Rebooting Pseudocontext Saturdays

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We create a pseudocontext when at least one of two conditions are met.

First, given a context, the assigned question isn’t a question most human beings would ask about it.

Second, given that question, the assigned method isn’t a method most human beings would use to find it.

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The dog bandana is the classic example. Given a dog, would most human beings wonder about the correct size of the bandana? Maybe. But none of them would apply a special right triangle to answer it.

Rules

Here’s the game. Every Saturday, I’ll post an image from a math textbook. It’ll be an image from one of the “Where You Will Use This Math!” sidebars.

I’ll post the image without its mathematical connection and offer five possibilities for that connection. One of them will be real. Four of them will be decoys. You’ll all guess which connection is real.

After 24 hours, I’ll update the post with the answer. If a plurality of the commenters picked the textbook’s connection, one point goes to Team Commenters. If a plurality picked one of my decoys, one point goes to Team Me. If you submit a word problem in the comments to complement your connection and it makes someone lol, collect a personal point.

Why?

  1. Fun. Teaching is a pretty serious occupation. It never fails to brighten my day when you all ping me with pseudocontext.
  2. Caution. My position is that we frequently overrate the real world as a vehicle for student motivation. I hope this series will serve to remind us weekly of the madness that lies at the extreme end of a position that says “students will only be interested in mathematics if it’s real world.” The end of that position leads to dog bandanas and other bizarre connections which serve to make math seem less real to students and more alien, a discipline practiced by weirdos and oddballs. Caution.

This Week’s Installment

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Pseudocontext Saturday #1

  • Calculating probabilities (49%, 185 Votes)
  • Calculating the area of the sector of a circle (41%, 156 Votes)
  • Finding the next term in a sequence of numbers (5%, 17 Votes)
  • Multiplying binomials (3%, 10 Votes)
  • Evaluating the quadratic formula (2%, 9 Votes)

Total Voters: 377

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(If you’re reading via email or RSS, you may need to click through to vote.)

I’ll update this post with the answer in 24 hours.

BTW. Don’t hesitate to send me an example you’d like me to feature. My email address is dan@mrmeyer.com. Throw “Pseudocontext Saturdays” in the subject.

Answer

Polls are closed. The commenters got rolled on this one, with only 3% having guessed the actual application. So one point goes to Team Dan.

Most commenters guessed “calculating probabilities,” which likely wouldn’t have been a pseudocontext. Humans wonder lots of questions about probabilities when it comes to darts, many of which are most easily answered with mathematical tools.

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But this is high-grade psuedocontext. Given a dartboard, few humans would wonder about the dimensions of a square that circumscribes it exactly. And even if they did wonder about it, none of them would name the radius r + 12. They wouldn’t even name it r. They wouldn’t use variables. They’d measure it.

The publisher included the dartboard as a means to interest students in special products. If you believe, as I do, that the publisher has done more harm than good here, positioning math as alien rather than real, what can be done? How do you handle special products?

Featured Comments

Amy Hogan:

Q #11: Pretend [certainly not a woman’s name] has no concept of darts, zero aim, and is liquored up at the bar anyway. What is the probability that he’ll hit a 20? Twice, with his eyes blindfolded?

Dennis Rankin:

Question: What percent of the dart board scoring area is red? white? blue?

Extension: Are the red, white, and blue percentages of area the same on an American flag?

Bowen Kerins:

Man, this “context” is an absolute embarrassment and wastes the time of students and teachers. This sort of thing is driven by textbook requirements for “full coverage” — some lessons have a useful “why” picture and description, therefore all of them must.

Awful.

Scott Farrand reacts to the commenters’ loss:

Now I see how to make the dartboard fit into our task. First we each randomly assign each the five options that Dan gave us to 4 of the 20 sectors of the dartboard, so that 1/5 of the sectors correspond to each option. Now all we need is a blindfold, and … let’s see if we can improve our results from 3% correct to about 20% correct.

Also, please enjoy this back-and-forth about the nature of pseudocontext between Michael Pershan, David Griswold, Sarah, and me. I know I did.

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

48 Comments

  1. Question: What percent of the dart board scoring area is red? white? blue?

    Extension: Are the red, white, and blue percentages of area the same on an American flag?

    • Extension: Are the red, white, and blue percentages of area the same on an American flag?

      Oo. Collect your point.

  2. I’m quite liking the idea of… find an nth term formula for the sequence 20, 1, 18, 4, 13, …. I think it’d be a touch on the tricky side, though.

  3. Zachary Sellers

    October 15, 2016 - 12:24 pm -

    My kind of textbook:

    Topic: precision vs accuracy

    Question: What precision (and/or accuracy) would be required to hit the string and make the dart board drop?

    Extension: how many darts would you need to sink in the wall under the board to catch the board and keep it from falling after snapping the string?

    • My first impression is that this is a pic I might use to discuss precision vs accuracy. I teach Statistics to community college students.

  4. If you’re a shitty darts player, what’s the best number to aim for?

    Does your answer depend on how shitty you are?

  5. Steven Alexander

    October 15, 2016 - 2:05 pm -

    You can tell at a glance that the dartboard is divided into a number of sectors of which kind?
    (a) a multiple of 5
    (b) an even number
    (c) a multiple of 3
    (d) a square number.
    Now tell _how_ you can tell at a glance.

    • You can determine the truth of all four at a glance.
      A multiple of five by seeing the evenly spaced blue or red sectors.
      An even number by traversing down the center of one sector across the diameter to the center of another sector (odd would result in going from the center of a sector to the border between two)
      Not a multiple of 3 (since we have an even amount of sectors)
      Not a square (since it’s an even multiple of five, and there are clearly not 100 sectors, which would be the first square that is a multiple of two and five.)

  6. Q #11: Pretend [certainly not a woman’s name] has no concept of darts, zero aim, and is liquored up at the bar anyway. What is the probability that he’ll hit a 20? Twice, with his eyes blindfolded?

  7. Who threw all those darts and missed? Someone from:

    Team Trump
    Team Clinton
    Team Stein
    Team Johnson

    Justify your selection.

  8. Laura Mckenzie

    October 15, 2016 - 6:27 pm -

    What’s the red dart person’s handicap if the blue dart thrower is statistically average and the ratio of darts is a fair game?

  9. What is the probability you will be invited back to play darts if you keep damaging their wall?
    …my favourite one is hitting the string and then having the darts in the wall to hold it up.

  10. How long has it been since the player has been to the eye doctor? How much will it cost to repair the wall?

  11. Mark Leadbeater

    October 16, 2016 - 5:14 am -

    Bob uses the quadratic formula to solve equations for the flight of his darts. His darts land as shown int he picture. Calculate the probabilities of the following.
    1) Bob doesn’t realise he’s living in a textbook pseudocontext and needs to get a life.
    2) Charles, his dart playing friend, tells him to hurry up and just throw the f***ing things.

  12. I can’t get past the trajectory of that bottom dart. Where they laying on their back when they threw it?

  13. How do I “win out” if I’m targeting a particular score, like 116 points? The last shot must be a “double” and you can take up to 3 shots per round.

  14. I’m glad that pseudocontext is back on the blog!

    In the past, I’ve always thought of pseudocontext through the lens of real-world math. And, as far as real-world math goes, the dart board painfully/binomial question fails painfully.

    But, looking at those overlapping circles has got me thinking differently. Because mathematicians have a cultivated set of questions that they learn to wonder and answer, and it’s different from what an untrained human would answer.

    (Christopher Danielson says that learning is having a new question to ask. In other words, to know math is to be able to ask more questions.)

    Looking at the “revealed” dartboard question makes me think strongly of contest math, where I feel as if I’ve seen this sort of question before. And, in that context, I think it might be better. There is a sort of playfulness with the real world that is distinctive of these pure-math problem sets.

    But I agree that something is wrong here, so let me posit that this is failed pure-math instead of failed applied-math. This pure math problem would be totally appropriate as part of a more advanced problem set, but the too-real picture sends the wrong signal. If you’re going to be playful with the real world for the sake of pure math (which I think is distinctive of mathematical culture) then you shouldn’t promise the real world with a high-res image. Maybe a cartoon dart-board would be better here.

    • This is interesting! I think this is one of the reasons CPM uses cartoon images exclusively. In many CPM questions, the context comes with an “isn’t this silly?” wink, exactly like you are describing here. I never connected it to contest math but you are right, that exists there too.

    • I think that is almost as bad. If we want students to make meaningful connections between math and the real world we should have quality meaningful contexts where appropriate.

      Does making it a cartoon and putting a little wink send a message that there is no real connection? I think it does.

    • I didn’t expect to find such a substantial thread in the middle of all of this mirth-making but here we are!

      One note I almost added to my definition of pseudocontext above is that textbooks have to opt into consideration. I almost referenced the problem about how many legs you see in a barn full of chickens and cows. It’s a puzzle, not an application problem, and the context exists only to serve the puzzle. Then there are situations where the textbook implicitly (and often explicitly) says, “This is how math is used in the world.”

      Those are the problems we’ll aim at in this series.

  15. I like to deal with special products in terms of patterns and the cool features we can see graphically…and then the fun puzzle of manipulating and transforming between “special ” products and “not special products”
    Then special products become a tool for exploring contextual quadratics

  16. Man, this “context” is an absolute embarrassment and wastes the time of students and teachers. This sort of thing is driven by textbook requirements for “full coverage” — some lessons have a useful “why” picture and description, therefore all of them must.

    Awful.

  17. If you were to determine the size of the corkboard square based on the given formula, how many of the darts shown would end up in the wall rather than in the cork square?
    What would be a sound experimental procedure for calculating the the ideal size of the corkboard square? What percentage of darts that miss the dartboard should hit the cork rather than the wall? How does your answer to the previous question change the size of the ideal square? Can you ever guarantee that it will be 100%? Is a square really the best shape for this, given the experimental pattern of missed darts? What shape would be better? What real-world reasons would cause people to use a square anyway?

  18. I can deal with psuedo-contexts in the classroom (I don’t use them). But what happens when the psuedo-contexts appear on the state graduation exam or on the SAT? How are we supposed to prepare students to do well on a test that we as math teachers find ridiculous?!

  19. Now I see how to make the dartboard fit into our task. First we each randomly assign each the five options that Dan gave us to 4 of the 20 sectors of the dartboard, so that 1/5 of the sectors correspond to each option. Now all we need is a blindfold, and … let’s see if we can improve our results from 3% correct to about 20% correct.

    • Not a great showing the first time around! But you folks will catch onto me pretty fast, I imagine.

  20. Wonderful first round, everybody. Not least because I won. (Though in a way, with pseudocontext, we all lose.)

    The conversation went deep and several of you posed questions that were deeper and more hilarious than the original.

    I added comments from Amy Hogan, Dennis Rankin, Bowen Kerins, Michael Pershan, Sarah, David Griswold, and Scott Farrand to the post. Many thanks, team.

  21. So Corby finds the formula for the area of the corkboard, then he, or is it she, puts in the value of r and calculates the area. Then has to find the square root of the area to find the side length. Almost there!!!!!!!!!!!!!!!!!!!(with a practical frame of mind)