There’s loads of stuff about dotty grids on NRICH at the moment. I’m trying to get my head round Pinterest, and figure out what content can be shared there, so why not check out my dotty grids pinboard? I’m sure I’ll add more to it once I figure out what can go there!
At NRICH meetings, we like to devote a little bit of time to working on some maths together. Today I’d like to share the problem that m’colleague Mike presented.
Mike provided dotty paper and circular objects to draw round, and invited us to investigate the convex hull of the points contained within circles of our choice. (The convex hull is what you get when you join together the points nearest the edge of the circle without being allowed any concave bits – imagine the circle is a rubber band on a pegboard and when you let go of it it springs round the outermost points.)
Mike threw a few possible questions at us, and then let us get on with it. This is my favourite type of maths investigation; very open ended, and no compulsion to work on something that someone else finds interesting, at the expense of exploring my own avenues. I started by centring my circles on a grid point, and exploring possible shapes. After a very short while, I got distracted and wanted to know whether I could create something in GeoGebra to help me. Others were busily discussing symmetry, whether shapes with different numbers of sides were possible, what happened as circles got larger, and much more. We even talked about practical applications of the mathematical ideas, approximating circles on a square grid such as pixels on a computer screen.
Alas, the meeting was over all too soon (not something you’ll hear me say very often!) and I had to get back to other things, but I saved my GeoGebra file to explore a bit more when I have the time. And if this starting point provokes any interesting questions for you, do let me know in the comments!
My second charity shop find is the marble puzzle that some of us played with at October MathsJam.
I picked this puzzle up in a charity shop somewhere down south while waiting for my brother to have a job interview this summer. I recognised the puzzle because my other little brother had it as a child and I used to spend hours playing with it after he’d gone to bed, when I was babysitting.
The object of the game is to use the Knight’s move from chess to swap all the blue marbles with the… well… I think they’re brownish, or maybe pink? Anyway, the marbles of the other colour. According to the box, 50-55 moves is average and 45 is excellent. I seem to remember I used to be able to solve it reliably, efficiently and quickly, but having played around with it again I have forgotten all the little tricks and skills I had as a teenager.
If I get any spare time in the next twenty years or so, then implementing a computer version of this puzzle would be an interesting programming challenge. It’s probably already been done, but it’s the sort of thing I can imagine rather enjoying having a go at for myself. Meanwhile, when I get bored of using my Tower of Hanoi as a stress-reliever that lets my mind wander, I have the Knights problem to occupy my hands too.
I have a good excuse for not blogging for a while – over the summer I was finishing my Masters thesis. Now that it is handed in, I’d like to get back into the habit of blogging, so I thought I’d do a short series of posts on my habit of finding mathematical stuff in charity shops. My other half is a record collector so I spend lots of time waiting for him while he browses record racks, and I use that time looking for geeky stuff in among the bric a brac and the books. It was actually from his record collecting that I got the idea of this series of posts; on his record forum they have a ‘charity shop challenge’ where people post about cool stuff they’ve found.
I found this in one of the charity shops in Ely, near Cambridge, some time ago. It looks hand made, and cost a couple of quid.
For those unfamiliar with the old problem, this is a Tower of Hanoi puzzle. The object is to transfer all the rings from one peg to another. You can only pick up one ring at a time, and you can never place a ring on top of a smaller one.
This shows the puzzle after a few moves have been made. (How many?) Altogether, there are nine rings. I did move all the rings successfully but not all in one go. This was a great find, because I’d been familiar with the Hanoi problem for many years, but actually having a purpose-built puzzle to play with it ‘hands on’ refamiliarised me with the task. If I was introducing the problem to kids, I’d want them to have something to manipulate. When we were little, we used to do it with the brass weights that went with the kitchen scales, as they were little discs of different sizes that stacked.
Have you met the Tower of Hanoi before? Have you used it in a classroom or masterclass situation? Have you ever found anything cool, mathematical and geeky in a charity shop?
Well I guess it must seem like all I ever do is go on holiday! Actually, work and life have been so hectic that the only time my batteries are recharged enough to blog are after I’ve taken time away. Check out the Cambridge MathsJam blog to see what else I’ve been up to in the past few months.
So I had the chance last week to spend a few days in the Canadian city of Toronto and the surrounding area. Here’s some of the maths I spotted.
The very first walk we went on took us past Canada’s Walk of Fame, where I snapped this picture of a ‘star’ who lends his name to a problem that causes arguments among probability enthusiasts the world over:
No wonder our students misuse the equals sign to be a “the answer is” sign!
Moving on, while stocking up on some important groceries I noticed that in Canada you can buy cereal with magnitude and direction:
Here are some facts about Niagara falls. I wonder what it’s possible to deduce from the picture.
And here’s a view looking pretty much straight down from the CN Tower. I wonder if it’s possible to estimate the height from this picture. Visit the CN Tower Website for all your CN Tower factoids.
Well after all that I’m pretty hungry. Can you estimate the calories in this picture?
Finally I was drawn in by this basketball court.
I’m sure it’s to put mathematicians off their game by distracting them with intriguing patterns!
I’m already thinking about where to go on holiday next, once my Masters thesis is submitted. Any suggestions of maths-rich holiday locations? Any favourite maths pics from where you are?
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.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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.
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
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!
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!
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.