Brian: To use this or any other heating element to model linear equations is pseudocontext. These devices do not dissipate energy in a linear fashion, and therefore to ask students to make such models is surely using the force of a teacher’s authority.

Wow, yeah, awesome.

Not for nothing, your comment goes the distance to help me explain the point of multimedia and my suspicion that multimedia inoculates pseudocontext.

Your textbook author writes the premise of the problem. Your textbook author also writes the conclusion – the answer.

It can make up any fanciful premise it wants. It can ask the student to apply any nonsensical mathematical operation to that premise in search of a conclusion. And then, in the back of the book, it can verify the conclusion.

But when David films the premise of the toaster regression and when he films the answer to the toaster regression and finds out *gulp* it isn’t linear, he knows if he does anything but change the problem, it’s a lie. It’s pseudocontext.

Multimedia keeps us honest, in other words.

To use this or any other heating element to model linear equations is pseudocontext. These devices do not dissipate energy in a linear fashion, and therefore to ask students to make such models is surely using the force of a teacher’s authority.

Using the same video to model exponential equations would not however be an example of pseudocontext as that is the true behavior of the device.

A final important question: Should a math teacher care given the fact that inconsistencies in a linear model can be chalked up to scientific error? Is is only the science teacher’s position to care about the pseudocontext with the problem?

Toaster Regression Data & Scatterplot: http://bit.ly/by1DB3

The great amount of scatter shows high levels of variability between toaster brands. I want to do this again with a bit more control for confounding variables (like let the toaster cool down before you start your next trial).

Interesting ideas. I think it totally makes sense that prior to having a concept of “pseudocontext,” we must have a definition of “context:” the essential reasons for studying math in the first place. Rather than a dichotomy, I see a spectrum of abstraction-application. Plenty of topics/problems have interesting applications AND are interesting abstract topics as well. Imaginary numbers, for example CAN be used as a very useful tool for modeling differential equations in electrical engineering; they are also a vastly interesting intellectual story of mathematicians wondering “what if” with NO connection to “reality”. We as teachers have a choice of if/how we engage the different reasons.

This is why I encourage my students to ask “why are we learning this,” or even pose the question myself. I think if we are afraid of the answer we might have to give, there’s a good chance we’re wandering into (our own) pseudocontext.

That’s a really nice distinction–real world vs. realistic–and thanks for making/relaying it. In the end, I wonder whether “pseuodcontext” is itself somewhat pseudo, in that there may never be any final agreement about what it means. If this is true, it isn’t a bad thing. Maybe it’s even a good thing, since it preserves discretion for the teacher & student. Maybe “pseudocontext” is a lot like art or the Supreme Court’s non-definition definition of pornography: I know it when I see it., where the addendum might be, and trust that you do, too, even if we don’t see the same thing.

Ultimately, I do think how we define the legitimacy of a certain context comes back to our respective definitions of mathematics…or if not mathematics per se, its purpose. (Or, to narrow this even further, its purpose in a school setting). Is math a tool to learn about *other things*, or is it an object of inquiry in an of itself: applied vs. pure? If you’re [predominantly] in the applied camp, then there’s the follow-up question of What’s the Point?: does the application actually add value to a student’s life/understanding of the world?

My point here isn’t to suggest which is the best–there is no such thing, I hope–but simply that there are any number of factors that influence our definitions of pseudocontext…and that’s a beautiful thing.

(Finally, a quick aside: I really want to thank everyone who participates in these blog discussions. When I started Mathalicious, my goal was to create the best content ever: the best math lessons ever written! Indeed, I was–still am, for sure–a bit arrogant in my ability to do that. But the more I read these posts, and the more I read the blogs of other teachers, the more I realize how ridiculous and utterly unfounded this hubris is. Participating in backs-and-forths like these feels at times like getting sanded down, power cleaned. Anyway, I know this seems a bit out-of-left-field, but I just wanted to put it out there).