What if the same picture was made with the graph tacked on but without the balls in the air? (I’ve seen this before as a way to “cheat” through the multimedia in a problem before.)

I’d sneak in an extra couple words:

Using multimedia that records an entire action makes it hard to create pseudocontext.

I’m with you, Dan. Not that images and multimedia make it hard to lie, necessarily, but that the lies of pseudocontext become that much more obvious when made visual. Their foundations are revealed as evidently false, which may or may not happen for students simply searching a word problem (as long as we’re talking about math) for relevant information.

There did seem to be a push for a while in textbook writing to try to “trick” the student by including more information than is necessary for the problem, which may be what has bled into the modern textbooks’ penchant for pseudocontext.

Is there somewhere a discussion on whether pseudo context is better than no context? I was trying to discuss the danger of pseudo context with some colleagues today and they were trying to convince me that pseudo context was better than no context. Being new to all these ideas I was at a loss to counter their arguments in a meaningful way. Can anyone point me in the right direction here?

Ta muchly,
Chris.

You can add “naturally arising questions” to your list of Annoyingly Difficult Concepts to Pin Down (reason/ curiosity), I think. In fact, don’t– because I’m 99% sure they’re the same thing. And therefore (once again) I find it suspect when folks speak so confidently in this conversation about what pseudocontext is and isn’t.

Consider this as an example. As a “wordie” with little to no intrinsic interest in math, no questions whatsoever naturally arise for me from the toaster regression. Have never done. (I do have a toaster, unlike a definition of pseudocontext.) Nor do I burn to investigate the bouncing ball, the jar of pennies, or the escalator, even if they’re prettily glowing on a video screen. I’m just not that into you, you know?

So if your definition of pseudocontext rests on “naturally arising questions,” every WCYTWT is pseudocontextual to me– because I don’t ask mathematical questions naturally.

And I’m pretty sure that this cretinous math reality of mine is the one to which you need to pay the most attention as a teacher. Your commenters have vastly self-selected as people with long and abiding interests in math. Your students will have done no such thing.

Given this, your argument may be that WCYDWT– the potent combination of multiple sensory input, the application of theory to objects kids see daily, teacher guidance, and proper questioning– is a superior tool for *eliciting* curiosity/reason/naturally arising questions. This, to my mind, is a far better defense for WCYDWT than to argue that multimedia facilitates curiosity, or scientifically verifiable problems, or learning a priori, given the voluble evidence that it doesn’t.

But even with this better defense, there are concerns you will need to answer.

First, the “WCYDWT as a superior tool” defense presupposes the creation of the catalytic presence of student interest, which you continue to dismiss, oddly.

Second, what then distinguishes WCYDWT as a tool from the plain old skills of good teacher? A skilled, sensitive teacher will *create* context– help me be interested in the possibility of a nine pound orange, perhaps even a heck of a lot more than a boring old toaster– since a Martian orange is quite interesting. (Admit it.)

Thirdly, if your corollary argument is that WCYDWT is “real world,” then why on earth aren’t you physically bringing this stuff into class– or the kids to it– instead of putting it one remove from the kids in a video? (Your historical schematic defense here– because it would be too complicated and messy for the typical math teacher– can be equally applied to learning Mac Pro and securing a digital projector in the inner city, so I don’t know how much water that holds.)

Finally, what about your argument that “math is its own context?” Why is the worth and value of math pinned only to what I can find in my bathroom? And haven’t you eliminated entire realms of curriculum (calculus?) otherwise?

That’s enough for now, I think.

So if your definition of pseudocontext rests on “naturally arising questions,” every WCYTWT is pseudocontextual to me— because I don’t ask mathematical questions naturally.

I think there’s a big difference between “naturally arising questions” and “questions which I might ask.” The first is about the question while the second is about the person asking it. Perhaps one the difficulties in defining pseudocontext is the fact that we are weaving the individual into the definition. I can always find someone who will not ask a mathematical question, but in my opinion, that doesn’t make the question pseudocontextual.

I think that a mathematical question which arises naturally, whether interesting or not, is not pseudocontext (if pseudocontext asks students to use operations that have nothing to do with the given context). For example, someone could ask the question “what is the arc a basketball makes as I shoot a hoop?” Though some (most?) people would not be interested in tackling the mathematics inherent in this question, the mathematics does exist. I can ask the question “what is the maximum height of a shoot I make?” This question asks for information people probably don’t care about, but there is mathematics involved in answering the question, and the mathematics used complements the way which the real world works.

On the other hand, the question “what is the highest point of a basketball if the ball moves in the motion of an absolute value function?” Would never be asked because basketballs don’t work like that. The question “what is the weight of this orange if the orange weighs nine tenths of its own weight plus nine tenths of a pound?” would never be asked because no one calculates the weight of an orange as a function of the weight of the orange. Pseudocontext asks people answer a question which would never be asked in the first place (and I mean asked by anyone, not just the “wordies”).

The question of “would someone reasonably ask this question?” is different from “would someone be reasonably interested in this question?” which is very different from “would my students ask this question?”

The first question is about the questions people ask as a whole. It does not rely on an individual asking a specific question, but rather on the collective curiosity of people in general. Again, you can always find someone who wont ask a question, but it’s a matter of whether or not the question would naturally be asked. If someone (anyone) naturally formulates the question (read: wants to know the answer as opposed to merely trying to ask a math question) then the context is authentic.

The second question really says nothing about context. As we’ve seen, someone will always be interested in answering a question. However, I see pseudocontext as whether or not someone would ask the question in the first place rather than whether or not someone wants to answer it.

The third question seems like a WCYDWT question. It is related to the first, but puts the focus on the specific audience rather than the question itself. If students naturally ask the question then they have satisfied the criterion for a problem to be not pseudocontextual. However, if the students don’t ask the question it doesn’t necessarily imply the problem is pseudocontext, merely that the information did not trigger the asking of a question.

This how I see the relationship between WCYDWT and pseudocontext. A WCYDWT problem gets students to ask a question which requires mathematics to solve. By virtue of the question arising naturally (and again, I’m defining ‘naturally’ as ‘without the sole intent of creating a math problem’) then it is not pseudocontext.

Dina:

Finally, what about your argument that “math is its own context?” Why is the worth and value of math pinned only to what I can find in my bathroom? And haven’t you eliminated entire realms of curriculum (calculus?) otherwise?

Calculus is usually where you leave pseudocotext behind in the traditional curriculum. Algebra is chock full of it.

@Z. Shiner Well said.

> “Try to commit that pseudocontext to a photo, though. It isn’t impossible, but it’s much, much harder.”

I think it would be even harder in video.

In a photo, you can cheat and “paste” the ball at all the right points. In a video, it’s much harder.

For one, there’s a large number of frames (100 per 4’ish seconds modulo NTSC/PAL) so it’s a lot more work to doctor a video.

Secondly, if the video is just a little bit shake, it becomes much harder to paste the ball each shot, because where it has to be pasted and at what angle depends on how the immediate context around the ball has changed on a frame-to-frame basis. Fudging this, I think, is something human visual processing is good at picking up.

(But I haven’t read the scientific litterature on visual perception, so don’t take me without some sodium chloride.)

[OT @Dan: where can I find a list of the markup I can use in your comment system? Could you put a link to that next to the Post-A-Comment field, that’d be swell :)]

@Dina and @Dan

I like the questioning of WCYDWT.

In my job I work with a few math teachers in middle school. Always, there seems to be pressure on the teachers to finish curriculum. The temptation is there to show and practice procedures often enough, with enough word problems mixed in, that the students will recognize when to use the correct procedure and apply it correctly. All this in an effort to be measured as competent in math.

Some students love this, they love putting numbers and letters into a machine and pressing buttons and dials until finally the right answer pops out. These students are considered smart and good at math.

When faced with an object in the real world these students do not connect it with math because math is numbers and letters and rules.

There are also teachers who struggle to find the math they teach in the real world. (I’ll raise my hand here).

I don’t see WCYDWT as a way to bring math to the real world. Just the opposite I see it as a way to explain why we invented math in the first place. We don’t have parabolas because they make cool graphs and have some consistent properties we have them because this is the way the world functions. “What goes up must come down”

I have seen through my own experience that when students start looking for patterns and asking questions about life that often those students who are “bad” at math lead the way. They lead because they aren’t looking for the right equation, but are just trying to figure out what they heck is happening.

Consider this, a math student looking at the basketball might start by trying to figure out the linear equation for the ball going up and the linear equation for the ball going down and push them together. The math curriculum s/he knows is y=mx+b. s/he might spend a long time and finally give up because s/he knows eventually the teacher will give up and tell the answer. Later s/he can memorize the new information.

The non-math student will see that the ball is moving in and arc and look for the high point and draw a nice curve up and mirror it down. With no calculations at all s/he will say the ball fails to go in because the left and right side of the curve have to be equal and his shot is just a bit off.

Then we can play with graphing calculators to regress to the correct equation. If we want we can try this with a bunch of canned examples and figure out what happens when we change coefficients.

The hope is that even wordies can see the symmetry and patterns so even if they don’t remember the exact math they will at least remember the keywords “parabola and quadratic equations” so their google search is a bit faster.

Just for fun historical conversation to Brendan’s post and other similar posts…

Algebra went through three main stages of development over several thousand years:

First – Rhetorical Algebra – found primarily in Arabic and Asian cultures (the solving of problems with written/spoken language)
Second – Syncopated Algebra – (the solving of problems with written/spoken language with symbolism and symbols taking an important place)
Third – Symbolic Algebra – created in the early 20th century (fully symbolic, and frequently) abstract

The way I look at what Dan is doing, would be to see similarities in algebraic development over thousands of years. Both of their chronology look rather similar to me.

It’s also worth mentioning that this historical perspective is something I teach directly to my 7th graders and we commonly use rhetorical, syncopated and symbolic to describe the stages we find ourselves in when solving problems.

Jason: “Using multimedia that records an entire action makes it hard to create pseudocontext.”

-I’d prefer to look into what multimedia will show clearly back to my students regarding their own problem solving regarding their own rhetorical, syncopated and symbolic steps they took. Not what it will prevent from creating (pseudocontext).

But it doesn’t go without saying, Dan, and especially to teachers, who are bombarded with the false promises of technology every single day. To throw around catchy vagueness like your multimedia statement puts the good work of WCYDWT at risk. Since pseudocontext, in your words, consists of problems that are “boring, irritating, and alienating,” isn’t your inoculation statement easily interpreted as another way of stating that multimedia is interesting, a priori? Seems that way to me. You’re going to require a few more (a lot more?) specific parameters on how to use multimedia to make your case, a la Jason Dyer.

As for student interest, you seem to continue to contradict yourself on the importance of such a thing in your protocol. Either it’s nice, conversation-clouding side-bonus, as in “I don’t place the premium on student interest”– or it’s necessary, as in “WCYDWT uniquely encourages students to invest in math problems, and therefore learn math.” We need to know which it is.

Why is this important? Because the question “What is pseudocontext for math?” in turn begets this much harder question: “What is a *real* context for math?” And that’s what I’m struggling with. If a real context is not defined by student investment, by what is it defined? If a real context is not delineated as something a student cares about, then what is it?

Which is where the orange problem comes in, and others like it. I understand that the nine-pound element is the answer. Have you never hooked student interest in solving the problem by foreshadowing the answer? You can still use it, in otherwords, to create real context (which I’m assuming here is, again, created via student interest). Isn’t my interest in the question, as a student, the thing that makes it real *for me*?

So much to think about in this (and the previous pseudo context threads).

My husband and I have our own still-fuzzy way of evaluating whether something one of our authors does is pseudo-context: if the scenario/example creates cognitive overhead because the learner is spending brain cycles thinking (even if barely at a conscious level)’ “yeah, but… What idiot would do it that way when you could just…” then there is a pseudo-context problem. We have not found arguments over what is and is not “real world” or even “interesting” to be as easy to pin down, so rather than say what IS a non-pseudo/good context, we just try to weed out the ones that make people furrow their brow over the scenario itself. We want furrowed-brow (scarce resource neurons firing) energy to be going to thinking about the problem, and any mental energy spent thinking it’s a totally bulls*** scenario is a learning hit.

Actually in our case, it may also be a financial hit… though this is wildly speculative, I bet if I mapped pseudo context use to book sales just within our series, I am pretty certain we’d see a relationship. The problem for us, though, is also that the authors who blow the context in this way also tend to have other, equally-damaging problems in their teaching approach.

We blame these problems (pseudo-context, especially) on the author’s lack of understanding of and/or respect for their
audience/learners.

Another way we think about pseudo-context is to use a filmmaking model… We do not care so much whether the scenario is real-world vs. fantasy; we care only whether the audience can suspend disbelief (if needed) and thus be fully engaged in what we REALLY want them to care about and focus on.

Films and novels can be as non-real-world as one could imagine, but what matters most for engagement is whether the world presented is internally consistent. Plausible can sometimes be more important than “relevant”, because once they are on the journey with you, relevance can emerge naturally as the viewer maps the not-quite-OUR-real-world experience into their own world. This sometimes happens only *after* and as a result of their participation, so we do not equate real-world with relevant for our books, as long as the learner can imagine how this thing-they-learn might actually morph or scale into something they COULD imagine using one day.

So perhaps part of the problem with pseudo-context is the violation of internal consistency… It asks you to imagine a world that does not make sense, period. A world where the fictional participants who have this “problem” are not constrained or curious, just… Lame.

Toaster example would not be pseudo-context for me… because it does not seem to inspire the “but only an idiot would actually do it that way” cognitive overhead. In fact, I love the toaster thing and I am not even sure why. All I know is it got me thinking about many different questions. And I am not Math Girl.

Oh, on the video/multimedia topic… Still thinking about that. I am thinking of many of the most horrific pseudo-context examples in our books (including some of my own cringe-worthy just-shoot-me code scenarios) and imagining whether multimedia could or would have stopped it. I *feel* like it would have, but I am not sure until I really have a chance to study them. *shudder*

Forgive the sentence fragment up there. I hit “submit” too fast.

Dina, I teach science and math, and my wife is one of the two science lab assistants at our school who sets up all the experiments for all the classes. The lab assistants work full time setting up our experiments and managing the resources of the laboratory. We regularly have to go and purchase things outside school hours so that experiments can be done.

Let us for a moment assume that the math teachers of our school were to as a group adopt the WCYDWT methods and do all those for which it is possible practically. We have 10 to 20 math lessons happening every period of the day (its a big school with 5 year levels and 4 70 minute lessons a day). To support every teacher doing WCYDWT practically would require the equivalent of the science prep-room for the math department as well. That is 1 or 2 assistants employed full time to manage and maintain the equipment and have the resources ready for class use.

That is simply less practical than our math department investing in 1 digital video camera, and 4 data projectors and laptops to go with them. It is also far more expensive: $30K for a lab assistant vs $10k for the mulitmedia needs.

Videos also have the advantage that if we find a one online, we can simply download and use it, no need to acquire new physical resources as we have all that we need. In other words a new video we find online is immediately usable in a class!

Editing videos is relatively simple in Windows Movie Maker which is free. Worst case you might need to get another free program to convert file formats so they are compatible.

Considering that a data projector and laptop combo can be moved from room to room, and used for far more than running WCYDWT lessons, the overall value (ie cost vs usefulness) of acquiring them is much greater than building a huge pile of physical resources.

In fact this is exactly what I have found! I purchased a data projector and use it in every lesson basically because I can do so much more with it being reliably available to me. If I want to do the “water tank fill” WCYDWT, I download and run the video. I can make the tank 2m tall on the wall and every one of my 28 students can see it clearly. We can get a ruler up there and make measurements that everyone can see as well. I have a data projector and a laptop. I don’t have a water tank and a hose. Makes multimedia a pretty clear winner in my book.

I purchased one because I wanted to always have it available. The school actually has 3 or 4 data projectors for the math department (hey I never need to borrow them so I don’t know).

I think you are misinterpreting “usefulness” in order to support your position. What is useful to you as a teacher? Isn’t it things that are “effective” in improving “learning retention” and “student engagement” etc? The quality of useful is determined by the circumstance of use, not by an absolute criteria.

The whole concept of “inherent educational value” applied to resources is a false one. Resources are only as good as their ability to be used to create good educational outcomes. A toaster is no better as an educational resource than a video of a toaster if the teacher attempting to use it as a resource is unable to draw educational outcomes from the lesson for the students.

Multimedia has the advantage of consistency of experience, lower long term costs (in time in particular), wider support from a funding perspective by governments, and in particular in safety for the members of the class.

Consider this: if I showed you a video of a flower growing in a language class what would I want to do with it? If I showed the same video in science would I want you to do the same thing? What would the differences be?

When I show a video of a ball throw in math I want students to think about the curve, and how that might be described mathematically (we are in math class after all). If we go out and shoot hoops, that is what their focus will be on, the physical elements. By controlling their experience I can appropriately direct their attention to the area I want them to focus on, and draw the science, math, language or whatever my lesson is about from that point.

I teach science, and sometimes experiments are good for teaching concepts, but sometimes distancing the student from the task is the best way of getting their attention on what you want them to learn. The difference between learning how to shoot a hoop, and the math that describes the motion of the ball.

There is actually a computer program that tracks the trajectory of a basketball throw. I haven’t read a lot about it, but it seems like it trains users to shoot at an optimum arc. Thee iPhone app graphs the trajectory. WCYDWT?

http://www.noahbasketball.com/optimal_arc.php