[PS] The End

This is completely subjective, but Peter Brouwer sent in the problem that I thought satisfied both halves of the working definition of pseudocontext in the most spectacular fashion. This is it. This is as bad as it gets.

From the June 2001 Math B New York Regents examination [PDF]:

Jo Boaler gets the last word:

Students do however become trained and skillful at engaging in the make-believe of school mathematics questions at exactly the “right” level. They believe what they are told within the confines of the task and do not question its distance from reality. This probably contributes to students’ dichotomous view of situations as requiring either school mathematics or their own methods. Contexts such as the above [pseudocontext], merely perpetuate the mysterious image of school mathematics.

That’s it. Thanks for pitching in.

I'm Dan and this is my blog. I'm a former high school teacher, former graduate student, and current head of teaching at Desmos. More here.


  1. Joe Henderson

    January 22, 2011 - 7:34 am

    This is great due to its cleanliness. Occum’s razoresque.

    So, where to go from here? I’m assuming that you’d argue for increasing contextual relevance now that you’ve established a marker. But again, who gets to define “the context”? The teacher? The student? Parents? Community? Some combination? Distant test-making bureaucrats? I’m looking forward to unpacking *that* process with you.

  2. M Barton

    January 22, 2011 - 8:40 am

    “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.

  3. gasstationwithoutpumps

    January 22, 2011 - 8:50 am

    I’m with M. Barton, but it may well be that Melissa and Joe are on the same team, and that you are computing the team score. Now the challenge is to design a game that is (somewhat) fun to play where the scores are complex numbers and where adding the scores makes sense. Perhaps we can come up with something in electronic circuits, where AC voltages are represented as complex numbers and adding and subtracting voltages is common.

    I’m not disagreeing with Dan that this is a weird pseudocontext problem. It’s just that it is so weird it could be fun to try make up a context for it.

  4. Curmudgeon

    January 22, 2011 - 9:54 am

    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/

  5. Tim

    January 22, 2011 - 10:06 am

    Wins + i (goals for)?

  6. Kate Nowak

    January 22, 2011 - 12:16 pm

    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.

  7. Karim

    January 22, 2011 - 12:35 pm

    To follow what Kate said, maybe Pearson itself is just one big practical joke: the educational equivalent of the Onion. Maybe in some mahogany-lined suite in midtown, an executive is chuckling to himself, I can’t believed they thought we were serious!

    Then again, maybe they are just a bunch of Dr. Claws (trying to kill Gadget, Penny and…Brain).

  8. Sue VanHattum

    January 22, 2011 - 12:55 pm

    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?

  9. Elizabeth S

    January 22, 2011 - 4:05 pm

    And just to throw in a plea for clarity of representation from the literary set — in my first reading of this question, I got the impression that Melissa and Joe were playing on one team *against* a team of complex numbers! Ack!

    – Elizabeth (aka @cheesemonkeysf on Twitter)

  10. Dina

    January 22, 2011 - 6:04 pm

    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.

  11. Chirs Sears

    January 23, 2011 - 5:27 am

    I graduated from a high school in New York. Now I know who to blame for the Regents exams. :-)

    I am a bit too tired to laugh out loud at this problem. I will manage it in a few hours.

  12. Daniel Schaben

    January 23, 2011 - 9:02 am

    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.

  13. Sarcasymptote

    January 23, 2011 - 10:05 am

    Ah, yes, I had totally forgotten about this problem. As my coworker and I aid our students in preparing for the Regents at the end of the year, and are reviewing complex numbers, I always see this question, and every time, I think, “What the hell are we doing?”

    I don’t know how much this is common knowledge, but the Regents in NY are ridiculous. Not only do we have questions like this, but the teachers grade their own students’ state tests. In a lot of city schools (thankfully not mine), there is a TON of pressure on teachers to grade the students very gently, because our rating as a school is based mostly on the performance of the students on the tests that we grade.

  14. John T. Spencer

    January 23, 2011 - 10:44 am

    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.

  15. Adam

    January 23, 2011 - 1:38 pm

    Also, complex numbers cannot be ordered…

  16. Peter

    January 23, 2011 - 1:51 pm

    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.

  17. R. Wright

    January 23, 2011 - 1:53 pm

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

  18. ClimeGuy

    January 24, 2011 - 6:24 am

    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.)

  19. Mike

    January 24, 2011 - 2:09 pm

    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.


    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?

  20. Bowen Kerins

    January 24, 2011 - 3:58 pm

    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.

  21. Chirs Sears

    January 24, 2011 - 8:03 pm

    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.)

  22. gasstationwithoutpumps

    January 25, 2011 - 8:48 am

    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).