It's nice to see you again!

 I've always claimed --- and at least once on this blog --- that I like having a job that combines research and teaching, but the last few years have really pushed this conviction.  It turns out that I like have a job that combines research and teaching actual students.  I'm not saying my 1014 and 1116 Zoom students were bots, but when you're talking to a picture of yourself and a lot of black rectangles on a screen, it's hard to tell the difference.  

 Lecturing to a large class is the closest most mathematicians get to being rockstars (and granted, it's not that close!), and it turns out I'd rather have a live concert in a small venue than a lot of downloads on Spotify.

It's a reflection of my tremendous luck and privilege that Zoom lectures are one of my big complaints about the last few years.  I recognise this, and my heartfelt sympathies go out to all those who've dealt with the much worse consequences of the pandemic.  I know this includes many of my students, and the effects are on-going, so I'll wrap this up before I get too wrapped up: it's been really wonderful to have the chance to talk with students again.  Human connection was one of the big casualties of defending against COVID, and it's fantastic to see this recovering.  Come to office hours --- it's great to have you back!

T&R

I think there are a lot of reasons that being a researcher makes me a better teacher, but it's not a one-way street: teaching is also good for my research.

Teaching reminds me that I won't be stuck forever.  I'll be honest, I remember being really confused by change of basis matrices when I first met them.  I also recall fighting hard with the Isomorphism Theorems for groups, but happily, both these topics seem completely transparent now.  I can simultaneously be sympathetic to my students' struggles and assure them that they'll get there eventually.  This is an honest assessment born of experience, not feel-good propaganda, and yet it's much easier for me to remember this in the context of teaching.  In research, you spend all your time dealing with the parts of the problem you don't yet understand, so it's really encouraging to have reminders that progress is possible.

Having to get to lectures on time also reminds me how nice it is to work on maths in a cafe!

A Flagrant Waste

A few weeks ago I was invited to speak at the Fenner Hall Academic Dinner, an event whose theme was a Joe Hockey quote about university research: "A Flagrant Wasted of Taxpayer Money".  I usually give talks on whiteboards, so it was a fun departure from the norm to prepare an actual speech.  Since I do believe pure research is important and that it needs some defense, I thought I'd give my remarks a bit longer life here:

In 1850, the British Chancellor of the Exchequer William Gladstone reportedly asked Michael Faraday what possible practical value electricity could have.  Faraday's answer was supposedly, ``Why, sir, there is every probability that you will soon be able to tax it." (nb: Wikipedia says this is probably apocryphal.)

This might satisfy Joe Hockey, but I'd like to offer some other thoughts on the value of research.

I'm a mathematician, and in some sense, this is a comfortable position to speak from.  Nearly every discipline of contemporary computing, science, and technology relies on mathematics to some extent, and it's an easy sell to point out that mathematics is the foundation of the internet, medical imaging, accurate polling, etc.

However, I'm actually a pure mathematician, and my work does none of that.  I study geometric topology, which means that I spend my research life studying knots and surfaces in three- and four-dimensional spaces.  It's fascinating, but it's not particularly practical.  For a small taste of what I mean when I say I study knots, consider the pictures on the screen. 

(There were four knot diagrams; I'll try to put theme here for fun at some point.)

Each one depicts a knot, in the sense that you could use this figure as a model to arrange a string in space. If you then glued the ends together and left them attached, you'd have a physical model of a mathematical knot.  Sometimes people assume that mathematics is only about equations, but knots are actually rather hard to study with equations; just as you might build a  loop out of string rather than steel, the mathematical study of knots allows loops that flop and stretch and rotate, not ones that stay rigidly in one place. 

It happens that if you build a string model of each of the knots on the paper, you'd actually have built two models that represent the same knot, in the sense that you could transform one of your loops to another without cutting and regluing.  My question for you is this:

Which two pictures represent the same knot?

The simplest question in knot theory is how one should go about answering this question, and some of my research builds mathematical tools to answer this question. 

Although the maths department at ANU is part of CPMS, mathematics lies at a strange intersection of the very practical and the artistic/philosophical.  It's easy to explain why applied mathematics should be funded, but I don't think this as interesting a question, nor one challenged  much in the public sphere.  In the case of basic research however, this is a very real issue; the Government has recently proposed a national research framework which assesses disciplines based on industry engagement and impact, criteria which leave much of pure mathematics out in the cold.  So today, let me restrict my attention to the chilly reaches of pure mathematics, the stuff that looks like maths for maths's sake.

The first defence of pure mathematics is the ``Not Applied Yet" claim: this work may seem devoid of application, but give it time ---year, decades, centuries--- and it will turn out to be useful.  There's some merit to this, as mathematics has an amazing track record of developing theories that end up being applied in unexpected ways.  

 It's important to fund pure research because it's not clear in the moment what will turn out to be useful.  I think this is broadly true, but I don't think it's the full story.  In particular, I'm reasonably confident that none of my research papers will ever contribute to building a better widget.  I hope my work helps us understand something about the universe, but I don't think it will ever solve practical problems.  So why do I ask the ARC for money and think that they should give me some?

It's a challenge to mount an articulate defense of ``curiousity driven research" especially in the face of competing budgetary demands for things that immediately affect peoples' lives.  Luckily, I'm not the first person to attempt this, so let me turn to Abraham Flexner, who was the first director of the Institute for Advanced Study in Princeton.

Flexner wrote an essay titled, ``The Usefulness of Useless Knowledge" in 1939. In it, he brushes away someone's admiration for Marconi, a Noble laureate and inventor of the radio, saying that the credit should go to the people who developed the fundamental theories, not simply applied the technology: He writes, ``Hertz and Maxwell were geniuses without thought of use.  Marconi was a clever inventor with no thought but use." 

The point here is not to turn up our noses at applied work --we certainly all live better lives as a result of it --- but to recognise the value of work which isn't motivated by application but which may someday turn out to be useful.  Flexner writes in the same essay,

I am not for a moment suggesting that everything that goes on in laboratories will ultimately turn to some unexpected practical use or that an ultimate practical use is its actual justification. Much more am I pleading for the abolition of the word “use”, and for the freeing of the human spirit. To be sure, we shall free some harmless cranks. To be sure, we shall thus waste some precious dollars. But what is infinitely more important is that we shall be striking the shackles off the human mind and setting it free for the adventures...
I'll make an easy concession, which is that much of the research done is not earth-shatteringly, world-changingly important. 

But some of it is. 

Sometimes, new ideas come along that change how we understand the world. Sometimes these do turn into technology, but sometimes they simply tell us something fundamental about the universe we live in.  Understanding that light is a wave and a particle is worthwhile even in the absence of applications.  General relativity, quantum mechanics, genetic inheritance ---these are triumphs of the human quest to understand the world.  And these ideas don't pop up in isolation. You can't selectively fund only the rare genuises (genii?) who will make the revolutionary discoveries. This is not just a detection problem, solvable with a better review process that screens out all the chaff--- but a reflection of how knowledge progresses, bit by bit, little by little.  I don't think my work will build a better widget, and I don't think that I'll revolutionise our worldview, but I do think I'm laying bricks in an edifice that may someday have a fantastic spire. 

So let's turn back to the knots in front of you.  Any guesses?

One thing you might note is that A and C are mirror images of each other, but in fact, they're not equivalent.  On the other hand, if you added a mirror image of B to the list, it would be equivalent to the original B.  So this is a question you can ask about a knot: can if be deformed into its own mirror image?  When the answer is yes, we say the knot is amphichiral, and when the answer is no, we have an example of a chiral knot: one with left- and right-handed versions.  This notion of chirality appears in many branches of science, with a famous example from chemistry being a molecule whose RH version smells like caraway seeds, but whose LH version smells like peppermint.

Algebra Redux

This is my fourth semester teaching Algebra 1, and I am a well oiled machine.  My lectures notes are all written, I have a large stock of old exams for revision, and I know the material so well that I can cover it twice as fast!

Oops.

As convenient as it is to teach the same course again, I find that there are some drawbacks to familiarity.  For the most part, my presentation is smoother and I'm more likely to focus on what's really important instead of getting sidetracked, but there is an increasing disconnect between how new the material feels to me and to my students.  I think good teaching requires some imagination on the part of the lecturer: you have to simultaneously understand the material and also understand what it's like to be confused by it.  In a calculus class, the exercise in duality was always clearly necessary, so I put some effort into maintaining both perspectives.  For upper level classes, it hasn't required as much effort, but apparently traipsing through group theory for the fourth time (sixth, if you count Stanford) changes that.

(Btw, if you're in Algebra 1 and reading this, we're only going about 10% faster, so don't panic!)

Ironclad Inflation?

Winter break is almost upon us, and all that stands between me and the end of MATH1014 is tomorrow's Examiners' Meeting.  In keeping with my impression that the teaching procedures are more tightly regulated here, all the department instructors meet before final grades are assigned.  We've already been asked to submit our marks to the department, but we're not allowed to make even final exam scores available to the students until the College releases the grades.  At the meeting, we'll have access to the statistics from previous year's courses: pass/fail rates, average scores, number of High Distinctions, etc.  While I think that institutional memory is a valuable resource, I'm less fond of the fact that this information is more than just an optional guide.  In fact, we're not only expected to make our averages match the previous years, but also to match the College's predictions of what our students should score.  This isn't a last-minute surprise; after our students had lower than average mid-semester exam scores, we wrote a final exam designed to raise their scores so that we wouldn't have to shift them too much at the end.  I'm not entirely comfortable with the process --and to be fair, I have yet to see it in action-- but I wonder if the difference from what I'm used to is more in style than substance.

At Stanford grades were inevitable "curved", which really meant "raised".  It feels different to make such adjustments in the privacy and comfort of one's own Excel gradesheet, but for the moment I'll set aside the University's firm hand in the process.  I think that the bigger difference comes from the fact that grades here are assigned as numbers on a scale of 1 to 100, with descriptors like "Pass" and "Distinction" added afterwards.  At Stanford, I could (try to) design an exam with an average score of 60%, leaving room for an exceptional student to get 90% or the weak student to pass with 35%.  I think there's pedagogical value to an exam which delivers the message, "If you can do two-thirds of these problems, you're doing fine, but I want you to know that there's lots of room to do better."  (A topic for another post is whether this message gets through or whether such an exam is just demoralising.)  I had more flexibility in writing my exams because I assigned letter grades on my own terms.

A drawback of this approach is that it lacks transparency; students understandably found it frustrating not to know exactly where they stood throughout the semester.  This invariably worked to their advantage (see curve vs. raise above), but I think an ideal grading scheme leaves no one guessing.  Nevertheless, I don't think the ANU system is an improvement; it offers the precision of a numerical grade, but the retroactive rescaling renders this number as arbitrary as a letter.  In fact, we can't release final exam scores precisely to prevent students from being able to calculate their grades.  I'm curious to see how the meeting runs tomorrow; I suspect that most semesters, it will be a bureaucratic hassle, but at least this once it has the appeal of novelty.

Three's a crowd

I've spent this semester teaching MATH1014, a course aimed largely at engineers and non-maths science students that combines linear algebra and calculus.  A class fitting that description was my bread and butter at Stanford, so I was looking forward to seeing how the ANU version compared.

As it turns out, the two classes aren't that similar, but the biggest difference in my experience here is the amount of contact I've had with my students.  My 1014 course has about 90 students (in contrast to Stanford's 50 - 60 students per section in Math 51), but I had vastly more contact with my students in the U.S.  I've usually enjoyed office hours as a chance to talk to students one on one (or at least one on ten) and in the past I've been somewhat proud of the crowds I drew.  I always provided chocolate, and I like to think that the math was useful, too.  While lecturers here hold office hours in some official sense, there seems to be little expectation that anyone will show up: I've seen a total of six of my students outside of class, and three of them were making up quizzes they missed.

As much as I like to think that my Stanford students just enjoyed my company, I suspect the difference stems from how assignments are structured here.  The 1014 students complete weekly Web Assign quizzes, but they don't have to turn in regular problem sets.  This is certainly more scalable than assignments that require eyeballs to pass over them, but something's lost, as well.  My life is certainly made easier by the fact that assignments are ready-made and waiting for me, but the medium shapes the kind of questions that can be posed.  Words like "show", "prove", and "why" are all off limits, whereas there have been several questions that ask students to round their answers to three decimal places.  I haven't done a computation involving three decimal places since such things were done on calculators instead of phones, so this seems very strange to me.  MATH1014 is aimed at engineers, and perhaps the assignments are calibrated to their needs; if your goal for your students is computational competency, then technology offers an efficient way to evaluate them.  However, I think that linear algebra is a great opportunity for students to wrestle with abstraction, and I wonder if the syllabus is shaped by what's easy to evaluate.

Midterm madness

I'm in the last stages of midterm limbo, the period between finishing writing the exam and beginning to grade it.  (The exams are currently sitting in an ominously large box just down the hall, so I really should be exiting midterm limbo and entering grading purgatory.)  Exam contents may not vary much by continent, but the exam administration here has a distinctly different flavor (or as I should write, "flavour") than what I'm used to.

In the U.S., final exams are generally run by the university rather than the instructor, but at ANU, this holds for mid-semester exams as well.  I was allowed to specify what week I wanted for my midterm, but the exams office chooses the day, schedules the room, and hires the invigilators.  I suppose there could be some advantages to this system, especially with respect to exam conflicts and make-ups, but I was unimpressed by the three (four?) weeks we waited to find out when our exam would be held; it makes a big difference to the lecture schedule to have a Thursday exam instead of a Monday one.

The exam itself is quite formal: students are told what time the room opens, down to the minute, and they have to present their ID's to the invigilators.  I've heard of this sort of identity check at big state schools in the U.S., but ANU is smaller than Stanford.  After the time starts, the students have an additional fifteen minutes to look at the test before beginning to write.  During this time, they're allowed to write notes on the official "Scribble Paper" (labeled thus!), but they can't start writing anything that they'll ultimately turn in.  I think this is a great idea, and I wonder how easy it would be to import to an American college without this tradition.

The cheeping of the exams, "Grade us! Grade us!" is getting louder, so it's time to get to work...