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!

Hi everyone, thought I’d start a blog to express and share more clearly and succinctly various mathematical ideas, thoughts, speculations and discoveries. I’ve got quite a few math videos at my YouTube site (called Insights into Mathematics at user: njwildberger) and in my MathFoundations series there I occasionally need a place to post longer streams of thoughts.

Might also share some of the admittedly mild adventures of a pure research mathematician, and answer questions that people might have. My mathematical journey has not been particularly standard, and I have been privileged to have stumbled across so much lovely mathematics. I would like to share some of that with you, as well as other speculations and opinions, a bit of personal stuff, some dreams and aspirations, and musings about this remarkable, difficult, and occasionally beautiful world in which we find ourselves.

Thanks in advance to my family for support, and to the Australian taxpayer who indirectly supports academics like myself, giving us a boost up, to see a little bit further. Hope you don’t hesitate to leave comments, ask questions, and share your ideas too.

Norman Wildberger, Sydney