This is a delightful way to revise Venn Diagrams with older students: A Venn Diagram of mythical creatures.
In the past I have commented on the QI forums that the difference between “negative” and “minus” was a good one and worth keeping*. I didn’t mean it to come across as self-congratulatory psuedo-intellectualism, though I’m aware that this is how it might have come across.
Listening to Stephen Fry’s ‘podgram’ on language, it appears that Mr Fry does not agree that pointing out the difference between “less” and “fewer” is worthwhile. To him, sadly, I am a pedant, attempting to impose a sort of lingustic-conservativism on the world.
Frustratingly, I agree with the majority of what Mr Fry discusses in his podgram; which is that language is a rapidly changing thing, and that what is aberrant in one generation will become established in another. Obliquely he suggests that language is ‘evolving’ though I am scared off that word having read John Gray’s Straw Dogs, and I think rightly so. The problem with calling something ‘evolving’ is that it somehow implies progress, and I do not think that this is the case for language all of the time.
My reason for believing that “minus” and “negative” should be more clearly demarcated in language is really for its utility in mathematics classrooms, and it is there where I believe the demarcation should be expressed and preserved. There are two separate concepts here: one an operation over two numbers to express (as a directed number) the difference between the two numbers; the other to express the direction of a number, ie whether postive or negative. They are difficult ideas, to be teased out. We tease them out poorly, and need to improve. They would be teased out more successfully if their use wasn’t so interchangeable.
If I am right, then the interchangability of “minus 5″ and “negative 5″ should not be considered part of the great evolution of language. Wittgenstein is right to an extent, when he claimed that the limits of my language are the limits of my world, and if I lose the ability to use two words for two separate concepts, I lose the ability to differentiate the concepts.
I do not wish to be a pedant, but I think that linguistic conservativism is the bathwater to a good many conceptual babies whose protection is worthwhile.
A quickie: here’s an interesting game from Canada where users have to find various interesting geometrical properties by eye and are assessed programmatically on their accuracy:
My score as about 3.03, having frustratingly crepty above an accuracy score of 3 with a shocking 9 in my final problem.
The following conversation in Metric Views catches the attention both for the interesting article and the subsequent comments.
Metric Views: Are our schools entrenching the ‘very British mess’?
The gist is that our schools reflect our current social muddle by teaching both imperial and metric measures and their relative magnitudes in school. In the article it is argued that the time and cost wasted on this is horrifying.
I have no love of imperial measures; I find it frustrating to have to remember how many pounds are in a stone, or ounces in a pound, or yards in a mile, and struggle to do so. I also find it difficult to convert between anything other than kilometers and miles. I know my weight in stone, but not in pounds, and certainly not in kilograms. I know my height in both metres and feet-and-inches. I am not sure that I can estimate volume in any unit with any degree of accuracy. It’s a horrible, muddy, confusing mess; that is undeniable.
I think my misgivings about the article are about the underlying idea that we should stop teaching both measures to achieve a feat of social engineering; by removing from the minds of the youth any conception of imperial measures, we would hasten the demise of imperial measures, which would be a Good Thing.
My difficulty is that feet-and-inches is such a good measure of height. I am 1.83m or 183cm, but neither is as satisfying as being 6′ tall, and neither is immediately conjourable in my mind. I don’t like Americans’ removal of ’stone’ as a measurement either; 13 (and a bit) stone is much easier to remember than… whatever number of kilograms or pounds I am.
Feet, inches, stones and pounds are good measures because they are useful. They give us a scale rooted in humanity and the measurement of humans, and allow us to compare ourselves with others accurately. I am not convinced that the removal of these measures in classrooms will remove their common use.
I should not be confused with someone in defence of a curriculum which monitors and assesses the knowledge of different weights and measures and their conversion. Conversion is a fairly dry arithmetical topic. However, there might be problems that involve imperial or metric measures (or even their conversion) which may contain some good mathematics. I would not want that potential to be excluded from the curriculum any more than I would want their being taught made compulsory.
My dad taught Philosophy and Sociology all of his professional life, and in his retirement continues to study and think about these subjects. He recently gave a talk about the work of John Gray to the Erasmus Darwin Society in Lichfield, Staffordshire.
John Gray is currently Professor of European Thought at the LSE, and has been an outspoken and controversial academic throughout his career. He has written about a great breadth of topics, but the thread of thought that ties his work together is his rejection of our contemporary belief in the progress of mankind.
The prepared text of my dad’s overview of Gray’s views is an excellent introductory text, with a good bibliography pointing towards further reading. I would strongly recommend this text to students as an overview of his thought.
In an ongoing email conversation within the ranks of the ATM on its purpose and voice within the uk educational establishment, one of our numbers recommended we read Lockhart’s Lament, an article posted on the website of the Mathematical Association of America by Keith Devlin.
Lockhart’s Lament is a a heartfelt plea to the beauty of mathematics, the place of mathematicians as artists, not engineers, and society’s complete miscomprehension of what mathematics actually is.
The article opens with a parody: what if society had the attitude towards music that it currently has to mathematics? Lockhart asks us to imagine a world where students learn musical theory without ever grasping what music is. In this world, students don’t hear music or feel it, it is a word used to describe a formal system, emotionless and austere. Perhaps a few get to understand, listen to and feel music when they get to university. If they try to describe their joy and amazement, people look at them blankly and conjour up memories of their tests on harmonic scales when they were at school.
For Lockhart, Mathematics is in turns the art of explanation and the music of reason. However, it is as poorly understood by modern western society as music is in his imaginary music-less world. Lockhart argues that “there is no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum.” Through standardisation and testing which puts the onus on memorisation over understanding and exploration, the subject is fundamentally undermined.
The breadth of Lockhart’s exasperation is great: from society to schools, to teachers, and universities, but most forcefully to the government and the curriculum. This, written in 2002 is ever more true. It is an unsettling prospect that the USA is further down the road of standardising the maths out of maths than we are in the UK. Perhaps, using them to see into our future we can change it. Reading this Lament strengthens my belief that we must try.
I hope you gain as much enjoyment, and as much fervour from its reading as I did.
The new Bowland Maths Website is the website of a new project which seeks to ground maths in an explorative, problem solving environment.
Bowland Mathematics seeks to develop meta-cognitive skills and promote an analytical, quantitative attitude towards problem solving. These goals are worthy, and important life skills, but they are difficult to measure cleanly. With curricula that separate the strands of mathematics in a way that encourages their their teaching to be separated also, and with testing that aims at accountability over intelligence, school mathematics has become ever more piecemeal and disconnected with reality. Bowland is an important project that seeks to reclaim some of the lost ground.
I urge, in the strongest possible terms, that anyone involved in mathematics education take this initiative seriously. I have no vested interest in the scheme, but simply I believe that it is crucial that initiatives such as this succeed and are built upon.
Below I review two media resources that are well worth a listen, for teachers, interested adults, and perhaps older students. These are not resources in themselves, but I am sure that educators will find stories and examples in these programmes that can have direct application in the classroom.
Cosmic Quest
Cosmic Quest This fabulous narrative history of human understanding of the Cosmos tells one of the greatest stories in the history of ideas. It is pleasingly compact, and easy to listen to. All the episodes are available to listen to from the BBC website.
In Our Time – Probability
Melvyn Bragg’s excellent In Our Time broadcast and podcast on probability last week was an excellent discussion of the history of probability with, among others, Prof. Marcus du Sautoy, who is always worth listening to! The podcast can be found here.
Dr Ron Knott in the Department of Mathematics at Surrey University is not a name I recognised, but reading his resume, I now realise that I have heard him talk a few times about Mathematics on Radio 4, both on Simon Singh’s 5 Numbers series, and in Melvyn Bragg’s In Our Time podcast.
I was looking for some information about exact values of trigonometric ratios, and came across his most informative site. I was extremely pleasantly surprised to discover that for some values the trigonometric functions give exact solutions in terms of phi, the golden ratio, among other information.
For example, did you know that the cosine of 27 degrees is exactly a half of the square root of (two plus the square root of (two subtract phi)). (One day when I finish writing my own equation display movies, I’ll write that out in a prettier way, Dr Knott’s website tries a little harder than I do). I love that the number 27, which clearly wants to be prime so much it tricks generations of children into thinking it is, the square root of two and the golden ratio are connected inextricably through the circle-based cosine function. Fantastic!
The whole page, indeed the whole of his site in general, is steeped in extremely interesting, and relatively accessible mathematics with Fibonacci numbers, Egyptian Fractions and so on and so forth. It’s mostly a site for KS4 and beyond (14 years old +), with most material for the older students. Some of it is not for the faint-hearted. However, it is a valuable resource for mathematicians of all hues, and well worth a look.
I recently stumbled upon Mr Barton Maths page of Essential Freebies, where I was delighted to discover Furbles was one of his essentials.
However, the real gem of the collection in my opinion is the free online PDF of a tribute to Martin Gardner, who was a spectacular mathematical puzzler, without whom the mathematical world would be much the poorer. You can download the ebook at G4G4.com.
