Maths I saw on my holidays

February 28, 2012 by

I’ve just come back from a lovely long weekend in the Netherlands. We stayed in Zwolle, capital of the Overijssel province, and also visited Ommen, Giethoorn, Zutphen, and we stopped off in Utrecht on our way home. Of course I kept my eyes open for maths while I was away! Some of the pictures are only mathematical in a very tenuous way but I hope you enjoy them anyway. Click on the photos for bigger versions.

First, the obligatory Dutch windmill shot. The sails look a little like a plus sign – that’s mathematical, right?

Next, a couple of floor patterns, one from a department store and another found on a pavement outside a supermarket.

There’s loads of maths in this photo of a cheese shop! How many cheeses? How many kilograms of cheese? How far would I have to jog to burn off the calories if I ate it all?

I loved these cute little mushroom shaped signs showing the distances to nearby places. Note the European comma where we would put a dot for the decimal point.

Right by the mushroom sign was a hexagonal cycle route sign. The world should have more hexagonal signs.

Next, a couple of clocks. I love clocks, particularly station clocks and clock towers with bells. I learned that the Dutch word for clock is ‘Klok’.

If you look very carefully at the packaging for the mini waffle I got with my cup of coffee, you’ll see a tiny diagram showing that it has a diameter of 4.5cm! Ideal if you want to compare waffle sizes between different cafes.

The waffle diameter cafe also had these brilliant salt and pepper pots. I’m not sure how you tell which is which. Is salt a 5 sort of condiment or more of a 3?

Our hotel lift pleasingly used the negative numbering convention for floors below the ground floor:

In the UK we have signs saying ‘No Under 18s’. In the Netherlands, they use a strictly less than < sign instead:

Finally, when I’m not being a mathematician I dabble in music. We saw a wonderful display of harmonicas in a shop window, including this fabulous six-sided harmonica :

Alas, the shop was closed so I couldn’t buy it.

What do you think of the photos? Which ones are the most mathematical? What maths have you spotted on holiday?

You are welcome to use and share these photos for non-commercial purposes, as long as you credit me and link to this post.

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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

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How to start a MathsJam

January 25, 2012 by

Take three pounds of mathematicians. Peel, remove any stones, and place in a pan with a little water and sugar. Bring to a simmer and stir occasionally. Test the jam on a cold saucer until it wrinkles when you push a finger through it.

Actually, that’s not how you start a MathsJam at all. I waited for some time for someone to create a Cambridge MathsJam, having seen the success of the ones in other cities and enviously following events on Twitter wishing I could be there. After attending the first two MathsJam annual conferences and having a jolly good time, I realised that everyone else in the Cambridge area was also waiting for someone else to start a MathsJam, and despite having the organisational skills of a very disorganised thing, I thought I’d give it a go.

I asked a few close friends if they’d be interested in coming along, and got in touch with Katie Steckles who organises the Manchester MathsJam. She provided me with oodles of advice from her own experience, together with some words of wisdom from Matt Parker of London MathsJam. Then I contacted the pub where we tend to go for pub lunches from work on the rare occasions that I’m allowed out of the NRICH office, and asked if they could reserve a couple of tables for us. “Probably between half a dozen and eight people I would guess”, I said when the landlord asked how many to expect.

So we had a venue, and a date. Now came the publicity! I sent a couple of tweets, and they were picked up and retweeted. Now that we had a venue we’d been added to the MathsJam website, so people started getting in touch that way. The close friends who had encouraged me to go through with this then invited everyone they could think of, and those people also mentioned it to their friends. I emailed the landlord: “Actually, it’s going to be more popular than we thought – maybe as many as a dozen or 15 people!”

While I was out shopping, I saw some bits and pieces – some of those wooden puzzles with rings that you have to disentangle, some playing cards, a set of dominoes, and I started building a MathsJam resource bag. I also stocked up on paper and pencils, chucked a couple of calculators in, and dug out one of my spare Rubik’s Cubes. Then yesterday evening I turned up early at the pub with my little brother in tow, got a drink and something to eat, and spread the maths paraphernalia out on the table so that people would know who we are.

“Is this the MathsJam?” “We’re here for the MathsJam.” “Hello, I’ve brought some maths!” The lovely thing was that people just sat down and started talking to each other. I’d prepared a sheet with a few NRICH problems to use to break the ice, and this proved to be a good idea, because once people were talking they started sharing other problems, card tricks, origami. I kept an eye on Twitter and read out some problems that were being worked on elsewhere, although we didn’t get round to sharing much of what we were doing. At one point, I counted 23 people in our corner of the pub, all working on maths and enjoying a drink! As people started to drift off at the end of the evening, I heard a lot of “Cheers, see you next month” and “I’ll bring you that problem I told you about”. I regret that I didn’t get the chance to talk to everyone and I didn’t catch everyone’s names, but I have high hopes that the people I didn’t spend time with will come back next month, and the month after, and the month after that…

A huge thank-you to everyone who made the first Cambridge MathsJam a success. Here’s to many more!

These are the problems I put out on the table at the start of the evening. We are building a collection of similar problems on NRICH and eventually they’ll have their own page. They should require no knowledge beyond A Level, and many can be solved using GCSE level content.
https://nrich.maths.org/249
https://nrich.maths.org/251
https://nrich.maths.org/279
https://nrich.maths.org/301
https://nrich.maths.org/327

The next Cambridge MathsJam will be Tuesday 21st February at the Castle Inn, Cambridge. Visit the website if you want to join the mailing list. To find a MathsJam near you, see http://www.mathsjam.com

ICT in the Classroom

November 18, 2011 by

This week’s MEd session was all about ICT in the classroom. We were asked last week to think about anything we could present on the topic for the other students, and I thought it made sense for me to talk about the way NRICH promote ICT in the mathematics classroom so I put together a few ideas for a five minute talk. It makes sense to write those ideas up here so they are not lost.

Firstly I made the point that as a web-based project NRICH is dependent on ICT to get its resources to its audience, but once they have visited the website, for a lot of the problems the teacher needs nothing more complicated than a board and pen/chalk to introduce the tasks. For others, there will be printable resources that can be handed out. But it is the problems with interactive elements that seem to raise the most interesting questions in terms of ICT use in the classroom.

One of my favourite interactive resources is Dozens. The interactivity allows students to keep trying, getting feedback if they get it wrong, and to generate unlimited examples at each stage. There are levels of difficulty corresponding to divisibility by two, three, four and six. Once students are confident with the mathematics, there is a final challenge to work on with pencil and paper. I like this resource because the feedback allows a whole class of students to work at their own rate without relying on the teacher to tell them how they are doing.

Charlie’s Delightful Machine also offers students the chance to work on unlimited examples that are different each time, but that’s not why I like it. The coloured lights in the problem are an enticing hook to draw students in, and the mathematics needed to completely solve the problem of when it’s possible to turn all four lights on is quite sophisticated. Again, there is the opportunity for students to work individually on the problem before the teacher brings the class together and draws out discussion points.

GOT IT is an old NRICH favourite. Again, there is the opportunity for students to work on their own and get feedback from the computer, perhaps studying the computer’s strategy and trying to work out why it works. One nice technique that we suggest in the teachers’ notes to several problems is to set a challenge, in this case something along the lines of “In a while, I will stop you and set up a game with a different target and maybe the numbers from 1-6 or 1-7, I haven’t decided yet. You need to have a strategy so that you can quickly work out the best way of winning my game, whatever it is.” This forces students to generalise, as they are being asked to come up with a way of solving an as yet unknown case.

Regular visitors to NRICH may have noticed a few videos appearing lately. This problem is a new version of something published on the site some time ago. There is a lot of power in showing children something in silence and inviting them to make sense of it. Of course, the teacher could demonstrate what’s in the videos on the board, but by showing the videos it removes the teacher’s position as the expert who knows what’s going on – “Shall we watch this video together and see if we can make sense of it?” rather than “I know what’s going on and you have to figure it out”.

Finally I mentioned that we publish problems where technology makes a solution more accessible, for example problems where graphing software, or dynamic geometry, or using a spreadsheet makes routine calculation or graph drawing or example creation much easier, and offers routes into the problem that pencil and paper methods wouldn’t allow. Whenever we do this, we try to signpost it in the problem and the teachers’ notes. I think part of a maths education should be learning to use such tools, which is why when I published Which List Is Which? last month, I included the data in a spreadsheet for students to download and manipulate.

There are other ways of using ICT in the mathematics classroom, and other reasons for doing so, that I haven’t mentioned. In fact, I’m sure there are ways that NRICH support and promote ICT that I haven’t thought of, so I guess the comment space below would be a good place to talk about anything obvious I have missed!

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.