“what is their total score?”
Why? Do you add scores in this sport for some reason? If team A scores 2 goals in soccer and team B scores 3, the score is 2-3, not 5. Adding scores is meaningless and weird.

How about this: “Melissa is playing a complex numbers game. Her score is 5-4i when it is increased 3+2i. What is her new score?”. This is still PS but at least it makes sense.

Is it wrong that I would absolutely love to play a game where the score is a complex number? Oh I’m drooling at the thought.

Can we do anything to this pseudocontext to make it real or is the process just too complex?

I kill myself.

Of course, there’s http://xkcd.com/849/

I realize this is going to sound urban legendy, but I know someone who knows the teacher who wrote this question. (NY has teachers come to Albany for like a few days at a time to write regents questions that are supposed to address weirdly-defined, strict learning outcomes. Or at least, they used to. For all I know Pearson writes them now.) And, the story goes she wrote this question as a joke. As in, as a lark she wrote something so bad and ridiculous that it would never be used. And then they put it on the exam.

Yep, bizarre question. On that sort of test, either give the addition straight up, or show a real context.

But I, too, wondered if someone could come up with an interesting game which would get scored with complex numbers.

Anyone up for the challenge?

Kate, I’ve participated in a few of those NYS item-writing sessions from the ELA/ESL end of things, and I wonder if your friend of a friend’s question was a response to finding them as Kafkaesque as I did– run by hired gun psychometricians whose state residency, classroom experience, and familiarity with, oh, the sun, all are in question.

You say this is the end . . . but isn’t it just the beginning. Now we need a mathematics curriculum that avoids pseudocontext. A curriculum that addresses all of the content needs of a state mandated set of standards, but in a fashion that allows for experimentation, reasoning, and conjecture. Something like Dr. Micheal Starbird’s book “The Heart of Mathematics” A curriculum that simultaneously addresses the needs of our students to pass that ACT, SAT, state, or federal assessment and makes mathematics relevant to their modern young lives. I am sure I am wrong, but I haven’t found that curriculum yet. Every text I have ever used is completely full of problems that fit your definition of pseudocontext. A random Sunday thought.

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.

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.

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.)

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.

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).