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Toaster Regression

David Cox has WCYDWT by the throat. He used digital video, Adobe AfterEffects, and MovieMaker to export a clever visualization of toaster times versus toaster settings.

Toaster Question from David Cox on Vimeo.

Not that he asked, but I wouldn't change a lot here. I'd rather see the data for settings one through four and use those to regress the eighth setting. By providing the seventh setting and asking for the eighth, he's made it easier for students to jump right into the math which makes it less likely that my remedial students will invest a guess.

I would have also sped up the first four videos (even more) because I want my students' impatient toe-tapping aligned to the question, "when will it end?" not before.

It's really strong work, though, and you're only going to see more of it from David because it just gets easier and easier to clear the annoying technical hurdles of video production. Soon he won't even notice them and it'll be as if there isn't anything in between the curriculum he can imagine and the curriculum he can create.

42 Responses to “Toaster Regression”

  1. on 28 Oct 2010 at 6:24 pmJen

    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! :)

  2. on 28 Oct 2010 at 7:04 pmDan Meyer

    Bring the tough questions, Jen. I’m still not resolved on the matter of pseudocontext, myself.

    My sense, though, is that it’s very, very hard to commit pseudocontext to video. If a question arises naturally from a video and if the answer to that question has also been filmed, I’m not sure the accusation sticks. I need someone to give me a counterexample here.

    Also, by asking students to guess how long they think the eighth setting will take (as is standard operating procedure with WCYDWT) you are encouraging them to invest in the problem. At that point, it doesn’t matter if we think the problem is pseudocontext or not. They will want to know the answer. They won’t let you move on. Very few students want to solve pseudocontext in the same way.

    I mean, check out Avery’s Greg’s example with the basketball [thx, Jason]. No one would solve the answer to the question “will he make the basket?” with parabolas or equations or symmetry. They’d just watch him to see if he makes it. But because a question has arisen naturally out of the image and because the answer to that question also exists, the students want to solve it, and like that, it isn’t pseudocontext.

    What do you think?

  3. on 28 Oct 2010 at 7:07 pmNumbat

    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)

  4. on 28 Oct 2010 at 7:37 pmTom

    Dan/Jen,

    I think that Jen is confusing psuedocontext with irrelevant or not-real-world. The problem with pseudocontext textbook examples isn’t that they don’t present real-world applications of Math in the sense of “someone does this for a living” but that the context and the question asked are not naturally aligned.

    The thing I like about Dan’s posts is that he takes curiosity for granted. You don’t convince kids to like Math by telling them Math will help them get a job or that Math will help them get a girl or whatever you think might connect to their real-world experience.

    You convince kids to like Math by showing them that Math helps them solve problems.

    Why do they want to solve problems, you ask. Because they’re human.

    If you don’t have faith in that last statement, everything falls apart.

  5. on 28 Oct 2010 at 9:26 pmJason Buell

    @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.

  6. on 28 Oct 2010 at 11:14 pmSam Critchlow

    One test we applied to pseudocontext word problems was replacing the “real world” unit with an imaginary one, and seeing if the problem suffered in any way. I would argue that you could replace the toasters with any sort of visual countdown, and the problem would be the same. There is nothing additionally compelling about toasters.

    The other flaw I see in this video (and by the way, it’s not a bad video at all, it just has some overachieving WCYDWT peers) is that there is no space for mathematical questions to arise organically. What the teacher/videographer is looking for is clear, and there is no extraneous information, no opportunity for creative inquiry. Just a countdown. Certainly better than most textbook examples, but still a good deal of handrails as to what is expected.

    Have we encountered the limit of what video can “sell” to a student here? I think we need to be careful that we don’t confuse screen-hypnotism (“math class was fun today because we got to watch a VIDEO!”) with mathematical engagement (“math class was fun today because we figured out this SICK problem!”).

    BTW, I think bringing the toaster in to class and letting the students mess around with it would be much more effective (and requires zero video production prep time).

  7. on 29 Oct 2010 at 4:39 amjg

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

  8. on 29 Oct 2010 at 5:26 amDaniel Schaben

    @ Jason and Tom I think you guys nailed it. I am finally because of this blog and listening to Cathy Seeley allowing my students to explore problems on their own. Allowing them to let their natural curiosity carry them through the problem. They primarily guess and check at first and the massive amount of time used in the classroom for each problem is almost painful . . . but they are using their own past experiences to explore and play. Yesterday a group of students of mine after spending 15 minutes to arrive at a correct answer with guess and check along with some other logic along the way asked me, “This is taking way too long, how would you have approached this problem MR. Schaben?” I had finally hooked them. Had I forced my methods on them to start with two of them would have listened. The rest would have counted ceiling tiles. I probably missed the point as well but thanks to everyone for getting me thinking more about this and how to make it work in my room. Great Toaster video! If vimeo is not blocked at my school I will use it.

  9. on 29 Oct 2010 at 7:38 amDan Meyer
    jg: Aren’t we supposed to not be the arbiters of truth here? Why isn’t there a video record of the correct answer?

    Good question, but the wrong place to ask it. Go check with David.

    Sam: I would argue that you could replace the toasters with any sort of visual countdown, and the problem would be the same.

    Really?

    The #1 setting toasts the bread for 1:40.
    The #2 setting toasts the bread for 1:52.

    is leagues more interesting to me than:

    1 snorfblatt corresponds to 1:40
    2 snorfblatts correspond to 1:52

    Go check out My Favorite Orange again and see if you think we’re dealing with the same kind of thing.

    Sam: BTW, I think bringing the toaster in to class and letting the students mess around with it would be much more effective (and requires zero video production prep time).

    Can you explain how this would work? How are all the students engaged and challenged by this one toaster?

    Sam: What the teacher/videographer is looking for is clear, and there is no extraneous information, no opportunity for creative inquiry. Just a countdown.

    This seems right on. There isn’t a lot of room to breathe with this one. But compared to the sort of textbook problem that offers you the same data in a table and puts a question mark in the box beneath #8, I’ll take it.

  10. on 29 Oct 2010 at 7:52 amDavid Cox

    The more I think about this, the more I realize that my choice to go with settings 1,2,5 and 7 instead of 1-4 was more personal than pedagogical. I wedded myself the the fact that there was an actual linear regression in there somewhere. I was curious and I had to find out for myself. And it became a great reason to put AE to use. (I ended up using Premiere Pro instead of MovieMaker, btw). I’ve been reminded that I probably should have shot each setting using a cold toaster as temperature plays a large role in when the thing actually goes off. So, I may be back at this again.

    Thanks for the feedback, Dan.

    I think Jason’s right about getting kids to ask the question. They don’t really care if the math is “real life,” they just want it to be interesting.

    Daniel,
    If Vimeo’s blocked at school, try viewing from home and just download the clip. That way you can view offline anyway.

  11. on 29 Oct 2010 at 8:14 amDavid Cox

    Oh and jg,

    here ya go.

  12. on 29 Oct 2010 at 8:16 amDan Meyer

    Cold toaster. That’s interesting. This is how you end up revising these videos for months.

  13. on 29 Oct 2010 at 8:42 amDavid Cox

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

  14. on 29 Oct 2010 at 10:20 amPaul Guilianelli

    Re: cold toaster

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

    I ran this problem with my kids today. They were into it, but still very much focused on “the right answer,” as in, “Mr. G., is this the answer?” I kept repeating that I was more interested in their problem solving process, but they were still concerned about being right. (Since this class was at 9am CST, I didn’t have the “right answer” anyway.)

    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. (Half were still in the, “This is sooooo hard, I need help” camp.) That’s why Im interested in the filming order, to see if that can explain the seemingly random fluctuations in toaster times.

    It appears that the digital readout gives us a false sense of precision and that the sliders and dials of the toasters of yore are a more honest reflection of the random elements of toasting.

  15. on 29 Oct 2010 at 10:29 amSam Critchlow

    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.

  16. on 29 Oct 2010 at 11:41 amDan Meyer
    Sam: 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.

    I’m curious if this video would fit anybody’s definition of pseudocontext and, if so, what that definition is.

    There’s an easy way to test this out. Play the video for your students. Take bets on the eighth setting. Write a few on the board. Ask them for an amount of time they know would be too high / too low. Then say, “cool,” and move on.

    Time and again, readers have pointed out that their kids revolt. (That effect would have been even more pronounced if David used settings one through four.)

    You go on to talk about separate categories of suspenseful videos and real-life videos. I’m not saying just toss any video up in front of your kids. I’m saying put a video in front of your kids that is simultaneously a) suspenseful, b) ripped from real life, and c) mathematical.

    You seem put off by how simple the video is and how easy it looks to produce, to which I reply, “Exactly!” Get on it, mate! Make something fun!

  17. on 29 Oct 2010 at 12:47 pmDavid Cox

    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.

  18. on 29 Oct 2010 at 1:15 pmDavid Cox

    Alright, settings 1-4 are up.

  19. on 29 Oct 2010 at 1:18 pmBill

    @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.

  20. on 29 Oct 2010 at 4:11 pmSam Critchlow

    Tried the video with two small Precalculus classes (11th grade) this afternoon. Quite the variety in both engagement and results (a couple kids decided to make exponential models for some reason–probably because we discussed exp. gowth equations last week–that seemed to fit even better than a line – are we seeing heating/cooling effects here?). We were doing some mathematical modeling, so I justified spending some time with this by using it as an opportunity to use statplot and various regressions on the calculator (led this after they had predicted and seen the answer).

    I had a few students hooked IMMEDIATELY by the problem, motivated by intensely wanting to know how long it would take for setting eight. Their enthusiasm helped the others get into it. The majority came along for the ride willingly. There were also a few who gave various versions of eye-rolling at the video. And a couple seemed slightly insulted by either a) being given an “easy” problem or b) being asked to figure something out without being provided the best method. Only one girl truly pushed back against the video/setup (thought it was contrived). These are 16 year olds, so I would imagine that slightly younger students (like yours, David) would push back a bit less. So I guess my worries about relevance/context/whatever (that I have clearly failed to articulate well) are exaggerated. I’m sure the love:hate ratio would be much lower with most written problems.

    Here are some quotes, tried to jot down in our quick debrief of the activity:
    “I really liked figuring out how to do it instead of being shown by someone else”
    “no one would ever do this. They’d just put the toast in at level 8″
    “wouldn’t level 8 burn the toast pretty bad?”
    “I wonder what level 100 would do to a piece of bread”
    “I didn’t know what to do at first; it was frustrating… it got better once I started trying things”
    “this would have been better if it was ANYTHING but toasters”
    “it was really exciting to watch the countdown”
    “I really wanted to be closest when we were watching the answer”

    So, with my small sampling, I’d say it went over just fine, not a home run for these kids, but certainly not a swing and a miss. Better than most of our warm-up problems.

  21. on 29 Oct 2010 at 4:26 pmSam Critchlow

    @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.

  22. on 29 Oct 2010 at 5:16 pmDaniel Schaben

    My perception of relevancy is usually at odds with the kids, but I did try my hand at this video problem stuff and made this problem. Is it pseudocontext? For most of my students around here it is not . . . however in a more urban setting maybe it is. My students got into it but they are the seniors. I haven’t tried it with the younger ones yet.

    I am sorry it does not have the answer as it is fall now and pivots are not running anymore. I had to start this up special for the video. Also did not include an in-bedded timer. I just had the students use their own timer. I won’t be able to get an actual video answer until next summer when it is irrigation season 2011.

    http://blog.esu11.org/dschaben/files/2010/10/Whene-will-my-pivot-come-back.m4v

  23. on 30 Oct 2010 at 4:43 amBrian Caine

    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.

  24. on 30 Oct 2010 at 4:53 amBrian Caine

    @Sam – I think I got overwhelmed with excitement and questions and read over your results post. Thanks for posting this.

    I think your exponential models might be onto something if the timing device in the toaster is embedded in the circuit. A non-ohmic device would not give a linear relationship… possible explanation?

  25. on 30 Oct 2010 at 5:38 amDaniel Schaben

    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.

  26. on 30 Oct 2010 at 12:11 pmDan L.

    On the topic of cold toasters:

    If the starting temperature of a toaster is a variable and is not controlled in the experiment, then one could collect data over many toasting sessions (randomize the experiment by toaster setting) to highlight the variance in the measurement.

    I hope I’m not too far off base with this suggestion, but, folks who teach engineering (as I do) love measurements of variance.

    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, …).

  27. on 30 Oct 2010 at 1:02 pmPaul Guilianelli

    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.

  28. on 30 Oct 2010 at 1:38 pmAndrew Nicholson

    Like the concept. I would repeat it with 1,2,4,8 settings. Stop the video after the first two, ask for guesses at the last two. Run the video forward past the third and then see how close the class could get on calculating the 8th setting. Lots of ways to think about the problem.

    Are the settings “time”, “darkness”, or abitrary points?

  29. on 31 Oct 2010 at 5:38 amKarim

    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.

  30. on 31 Oct 2010 at 7:41 amgasstationwithoutpumps

    The nickel-chromium wires commonly used as heating elements are indeed non-ohmic, in the same way that incandescent light bulbs are. Basically, the resistance of most metals increases when the metal is heated. An ohmic device would have a constant resistance.

    NiChrome wire has a resistivity of about 1.1E-6 ohm-meters at room temperature. (To get the DC resistance of a wire in ohms, multiply by the length of the wire and divide by the cross-sectional area.) The thermal coefficient of resistivity for nichrome is about 0.4E-3 C^-1, that is, the resistivity is about 1.1E-6 (1 + 0.4E-3 (T-25)) at temperature T in degrees C. (Note: different formulations of NiChrome can result in slightly different numbers.) Nichrome is popular for heating elements because it has a high resistivity, a low thermal coefficient of resistivity, and a high melting temperature. (http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/rstiv.html#c1)

    The toaster or light bulb connects the resistance to a voltage source, and the power dissipated is V^2/R. The temperature rise is roughly proportional to the total energy (ignoring losses due to convection, radiation, and conduction, which are going to be quite high for toasters and light bulbs), so is the integral over time of V^2/R(t).

    Initially, the resistance is low and the power dissipation high, so temperature rise is rapid. As the temperature rises, the resistance goes up and the power drops. Eventually, the losses (which we weren’t modeling) match the power in, and the temperature stabilizes (unless the metal melts first, as in a fuse).

    You can get semiconductor devices with a *negative* thermal coefficients. These are used as “inrush current limiters” in series with a device that would normally have a big current spike when turned on. These start at a fairly high resistance (0.2 to 200 ohms) and quickly warm up and drop to a low resistance (a hundred times lower than initial).
    http://www.ametherm.com/inrush-current/inrush-current-limiters-pcim.html

  31. on 31 Oct 2010 at 7:43 amDan Meyer

    I really appreciate the vigilance on the pseudocontext issue, but if any of the following conditions apply …

    1. the students wonder a question without coercion.
    2. the students want to know the answer to that question without coercion.
    3. the teacher doesn’t coerce the students to use unnatural math to resolve the question.

    … I can’t rightly call it pseudocontext. In Dave’s case, all three of those conditions apply

    (Which, it has to be said, we only know because several of you guys were generous enough to field-test the material in your classes the very day Dave posted it and report back. This math-ed-blog community really outdid itself with this one.)

    Karim is highlighting a different (but no less important) dichotomy from pseudocontext / not-pseudocontext. I think it’s better described by Frank Noschese‘s “convergent media” and “divergent media” and the follow-up.

    Along those lines, Dave’s toaster regression is ten times better than the identical regression problem in my algebra textbook but it doesn’t diverge much — unless it leads into this Ohmic stuff, which I don’t understand at all. But so what? If I have to assign that regression problem, I’m assigning Dave’s, because it’s ten times better and way less coercive. And it doesn’t teach kids that the math always works to the exact answer in the back of the teacher’s edition.

  32. on 01 Nov 2010 at 4:50 pmMarcia Weinhold

    A useful term might be one we adapted from “Realistic Mathematics Education,” a way of thinking about contexts first developed in the Netherlands. They make great use of what they call “realistic contexts,” and the criteria are (1) that students are easily able to think in the context, (2) the context supports the mathematics that students can learn in the context (that is, it doesn’t break down as a context if students choose weird numbers to test their hypotheses) and (3) the context can be expanded to support further learning of related mathematics.
    So the context of the toaster is clearly intriguing because of its presentation. It may not be “real-world” but it might be “realistic.”

  33. on 02 Nov 2010 at 2:43 amKarim

    @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).

  34. on 02 Nov 2010 at 2:19 pmSam Critchlow

    @ Karim -

    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.

  35. on 12 Nov 2010 at 9:40 amJason

    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).

  36. on 12 Nov 2010 at 10:00 amDan Meyer

    You guys kill me sometimes you know that?

  37. on 12 Nov 2010 at 4:36 pmJulia

    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?

  38. on 12 Nov 2010 at 5:40 pmChris Sears

    I taught a lesson on exponential modeling with some pictures of a teapot of boiling water cooling. It is in a similar mold. You can get a PowerPoint of the pictures on my blog post.

  39. on 13 Nov 2010 at 11:46 amBrian Caine

    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?

  40. on 14 Nov 2010 at 8:59 pmDan Meyer
    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.

  41. [...] owe Brian Caine a debt of gratitude for flipping my switch on the question of "what is multimedia doing for us, [...]

  42. [...] Okay, so if you let the toaster cool down in between rounds, it is (more or less) linear. (Contra Dave's experiment.) [...]