We live in interesting times; a good thing—so far at least. One of the momentous waves of change which is just now starting to roll over universities and academics around the world is a whole new online way of learning, accessible from essentially anywhere, for free. This will have a deep and profound effect on academic life. The Australian recently ran an article on this development, featuring my friend and colleague Chris Tisdell (Google: Tisdell, seismic shift, education to access the article). 

Increasingly you can log onto YouTube, or iTunes U, or other repositories, and start learning about anything you want. While in many areas the offerings are still in a scattered and embryonic form, the amount of material and resources is increasing exponentially, and the process seems clearly irreversible. More organized courses called MOOCS are using platforms such as Coursera, EdX, Udacity and others to train tens of thousands of students (how successfully is still a question). Other platforms are being established as you read this.

No amount of feet dragging by academics, textbook publishers, college administrators, and other entrenched interests will likely stop this trend. The reality is that universities as sole repositories of high-end knowledge and learning is coming to an end. Academics like myself will have to adapt or be prepared to go the way of the harness and carriage-makers a hundred years ago with the advent of the motor car. The lesson is clear: change, or be made irrelevant.

Right now, I have about four tutorials a week in first year Linear Algebra or Calculus. I find myself saying the same things as the lecturer in the next room, and that I have said dozens of times in the past. The same scenario is repeated with little variation in thousands of colleges around the world. This is an unsustainable situation, much as perhaps we would like it to continue, as our jobs largely depend on it.  The reality is that having thousands of essentially identical first year tutorials/classes around the world on, say, “the derivatives of inverse hyperbolic functions”, or “how to apply the normal distribution” is increasingly a situation approaching its use-by date. Clearly it is vastly more logical and practical for a few people to develop the lessons really well, and put them on YouTube for anyone, anywhere to watch whenever they feel like it. Once that happens, and students can access easily the information they need, thousands of academic jobs almost immediately become redundant.

The teaching role of universities, especially for large popular subjects, will inevitably change from providing primarily learning content to providing primarily assessment, support and certification. People will pay to get a certificate of achievement. They will no longer be so willing to pay to get instruction that they can easily get for free online. No doubt there is a social aspect of going to university; meeting other young people, playing cards or soccer during lunch hours, and chatting to your university lecturers. Attending a class can be a positive experience. But it can also be rather lukewarm: some college level lecturers are not stellar teachers, have ordinary communication skills and little real training in education. Once the choice between a mediocre live lecture and a high production video with powerful graphics and an entertaining dynamic expositor is available, I think we all know where most students will go. The core idea that universities and colleges primarily provide instruction, and rather high priced instruction at that, has to change.

Many of my academic colleagues will, quite understandably, be upset at this development. My own efforts at posting lots of mathematics videos online at YouTube (my channel is called Insights into Mathematics, user: njwildberger, check it out!), along with those of Chris Tisdell (his channel is called Understand Mathematics, user: DrChris Tisdell, check it out!) are seen by some of our colleagues as competitive with the traditional lecture format. But the reality is that the changes that are coming are made inevitable by the technology at hand; the question is only whether one is willing to embrace them and move forward on the train, or stand still like deer in the headlights. We see ourselves as potential bridges to the future: establishing UNSW as a key contributor in providing quality mathematics instruction to the world, along with more established and well-funded players like MIT, Stanford etc.

For centuries universities have been elite institutions catering to the sons of the rich to ready them for positions of power and privilege. In the twentieth century that scenario gradually expanded, allowing first women and then more and more middle class and even working class students into the college and university framework. While this has been a great contributor to the rise in equality in the Western world, still most of the rest of the world was excluded from the process, as the high-end educational institutions were concentrated in mostly well-to-do Western countries. The current technology supports a massive expansion of knowledge into the third world, as well as empowering ordinary people, young and old, rich or poor, to learn, learn, learn, as long as they want to! It will be one of the really big game-changers in the brave new world of tomorrow. Education is a killer application for the internet.

Every so often my thoughts turn to the curious economic situation we have built for ourselves here in the Western world, over the last century or two. The greatest building and economic boom in history has been hurtling along for some time now, but there are some serious issues that are starting to become difficult to ignore, even though we might like to. Most of these revolve around a four letter word which is the cure to all evils to some, and the cause of all evils to others: debt.

Around two hundred years ago, governments in Denmark and Germany started raising money for building projects by issuing bonds: asking people for loans which would be paid back over a long time period (say 30 years); which paid a generous rate of interest; and which were backed by the buildings themselves. Since then, the bond market has blossomed into a huge industry with many variants, but the basic principle remains constant: the public/government borrows money from rich citizens to spend on needy projects, and promises to pay back, with interest, sometime in the future. The rich citizens win–they get a sure and safe return on their money, often with a better return than they could otherwise expect from other investments. The public wins: they get shiny new buildings, roads, airports, armies, hospitals or schools, and generous pensions for civil servants. The governments win: they get credit for providing essential services, a happy populace, and money to spend promoting re-election. It seems like a win-win situation: but are there really no losers?

Unfortunately, there often are. They are called children, grand-children, and great grand-children. Boiled down to its essentials, the basic equation is this: we borrow from our children to support a lifestyle we otherwise could not afford. The beauty in this arrangement, from the point of view of the generation making it, is that the approval, or even the awareness, of the future generations who will pay for it is not required! With the the modern financial, legal and government set-up, we are able to simply legislate monies into existence. This is a modern kind of financial sleight of hand, as if from the Ministry of Magic in Harry Potter world. Need an extra few billion dollars? Wave the legislative wand, and lo!, it is there.

In time, the borrowing inevitably starts to escalate: it is all too easy. We find ourselves always requiring more money, each time to cover not just our bigger and bolder plans, but also the interest charges of our previous loans, which grow more and more onerous. Once we get used to the idea of an inevitable “national debt”, we don’t resist when the politicians jack it up yet a little bit more—in our best interests, naturally.

After a while, successive generations become simultaneously victims and beneficiaries of the system, which comes more and more to resemble a Ponzi scheme (first-in get paid, last-in lose). Each new generation, as they become conscious of the state of affairs, looks backwards with annoyance at the previous cohort for having gotten them into the mess, and then looks ahead at ways of passing the buck to the next lot. Government debt getting into stratospheric levels? Let’s just sell off a bit more of the public domain; our roads, our power systems, our hospitals, our airports, our public lands. Still not enough cash to support all the lavish entitlements and retirement plans that we all so richly deserve? Borrow more money from our ever-more-wealthy wealthiest citizens, or perhaps the Asians, or just print some more.

With the baby boomer demographic bulge moving into retirement, and life expectancies still on the rise, many commentators are now starting to realize that the pressures on the current generation of young people to keep the system going is shortly going to become acute, perhaps even unmanageable without serious alterations to expectancies.

Governments are raising retirement ages, pension plans are, in many places, starting to look wobbly (my own UniSuper superannuation scheme here in Australia has started making rumblings about not enough cash to meet future committments), and while the mainstream media maintains their reluctance to acknowledge and assess the problem fairly, more and more discontent is registering in alternate forums on the internet. The prospect looms of future action by younger people to rein in the lavish retirement packages that the baby boomers have lined up for themselves. What legislation can give, legislation can take away.

In the university sector here in Australia, I joined a few years too late to enter the “old superannuation scheme”, which overly generously allows my older colleagues to have retired in their late fifties with essentially full salaries as pensions. My cohort will have to work well into their sixties to receive possibly only 1/2 to 2/3 of our salaries. Does this seem fair? Well not to me, but then again to an already-retired person it is probably quite acceptable. And future generations? What kind of weight will my daughter’s demographic group have to collectively bear? It’s a question.

My conferences in Novosibirsk and Baska (on the isle of Krk in Croatia) are over and I am in Zagreb for a couple of nights before my flight home. Novosibirsk is in the middle of Russia, the largest city in Siberia and third largest in Russia itself. It straddles the Ob river, one of the longest in the world, and was built largely to support the Trans-Siberian railway bridge built more than a century ago across the river.

A typical walk in the Academic town between Institutes

The Geometry conference was held in a satellite suburb called something like “Academic Town” in Russian: a sprawling complex of Scientific Institutes set in a forest, with pleasant walking paths through the birch and pines. The northern forest reminds me of Canada, and so I enjoyed the walks I took, several with my new young friend Maxim, a student of Alexander Mednykh, the mathematician who had invited me.

The Ob sea: man made reservoir outside Novosibirsk

Maxim showed me the Botanical Garden and local Russian Orthodox church, all in wood, set amongst the trees, as well as the Ob sea/reservoir, an impressive man-made lake which is hard to see across it is so big.

Sasha Mednykh and me in downtown Novosibirsk

Alexander and his son Ilya took me on a nice walking tour of Novosibirsk itself, showing me the theatre, train station, some interesting squares, and even a semi deserted fair with a Ferris wheel we took a ride on, to get a good view of the Ob and the longest subway bridge in the world over it.

Main street in Academic town, Novosibirsk

I stayed in a pleasant student dorm which had a few floors acting as a hotel: there were not a lot of accommodation options. The weather was lovely, end of summer warmth and blue skies.

The conference itself was fun, particularly the dinner with lots of toasts, of which I gave one! I met some interesting people; one of my neighbours was from distant Yakutsk, a name familiar from years playing Risk, but of which I know rather little, and he told me about life there. Evidently the permafrost spews out slabs of ice from the ground erratically in places, even in the middle of the summer!

I also had some good mathematical discussions, one in particular with I. K. Sabitov from Moscow, who had proven the lovely theorem that given a polytope in three dimensional space, there is an algebraic equation satisfied by the volume whose coefficients depend only on the quadrances (squared lengths) of the edges. However for complicated polytopes this polynomial can be exceedingly large. (For a tetrahedron the relation goes back to Tartaglia, and it was also discovered by Euler.)

A view from Novosibirsk

One small aspect of Russian life perhaps worth retelling: in the small minibus that Maxim and I took to go into the city, passengers pass up their fares to the driver hand to hand via other passengers, even from the back, till it gets to the driver. He dispenses the various lots of change and it snakes its way back, each person taking the amount due him/her, passing the rest on. Can I imagine this happening in a big city in Australia or America? Not really, to tell the truth. It says something about the level of trust people have with strangers. Perhaps an aspect of the `quality of life’ that doesn’t get as much attention as it should.

Downtown Novosibirsk: The center of Russia

Alexander Dimitryovich, as he is known by his students and colleagues (Russians use the first two names as a sign of respect, and the second name is that of the father) has a small office which is quite often full of students, so there is almost no room to turn around. There was a nice feeling of working together there, and I had some chance to exchange some interesting thoughts about hyperbolic geometry, a subject I have developed a deep interest in. (My Rational Trigonometry extends to the hyperbolic setting, and gives a rather new, and of course exciting, view of this classical subject.) I was happy that Alexander Dimitryovich was willing to entertain using my algebraic reformulations to make some computations.

So… I am eager to go back to Siberia. Next time I will perhaps try to stay a bit longer, and get further out of town, perhaps to the Altai mountains which are not so far away.

Mathematics throughout its history has wrestled with a major schism: between the discrete and the continuous. In the earliest times this was the difference between arithmetic and geometry.

Arithmetic ultimately comes down to the natural numbers 1,2,3,4,5,6…. These are so fundamental and familiar that most ordinary folk don’t see much point in `defining’ them. But us pure mathematicians like to ponder such things, and it is fair to say that the issue is still open to further insights coming from programmers and computer science. One way or the other natural numbers are symbols that we write down to help us count; the number of apples in a bushel, children around a campfire, stars in the sky.

Geometry, on the other hand, ultimately comes down to points and lines. It is not so easy to say exactly what these are. In the 19th century mathematicians started to acknowledge that the bible of geometry—Euclid’s Elements—didn’t deal adequately with this issue. Things were okay as long as you just assumed you knew about points, lines, and the plane; and accepted various physically obvious properties they satisfied.

Fortunately Descartes and Fermat some 300 years earlier had constructed a framework—the Cartesian plane—which allowed geometry to become subservient to arithmetic: a point is an ordered pair [x,y] of numbers, and a line is an equation of the form ax+by=c. This was a wildly successful conceptual leap. It allowed algebraic techniques to bear on higher order curves, like conics or cubics, gave a straightforward and uniform treatment of many geometrical problems, and led to the development of the calculus.

But there was a heavy price to be paid for this arithmetization, which was mostly unacknowleged for centuries. The precise and logical arithmetical form of geometry which Descartes’ system gives us has curious aspects that diverge from our everyday physical experience. No longer do two circles which pass through each others centers meet. We cannot guarantee that a line passing through the center of a circle meets that circle.

This explains, I believe, why Euclid shied away from an arithmetization of his geometry: he knew that standard geometrical constructions yielded `irrational’ numbers whose arithmetic he did not understand. The Greeks’ numerical system was cumbersome compared to our Hindu-Arabic system, they had no good notation for algebra, and they considered geometry more fundamental. So ultimately Euclid choose to consider a `line’ as a primitive object which need not be defined, and carefully avoided using distances and angles as the main metrical measurements. For him, logical purity trumped practical considerations.

Modern geometry has steered away from the concern and esteem for rationality of the ancient Greeks. The trigonometry we ostensibly teach in high school texts is logically half-hearted and involves a hefty amount of cheating; at the research level we have resorted to simply walking away from this challenge. Euclid would be appalled at the sad state of affairs in modern geometry, and would find it inexplicable that the majority of educated people have almost no understanding of this beautiful subject!

The problems with irrationals have been around for two and a half thousand years, and are still with us, whether we like it or not, acknowledge it or not. Deep at the bottom of modern mathematics lies a gnarled and warted toad: the lack of a true understanding of the continuum. Many (but not all!) modern mathematicians will view this statement with skepticism. We like to believe we understand the continuum—in the context of `real numbers’—and have faith that the definitions involving Dedekind cuts, Cauchy sequences, or just axiomatic assumptions, deal adequately with the problems. Unfortunately, they do not.

In my opinion, the continuum is actually much, much more complicated than mathematicians think. Our current view of the continuum is analogous to the simple-minded model of the heavens that ancient, and not so ancient, peoples had: that we live surrounded by a large celestial sphere on which the stars are pinned, and on which the sun, moon and planets move. For better or worse, the true celestial story is vastly richer and indeed more interesting than this, and so it is with the continuum.

Modern mathematics has accepted a confusion which has spread its poisoned tentacles into almost every aspect of the subject. By accepting the logically dubious, we come to accept also that some parts of mathematics are just inherently vague and obscure—that logic has its limits, and beyond that is a kind of no-mans land of convenient but arbitrary assumptions. Mathematics loses its certainty, and descends into shades of grey. This shrugging away the bounds of careful reasoning at the research level also naturally affects the integrity of mathematics education.

The reader will want some initial evidence to support these statements. Look in any modern Calculus textbook in the introductory section which purports to establish, or review, the fundamental properties of `real numbers’. Almost all resort to waffling or unwarranted assumptions, with a few honest exceptions that admit to the lack of proper foundations. Then consider how the modern computer programming community deals with `real numbers’. What you find is that they don’t, because they can’t; the rigour of their machines interferes with wishful thinking. Instead, the programmers work with floating point representations or rational number computations, which are light years away from working with `real numbers’.

So let me put some of my cards on the table: I propose to shine a clear light on this whole issue, to expose the logical confusion that underpins modern analysis, and most importantly, to provide a way forward that allows one to envision a time when mathematics will be logical and true, without waffling and the ugliness that naturally follows it around.

In my MathFoundations YouTube series at http://www.youtube.com/course?list=EC5A714C94D40392AB&feature=plcp I will tackle the detailed mathematical aspects of this campaign. In this blog I hope to provide some overall framing and discussion of both the mathematical and the sociological aspects of this unfortunate delusion—a delusion that has got its stranglehold on mathematics education, as much as research level mathematics, since the beginning of the twentieth century. We need to move into a bigger, brighter, more honest space.

I am flying on Monday to Novosibirsk in Siberia, and then a week later to Krk, an island in Croatia. The occasions are two interesting geometry conferences, where I will meet old friends, make new ones, and talk about my recent work. Travel is one of the joys of my life, and even a long plane flight is only half a burden, knowing that it ends in—a new place! It’s a chance to step out of the groove, in a small way, for a while.

The life of a research mathematician, if the truth be told (and I want to do that), is outwardly not very exciting. The routine is piled on thick; calculus and linear algebra lectures don’t change much from year to year, writing papers is rather boring and tedious, and the stable, conservative environment of most universities means that one’s working life is like a sheltered cocoon. Jobs are hard to come by, so career mobility for most of us is limited, and we get pretty familiar with our stomping grounds after a few decades.

The inward life, however, is a totally different story. Mathematics present us with such a rich and fascinating framework of ideas, concepts and challenges that I can think of no work that would give me even half the satisfaction I get from being a research mathematician. I would not trade my job for any other, not in law, politics, business or medicine. The search for patterns, the hunt for a key equation, the challenge in proving what our intuition tells us loud and clear must be true…until we find out a week later that our intuition was wrong and the story goes in a quite different direction! What a happy life to be able to think about interesting things; I am very grateful, and one of the purposes of this little blog is to share some of the rich ideas I get to think about, with you all.

At both conferences I will be talking about the Rational Trigonometry of a Tetrahedron,  albeit in somewhat different forms. You are all familiar with ordinary trigonometry (the word means the study of triangle measurement); with those angles and lengths, cosines and sines, and a whole raft of complicated formulas. It’s useful, but not altogether pretty.

Hopefully you have heard that there is now a new and better way of tackling the whole subject, called Rational Trigonometry, discovered about 10 years ago by yours truly. This story of a completely new way of thinking about a very classical subject is rather interesting; as you might expect such a bold departure from tradition doesn’t occur very often in elementary mathematics. Is it really better? Does it make computation simpler? Does it lead to a lot of beautiful new mathematics? Yes, yes and yes. And the crucial question: is it what we ought to be teaching our young people in high schools?? Definitely yes!

I will be telling you more about this discovery of mine, and the reaction that it has gotten from colleagues etc. in due time. An interesting consequence is that new doors have been opened to me by this understanding; vistas and trails that lie before me now, undreamt of a decade ago. The possibility of a new approach to the entire subject is starting to emerge as the fog slowly settles—a more careful, honest, logical and beautiful mathematics, more closely aligned to computer science.

The consideration of “revolution” in the context of mathematics is to many practitioners unlikely and even heretical. The safe confines of academia enclose an even more secure installation of pure research mathematics, where orthodoxy, accepted practice and authority largely rule.

Perhaps my whole life has been building up to the realization that, even in mathematics, there are true paths, and false paths, and paths in-between, and that ultimately only I can decide which is which—for myself. If you allow me, I propose to take you on some little mathematical journeys, and show you new possibilities for thinking. Then you too can decide what is true, what is false, and what is in-between—for yourselves.

Thought I’d give you a window on my week to come, since it is early Monday morning—so you can get an idea of what mathematicians do. Today is my big teaching day, I have four tutorials, all on first year calculus. I am handing back the tests the students wrote last week, and will try to convince them they must make more effort writing coherently and expressing their mathematical thoughts more carefully. Our first year students often don’t have a lot of practice in writing mathematics. Sometime during the day I will go for a swim at the Uni pool to make four hours of teaching go smoother.

Tomorrow I will chair a meeting of our Learning and Teaching Committee, we need to get going on mentoring new tutors, thinking about a L&T seminar for later in the term, and putting one of our colleagues forward for a teaching prize. I will also talk with one of my graduate students, Nguyen Le, on how we can construct a planar triangle over the rational numbers that has incenters in both blue (Euclidean), red and green (relativistic) geometries simultaneously. She has been working on triangle geometry, and we would also like to start writing a paper on some extensions of classical results involving Gergonne, Nagel, Incenter points etc.

I will need to schedule a list of mentors to oversee our new casual tutors (the School of Mathematics and Statistics employs outside help to also tutor some first year classes).

On Wednesday and Thursday I have three hours of my Algebraic Topology class. I need to have marked the problem sets before that—something else to do on Tuesday—hand them back, and give lectures on a review of free groups and non-commutative groups, and also on applications of winding and turning numbers (for example the ham sandwich theorem!) Nguyen will video one of those classes for me, and at some point, probably the weekend, I will edit it and post it on my YouTube channel (user njwildberger).

Friday I don’t have classes but I want to spend a good chunk of time with my Masters student Sharanjit Paddam who is working on rational views of classical tesselations of the hyperbolic plane. I am heading overseas the following week, to Novosibirsk (Russia) and Krk (Croatia) for two geometry conferences, will be gone three weeks, and want to make sure his project is on track.

I also need to spend some time preparing the talks I will give at these conferences. But what I also would like to do, in fact what I really want to do, this week is: think about the group structure on a cubic curve (one step up from the conic sections like ellipses, parabolas and hyperbolas) in terms of ternary forms. I have this reformulation of algebraic geometry kicking around in the back of my mind, which starts by reconsidering the basic nature of an algebraic curve; not as a set of points, but rather as… well something both more complicated but simpler to deal with. I`ll explain this at some later point when I understand it better, but suffice it to say for now that I know there are interesting newish algebraic objects which I call `pods’ floating around under the noses of classical algebraic geometers. I have been putting together Geometer’s Sketchpad worksheets that allow me to explore cubics and play around with group and pod structures.

The other thing that has been on my mind is the Cayley transform, something that I thought quite a lot about when I was a graduate student, and post-doc at Stanford. I reckon that we need a more algebraic theory of Lie groups, and I have known for a long time, although I haven’t written anything on it, that the key is replacing the exponential map with the Cayley transform, suitably interpreted.

Jack Hall did a Master’s project on related issues with me some years ago. He has finished a PhD at Stanford under Ravi Vakil, who curiously was also briefly a student of mine: he sat in on my first year algebra course at the University of Toronto when he was a precocious high school student. Jack will be taking up a position in Canberra next year, it will be nice to have him as part of the mathematics scene.

Anyway, the Cayley transform is a lovely thing, ought to be more widely appreciated and used. I would like to extend Rational Trigonometry to the Lie group setting, a big project of course, but something that I like to contemplate now and then.

I have also been feeling guilty about not posting any more videos in my WildLinAlg series for quite some time. If I can make time, I might put together a new video on the linear algebra of cubic splines. Not sure if I will get around to that though.

So that is my week, coming up! Lot’s of fun, and interesting.

The London Olympics are over, what a success they have been. Many Australians however are pondering the performance of our athletes, as we only placed 10th.

Only placed 10th?? By whose count?

The current rankings, which are published by most of the international media, rank countries by order of Gold medals won. How can this be at all sensible or fair, since it ignores silver and bronze medals entirely? If you are going to give out prizes, those prizes ought to be worth something.

There is another popular ranking system which we also see; which tallies the total number of medals won. This is also patently simplistic. Why should a bronze count as much as a gold?

The uncritical acceptance of these two simple-minded scoring systems around the world reflects an astonishing mathematical naivety. Surely we can do better in the 21st century! For a country like Britain to put on such a complex, dazzling show is inconsistent with its media performing just a cursory back-of-the-envelope calculation to determine rankings.

More than a hundred years ago, a fairer ranking system whose proposed by the English press: each gold is worth 5, each silver is worth 3 and each bronze is worth 1. So using a little bit of multiplication and addition, we get a much clearer and more equitable picture of how different countries performed—in total. Of course one can argue about the weightings, but I personally think these are quite sensible.

With such a weighted ranking system, all medal winners contribute to their countries standing, but on a scale that is reflective of the different levels of achievement. If we had adopted this, silver and bronze medallists would be happier knowing that their win has contributed to the national account, and we would see fewer dejected  athletes having just placed second or third in the world!

My calculations of the Weighted Ranks for the London Olympics are available at http://www.maths.unsw.edu.au/news/2012-08/weighted-olympic-rankings-london-2012-n-j-wildberger. Australia places 8th, not 10th; while Great Britain gets beaten into third place by Russia—perhaps this is the reason the British press don’t promote this fairer system this year!;— Spain goes from 21st to 14th, and Canada goes from 35th place to 22nd.

Congratulations to all our athletes for their excellent 8th place in a highly competitive meet. Let’s hope that the Olympics Committee can consider instituting this weighted ranking system as the gold standard for Rio, or that at least the international press can think about adopting it. A little mathematics can go a long way!

On Monday, after my early 9 am class, I drove up to Sydney’s North Shore all the way to Wahroonga. I was slated to give a talk at Knox Grammar School to 70 or 80 Heads of Independent High School Maths Departments from around NSW, on the subject of “National Curriculum Issues and opportunities for revitalizing geometrical thinking in the classroom”, which I admit is a rather long-winded title.

I had been invited by Joshua Harnwell, a teacher there, who I had met in an earlier Board of Studies meeting, also about these ghastly new mathematics curricula proposed by ACARA, the national group entrusted with coming up with a syllabus for maths education around the country.

Knox Grammar is a lovely old-money private school set in the affluent suburbs of Sydney. There are a lot of such around, quite a difference from Canada where I grew up, where almost everyone except close friends of the Duke of Buckingham (or some such) just went to the nearest local high school, which was invariably a public school. Meaning it was free, and open to all who lived in the area. Although Australia prides itself on its egalitarianism, there is in some curious way quite a lot more class differentiation here than in Canada where I was raised, although perhaps I am just getting older and wiser to such things. In particular there are so many private schools in Oz that the public school system languishes a bit for funds, and there is an unhealthy divide educationally. Growing up where I did, my default view is that governments have an obligation to level the educational playing field as much as possible.

Not to say anything against Knox, a lovely school; and they put on a really fine lunch for us in a high-ceilinged glass-enclosed foyer, with a fine view of the spacious grounds. I had pleasant conversation with some high school heads over our salmon, talking about the merits of GeoGebra and other dynamic software packages.

After lunch I gave my talk, recorded for posterity since I had brought my trusty Sony video camera with me. So in case you weren’t there, and you are interested in the topics, which are quite important from my point of view, you can find the video and pdf of the talk at http://www.maths.unsw.edu.au/news/2012-08/national-curriculum-talk-norman-wildberger-knox-grammar.

One of the side points I made was that we need to rethink, or rather the media ought to rethink, the ranking system used to score the Olympics. Turns out that was related to some of the geometry I talked about, and maybe next time I’ll tell you about that.

In any case I enjoyed the opportunity to talk to high school teachers about something important; it feels good to get out of the ivory tower of academia every so often.

I’ve been thinking about what are the essentials of mathematics education, summarized in a few lines? [This has been motivated by preparing for the AIS talk on Curriculum Issues and Geometry that I am giving on Monday.]

I’ve decided that good mathematics education requires a three-way balance between

1) conceptual understanding

2) algorithmic computation

3) problem solving.

Suppose we are learning about arithmetic with fractions, as so many of us do around the age of 10 (and for many years afterwards, if we don’t understand it properly to begin with!) The conceptual understanding is to have a good feeling for what a fraction is, maybe several different ways of thinking about them: parts of a pie, a point on a number line, the result of the division of one natural number by another, and how these different view relate. The algorithmic computation skills are being able to add, subtract, multiply or divide any two fractions automatically. The problem solving is more open-ended, and tackling a variety of problems we have not seen before; by first putting them into a conceptual framework, perhaps then applying some computation, and then properly interpreting the results. For example, that might be: if twelve people divide a bill totalling $165, and each person has only twenty cent coins (yes rather unlikely!) then how many coins must each person front up?

[Note that you can tell this is an Australian problem: we have twenty cent coins here, not quarters. Our twenty cent coins are bigger, though.]

After chatting about this topic with David Houghton, a high school teacher who is spending the year teaching here in the School of Mathematics and Statistics at UNSW, I’ve realized that a big part of the problem with mathematics education around the world might be that: the second piece of the picture—algorithmic computation—is a lot easier to teach than either the first—conceptual understanding—or the third—problem solving.

So students get more of that second slice, and not enough of the first or third parts of the pie. It’s like getting too much cheese with the pizza, and not enough crust, or other toppings. This can’t be a very original idea, but sometimes admittedly simplistic categorization like this can help us structure our understanding.

How does it explain what it is going on? Common sense suggests that there will be a natural inclination for weaker, or less prepared, teachers to concentrate more on algorithmics (how do you divide 16/3 by 8/5?) rather than concepts (what does 16/3 divided by 8/5 actually mean?) or problem solving (if an alien requires 8/5 of a kilo of asparagus a day to survive, how long will she survive on 16/3 kilos of asparagus?)

Assessing algorithmics is more straightforward than assessing understanding or problem solving; for example it can be done by multiple choice tests, or even with these on-line automatic maths learning programs. Students conspire in this too: they are reassured by the cut-and-dried aspect of an algorithmic approach, and it limits the amount of effort they have to put in. They, and some of their teachers, might even come to think that this is all that mathematics is.

Week after next I will be giving a talk to 60 or so high school Heads of Math Departments from around NSW (that’s New South Wales, the state of Australia in which I live, containing prominently Sydney, Wollongong, Newcastle, and smaller towns) on a rather contentious educational issue currently here in Australia. The government has decided that we need a national curriculum in mathematics to replace the current hodge-podge of state curricula, which probably makes a lot of sense. Unfortunately the current Draft of the new maths curriculum for advanced maths (Years 11 and 12) is deeply inadequate, and I will be talking about that.

I won’t bore you with the details, [if you are interested, you can view our detailed report at http://www.maths.unsw.edu.au/news/2012-07/schools-response-draft-senior-mathematics-curriculum-acara ] but there is an issue here that is of more general interest, and that is the role of geometry in maths education. Geometry used to be the core of higher training in maths (here in Australia we use this abbreviated form often instead of “mathematics”, and I will be following that a lot of the time), with Euclid’s Elements the main text until the beginning of the 20th century. Around that time a steady decline in the amount of geometry taught in Schools began; with the hefty and careful sequence of Euclid’s Propositions slowly giving way to ever briefer descriptive approaches; the use of instruments and drawing diminished; and the emphasis on strict, perhaps overly rigid, logical presentation was replaced by a more flexible view. Linear algebra expanded to fill the void. These days Euclid has well and truly been relegated to historical interest, and the subject re-badged with titles like “Shape, space and measurement”. Students memorize terminology and facts, but problem solving and an appreciation for proofs and logical structure has diminished markedly.

In 2001 the British Royal Society proclaimed that “We believe that geometry has declined in status within the English mathematics curriculum and that this needs to be addressed. It should not be the ‘subject which dare not speak its name’.”

Ironically the decline of geometry in schools was accompanied by the development and rise of key geometrical mathematical subjects of the 20th century, such as differential geometry, algebraic geometry (which used to be called projective geometry), topology, and linear algebra/functional analysis. While maths students spend less time on pure geometry, the physics community has slowly but steadly, starting with the pivotal work of Einstein, come to appreciate the close synthesis between geometry and physics. This is a turning full circle, since much of geometry originally was motivated by astronomical interests.

Is any of this likely to change in our new millenium? In fact I think it is already doing so. Young people are visually oriented, and so geometry appeals to a large cohort of students who are bored by algebraic manipulation and large amounts of numerical data. Computer graphics, video games, 3D movies all clearly require spatial understanding. But I think the key drivers are, and will increasingly be, the remarkable new dynamic computer software programs such as Geometer’s Sketchpad, C.a.R., Cabri, GeoGebra and Cinderella, that allow you to create 2D, and even 3D, constructions on a computer and then manipulate inputs to see how relationships change and are maintained, and the newish construction sets that make it easier to physically make models of interesting geometrical objects and explore them, such as Zome, polydron and Frameworks, and many others (for a good list, see http://www.ics.uci.edu/~eppstein/junkyard/toys.html)

If you feel like experimenting with dynamic software packages, check out the free programs C.a.R. (a one-man creation: thanks to Rene Grothmann) which is called Z.u.R. in German, or GeoGebra. But all the programs I listed are really quite special!

In my thinking, there can be no better way to interest students in geometry, and perhaps mathematics, than playing around with such programs and construction sets. Perhaps mathematics education will turn around, and educators will come to realize that stripping geometry from mathematics renders it bloodless and potentially tedious.

In Australia, I hope that the people in charge of curriculum design (ACARA) wake up to the fact that a high school Year 11 and 12 core mathematics course with mostly calculus and statistics, but no geometry, is a disaster for prospective engineers and scientists, and will drive away a lot of students who otherwise might be attracted to the subject.

There was a good reason that geometry was the heart of mathematics for more than 2000 years!