to feature led to the idea of using sports graphs, and thus linking our theme on stats with the ongoing sport project. (Visit the NRICH site in October for lots more on stats.)

When teaching about handling data, I believe it is important to use real data and ask interesting questions. I also believe that if a theme such as sport is tacked on to a problem just to give it a context, kids will treat the problem with the same disdain that an adult would if presented with something artificially bolted on. I really hope that Charting Success has avoided these traps – the graphs and diagrams are ones that are genuinely used, so the maths is embedded within the sport rather than being an add-on, and I think enough people are interested in sport that there are interesting questions provoked by these representations.

I’m hoping to put together a follow-up problem early next term with other interesting representations used in sport. If you have any ideas about suitable graphs that are used in sporting contexts, leave a comment. And of course, feel free to comment if you want to agree or argue with anything I’ve said!

The Pizza survey – part 2

August 22, 2011 by

Here is the first instalment of the eagerly awaited results to the pizza survey. Go and read part 1 for the context.

As of this afternoon, there were 504 respondents. I removed one duplicate and one nonsense response, and have done my best to interpret everything else.

In answer to the question “Are you a mathematician?” there were 217 unambiguous “yes” answers (either yes or Y), and 225 unambiguous “no” responses. In addition, there were 7 responses along the lines of “yes ish” or “yes but”, 42 responses that I interpret as being positive, and 11 responses I interpret as being negative. In the next paragraph, I will outline my interpretation process for these.

I was rather generous in assigning people to the group “mathematician” rather than “non-mathematician”, so I gave people the benefit of the doubt. Anyone responding “Depends how you define mathematician” was definitely enough of a mathmo by my standards. I accepted physicists and computer scientists as long as they showed some desire to be counted, for example those who said “I’m a physicist – is that close enough” counted as a yes, but the respondent who said “No, I’m better than that, I’m a physicist” was counted as a no. All those who said they were studying to become a mathematician were included in the positive responses.

Of the 225 unambiguous “no” respondents to the mathematician question, 11 also answered “no” to the pizza question, and 185 answered yes. The other responses will have to wait until I have time for more detailed analysis. Of the 217 unambiguous “yes” respondents to the mathematician question, 12 answered “no” to the pizza question, and 185 answered “yes”. So my preliminary findings are that no matter how many respondents you have, 185 will always be unambiguously in favour of pizza. Alternatively, it seems that I have found that for those who follow me on Twitter or who follow someone who is likely to retweet a silly twitter experiment that I made, whether they are mathematicians or not makes very little difference to their pizza loving.

If there is enough demand for it, I’ll sift through the rest of the spreadsheet and analyse some more. And if you ask very nicely, I might post some graphs too!

The Pizza survey – part 1

August 20, 2011 by

This is the first part of the story of the pizza survey I set up the other day. I don’t know a great deal about conducting studies, but I know that things like aims and methodology should come before the results, so part 1 will be about those sorts of issues, and then I’ll blog some results later this weekend.

Wednesday was an odd day at work. Tuesday night I’d driven up to Scunthorpe after work to watch Scunthorpe United lose (the intention was to see them win but these things don’t always work out as planned), and I’d got back quite late. So on Wednesday I was a little daydreamy as I worked my way through some fairly routine tasks on my to-do list. I paused for lunch as usual, and went up to the cafeteria to buy something. They had the new pizzas they’ve started doing; they used to do pizza-type things that were actually a half-baguette with toppings on, but now they have actual thick pizza base with peperoni, bacon, mushrooms and other delights. (More on toppings in the results blog later.) I idly wondered whether mathematical output of the building would improve with the improved standard of pizza, and then realised that I was extrapolating from my own experiences of mathematicians as pizza-lovers and assuming that all mathematicians were like my friends and me. Thus the survey was born.

I thought long and hard about the questions. Obviously I had to ascertain whether people were mathematicians and whether they liked pizza. Some people found my choice of free text boxes rather than yes/no buttons an odd one – this was quite deliberate. I expected most people would be happy to type in “yes” or “no” (and indeed, I can now do some interesting analysis about the proportion who capitalised, chose just to use Y/N, added emphasis such as “f*** yeah!” …) but I suspected that some would want to tell me a little more. I decided before the survey went live that anyone typing “It depends how you define mathematician” is probably enough of a mathmo to be counted in the yes camp.

Ultimately of course, it would be nice to see whether the pizza-loving is more prevalent among mathematicians or non-mathematicians, but as my data collection relied on Twitter, and as my Twitter followers are mostly maths or maths-ed people, I suspect that my results will consist so overwhelmingly of mathematicians that it will be difficult to make any significant conclusions.

Peter Rowlett has preserved some of the Twitter conversation from Wednesday afternoon on his excellent blog. Come back later this weekend for the first results from the survey. There may be pie charts!

Recent workshops

May 27, 2011 by

It’s been a while since I blogged – sorry if you hang on my every word and have missed me, although I don’t think that applies to very many people!

I promised a couple of people a while back that I would share some of the stuff I’ve done at recent conferences and seminars. Here is a powerpoint presentation from a seminar that Charlie and I gave in Nottingham earlier this month. In the session, we aimed to discuss and explain some of the thinking behind a couple of our recent tasks on the site, Opposite Vertices and What’s Possible. In preparing for and giving this talk (as well as our sessions at the ATM and MA Easter Conferences), we’ve been talking a lot about our philosophy of maths teaching and what the secondary NRICH resources should be like.

A few thoughts have emerged from all of this. We have to balance the needs of different audiences when we’re writing rich tasks. Students who come to the site on their own need to be presented with something that gives enough help for them to get started but without giving the game away and robbing them of the joy of mathematical discovery. Teachers who come to the site might prefer to look at the teachers’ notes to see how we think the problem should be used in the classroom. But these can’t be too long, as teachers are busy people. And yet each problem can be used in a variety of ways, for different groups of students, and it’s a shame if our teachers’ notes can’t capture that.

We are starting to experiment with creative ways around these difficulties. One idea is to hide certain sections of a problem or the teachers’ notes, so teachers or students using the site are presented with the bare minimum input but can choose to click for more. We have tried adding video to Teachers’ Notes, as we think it’s probably easier after a long day’s teaching to watch how a lesson might evolve rather than reading a long chunk of text. Of course, it’s not possible to add video support to every single problem, but I get the feeling that if we can share our thinking like this for a few of our problems, it will be easier for people to “get” NRICH and what we’re about.

Why teach maths?

April 4, 2011 by

Why do we teach maths in schools?

a) To create the research mathematicians of the future
b) To empower ALL of our children to take their place as mathematically literate members of society
c) To instil in our citizens an appreciation of mathematics as a thing of beauty and truth

If we create a mathematics curriculum that allows everyone to reach a minimum standard of mathematical understanding (functional numeracy, perhaps) but also allows a generation to leave school without any appreciation for the wonder and pleasure of doing mathematics, then we have failed. But at the same time, if the system identifies and nurtures superbly talented mathematicians who go on to win Field’s Medals, while allowing some children to slip through the net and leave school innumerate, we have also failed.

If we work towards c) however, and see the job of school mathematics lessons as teaching all children to think mathematically (and to understand what we mean by thinking mathematically), I think we will go a long way to achieving the other two objectives – they needn’t be mutually exclusive. In classrooms where high-level mathematical reasoning is the norm, a good level of mathematical literacy becomes the currency for convincing others of your ideas, so pupils are given a motivation for wanting to become more skilled in mathematical procedures. If thinking mathematically is the expectation, those children who enjoy the pedantic* process of convincing themselves and others of the truth of a conjecture will discover themselves to be mini-mathematicians and will be more likely to embark on the process that could lead them to fame, fortune and Field’s Medals. As soon as our curriculum aims to do anything other than exposing young people to mathematical thinking, we risk doing at least some of the children in our care a great disservice.

*I do not use this term in a pejorative way. I took great delight in being exceedingly pedantic throughout my secondary school career.

Birthday, again.

March 31, 2011 by

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

March 22, 2011 by

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

March 2, 2011 by

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?