# Category Archives: mathematics: general

## Too innumerate

In case you missed it, the New York Times has just published an opinion piece aptly named “Is Algebra Necessary?”. The title basically says it all: the 3 pages of the article boil down to the argument that math is hard and students don’t like it, so we should cut it out from most school and university programs, replacing it by “citizen statistics” or something like that.

Think of math as a huge boulder we make everyone pull, without assessing what all this pain achieves. So why require it, without alternatives or exceptions? Thus far I haven’t found a compelling answer.

Let me try to help. A friend has linked to this page describing the new British NHC policy on availability of testing strips to people with diabetes. I don’t have diabetes myself, but I know folks who do, so here’s a little bit of background based on what I know from talking to them and reading up on it.

When you have diabetes, your body either does not produce insulin at all or does not produce enough of it to metabolize sugar properly. To maintain blood sugar levels within an acceptable range, diabetics need to monitor strictly their food intake and inject insulin when necessary. This is a tight balancing act. Not enough insulin, and your blood sugar level skyrockets, leading to serious complications. If you inject too much, your blood sugar level drops too low, with diabetic coma and death as a possible consequence. So, the amount of insulin has to be just right, relative to your metabolism and food intake.

How do you know when your blood sugar level is within an acceptable range? Mostly, you don’t. You can only find out by testing. That’s where the test strips come in. You prick your finger, smear the blood drop on a test strip, put the strip in a meter, and read the results. Each strip can only be used once. Frequent testing is essential, not only to know whether your blood sugar level is acceptable at a given time, but also to figure out how it depends on your food and insulin intake over time, for example how quickly it increases right after a meal and decreases afterwards. With that knowledge, you can better time your meals and insulin shots.

Fortunately, I have not had the experience, but here’s what the Female Science Professor had to say about it:

I had a little booklet in which I recorded my blood sugar level, but I also started keeping track of the results in a spreadsheet and I graphed the results every day. I got interested in the shape and magnitude of some of the blood sugar highs and lows, but my initial sample spacing (in time) was too rough to get a satisfactory graph of these spikes, and there were other aspects of the data that I didn’t understand when I did the minimum number of recommended tests.

So, despite my loathing for jabbing myself in the finger with a sharp object, I started collecting more data. I tracked the blood sugar spikes so that they were defined by more than one point and I could really see their shape and I was certain of their maximum values. I collected data day and night. I dreamed of a device that could provide a continuous readout of my blood sugar and make perfect graphs. Even with my primitive data collection techniques, however, I made beautiful graphs and I did things with the graphs in terms of how I analyzed them over different time periods and how I displayed the data. I was obsessed with these graphs.

Filed under mathematics: general

## Women in math, and the overhaul of the publishing system

If you have not yet heard of the Elsevier boycott, you have a lot of reading to catch up on. I’ll wait. I’m not likely to miss traditional commercial publishers when they’re gone, which could well happen within the next decade or so, especially if they and their agents keep asking for it. Think whatever you want about the Cost of Knowledge website, but open access journals have already gained a lot of ground, we have taken charge of the dissemination and advertising of our own research on the internet, and good luck to any journal that tries to stop authors from placing their articles on publicly available webpages and preprint servers.

The better question is: do we still need journals, be it commercial or any other kind, and if not then what will replace them? Among other possibilities, open web-based evaluation systems have been proposed.

This post suggests that a web-based evaluation system would be good for women, the idea being that “women don’t ask” and therefore they are less likely to, say, submit a paper to Annals. I see it exactly the other way around. I’ve talked about some aspects of it already, but not all, and in any case it never hurts to say something more than once, especially when you’re female.

This is not to say that I’m against discussion boards for mathematicians on the internet. I’ll be very happy to have them, as long as they’re not mandatory for everyone and don’t drive out those parts of the current system that function reasonably well. We need more options, not fewer. For instance, I rather like the idea of “evaluation boards” to which authors could submit arXiv papers for validation, without the boards ever pretending to “publish” or “disseminate” the papers. That, if done right, would preserve the advantages of the current system while losing most of its disadvantages. (And it should work just fine for women, I think.)

Now, the details. (This is another one of those long posts. Sorry.)

Proxies. It would be really, really nice if we just evaluated everyone based on the actual merit of their work:

To fix the academic publishing mess, researchers need to stop sending their work to barrier-based journals. And for that to happen, we need funding bodies and job-search committees to judge candidates on the quality of their work, not on which brand name it’s associated with.

Happily, there are signs of movement in this direction: for example, The Wellcome Trust says “it is the intrinsic merit of the work, and not the title of the journal in which an author’s work is published, that should be considered in making funding decisions.” We need more funding and hiring bodies to make such declarations.

If we all did that, there would never be any need ever to worry about either publishing or gender bias. We’d love to be judged purely on merit. Also, everyone should get a pony.

Filed under academia, mathematics: general, women in math

## 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.

## 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.

## Is collaboration making us smarter?

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.

Filed under mathematics: general

## 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.