On Sat, Nov 3, 2012 at 5:04 AM, Robert Hansen <email@example.com> wrote: >
> > Maybe you only hear what you want to hear. >
Maybe, or maybe Devlin has said lots of things over the years and maybe I'm as faithful a reporter as you are.
Knowing how to breath and blink your eyes is also required for mastery of algebra.
That doesn't mean breathing is algebra, nor that arithmetic is algebra.
To say you need to be good at arithmetic to be good at algebra is not to say that arithmetic is "really math".
Someone who is good with the four operations and uses them daily in business, is not thereby considered a "mathematician" in our culture.
You can be brilliant with Excel. That won't make you qualified to teach math.
Except maybe you could teach arithmetic which goes by the name "math" in the lower grades.
>> >> It's somewhat endemic in the culture, I'm surprise you haven't noticed. > > What I have noticed is that people (mathematicians are people) tend to take arithmetic for granted. They have very little (usually none) experience with the processes that occur with children learning arithmetic. And I don't mean "scientifically how" (which is silly to even suggest that we are even close to that yet). I mean that they simply have not witnessed first hand (as a teacher) the phenomena (which occurs over a period of 3 to 5 years) of children learning arithmetic. >
Lets grant you that. In our culture, mathematicians are kept apart from small children for the most part.
The don't have these opportunities to study the learning process and it's not considered their professional responsibility to do so.
A lot of mathematicians wind up teaching in universities or maybe with the NSA.
> >> >> The comic book math mensch can visualize hyper-dimensional topologies >> rotating with ease but can barely balance a check book, because mere >> addition is boring/tedious/error-prone and he never bothered to get >> good at it (too busy getting good at math).
>> >> Being good at arithmetic means you're a well oiled machine, like fast >> with an abacus, maybe you have lots of mental tricks and can add the >> grocery bill in your head. > > When have I, or anyone for that matter, described "being good at arithmetic" in that manner? Being good at arithmetic means knowing numbers, addition, subtraction, multiplication and division and how to apply them in context. You can't accomplish that without becoming familiar with it yourself. In that process, some people get very good at calculation, most get reasonably good. >
I'm trying to clarify / represent the view that mathematics and arithmetic are culturally distinct, even if the latter is a stepping stone towards mastery.
Arithmetic has a lot of "twitch game" or "reflex" aspects.
You've memorized your times tables, done lots of practice, and now, congratulations, you're a cheap imitation of a $10 calculator.
You still know barely if any math though, is how the thinking goes.
Now, I think there are ways to teach arithmetic that promote more comprehension of math than others.
Learning about different bases for example, like binary and hexadecimal or time (mixed base).
Learning about mappings, say of characters to Unicode patterns in bits, and about functions in general...
New levels of abstraction and generality start to enter in. And fortunately, even bad K-6 curricula are *not* just about teaching arithmetic.
This has arithmetic in it, but also a lot of math. An awareness if types is building up:
cubocta :: (Integral a) => a -> a cubocta 0 = 1 cubocta x = 10 * x * x + 2
Not Python for a change.
> > >> Good for you (applause!). But that's more >> under the heading of "salon trick" or "impress your friends at >> parties". Math means knowing lots of theorems and history and >> applications and... > > And music must mean knowing lots of songs and the history of music. A million musicians would disagree with you, if they even bothered. >
Being a musician does mean having a lot of cultural awareness, yes.
I've been hanging out with a retired violin player who was based in Milan and flew around Europe playing chamber music and so on. She specialized in music from around the time of Louis IVX.
If you're a musician in a rock and roll band, you probably know a lot of music history as well.
Frank Zappa was classically trained and knew a lot of music theory.