Comments on: [PS] The End

By: Pseudocontext « The Death of Thinking

Pseudocontext « The Death of Thinking — Fri, 28 Jan 2011 20:50:56 +0000

[…] introduces the series here, and some of the examples that follow in subsequent posts such as these are mind-numbing. Comments RSS feed LikeBe the first to like this […]

By: gasstationwithoutpumps

gasstationwithoutpumps — Tue, 25 Jan 2011 16:48:25 +0000

I think that the question of proper ways to present complex numbers to kids so that they see the usefulness of them is an interesting one. (I used the plurals there deliberately, as I’m sure there are multiple ways and that different ways work best with different kids.)

I’m trying to remember how I introduced my son to complex numbers, and I really can’t remember. I don’t think it happened once, but gradually over the past 4 years, to the point where he is now comfortable using them as phasors in AC circuit analysis, where a sinusoid A*cos(omega*t+p) is represented as A*e^(i*p) (the frequency is held constant during the analysis of linear circuits, so only amplitude and phase need to be represented explicitly).

I think that his geometric understanding of complex addition, multiplication, and e^(i*theta)=cos(theta)+i*sin(theta) will make it much easier for him to learn trigonometry (which he is starting at high school tomorrow).

By: Chirs Sears

Chirs Sears — Tue, 25 Jan 2011 04:03:22 +0000

It's a shame that the pseudocontext conversation is winding down. I just read a passage from the book Montessori: The Science Behind the Genius that calls out pseudocontext, not by name, in a 1917 textbook by Edward Thorndike. I wrote about it on my blog. Now that I am more awake, you're a good number one, Dan. (Lost reference.)

By: Bowen Kerins

Bowen Kerins — Mon, 24 Jan 2011 23:58:18 +0000

To “R. Wright”: there are a few ways to play complex numbers, and one of them is algebraic, having to do with how numbers get constructed. Negatives are needed to solve x + 7 = 5; fractions are needed to solve 3x = 2; irrationals to solve x^2 = 2; complex numbers to solve x^2 = -1. And, surprisingly, that’s the end of the chain: any polynomial equation with real or complex coefficients must have complex roots. That, to me, is the legitimately algebraic way to play the topic, but most texts just add, subtract, multiply, divide, then get out, and nobody learns a thing.

The other way to play complex numbers is to look at the relationships between the algebra and geometry of them with complex numbers in the plane. For example, the fact that the sum of a complex number and its conjugate is real has a lovely geometric representation, and there is an interpretation of both addition and multiplication that have long-term purposes. Most legitimate applications of complex numbers (in analysis, especially in stability of systems) require knowledge of the complex plane, so why not teach it alongside the algebra?

To Daniel Schaben: consider the above description of our Algebra 2 complex numbers chapter a shameless plug for the book series linked by clicking through. I can’t claim to rid the world of pseudocontext, but I think we did alright, and our approach to linear inequalities (and, actually, graphing equations in general) is ridiculously similar to what you wrote here: http://blog.esu11.org/dschaben/2010/10/27/why-do-we-graph-linear-inequalities/

Thanks Dan. I suspect we’ll see more pseudocontext on this blog someday, it was too good a topic.

By: Mike

Mike — Mon, 24 Jan 2011 22:09:56 +0000

Please consider that some games might be scored in a non-ordered field. Take for instance the computer game series Civilization which has many means of achieving victory state that cannot be directly compared to each other (a successful moon program vs military/diplomatic take-over).

I realize I’m late to this dicussion, but while catching up on my RSS feeds, I came across this comic which puts an interesting twist onto an old game and gives actual context to the above problem.

http://www.smbc-comics.com/index.php?db=comics&id=2131

Suppose you could play the new game in teams that sum or multiply up? It actually might be a tactical advantage to lose at some times. And if I had a class of bored students needing some practice at calculation and bored of lecture, why not pose a group experiment?

By: ClimeGuy

ClimeGuy — Mon, 24 Jan 2011 14:24:20 +0000

I tried to resist commenting on this again. But because I havent done a very good job in nailing it before, I’ll give it one more shot. Pseudo means false, fraudulent, pretending to be something that it’s not. IMHO, your problem above is contrived (artificial, forced, laboured, overdone) but its not pretending to be something that it’s not. (In this case, a silly game.)

By: R. Wright

R. Wright — Sun, 23 Jan 2011 21:53:01 +0000

It has always baffled me why anyone would spend more than 3 minutes talking about complex numbers in an algebra class.

By: Peter

Peter — Sun, 23 Jan 2011 21:51:02 +0000

Saturday Morning Breakfast Cereal recently came out with a RPS variant that has imaginary points: http://www.smbc-comics.com/index.php?db=comics&id=2131#comic . It doesn’t explain how you’d get negative imaginary points, but it’s not completely out of the realm of possibility.

By: Adam

Adam — Sun, 23 Jan 2011 21:38:20 +0000

Also, complex numbers cannot be ordered…

By: John T. Spencer

John T. Spencer — Sun, 23 Jan 2011 18:44:04 +0000

Wow, and I thought I was the only child who grew up playing Complex Integer Challenge 2000. See, it involves complex numbers, drinking and steal cage matches. Much more fun than you might imagine.