History of Mathematics
I have been reading the following site last few days: http://wwwgap.dcs.stand.ac.uk/~history/index.html I think the site is excellent. Learning maths (and other subject) by tracing its historical development is interesting for me and I think children may like it.
Rick Tang
August 10th, 2005
One big plus is seeing how it actually relates to something. A big problem of mine when studying math was why it mattered in the least or how it was at all useful.
I am Jack's infinite id
August 10th, 2005
Why does it have to be useful? Is art "useful"? Plebeian!
No, but art can be naturally enjoyable. Memorizing calculus tables is too, if you are some kind of mutant.
I am Jack's infinite id
August 10th, 2005
"Memorizing calculus tables" is to mathematics as "painting by numbers" is to art.
I love math. I was top of the class in every math class I took. Unfortunately I hated institutionalized schooling, so I haven't accomplished near my potential, something that always makes me sad. I am constently trying to figure out how to motivate my children, if they show similar tendencies.
I like the historical context idea, I know it would make me curious. I always want to expose them to higher concepts at a much younger age. I think it'll be surprising what they can grasp.
Jeff Barton
August 10th, 2005
It was just an example of something mundane in math...
Hmmm, tour the Sistine Chapel, or work out the slope of that line?
I am Jack's infinite id
August 10th, 2005
Two simple examples:
1) Sine/Consine/Tangent..
Memorize the definition of dividing the length of different side of a right angle triangle, vs how to mearsure the arc length within the circle. I think learning through the second option is more fun.
2) Logarithm tables to help calculation vs logarithm function and natural logirithm.
Don't you think it's fun to see both perspectives?
Rick Tang
August 10th, 2005
There's as much beauty in something like e^(i * pi) = 1 as there is in the Sistine Chapel. I suspect you're making the all too common error of confusing "calculation", "arithmetic", and "mathematics".
> I like the historical context idea, I know it would make me curious.
I recommend _The music of the primes_  I think that if it had existed when I was a student, it might have given me insight in the world and lives of professional mathematicians.
Christopher Wells
August 10th, 2005
> insight in the
insight into the
Christopher Wells
August 10th, 2005
I'm with Mat; I didn't need any sort of context to click with mathematics.
The simplicity of algebra (groups, rings etc.), the attention to detail of analysis (limits, calculus) right through to more applied subjects such how to solve differential equations both analytically and numerically.
These are the kind of things that used to amaze me, but most people go "so what?": * there are the "same number" of even numbers as there are integers in total * there is no general solution for a quintic polynomial equation * threedimensional space will not support an algebraic field, but two and four do * Euler's equation * the Axiom of Choice
Sorry, I'll stop now.
I think the sistine chapel vs. slope of the line example is slightly off. The sistine chapel is a once in 100s of years artistic accomplishment. Calculating the slope of a line is more like checking out the new paint job on the local park bench. One of the major differences is the initial accessability of art. While both art and math reward deep study, art is easier to access for the neophyte. Math demands work before it reveals its beauty. I highly recommend http://www.amazon.com/exec/obidos/tg/detail//014014739X/00267435927776811?v=glanceAlso the presentations online at claymath are good  although production values are a bit low. Perhaps they need some help from the video arts!
FaLing@Orbiz.ch
August 10th, 2005
I'm reading Music of the Primes at the moment, and it's not bad at all... I tend to buy books on impulse, but sadly the bookshops round here don't carry much in the way of "popular science", so I have to take what I can get which can result in some pretty poor results. This one has turned out to be a winner, though.
"there are the "same number" of even numbers as there are integers in total"
And for every integer there are an infinite number of fractions, so although there are infinite integers there are *more* fractions; they're both infinite in number, but one of the infinities is bigger than the other. Crazy!
Then who design those maths textbooks?
They are BORING.
Rick Tang
August 10th, 2005
s/fractions/irrational numbers/
Of course there are the same number of (rational) fractions as there are integers. Crazier still!
How about Zeno paradoxes?
Any maths textbooks mention them?
I am confident there are, but the textbooks I used are boring.
I am naturally inclined in maths/music/chess etc, so it's not too bad on me. But for others ...
Rick Tang
August 10th, 2005
Sorry, having a bad day, and cocked it up AGAIN. I think I was getting over excited at the distraction from arguing about fat people. :)
There are the same number of rational numbers (i.e. fractions) between 0 and 1 as there are integers  an infinite quantity of both. And oddly, there are as many rational numbers between 0 and 2 as there are between 0 and 1. However, compared to how many irrational numbers there are, all those inifinities are positively miniscule (although still infinite.)
Right, I don't think I got it wrong this time, but I'm sure you can see why I might have got confused in the first place. :D
"How about Zeno paradoxes?"
Zeon's paradox is quite simple. Here's my rough retelling of it:
Two people, we'll call them A and B for simplicity, are having a race. B is cocky and convinced he'll win, so he gives A a short head start.
The race bagins, and in no time at all B has reached A's starting point. However, A is no longer there, and although he hasn't progressed quite as far from where he started as B has, he's still made some progress.
On the race goes! B has now reached where A was when B got to A's starting point, but again B has inched on slightly. The gap is closing, but it's still there...
Every onward presses B, and by now he's reached where A was just a moment ago, but A has still managed to move on a fraction. The gap is even narrower, but it's still a gap.
In fact, no matter how quickly A gets to where B was, B will always have moved on slightly. The gap will decrease with each "turn", but A can never quite catch B!
I'll leave you to ponder on what the flaw is for a while while I go and have a smoke. Enjoy!
Zeon? Zeno!
Maybe I'll go and have a lie down...
Abstract mathematics is beautiful to me too. It's like a game in your mind. But even though it appealed to me in the absract, it didn't stay abstract. I remember when I was taking differential equations and linear algebra I started seeing relationships and formulas everywhere in day to day life, my brain would even start to turn random things (table and chairs) into formal systems.
BUT, none of this made me like school any more. I had high marks in all of my classes, it was just that I didn't care about marks at all, and I resented the fact that I "needed" some kind of acknowledgement on paper from an institution that really irked me. (I have a thick anarchist streak.)
I think what I would like to do for my kids is help them overcome any antiestablishment distractions with a sense of gratitude for the opportunity to study full time.
Jeff Barton
August 10th, 2005
I appear to have swapped the roles of A and B halfway through. That's it, I'm calling it a night! Lost is on in about 45 minutes, I need to go buy cigarettes, and I've wasted too much time writing incomprehensible gibberish on here for one night...
But yes, abstract maths is fascinating, beautiful, aweinspiring, mindboggling, but sadly not as accessible to the layman as some pretty pictures are. A real pity, but I think it's more than worth trying to overcome the initial hurdles. Pick up some decent "popular science" books (pretty much anything by Ian Stewart is a good starting point) and soon you'll be mesmerised by all the hidden wonders. :)
'night all!
I'd like to go a step further  math books should probably be largely historical. Maybe a SERIOUS history article preceding the straightforward presentation.
I'm thinking of Salus's _Handbook of Programming Languages_, which had the Ritchie article about C's evolution from BCPL, inviting you to critique the decisions, before someone else explained C.
But of course, it's hard to expect serious history that isn't misleading.
> but I think it's more than worth trying to overcome the initial hurdles
I knew a couple adult students at a community college that wanted to be engineers (I'm not sure why). Math, though, was something of a problem for both of them. One of them had to start in prealgebra math classes. Both of them had to struggle hard to comprehend each new truth. It took them years and years, but they never quit. One of them actually created some strange mental process to teach himself new things. I had a statics class with them (I, too, mistakenly thought I wanted to be an engineer), and they did okay (middle of the pack). It was humbling to see people so commited to reaching a goal.
I would never hire either of them to build me a bridge though. ;)
Jeff Barton
August 10th, 2005
Rick Tang sez : "How about Zeno paradoxes? Any maths textbooks mention them?" Better yet  there's a lovely little book called "Zero : the Biography of a Dangerous Idea" that traces the history of the concept of "zero". Essentially, it talks about why zero is such a subtle and, yes, dangerous idea  how it lies at the intersection of math, philosophy, and art  and how var23Eious civilizations dealt with the concept of zero and its evil twin, infinity. Loads of fun, a great little read, and has a nice section about Zeno's paradox in the bit about the Greeks' problems with zero and infinity (both as a philosophical and as a mathematical problem). I'd give this one to schoolkids to read .. I'm not a math buff, but it definitely made me want to be one. http://www.amazon.com/exec/obidos/tg/detail//0140296476/qid=1123704167
Snark
August 10th, 2005
In fact, people letting you peek behind various curtains, like the Battlestar Galactica commentaries and Gordon Ramsay's shows, are important in the same vein.
Tayssir sez : "I'd like to go a step further  math books should probably be largely historical."
I couldn't agree more. Mathematics is often perceived as sterile and lifeless because it's so often presented just that way. It's taught as "here are the rules" with none of the sense of how and, more importantly, why these rules were derived. Much of the joy of mathematics lies in it2DDs internal consistency : each rule can be derived from a previous rule, all the way back to first principles ... and each was, by someone.
Someone had a moment of inspiration, or spent weeks or months poking at the numbers, or reread a source that they hadn't encountered before, and discovered a new and elegant way of expanding the set of mathematical "tools." Teaching that process can only be done by example  if a student is given the historical context, and introduced to the very human ambitions and arguments that undoubtedly accompanied each discovery, he will feel part of the process of discovery that's so central to mathematics.
Snark
August 10th, 2005
++There's as much beauty in something like e^(i * pi) = 1 as there is in the Sistine Chapel.
Maybe, but there's no way you can honestly believe it is as obvious and comprehensive to the masses. There's no beauty in the formula itself unless you just like looking at cryptic sequences of characters and symbols.
The beauty is in the application. You need very little knowledge aside from what is literally presented to you to appreciate art.
I am Jack's infinite id
August 10th, 2005
trollop
August 10th, 2005
Is there an easy intuitive explanation for why e^(pi i) = 1? Sure, if you look at the power series e^ix = cis, and then if you plug in pi you get your answer but that's not really intuitive and elegant (to me at least).
28/w
August 11th, 2005
"The beauty is in the application"
I find that the beauty lies in the fact that the equation contains the concepts of logarithms, exponentiation, imaginary numbers, negative numbers, and pi, a collection of seemingly unrelated cornerstones of mathematics. It may not be intuitive, but that doesn't make it any less elegant...
Reading a novel assumes a great deal of background. It assumes you understand the language it's written in, and have enough of a background to empathize with or enjoy what's in it.
There are many novels people rave about that I can't begin to see what they're so interesting. I think this is similar to anything within mathematics.
Incidentally, about history, I'm reminded of someone's link to a Feynman anecdote when he was eavesdropping on someone's conversation during lunch at college. These girls were talking about crocheting, and taking "deep" math theorems for granted. (One advantage of diversity; masculine guys don't necessarily have the best ways to visualize things..) This is honest history to me, and I think meaningful.
"I think this is similar to anything within mathematics."
All too true, and a great shame. I think I was lucky in that I had a teacher early on in my school career who made the subject genuinely interesting  I was about 9 or 10 and he started introducing us (in a very general sense) to things like inifinite series, and while some people came over with a glazed expression it stirred something in a handful of us. (To my knowledge, at least four of the people in my class went on to do degrees in some field of maths.)
All too often school maths classes focus heavily on arithmetic, rote learning of a handful of mundanane formulae, etc., and never touches on the more esoteric areas of the subject, the background as to why or how these things were arrived at, or anything that might spark the imagination of a child. You're just told stuff like "the quadratic formula is negative b plus or minus the square root of b squared minus 4ac all over 2a", and it's no great surprise that popular opinion holds that maths is dry and boring as a result...
