## Insurance Against Pseudocontext

I have to disagree that this is pseudocontext under the argument that (as I posted on the original CarTalk thread) that a trucker actually came to me with this problem a couple of years ago. He wanted a dipstick method that would give him more accurate fuel readings than his fuel gauge (I guess they’re universally inaccurate?). If a member of a profession brings you a real-life problem from their “world,” I think that has to disqualify the problem from being labeled pseudocontext.

Does anybody have a problem with this? I don’t have a problem with this.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Joe Henderson

December 15, 2010 - 2:36 pm -

Dan, perhaps I need to go back and reread your definitions from earlier, which I have lost at this point in the chaos that is the end of the semester…

But to me, I think this question is definitely a start. The larger question, which I have raised over and over here is who’s “world” is the one providing the context? I get the professional’s world and love the idea, but how does that then relate to the world of the student? There’s a translational piece there that’s really interesting to me, and is something that I’m still missing in your definition.

How do students take ownership over the context?

Whenever you throw out the WCYDWT, I find myself wondering who exactly the “you” is. I so wish you’d reframe it as WCYSDWT. What can your students do with this? That would be so much more interesting.

Am I misinterpreting here?

2. #### Peter

December 15, 2010 - 2:40 pm -

Shouldn’t the conclusion be that a problem from “the real world ™” might be as bad as pseudocontext?

Which would simply indicate that not all real world problems are useful for teaching mathematics (just as not all books are good for a literature class).

3. #### John T. Spencer

December 15, 2010 - 3:10 pm -

Dan,

When I was a kid, I remember my dad using the formula for the area of a circle and then dividing it by the cost to determine whether or not an Extra Large was a better deal than a large. Before the days of the iPhone, he would carry around a tiny pad of paper just for situations like that.

This problem was a joke, because anyone who eats pizza knows that regardless of the size of the slice, you’ll still usually eat 2-3 slices.

Now, the problem was definitely real-world and my dad was a man with a finance degree, so that probably explains the insanity of it. I don’t think it fits the definition of pseudo-context, but it is still a ridiculous math problem.

4. #### Brian

December 15, 2010 - 4:54 pm -

If anybody does have a problem with this, then I think we’re missing the point of the term “pseudocontext” to begin with. Context is context, even if it seems weird to those of us who wouldn’t normally encounter a given scenario.

However, it’s likely the case that our students also have not encountered these scenarios, which is why giving the proper framing is key to them. How can we wash away any stains of potential pseudocontext when the situation is, in fact, completely contextual? Also, does it matter if we do that or not?

5. #### Donovan from VI

December 15, 2010 - 5:47 pm -

Most real world problems like these could be tweaked to make investigations with assessments for students. I’ll be trying a modified version of the oil tank problem with my geometry students and another version with my advanced algebra students.

The pizza problem isn’t completely ridiculous and could be salvaged. I’d like to see what I can do with it for my geometry class.

6. #### Benoit

December 16, 2010 - 5:19 am -

@John T. Spencer:

the problem of comparing the cost per square inch of the pizza is not a ridiculous problem. You may think that it doesn’t matter but that doesn’t mean that everyone think the same way. In particular, I know some very skinny people (and I was one of them) for whom the calorie count is important. Some times, I was trying the calorie yield for my dollar.

Donovan from VI says the pizza problem could be salvaged… it doesn’t need to be. It is perfectly good despite what John T. Spencer says.

7. #### Perdita

December 16, 2010 - 5:46 am -

Seems like a good axiom for whether something’s pseudocontext, but I’m still not convinced that pseudocontext is necessarily, or even usually, a bad thing. Lack of student engagement with a problem is a bad thing, but is this even correlated with the use of pseudocontext? And if so, is it only because the teacher is irritated by the pseudocontext (as Dan obviously is), or does the correlation still exist once you control for the teacher’s feelings about the problem? I’d like to see experimental results on this before devoting effort to eliminating pseudocontext from textbooks etc. Anecdotally, my son adores Descartes’ Cove, which I’d say proudly embraces pseudocontext (typical q: A horse is tied to an exterior corner of an octagon-shaped castle with a rope that is 15ft long. If the length of each side of the castle is 20ft, what is the grazing area, to the nearest square ft, of the horse?).

8. #### Michael Paul Goldenberg

December 16, 2010 - 6:10 am -

I don’t think ‘pseudocontext’ as Jo Boaler uses it in WHAT’S MATH GOT TO DO WITH IT? means that if a problem isn’t ‘real’ that it’s bad (and if she DOES mean that, I would disagree with that notion). What I think she’s talking about is the sort of problem that purports to be ‘real world’ and then proceeds to make students have to abandon what they actually know about the real world in order to be able to engage with it. One such example:

‘Joe can do a job in 6 hours and Charlie can do the same job in 5 hours. What part of the job can they finish by working together for 2 hours?’

The problem is that two people working together in the real world rarely work at the same individual rates as they would working alone. Kids are told to forget about that.

Another of her examples is, I think, even more on point:

‘A restaurant charges \$2.50 for 1/8 of a quiche. How much does a whole quiche cost?’

If kids are expected to forget that it’s highly likely that buying the whole quiche will cost less than \$20, despite the fact that 8 x \$2.50 = \$20.00, then we’re telling them that context really doesn’t matter. But if that same kid answers the infamous army bus problem (‘An army bus holds 36 soldiers. If 1,128 soldiers are being bussed to their training site, how many buses are needed?’) with ’31, remainder 12,’ then experts crow that American kids don’t really know how to do story problems. Catch-22 anyone?

All that said, there’s nothing wrong with thought-problems or gedanken problems, as Einstein called them. They’re extremely valuable in physics and many other contexts. Suspending our disbelief about situations in order to focus on one or two specific variables can be useful and sometimes essential to solving things. It can certainly help make a problem manageable.

Further, there are many fascinating and powerful mathematical problems that have zip to do with ‘the real world.’

I think the point would be not that one can’t give such problems, but that one need do so sparingly and honestly. Don’t pretend to be giving a ‘contextual’ problem that really isn’t. Don’t give problems where the ‘context’ doesn’t really have squadootch to do with the math and vice versa. Don’t lie to kids. Don’t insult their intelligence. Be straight when you are asking them to do a ‘thought-problem’ and make sure the context, even if artificial, really does inform the mathematics.

By the way, I think the pizza question above is perfectly meaningful. I don’t see why it would be ‘pseudocontext’ at all.

9. #### Sue VanHattum

December 16, 2010 - 6:54 am -

John, you called that problem ridiculous in a roomful of mathematicians. ;^) Like many of the commenters above, I like that problem. It makes you think about the difference between the way sizes are stated (by diameter) and what really matters, which is the area. I bet it was embarrassing, though, to have your dad pull out his little pad of paper when you were at a pizza stand!

I like what Michael said above. The danger of pseudo-context is that it will convince kids not to use their common sense. Perdita’s comment points to one answer to that problem – use a fantasy world when you have to make up silly problems. As soon as I see the word ‘castle’, I know I’m in a fantasy land, where things are likely to work in strange ways, and I have to trust that things work the way they’re stated in the problem. I have to set aside some of my common sense because the rules may be different in this world. (Although I like the problem of finding the grazing area, even when it’s given a more pedestrian setting.)

Here’s a cool post on using fantasy when you feel a need to give students problems with silly storylines. (I may have mentioned it before…)

10. #### John T. Spencer

December 16, 2010 - 8:01 am -

I concede :)

I was a history geek in a family of math geeks. I am in the minority on this one and readily admit that I have used a similar problem in my class (cost per ounce / and “eat this not that”) I just remember being horrified as a child when my dad actually began doing the math right there in the pizza parlor.

11. #### Perdita

December 16, 2010 - 9:19 am -

@Sue: evidently you don’t live in Scotland, plenty of castles here :-) Seriously, though, I think your point is good, and I like that blog post – thanks for linking.

12. #### Dan Meyer

December 16, 2010 - 10:46 am -

Joe: Whenever you throw out the WCYDWT, I find myself wondering who exactly the “you” is. I so wish you’d reframe it as WCYSDWT. What can your students do with this? That would be so much more interesting.

Crazy! Two years ago, that was the original title of these features, but I found the acronym didn’t roll off the tongue quite as well as WCYDWT. Seriously, though, is the difference substantial or symbolic? How would the three WCYDWT lessons I posted last week change under WCYSDWT?

Joe: There’s a translational piece there that’s really interesting to me, and is something that I’m still missing in your definition.

Certainly, real context can be unengaging and unchallenging. It can be poorly represented and poorly examined. Those are huge problems (even larger than pseudocontext itself, by my estimation) but they aren’t the same problem as pseudocontext. They don’t distort reality or punish students for their foreknowledge of a given context. For these reasons, you aren’t going to find the prescription for translating real context within the definition of pseudocontext.

Peter: Shouldn’t the conclusion be that a problem from “the real world ™” might be as bad as pseudocontext?

I’m afraid there’s too much to unpack within “bad.” As unengaging as pseudocontext? Yes. As distorted as pseudocontext? No.

Brian: However, it’s likely the case that our students also have not encountered these scenarios, which is why giving the proper framing is key to them. How can we wash away any stains of potential pseudocontext when the situation is, in fact, completely contextual? Also, does it matter if we do that or not?

No joke, y’all are bringing the roof down here. Great stuff. The “proper framing” is where I’d like to take this blog next, after the dust settles from our deconstruction of pseudocontext. Unfortunately, that framing is going to be even harder to classify than pseudocontext.

Perdita: Lack of student engagement with a problem is a bad thing, but is this even correlated with the use of pseudocontext?

You suggest a good study. Just guessing here: advanced students who have already formed strong conceptions of math’s application to the world aren’t going to be bothered by pseudocontext. Those students might perceive pseudocontext as some delightful little riddle.

I’m defensive of the students who don’t have that strong conception, who are still trying to determine math’s application to the world, if it even has one. And then their teacher tries to pass pseudocontext as context and they have their answer.

Perdita: Anecdotally, my son adores Descartes’ Cove, which I’d say proudly embraces pseudocontext. “A horse is tied to an exterior corner of an octagon-shaped castle with a rope that is 15ft long. If the length of each side of the castle is 20ft, what is the grazing area, to the nearest square ft, of the horse?”

Someone’s gotta explain to me how this is pseudocontext under the working definition.

13. #### Perdita

December 16, 2010 - 2:06 pm -

Oh all right :-) Flatly untrue. Who’d build a castle (not a tower, mind) that was octagonal without taking advantage of the opportunity to have towers at the vertices? It’s also ridiculously small for a castle – area of the whole castle (unless I’ve made a mistake, it’s late here!) is 800/tan 22.5 degrees i.e. < 2000 sq ft. Now, if they'd based the q on the Castel del Monte in Apulia
http://en.wikipedia.org/wiki/Castel_del_Monte,_Apulia
it'd've been interesting in every way ;-)

But if you want to defend that one, I could pick any number of other examples from Descartes' Cove.

14. #### Peter

December 16, 2010 - 3:00 pm -

@Dan: I agree — that’s what I tried/failed to convey with my second sentence :(

Pseudocontext is simply not the only way a problem can be unsuitable for teaching. As you wrote above: real world context can be “unengaging and unchallenging”.

I will defend my comparison with a literature class at all costs, though ;)

15. #### Michael

December 16, 2010 - 3:20 pm -

Perhaps necessity is the mother of invention and therefore pseudocontext need not apply. That is how we came up with much of our math anyway. We needed to know something and we needed to discover (invent) how to find the answer to our need.

I think pseudocontex occurs when this “need” is artificially (sometimes painfully) forced upon the audience. If I don’t care to know something, there is I have no need, necessarily.

Therefore, can pseudocontext be relative to the observer? One observer can have a need, while another may not, thus creating pseudocontext for the second observer?

16. #### Dan Meyer

December 16, 2010 - 3:33 pm -

It’s true that nobody needs the solution to a pseudocontextual problem. But just because a fourth-grader doesn’t need the solution to an astrophysics problem doesn’t make the astrophysics problem pseudocontext. There we’re talking about the myriad ways we can sully real context.

17. #### Perdita

December 17, 2010 - 12:18 am -

Yes, that’s interesting speculation. Here’s my counter-speculation: students not already enamoured of maths will tend to be engaged by problems that have personal relevance to them (and to a lesser extent those that have personal relevance to people who have relevance to them); however, there won’t be any difference between how engaged they are by problems that have relevance to someone they don’t especially care about, and how engaged they are by problems that have relevance to nobody. E.g. unless the student has a truckdriver in the family, the petrol gauge problem will be no better than a pseudocontext presentation of the same material.

18. #### Perdita

December 17, 2010 - 12:31 am -

and of course, if the student’s truckdriving friend or relative has a reliable fuel gauge or a different pattern of refilling, you may be out of luck there too…

19. #### James McKee

December 17, 2010 - 6:59 am -

As I read through the posts, there were several that addressed the use of the word “world” in my original post. People have observed (correctly) that the context of any problem may not be engaging to all students.

I am not hopeful that we will be able to generate problems that are universally engaging to all students, and that, for any problem, there will be a subset of our students that don’t find the context engaging. I’m afraid that if we try to work that into a definition of pseudocontext that every problem will be pseudocontext.

But it’s OK if some students don’t find the problem engaging, because some will, and part of what I see as my job is to provide students with snapshots of how mathematics gets to play in everybody’s backyard, and that regardless of what my students “grow up to be,” there will be mathematics that is relevant to their professional “world.” If they’re lucky, it might be engaging (read “fun”) mathematics.

20. #### Dan Meyer

December 17, 2010 - 10:15 am -

Perdita: unless the student has a truckdriver in the family, the petrol gauge problem will be no better than a pseudocontext presentation of the same material.

Seems like a stretch. It can’t be the case that all of Alex’s students have long-haul truckers in the family. There are myriad methods for making unfamiliar context familiar. There are far fewer methods for transforming pseudocontext into real context.

21. #### Perdita

December 17, 2010 - 11:13 am -

It can be the case that Alex was enthusiastic about the problem and his enthusiasm was infectious, though – might not transfer to a different teacher or an everyday practice. The plural of anecdote is not data, as they say.

22. #### Dan Meyer

December 17, 2010 - 11:21 am -

But a single anecdote does disprove an absolute. Coming up, we need to talk about methods by which real context — even context with which students aren’t directly familiar — can be made engaging and challenging and immediately real. We’ll certainly file “teacher enthusiasm” among those methods.