Category Archives: mathematics: general

Let’s overhaul the seminar!

With all the talk in the blogosphere recently on how we should overhaul the science publishing system, I started thinking about what else might be in need of an update… and, well, isn’t it time to rethink the weekly ritual of the seminar?

The research seminar as practiced today has been around for centuries. Its format goes back to the days when the primary means of disseminating scientific research were journal publications and handwritten letters exchanged between scientists – that is, of course, unless you were lucky to attend a seminar given by a visiting scientist who would tell you all about his recent research, some of it not published yet! So now you knew about it! We don’t have go go back all the way to the 19th century for this, either. The 1960s or 70s would do quite nicely.

The seminar has survived in an almost unchanged form since then: the introduction of the speaker to the audience, the lecture, thanking the speaker for the first time (also known as “the first clap”), the question and answer period, thanking the speaker again (the second clap). There’s sometimes a dinner in the evening.

But the times, they are a changin’. The internet takes care of the dissemination well enough. The lecture has since been discredited as a teaching tool and many groups of educators seek to lose it. Do we still need the seminar, then? Or should we try to reorganize it?

Here are my modest proposals.

1. Modernize the seminar taking current pedagogical research into account. Why must we stick to the outdated lecture format? Let’s make it interactive! The audience could be asked to read the introductory materials ahead of time. Upon entrance, they will be given clickers and prepared worksheets. The speaker will then guide them through a series of hands-on activities that will enable them to discover for themselves, say, the Langlands fundamental lemma. Upon completion of seminar, the participants will be able to derive fundamental lemmas, or something like it. They will also be given Math Blaster points (displayed on an internet scoreboard) that they could elect to convert to their choice of Dungeons & Dragons or World of Warcraft points.

Let’s not forget real-world applications, either. If you proved Poincare’s conjecture, that’s nice, but we also need to know how this will influence the development of the next generation of iPods. Please try to explain it to us in about 10 minutes. A multimedia presentation would be great.

But if we’re not ready to lose the lecture altogether, then here’s another option.

2. Follow the Khan Academy model. The Khan Academy videos feature – you’d never guess! – a guy lecturing. On a blackboard. With the electronic equivalent of colored chalk. In a fairly monotone voice. The explanations of math topics are very good, but not necessarily head and shoulders above what I see around here.

The difference is that Khan has found the perfect format for the lecture: a 10-minute single-topic video clip that can be watched at home. There’s no need for the lecturer to hold your attention for 50 minutes without interruptions. If you missed some part of the explanation, just rewind the clip and watch again. The instructor does not have to vary the pace, insert jokes or resort to gimmicks, or deal with classroom discipline issues. He speaks in a normal conversational voice, without having to project across a large classroom. He does not have to struggle in class with the AV and IT equipment, generally at the development stage of MP3 players before the iPod. He does have to set up the software and recording equipment, record, edit and upload the video, but all this happens off screen. All we ever see him do is explain the math, simply and naturally.

So… why wouldn’t we follow suit? Instead of travelling to conferences and seminars (think TSA screening lines, or the middle seat with broken in-flight entertainment system on a transatlantic flight) we could simply record a series of videos explaining our work and post them on YouTube. We could even upload them to a central, Math Overflow type website that would award Math Blaster points to logged-in users.

There could be problems, naturally. For example, not everyone was born with the gift of the golden voice. Some of us have less than perfect enunciation or even a foreign accent. Would we start hiring voice-trained actors for our videos, and would that be an NSERC-eligible expense? Come to think of it, why not also hire someone better looking, given that we (women) are constantly judged on our appearance even in contexts that should have nothing to do with it? Or why not just hire a male actor to narrate my presentations, in a Remington Steele fashion? Personally, I’d love to hear James Earl Jones explain this paper to a YouTube audience.

Disclaimer. Since apparently a lot of people have trouble recognizing irony when it’s being employed by a woman: no, I’m not actually making these proposals seriously. The YouTube videos might be worth trying for those so inclined, but I’d never want it to become a de facto professional obligation. There are legitimate reasons for wanting to avoid YouTube celebrity.

I’m aware that some of our conference lectures get videotaped and then posted on institute web pages. There might even be a few of mine around. In terms of effectiveness, it’s taking a presentation in one format and converting it to another one, with a lot of compatibility issues along the way. Also, the videos don’t tend to propagate and generate notoriety in the same way as YouTube clips, or at least I have not seen that happen.

I do think that there are points to be made by juxtaposing the way we learn our own trade with the way we propose to teach it to others.

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Random thoughts on publishing and the internet

I doubt that there is anyone reading this blog who does not also read Tim Gowers, but in case you missed it, here’s his blog post proposing a hypothetical alternative publishing model in mathematics: essentially, a massive website combining the functionality of arXiv, Math Overflow, and more. There is also a revised (mostly scaled down) version, where the website would mostly serve as a venue for exchanging constructive feedback.

I’m old enough to remember the days when most math departments had pre-printed postcards with requests for journal offprints. (“Dear Professor [fill in the blank], I would be most grateful if you could send me an offprint of your article [fill in the blank] that has appeared in [fill in the blank]“. That’s what the offprints were for, mostly. They also looked much better than a manuscript typed on a mechanical typewriter, with handwritten math symbols.) Scientific journals actually served to disseminate information back then – checking new issues in the reading room was an important part of keeping up with recent developments. Ah, the good old times.

Dissemination is in our own hands now. I usually check the arXiv every day, but it’s been years since I last bothered with the current journals in the library, other than to look up published versions of papers that I’d already seen as preprints. Of course we will want to take ownership of the rest of the publishing process: the record-keeping, the peer review with its twin goals of debugging papers and evaluating their merit. These are functions that are worth keeping. I do use the library on a regular basis for older articles; I’d rather cite a stable, debugged journal article (where possible) than a preprint that could get replaced or pulled down tomorrow; and, as inaccurate as it can be to judge papers by the journals they appear in, I’d rather have such (approximate) marks of the quality of my work in place than leave it to each year’s departmental committee on merit pay increases to try to figure out all over again what I’m doing and why it’s supposed to be important.

It’s clear enough that any alternative publishing model will likely be internet-based, with interactive components possibly similar to Math Overflow or blog comment sections. It has also been noted that women have significantly less visible presence on MO than they do in research mathematics overall. One might ask, therefore, whether switching to an internet forum-based model of publishing could have the side effect of alienating women mathematicians and driving them out of the field.

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Is collaboration making us smarter?

This sounds about right:


Back to our puzzle about reasoning: why does it lead to better performance in the context of group discussion? To provide a proximal explanation, we would have to look inside people’s head: what happens there that make them better at solving problems when they discuss with others than on their own? In future posts, I will suggest such an explanation. But for now let’s focus on the ultimate question. Whatever the psychological mechanisms are that make people better reasoners in group discussion, why do they work that way?

To answer the ultimate question, we can turn to a suggestion made by Dan Sperber in a couple of papers from the early noughties. His idea, in a nutshell, is that reasoning evolved for argumentation: so that we can convince others and to examine the arguments they offer us. Reasoning would be adapted to work in dialogue, when people exchange arguments, and not within the confines of a solitary mind. Just as human lungs work better in normal atmospheric conditions because they are designed to work in these conditions, reasoning works better in group discussion because it is designed to work in such a context.

The last time I wrote a single-authored paper was back in 2000; everything since then has been collaborative work. I’m not sure how much longer this will continue.

The functional aspects of collaboration are obvious: the wider range of collective expertise, the complementing abilities and skills, the sharing of work. There is the camaraderie between coworkers if the chemistry is right. Of course, there are also collaborators who can’t agree on anything, insist that the paper be written their way or no way, or at the other extreme, who won’t answer email for months.

All other things being equal, though, I’ve noticed that my own thought processes seem to work better when I’m collaborating with someone else. This is not just a matter of receiving feedback from collaborators and benefitting from their contributions. It’s more subtle than that. It’s that, somehow, my own brain shifts gears sometimes and finds more effective ways of thinking about the subject when it knows that I’ll be discussing it with an actual live person soon. I have no idea what this does to my IQ – I actually don’t even know my IQ, never took the test – but the effect is noticeable enough, consistent, and can’t be attributed to anything else that I can think of.

I’d be very interested to know what everyone else thinks.

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Math < reality.

John Allen Paulos explains how the bell curve works. The bell curve, you will recall, is given by the equation

\displaystyle{y=\frac{1}{\sqrt{2\pi \sigma^2}}\exp\Big(-\frac{(x-x_0)^2}{2\sigma^2}\Big)},

where x_0 is the average value of the variable (the peak of the bell curve) and \sigma^2 is the variance. Paulos’s point is, apparently, that small differences in the values of x_0 and \sigma can lead to extreme imbalances at the far ends of the curve (\int_R^\infty y dx for large values of R). Here is how this might manifest itself in practice:


The corporation’s personnel officer notes the relatively small differences between the groups’ means and observes with satisfaction that the many mid-level positions are occupied by both Mexicans and Koreans.

She is puzzled, however, by the preponderance of Koreans assigned to the relatively few top jobs, those requiring an exceedingly high score on the qualifying test. The personnel officer does further research and discovers that most holders of the comparably few bottom jobs, assigned to applicants because of their very low scores on the qualifying test, are Mexican.

She may suspect bias, but the result might just as well be an unforeseen consequence of the way the normal distribution works.

Yes, really. Of course, Paulos chose the direction of the imbalance at random. He says so right in the article.

There’s a way of misusing mathematics that goes like this: start with a mathematical model, often a probability distribution or a differential equation, that looks reasonable enough in typical circumstances. Then assume a very specific set of circumstances, for example making one of the parameters abnormally large, plug this into the general purpose equation, manipulate it for a bit, and draw the conclusions. QED, or something.

What’s missing from it is a level of mathematical maturity. In my experience of teaching undergraduate mathematics, manipulating exact formulas is the easy part for most students. (Relatively speaking, of course, but whatever.) The hard part is the inequalities, approximations, error estimates. You no longer have an exact equation that can be rearranged every which way and still remains equivalent to the original one. If you move around and rescale the terms in an approximate formula, the error might still be acceptable, or it might not be, and you can’t always tell which is which by just backtracking through an automated series of algebraic manipulations. You actually have to understand what’s going on.
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Just what I needed.

Yesterday I tweeted a link to a web page that produces instant drabbles. (Don’t ask.) There should be a similar page for instant math papers, I thought. We could all use an extra paper or two when we apply for a grant, come up for promotion, or (shudder) go on the job market. In response, Ahavajora suggested this link. So, I tried it. Here’s my brand new instant math paper:

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The mysterious effectiveness of trivial pursuits

This is probably not worth posting about, but I’m curious to see if anyone will respond.

I usually take notes while reading a math book or article. There may be obvious utilitarian reasons to do so, for instance I may want to reorganize a proof in order to understand it better, fill in some gaps, work out the details of the omitted calculations, and generally rewrite selected parts more to my liking. No, I didn’t really expect that you’d be surprised to hear any of that.

What’s more interesting is that I also seem to benefit from the physical process of writing out my notes in longhand, the old-fashioned way. This summer I spent some time reading a paper that, in addition to proving a significant result, is a perfect example of how I’d like all math papers to be written. Aside from a couple of minor calculations that I wanted to work out in detail, there was no need to add, clarify or reorganize anything whatsoever. Even so, I still found that copying bits and pieces of the paper by hand helped me understand it better, as if the motor activity fired up some part of my brain that would otherwise remain disengaged.

It was not the first time I noticed this, and I don’t think that this is just me, either. I even have a vague recollection of reading a popular science article that said something to that effect. (That was a long time ago and I can’t find it now.) It works in other settings, too. You might expect that taking notes during a lecture would divide my attention and leave me less able to focus on the subject matter, but actually it’s quite the opposite. This is of course assuming that I can follow the lecture, at least in principle; some topics are too far out of my reach, and some lectures can’t be saved. But I digress.

Now, here’s where I have a question, especially to the readers under the age of 30 or so. I’m told that handwriting is out of fashion these days, all but replaced by typing and texting. This article reports that the younger generation in China loses the ability to write Chinese characters by hand because they don’t have much use for handwriting except to sign the back of their credit cards. Over here, I suppose it helps that we have a somewhat less complicated alphabet, but it’s still true that there are fewer and fewer reasons to write anything in longhand. Even signatures might become obsolete eventually, replaced by PINs, passwords and biometric technologies.

This isn’t going to be a “kids these days” rant. I’m not particularly eager to go back to the days when “cutting” and “pasting” meant using scissors and glue, respectively. I don’t really miss the logarithmic tables, either, in case you were interested. Nor would I want to type all of my papers the way I typed my Master’s thesis, on a borrowed mechanical typewriter with no word-processing capabilities and no math symbols. Those had to be filled in by hand, in the spaces you’d have to leave between the typed letters and numbers. For corrections, I used the white-out fluid that a colleague had brought from a trip abroad. Also? The wired kids these days might not even know the literal meaning of the expression “carbon copy”, or they might look at old issues of scientific journals and wonder about the instruction for authors to submit the “original copy” of the manuscript. I could explain it all to them in more detail than they’d ever want.

But I said I had a question. It’s this: if you didn’t grow up doing a lot of handwriting, if it’s typing and texting but not longhand that feels natural, do you still take handwritten notes when you read a math paper, for reasons similar to those I described? If not, is there something else that you do instead? Taking notes in TeX, for example, does not do it for me. Obviously I use TeX to write papers and exchange notes with collaborators, but typing in TeX distracts me from thinking about the mathematics involved, whereas writing in longhand helps me focus on it. Does anyone here see it differently?

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The girl who played with Fermat’s theorem

I finally got around to reading Stieg Larsson’s Millennium trilogy over the last couple of weeks. In case you too are late to the party, here’s a New York Times article about Larsson, his books and his legacy, and here’s the trailer for The Girl With The Dragon Tattoo, the Swedish movie based on the first book in the series.

The best thing about the trilogy is its feminist angle. The villains are “men who hate women” (the Swedish title of The Girl With The Dragon Tattoo), and here Larsson has a point of view that’s all too rare in mainstream popular culture. His female characters aren’t just props against whom crimes can be committed so that the action could advance. They’re actual human beings who have agency, fight back and take control of their lives, even as they remain damaged by the experience. Larsson does not romanticize domestic or sexual violence – it’s not about love or sex, it’s about control and humiliation – nor does he spare the legal and welfare systems that let the victims fall through the cracks too often. (The Robert Pickton case comes to mind, for several reasons.)

Parallel to this, and not entirely unrelated, is the nagging sexism in the workplace, the media, and the society at large:

She had been the first journalist to pounce on the story, and without her programme on the evening that Millennium released the scoop, it might not have made the impact it did. Only later did Blomkvist find out that she had had to fight tooth and nail to convince her editor to run it. [...] Several of her more senior colleagues had given it a thumbs-down and told her that if she was wrong, her career was over. She stood her ground, and it became the story of the year.

She had covered the story herself that first week – after all, she was the only reporter who had thoroughly researched the subject – but some time before Christmas Blomkvist noticed that all the new angles in the story had been handed over to male colleagues. Around New Year’s Blomkvist heard through the grapevine that she had been elbowed out [...].

This is stuff that I normally only read on feminist websites. I’m not used to seeing it in #1 New York Times bestsellers.

The first book in the series, The Girl With The Dragon Tattoo, is also the best one and I’ve caught myself wishing that Larsson had stopped there. It feels like a cop-out when we learn in the third book that “All The Evil” (Larsson’s term) was really the work of a few deranged individuals overstepping legal boundaries and that the negligent legal system of TGWTDT just needs a good kick to snap back into place. If only it were so simple.

The worst thing about the series is the mathematical interludes in The Girl Who Played With Fire. We’re told that Lisbeth Salander, the goth hacker played by Noomi Rapace in the movie, is also a puzzle-loving math genius who solves Fermat’s last theorem, or thinks she does, in a passage that Tim Gowers singled out for attention some time ago.
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