Let H be a load of hogwash.

Let H be a load of hogwash. By which we mean, of course, that H is an unbounded category of fuzzy schemas, expressed in the first order language of obfuscation with only countably many incompleted disjunctions.

Now take the space L of all cohomological Aleph one completions of H, partially ordered by increasing complexity—the de-facto mathematical convention in the beginning twenty-first century, but we spell it out for grad students—and consider the set N of all normalized functors from L to its contragradient.

The model space of N clearly has an adelic inductive boundary, which we denote by N_infinity. Let M be the infinite unstable tensor product of Aleph squared many copies of N_infinity, and take G to be the stable homotopy group of the measure zero projection of the affine homological dual of the K theory retract of M upon its enveloping quantum C* algebra.

While there are many fascinating questions arising from the inverse scattering problem of the functorial pair (L,G), we are naturally interested in considering the projective Hom groups of M into the space of all transcendental harmonic twistings of G mod its radical.

 Assuming the Axiom of Unrestricted Freedom with NP dominance, the associated cardinality of all semi-stable injections of H into the perverse sheaf of pseudo-differential connections of the cotangent bundle T(L,G) ought to be wildly inaccessible, making the whole subject a bonanza for further investigations and grant applications. Which of course goes to show yet again that ZFC is indeed finger-licking good. 

Just some thoughts I had the other day, which i thought I might share with you.

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The real information revolution

Johannes Gutenberg’s introduction of the printing press around 1450 was one of the defining moments of the modern age, ushering in a new era where knowledge could be cheaply reproduced and widely distributed. Since then the printed word has come to dominate our understanding of what information is.

Whether it be a book, a pamphlet, a newspaper or magazine article, a letter, a legal document, or these days a pdf or ebook, we have been completely ingrained to understand that information is printed information, and that to learn something means more or less to read it and understand it. This is the bedrock of the educational system, including of course tertiary education; with its heavy reliance on textbooks, libraries and learned journals.

All true, until now.

In only the last five or perhaps ten years, a new paradigm is suddenly upon us, sweeping through the modern world like a wildfire fanned by the deep untapped desire of people to learn by watching and listening, not reading. We are talking about video, my friends: most notably YouTube videos, but also of course iTunesU, Vimeo, Coursera, OpenLearning etc. Young people increasingly go to YouTube as a default if they want to know something; now already the 2nd largest search engine in the world—next only to Google—and moving quickly to become one of the prime repositories of really useful knowledge on the planet.

For one-sentence knowledge, the printed word will remain king. What is the circumference of the earth? Who was the president after Lincoln? Where was the first mammoth discovered? For such tidbits of knowledge, the printed word is optimal. For large-scale knowledge, the printed word may also be harder to replace. But for everyday middle complexity information, which requires, or at least requests, something of an explanation, video will rule.

How do I fix my lawnmower? Who were the greatest conquerors in history and why? What is Rational Trigonometry, and why is it so superior? Where are the best surfing spots in Sydney? What is the best way of chatting up a girl? For this kind of important info, and much, much else besides, most of us would rather get the answer from a person, using a combination of audio and visual representations. Video cannot be beaten here in my opinion.

While MOOCS and all kinds of fancy e-learning systems are much the rage in tertiary education these days, it is useful to keep in mind that the key ingredients are almost always the videos themselves. We are returning to the rhythm and logic of an earlier vocal tradition, where knowledge was memorized and passed on from father to son, from mother to daughter, from leader to followers—by talking, explaining, showing. This is far closer to our biology than the current arcane system of letters and numbers that form our printed sentences, like this one. If I was reading this out loud on a video, then my emphasis, pauses, expressions and posture would convey just as much, maybe more, than the words themselves. As it is, you have only the words.

Video as information is an idea which may well prove to be more interesting and important than video as entertainment. It is happening now, as we speak. When I started posting math videos on YouTube in 2007, most of my colleagues thought it was a strange use of my time. Don’t academics spend all of their energy writing furiously to continuously augment their all-important list of printed publications? What’s the point of posting videos that you will get little academic credit for?

Some of my colleagues probably still feel this way, but I bet they are a lot less confident now. They are perhaps starting to acknowledge something that students have long known—that even interesting and pretty mathematics may be difficult or painful to learn from an article or book! And some of them are starting to realize that if you don’t join the video revolution, your work runs the risk of being left behind, forgotten and unused, no matter how good it looks officially on a CV.

A salutary story for me: when I was a graduate student at Yale, I had a desk in the annex of the library on 11 Hillhouse Avenue; a somewhat dark and hard-to-find room in the basement which was stacked to the rafters with ancient math journals (for which there was no more room in the main library upstairs). Late at night, bleary from too much mathematical pondering, I would pull down a volume from on high and have a look into journals from the 1800′s. Creakily the dusty tome would relinquish its grip on its neighbours, having been unmoved in at least half a century: then I would skim these lovely, elegant articles, thinking—why is no one reading this great stuff anymore?, and— is this what will happen to my work once I am gone?

This need not be the future of today’s mathematicians. Well-presented videos of interesting topics embodying deep understanding will be regarded like gems of classical music to future generations of students and scientists, is my guess. Maybe this is a tad poetical, but I really do believe in the huge potential for broadening understanding and interest in the general public towards mathematics—that most beautiful of disciplines!

So, young mathematicians, take my advice—by all means play the game of oft and repeated publication in learned journals, but also spend some time developing your skills at explaining and presenting your knowledge and work through videos, so that your ideas will be accessible, useful and engaging to a wide spectrum of listeners. It is the future of publication, as much as it is the future of knowledge distribution.

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Politics and the two great fears

Australia is two weeks away from another national election, featuring Prime Minister Kevin Rudd (Labour) versus Tony Abbott (Liberal). I won’t bother you with the details, which almost everyone agrees are not that compelling. But it does provide an opportunity to muse about the broad division in politics between the left and the right—a schism that seems pretty universal, across the western world at least.

These two positions are natural responses to the two great fears that have dominated political thought for thousands of years. The first is the fear of the rabble—the unwashed hordes—the lawless thugs—violently taking what we have struggled hard to reap, build and create. The second is the fear of the leaders—the aristocrats—the powers in charge— stealthily taking what we have struggled hard to reap, build and create.

There are good historical reasons for both fears.

In times gone by, the most efficient way to amass great wealth was simply to steal it. Get yourself an army, the more impoverished and violent the better, offer your men pillage, rape and looting, and off you go up and down the countryside, taking whatever you please. If you are extremely successful, you will eventually be hailed as a great conqueror (Alexander the Great, Cyrus the Great, Caesar, Attila, Genghis Khan, Timor the Lame, Pizarro, Cortes, Napolean, Queen Victoria, Hitler, etc etc)—at least for a while, until your fragile imperial structures collapse. Or not—if you are competent and perhaps lucky.

On a smaller scale, the peasantry can simply revolt without the coherence of an army. Sometimes people just get fed up with suffering inequalities and injustices at the hands of their betters. With these kind of insurrections a lack of clear direction might mean that countries descend into civil disobedience, lawlessness, and lack of respect for property that can last generations.

At the other extreme, there is an almost opposite fear. In the absence of social upheaval or external military conquest, there are good reasons to worry about the abuse of power by those in charge within the society. This can take the form of a feudal hierarchy, where a small  cohort live the life of luxury at the expense of the peasantry, which is almost everyone else, or in the more modern form of a dictatorship, where a strong and ruthless leader at the head of a vested minority seizes power, brutally eliminates any effective opposition, and sets about taxing and bleeding the citizenry. Both situations have been historically very common and are still with us today.

Another more subtle variant occurs in our modern democracies, where the rule of law forbids the more extreme forms of exploitation of the masses, but where the ruling classes have none-the-less figured out how to slowly and surely tighten their grip on power, and accumulate ever more wealth and influence at the expense of the proletariat. The techniques are well-known and indeed obvious: the wealthy and powerful go to the best schools, meet the right people, obtain the positions of decision-making, and then naturally steer the legal structure in directions which favour them and their class. This kind of insidious transfer from the poor to the rich seems almost to be a kind of natural law in stable economies. See the history of almost any western country since World War Two.

So which side of politics is someone likely to be on? Usually it’s a pretty good guess, if you can find out the income and assets of the person in question. Belong to the top 20%? Then you are most likely a die hard Republican (USA), or Conservative (Canada) or Liberal (Australia) etc. Belong to the bottom 30%? Then you are most likely a die hard Democrat (US), Liberal or NDP (Can) or Labour (Aus) supporter etc.

And if you are in the middle somewhere? Then you probably and rightly fear both the poor and the rich taking your fair share. In that case you will be pulled and pushed by both groups—the right wingers trying to convert you for patriotic reasons or for fear of outside groups, and the left wingers trying to convert you for moral reasons. And there will be other parties, like the Greens here in Australia, that try to occupy more of that middle ground, but who find it very difficult to actually gain power, without either the support of the rich or the working classes/peasantry.

Naturally this is all very simplistic, but sometimes simple explanations have something to tell.

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The essentials of effective tertiary mathematics education

I am in Austria right now; having spent a week in Innsbruck visiting Hans-Peter Schroeker and Manfred Husty, I am now in Graz at the Technical University, and I am talking with Anton Gfrerrer, Sybille Mick, Johannes Wallner and Johann Lang of the Institute for Geometry, and giving some talks. Central Europe has a long and distinguished history of excellence in research and teaching of geometry. Here in Austria students learn Descriptive Geometry, Hyperbolic Geometry, Projective Geometry and CAD systems for visualization: if only Australian students were exposed to half as much!!

Its a pleasure to be in this part of the world, both personally and academically. My father is from Austria, born in Linz, and my mother from Liechtenstein, and so I have been travelling to this part of the world on and off for many years visiting relatives. Here is a picture from Triesenberg in Liechtenstein, where I stayed for a few days visiting my aunt Maria, who lives a few kilometers down the mountainside.

What a beautiful area of the world! Photo 15-05-13 8 00 55 PM

Here is a shot from a town in Austria called Landeck where I overnighted on the way to Innsbruck. Some lovely walks around the countryside there.Photo 19-05-13 2 28 59 PM And I suppose I better show you a mountain, of which there are many around, but sadly the weather has not been entirely cooperative for sunny photos. This taken from the train on the way to Graz from Innsbruck. Photo 21-05-13 11 42 30 AM

Today I want to tackle the challenge of succinctly summarizing some essential features, in my opinion, of effective tertiary level mathematics education. This follows naturally from my last blog on MOOCS. Of course there is much to be said here, but suppose we had to just write a paragraph or two: an excellent exercise for focusing one’s thoughts.

Putting together an effective university level mathematics course requires:

1. A prior solid understanding of the mathematical content of the subject, and its connections and application to other areas within and outside of mathematics.

2. A carefully chosen syllabus that lays out a logical sequence of topics: not too many, not too few, pitched at the right level for the intended audience.

3. A written text, either notes or a book, which covers in detail the syllabus of the course, including a wide variety of examples. This can of course be online.

4. A series of lectures, given either live or via video, which explain the course content but perhaps do not go into quite as much detail as the written notes. These lectures should be obviously accessible, interesting and useful to the students for learning the material.

5. A complete and comprehensive collection of exercises for students to attempt. These should help students gain familiarity and mastery of the course content, to develop problem solving ability, and to spark further interest in other aspects of the subject. The exercises may possibly be organized into various levels of difficulty if appropriate.

6. A carefully prepared set of worked solutions to many of the exercises, and summary answers to the rest. This could be either in written or video form, or both.

7. A mechanism for grading student work at solving exercises and writing up solutions, and providing a reasonable level of feedback on their written work.

8. Effective and fair tests and final exam, that motivate students to study, review and ultimately absorb the material.

So if you can manage to incorporate all these aspects in a coherent way, you will for sure have an effective mathematics course.

And how much of this could be done on-line? Almost all of it, with the important exceptions of 7, and possibly 8. This is the key challenge in setting up online courses in mathematics—(in fact also of regular courses at university level!)—how to provide good feedback to students on the exercises that they ought to tackle.

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MOOCs and TOOCs and the role of problem solving in maths education

A quick quiz: which of the following four words doesn’t fit with the others??

Massive/Open/Online/Courses

We are going to muse about MOOCs today, a hot and highly debated topic in higher education circles. Are these ambitious new approaches to delivering free high quality education through online videos and interactive participation over the web going to put traditional universities out of business, or are they just one in a long historical line of hyped technologies that get everyone excited, and then fail to deliver the goods? (Think of the radio, TV, correspondence courses, movies, the tape recorder, the computer; all of which held out some promise for getting us to learn more and learn better, mostly to little avail, although the jury is still out on the computer.)

It’s fun to speculate on future trends, because of the potential—indeed likelihood—0f embarrassment for false predictions. Here is the summary of my argument today: MOOCs in mathematics are destined to fail essentially because the word Massive is intrinsically unrelated to the other words Open, Online and Courses. But, a more refined and grammatically cohesive concept: that of a TOOC, or Targeted Open Online Course, is indeed going to have a very major impact.

When we are teaching mathematics at any level, there are really two halves to the job. The first half is the one that traditionally get’s the lion’s share of attention and work: creating a good syllabus with coherently laid-out content, which is then clearly articulated to the students. The other half, which is almost always short-changed, and sometimes even avoided altogether, is to create a good set of exercises which allow students to practice and develop further their understanding of the material, as well as their problem-solving skills. In my opinion, really effective teaching involves about equal effort towards both halves; again this is rarely done, but when it is, the result usually stands well out above the fray.

Here are some examples of mathematics textbooks in which creating the problem sets probably occupied the authors as much as did the writing of the text: first and foremost Schaum’s Outlines (on pretty well any mathematics subject), which are arguably the most successful maths textbooks of the 20th century, and deservedly so, in my opinion. Then come to mind Spivak’s Calculus, Knuth’s The Art of Computer Programming, Stanley’s Enumerative Combinatorics, and no doubt you can think of others.

Good problems teach us and challenge us at the same time. They are the first and foremost example of Gamification in action. Good problems force us to review what we have learnt, give us a chance to practice mundane skills, but also give us an opportunity to artfully apply these skills in more subtle and refined ways.  They provide examples of connections which the lecture material does not have a chance to cover, they give students a chance to fill in gaps that the lectures may have left. When combined with a good and comprehensive set of solutions, problems are the best way for students to become active in their learning of mathematics, a critically important aspect. When further combined with a skilled tutor/marker who can point out both effective thinking and errors in student’s work, make corrections, and advise on gaps in our understanding, we have a really powerful learning situation.

Here is where the Massive in MOOCs largely kills effective learning. It is the same situation as in most large first year Calculus or Linear Algebra classes around the world. Officially there may be problem sets which students are exhorted to attempt, but in the absence of required work to be handed in and marked, students will inevitably cut down to a minimum the amount of written work they attempt. In the absence of good tutors who can mark and make comments on their written work as they progress through the course, students don’t get the feedback that is so vital for effective learning.

Once you have thousands of students taking your online maths courses, it becomes very challenging to get them to do problem sets and have these marked in a reasonable way. The currently fashionable multiple choice (MC) question and answer formats that people are flocking to can go some small way down this road, but rarely far enough. Students need to be given problems which require more than picking a likely answer from a,b,c or d. They need to define, to compute, to evaluate, to organize, to find a logical structure and to explain it all clearly. This is practice doing mathematics, not going through the motions!

When we are planning an open course for possibly tens of thousands of students from all manner of backgrounds, the possibility to craft really good problems accessible to all diminishes markedly. There is no hope of giving feedback to so many students for their solutions, so all we can aspire to are MC questions that inevitably ride on the surface of things and don’t effectively support the crucial practice of writing. Learning slips into a lower gear. Such an approach cannot be the future of mathematics education. Tens of thousands of students going through the motions? They will find something more worthwhile to do with their time, like just watching YouTube maths videos!

But a slight rethinking of the enterprise, together with some common sense, can perhaps orient us in a more profitable direction. An education system ought to make enough money to at least fractionally support itself. People are willing to pay for something if it has value to them, and they tend to work harder at an activity if they have committed to it monetarily. All good technical writing has a well-defined audience in mind. These are almost self-evident truths. What we need is to think about crafting smaller, targeted open online courses, that generate enough income to support some minimal but effective amount of feedback on students’ work on real problem sets. By real I mean: problems that require thinking, computation, explanation.

Can this be done? Yes it can, and it will be the big education game changer, in my humble opinion. We will want to stream people into appropriate courses at the right level. Entry should be limited to those who have enough interest and enthusiasm to fork out some—perhaps minimal, but definitely non-zero!—amount of money, which hopefully can be dependent on the participant’s region; and who can pass some pre-requisite test. Yes, testing for entry is an excellent, indeed necessary, idea that will save a lot of people from wasting their time. Having 300 people from 10,000 pass a course is not a successful outcome. Better to have targeted the course first to those 1000 who were eager and capable. Then you get a lot more satisfaction across the board, from both students and the educators involved.

A major challenge will be how to provide effective feedback for written work. Relying exclusively on MC exercises should be considered an admission of failure here. If and when this challenge is overcome, TOOCs will have the potential to radically transform our higher education landscape!

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A radical birds-eye overview of algebraic geometry

Let’s talk about a rich and fascinating branch of mathematics called algebraic geometry. The subject has its beginnings with Descartes’ realisation that geometry could be approached algebraically by first introducing coordinates. In this way points become pairs, or triples, of numbers; lines become linear equations; conics become quadratic equations etc., while relations between objects can be encoded and studied purely algebraically.

In this brief note, I want to outline a somewhat radical birds-eye view of the subject, without getting into details. I probably should qualify my expertise here, in that I am not a professional algebraic geometer in the usual sense of the word. Nevertheless I have been studying the subject from a new point of view for more than ten years now, and have arrived at some rather novel understandings of what the subject is about. So what follows is my ten-minute take on algebraic geometry.

The most essential fact of the subject is that it is divided equally into two interlocking areas, the affine theory and the projective theory. The former theory rests on a vector space over a field, the latter theory rests on the associated projective space of lines through the origin. Neither is primary, contrary to popular belief; they are equal partners, and pretty well all aspects of the subject have both an affine and a projective version.

And what field are we working over? Certainly the rational number field is by far the most important, but finite fields are also  interesting, as are various extensions of the rationals,  for example the complex rationals obtained by adjoining a square of -1. But the truest theory is that which applies across the board to all fields (with the notable exception of  fields of characteristic two, which ought not to be called fields!) Note that the usual ‘field of complex numbers’, built on the so called ‘real numbers’, must be avoided at all costs if one aspires to be logically careful; it is a fantasy arena in which almost all our dreams come true, at the cost of abandoning our hold on mathematical reality and diminishing the natural number-theoretical richness of the subject.

Returning to the large-scale organization of the subject, there is a complementary and largely independent subdivision of the subject into various layers depending on the complexity, or degree, of the objects and operations involved.  The main distinction is between the first half–the linear theory, and the second half–the nonlinear theory.

The linear half of algebraic geometry is the more important half, and it goes by another name: linear algebra. This is the study of points, lines, planes and their generalizations and relations. The nonlinear half is itself divided roughly into two halves: the quadratic theory and the non-quadratic theory. The quadratic half is again more important than the non-quadratic half, and is occupied with conics and their associated metrical structures, namely bilinear or quadratic forms.

The non-quadratic half of the nonlinear half is again roughly equally divided into the cubic/quartic half and the higher degree half. Degrees three and four seem to be naturally linked, and support structures that don’t easily generalise to higher degrees. Although one could keep on subdividing, it seems reasonable to lump degrees five and higher into one-eighth of the subject.

I ‘ll try to figure out how to make a table to summarise the situation. But at least you get a sense of the various natural compartments of the subject, at least along the lines of how I see things currently.

What one studies in each of these areas is another important matter of course, but one that seems secondary to me to the basic subdivisions described here. Perhaps this rough guiding framework may provide a simple-minded but helpful orientation to the beginning student.

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Recent LT grant application

This semester I have been on Long Service Leave, so I am off the hook for teaching, and can spend more time with my graduate students Ali Alkhaldi and Nguyen Le, do some investigations into hyperbolic geometry and related issues, make more videos, and do some travelling. Ali is in his fourth year of the PhD, and is writing up his work on the parabola in hyperbolic geometry, which is now blossoming into a major re-evaluation of this subject, with dozens of new theorems. Nguyen is in her second year, and is making good progress on various aspects of Euclidean and relativistic triangle geometry, at this point related to the Incenter hierarchy.

Another main pre-occupation in the last few weeks is a large LT (Learning and Teaching) grant that Chris Tisdell, Bruce Henry and I have applied for, in conjunction with other colleagues here at UNSW, and some other universities in Australia. The country’s chief scientist, Ian Chubb, has organized a largish pool of money to be allocated to projects that improve teaching of maths and science in secondary schools in Australia, and our project proposes to address this by creating online professional development courses for high school teachers that teach them more mathematics and science.

Teachers are exposed to lots of in-service development that addresses the educational side of things: how to improve student learning, principles of effective pedagogy, teaching strategies etc. But in mathematics I think the greater problem is that not enough high school maths teachers understand the content of the subject well enough. We hear lots of anecdotal stories of Principals hiring Phys Ed teachers to teach mathematics because of shortages: after a quick 6 week training period the hapless new teacher is expected to inspire and motivate his/her students in a subject many of them already find difficult. Clearly not a very good situation.

Our idea is to make some high quality online courses that lay out the new Australian maths and physics curriculum in an engaging way for Years 11 and 12 teachers. The lectures for these courses would be make freely available on YouTube to anyone. With enough resources, we are hoping to put together videos and other materials with cool graphics, animations, demonstrations etc that will inspire high school teachers (and students too).

In addition to these courses, we hope to also organize a (YouTube) video library of Year 9 and 10 maths subjects, also aimed for teachers, that systematically presents the subject in a careful and fun manner. Let’s hope we get funded! If we do, I will probably be spending more time in the next few years making videos and online courses, and less time doing traditional teaching, which would be a nice change. [As you probably know, I like making YouTube videos!]

On another front, I will be heading overseas soon, visiting fellow geometers at the Universities of Innsbruck and Graz in Austria, and then on to Toronto via Florida to visit my family and friends. I will stop at my old alma mater the University of Toronto to talk maths with my friend Joe Repka and others, and I usually give a talk there.

My daughter Ali loves going to Canada, and she is taking some extra weeks off school to make the trip worthwhile. I will try to give her some personal maths instruction to make up for her lost classes, but that is not always easy! Right now we are talking about Pythagoras’ theorem, and I am trying to get her to see that it is really rather remarkable.  I think that possibility is not emphasized in school: that we are here witnessing a small miracle: make a right triangle, carefully draw three squares on the three sides (graph paper is essential for this), compute the areas of those squares, and then notice that the sum of the smaller two equals the third. And if you try it with a triangle which is not right, it doesn’t work! Isn’t that amazing?

 

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