Math curricula

The Ontario Government is getting into a frazzle about the failure rate of public school students in mathematics, offering bonuses for teachers doing skills upgrading.  Fingers have been pointed at the teachers' math skills, their math teaching skills, the curriculum, and other possible culprits.

There is, of course, a long-term issue with teachers and math skills, in that someone with good math skills coming out of university can find more attractive areas to specialize in than teaching.

However, I have spent the past several years watching my daughter deal with the curriculum, and I do think that there is a problem with it: it fits the needs of almost no children. It flits from topic to topic in such a way that students who have difficulty do not get enough drill to master the issues; but it also allows for no real exploration of the issues by students who are good at math, for the same reason.

They will have a two or three week unit on series, for example, which could be extended to any number of interesting things to explore; but then there will be a sharp break and they'll move on to an equally short unit on, say, geometry.

And while we're on the topic of math skills in the general population, maybe we should ask: what skills?

A few weeks ago I was on public transit and saw an ad designed to popularize the idea of a need for math skills. The example problem it gave was a simple word problem of the sort which requires conversion from words to a simple polynomial and then solving for x. It was soluble by someone with high-school math trivially. But I thought: for someone who does not work in the sciences or engineering, where is that type of skill necessary?  Certainly not in everyday life; nor, I think, in accountancy or other similar fields, where numbers tend to be delivered and manipulated in quite a different way. So for a student with an interest in (say) history and languages, there's no obvious reason for this sort of skill to be considered a critical one: yet the provincial curriculum makes it important. In contrast, estimating probabilities and risk in a back-of-the-envelope manner is probably more important in day-to-day life; humans are notoriously bad at it. Understanding some of the underlying issues around encryption (such as the difficulty of factoring large primes, or the ease of doing frequency analysis) is becoming a generally useful skill set (not the ability to do it, but the understanding of what is hard and what is easy).

Students who have the potential to be good scientists, mathematicians, engineers, and the like need a far more challenging, ambitious, and interesting curriculum than we are giving them. (A genuinely competent 17-year-old should be able to handle Hardy's Course of Pure Mathematics if properly prepared.) Students who are not going there need only a fraction of what we are shoving down their throats. (When was the last time I really needed, say, trigonometry at the high-school level? I don't have to measure very many things by angle-and-one-side mechanisms; either I need more -- Fourier analysis, full-blown trigonometric functions, etc., or I need none.) And the relatively rare students who have the ability but are interested in everything -- literature, history, philosophy, and languages as well as maths and sciences -- can handle the more difficult stream and benefit from it.

Isaac Asimov

Asimov is back in the news, largely because he made some predictions about 2014 in 1964.

In 1964, Asimov also published his second collection of robot stories, The Rest of the Robots.  Aside from the novelization of Fantastic Voyage (1966), his next book of science fiction was The Gods Themselves in 1972, and he did not resume novel-writing an a more regular basis until 1982, with Foundation's Edge, when he started to extend the Foundation series and then join it up with the robots stories. (He did publish many mystery short stories about the Black Widowers during the period.)

I grew up reading science fiction between about 1968 or so and 1978 (arbitrarily chosen as a terminus when I entered university and "was grown" and my reading patterns changed).  So when I was growing up Asimov was basically a science (and pop history, etc.) writer who used to write SF. (The Gods Themselves appeared just before I was in high school, and included a relatively refreshing attempt to imagine genuinely alien aliens, although it was otherwise very Asmovian -- and relatively dull -- in its human parts of the story: see Jo Walton's review of it.)  For me, the definitive Asimov stories were from before 1960; and I think that this is still true.

He had been assigned a place for some time as one of the three most important living SF writers, along with Clarke and Heinlein.  But it wasn't for the depth of his characterisation (nonexistent), or even the complexity of his plots.  And despite what's sometimes said, it wasn't particularly representative of "hard" SF: his two most characteristic innovations, the positronic robots and the discipline of psychohistory, are both handwaves which are, under any close examination, completely implausible.  (The positronic robots are doubly implausible, once at a physical level (how do we avoid a matter/antimatter meltdown?) and once at an AI level (how do we encode complicated concepts like the three laws in the first place?).) Instead, both of them provided a broad setting for stories which used plot twists to create surprise endings or plot reversals.  The same was true of the time travel in The End of Eternity.

His later turning of his earlier stories into episodes in a broad future history with R. Daneel Olivaw as a linking character was misconceived, although they sold well.  The patching-up was clunky and extraneous to the themes and structure of the earlier stories.  And he ended up relying heavily on telepathy as a plot device (he had already introduced it in the Foundation series with the Mule, and the Robots stories with "Liar!" (from 1941)), which moves the work firmly away from Science Fiction and towards Fantasy, regardless of the Campbellian grandfathering of psionics into SF.

In addition, the optimism of the post-war America which had permeated the earlier stories was gone.  (Foundation and Doc Smith's First Lensman, from this distance, both look more like the 1950s than the future.) If psychohistory had been held out as a good thing (assuming it were possible) in the early stories, the later ones are less sure. The absence of a leisure society with the help of robots was explained away by making the Spacers a "dead end" (to explain (away) the Empire as Asimov had originally shown it).

I can't think of any ex post facto attempt to tie everything together that has really worked -- it was one of the major problems with Heinlein's later books, from The Number of the Beast on, as well.  If you start with a plan to connect everything, that's one thing.  Patching in an odd work (like the incorporation of Triplanetary after the fact by rewriting it) can work.  But grafting differently-conceived works, and, even more, universes, together is an invitation to trouble.

There are problems with Asimov's work which were more-or-less invisible at the time but became more obvious as it aged.  It was thoroughly sexist, in a way that blended into the background in the 1950s, but became glaring by the turn of the century; and this never really went away, even in the later books, although he added more realistic women in the later books.  (Asimov's personal sexism  / harassment patterns play into this.)  What felt like purely neutral, invisible, prose at the time has also (inevitably) taken on a patina of age.

His influence has long been integrated into the field.  It's hard to remember now, but the idea of robots was less than 20 years old when he wrote his first positronic robot story (1939 for "Robbie" -- Capek's R.U.R. dates from 1920) and the model of non-threatening robots he introduced has long been drawn on in other works, some closer (there are relatively many authorized stories in the "early" robots universe by other authors) and some more removed from his original model.  And it's possible to argue that the many all-human space kingdoms / empires of later space opera draw some of their inspiration from the background to the Foundation books. Psychohistory is less of an influence, aside from works such as Psychohistorical Crisis which are direct homages, although one can see Dickson's William of Ceta as someone making a failed attempt to use a similar discipline (critiqued towards the end of Dorsai! by Donal Graeme).

I suspect that Asimov's works have "aged" (in the sense of becoming more alien) about as much as they're ever going to -- what survives now will continue to be read, more or less as the best of Verne's novels do -- at least by those interested in the roots of SF.  This is assisted by the fact that much of his better work was short stories, which show off his plot twist model far better than longer works (even the early "novels" in the Foundation series are really collections of novellas).  (One of the problems with his later work was that he took to writing longer novels which showcased his plotting techniques less effectively; they're loosely episodic in structure some feel a bit as though they are held together by duct tape.)