[PS] Midterm — Solution Key

[see midterm]

1. Which of Will’s commenters has suggested a pseudocontextual problem?

Very few of them, it turns out. Which isn’t to say that all of the suggested problems are good problems, just that our worst tendency in these discussions is to conflate the term “pseudocontext” with “problems I don’t much care for.” Pseudocontext uses the full authority of the teacher or the textbook or of grades to force a connection between The Math and The Context that doesn’t naturally exist. This is a separate matter from “Do professionals really use math in that way within that context?” or “Will my students care at all about the math, even though it exists naturally in this context?”

Spot check me here but I find it pretty easy to divide the suggestions at Will’s place into four categories:

Valid Context, Of Inherent Interest To Basketball Professionals

These questions are ideal, and really hard to find. Will is probably better off asking a basketball player, a coach, or a sportscaster because they won’t waste his time (or, especially, theirs) with pseudocontext.

Glover:

You can probably set up different metrics – basically giving different weights to things like free throw percentage, scoring, rebounds, steals, etc. – and compare players and be able to claim “I can prove that Kobe is better than Lebron” or whoever else you want to compare. This would also extend to comparing teams and stuff.

Have you seen John Hollinger’s formula for player efficiency? It’s gargantuan. Have your students create their own, balancing factors however they choose, checking the results against their intuitive sense of the best basketball players. (ie. “I know Kobe’s the best but my formula is coming up with Lebron.” So you rebalance.)

George Woodbury:

There are some probability applications. If a free throw shooter has a 75% chance of making a free throw, what are the chances that he or she makes 2 free throws? 1 of 2? 0 of 2?

Pair with video clips of different shooters at the line. Given their overall average, place bets on which players will hit both, which will hit one, and which will miss both. Then show the answers.

Valid Context, Uninteresting To Basketball Professionals But Interesting To Students If We Develop It Well

Do basketball players need to know anything about 3D vectors? When they go for jumpshots are they converting force to acceleration and then solving the quadratic formula to see if the ball will land? No, but these problems are of interest to students. Especially if we visualize them well, if we present them in a format that appeals first to intuition (“do you think it goes in?”) that only later uses math to formalize that intuition.

Angie B:

Dan Meyer’s dy/dan blog had this on it.

wmchamberlain:

How about studying the lines and shapes on the court? A little bit of geometry there.

I like the challenge here. There is a lot of geometry in the lines and shapes on the court. But the hard work remains. What question do you ask? What problem do you pose to get students diving into the geometry of a basketball court, wondering “what kind of shape is the three-point line anyway?”

One idea: I draw two points on a piece of paper. The line between them is the baseline on one side of the court. Can you draw a scale replica of the rest of the basketball court. Bonus: do it with a compass and straightedge alone. Extra bonus: write a program that accepts two mouse clicks and does the rest automatically.

Valid Context, Uninteresting To Basketball Professionals, Also Uninteresting To Students

It’s within your professional jurisdiction to ask these questions, but these questions only appeal to math teachers. Not basketball professionals, and certainly not students. But they aren’t pseudocontextual. These problems have an inherent connection to basketball. If anybody is selling a term to describe these problems (ideally something more descriptive than “lame”) I’m buying.

Lyn Hilt:

Incorporate geography…calculating traveling distances/methods to visit arenas and attend games.

Jason Kornoely:

If it takes one player an average of 15 seconds to dribble across the field, how much time would it take for a team of x to finish?

Rhett:

[What is the] volume of skin making up ball assuming skin is 1/16 th thick?

Pseudocontext

Jason Kornoely:

Determine the hypotenuse using the distance from the free throw line to the center of the hoop and the height of the hoop.

This was the only response I could call pseudocontext with absolute certainty. The Pythagorean theorem has no use or meaning here. The teacher is imposing the theorem on a context that doesn’t want or need it.

2. Create a math problem in response to Will that would be pseudocontextual.

Chris Sears:

Kentucky and UCLA have appeared in the NCAA Division I men’s basketball tournament 80 times, with Kentucky appearing 8 more times than UCLA. How many times has each team appeared in this tournament?

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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.

14 Comments

  1. Category 3 as “extended/other context” maybe. I can see those questions being interesting to other professionals, the people who design/manufacture of the ball for instance.

    But I can’t help thinking: whose responsibility is it that we come up with these questions anyway, and why are we doing it? I agree that pseudo-contextual questions are just ugly and horrid and should be avoided. But if we are choosing between mathematical and “real-world” context, why isn’t mathematical context enough?

  2. So pseudocontext requires mangling reality to impose a math construct? (imagining a hypotenuse that isn’t there, or distorting the flow of information to construct a system of linear equations)

    Also, at a bit of a loss here; did my submission to the previous post get missed or do you see it as something other than pseudocontext? (My post was a bit messy but I did have a problem written in there.)

    http://blog.mrmeyer.com/?p=8447#comment-269665

  3. Yesterday, I met with the authors of the problem I submitted. I wrote a blog post about it. I tried to ask about pseudocontext, but I couldn’t find a way to do it without being a jerk. They are both very nice people who but a lot of effort into their books. They did tell me that their College Algebra book has more relevant problems. I will have to check it out.

  4. I guess that’s part of why I keep coming back to trying to imagine the process in which these kinds of problems get written. We should be able to critique these problems in a way that deconstructs where these things come from, not one that just points the fingers at particular authors or even just specific problems.

    I presented so many pseudocontextual systems-of-equations problems this week that my head is spinning. Textbooks, worksheets, you name it, they’re all over the place. So why are there so many of them? Where did they come from? Are they a disease or just a symptom?

  5. I have a question about the pseudocontextuality (that’s a mouthful) of the hypotenuse question:

    “Determine the hypotenuse using the distance from the free throw line to the center of the hoop and the height of the hoop.”

    The standard height of a basketball hoop is 10 ft. It is quite easy to get a estimate for how far it is from the free-throw line to the point on the floor directly beneath the center of the hoop (“pacing it,” for example). One can then use the pythagorean theorem to determine the distance from the three point line to the center of the hoop (in the air).

    This can be made into a good case that the problem in question is not pseudocontext. It is true that the question “What is the distance between the free-throw line and the center of the hoop (in the air)?” is not important to basketball players, but that (if I understand correctly) is not the issue of pseudocontext. The issue is whether the physical situation has anything to do with the mathematics of the question. Here the physical situation is relevant, for it is common knowledge that the height of a basketball is 10 ft and it is easier to measure distance along the floor than distance through space. In this situation the pythagorean theorem allows us to make the easier measurement and then compute the distance that would be more difficult to measure.

  6. When people envision teaching the free-throw problem, is the emphasis on the basketball, or on the math skill? From the would-be author’s perspective, it seems there are three ways of approaching a problem. For the most part, these are pretty similar to Dan’s four categories:

    A. Here’s a real-world scenario. What math can I get out of it?. This is
    B. I have to teach [topic x]. Where does it occur in the real world?
    C. I have to teach [topic x]. How can I fit it into a basketball context?

    According to my understanding of Dan’s definition, problems of pseudocontext become more likely as you move from A to C. This makes sense, since the funnel becomes more narrow; A is wide-open; C is fairly closed. (@Aaron: regarding calculating the hypotenuse, if students asked why they were doing it, how would you respond?).

    On the flip-side, what happens with topics that simply don’t occur naturally: either in the real-world, or as methods of solving not real-world but potentially interesting anyway questions? (Simplifying polynomials, for instance). If a skill can’t fit into A, B or C, what do we do about Category D: I have to teach [topic x], and the only way to do it is to teach [topic x] ?

  7. I’m starting to get the distinct impression that I need to go to the original source before the definition of Pseudocontext is going to make sense to me. Right now it feels like I’m inheriting too many second-hand interpretations.

    Also, I think if this is going to make any sense via second-hand examples, there needs to be a name for the irrelevancy factor as well, and it needs to be highlighted along the way as such. If there are two axes along which something can be messed-up, but we only ever talk about the Pseudocontext axis when we give examples, it’s going to be really easy for people (ie. myself) to misinterpret and confuse the two.

    I guess I just suggested “irrelevancy factor” as a potential name by accident, but I’d gladly hear a better suggestion.

  8. Julia: I agree that pseudo-contextual questions are just ugly and horrid and should be avoided. But if we are choosing between mathematical and “real-world” context, why isn’t mathematical context enough?

    Great question. I suppose students enjoy seeing math applied to the world around them. I don’t begrudge them that. Math has made my own real-world context a much happier, clearer, more interesting, and more profitable place. The mathematical context has to be enough, though. If we go into curriculum development on the notion that “we can’t teach [x] unless we find a real-world hook for [x],” we’re about to do a lot of damage.

    josh g.: Also, at a bit of a loss here; did my submission to the previous post get missed or do you see it as something other than pseudocontext?

    Yours was the definition of pseudocontext. It was too long to excerpt, though.

    Aaron: The standard height of a basketball hoop is 10 ft. It is quite easy to get a estimate for how far it is from the free-throw line to the point on the floor directly beneath the center of the hoop (“pacing it,” for example). One can then use the pythagorean theorem to determine the distance from the three point line to the center of the hoop (in the air).

    You could also construct a rectangle using the two dimensions given and ask for the area. But in both cases, we’re using dimensions from basketball to ask a question that has nothing to do with basketball.

    Karim: If a skill can’t fit into A, B or C, what do we do about Category D: I have to teach [topic x], and the only way to do it is to teach [topic x] ?

    Right. This is a whole ‘nother discussion and, since classroom teachers don’t usually have the luxury (like you and I do) of waiting for [A], it’s worth having. But maybe not here. Short answer (for me) I guess is that I try to locate the pressure point between [skill x] and [skill x – 1]. Then the students develop [skill x] to alleviate that pressure. Makes total sense, right?

  9. Would it be a reasonable extra credit opportunity to ask students to identify psuedocontext problems in their practice/assignments and create better alternatives?

  10. Dan wrote: The mathematical context has to be enough, though. If we go into curriculum development on the notion that “we can’t teach [x] unless we find a real-world hook for [x],” we’re about to do a lot of damage.

    You’re narrowing the field of possibilities. Curriculum development means putting together ideas that will help teachers do the best possible job of teaching a particular math idea even a mundane one like the associate law. It’s not about finding a specific real world or mathematical hook. Its about finding an appropriate, engaging hook (multimedia etc.) so that the math becomes a reasonable thing to do. When I was teaching advanced classes in a private school in NYC almost any context (within reason, of course) worked. These were kids that loved to e.g. discover the formulation of the quadratic formula and they would yell at me if they thought I was jerking them around; which BTW was a strategy I actually used to see if they were getting the point of the lesson.

  11. This reminds me of a similar exercise with my students.

    STUDENT EFFICIENCY FORMULA

    I created a fictional student and marks based on assignments and test we had completed already.
    ex.

    Algebra test: 15/20
    Algebra Assignment: Level 3 (based on a rubric)
    Homework Checks: 9/10
    Angles Test: 30/32
    Angles Poster (take home): 4/5

    After presenting these numbers I said, “Any Questions?”

    many students: “What’s her average?”
    one or two: “What’s her mark going to be?”

    Taking a closer look each of my classes came up with the idea of WEIGHT (or worth). Homework checks (90%) is a classroom management technique and not necessarily a significant assessment. The take home poster (80%) was a weekend assignment used to practice terms not checking for knowledge. Algebra Test was huge.

    The students came up with their own FORMULA…their own WEIGHTING…and then used their formula applied to their marks and calculated. Excel came into play.

    Instead of a Player Efficiency formula, this was essentially a Student Efficiency formula.

    (sorry i’m late…i’m catching up on my starred google reader items!)