Santa Cruz Sentinel, today:
The City Council will consider a proposal today to establish a citywide pay-by-cell phone system that would allow motorists to start, finish and extend time for meters or fee-based parking spots. [..] Consumers would pay a fee of 35 cents per transaction, or 25 cents for frequent users if they are willing also to pay a monthly access fee of $1.75.
Is pseudocontext a failure of imagination or is it a symptom of laziness? Because this sort of thing just isn’t hard to find.
Z. ShinerOctober 26, 2010 - 5:35 pm -
Thanks to you I ended up spending a good chunk of this evening thinking about pseudocontext… I agree that it is definitely easy to see mathematics in everyday life. However, I am starting to wonder if the problem isn’t in finding math which naturally stems from reality, but finding the reality which naturally brings about specific mathematics. I wonder if pseudocontext is a result of textbook manufacturers being tasked to find realistic problems which fit to certain mathematical concepts rather than finding mathematics which naturally arises from the real world. When we work to design mathematics in a vacuum, without careful planning, our applications to reality are bound to be lacking.
It’s easy to pick up a newspaper and find a math problem. It’s much harder to be given a math problem and find an appropriate newspaper. I have no doubt that mathematically inclined people are great at the former, but it seems to me that the cause of pseudocontext is the latter.
John T.October 26, 2010 - 7:07 pm -
I agree with Z. It’s easier to cook a good meal for friends than it is to plan a menu for 300 people night after night and turn it all into a great business. The most difficult thing for me to do after I’ve thrown out the text book because the problems are bad is to plan something awesome that somehow incorporates the standard. I am more and more successful with more and more experience but I want to write a big text book as much as I want to open a restaurant.
Bill HOctober 26, 2010 - 7:43 pm -
This pseudocontext thread has definitely raised my awareness and made me think twice when I write problems for my eighth grade class. But this last example of pay schedules for parking in Santa Cruz made me think again. Is the goal necessarily to find the most authentic context? This particular example may be authentic, but I’m not convinced that that would make it any more compelling to middle schoolers than the pseudocontext problems that we’ve enjoyed ridiculing.
HalOctober 26, 2010 - 10:15 pm -
I’m with the other commentors on this one. When I first glanced at this post on my feed, I had to read it twice to find out if this was actual data or if it was supposed to be another example of silly pseudo-context. I mean, I agree that calling measuring the angle of a piano silly, but this question, taken out of context is just as removed from the students.
But there are plenty of ways to pose the same question in a context that the students WOULD be able to connect with, I think even better than if you printed an article about a far away place from the internet:
It costs $200 for a season pass lift ticket, and with this pass, you get discounted ski rentals: $10 per day. A day-pass plus rental costs $45
This type of question writes itself. Others don’t so much.
David LOctober 27, 2010 - 4:34 am -
I’m not ready to laugh at this one.
While it might not be age-appropriate for MS or even HS students, change the problem to downloads of music(pay per song) versus a monthly plan(unlimited downloads), or a similar problem like ‘pay per text message versus a text message plan’ and you’ve got something.
Comparing different plans and rates, determining the ‘best buy’ and being able to defend your choice – that’s valuable, appropriate to many different levels, and will be found in almost all MS standards across the country.
Daniel SchabenOctober 27, 2010 - 5:11 am -
Good post and I do a similar problem with my students right before I work on simultaneous equations. Some are motivated to do it and some are not. Some see the connection between the traditional problem and the one above and some do not. I will throw in my two cents here. I want a real life example for every topic I teach the students in my class I want a problem that totally gets them motivated, but even my best efforts at this still come up short as it still doesn’t reach all the students. I have a ton of linear examples. They are everywhere. I have a ton of quadratic examples, and I have a ton of exponential examples in the form of biology and finance. For trigonometry we can talk modeling average temperature of cities based on time in months. But even with my best efforts of placing context with at least one problem for each area of math that I teach, in the end we are preparing them for the ACT, SAT State tests etc . . . and the problems on these tests are all psuedocontext so are we going to change the tests that we have to prepare them for? I do not think that is going to happen. So in the end we have to some how make all students motivated to solve psuedocontext. There is no way around it and the only way to have them motivated to solve them is to do them and some how use all the tools at your disposal to get them to see math as its own subject and be motivated to solve x^2+2x+1=0 just because they can. Get them motivated to say “well that problem was easy, I hope there are more of those”. And I think the key to that is to let them play with the problem first let them use guess and check . . . here come my students.
Kris KramerOctober 27, 2010 - 6:29 am -
I’m with Z on this one. Yes, math is all around us; that’s not the challenge. The challenge is saying I have to teach polynomials to the nth degree and maybe even divide them. Or find an example of taking the square root of a negative number in “every day” life. It’s not always so easy to find the math that the state standards and curriculum say have to be taught.
Dan MeyerOctober 27, 2010 - 6:33 am -
Good stuff here. Two things.
Right. That’s much harder. But also, if we pursue that task with any kind of religious conviction – “I’ve gotta find some real-world context to motivate quadratics!” – we allow ourselves all kinds of fictions about math and the real world. Like, “This is fertile ground for a discussion about the quadratic formula!” when it isn’t that.
Our first rule is “do no harm.” So if we can’t find any kind real-world context, we teach the math straight and feel no guilt. But we should also pursue exercises –Â in the case of this post, “writing down math I see around me as soon as I see it” – that make it easier for us to witness the math around us when it occurs.
So perhaps two issues then: pseudocontext and age-appropriateness. My first thought was that this is legitimate context for systems of equations. I stand by that. It’s a separate deal, though, to say, “you should use this in a class of people who have never paid for a parking meter in their lives.”
MichaelOctober 27, 2010 - 9:51 am -
So, pseudocontext applied in the real world is a ruse to rip off consumers and pseudocontext applied in a textbook is a ruse to convince students they can apply the math they learn in the real world.
So, students never get to see when they should apply their math to prevent from getting ripped off.
sylvia martinezOctober 27, 2010 - 10:01 am -
The question brought up by this news item seems to be, “is it worth it to pay the monthly access fee?” And you don’t need systems of equations for that.
So if you use a question like this to make the kids use equations when there is a simpler way, isn’t that imposing psuedo-solutions?
NumbatOctober 27, 2010 - 11:26 am -
@Sylvia I agree and some might argue that an inductive approach to the solution a) is more efficient and b) displays a far better understanding of this thing we call math than blindly applying sim equations.
Now the real challenge becomes finding a problem which is both real and not easy to solve without sim equations.
Dan MeyerOctober 27, 2010 - 11:48 am -
@Sylvia, I can’t get too worked up over students using a table to solve this. Say they do. It’s going to be a trivial matter (now that this context has been established) to adjust the coefficients to make guessing-and-checking supremely annoying, which will leave a door wide open for systems.
ScottOctober 27, 2010 - 3:32 pm -
LanaOctober 27, 2010 - 6:17 pm -
I had my students research a real event and make their own problem based on what they find. For “work problems”, for example, a group of kids made up a pretty funny problem that involved cupcake eating contest which actually happened at a nearby school. They found out the names of participants, teams, number of cupcakes eaten, how fast – everything, and incorporated all of that into a word problem.
This has worked for me with varying success – sometimes, they get really excited about what they find. Other times, they just don’t care, and I’ve learned to accept it and move on.
Chirs SearsOctober 27, 2010 - 7:05 pm -
@Sylvia I’m with you on this one. Why don’t we just solve the inequality .35x > 0.25x + 1.75?
@Numbat I went to a talk two years ago where an instructor had built a table top demonstration building with three rooms. He used light bulbs as a heat source. Using thermal conduction to set up a system of equations, he calculated equilibrium temperatures for each room. He would then measure the temperatures directly. I can’t find the conference program on-line (KYMATYC must have taken it down), so I can’t give the instructor credit. His target audience is two year, non-transfer students (AAS). They have no tolerance for pseudocontext.
sylvia martinezOctober 29, 2010 - 10:38 am -
@numbat I don’t even see that a table or guessing or even an equation is necessary. The problem is easily solved by just stating that the difference in transaction cost (10 cents) is not worth the fee until you park 17 and half times (1.75/10), so the answer is that if you park 18 or more times a month, the fee is worth it. If not, not worth it. Seems like this problem is a good intro to the concept of “break even” which is a real thing that people deal with in real life all the time.
@dan As you say, if you give the leeway to solve the problem with multiple means, you are naturally going to see solutions that don’t involve systems of equations, which may defeat the purpose of using a simple (yet real) problem like this to do systems of equations.
So I’m curious, are you saying you would you pump up the reality and make the problem more complex to force the issue? At some point, some people will give up on tables and that would be a good time to talk equations – but at this simple level, I don’t think anyone would get to that tipping point.
In other words, would you agree that your anti-pseudocontext stance doesn’t mean that you have to find an actual verbatim problem, but instead, you are advocating that the context would appear to be real and within the student’s realm of experience. Would you fudge the actual circumstances a bit to build problems that are more complicated to create annoyance and induce students to beg you to help them do it in an easier way than guessing or tables? Would you start with this problem and then ramp it up? Start with a harder problem and use this as practice? Introduce equations only when they beg for an easier way?
I’m mostly interested in hearing how you think this would unfold with kids.
Dan MeyerOctober 29, 2010 - 11:30 am -
Yeah, this is basically it. I wouldn’t feel great about it, but I’d fabricate a news clipping from another city where the difference between surcharges wasn’t a multiple of ten and then another one where both plans featured a flat monthly fee. The pseudocontext is obviously picking up speed but it’s mitigated by a) the reality of the first example and b) the students’ confidence with the first two examples.
And I’d still keep an eye out for a better example of systems of equations.
DaveOctober 30, 2010 - 10:53 am -
I’m not sure about this. The implied algebra problem (finding the break-even point) is reasonably interesting and authentic. But an individual driver would have no need for this information. She would simply calculate the cost of each option, based on how often she expects to park downtown, and select the cheaper option.
JimPOctober 31, 2010 - 9:41 am -
What is the half-life of context decaying into psuedocontext? What are some of the intermediary steps?
Bill BradleyOctober 31, 2010 - 10:43 am -
@JimP Pretty short half-life I’d say. Most of the even vaguely relevant problems are starting to smell pretty awful by the time the book is published and delivered. Give it a couple of years in a classroom and they stink pretty badly.
Dan MeyerOctober 31, 2010 - 12:17 pm -
But I don’t know exactly how many times I expect to park downtown. I could throw out a few numbers that seem plausible, but more helpful for me is to know this break-even point. I’m not sure how many times I’ll park downtown but I know it’ll be less than 17, so I’ll pass on the monthly access fee.