You're reading: Columns
Geometric nightclub by Fred Mafra
Hints about Relatively Prime
Although we all know Samuel Hansen will do a fantastic job telling stories behind mathematics, provided you donate some money to help him do so, we don’t entirely know what those stories will be. Here are a list of tweets I’ve seen on Samuel’s Twitter stream about this, giving hints.
help me tell people how crickets led to a better understanding of Kevin Bacon through math
Why does it seems that 20% of your friends get you 80% of your news?
Can slime molds make Steiner trees?
Networks are the basis of our social lives & what I’m trying to leverage to support my Kickstarter Relatively Prime
True story: It take 362 pages to prove 1+1=2. Find out more by Supporting the Relatively Prime Kickstarter.
Can a war over a math discovery make a country to fall years behind in science?
Ever wonder what how a river and logic could both lead to the same mathematical discovery?
Wondering if you can musically represent a function? Support Relatively Prime and I will have the chance to answer
Parallel lines intersect at exactly two points! Find out why this is true by supporting Relatively Prime.
The shape of the Internet is hyperbolic. And if you want to know why support my kickstarter Relatively Prime
Plus there’s the original hint on the Kickstarter page:
is it true that you are only 7 seven handshakes from the President, what exactly is a micromort, and how did 39 people commenting on a blog manage to prove a deep theorem … With each episode structured around topics such as: The Shape of Things, Risk, and Calculus Wars, Relatively Prime will illuminate each area by delving into the history, applications, and people that underlie the subject that is the foundation of all science
So if you haven’t got the message yet, find out about these stories by supporting Relatively Prime. My previous post focused on ways you can help as well as donating, including a ready-written blog post for you to put on your blog.
Relatively Prime is failing; what you can do to help
Trying to study mathematics without the human stories is like reading a typed transcript of a Rolling Stones concert. The Relatively Prime project will throw the mathematics television out of the hotel window.
On Friday as I was going to bed I sent a message to Samuel Hansen, try to think of anything we could do to promote Relatively Prime on the Math/Maths podcast this week. By Saturday he had five people who had sent him audio recordings of their reasons for supporting, which he had edited into a 1 min advert for the Kickstarter fundraising project (the quote at the top of this article is from one of those funders). We then had a chat in which I asked Samuel about the project, what sorts of stories he was going to tell, what made it different from other podcasts he does. The result is in the latest Math/Maths Podcast 57 and has been released as an 8 minute audio piece through the acmescience podcasts. Listen to the funders’ reasons and our conversation here.
I’ve become quite vexed with the process of promoting this. If you look at Samuel’s twitter stream you can see him tweeting intriguing questions that will be answered by Relatively Prime. A sample of three:
Can a war over a math discovery make a country to fall years behind in science?…Can slime molds make Steiner trees? I’ll tell you if you support my Kickstarter Relatively Prime…help me tell people how crickets led to a better understanding of Kevin Bacon through math
I thought this was a neat idea and ReTweeted these when I saw them. This, and other marketing Samuel is doing, is attracting interest and, as the graph of donations, cumulative amount and time shows, there is a steady increase but it simply isn’t increasing quickly enough (click to enlarge).
It amazes me that 93 people have so far donated $3,793, but we have a problem. If the whole amount isn’t raised, all donations are cancelled. That’s the way Kickstarter works. So it’s a question of reaching new audiences. Samuel can keep tweeting, and I can keep tweeting on his behalf, but the message isn’t getting further out. I already posted on this blog about the project – “Why I supported Relatively Prime and you should too” – and according to Blogger some 250 people have viewed that post. So I’m running out of new audiences to reach.
This is where you come in. If you haven’t donated to Relatively Prime, please consider chipping in some money. Small amounts sum to larger amounts, so even a small amount will help. There are only eight days to go.
If you have donated, please have a think – is there anything you can do to help promote the project. Post a message on whichever social networks you use. Put a message on your blog. It doesn’t have to be as elaborate as this one! Here’s some text you could use for a quick blog post:
Relatively Prime: Stories from the Mathematical Domain
Relatively Prime will be an 8 episode audio podcast featuring stories from the world of mathematics. Tackling questions like: is it true that you are only 7 seven handshakes from the President, what exactly is a micromort, and how did 39 people commenting on a blog manage to prove a deep theorem. Relatively Prime will feature interviews with leaders of mathematics, as well as the unsung foot soldiers that push the mathematical machine forward. With each episode structured around topics such as: The Shape of Things, Risk, and Calculus Wars, Relatively Prime will illuminate each area by delving into the history, applications, and people that underlie the subject that is the foundation of all science.
I think this could really be an amazing project, but it can only happen with your support. So please, if you can, support it financially, or please twitter, tumblr, reddit, blog, or any other thing about it – you can use the nice link http://bit.ly/relprime
Please just take that block of text – title and 2 paragraphs – and paste it on your own blog. It should only take you seconds and by doing so you will help break the message out of the same circles and reach new, interested people.
Plus, if you have any ability to get something written or an audio interview released though any sort of outlet in the next week please contact Samuel Hansen and give him the opportunity to talk about his project. You get some interesting content for your podcast/radio show/magazine/whatever and you’ll be helping Relatively Prime.
The unplanned impact of mathematics
Time and again, pure mathematics displays an astonishing quality. A piece of mathematics is developed (or discovered) by a mathematician who is, often, following his or her curiosity without a plan for meeting some identified need or application. Then, later, perhaps decades or centuries later, this mathematics fits perfectly into some need or application.
Maths at the East Midlands Big Bang Fair
Recently I was invited to take a mathematical puzzles stall to the East Midlands Big Bang science fair. This took place in Nottingham yesterday. I gathered a few friends from the Nottingham MathsJam group, which I run, and we planned what we could do with a stall. We agreed a list of puzzles we could put together and run. We felt it was important to have solid, physical puzzles and games that would attract people to the stall, including making use of the floor area, as well as more advanced and intriguing items and a takeaway sheet. I wanted the takeaway sheet to provide some advice on problem solving techniques as well as some puzzles to try. There were various extra constraints as well as what we could physically make with no budget, including the difficulty of catering to the wide age range of those attending: 9 to 19!
We met a couple of weekends ago and agreed a set of puzzles, tried them on fellow MathsJammers at the monthly meeting last week and have spent the last week or so making bits and pieces ready for the fair yesterday (particular thanks in that regard are due to John Read and Kathryn Taylor). We called the stall “Solving it like a mathematician”. For big, attractive, fun we had Latin squares with giant playing cards, a puzzle involving arranging tokens inside a giant circle (a hula hoop) and matchstick puzzles with giant matchsticks (bamboo canes). For hands on activity we were making Möbius strips. The more in depth tabletop exercises included: Buffon’s needle for estimating pi (we got 3.78 from 141 throws); a ‘wisdom of crowds’ guessing how much rice is in the jar and rice on the chessboard exponential growth combo; and, the fifteen puzzle and how to tell if an arbitrary position can be solved. Each puzzle had an advice sheet and these as well as the handout are available on a page on my website.
I have been unwell recently so I took a lighter load than I might have for the day. I helped set up the stall and stayed for the first hour, in which not much was happening, then left until the afternoon. Here is a picture of the stall, ready to go but sans visitors.
After the first hour, I left the stall in the capable hands of John Read and Ian Peatfield for the morning. We had agreed a kind of shift system – I didn’t want everyone arriving first thing and us all getting tired mid-afternoon. I went and found a cafe for a quiet read. When I returned after lunch Ian had finished his stint, Alex Corner and Noel France had joined John, and the stall was abuzz! Here is a photo.
Apart from the combination of bamboo cane ‘matches’ and plastic plate ‘coins’ for some of the oversized puzzles leading to a plate spinning class, everything was going as planned. Soon we were joined by Kathryn Taylor and the five of us spent the afternoon rushing around after wave-upon-wave of pupils. That every few minutes another pupil was dragged away from the stall, “put that down now, we’ve got to leave”, by their teachers was, I think, a sign of success. Here’s one more picture from the afternoon.
Overall, I am very pleased with the stall we made and the team who ran it. My first science fair and a very pleasing experience indeed. I hope some of our visitors saw some interest in mathematics and the couple of hundred who took the advice sheet might learn something about approaching problems. Now, to find somewhere to store my new ‘puzzles stall kit’ for next time!
Congratulations should go to David Ault and his team for organising the fair which, as far as I can tell, went very smoothly.
Developing mathematical thinking – a generational problem?
We were sent a link to a blog post by Katie Steckles for the Math/Maths Podcast a couple of weeks ago. I’m preparing for the recording of episode 52 in a few hours and I thought I would share my thoughts on the topic here.
The blog post quotes another, ‘The Mathematics Generation Gap‘. This starts with “Profs do not know how their students were taught mathematics, what their students know, what their students don’t know – and have no idea how to help their students bridge those gaps.” This makes me think of the document written by MEI and published by my employer with others, “Understanding the UK Mathematics Curriculum Pre-Higher Education – a guide for Academic Members of Staff“. The problem this looks to address is that “it is not always clear what mathematics content, methods and processes students will have studied (or indeed can be expected to know and understand) as they commence their university-level programmes”.
However, the main thrust of the article is on what is called “The arithmetic gap”: “profs over a certain age (and some immigrant profs) were drilled in mental math;… students under a certain age haven’t been. Some implications of the arithmetic gap are familiar: profs who can’t understand why students insist on using calculators; students who can’t understand why their profs are so unreasonable. …” The article goes on to talk about analogue clocks and even Google Maps as forming a difference in understanding and approach between students and their professors.
The blog post Katie sent a link to, titled ‘“The Mathematics Generation Gap”‘, talks about “mental arithmetic tricks”. I don’t want to quote the whole thing here and stop you going to read the other post so I’ll take out a lot of the detail (…), but it gives an example: “to multiply any single digit number by nine, just add a zero to the end and subtract the number… Then, it’s easy to generalize, 9 times any two digit number is the number with a zero attached minus the number… Then extend further … This can be generalized further… This also leads directly to the proof…” Then we come to the main argument:
How do you discover this rule, and learn how to take it to a proof, without rote exercises that force you to search for shortcuts? I understand that the response to all of the above is to use a calculator instead, these tricks aren’t needed if you have a calculator at hand, but that isn’t the point. The point is that these exercises lead to additional insights, proofs, etc. and those insights are critical for more advanced insights and more complex proofs.
The inductive type reasoning that emerges from these exercises is valuable in many settings — I’d guess learning to find patterns is a skill that is useful beyond pure mathematics — and I worry that an over reliance on calculators will erode the development of these skills. I am absolutely convinced, for example, that forcing people to do econometric and statistical exercises by hand develops intuition that you cannot get any other way, and this is a key to moving on to doing proofs.
A related area is whether to allow use of computers for solving advanced mathematics. At work in January we ran the HE Mathematics Curriculum Summit, the report of which is now available. This included a debate on, basically, whether students should be expected to use memory, acquire subject knowledge and demonstrate technical fluency, or whether the computation part of mathematics could be left to computers, leaving the students to worry about when and why a particular calculation is used. However, the compelling arguments for me of students performing mathematics by hand there lay in understanding what a computer would be doing and what its limitations would be, whereas the arguments in the blog post seem to be that performing mental arithmetic develops other skills that a mathematical thinker ought to have.
What is my view? Certainly the point isn’t finding the numbers; if it were a calculator or computer can be used for certainty. Having said that, there are other areas of mathematics that are well suited to developing this mathematical thinking. I appreciate the desire to encourage pattern searching, logical reasoning, abstraction and extension, but I’m not sure forcing students who haven’t been brought up on mental arithmetic to do such tricks is a productive way of doing so. If everybody has a phone or calculator in their pocket that can solve the question in a millisecond, then forcing them to not use that device and do it by some mental trick instead is just going to put people off, I would say. Beyond this, a lot of people have a genuine anxiety, or some even a disability that can produce a panicked reaction when faced with numbers. Doing something in a non-numerical area might be much more effective. Tilings seem to be a good option, and at work we are running a workshop at Greenwich in a couple of weeks, led by Noel-Ann Bradshaw and at which Katie is a presenter, on using problems, puzzles and games to develop mathematical thinking. Areas such as these can be used to develop the same skills but don’t have the hangups of mental arithmetic. In fact, I have a group of people coming round this afternoon to plan our stall at a local science fair in a couple of weeks. I intend our stall to be themed around using puzzles for developing problem solving skills. Beyond this, mental arithmetic forms part of a number of magic tricks for which a calculator would give the game away, so perhaps encouraging students to play around with this sort of thing may give a motivation to learn some mental arithmetic tricks. (Of course, this all depends what topic you are trying to teach.)
Overall, I think the battle is lost – the distinction between profs and students is not as clear as this article would have it because plenty of (and increasingly many) lecturers will have been brought up on calculators as well. I agree there are differences between how lecturers and students approach mathematics, some of which will be generational due to the increasing availability of technology; some will be due to the lecturers being unusual (perhaps more capable and motivated than average) students in their day. Still, if the aim is to develop a mathematical topic, using modern tools to make this more efficient is a good thing; if the aim is develop mathematical thinking I think there are more interesting approaches for developing the kinds of skills the blog post author would like to develop.
The blog post ends: “But what is your view on all of this?” Katie has sent me her view for the podcast and she may choose to repeat it in the comments but because I have been sent it for one purpose I don’t feel I should copy it out here. Perhaps you will share your views in the comments.