Welcome to Mathland; Leave Your Common Sense at the Door

 “To do well in math class, children know that they have to suspend reality and accept the ridiculous problems they are given. They know that if they think about the problems and use what they understand from life, then they will fail. Over time, schoolchildren realize that when you enter Mathland you leave your common sense at the door.”

Jo Boaler. What’s Math Got To Do With It? Penguin Books, 2008.

Dan is at it again. Making me think hard, when all I want to do is collapse on the couch and watch Law & Order: SVU. His post about pseudocontexts got me excited to participate in his problem-hunt assignment. I emailed my (somewhat randomly selected) problem from a well-known publisher, who will remain nameless, to Mr. Meyer and was a little surprised to hear back from him that my problem was not, in fact, a pseudocontext.

O-kay. Either I completely misread Dan’s post (and Jo Boaler’s book), or the way Dan and I are thinking about this is different. So, what exactly is a pseudocontext?

The way I see things, a pseudocontext is a way of framing a problem that doesn’t make any sense, either with respect to what we know about the world, or with respect to what we know about this particular problem.

Pseudocontext can be a context that is completely arbitrary to the problem. In other words, it doesn’t really matter if you’re dealing with volleyballs, or roller skates, or Fraggles. As Dan puts it, “If you can replace the real-world units with “mystery number” or nonsense units and not lose any kind of interest or engagement you’re dealing with pseudocontext.”

In addition, I think that pseudocontext can also be found in the “make-believe contexts — contexts that students were meant to believe but for which they should not use any of their real-world knowledge” (Boaler) that are far too commonly found in math textbooks. For instance, a problem where you have a plethora of detailed information, but you don’t have the one obvious fact (that you would have if you were operating within the bounds of normalcy) which the problem is asking you to find.  As an example:

 

Okay, in what reality am I where I know 1) the cost of the meal, 2) the percentage I paid in sales tax, and 3) the percentage I paid in tip, but I do not know the total amount I spent each night? This to me is a pseudocontext, because it places me to a world that doesn’t match up with my real-life experiences.

Now I will admit that the mathematical calculations I’m going to need to perform do make sense in the given context. Perhaps some of you would argue that this means that this is a non-example of pseudocontext. I say, the context itself is unrealistic. I’m having a hard time differentiating between this example and the classic age problem—where I am four years more than half my sister’s age (or whatever)—in the sense that both problems withhold information that you would have access to if you were in that situation.

Sometimes a problem can become a pseudocontext because it is poorly written. I had a great example of this come up in my Math 2 class last week. Students were looking at a Markov chain involving taxi cabs, all of which started out each morning in the garage at Northside. After looking at some interesting patterns showing how the taxis moved around the city, students were asked where all of the cabs would wind up eventually. Logically, many groups said that they would wind up at Northside, because that’s where the garage is. Well, yeah[1]. Of course, that wasn’t what the writers of the problem intended when they wrote it, so I had to do some backpedaling and rephrase on the fly. In this case the context of the problem was ignored in the wording of this particular question, thereby rendering the context irrelevant to finding the answer. If you can—or have to—throw out the context in order to solve a problem, you have a pseudocontext rearing its misshapen head.

What’s the big deal? Why is everyone so down on pseudocontexts? Well, other than the fact that they are simply lame, one of my all-time favorite math people, Avery, points out that “pseudocontext can also change what is being taught/assessed. This is bad news if you think you’re teaching/assessing one thing and it turns out you’re teaching/assessing something completely different.” There was a problem about chickens in my former school’s textbook that always left my city-kid students scratching their heads. Not about the math involved, but because the context of the problem was so unfamiliar to them. I’m not saying that we need to remove any unfamiliar situations from our activities, but when we use these types of problems to assess student understanding we are setting students up to fail for the wrong reasons. It’s brutal to realize that a student bombed a quiz/test not because he/she didn’t know the mathematical concepts, but instead because that student had a misconception about the context of the problem.

Put this way, the use of any context at all sounds a little intimidating. Maybe we shouldn’t use contexts at all. But contexts, used appropriately, can be really awesome. See Amanda’s super-cool lesson on finding the height of the water tower next to her school. Personally, I think Amanda’s lesson satisfies Jo Boaler’s criteria for good use of context, that is, “A realistic use of context […] where students are given real situations that need mathematical analysis, for which they do need to consider (rather than ignore) the variables.”

In a comment on Dan’s post, Ben Blum-Smith summed things up nicely. “The real test of whether a math problem is “relevant” is not “do you use this in ‘real life’,” whatever that means, but “do you want to solve it?” It’s not that you want to solve it because it’s relevant; wanting to solve it is what it means to be relevant.” A good context may or may not encourage students to work on a problem. A pseudocontext will disengage many (if not all) students, because it forces them to disassociate themselves from the world for the duration of the problem. It makes what they already know worthless. And feeling like what you bring to the table has no value is not my idea of a positive learning environment.  


[1] This is one of those moments when I realize my students are much, much smarter than I am. Having been well trained, I went into automatic math-question-translation mode and overlooked the obvious.

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One thought on “Welcome to Mathland; Leave Your Common Sense at the Door

  1. Pingback: dy/dan » Blog Archive » [PS] Pseudorejects

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