I must be something of a geek. I’ve lately been reading popular maths books for no other reason than for enjoyment (and to look cool on the bus). In particular, the books of Ian Stewart – I just finished “Taming the Infinite: The Story of Mathematics from the first Numbers to Chaos Theory”, and can recommend it highly.
This book is exactly as it says on the tin; a roughly chronological history of the developments of mathematics, from ancient Arabia to frontiers of modern maths. What made the book so enjoyable is the way the history of each branch of mathematics is related. Stewart doesn't flit from one dry area to another, all equations and definitions – he tells of each development through the biographies of the people who developed them, the historical and geographical context in which the discoveries were made, and – crucially – relates these discoveries to how they are applied in the real world.
One of my chief issues with the way maths is taught in schools and universities is the “abstractness” of it all. I don’t refer to the maths itself – for instance, complex analysis or higher-dimensional topography, which is pretty abstract by nature. I refer instead to the lack of context provided along with the theory, at all levels, from primary school to third level.
It starts well; in primary school we are taught the numbers, then addition and subtraction by relating them to physical things. “The number two represents two apples. If I receive another two apples, then I have four apples. Likewise, if I eat two of these apples, two are subtracted and I am back to two apples.” This is intuitive. Likewise, multiplication and division were related (to me anyway) by the concepts of lots of apples. “If a packet of apples contains four apples, and I buy four packets of these, then I now have sixteen apples.” Obviously, the simplistic storyline was soon abandoned and the theory of long multiplication and division introduced, but always in the background, what you were doing was manipulating apples. From the humble introduction you were aware of what this arcane manipulation of numbers could be used for in real life, in the real world.
It is only in secondary school that it all goes wrong really. The syllabus wades into trigonometry with nary an introduction. “If this triangle has an angle of 30 degrees, an adjacent side of length 10, what are the other two angles in the triangle if the hypotenuse has length twice the size of the opposite?” You’d be forgiven for answering: “Who cares?”
But given the three layers of context (historical, personal, applications) would, I suggest, make it much more interesting and relevant. On trigonometry, Stewart writes of the origins of this subject, Greek astronomers around 300BC used basic trigonometry to estimate, accurately enough, the distance from the Earth to the Moon. Ptolemy of Alexandria (of map-making fame), around 150AD, used trigonometry to develop a model of the planetary cycles in the night sky. The major advances in trigonometry were by Arabian and Hindu mathematicians in the first millennium AD – where they advanced trigonometric concepts to spherical geometry. These ideas came into their own around 1400AD when they were adapted for use in navigation – the Astrolabe was the principle tool by which ships set sail from Europe and were able to navigate their way to far-flung corners of the globe. Trade flourished, knowledge accumulated, slaves abducted, cultures met, and the world would never be the same again. In modern times, trigonometric principles are used in the engineering of buildings, in mobile communications technology, modern weaponry and GPS systems.
If such a brief prologue were given, by imaginative teachers, before every new topic was broached – before the nitty-gritty of the theory and the tools – then these subjects would be far easier to study. There is a joy in solving such problems, but I believe it is augmented further if the pupil is satisfied that the learning and exercises are not just for the sake of it - not just abstract logical walkabouts, like brain training - but can, and has been used in the great wide world to achieve amazing things, moral or not.
The list goes on. I’d suggest three contexts, the historical, the biographical and the applications, that should preface or underlie each branch of mathematics.
The historical:
The flourishing of maths in the Greek empire; the development of numbers and impact on commerce (without numbers there can be no economy worthy of the name – no debts, no fiat currencies, no credits); the use of maths to design Greek temples (think of the columns of the acropolis); the incredible strides made of Arabian empires in the first millennium and then the sudden stagnancy in the sciences in these regions as Islam closed the minds of its best and brightest (“algebra” is named after the Arabic phrase “Al-Jabr”); the golden age of exploration and their trigonometric navigation aids; Galileo's heliocentric theory and victimisation by the Inquisition (the pinnacle of the European dark ages); the Enlightenment – when a broad flourishing of maths began, and hasn’t yet ended – when Newton, Gauss, Leibniz all threw light on light, gravity, the language of the physical world; where maths was subsumed into design, and the Industrial Revolution was brought into being; where this design was bought into the world of weaponry, where Colt mastered his designs using Newton’s calculus, where Alfred Nobel cracked the chemical compounds needed to cause explosions; where the two World Wars used maths as a tool for more ingenious and deadly advances – the fluid dynamics of submarines, the aerodynamics of spitfires, the Brownian motion of diffusing mustard gas, the development of nuclear weaponry, the development of ever more elaborate coding and encryption systems, and the means to crack them; the twin forces of military might and economic girth heralding the era of the superpowers; Einstein’s relativity and the era of space exploration; the maths of medicine cracked polio, partial differential equations are used to model the spreads of epidemics, the world saw an unheralded advance in life expectancy, the use of statistics in evaluation effects of new drugs; with increased longevity came materialism, maths-based techniques raised efficiency in production to new levels, sustaining (most of) the word’s population in plenty and concurrent environmental destruction; the Information revolution, buttressed by the physical sciences, arguably making the world smaller, the end of the era of the nation…
The biography:
Talk of Pythagoras, the first expressed belief that numbers and maths are the language of the universe; the Pythagoreans who believed they had found a rigorous mathematical theory of the universe – one of these, Hippasus, discovered the existence of the square root of two which completely destroyed their theory – he announced is discovery to them on a boat sailing across the Mediterranean, they were so incensed they threw him overboard and he drowned; Archimedes, foiling the Roman army almost singlehandedly (using his “Law of the Lever” he invented the catapult, he also used the geometry of optical reflection to direct the suns rays onto Roman boats so that they caught fire); running through the streets naked shouting “Eureka!” when he discovered his method for determining the volume of irregular shaped objects in the bath; talk of Hypatia of Alexandria (~400AD), the first great female mathematician – depressingly she was inevitably accused of witchcraft and was murdered by a mob of Christians, wielding, strangely, oyster shells, with which they hacked her to death; Leonardo da Vinci, who introduced the base-10 Arabic numerical system to Europe (in use today*) and, among dozens of ingenious inventions, somehow invented the helicopter and contact lenses back in 1200; the intense unpleasantness of Isaac Newton, and his unsurpassed genius; the eight successive generations of the Swiss Bernoulli family, all mathematicians of note; Johann Kepler, who refined Copernicus's idea that the Earth revolves around the sun, and who’s mother was accused of being a witch, but was released because the authorities didn’t follow the correct procedures for torture; Galileo, who proved Copernicus's helio-centric theory, and who died under house arrest imposed by the Inquisition; Sofia Vasilyevna Kosalevskaya, another genius and female, whose father – in order to save money – wallpapered her nursery with pages from old calculus books, who taught herself calculus from her bedroom walls, who was never admitted to any university to study because of her gender, who entered an essay to a competition of the Academy of Sciences in 1886 and won, the jury finding the essay so brilliant that they increased the prize money; the artist Maurits Escher, who created astonishing art from his studies of hyperbolic geometry and studies of infinity, and was referenced by the Flight of the Conchords in a song “Inner City Pressure” - “Your perspective’s all messed up, like a painting by Escher”; Andrew Wiles, who proved the four hundred year old problem known as “Fermat’s Last Theorem”; the Irish mathematician, William Rowan Hamilton, who, walking with his wife along the Royal canal in Dublin in 1843, had the sudden inspiration for “Quaternions”, four dimensional complex numbers, and immediately carved the his profound equation into the stone of Brougham Bridge (still there today) – his inspiration (and tens years of subsequent perspiration) ultimately led to modern computer games, among other things; Godel, whose Incompleteness Theorem changed the philosophical discourse on maths, whose mentor Mortitz Schlick was murdered by a student for being Jewish (Germany, 1930s), who left Nazi Germany for the USA, who had two breakdowns, and in a terminal paranoia about being poisoned, starved himself to death…
Finally, the most interesting context, the applications:
Trigonometry – navigation, GPS, mobile phones.
Number theory – commerce, currency, debt, credit, trade, codes and cryptography (modular numbers).
Geometry – industrial design, astronomy, CGI graphics, 3-D cinema, art (the development of portraits in Renaissance Italy was founded upon a study of the geometries of the human body).
Logic – Boolean logic in internet search engines, computer programming languages, the scientific method
Algebra – everything! Apropos Stewart: “Algebra…is the mathematics of symbolic expressions…equations can be solved to represent unknown quantities in terms of known ones.” As such we use algebra in every facet of science, it is the fundamental tool in manipulating the world.
Calculus – the design of airplanes and spacecraft, design of bridges, meteorological predictions, design of formula one cars, studies of populations, construction of bridges, computer aided design, nuclear power and weapons.
Probability – winning at poker (game theory), avoiding nuclear holocausts (game theory again), medical trials of new drugs (statistics is a branch of probability), social sciences and anthropology (statistics again)
This is just the three contexts I'd suggest for teaching maths in secondary school. Maths is a human-invented tool, not unlike language, for manipulating the physical world and advancing our understanding of the universe and our place in it. We do not teach languages (say English) by focusing relentlessly on the tools of language – vowels, syllables, letters, syntax, verbal constructs – while important, we focus on studying great works in the various branches of language (drama, essays, novels, poetry) and are encouraged to use the tools of language to comment upon or create our own such works. While the tools are important, it is always in the penumbra of the great human achievements that we study these. Maths should be no different.
It should be more applied, with more context given (in the areas I suggest above – historical, biographical and areas of application). Having read Ian Stewart’s book and others like it, I am convinced that the development of mathematics is inextricably bound with the development of civilisation. To this student of maths, this was unfortunately never conveyed to me or my classmates.
Some links:
Scary stuff
www.irishexaminer.com/ireland/half-of-maths-teachers-unqualified-112420.html
Mind-bending paintings by Escher:
http://en.wikipedia.org/wiki/File:Escher%27s_Relativity.jpg
http://en.wikipedia.org/wiki/Base_10#Alternative_bases
* The Base-10 Hindu-Arabic number system (representing all numbers with 10 numerals, 0,1,2,3,4,5,6,7,8,9) that Leonardo popularised seems very natural to us now, indeed it seems inevitable; but it wasn’t. Base-10 is thought to have developed, obviously enough, because we have ten fingers and used to count on our fingers. It is thought that the Maya civilisations used base-20, counting on fingers and toes. Other civilisations had different bases, 2, 4, 5 and 12 being prevalent. The Babylonians used base-60, and this is why 60 is such a prevalent number in mathematics today – 60 minutes in an hour, 60 seconds in a minute, 360 degrees in a circle… Indeed it is sheer coincidence that the base-10 prevailed, although it does seem the most efficient counting system but that might be wisdom after the event. I often wondered (as I said, I may be a geek) whether there were any patriarchal societies with base-21 counting systems, counting on fingers, toes and another appendage?
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