[PS] Pseudorejects

These are submissions I received that didn’t seem to fit the criteria. This isn’t to say they’re great problems. This isn’t to say that I’d throw water on any of these problems if they were on fire. This isn’t to say that they’ve even represented or examined their context well, just that the context itself isn’t pseudocontext.

Breedeen Murray

McDougal Litell’s Math Course 1:


Barbara Panther

McGraw-Hill’s Total Math – Grade 6:

Bill goes to a farm and sees cows and chickens. He counts 6 heads and 18 legs. How many of each animal does he see?


Melissa Griffin

Haese and Harris Mathematics for the International Student.:


Jeff Bowlby

McGraw-Hill’s Algebra 1:


Phil Aldridge

EdExcel International’s Longman Mathematics for IGCSE Book 1:


Iain Mackenzie

Scotland National Examination:


Christine Lenghaus

Australian Year 12 Exam:


Amulya Iyer

Pearson’s Algebra 2:


Steve Bullock



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. I think that the dartboard problem is seriously flawed, though perhaps not in the pseudocontext way. The error in dart thowing can be modeled with a normal distribution of positions, but not well by a uniform distribution. Thus taking the ratio of areas tells you very little about ration of hits to misses, unless the standard deviation for the normal distribution is very much larger than the target size (OK, I’m a bad enough dart thrower that it might apply to me.). The problem here is that they said that all the darts hit in the triangle, which immediately removes that save. If you have a normal distribution so tight that 100 hits in a row are in the triangle, then the standard deviation has to be so low that almost all of them are in the circle.

    If they had said that 100 out of 100,000 throws had landed in the triangle, then maybe the uniform distribution over the triangle is a good enough model.

  2. The last problem makes me feel sick. Teaching in a school where nearly half of the students are non-native speakers, I have become overly conscious of the vocabulary I use. When I see a problem about unicorns which uses the words Styria, Ottokar, Bohemian, ducats, cubit, and gilded alarms start ringing.

    Then again, for all I know, this could be from a 500 year old European text (where and when cubits were still used and ELLs didn’t yet exist). Somehow I doubt that’s the case though.

  3. There are several different and distinct ways in which I find these kinds of problems troubling. One is, as Z. points out, how heavily over-weighted they are in favor of native English speakers, who do not constitute a majority in many of our classrooms. Another is the way in which they try to place restrictions on the kinds of interpretations students’ must make in making sense of the problems (the cows and chickens problem fit this case).

    It seems to me that if you have to distort the presentation of a problem to this extent, you are probably not doing the required mathematics any favors either.

  4. The problem about heads/legs of cows/chickens strikes me as pseudocontext, since it involves “operations that have nothing to do with the given context.” Why would you possibly count the number of legs and heads at all, especially if you are interested in the number of animals?

    gasstationwithoutpumps: I’m a pretty terrible dart player and I don’t doubt that if I kept throwing until 100 of them eventually hit the board, they’d be close to uniformly distributed. But I agree that using a uniform spread here doesn’t match the context well.
    How about this? Still a silly context but the math-model is more realistic:
    Your friend wants a round chocolate-chip brownie today. But you only have some triangle-shaped baking dishes, for making a pie-cosahedron:
    So you bake a triangular brownie, cut out the biggest-possible circular chunk from the center for the friend, and keep the rest for yourself. (a) Is it a fair split? What proportion of the brownie will be part of the final round brownie? (b) If you put about 100 chocolate chips in the full triangular-dish recipe, about how many do you expect will be in the round portion?

  5. Gilbert Bernstein

    January 15, 2011 - 2:35 pm

    Not that it’s terribly compelling for a high school student, but the Pearson’s Algebra 2 3d coordinate problem is something I might actually do while debugging 3d graphics software.

  6. Pwolf,

    Different city or different state. Tax rates are not uniform around the country. The big problem with the question was that it is not clear whether the “dinner bill” includes tax or not (usually they do) nor whether it includes service charges or not. So the question suffers mainly from vagueness, not from pseudocontext.

    A more realistic version of the question would be something like: You have a $35 charge on your credit card for a business dinner which you want to be reimbursed on, but you are not allowed to be reimbursed for alcohol and you had a $3 beer with the dinner. Compute how much you are allowed to be reimbursed, given that the sales tax is 9.25% and you tip 15%. (The tax and tip attributable to the alcohol purchase are also not reimbursable.)

  7. Right, it’s a lousy problem for plenty of reasons, but, for whatever it’s worth, I don’t think it’s pseudocontext. By definition, the problem has to be attempting to represent reality before we can accuse it of misrepresenting it.

  8. gasstationwithoutpumps: I think your solution might be cheating: all these problems would be better with alcohol. ;)

  9. While these problems don’t immediately jump out at us as detrimental to a student’s understanding of how math applies to the world, they don’t do anything to help. One way to redeem many of these is to retool them so that the connection between the math and the context is more clear.

    I agree with Dave as well. To get someone to properly think about the problems, you basically would have to tell them “forget the unicorns and the cubits,” “forget that it’s a lightpole,” or “forget why the taxes are different.” It’s like, though the context isn’t patently false, it still somehow obscures the underlying math. In order to understand the problems, the context needs to be stripped away. In a “good” applications problem, it seems, the math derives naturally from the context.

    I don’t know if this has been addressed before, but I think that if we’re trying to get students to agree with us that math describes the world, we should be wary of giving them problems that require a leap of faith or willful suspension of disbelief if we can help it. (“Tomas and Troy come across a water hose and turn it on. Instead of spraying each other, they decide to do math with it.” Get real Mr. Wolf, they’d say. By the way, the water hose “problem” has already been done so much better on this blog, it’s almost not even fair to pick on it.)

    All this to say, at some point in my career I’ve given a problem like every last one of these to my kids, and even though I can’t articulate it for all of them, I am starting to see why I won’t anymore (except of course, when they show up in EOC review packets…)

  10. The tax problem could easily be cleaned up if we were comparing final sale price. Suppose a sweater at store A cost $30 and is 20% off then you use a 5% off coupon. At store B you bought an identical sweater that retailed for $25 at 5% off then you used another 5% off coupon. Which store did you pay more for a sweater?

    The question, why the difference in percentage can easily be explained by the fact that you clearly went to different stores; something students have analog experience with.

  11. Okay, normally I hate it when people leave comments saying “check out the cool blog post I just wrote on this very topic!”, but since I happened to write this several months ago, when I was first Pseudo-rejected, I guess it’s not so bad…

    This was my response to, and reflection on, what I think Pseudocontext means. It’s a slightly different take than Dan’s, though not quite as succinct.


  12. Robert E. Harris

    January 23, 2011 - 9:07 pm

    The legs problem makes me think of a puzzle my bridge friend Bill Yancey put to me one night, years ago.

    A farmer sends his son to market with $100 and tells him to spend the $100 to buy one hundred animals, a mix of chickens, pigs, and cows. Cows cost $10 each, pigs are $3 each, and chickens are $0.50 each. How many of each kind did the boy buy?

    Maybe I have got the details mixed up somewhat, but that’s the general trend of it.

    I worked it out in my head as I drove home; there is some sort of small mental trick that occured to me that made it simple. What I have forgotten.