[PS] Context That Is Flatly Untrue

I’m clearing out my inbox, trying to tie a bow around this pseudocontext thing. Here are three problems that satisfy the first half of the working definition of pseudocontext.


CollegeBoard’s SpringBoard Mathematics with Meaning – Algebra 1:

One thing there: according to FIFA’s Laws of the Game, the dimensions that satisfy the math problem – eighty yards by eighty yards – aren’t legal:

I suppose you’re just hoping no one in the class knows you can’t have square soccer fields.

David Cox

McDougal-Littell’s Mathematics Concepts and Skills (Course 2):

Things don’t fall at constant rates. That isn’t what things do.

Zac Shiner

McDougal-Littell’s Algebra 1: Concepts and Skills:

Regardless of how long someone’s been running or how tired they are, they will always move at a constant speed of 200 meters per minute when running up hill and a constant speed of 250 meters per minute when running down hill. Since runners can only run at two different speeds, there is clearly no acceleration nor deceleration – just instantaneous jumps from one speed to another which coincide with the instantaneous changes between the only two slopes in the math world.

A quick aside to the pseudocontext: the problem asks the student to “write an algebraic model” but not before it gives her the “verbal” model. I’m not always certain how helpful to be in these situations but I know you have to be less helpful than that.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. This feels like it has the makings of a joke book, the soccer field in particular. Are we sure about the parachutist, though? A skydiver certainly doesn’t fall at a constant rate, but isn’t the purpose of a parachute to *prevent* acceleration?

    Re: the uphill/downhill constant speed problem, it turns out that it is possible, but you have to be wearing Sketchers Shape-Ups.

  2. Regarding David Cox’s “falling” questions: I completely agree that question #4 (a falling rock), is forcing the student to use bad model (constant velocity) in this case.
    However, question #3 (parachute): the point of the parachute is to get the person to fall at a constant rate. It’s a good model to apply to this problem. Running the numbers:
    1200 (feet per minute) = 6.1 m/s
    Question #3 does not say whether the parachute is open or closed. For an open parachute, I found a value of 12 mi/h (5.4 m/s) quoted, so the book’s number seems reasonable for an open parachute.

  3. Skydivers do generally reach their terminal speeds and fall at (pretty much) constant rates for appreciable portions of the trip. The ridiculous part is that this is less than 14 miles per hour.

    The one about the runner isn’t an unreasonable assumption – it doesn’t take long for a runner to get up to speed and to stop. Maybe the solution is always stating our assumptions. If we only did problems in algebra

  4. …one that were completely doable with those skills, we wouldn’t ever do much. Let’s state the assumptions all the time, though, so that kids get used to the approximate nature of pretty much all models.

  5. The soccer problem can be easily rescued by adding the additional constraints (and perhaps doing a follow-on where the coach buys more marking material).

    Parachutists DO fall at a constant velocity after a while (look up “terminal velocity” on wikipedia or http://hypertextbook.com/facts/JianHuang.shtml ) when the drag force matches the gravitational force. For a skydiver without a parachute, it is about 55-90 m/sec. The only problem with 1200 ft/minute (other than absolutely stupid units) is that it is too slow for freefall without a parachute. I think 6 m/sec is a bit fast for a terminal velocity with a deployed parachute, though it is in the right ballpark (I think 5 m/sec is more typical, but I’ve been having a hard time finding a good source.)

    Rocks dropped from cliffs don’t reach terminal velocity soon enough for a constant-velocity model to be sensible, so that problem is not rescuable.

    The running model would be better done as a bicycling model, since bicyclists at a constant effort do come closer to constant velocity uphill and downhill, much more so than runners. Ignoring the acceleration and deceleration at the transitions is fairly reasonable for bicyclists in hilly terrain. Bicyclists also typically have bicycle computers that give total distance and total time, which runners rarely have available without measuring on a map (pedometers count steps, which is not at all the same measurement).

    So none of these problems are great, but minor tweaks are all they take to be reasonable.

  6. As Damon noted, 1200 ft./sec. is a reasonable terminal velocity for a skydiver under canopy, hence their usage of “parachutist”.

    I think their intention with the rock problem is also to deal with terminal velocity, though the question is poorly phrased: they don’t state that it falls from the top of the cliff to the bottom of the cliff in those 15 seconds at 4.5 meters per second, but it would be easy to assume from their wording. Also, 4.5 m/s seems a bit slow for the terminal velocity of a rock, even an extremely porous one.

  7. @gasstationwithoutpumps: The terminal velocity of a skydiver under canopy is a function of many variables, including the size of the canopy, the type of canopy, the mass of the skydiver, and the air currents. 6 m/s is certainly reasonable.

  8. Just as an FYI, you can have a square soccer field.

    Your claim about 80×80 being an inappropriate solution is fine, but… you can have a field that is 100×100 according to the very laws of the game that you linked to and printed. Maybe delete the last bit of snark?

  9. @Timfc: The rules state the touch line must be greater than the goal line. So there’s still an issue.

    I feel a little bit like we’re Star Wars nerds dissecting the size of the Millennium Falcon. A square soccer field is strange enough students might balk, but the other two make assumptions that are at least ballpark.

    Any kind of modeling problem requires assumptions. The running problem presumably is discussing an average — the moment to moment speed is so chaotic as to be useless — and the runners I know do make assumptions along those lines when they want to track their progress. What’s more egregious is calculating the “number of miles you ran uphill and the number of miles you ran downhill” condition; almost certainly that would be found via other means.

  10. What kills me about problems involving running is that I am an avid runner and I often use math. However, it is nothing like the book. I’m more likely to use:

    1. If I’m generally running at a ( ) pace and I slow it down by about ten percent in the middle, what will my overall time be? (I would use this on my long runs) And will I make it back to see the Forty-Niners blow it yet again?

    2. If I am going to run the canal (a diagonal) and it covers an area that is 10 miles by 10 miles, how far will I be running (using the Pythagorean Theorum)

    3. If I am running at ( ) pace and I speed it up to ( ) pace how long will it take for me to reach the park with a restroom?

    4. I’m generally running 35 miles a week. My marathon is in six months. If I want to “break in” my shoes before I run the marathon, what will be the best times for me to buy new shoes (if I’m going by the rule that the most I should ever go is 500 miles before changing a pair)?

    I used other math, too. Often, I was surprised by the amount of estimation and algebra I used. However, it never looked anything like what I’ve seen on a textbook. I never ran a hill and calculated a slope. I never figured out constant change of speed. I never calculated the ratio of uphill and downhill moments. (If anything, I simply tried to predict how I would handle the uphills at different parts of the run)

    What kills me about sports-related questions is that they are so easy to fix. One phone call to a runner or a soccer coach or a basketball player could fix some of the awful pseudocontext problems.

  11. Ooh! Love John’s reply. John, if you put some numbers in those parentheses, I’d give those problems to my (adult) students.

    Although rocks may hit a terminal velocity, that problem is dangerous, because it reinforces a wrong idea about motion. The parachute feels different to me. (But I’d make sure to have a footnote describing the acceleration due to gravity that happens before you get to a terminal velocity.)

    I’d love to collect problems that come from our collective experiences of using math. The one thing that sticks in my mind is higher level – when I was looking for a house to buy, I kept having to derive the formula for the monthly mortgage payment. I didn’t have internet at home in ’98 to look it up, and I kept losing my scraps of paper where I’d write it down.

    Who else has used math outside of schooling and playing?

  12. My use of running contexts was often of the form:

    1) I need to run X miles and I have Y minutes. How fast do I run? How do I ensure that I’m running that fast without mile markers.

    2) I have an hour, how far should I run knowing that my way back is always slightly faster than my way out?

    3) Is it better to run 5 miles at a fast pace or 8 miles at a slow pace? How do you factor in, “can entice a friend to come for the 8 mile run and thus have social conversation?” into the equation?

    Mostly, it’s (1) though.

    But, no, the running context is really silly.

    Also, what if the field is 89.9x90m?

    It’s not square mathematically, but it is practically. I was a ref and one of the area fields could only have 90m of length, but, the coach thought she had the best conditioned team in the area and so marked it out as 89.5m, but how could we know without a 100m tape?

  13. The one area that math has enormous potential for use in the classroom is gambling and sports modeling but I am always wary of using it in the classroom. I am an avid punter ( I don’t use the term gambler because I believe I have an edge) and the amount of math that I use in my modelling is substantial.

    As a math investigation, building a sports betting model would be a very powerful project and one that would engage a large percentage of the class, but I fear the repercussions.

  14. Sue — why not ask your adult students if any of them run, and put their pace into the problem? Same with the soccer field. “Any team members here? Are soccer pitches ever square?” If no runners or soccer players in the room, might be worth the time to ask what sports they do, and build a problem around that. (The running problem above can be changed to practically any self-propelled activitiy). If you can get pseudocontext insurance from people outside your classroom, why not get it from people inside your classroom?

  15. One commonality I’ve seen with the best (worst?) psuedo-context problems is that they *interfere with real-world knowledge*. They either use numbers that are impossible or require methods that defy common sense.

    Math is a tool designed to complement or extend real-world knowledge.

  16. There is a question about a soccer field in Thomas’ Calculus that I use in class every time I teach Calculus I. The soccer field is put inside a 400 meter oval running track of unspecified dimensions. The students have to find the dimensions of the field that maximize the area. The length of the field is exactly 100 meters, and the width is 63.7 meters. I like this problem because it gives the exact length of the straight of a track and legal field dimensions.

  17. Maybe having a chapter (in my “perfect” textbook) called “solving dumb math problems” and asking kids what makes the problems dumb may lead to a discussion that motivates thinking about what makes for smart or real math problems.

  18. Most runners I know don’t think about speed in miles per hour or m/s or anything “normal” to the textbook writers. We think in terms of seconds per lap or minutes per mile. I know a 6-min mile pace, 7-min pace, etc. If I want to speed up for some reason, I’ll switch paces. I’ll also find some mile markers and use elapsed time.

    Similarly with basketball. I know from experience the power and force to sink a 20-ft shot, but couldn’t for the life of me tell you any numbers.

    The problem is in taking those numbers to the student. It’s easier to frame a question that is directly formulaic. When you have to come up with 80 items for section 8-3, and 80 more for 8-4, you cheat a little.

    The textbook selection process takes a huge toll on reality, as well. When you have to satisfy every possible special interest anti-discrimination group, you eliminate questions rather than rewrite in a fantastic and useful way. If your question relies on “insider” knowledge, it’s right out. If your question requires thought, it’s out. If you don’t have enough women v men, you drop the male questions or you rewrite the names. Ditto for every possible situation and question. In the process, things like “realistic terminal velocity under canopy” are ignored in favor of spending time making sure that there is a Maria and Jin Lian for every LaDanian and Steve.

    Unless we scream loudly enough for the Texas schoolboard textbook selection committee to hear it, nothing will change.

  19. Having been through the textbook selection process as a teacher, and now the textbook writing process, I can say that much of what Curmudgeon says is valid, but not all of it. There’s no reason as a writer that I can’t balance the names between male and female or between different ethnicities. It stinks that we have to use “number cubes” instead of “dice”, but if those small adaptations can get the book through a few more doors, great, and it doesn’t change the level or quality of the math.

    I’m curious what you mean by “insider knowledge”. Recently, I tried to write a problem about the probability of winning a game of tennis given the server’s was 60% likely to win any given point. The problem included all the details about the point scoring of tennis (0, 15, 30, 40, game, except 40-40 is deuce… etc). Another author pointed out this didn’t make much sense to him, because he had no experience with tennis, and would be forced to learn the ridiculous scoring rules of tennis in order to solve a problem about probability. For him, tennis was a useless context (not really a pseudocontext, since it was being used legitimately).

    It led to a better problem: while still about tennis, either player wins when they win 4 points, with the rule that they must win by 2. This is much more understandable and has less “insider knowledge”.

    So that’s what I agree with, but I couldn’t disagree more that books can’t have questions that require thought, or that we can’t use realistic contexts. I’d like to think every problem in a good text requires thought — otherwise, why is it there? And I can only speak for our series, but we use many real contexts legitimately. We take students through the entire process to construct and understand the formula for the monthly payment on a car loan, starting from numeric example and working across different chapters of different books, including topics such as recursive rules, generalizing from repeated calculation, and sums of geometric series. (Our books use other contexts, but that’s the one I think is really great.) Inclusive textbooks and good, quality mathematics are not mutually exclusive.

    I am happy that we published a full high school series without ever once writing 80 problems for a section. 10-15 good problems is more than enough, and the best number of problems to work through is still 1.

    – Bowen

  20. Bowen: It led to a better problem: while still about tennis, either player wins when they win 4 points, with the rule that they must win by 2. This is much more understandable and has less “insider knowledge”.

    FWIW, this is called no ad scoring, which is legitimate, so you’re totally in the clear here.

  21. Dan wrote:

    Yikes. It’s kind of a depressing mindset that sees inclusive textbooks and real context as mutually exclusive.

    Well, I’m sure that’s just because, as Stephen Colbert often points out, Reality has a well-known liberal bias.

    – Elizabeth (aka @cheesemonkeysf on Twitter)

  22. Wait, what do you mean about “no ad scoring”? Just keeping track of points, like “I won that game 7-5”? I’ve never seen that in action, the closest thing I’ve seen is the way tennis tiebreakers work (score 7, must win by 2… same thing for final sets at Wimbledon) and something we used to call a “pro set” (score 10 points, must win by 2). Never heard of it for regular tennis scoring, as much sense as it makes!

    One other cool thing about the tennis problem is there’s a rational function whose input is p, the probability of winning a point, and whose output is the probability of winning the full tennis game:

    G(p) = [p^4 (1 – 16(1-p)^4)] / (p^2 + (1-p)^2)(p^2 – (1-p)^2)]

    It’s not easy to come up with this, but it’s possible by analyzing the different ways to win (better to use “q” for “1-p” initially).

    And G has the “right” behavior, with domain 0 <= p <= 1: G(0) = 0, G(0.5) = 0.5, G(1) = 1… it actually looks kind of logistic.

    A nice example of a messy function with behavior that matches its context.

    Maybe the World Series should become "win by 2". That would be sweet.

  23. This post seems to digress from the whole concept of “pseudocontext”, into an argument about “what numbers are realistic”. Yes if the student takes the extra step of calculating or looking up what numbers are realistic (tennis scores are one thing, but who – other than skydivers – knows how many mph one is supposed to fall at terminal velocity??), then they’ll get a nice feeling if the math problem had realistic values. But I think that’s a small victory. The larger victories will be getting the students to realize – at every step of the process – that this math problem is a realistic one that could be / would be / is used by real people to solve real problems.

    So fine, for tennis, use the tiebreaker example. For skydivers, do a bit of estimation but get them to calculate something you might have to calculate anyway (such as how many seconds to freefall until you should pull the cord).

    In that sense, I agree with curmudgeon. Don’t waste time trying to come up with more realistic numbers. Spend that time trying to come up with more realistic problems. If the student questions the numbers, just say you’re a 800lb robot parachuting out of a plane in Mars!

  24. J Chak Don’t waste time trying to come up with more realistic numbers. Spend that time trying to come up with more realistic problems.

    This seems like another false choice. Why can’t we have both? I know my students would prefer it.

  25. False numbers fail the student because there is no possible way to judge the rightness of your work. If you have a problem that involves a 75 foot tall woman, what kind of suspension of disbelief is needed here? Is it really that hard to switch things around so that the numbers work? If you can’t make the numbers work, then the problem isn’t rooted in reality.

    “A woman race car driver (because we need to placate the publisher’s mandates against men race car drivers) travels 500 miles on a 1/4 mile race track in 20 min. How fast is she going?”

    Pretty simple to fix, no?

  26. “500 miles in 20 min.” That’s funny.

    Making good textbook problems is hard, and worth celebrating when you see one.
    Of the things that bug me when I see a bad math problem, the issue of pseudo context is a very big one, and worth articulating. Dan’s work on this has been really great, and I appreciate the attention it has gotten.

    There is a survival skill that kids pick up very early, and they learn to be really good at it. Without it, they would never survive school. I call it “figuring out what the teacher wants”, but it doesn’t only apply to teachers. The Mommy rules are different from the Daddy rules, and the way you behave at church is different from how you behave at the supermarket.
    What you do in one of these pockets would look totally ridiculous from the perspective of the other. When you run track, you pretend that it is important to get to the end of the track fast, but if you could get to the finish line faster by running in the opposite direction, you agree that that doesn’t count. In math class, you want to get the answer to 23×45, but if the best way to get there is using your calculator, we all agree that this doesn’t count. In history class, if the best way to find out who “the Red coats” were is to look it up with Google, we all agree that this doesn’t count.

    I don’t particularly mind that math class is its own universe with its own rules. Yes, the rules are arbitrary, but so are the rules of running track.

    What really bugs me is when I see someone spending her break between classes solving SuDoKu puzzles with gusto and on her own accord, then walking into the high school math classroom and telling her friends “I hate math!”
    I bet she’s doing more “math” thinking in the five minutes of SuDoKu puzzle than she and her class mates do together in the whole hour of math class.
    What have we done to “math class” so that it reliably wipes out any interest and joy the students might have in problem solving?