Teaching maths: Do concrete examples any good?

[Argos]

Interesting article on the NYT (http://www.nytimes.com/2008/04/25/science/25math.html)

(...)many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn. That idea may be wrong,(...)

[Moose]

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.

[Delvo]

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.

[Chuck]

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.

[tdvance]

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.

[Disinfo Agent]

The idea is that making math more relevant makes it easier to learn. That idea may be wrong,(...)Regardless of whether it makes math easier to learn, it sure makes it more interesting.

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.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.

[peter eldergill]

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.

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 :D

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

[Neverfly]

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.

[Moose]

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

It would be. That helps. Thanks.

[tdvance]

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 :D

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

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

[Moose]

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

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.

[tdvance]

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.

[Ilya]

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 :D

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.

[Tobin Dax]

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

[sarongsong]

The Gigantic Book of Teachers' Wisdom (http://skyhorsepublishing.com/details.php?TitleID=89)
(777 pages of relevant quotations re teaching.)If a child can't learn the way we teach, maybe we should teach the way they learn.
---Ignacio Estrada

[Ilya]

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.

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.

[Tobin Dax]

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

But then, I took differential equations course for fun.
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.)

[SpaceShot]

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.

[SolusLupus]

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.I'd probably be one of these people.

Oh, it's not because I'm bad at math. But I'm bad at using it on the go. I cannot see numbers in my head very well, and for some reason I have troubles dealing with single digit numbers subtracting from double digit numbers. (Oh, I mean, 105-5 is easy; but 100-13, I need a moment to think).

When I'm stressed out and would have been handling a register all day, I'd probably be more in a stupor.

[Ilya]

The few times in my life I worked as a cashier were all dismal failures -- not because I can not do arithmetic in my head (I do that very well), but because I can not hold my concentration for long, and start handing out wrong bills without noticing. I knew just how bad I am at this job (and should never ever work at anything like ER!) when at the end of a day I handed the stack of $20's to store's safe manager, and there was a $50 stuck in the middle.

[Larry Jacks]

People learn in different manners. Some are good at listening to a speaker while others require a more hands-on approach. Many education studies suggest that boys learn better with a more hands-on approach while girls often learn better via other presentation techiques. One size does not fit all.

Personally, I became bored with my high school advanced math classes because the teacher didn't know or didn't relate any real world examples to the class. In fairness, many teachers went from being a student in classrooms to being teachers in classrooms without ever gaining much real world experience. My high school teachers simply might not have known of any good examples.

A year after graduating high school, I went to school to learn how to be an electronics technician. Suddenly, we were using the same math but with real world examples. It clicked.

The types of examples can vary widely depending on the age of the students and their skill level. It could be as simple as calculating the wall area of a large room to determine how much paint is needed. It could be calculating their taxes based on specific incomes and sets of deductions. It could be from the sciences, space exploration, sports, politics (e.g. surveys and polling), economics, etc.

When I taught school for a year, I did have students ask me how they were ever going to use math. Looking back, I'm much better equipped to answer that question today than I was at age 26.

[SpaceShot]

I wish kids and adults understood math better for two different reasons.

1) Grades
I saw this when I was in school and now that my fiancee is student teaching. Kids don't get that losing points adds up to dropping grades. They don't get that getting a D on a quiz is infinitely better than ditching it and avoiding it and getting a zero. Look, the D sucks, but averaged with A's, it's a lot better than the zero.

Kids don't turn in their homework because "it's only a few points". But they are FREE points (usually). And again, waaay better than a series of zeroes littering the gradebook. Assuming 59% is an F, that's still way more points than a zero, and it drags down the A's earned in tests and other work.

Sure there are a lot of reasons kids fail in school and get poor grades, but I really feel like they should be taught how grading works. It is an important math lesson and prevents the notion that you "can make it up later". Points earned now are better (and less pressure packed) than points you might earn later.

Start in math class and then even have the English teacher show the class how she comes up with the mysterious A, B, and C grades.

I would have math assignments that involved calculating grades and GPA's. I think it (should) really open someone's eyes the power of the zero in an average and the difficulty in averaging high scores when you give points away.

2) Sales
"Buy one, get one 50% off". This is, at best a 25% off sale. And that's FINE. I often think "hey those prices usually don't waver much, so 25% off is a better deal than normal".

But I think many people focus on the 50% off and forget that (usually) it's 50% off after a full price purchase of equal or greater value. And maybe you would have gone in and bought one of something. Now you've bought two.

Used properly, retail sales can be a great way to save money... but it is rarely so on impulse alone.

[Grizzle]

A year after graduating high school, I went to school to learn how to be an electronics technician. Suddenly, we were using the same math but with real world examples. It clicked.

At the uni, due to being somewhat ahead of schedule along some tracks but on schedule on others, I got the theoretical and practical treatment of certain branches of math almost simultaneously. That was a little weird.