Ok, I have a question. I love this idea and would for sure use this in my class to make things more interesting and engaging for the students. However, I tend to get confused as to how this is ‘real life’ math. Maybe I am understanding this incorrectly, but I was under the impression we are trying to make the math relevant to the students. What student or person would do a regression for a toaster in real life? I agree students would rather solve this problem vs a problem on a worksheet because of how it is presented, but does it start to become a ‘psuedocontext’ type problem that is simply more visually appealing? Again, I LOVE the video and the problem (so I PROMISE I am not trying to be critical or mean!), but just having a slightly difficult time trying to sell this to a class that this will help them outside my classroom. Or do we just kind of admit that people wouldn’t do this in real life but continue giving the more engaging presentations because we know we have to teach the topic one way or another and it might as well be as interesting as possible? Thanks…love the blog and love all the great ideas from all the wonderful teachers! I so wish I had been able to teach at a school with these kinds of teachers! :)

If I’m right about this, the context comes from the student being invested in the question “when does it end?” rather than being invested in the toast?

(please be gentle if I completely missed the point, I’m a noob)

@Jen

I think if a student asks the question, that’s all the context you need. The only way you blow it after that is by forcing some unnatural method onto them.

like Tom said, the problem isn’t that textbook questions aren’t “real world.” It’s that nobody would ever ask those questions and even if they did, they wouldn’t solve it in the way you’re forced to solve it.

Aren’t we supposed to not be the arbiters of truth here? Why isn’t there a video record of the correct answer?

Oh and jg,

here ya go.

Yeah, no kidding. I’d hate to have to sift through all your footage. I’m gonna need my own server soon.

Quick disclaimer: my ramblings are really to help me make sense of why this problem that for some reason doesn’t sit as well with me as most of the other WCYDWTs. I love the fact that teachers are sharing their work here for the world to see.

@Dan, Yes, I suppose that one toaster would be *ahem* hotly contested by the many students, if you brought it in. I can still envision small groups watching and timing as one kid starts it at the different levels. Looking back at the accelerated timeline, and this might be too time-consuming (but, I would argue, you’d hook them more deeply: doing>watching a video>reading in a textbook). Depends on class size, dynamics, and how you set it up. Maybe the trump card is once they’re come up with their answers, pull the toaster out at make yourself a bagel at setting 8, with the clock running.

Maybe this video doesn’t fit your definition of pseudocontext, Dan, but like Jen I struggle with the relevancy, and wonder to what degree the students are hooked by curiosity vs. wanting to just get through the problem. It’s great to have faith in humanity’s curiosity, and I think its motivating power is generally underestimated. This can’t be a free pass to put just ANY suspenseful video in front of them, right? Several posters (Tom, Jason, Daniel) have said something to the effect of “it doesn’t matter if its relevant, as long as the kids want to solve it.” I think anytime you videotape something in the real world and show it to your students, you are essentially making an argument for its relevance, no? If not pseudocontext, pseudo-relevancy?

@David, I don’t mean to come across too negatively about this video. It is well-produced and exactly the type of thing I notice and think about as a somewhat nerdy individual and math teacher. It leaves me curious about a few things. 1. How many hours went into production? 2. How did students respond to the problem? Did they find it interesting? Did they find solutions independently, or did they need some help? 3. How long did you take in class for this problem? Bottom line, if the kids were truly psyched and curious about what you made, its a success. Was this done for a data fitting class, or linear equations, or what?

Thanks, all, for a quality discussion.

Paul: I’m curious, what was the order in which you filmed the toasters?

I filmed them in order 1-8. I wanted to wait till last to set off the smoke alarm.

Paul: 3:38 is significantly longer than my guess of 3:30, and the low 30s guesses of those students who had come up with answer by the end of class.

You’re 8 seconds off–less than 4% error. I’d say that’s pretty close.

@Sam: I haven’t done this problem with my kids yet. In order to stay true question being asked by the phrase “What Can You Do With This?”, I honestly wanted to throw this out for y’all to pick apart before I tried to do it with my students.

I teach middle schoolers and we are in the middle of dealing with linear relationships. The linear regression is ideal for my class. The more linear, the better at this point. However, I see the benefit of going with settings 1-4 (which is rendering in Vimeo as I type this) because it opens the door for a whole new discussion.

@Jen

My understanding of pseudocontext is that a “real world” context is forced into a question in an unnatural way.

What we have here, is a question naturally being asked by a series of visual cues.

I don’t think this is the same as pseudocontext, because the question naturally occurs without having the teacher directly ask it.

@Dan – yeah, perhaps pseudocontext isn’t my hesitation here. Plenty of my students were pulled in by it, so perhaps I am worrying about things that matter to me much more than them (my perception of relevancy). I have no problem with purely abstract problems, or with great applications. It’s the grey area in between that sometimes worries me, situations when you COULD use math to figure something out, but no one actually WOULD. I think this gets at my hesitation with this one better than my previous attempts to explain Really, it was an initial reaction, and the problem has grown on me a bit.

My question about the amount of time required to make it you took backwards, I think. I wasn’t implying it was thrown together or low quality; quite the opposite. I imagined it took many hours, and was trying to get my head around cost/benefit for doing this kind of video work myself.

Hoping for instant gratification here from someone who has plugged in this data.

I am one of the many physics teachers who is not actually a physicist by profession. So early in my physics teaching career I made the mistake of having my students determine the resistance of an incandescent bulb. They were to collect data regarding the potential drop across the bulb at varying currents created by a replacing known ohmic resistors in a simple circuit.

Problem (already obvious to some): Incandescent bulbs are not ohmic. I.E. they don’t obey Ohm’s law, and therefore do not generate a linear trendline. Lesson failed to teach intended topic, but did present us with a different problem.

So did you end up with a linear relationship?

I would have guessed the heating elements were non-ohmic. Just trying to move this great video into a new direction for use with my students.

When I introduce this problem to my algebra class who are in the middle of linear functions and have done a bunch of linear regression already they will probably not see anything but linear. I may also introduce it to my Senior trig class and see if they can recognize the exponential. Thanks for the posts I really can’t wait to try this with my students next week.

Dan L.: I would be intrigued to know if the toaster temperature is, indeed, a noise variable in the measurement. This is especially interesting as the toaster has “quantized” settings (1,2,3, …).

I’m interested in this as well. (Btw, thank you for concepts like “noise variable” and “quantized.” As I was writing an earlier comment, I was grasping for terms like these.) I left my figuring at work (in the form of a post-it stuck somewhere on my paper-strewn desk), but in the initial data (1,2,5,7) I was interested to see the time variations between settings expand, contract, and then expand again.

Brian: I would have guessed the heating elements were non-ohmic. Just trying to move this great video into a new direction for use with my students.

Any more on this? Sounds plausible in that “someone used jargon I don’t know” kind of way.

PBS Frontline did a great documentary a while ago that explored the consequences of the increasing digitization of our world. In “Digital Nation,” Arne Duncan and journalist Todd Oppenheimer each had very different opinions about the role of video games in classrooms. Duncan was all for them, citing their ability to get kids excited. Oppenheimer was less sanguine, and argued against confusing entertainment with learning.

Vis a vis psudocontext, then, perhaps an important question might be: “What’s the point?” The toaster video is cool, so what’s the point?

Jen argues that it’s not “real” insofar as it doesn’t provide for much application or insight beyond itself. Put another way, will students walk out of class having learned something important, or simply having done something fun? If you believe that math is a primarily a tool to understand the world around us–that its value is in its application to real life–then the toaster problem can rightly, I think, be called pseudocontext.

On the other hand, many might argue that the *point* of mathematics is the act of exploration itself: that we send a man to the Moon because it’s there. In this sense, pseudocontext no longer has any inherent definition, but is determined by the students’ reaction; if they ask follow-up questions, i.e. if they’re curious, it’s authentic. If not, it’s contrived. Here, if students are excited, then the toaster problem is authentic.

Ultimately, I’d suggest that each answer is wrong, unless both are right. Indeed, the history of mathematics itself is characterized by the interplay between application and abstraction, between “How much water is in the tub?” to “What might what a fourth dimension look like?” Accordingly, an engaging and legitimately educational math class will hopefully have elements of both: we’re going to explore how to determine the best Verizon cell phone plan (directly applicable), but also when Dan will exit the escalator.

That said, it’s probably very difficult to find that balance. Personally, I lean more heavily towards the application side, in the same way that, for me, I contemplate the stars in order to better understand my own life. And so my concern with a curriculum based on WCYDWT would be: WCYDWT? How will you apply this lesson tomorrow? Will it measurably improve your life, and your ability to navigate it?

Finally, lest this be too abstract, here’s how it plays out for me. I currently have around 140 in my Mathalicious lesson queue. These run the gamut from “Is dizziness linear (spins vs. walking distance)?” to “Where should cell towers be?,” from “Hot Pocket temperature vs. microwave time” to “The Matthew Effect.” I think each lesson is worth writing, but in each pair, I’ll almost certain prioritize the second. The reason is because, while the dizziness and Hot Pocket problems are fun, the other lessons may have longer staying power. They may elicit a less excitement on the front-end, but might matter more in the long run. But again, that’s just me. That’s just how I define the paramount purpose of mathematics and, by extension, pseuodcontext.

@Marcia

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).

I had my students go home and time their toasters on 5 different settings.

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).

I have a water heater (what are they called in English – the electronic kettles?) with adjustable temperature. I think I could get interesting data out of manipulating temperatures and water levels but it’s difficult to foresee the level of complexity. Won’t I get three-variable stats?

This video and discussion has been stuck in my mind since the first time I saw it. I have been watching Dan’s explanations and examples of pseudocontext and feel that I now have at least a slight grasp of the concept, enough to venture this idea.

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?