Interesting article on the NYT

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I'd suggest, from my own struggles learning calculus and especially linear algebra that concrete examples are essential for many. I've griped about this before, but in college, I'd been expected to learn eigenvalues. Okay, but the professor never explained what you use them for. I've since learned that there are applications for them "in computer graphics", but again, that doesn't tell me very much.

I never successfully learned how to compute an eigenvalue.

It was much the same in linear algebra. Vector rotations (even higher dimensional vector rotations) and the like weren't a problem for me. But then 2/3s of the course became one long "here's a property of matrices, prove it's true".

My mind really doesn't work that way. I can follow a proof to validate it, but I can never seem to figure out how to get "there" from "here".

Show me a property, prove to me it's valid, and I can grind it. And that's all I really need to be a prog. I'm not out to develop new algorithms, but to apply the good ones to the (non-cutting-edge) problem I happen to be working on.

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In Fallout 3, 'happiness' is a warm junkyard dog and a loaded gun. It's mostly the loaded gun.
- Moose's one-line review.

"your going to regret that one. You are now a colonoscope...
- Chrissy, corrupting PraedSt's wish.

The experiment was awful. They essentially taught one group A and another B where B was the basic abstract concept behind something and A was an application of the concept. Then they tested both groups on C, a separate application also based on abstract concept B. Amazingly, the group that started at B had an easier time going from B to C than the group that started at A had going from A to B to C. Leave it to the world of "education" to come up with something so stupid.

All examples should involve the mixing and pouring of concrete so that even if the value of the example is questionable, there will still be the opportunity for a good pun.

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ok, sounds like the analogous of "2+2=4, now what's 3+3" versus "this is how you add" without giving an example. You need both. Also "concrete examples" don't have to be "everyday stuff" all the time--especially that "fake" everyday stuff often found in textbooks that nobody really does--say you wanted to put up a christmas tree of dimensions xyz and needed to know how thick a bolt to hold it up assuming w inches of tree below the bolt, the tree weights t, whatever... but instead, examples of how the math really is used in practice.

A really bad elementary school math example I once saw was "blue stars have x temperature. Red stars have y temp. etc. etc." and even mentioned "purple stars" (????) but even ignoring that silliness, the final question was, "you see this many blue, this many yellow, etc. in the field of view. What is the total temperature?" (average temperature would be meaningful at least, and I admit, this was probably a book at a level before "average" was defined, but still--I'd think even an intelligent kid would say, "gee, that's stupid" and be turned off!)

That's what happens when teachers say, "we want scientific applications of the mathematics" and textbook writers hurry to find as many as they can with little vetting, and stuff them in.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

Regardless of whether it makes math easier to learn, it sure makes it more interesting.

There is that danger as well, of course. You can't make everything practical unless you lie to the students. A lot of mathematics is of the "practice makes perfect" variety. Sorry, but it just is.

Of course, when you offer your students too many practical examples they typically start to get nervous at the prospect of having to become applied mathematicians themselves, and ask for more abstract examples. At least in some teaching levels.

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I think eigenvalues aid in the solutions to certain types of differential equations...what the solutions to the differential equation might be useful for I don't know

They can also replace a transformation matrix with scalar multiplication (I think, I'm going by memory), which is definately useful for computer graphics.

Pete

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PJE

There's so much I don't know about astrophysics. I wish I had read that book by that wheelchair guy.

This was mentioned in the Ant and Rope thread too.

It seems to me that kids already fail to recognize how they will use math in the real world under real world applications.

Word problems are designed to encourage kids to recognize how to use the math they learned to solve real world problems where you are not handed a formulas and equations and told which numbers to stick where.

It would be. That helps. Thanks.

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In Fallout 3, 'happiness' is a warm junkyard dog and a loaded gun. It's mostly the loaded gun.
- Moose's one-line review.

"your going to regret that one. You are now a colonoscope...
- Chrissy, corrupting PraedSt's wish.

probably the most fundamental use of Eigenvalues is in Quantum Mechanics--the theory is that EVERY measurement is an eigenvalue.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

Which leads to the question: what is an eigenvalue?

I'm looking at the wiki page now, and if I'm understanding it right, the whole eigen-nomenclature is another way of saying: "higher order pivot", where your eigenvector or eigenplane stays put while the universe transforms itself around it.

If I'm understanding the concept right, in CG, you'd compute eigen-whatevers if you were doing transformations of objects in context of a fixed camera reference point, such as moving a game world about the camera.

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In Fallout 3, 'happiness' is a warm junkyard dog and a loaded gun. It's mostly the loaded gun.
- Moose's one-line review.

"your going to regret that one. You are now a colonoscope...
- Chrissy, corrupting PraedSt's wish.

I understand an Eigenvalue to be a single number that represents the action of a matrix (linear transformation) on a single dimension of a space. For example, a 3x3 matrix that deforms a 1x1x1 cube into a r x s x t rectangular box has for its Eigenvalues r, s, and t (assuming the sides of the cube were parallel to the directions of the axes the matrix works on).

With quantum mechanics, the matrix would show how nature changes the state of the system under consideration, and the eigenvalues would be various possible measurements on the system--an observation would produce one eigenvalue. (still is a single dimension of a space, but an abstract state space of a system rather than physical space we live in).

ETA: the Eigenvalues of the 3x3 matrix above could as easily be r, -s, t, or -r, -s, -t, or, etc. if it "flips" some of the directions.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

As I recall from differential equations course in college, just about every problem the professor gave us involved real-world applications. Actually understanding why force of gravity drops off as square of distance, but escape velocity drops off as square root of distance was rather cool.

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That is rather cool. The problem is that it's only cool to us. Most people don't consider these real world examples, but rather things they have to learn.

The Gigantic Book of Teachers' Wisdom
(777 pages of relevant quotations re teaching.)

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OK, I agree. Other examples involved things like ice cubes melting, spring stretching and contracting, and a protozoan respiring. All of which are fairly arcane.

But then, I took differential equations course for fun.

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Fiction has to be plausible. Reality is under no such constraint.

I use those as real-world examples in physics, and they are for that class.

You're insane, do you know that? Insane a good way, but insane none the less.
(Of course, I had a horrible prof for Diff EQs, and I know that has a good amount to do with my dislike.)

I tend to think that early arithmetic is only going to be learned by rote and repetition. Do enough addition, subtraction, multiplication, and division and you just begin to memorize a lot of it.

Now, I do think many people, including the children we educate in school, need concrete exampls so they can take a real interest in this stuff. There are tons of people who just don't see the relevance, and most of them are the ones who I hand a 5 dollar bill to pay for something that is 4.07, and they punch in 50.00 by mistake and are completely at a loss when the cash register says 45.93 is the change. In one bizarre case the clerk was even about to hand me the 45 dollars and change. Usually, though, they stand there completely baffled, looking for a manager to come void the transaction so the magic oracle can tell them the correct change.

And yes, sometimes in cashiering, you have to get the transaction right so the drawer and merchandise match up, but this is not one of thse cases.

When it comes to higher math, such as calculus, I flunked hard out of those classes. I am not sure if I was simply burned out of school, having trouble with concrete application, or simply incapable of understanding the abstractions then. I am curious if I could take it back up now.