September 27th, 2007
Daniel Tammet has an extraordinary relationship with numbers. Born with Asperger’s Syndrome, Daniel has an extremely kinaesthetic relationship with number, and has a host of other impressive mental abilities, such as his extreme aptitude for learning languages rapidly. His book Born on a Blue Day is a wonderful memoir of his life to this point, and well worth a look both because it is a great read, and because it offers an interesting insight into Asperger’s Syndrome.
Mathematically however, it is even more interesting. Daniel suffered at school because his mind was creating associations between numbers and other concepts, and between the numbers themselves in creative and unexpected ways. He and his parents had the courage – or perhaps just the necessity – to persist in working in and with those associations. They have served him well; his numerical mental dexterity is far beyond what almost anyone else could muster.
Schools are inevitably ‘one size fits all’ institutions to some extent. Schools are mass-education institutions; it is not possible to ‘personalise learning’ precisely for individual students. That is not to say that the Personalised Learning initiatives from the government are bad things, far from it; merely that because often a teacher is dealing with about 30 students, it is not possible to tailor learning exactly to the needs of every student.
In mathematics classrooms, the tendency to make uniform what is not becomes more apparent. In maths teaching there is a tendency towards drill and rote. It can be seen clearly in textbooks, and it is confirmed by asking any reasonable cross-section of society to recount their experiences of maths at school.
It would be absurd to argue that allowing students more freedom to explore the associations between numbers and to create their own understandings of them will most students become as adept at maths as Daniel. Nevertheless the root of his understanding of number comes across as creatively based rather than based in algorithms and routines. To me, Daniel’s story is compelling evidence in support of the theory that creativity and imagination are central to the development of young mathematicians.
As a foot-note, after reading Daniel’s book I began to reflect on the ways in which I perceive numbers. This directly led me to expand my own concept of the structure of numbers by creating the Primitives concept.
September 22nd, 2007
BBC News today reports that "Online marking systems ‘faulty’". Prima facie evidence is that with the growth of online marking there has been a corresponding increase of complaints about grades from teachers.
Exams are an inexact method of assessing students’ abilities. Teachers know their students thoroughly and are able to gauge with a very good degree of accuracy how they should perform in exams. When differences between expected performance and actual performance start become too widespread, then there is a problem with the examination system.
John Bangs, head of the NUT is reported as saying "They are not able to annotate the scripts by hand, there’s a time constraint and you can’t take into account youngsters who do quite a lot of writing and don’t fill in the standard box that online marking demands. So legitimately there’s a question whether or not online marking is missing some of the achievements of youngsters." There is also reported a trend towards less well paid, less well qualified examiners.
Technology has a worrying tendency to make things more uniform than they might otherwise be. Marking an exam can be a complex business, and it seems reasonable to contend that someone whose performance is good but unusual could be at a disadvantage in the new marking regime.
September 20th, 2007
Scientific-Computing.Com’s recent blog articles "Beyond the Prisoner’s Dilemma" and "Global warming and the Prisoner’s Dilemma" are interesting examples of using the logical structures of game theory as starting points for logical thought at younger levels.
The blog conversation starts with the big environmental issue of global warming and this friendly video where the protagonist explains the application of minimax to the global warming debate. He argues, in a nutshell, that given uncertain future consequences of humanity’s impact on the planet, and given also a choice of decisions about how to act against those potential consequences, it makes the most logical sense to exclude the catastrophic choice of doing nothing to prevent global warming by acting as if global warming were a certainty.
I like how the minimax principle here facilitates students’ understanding by offering a powerful structure for considering different possibilities. Because minimax is so clearly structured and relatively easy to grasp, it is the sort of idea, like the Venn Diagram, which ought to be a constantly recurring feature of students’ education. In Mathematics, it bears close resemblance to the Carroll diagram, which is a similarly undervalued structure for understanding issues.
Primitives represents numbers in terms of their prime factors, offering unusual insights into their structure. Numbers are presented as nested sets of small black dots. Three is presented as a blue circle enclosing three dots; a ‘set of three’. Six is presented as a set of three sets of two dots, or as a set of two sets of three dots.
When a number has many different prime factors, such as 30 = 3·2·5, the factors can be rearranged to offer different images of the same number.
Primitives is tailored for classroom use and perfectly suited for use with an interactive whiteboard. Hopefully older primary students and secondary students will find this application interesting.
Primitives is companion software to an article published in March 2008’s Mathematics Teacher Magazine.
Posters based on the Primitives software are now available from the ATM online shop.
Grid primitives combines the ‘Primitive’ visualisations with the Sieve of Eratosthenes. The Primitive visualisations of the first 42 numbers are arranged into a six column grid that is often used when finding prime numbers by elimination using the Sieve of Eratosthenes algorithm. Click on the image or here for a larger image.
The visualisations are reduced in size to conform to the size of the grid, and behind them the pattern of multiples of 2, 3, 5 and 7 are each highlighted in the colour corresponding to the respective prime number. The prime numbers are particularly highlighted by an 11-pointed star behind each one.
The choice of a six column structure may lead students to question why after the bottom row, primes only appear in the first and fifth columns, or to put it mathematically, why all primes other than 2 and 3 are in the pattern 6n±1, where n is a positive integer, but that not all such numbers are prime. In each column, 25 and 35 disrupt the apparent pattern of primes. Studying this may lead students to futher questions about prime numbers for further study.
September 19th, 2007
Firaxis Games, the makers of one of the greats of computer gaming Civilisation, discuss on their website the growing trend for computer games to be used in the educational arena. It is encouraging that educators are starting to understand the potential of technology to educate, though I suspect that the use of commercial games as educational tools is an transitional step before bespoke educational games begin to be produced with production values that begin to approach those of commercial games.
One of Firaxis’ contributors Kurt Squire proposes Civilisation as a good model for learning about World History. There is an interesting tension here. On the one hand, a game like Civilisation engages students in such a way that they build a sophisticated model of the game in order to succeed at it. That is good educationally to the extent to which the game models genuine historical processes. It is not clear that the ‘history’ that Civilisation presents is particularly convincing.
While Kurt Squire argues that Civilisation “represents world history not as a story of colonial domination or western expansion, but as an emergent process arising from overlapping, interrelated factors”, it does still give an essentially American – or at least New World – view of history. Land is virgin territory until moved into by the great civilisations; pre-colonial Afrians, native Americans, native Australian Aborigones do not have a story. Intellectual and technological progress happens linearly; the Middle ages and the loss of Roman and Greek learning cannot happen. There is no potential for a European type of historio-political scenario; states are the size of continents.
On the other hand, if one ignores the problems with the historical model, it does offer a ‘big picture view’ of history. Could such a grand model of historical processes be so readily expressed without the means of technology? Certainly the answer is yes, though it would take an extremely talented teacher, and those are notoriously thin on the ground.
The pipedream is for someone to create a game with production values on a par with Civilisation, but which takes as its starting point an historical model that aims at accuracy. This of course, is rather like desiring an historically accurate documentary that looks and sounds like a Hollywood movie, but there will surely be moves towards higher production values in educational software in the future.
September 15th, 2007
The Independent quotes Dr Kelly on the SATs sat by every student at 7, 11 and 14: "Testing every child has, overall, a negative on the learning outcomes and attitudes of children. Repeated practice tests reinforced the low self-image of the lower-achieving students. The feedback from teachers often hurts children’s feelings rather than helping them understand their weaknesses. Children often responded by reducing their efforts towards further learning and focussing on performance in tests."
It is encouraging to hear Dr Kelly say this, which seemed apparent to me and to many of my colleagues as soon as we started training to teach. It is important that he says it too. The more high profile criticisms of the current testing regime, the more the government will have to listen.
The current testing regime comes from an array of misguided notions about the appropriate means of assessing schools, teachers and students. It is easy to think that those who argue against the current regime are against testing per se. Not so; the criticism of the testing regime for younger students does not imply a criticism of the idea of testing at 16 and 18.
Richard Garner disputes Dr Kelly’s argument that ‘rising achievement’ can be explained by examinations having been made easier. Garner argues that the rise in perceived achievement is explained by teachers becoming ever more adept at coaching for exams. I’m inclined to think that a little of each is more or less right. I discuss a similar idea in my post Playing Politics with Education.
Dr Kelly has a host of other things to discuss in Making Minds too, from the hours that schools open to the ages at which language acquisition is most acute. It is encouraging that research into neurological science is starting to have an impact on educational thinking. I will post more on these ideas once I’ve got my copy!
Bird Idol is an interactive song building program that was designed by Dr Karen Spencer (University of St Andrews) and Dr George Lovell (University of St Andrews). The game allows you to build a song out of 9 different sub-units (also called syllables) and pretend you are a male bird. These syllables have been recorded from real birds and players get the chance to construct a real canary song. Players can compete against one another for the affections of a female canary. In order to win the player must choose a song that best attracts the female. This can be achieved by choosing song characteristics that reveal the quality of the male, specifically, a selection of syllables that are complex and energetically demanding to perform.
This program was designed to introduce the basic concepts of pitch, frequency and song complexity to younger students. Older students have found this useful in understanding the ideas behind sexual selection and the evolution of mate choice. It is aimed at students and teachers alike, and was showcased at the Charles Darwin Award Lecture given by Dr Spencer as part of the British Association for the Advancement of Science’s National Festival of Science in York (9th – 15th September 2007).
Bird Idol was featured on BBC Radio 4’s The Material World on 13th September. Ahead of her lecture, Dr Spencer spoke to Quentin Cooper about her research, and played Bird Idol to illustrate the point!
The ptolemy.co.uk implementation added rich graphical content, animation, and an interactive interface design in a fun and lively way, in order to capture and retain students’ attention and to add humour.
Instructions for Use
A set of simple instructions are availble in PDF format in the Bird Idol Instruction Manual. These include both a getting started guide and some clues as to how to maximise your male canary sexiness!
If you foresee situations in which you may want to use Bird Idol in a situation where there is no internet connection, then you can download the AIR version.
If you have never installed an AIR application before, you will need to first install the Adobe AIR Platform. This only needs to be done once for all Adobe AIR applications. Once the the AIR platform is installed, you can open the downloaded file BirdIdol.air, and you will then be presented with the dialog on the right.
At the time of publishing, Adobe AIR was extremely new and publishers could not be ‘verified’. When installing, Adobe AIR warns users that the application may access your internet or hard drive. I can assure you that Bird Idol does neither.
September 10th, 2007
Today Ed Balls, Secretary of State for the Department for Children, Schools and Families announced a new ‘national consultation’ called Our Children, Time to Talk.
Am I wrong to be cynical? While I am sure that there is much good that can be done by the government for the children of this country, I don’t believe that they will take on board ideas about reforming the educational system.
When it was mooted that A-Levels be abandoned in favour of a Baccalaureate system, one headline from the Daily Mail saw the policy publicly abandoned. The adoption of a Baccalaureate was a progressive, exciting proposal worth serious consideration. It was abandoned at the first sign of reactionary scepticism. Hockerill Anglo-European College is a model of how adoption of the International Baccalaureate over A-Levels can dramatically inprove academic achievement.
I can’t help having the impression that when a government says it “wants to listen”, it already has a very strong idea about what it wants to hear. I have no doubt that the government will listen to anyone with a safe, mediocre idea that can be sold as change. I really hope that I am proved wrong.
September 3rd, 2007
The government is upset that improvement in educational achievement is stalling. The Conservatives are considering proposals to ‘hold children back’ at primary school if they fail to achieve a particular standard. Neither party takes any serious notice of what serious educationalists say. Will politicians ever realise that playing politics with education serves only to undermine the system?
Since the introduction of SATs, primary schools in England are measured by their KS2 results. Though in the first instance it was never meant to be this way, increasingly so too are the students. So worried are schools about those tests that almost all of Year 6 is given over to preparation for tests. In those years where SATs are not taken, teachers take a very different approach to educating youngsters. In Year 6, they essentially become exam factories. In KS3 at secondary school, a similar pattern emerges. In years 7 and 8 students and teachers are ‘free’ to concentrate on learning, but in Year 9 everything is again given over to SAT preparation. It is not unreasonable to argue that in Year 6 and Year 9 are probably the years in which students actually learn the least. Learning should not be identified with recall and with ability to understand exam questions, particularly at such a young age.
“Improvement in educational achievement is stalling”. When new educational systems and testing regimes are introduced, it takes teachers some time to work it all out. When tests are introduced, it takes teachers a few years to really understand how best to prepare their students for those tests. During that period, the quality of test preparation goes up and up from a low initial level. The government can point to this and call it ‘improvement’ in the standards of education. Eventually of course, it will level off. Suddenly, there is no more improvement, and the system is in crisis. The government’s response? A new educational system or testing regime is needed! It would be funny, if it wasn’t playing politics with our nation’s education system.
The most worrying aspect of the Government’s approach to education is that they appear to understand education only through the lens of testing.
“If students are failing, hold them back”. Listening to radio 4 this morning, every educationalist agreed that the Conservative party have not taken any serious educational advice on this issue. The Conservative’s response to a failing student in Year 6? To make them go through another year in which learning is put on hold in order to improve performance in tests, despite it having clearly not worked the first time around! There are other problems with the Conservatives’ policy too, such as labelling students as failures and the logistical nightmare of requiring more primary school spaces to cope with the numbers held back, but for me the biggest problem is that it shows that the Conservatives too understand education only through the lens of testing.
In economics, in 1997 a newly elected Labour government was the first to realise that the way to ensure fiscal stability was to divest the responsibility to decide baseline interest rates away from the political arena. Let the Bank of England decide on the correct rate. By ceding power the government made the economy more stable and more effective. There was no more political squabbling about the level of the interest rate. In education, while England’s politically driven education system moves from crisis to crisis, no similar level of discontent is felt north of the border, where education is essentially governed by the General Teaching Council of Scotland.
Will politicians ever come to understand that playing politics with education might actually be a bad thing?