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You mentioned you learned linear algebra. Don't you think you can understand linearity of everyday quantitative relationships better because of it? Same for probabiity, which I think you mentioned also.
originally posted by: TycoonBarnaby
If I wasn't taught the basics of mathematics throughout my younger years I never could have earned my PhD in Mathematics. Are we supposed to just ignore students who might someday go farther than YOU have with mathematics because YOU don't see the point?
From rockets to stock markets, math powers many of humanity's most thrilling creations. So why do kids lose interest? Conrad Wolfram says the part of math we teach -- calculation by hand -- isn't just tedious, it's mostly irrelevant to real mathematics and the real world. He presents his radical idea: teaching kids math through computer programming.
Dividing polynomials, though I prefer to use the more simplified version of synthetic division over long division most of the time.
My main issue with those who claim we should never teach students to memorize mathematics is that they rarely bring up the fact that many other subjects also require students to memorize information and regurgitate it on tests. History comes to mind as an easy example of this.
Its really a question of relevance, is it relevant to you, and for many people the answer is a definite NO.
And those who can only remember things after they understand the point of it, being the other half, and likely the way you learn, the meaning of it, or even the use of it, before you can bother with it.
originally posted by: ChaoticOrder
a reply to: jedi_hamster
yes, i'm a programmer as well. 3d engines, FFTs, image processing - been there, done that. and while i agree that there are many things that are best left for research on as-needed basis (we can't be experts in every field possible), there are some things that one just should know - and sorry, but i can't imagine blaming not remembering a multiplication table on anything else but lazyness. such simple things that you prefer not to remember have to slow you down quite a bit during your work as well.
Yes I am lazy, I am a completely lazy programmer. Laziness is often cited as one of the most important traits of a good programmer. See: The Three Virtues of a GREAT Programmer or Why Good Programmers Are Lazy and Dumb. The fact is, I cannot remember the last time I needed to calculate 7x12 in my head, so I see no reason to remember a table of numbers I'll hardly ever use, especially when I have a calculator handy 99.9% of the time. Of course I can remember many of the smaller multiplication answers, but that's only because I do tend to multiply smaller numbers in my head often. I remember what I need, nothing less, nothing more.
Of course you can understand how particles will move through space simply by understanding how particles interact with each other. First of all you just need to know how they gravitationally react to each other, and as we know the force of gravity gets exponentially stronger as two particles move closer together. Then we also need to take into consideration the electric forces and the nuclear forces, etc. Of course we can use equations to help us understand all those different forces but it would be extremely complicated if we tried to combine all of those equations into one equation which described everything about particle movement.
also, i think you're quite far from understanding the idea behind math at all. i also prefer to think like a programmer - because i am one. but you can't simply imagine how a stream of particles will move in space just by knowing the rules being the building blocks of their movement. you'll get it simple - but you'll have to perform so many iterations of your simple math and traverse such a tree of dependecies to get to the end result, that by the time you'll get there, you'll forget what was it that you were checking. computer will do it - sure. you won't.
on the other hand, someone able to think like a mathematician will be able to imagine the flow of said particles just by looking at a single, complicated equation. why do you think scientists all over the world use such complicated math? are they a bunch of morons, unable to come up with anything better? far from it - it's just a natural way for them, and a natural way for a human in general. you can of course research new problems using computers, you can create models and optimize them using methods you've described - but that's not how things are done, and it's not optimal.
We often start with mathematics because it's the most fundamental way to describe complex systems. By studying such equations and manipulating them we can often make other important realizations. Like I said in my previous post, math is much more abstract and is better at conveying abstract ideas. Clearly it wouldn't make much sense to start by describing physics in C++ and then convert it into math equations at a later point in time. We start with the most fundamental mathematical descriptions and then we can implement those ideas into computer code by quantizing the equations into finite computational problems.
as for lazyness, a programmer lazy in a sense implied by Larry Wall, would remember that damn multiplication table
as for particles, you cannot visualize it in your head without understanding the complex equation behind it. you can either focus on the building blocks (each of the forces you've mentioned), calculate each separately, sum the results, and do so step after step, just like a computer would, just thousands times slower - or you can imagine the forces at hand translating into function graphs in three-dimensional space, as long as you can imagine those functions.
the education system you're proposing would produce people that would be only able to imagine basic operations - because those would be the only ones they would be able to perform.
do you have any idea what kind of flexibility a language like forth, or even tcl, gives? judging by your posts, doubtful.
I saw a thread the other day about how kids are learning ridiculously tedious methods to do math and it got me really thinking about how flawed the overall way we teach math actually is.
No, they would remember parts of the multiplication table they actually use regularly. Even if I did remember the entire multiplication up to 12 or x13... you do realize there are about an infinite number of larger numbers which I will often need to multiply at some point. And in practice most of the multiplication problems I need to do are with very large numbers, some so large most people wont even encounter such large numbers in their life time.
I'm not exactly sure what you're saying here. My whole argument is that understanding the equations and being able to visualize them is far more important than simply remembering them. Clearly it is much easier to understand how something works when you break it down into its component parts, by visualizing each force separately instead of trying to combine them all into one equation or one function graph. It also happens to be the best way to write code, so that's it's clear what the program is doing, instead if trying to combine all the forces into one programming function.
In the education system I propose people would be able to imagine the complicated 3D function graphs you talk about because they could be visually represented by a computer and they could play around with the variables in real time. On a related note, obviously the education system which produced your grammar and punctuation skills did a very poor job.
Regardless of the flexibility those languages offer, there is nothing ambiguous about the syntax, because if there was the compiler wouldn't be able to interpret the code. That is not the case for mathematics, complicated equations often do require some degree of reasoned guess work to figure out the real interpretation.
ok> 1 1 + .
ok> : 1 2 ;
ok> 1 1 + .
example: 7x57 - it's 7x50 + 7x7, which is 7x5x10 + 7x7 - and that's something you can solve in seconds at most in your mind, by only knowing the answers for up to 10x10.
my point is that visualizing the components doesn't equal to being able to see the bigger picture. you have to be able - at some point - to wrap your mind around the whole problem, and letting the computer visualize it for you just because you prefer to operate on a simpler level isn't the answer.
i'm far from saying that computers shouldn't be used more in the education - quite the contrary. but they should be used as a tool to teach people how to understand complex problems and how to solve them ON THEIR OWN - not to free them from thinking about the complexities.
some languages let you redefine the syntax - or almost anything actually.
bad C++ code is a clusterf... no math equation can match, that's for sure.
This is just complete nonsense. Being able to visualize the equation and manipulate variables in real time gives an extremely deep understanding of how it works. Some times just looking at the equation isn't enough, especially complex equations. You may have to see what it outputs and graph those outputs, which is exactly what a function graph does.
Even if it's hard to read and understand for a human it's still not ambiguous to the compiler, so my point remains.
Well written C++ code, even if thousands of lines long, is still easy to interpret and understand. Thousands of lines of math equations is not easy to understand by any standard. When it comes to complex and length equations, it's just not easy to digest for most people. As an example, when I need to learn a new mathematical concept, I will often start by reading the wiki page on the subject, and a lot of the time, the page will contain some complicated equations that I can barely understand. However, that's never really a problem, because by reading the text on the page and looking at the useful charts often sitting to the right of the text, I can usually understand what the equations are doing without even fully grasping the equations. Then I can write some code which does exactly what I want, just by understanding the concepts involved, I can turn those concepts into code, which solidifies my understanding of how the math work.