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Here are three problems that satisfy the second half of the working definition of pseudocontext. I cop to a lot of guilt at the end of this post.
 


@sophgermain

Prentice Hall's California Mathematics — Pre-Algebra:

 


Kevin Krenz

McGraw-Hill's Mathematics: Applications and Concepts, Course 3:

 


Wing Mui

McGraw-Hill's Algebra 1:

Can you spot the common problem?

The author has fit an equation to a context that doesn't want or need it. Does anybody think the elephant/grizzly problem would be any less engaging if we just described "a mystery number, five sixths of which is 25?"

And I'm guilty. When it comes to systems of equations, I shake a handful of coins and ask students to write down how many coins they think they hear. We trade guesses and I tell them "40 coins." The best guesser gets some love from the class. Then I ask them to tell me how much cash they think I have in my hands. They ask me for the denominations of the coins. ("Nickels and dimes.") We trade guesses again, reveal the answer again ("$2.75″), and congratulate the winner again. Then I ask them if they think there's more nickels or dimes and why. Then we figure out the answer exactly, first by guess-and-check, then by systems of equations as I introduce an unmanageable number of coins. Then I confirm the answer visually.

But seriously: nothing inherent to a handful of nickels and dimes would lead a student to formulate and solve this system of equations.

n + d = 40
5n + 10d = 275

Nothing. Our arrival at that system of equations was painless only on account of a lot of coy teacherly showmanship. Does that theatricality — the shaking coins, the cocked eyebrow, the dramatic pause before the question, none of which is included in the problem as written in the textbook — inoculate the pseudocontext? Am I absolved if I don't pretend this is (as elephant/grizzly puts it) "when you're going to use this?" I'm not sure. I'm only sure that implicit in my use of pseudocontext here, whether I've inoculated it or not, is the admission that I'm empty-handed when it comes to a real context for systems of equations. I'm admitting that we only use this stuff in silly games.

35 Responses to “[PS] Operations That Have Nothing To Do With The Given Context”

  1. on 08 Jan 2011 at 8:12 amChristine Guest

    When I was sewing a pleated skirt, I wanted to use the entire length of cloth, but fit my hip circumference, then my waist by easing each pleat to fit. I had to figure out an equation with number of pleats, and optional space between the pleats. The folded cloth had to equal my hips, the unfolded the cloth width was my only set point (60 inches). I made a mock up to see approximately how many pleats looked pretty, then drew a sketch to organize my thoughts about where to put my variable. I also had to choose if i wanted box pleats, or knife pleats. i wasn’t matching plaids. It took a while, but my skirt turned out well without wasting fabric.

    I learned how to feel comfortable with variables in Algebra, Chemistry and programming classes. But I found my own contexts to use them. If your students get their skills down in a fun way, it’s not cheating, but do encourage them to keep their eyes open for examples.

    Chemistry, sewing and knitting are awesome for this, so is enlarging recipes, especially when you are low on eggs.

    Math books are notorious for heavy handed applications, but how to books that include math are great.

    Hope you have a great book budget,
    Christine

  2. on 08 Jan 2011 at 8:26 amDan Goldner

    Also guilty (and I didn’t even provide theatrics). I just spent a chapter teaching transformations of parent graphs. The hook was that they’d be able to sketch graphs of complex-looking equations quickly, which interested maybe 4 of them. An application at the end of the chapter was Will it hit the hoop?, which had everyone riveted, but no one wrote an equation. Most waited for someone else to figure it out. (One person solved it quite neatly by symmetry.) *I* know that later on, a million things will be easier if you can just see and understand the relationship between y=x^2 and y=(1/3)(x-3)^2+2. But that doesn’t count for many of them. Result: 20% found the mathematical context interesting enough to really figure out how these work. 80% dragged themselves through it enough times to get some rote patterns down that will get them enough points to pass. I think next year I will take away the graphing calculators for this unit and ask some of them to figure out how to move a graph to the left and others to explain what y=f(x)+5 does (I’ll give some of them a particular f) and why. In other words: more “math is a context.”

  3. on 08 Jan 2011 at 8:27 amAllyn Harris Dault

    I’m not a math teacher (though I almost went down that road), so I hope future comments have a better answer. But I wonder if you’re only going to be able to teach the idea abstractly, then come up with ways that it relates to “real math”. I’m thinking about finding a police suspect based on eyewitness descriptions. One person spots a license plate, another sees brown hair but can’t identify height, another sees sandy blonde hair but knows the suspect is 6’3″-6’6″. You use systems to understand the constraints that narrow the possibilities (you’re not looking for a short punk with kool-aid hair), but you can’t get to a particular answer like how fast the grizzly bear runs.

    Hope that makes sense.

  4. on 08 Jan 2011 at 9:11 amJ. D. Salinger

    Sorry but I find the whole context issue a red herring. The marble problem is a good one because it requires students to be able to take English descriptions and express them mathematically. That’s the reason for the problem, that’s the reason students should be told they are doing the problem. Making the problems “relevant” may be an issue for all of you, but for the many students, if they are given adequate instruction, explanation and opportunities to practice, they tend not to care whether it’s relevant or not.

  5. on 08 Jan 2011 at 9:37 amJen

    I rather have to agree that while *I* enjoy looking at your problems more than the book problems, I’m not sure that it’s more than that they’re visually attractive and especially that they make your enthusiasm clear. Same with the coins.

    On a related note, since my kids were little, I spent a lot of time begging that the “gifted program” frustrate the kids more — that is, teach the kids at a level that requires hard work and repeated attempts on their part. That skill of perseverance is the one skill I’d wish for if there were a magic teaching genie.

    Now, if your problems give less help and the context builds enough interest to suck the kids in to work hard, then I think that’s huge. But the truth is that a lot of math concepts didn’t really sink in for me until I’d done the standard algorithms in a rote way many, many times. In calculus in HS I remember feeling at each test that I just barely had the concept down, was just barely choosing the right steps to complete the test problems.

    And then, oddly, in the next unit, I’d be blithely using concepts that I had thought I didn’t really know. By two or three “units” later, I would realize that I then understand them — not just how and when to do them, but why. I’m not sure that that can or should be taught initially — that is, waiting until you get a more conceptual understanding before moving on. I think that some of it is just developmental and practice oriented. Just like I can’t really carry a tune…unless I’ve heard that tune about a million times.

  6. on 08 Jan 2011 at 9:37 amJen

    Skill of perseverance? Maybe I’d make that habit of perseverance if I could rewrite it!

  7. on 08 Jan 2011 at 9:42 amlouise

    The day you are being chased by a grizzly bear from the west and an elephant from the south and you can’t figure out which way to run, then you’ll be sorry that you didn’t figure out this problem ahead of time (and “faster than your hiking buddy” is not a solution)…
    Indeed, putting simple problems “in context” is silly, which is why I think Physics and algebra should be taught together. After all, the main impetus for mathematical education has been how to kill the most of the enemy – predict where the trebuchet will send the rotted corpse, predict where most of the enemy will be, predict which enemy has the most loot to grab, break their codes, get to the treasure first – but we’re too polite now to acknowledge that. DARPA doesn’t fund computing and robotics because it cares about safety.

  8. on 08 Jan 2011 at 9:56 amMark

    In physics, systems of equations are everywhere.

    Just look at circuit diagrams. Usually, the goal is to figure out the voltage level or the amount of current flowing through the circuit to make sure that it doesn’t overheat or explode on you.

    I’m going to try to integrate physics and chemistry into algebra as much as I can (or, at least, work in tandem with the chemistry and physics teachers) because, particularly when I was in high school, that was where the pseudocontext really gave way to the real-life context.

  9. on 08 Jan 2011 at 10:13 amDarren Draper

    Dan: I think what Mark is trying to tell you is that your last sentence should really read:

    “I’m admitting that [98% of normal adult humanity] only use this stuff in silly games.”

    J.D.: Is that your real name?

  10. on 08 Jan 2011 at 11:16 amDan Meyer

    If someone wants to advance an argument for eliminating contextually relevant problems from our math curriculum, I’m open to it. Certainly, I’d rather we did away with the extensions section of our textbooks if pseudocontext and poorly-represented, poorly-examined real context were our only options. That seems like a long shot, though, so the next argument I’d like to see advanced is the one where I’m supposed to teach contextual relevance but it’s a waste of our time to debate and blog about the best way to teach it.

    Darren: I’m admitting that [98% of normal adult humanity] only use this stuff in silly games.

    Right, I think that’s accurate. The real no-man’s land is when there are useful applications of the math concepts we teach in primary and secondary school but those applications are only useful to grown-ups in a highly specialized professional context.

  11. on 08 Jan 2011 at 11:49 amBryan

    Math education seems to have a serious problem here. When we are so restricted by standards, tests, and national comparisons of both that teaching the practical applications of mathematics has become a “long shot,” I can’t help but ask myself “why am I even doing this?” My answer, I suppose, is that I hope to help young kids become more confident in their numeracy and to become patient and persistent in solving problems. All of these goals can be accomplished via contexts that are far more interesting than systems of equations.

    I know many people won’t agree with me here, but why are we so concerned about student performance on tests that measure their competency in relation to a bunch of skills that they likely won’t ever need or use again? Sure, frowning on standardized tests is a tired debate, but I can’t see how math education can progress until we do something about it.

    I’m currently planning a project on cryptography, code-breaking, and internet security which has a ton of math in it (combinatorics, frequency analysis, modular arithmetic, etc.) but many of these are untested topics and, as a result, I feel like I am wasting my kids’ time. To me, that is very telling about the current state of math education.

  12. on 08 Jan 2011 at 12:37 pmZach

    It gets worse in upper-division math courses. As I am studying to teach math, I’ve taken Calculus (all three semesters), differential equations, linear algebra, abstract algebra (but did I expect to really learn any applications there… no, I guess not), and now combinatorics (which may prove to have something interesting, but right now I see pseudo-context problem one after another). Everything is devoid of REAL-world meaning. Is Euclidean geometry simply meant to be abstract and “out there” for most people (and, no one has really even heard of hyperbolic geometry, unless you’re a mathematician)?

    Yet, it is interesting to me that G.H. Hardy said “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” (See http://en.wikipedia.org/wiki/G._H._Hardy) And yet, his stuff is used, as Wikipedia states, in real-world contexts. So, math certainly can be applied, but HOW? If we are going to stick to “just learning math” in math class and not having to learn about biology, chemistry, physics, economics, art, music, and other contexts as a result, then I think the best we can do is pseudo-context problems. But, I do like the idea of learning about other things in math class. That’s part of what makes math interesting to me–someone found a theory and then someone else, somewhere down the road, found a use for it. That’s the story of Calculus :)

  13. on 08 Jan 2011 at 3:51 pmAnonProfessor

    I think the commentators are right. I suspect the list the topics that are required to be covered in K-12 education is poorly thought-out, and we could put together a much better math education that would be more useful to many of these students. I would love to see schools teach students basic numeracy and literacy skills and basic statistics, and skip many of the topics that are currently covered.

    However, I realize that the topics that must be taught are largely beyond the control of math teachers. So there are really two parallel topics for discussion: (a) for teachers who are required to teach a set list of topics and all-but-required to teach to the test, how can we help them make their classes as effective as possible, given the constraints they are working under?, and (b) if we think more generally about how to improve mathematics education, what changes would make it more relevant, effective, and useful for students?

  14. on 08 Jan 2011 at 4:52 pmZeno

    Your theatrics are a good way to keep the students’ attention and prevent them from falling asleep, both good things. In my opinion, we shouldn’t try so hard to pretend that every problem is an “application” from the “real world.” Many of the problems are just to instill some technical proficiency. I also dislike it when textbooks cite the elephant/grizzly problem as an example of a “wildlife management” or “ecological” application. Bogus applications undermine a text’s credibility.

    At the same time, of course, I love it when genuine applications lend themselves to classroom activities or homework (like the student I had who used a linear equation to compute omitted values from an enlargement chart at his workplace in a photo shop).

  15. on 08 Jan 2011 at 6:15 pmGavin

    The nickels-and-dimes question is not an example of a problem that anyone is going to solve in real life, yet I think it’s a good type of question for students to solve. (The theatrics makes it better, too; thanks for the idea.) That’s because students, at that stage of their learning, don’t expect that you _can_ solve the problem of how many nickels and dimes there are using mathematics. I like to harness that element of surprise and to encourage them to feel like they’ve learned something significant. (It helps that that’s how I felt when I learned that sort of thing for the first time.)

    In a way, though, the nickel-and-dime “application” is genuine. People care about solving resource conflicts, preventing electrical circuits from overloading (as mentioned above), etc. The spirit of the application is there, but boiled down to a context that students can understand.

    However, ideally, we would find more substantial problems (like electrical circuits), get them to _learn_ the context, guide them through the maths as much as necessary, and leave them with a sense of appreciation for the application and the mathematics. It’s hard for a textbook to do this; the activity needs to be targeted at a particular class. Textbooks can and should give teachers ideas for such ideas, however.

    At a conference, I saw a teacher describe an activity by which basic trigonometry was used to determine the school’s latitude from a shadow. Great activity, but from memory it only works on two days a year :)

  16. on 08 Jan 2011 at 6:42 pmDavidC

    Let’s not overstate the importance of context/applications. Math can be for its own sake. It’s not like we insist that music be “useful” before we ask anyone to appreciate it (http://www.maa.org/devlin/LockhartsLament.pdf).

    Irrelevant applications pretending to be relevant, though, that truly is bad. (Applications that aren’t even pretending to be relevant are another matter. Sometimes it can be nice to make the applications extremely silly, so no one thinks you’re trying to pull one over on them.)

  17. on 09 Jan 2011 at 2:15 amGuy Hed

    back to “just a silly game”:

    I think games are great context for math. I myself use them (in written activities) quite often. I the the students are interested and enthusiastic about the game (here the teacher’s showmanship plays an important role) than it is a relevant context, and if it presents an interesting problem, than it is not “silly”.

    Not long ago I came upon a great problem (or so I hope) for middle school students (with dynamics, attractors and such). I just couldn’t find any “excuse” for the problem. I nearly gave it up, until a colleague suggested the problem back to me as a competitive game. Somehow, seeing the problem at this context, made it much more interesting even for me. (Now I just have to sit and write it down…)

  18. on 09 Jan 2011 at 7:30 amAdam Glesser

    There is no problem with there being a huge difference between the examples you use to introduce a topic and the exercises you assign to help students master the topic.

  19. on 09 Jan 2011 at 10:10 amCarl Malartre

    Any references on the historical context in which systems of equations were discovered?

    Carl

  20. on 09 Jan 2011 at 12:23 pmYaacov

    Linear systems get used quite a bit in optimization. Cost optimizations are the traditional question to ask, but you can optimize lots of other things like fuel use (is it more environmentally friendly to send the basketball team to the game by bus or in cars?).

    Surveyors also use linear systems to figure out coordinates for planting survey markers. They often know the starting points and directions of the lines that form two sides of a lot and need to put a survey marker at their intersection. This requires converting angles into slopes, which may be a hurdle.

  21. on 09 Jan 2011 at 12:32 pmYaacov

    Another source of inspiration:
    http://climate.nasa.gov/keyIndicators/index.cfm#seaLevel

  22. on 09 Jan 2011 at 12:44 pmMarty

    Now i want students to learn and be great at math as much as any one. And I feel like I bust my backside to do it, but one thing I think that hasn’t been mentioned so far is the emotional side of learning. We aren’t teaching robots who feel nothing no matter what is happening. We also know that, whatever we teach, a massive amount of it will be forgotten extremely quickly.

    But what we can leave for a very long time is an impression. People often remember how they felt about something rather than the specifics of what happened.

    Trying to give an emotional feeling when learning math is unbelievably important, I feel. Yes the math is hugely important, but we are not teaching the content, we are teaching the person. I want my students to have the knowledge as well as the positive feelings that come from enjoyment, perseverance and genuine achievement. So put on a show. Make the kids enjoy it and do some great math.

    Personally, my ‘comedy filter’ is:
    1 it has to be real and authentic (sort of thing Dan tries to do)
    2 if I can’t do 1 (which is most of the time), then it has to be a good and compelling puzzle/challenge.

    Sometimes I can’t do 2 too

  23. on 09 Jan 2011 at 6:48 pmChristine

    I have a set curriculum and it is inquiry based, but since I am teaching Algebra II and Pre-Calc I am really having a tough time selling my kids on this stuff. There are a lot of problems that sound so ridiculous. I am about to finish my semester and I am desperately looking for different problems for next semester and for possible extra problems for the next two weeks. I am doing systems of equations and inequalities now. Does anyone have anything interesting for those topics?

    I also had trouble with the real life connections to parent graph transformations and the connections to system of equations word problems that are not just linear or parabolic.

    Could people please respond back (cfrench@tcsdk12.org) if you have materials that would be good on these topics?

  24. [...] the pseudocontext problem I submitted appeared on dy/dan! Woo! I guess it really speaks more of the textbook’s weaknesses than my ability to pick out [...]

  25. on 10 Jan 2011 at 2:13 pmDave

    Is it just me, or is the “WHEN am I ever going to use this?” above the middle problem like getting punched in the face? I don’t think psuedocontext is outright evil, but to put -that- problem under -that- heading is pretty egregious. I’d love to see a response from the authors of that text.

  26. on 10 Jan 2011 at 9:24 pmAaron

    It seems to me that what we want is for students to be able to figure out problems that they want to figure out. Suppose that I, as a teacher, have two options:

    1) I can present the coin problem described in the post in a way that actually engages the students and places them in a position of wanting to figure out the answer.

    2) I can present a completely legitimate application that does not place the student in the position of of wanting to figure out the answer.

    Without hesitation I would choose the first. My primary goal is to have my students thinking clearly and working well. Authentic applications of the “material” is of secondary importance. I am a new teacher and maybe this will change with time, but for now I don’t worry at all about good contextualized problems. When I find excellent contextual problems I use them, but if I don’t have any good contextual problems for a topic, I don’t worry about it. I’m much more focused on whether I have material that is likely to substantively engage my students’ reasoning and problem-solving faculties.

    By the way, it would be fantastic to have a good on-line database of excellent contextual problems. It would be nice if you could:

    1) Upload your favorite problems.
    2) Tag problems by topic, area of application (for example, physics), etc.
    3) Give thumbs or thumbs down to problems and see next to the problem the number of thumbs up and thumbs down (it may even be nice to be able to comment on each problem).
    etc.

    Anything like this out there?

  27. on 10 Jan 2011 at 9:56 pmBowen

    I don’t take much issue with the (pretty obviously fake) contexts in these problems. But what I do take issue with is the removal of probably the most important part of the first two problems: coming up with the equations and expressions!

    The statement “then Bill has 2m+4 marbles” is goofy — why just drop this in, when you could ask students to build the expression? It’s hard to really know the purpose of isolated problems like these, but what is the purpose of this marble problem? It can’t be the translation from context to mathematics, because that part is just auto-done in the problem statement. So what is the purpose?

    The second one has the same problem. “You can write 25 = 5/6 * s”. Well, then, let the kids write that! This is an important mathematical piece, why bypass it?

    So the purpose of the problem must be to analyze the different methods of solving 25 = 5/6 * s. This is better done without a context.

    Also (nitpicky) use “g” for the grizzly’s speed, since there are two potential uses for “s”. And I agree with Dave’s recent comment: there’s no freakin’ way this qualifies under the “WHEN am I ever going to use this” flag. What kid reads this problem then stands up and says “Holy crap I have figured out my life’s calling, and it is running footraces between elephants and grizzly bears!”

    I don’t mind the third problem. It’s clearly an exercise rather than something truly useful, but the students have to do the mathematical legwork to solve the problem.

    At some level, I think you have to use some kind of lame context to introduce concepts as they’re built. “Alice has 3 fewer apples than Bill. If Bill has b apples, how many does Alice have?” It’s lame, but it gets the point across, and ideally you teach kids how to do it using mathematical thinking skills (in this case, use numeric substitutions for b until you can see the pattern in the calculations, then get b-3 and not 3-b).

  28. on 11 Jan 2011 at 12:06 pmPerdita

    What’s the point of all this guilt and all these accusations? You don’t AIUI have any evidence beyond intuition that there’s anything wrong with teachers who want to using these problems. You don’t like them, fine. You feel able to create problems you like better and you are willing to share them with other teachers, fabulous! Now everyone has a wider choice. But couldn’t you do that positive thing without the jeering?

  29. on 11 Jan 2011 at 10:05 pmclobbered

    HD TV sets come in a 16:9 aspect ratio and are sold in diagonal sizes. Calculate the height and length of a 42 inch HD TV.

    (Yes I really had to do this in real life, long story)

  30. on 12 Jan 2011 at 7:54 amDan Meyer
    Perdita: What’s the point of all this guilt and all these accusations?

    “Guilt,” “accusations,” and “jeering” — I don’t know about any of that. I am trying to stigmatize curriculum that I know (intuitively) is corrosive to students whose mathematical conception of the world is still in formation. (Which is to say, most students.) I’m all for wider choice. These problems should be available in boutique bundles which parents can purchase for their advanced students. They have no place in a textbook that’s assigned to every student in a district or state. Those students have no choice.

  31. on 15 Jan 2011 at 10:23 amBen Blum-Smith

    (Have only briefly scanned rest of comments.)

    It seems to me your characterization of the nickels and dimes as pseudocontext shows that the definition needs further development. That is a great problem set up a great way, and nobody comes away from it with the lesson that you have to check your common sense at the door of the math classroom (Jo Boaler’s point about the problem with pseudocontext). Essentially, it is a math-as-its-own-context problem that has been spruced up with cuteness and theatricality to make it more fun. In this way it makes me think of Littlewood’s man-and-lion problem, although technically that one is more on the “flatly untrue” than the “nothing to do with the given context.” But that’s a great problem, that would lose something if it were stated in terms of two points that are allowed to move at equal speeds within a circle rather than a man and a lion in an arena. What it would lose is cuteness and theatricality. Same with your coins.

    This is reminding me of an earlier definition of pseudocontext that you offered long ago, and that I think captures something important that is absent from the current working definition: would the problem lose anything in terms of engagement if it were removed from the context? Answer in the case of nickels and dimes: yes; thus, not pseudocontext in this sense.

    I hear you as saying that cuteness and theatricality are an admission of a lack of real motivation. I don’t think this is right. First of all, just because you’re taking a worthwhile math problem and adding a little theatrical fairy dust doesn’t mean the same skills can’t also be motivated in your favorite WCYDWT way. (See discussion of IMP and linear programming below.) Secondly, I think it’s worth cultivating an appreciation of fairy dust for those kids (I think it’s more than we tend to think) who are going to respond to that.

    Now we’re getting into an issue that I feel bears a lot more thought – there’s a distinction waiting to be made and I don’t know what it is yet. I know that certain kinds of cutification and theatrification really rub me wrong. For example, much as I love Winnie Cooper, something does not sit well with me about Danica McKellar’s series of math text books. They sort of feel to me like they operate from the belief that math is actually boring and threatening unless it is dressed up in fluffy costumes made from plush Mickey Mouse dolls and cut-up copies of 17 magazine. (To be fair, I also think they’re kind of brilliant on this exact same tip. People have been doing this let’s make math unthreatening by dressing it up all cute thing for quite some time, and never has anyone ever done it so effectively. The most recent one, Hot X: Algebra Exposed, is the best yet.) But meanwhile, there’s a completely different kind of cuteness and theater that I experience as like 100% authentic and intrinsic to the discipline of mathematics, as opposed to being grafted on to cover it up. Consider almost any of the mathematical work of John Horton Conway, the inventor of Conway’s Game of Life, rational tangles (pdf), coauthor of Winning Ways for Your Mathematical Plays and The Symmetries of Things, and generally major mathematical beast of the 20th and 21st centuries. Everything that he does has this striking quality of winsomeness. The math itself has this quality but it is also often set in terms of adorable and whimsical settings that bring this out the more so. I just recently bought The Symmetries of Things, in which among other things they provide a proof of Euler’s formula F-E+V=2 that begins like this:

    We can copy any map on the sphere into the plane by making one of the faces very big, so that it covers most of the sphere. [figure] We’ll think of this big face as the ocean, the vertices as towns (the largest being Rome), the edges as dykes or roads, and ourselves as barbarian sea-raiders! (See figure 7.1.) p. 83

    Another example is all the work of the great logician Raymond Smullyan. I maintain (with no real basis but my own love of it) that his book What Is the Name of This Book? is the greatest collection of logic puzzles ever written. Every problem in it is a complete contrivance. (E.g. all the problems set on the Island of Knights and Knaves.) But they’re ALL ADORABLE (and theatrical). They played a significant role in shaping my own relationship to mathematics when I was only 8 or 9, i.e. before it was well-developed.

    My point is this: somehow these authors are capturing something that is authentic to the mathematician’s disciplinary practice with the way in which they are adding “context” full of theatricality and cuteness (what for short I’m calling “fairy dust”) to their mathematical thoughts and problems. Strip the thoughts and problems of the fairy dust and they lose something. Here the fairy dust can be “untrue” or can “have nothing to do with the given context” but neither of these features stops it from doing its job. It’s a totally different job, having to do with a different part of the aesthetics of math, and furthermore I don’t really think there’s any conflict of interest with having students understand math as useful and sensible and powerful for analyzing the physical and social world and all those things that WCYDWT-typed contexts are great for; so I think contexts of this kind aren’t really subject to the “flatly untrue” or the “operations that have nothing to do with the given context” tests unless they’re falsely selling themselves as being about what makes math useful outside of math.

    I think that more or less the same (or at least a closely related) job is being done by how you set up the coin problem in your class. So the failure of the operations to be forced on you by the fact that these are coins does not (at least, should not) make this pseudocontext. It would only be pseudocontext if you were pretending that the skills in question were going to help them make change. Since you’re not acting like this, there’s nothing to inoculate against. If one were to call the coins (or the barbarian sea-raiders, or the knights and knaves) pseudocontext, then that strikes me as a sort of exacting ascetic doctrine that says that unless a problem is like 100% realistic, it has to be set in a purely mathematical environment with no theater or fantasy, even for fun. This isn’t authentic to how math is practiced. For anyone like me, raised on Smullyan, it is full of theater, fantasy and fun.

    Again, ultimately this is about how students come to learn what math is all about. The goal of the word “pseudocontext” is to stigmatize problems that alienate students from the subject by forcing them to go to la-la land while pretending that you’re teaching them something useful. If a problem is not sold as a real-life use but as a cute fun possibly fantastical puzzle, then instead of alienating them from the subject maybe you’re cultivating their appreciation for puzzles and mathematical cuteness. I guess what I’m saying by bringing up Smullyan and Conway is that I think maybe the contrast “real context” vs. “only silly games” is an unfair derogation of the whimsical, the fantastical and the theatrical. They are a vital and central part of mathematical life.

    Btw, the IMP curriculum’s take on motivating systems of linear equations is linear programming. They have a unit about a cookie store trying to decide how much of what different kinds of cookies to make to maximize profit. It’s a little contrived, but only because it’s a simplification of the type of situation companies actually face, so the information you know makes sense to know and the information you’re looking for makes sense to want to know. (I think therefore that it’s not pseudocontext by either definition?) I’m sure you could find some industrialist who would explain to you an actual authentic business application of linear programming and it would look basically like the IMP problem but with way more constraints and uglier numbers. I’d actually love for you to do this because I’d eagerly await the WCYDWT that you’d come up with, and it would give a good reason to solve a system.

  32. on 17 Jan 2011 at 6:40 amDan Meyer
    Ben: It would only be pseudocontext if you were pretending that the skills in question were going to help them make change. Since you’re not acting like this, there’s nothing to inoculate against.

    Right. I intended the coins example to pull out this kind of definitional nuance. Same with a lot of math riddles or even this problem here, if it isn’t pretending to represent context, it may have other issues, but it can’t reasonably be accused of pseudocontext.

  33. on 19 Jan 2011 at 6:45 pmCathy

    Wow, some great responses and comments about pseudocontext. I don’t think we are trying to create good/bad or right/wrong categories but this discussion gives us the opportunity to reflect on how we teach math through problem solving. I appreciate the comments that have been posted here as they help me clarify what I believe and my goals for students.

    When I work with teachers I often share your video, ‘Math Curriculum Makeover’ so we can begin discussing and reflecting on classroom practice. One of these teachers recently passed on this problem which was produced by an exam bank. Pseudocontext?

    - A full set of teeth for an adult consists of 32 teeth. Brian and Sampson collided while skiing and both were injured. After the skiing accident Brian found that he had lost 3/16 of his adult teeth. Sampson found that he had lost 1/3 as many teeth as Brian had. Determine how many teeth Sampson lost.

  34. on 11 Feb 2011 at 5:41 pmBilly Wenge-Murphy

    Well, clearly, if you see a grizzly bear running after an elephant (both native to Africa, as any biology student knows) you’re immediately going to get out your pen and paper and start doing some math

  35. on 25 Feb 2011 at 7:02 pmChuck Hayward

    Stumbled upon your site recently and immediately started borrowing your teaching ideas. I used the coin trick in class today and literally got applause from my students as we revealed that solution to the systems was correct. Applause at one of the lowest-performing schools in Philadelphia. That says something about your teaching ideas to me. Thanks.