In response to: Language makes maths a rough ride for students (http://www.eastandard.net/mag/mag.php?mnu=details&id=1143976743&catid=316)
Every day at 2:15, I go to get my brain smashed by a sledgehammer. Who knew two numbers and a letter could have this effect? 51H has five one hour classes per week. There's one other notable category of classes that share this kind of scheduling at Stanford: languages.
Unlike lecture based courses, where it is more beneficial to have a day for newer material to sink in, languages are better learned through constant exposure, as often as possible, so less time is spent going over what may have been forgaotten between installments.
So why 51H? Why not the other math courses? The nature of 51H is such that it is like learning a new language, the language of proof. The standard 50 series focuses on techniques for manipulation of numbers and terms, whereas 51H is designed to look at the language from where these manipulations are derived. This is a language with it's own set of gramatical rules, based on a strict logic not to be broken. Interestingly enough, the rules for manipulations of numbers are largelt expressed without numbers, inorder to express generality. (I miss 4. It was such a pleasent number) Similarly, if you were studying the syntax of a language, the terms like noun and verb would be far more useful than the specific instances table and kick.
But does this kind of language actually fall under the language acquisition mechanism, as Chomsky views it? Or under the Boroditsky-an model, does the learning device have to create a new program for learning Proof, or can it fall largely under the previous systems used for language?
John Hawks posted in his blog on just this subject (http://johnhawks.net/weblog/reviews/brain/function/math_localization_language_2006.w) citing a recent study showing subjects suffering with aphasia, similarly had trouble with calculation. His hypothesis is that math is expressed through language, and this would account for the proximity of the centers of the brain.
I personally would like to think, that math can exist as a language unto itself, but it would follow without roots in a language of productive human interpretation, where would such a language spring from? Or if it is inherent in the universe, from where would it be percieved?
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2 comments:
My post from last week was actually somewhat similar to this. I think that it's possible to create a hierarchy of languages, where stuff like math or programming languages would fall on the more structured side of the spectrum. What I wonder is - do you think that it's necessary for a language to have a reference to something beyond itself in order to be considered a language? I agree with your post 100%, but I've been really bogged down with the question of whether a language has to be a symbolic representation of the outside world, or whether a language can just have meaning in and of itself without reference to anything in the real world specifically.
Ok, that wasn't very clear, but the point is that numbers and mathematical processes don't actually reference any specific thing in the real world besides the numbers and processes - so is it necessary for there to be an empirical connection for something to be a language?
If you are talking about notations, I agree that math is a language in and of itself that you must learn through translation and practice. What about math vocabulary, however? For example, what about the term "variable" in another language? Wouldn't the mathematical language used in each country be closely related to that country's language, then? If so, assuming language shapes thought, do you think the language math is explained in has any influence on how or from what approach a listener might look at a math concept?
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