Author Topic: Continuance of the INfinity Proposal  (Read 21721 times)

Offline Numsgil

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Continuance of the INfinity Proposal
« Reply #60 on: October 05, 2008, 08:58:12 PM »
I liked math up until I got into real analysis and topology and abstract algebra.  Calculus, linear algebra, differential equations, and numerical analysis were ether things I could visualize or memorize algorithms for.  They made a certain amount of quirky sense and built on things I'd learned before, so it was fun and reasonable.  I hated proofs but they were only like 25-30% of the material.  Then when I got to the last 4 or so classes to finish the major they totally change gears.  You start almost completely from scratch (not really building on previous knowledge.  In fact, previous knowledge can handicap you since it can lead to faulty reasoning).  And it's 100% proofs!  Yech!  If I had understood that up front I'd probably have aimed for applied math.  All it would have taken is an advisor telling me "see, these last four classes here?  All you do is proofs." sometime in my freshman or sophomore years.

Offline abyaly

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Continuance of the INfinity Proposal
« Reply #61 on: October 05, 2008, 09:17:43 PM »
Heh. Seeing a little bit of abstract algebra in high school is was prompted me to major in math. I specifically picked classes at the beginning just to meet the prereqs for abstract algebra and then number theory as soon as possible. Proofs are interesting because you generally aren't given the answer before you're given the problem. Too many people in too many fields give away the answers to good problems. In many areas (but math especially) a reliance on being given answers is a bad thing.
Lancre operated on the feudal system, which was to say, everyone feuded all
the time and handed on the fight to their descendants.
        -- (Terry Pratchett, Carpe Jugulum)

Offline Numsgil

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Continuance of the INfinity Proposal
« Reply #62 on: October 05, 2008, 10:02:18 PM »
Actually abstract algebra was my favorite of the late level courses.  It was just as abstract as everything else, but we started slow enough that I could brute force memorize things (abelian and groups and things), which I guess is the only way to learn that sort of material.  Plus, with abstract algebra at least, I could see some real world uses.  You can do rotations on a cube and understand that there are really only n transformations.  Mostly I was just irked that I'd spent like 15 years of my life learning what I thought was math, only to be told that at best what I knew would be a hindrance.  I hate memorizing things (I failed a lot of spelling tests in school because I felt like memorizing the spellings would be cheating), and that's really what you have to do when you deal with that stuff.

For instance, in AP physics I had to memorize very little.  Most equations are just transformations of other equations, so if you understand the basic principles and a memorize a few key equations (or put them in your graphic calculator's memory (I don't consider this cheating, my calculator is my brain's external hard drive )), you can derive anything you need during the test.  I think during one quiz on relativity I managed to derive E = m c^2 (though I think the question was on time dilation so this wasn't all that useful).  But there's no deriving what an group means exactly.  You can understand the identity principle, understand the idea of an operation, of inverses, of associativity, and yet be no closer to understanding how to answer a test question: "prove that natural numbers under addition form a group".  It's just vocab word after vocab word.  Physics was like that, too, but I had an easier time of that for some reason.

You could probably teach abstract algebra in elementary or middle school.  There really isn't a lot of prereq knowledge required, beyond the ability to understand how proofs work.  And there's nothing inherently difficult about the material beyond memorization, which is what kids that age do for school anyway.  You could have applied math, which leads to calculus and engineering, and pure mathematics, which deals with proofs.  Geometry like it's taught in early highschool would probably be a good intro course for pure math, since that's essentially what it is right now anyway.  I would have received it a lot better if I was 10 years younger and it wasn't the last few classes before graduation.

Offline abyaly

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Continuance of the INfinity Proposal
« Reply #63 on: October 06, 2008, 08:12:58 AM »
I guess you didn't have a good teacher for that stuff, then. It's possible to do quite a bit without using fancy words. In one of my undergrad classes our professor intentionally didn't tell us what most of the things were in order not to distract people from the important things. Eg: "Show that if a1, a2, a3, ...  is a number sequence such that for any positive number c there is a natural number N such that if n and m are greater than N, then |an - am| < c, then a1, a2, a3, ... converges." was one of the problems, but in that class we never used the term cauchy sequence. Avoiding new words does tend to make things a big longer, though.

I'm sorry your classes were focused on memorization. That is not a good way to learn the material.
Lancre operated on the feudal system, which was to say, everyone feuded all
the time and handed on the fight to their descendants.
        -- (Terry Pratchett, Carpe Jugulum)

Offline Peter

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Continuance of the INfinity Proposal
« Reply #64 on: October 06, 2008, 02:27:06 PM »
Good teachers can help a lot. Especially in math. This reminds me of a math teacher that, well tried to explain differential equations  
He just ignored any questions with, you're supposed to know that or that's not important. He didn't really explain anything but just start writing the whole calculation down. And often it came to mind with him, that he made another mistake, wipes a part away. And starts doing something else he hopes that works.   , strange experience that was.

He didn't did it every class that extreme, but it was sure he wasn't a good math-teacher. I'm not sure anyway how he did exactly maybe he did improve. I skipped those classes after a while.

Fact was that most failed math that specific semester, I was then one of the lucky ones that passed it.  

That wasn't becouse it was all hard, the classes (math) from him really was confusing.
Oh my god, who the hell cares.

Offline Numsgil

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Continuance of the INfinity Proposal
« Reply #65 on: October 06, 2008, 02:36:57 PM »
Quote from: abyaly
I guess you didn't have a good teacher for that stuff, then. It's possible to do quite a bit without using fancy words. In one of my undergrad classes our professor intentionally didn't tell us what most of the things were in order not to distract people from the important things. Eg: "Show that if a1, a2, a3, ...  is a number sequence such that for any positive number c there is a natural number N such that if n and m are greater than N, then |an - am| < c, then a1, a2, a3, ... converges." was one of the problems, but in that class we never used the term cauchy sequence. Avoiding new words does tend to make things a big longer, though.

I'm sorry your classes were focused on memorization. That is not a good way to learn the material.

It would have helped, but even explained out in words makes it hard to understand.  It's like having the Pythagoras theorem explained as "In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs".  You can't just gloss over a sentence like that and understand what it means.  You need pictures and notation.

Quote from: Peter
Good teachers can help a lot. Especially in math. This reminds me of a math teacher that, well tried to explain differential equations  
He just ignored any questions with, you're supposed to know that or that's not important. He didn't really explain anything but just start writing the whole calculation down. And often it came to mind with him, that he made another mistake, wipes a part away. And starts doing something else he hopes that works.   , strange experience that was.

He didn't did it every class that extreme, but it was sure he wasn't a good math-teacher. I'm not sure anyway how he did exactly maybe he did improve. I skipped those classes after a while.

Fact was that most failed math that specific semester, I was then one of the lucky ones that passed it.  

That wasn't becouse it was all hard, the classes (math) from him really was confusing.

I had a teacher like that for a discrete math class in computer science.  I knew most of the material already, but the way he explained things, it sure didn't feel like it

Offline abyaly

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Continuance of the INfinity Proposal
« Reply #66 on: October 06, 2008, 04:55:29 PM »
But once you grasp what something means when decompressed from the notation, you can understand any mathematical statement.
Lancre operated on the feudal system, which was to say, everyone feuded all
the time and handed on the fight to their descendants.
        -- (Terry Pratchett, Carpe Jugulum)

Offline Numsgil

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Continuance of the INfinity Proposal
« Reply #67 on: October 06, 2008, 05:27:20 PM »
Yeah, most math things are actually pretty easy to understand.  The hard part is just presenting the idea in a way that someone can understand it.  Pure math hasn't quite gotten as far as basic arithmetic in this regard.  English (any human language for that matter) is an ill fit for mathematical ideas, because it's not always clear what's a loaded word (group and set in a mathematic sense mean something very different from English, where they really mean the same thing), and what's just filler to make it grammatically correct.  Presenting ideas in a way that's easy to understand isn't easy, of course.  I'm not faulting anyone for the current state of affairs.

It's like word problems.  I hated word problems in Elementary school because they had to use loaded words for operations.  "difference" meant subtraction, "and" means addition, unless you're talking about some other field of math like sets (where it means intersection, which is counter intuitive because addition is a constructive process that makes something bigger, and intersection is a destructive process that makes it smaller <-- an insight in to how my mind works).

I don't have a better solution, though.  The set building notation isn't all that much better than raw English, for instance.  Upside down A's and backwards E's are just as confusing.  It's taken calculus hundreds of years to be as digestible as it is (and it still has a ways to go, I think).  So hopefully in hundreds of years they'll figure out better ways of presenting the material that isn't so obtuse.
« Last Edit: October 06, 2008, 05:28:41 PM by Numsgil »