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.

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.

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.

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.

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

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).

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?