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Razib over at Gene Expression offers some thoughts on the algebra issue, in which he suggests some historical perspective:
The ancient Greeks were not unintelligent, so the fact that many of us (rightly I believe) take symbolic algebra for granted as a necessary feature of our cognitive landscape is something to reflect upon. Maths that we assume to be fundamental elements of our mental toolkits would have been beyond the very conception of the most brilliant minds of our species over one thousand years ago.
I'm not really happy with this, because it seems a little too Steven Pinker-- "our ape-like ancestors on the savannah didn't need algebra, so we never evolved the brain module for it..." I'm not really comfortable with the claim that the relatively late development of an idea indicates that it's somehow counter to our brain chemistry, and thus ok for people to not understand it.
I originally planned to note that Newton's Laws post-date the invention of symbolic algebra, and that doesn't make them incomprehensible to normal humans. This quickly runs into the previously noted problem that many people do have a hard time grasping Newton's Laws...
I thought of another example, though, after a conversation with Patrick Nielsen Hayden at Boskone. In the course of explaining the origin of the quote "Plot is a literary convernstion, story is a force of nature," he made reference to (I'm paraphrasing here) the fundamnetal narrative connectors of medieval literature, which are "And here's another thing..." and "And I forgot to mention that..."
(More after the cut...)
It's a description that rings pretty true, based on the medieval lit course I took in college (scholars of medieval lit are welcome to object in comments), and points out that our modern ideas of narrative structure are also relatively recent. As Patrick noted, these are the same story-telling conventioned used by five-year-olds today, but most adults move past them fairly quickly.
The idea that a story is something more than a series of events presented in the order in which they occur to the teller is arguably a fairly recent development (of about the same age as symbolic algebra), and yet, nobody expects students to struggle with it. An even more recent development is the idea that an imaginary-world story can be told without first spending a chapter on establishing the provenance of some fictitious manuscript that serves as the source-- that didn't really change until the early 20th century, and nobody expects students to have trouble picking it up. In fact, people who can't cope with the modern conceptions of fiction are widely considered to have problems...
I don't think you'd find literary scholars trying to excuse reading comprehension issues with references to cognitive development, and the relatively recent adoption of modern literary conventions. In fact, people usually go in the other direction-- faculty teaching older works take some time to explain that narrative convetions were different in the past, and treat our current set of conventions as the natural state of affairs.
I don't think this is an iron-clad argument-- for one thing, works like The Iliad don't use the "And another thing..." transition to the same degree as later works. But it's a non-science illustration of why I'm uncomfortable with attempts to excuse struggles with mathematics by reference to history-- there are lots of other subjects where we wouldn't attempt to make a similar justification for student difficulties, so why do we go to such lengths to excuse struggles with mathematics?
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Leon Wieseltier had a great line reviewing Dennett in the NYT book review the other day:
"The idea that a story is something more than a series of events presented in the order in which they occur to the teller is arguably a fairly recent development (of about the same age as symbolic algebra"
So symbolic Algebra is as old as Euripides? Aristotle's categorization of comedy and tragedy did not come out of thin air. But I do agree with the thrust of the arguement.
Symbolic algebra did not come out of thin air either - it really does make modern people better at designing things, better at understanding things, better at making decisions than the ancient Greeks. Not because we are somehow smarter at the basic level, but because we have tools which give us intellectual "Leverage".
That is the real tragedy of that algebra opinion piece - the writer ignores that people make decisions all the time, on whether to take a loan, on what kind of loan, on whether to invest, on what the job climate is like in an area. If a person doesn't have the background in proportionality and decides for example, that the fact that there are half the jobless in some city so she wants to move there, without realizing the city is 25% the size, so there is higher unemployment per capita, there is a cost to her and to all of us as a whole. This doesn't mean she would be writing down X = (U1/P1) / (U2/P2) and checking if it is greater than 1 explicitly, but to just consider it internally.
That is the hidden cost of not using the tools we have today including algebra. As a society as we make more and more bad decisions like this the costs go up, until the levies break...
Leon Wieseltier had a great line reviewing Dennett in the NYT book review the other day
great lines do not necessarily equal substantive lines! :)
I think the particular way we understand mathematics is adapted to what our brains can handle. Many people (most of whom I suspect are human beings) in fact do not find algebra hard, and some find it very easy, which suggests that there is _some_ part of the brain that finds symbolic manipulation congenial.
The "and another thing" model of medieval lit is reductive but useful; quite a few texts use it, and quite a few do other things. I'd suggest, though, that in some texts strung together that way, there's a good sense of purpose behind the sequence of events, which storytelling five-year-olds rarely have. Suspense, foreshadowing, and character development preexist the eighteenth-century novel in the western tradition....
Also, some twelfth- and thirteenth-century inhabitants of Western Europe clearly had little problem finding nuance in Vergil, Statius, Cicero, etc., in a manner similar to C20/21 readers, despite the differences in narrative convention.
Markk: That is the real tragedy of that algebra opinion piece - the writer ignores that people make decisions all the time, on whether to take a loan, on what kind of loan, on whether to invest, on what the job climate is like in an area. If a person doesn't have the background in proportionality and decides for example, that the fact that there are half the jobless in some city so she wants to move there, without realizing the city is 25% the size, so there is higher unemployment per capita, there is a cost to her and to all of us as a whole. This doesn't mean she would be writing down X = (U1/P1) / (U2/P2) and checking if it is greater than 1 explicitly, but to just consider it internally.
Some of the problem, I think, is that people are sort of fuzzy about what counts as "algebra," probably because it's so badly taught. There are lots of situations in which people make calculatios that are essentially algebraic ("I have ten dollars. How many cheeseburgers can I get and have enough left over for a soda?") without realizing that they're doing algebra.
This is probably a symptom of bad math teaching.
Also, congratulations to Markk, whose comment was #666 in the database. Hail Satan! Or something.
greythistle: The "and another thing" model of medieval lit is reductive but useful; quite a few texts use it, and quite a few do other things. I'd suggest, though, that in some texts strung together that way, there's a good sense of purpose behind the sequence of events, which storytelling five-year-olds rarely have.
My knowledge of medieval literature is pretty much limited to that one class back in college, plus a few odds and ends picked up from other people on Usenet or whatever, but it was certainly instantly recognizable when Patrick made that comment. From Kate's comments on reading Genji, it almost seems like it could be applied to that as well.
It's an oversimplification, I'm sure, but it's not without cause.
Also, congratulations to Markk, whose comment was #666 in the database. Hail Satan! Or something.
"Your lighthearted salute has been noticed. Consequences will follow."
666 how quaint ... I feel like I should write some Ramanujan style comment - oh 666 that is a very interesting number ...
but its not coming ... hmm 6 * 111 = 9 * 74 = 18 * 37 = 2 * 3 * 3 * 37 kind of interesting it is the second triplicate multiple of 37, I wonder how many ... wow didn't realize but its obvious ... 37 divides all triplicates since it divides 111. See:
111 = 3 * 37
222 = 2 * 3 * 37
333 = 3 * 3 * 37
444 = 2 * 2 * 3 * 37
555 = 5 * 3 * 37
666 = 2 * 3 * 3 * 37
777 = 7 * 3 * 37
888 = 2 * 2 * 2 * 3 * 37
999 = 3 * 3 * 3 * 37
Well there is a dumb little thing special about 37 that I learned from looking at 666 anyway.
How about written communications? Although we've had writing in some form for a lot longer than algebra..
(a) They are both short on evolutionary timescales,
(b) Historically, not many people were able to read/write, and
(c) Reading is a lot more complicated.
The problem with all these examples, though, is that mathematics and logic are fundamentally different: they are modes of discovery rather than expression. The novel gives the author new ways of saying things, but the novel does not (in principle) give the author new ideas. Math, on the other hand, allows you to get from 1.795372 and 2.204628 to arrive at a new thing: 4.
The source of "Plot is a literary convention, story is a force of nature"? That's easy: me.
The source of "Plot is a literary convention, story is a force of nature"? That's easy: me.
I knew that. Patrick was telling the story of how it came to be said (which, come to think of it, I don't think he ever got to finish...).