Mike Manganello offers a useful critique of Car Talk, pseudocontext, and WCYDWT:
I can certainly accept working definitions that require clarification, but the Car Talk problem confuses the issue [of pseudocontext] (at least for me). I’ve only done a little tweaking to the Car Talk problem:
“The fuel gauge of an 18-wheeler is broken, so the driver decides to check the gas level of his cylindrical gas tank with a dipstick. When the level of the gas measures 20 inches high, the tank is completely full. What will the dipstick measurement be when the gas tank is one-quarter full?”
Based on the working definition of pseudocontext, this problem fails on both counts. It completely ignores reality: Why wouldn’t you just fix the gas gauge? Then the problem asks for an irrelevant measurement: Why would we need to know that the tank is one-quarter full?
I assumed the trucker wanted to know one quarter of a tank (rather than four fifths) so he’d know when he had to refuel. An arbitrary number maybe (why not one fifth of a tank?) but not irrelevant. As for ignoring reality, I know more about mid-century Russian architecture than I do about long-haul trucking, but it seemed plausible to me that the trucker couldn’t waste time fixing the gauge in the middle of a run. In both of these cases, I deferred to the authority of the radio hosts. If either of Mike’s complaints were valid, why wouldn’t the hosts have echoed them?
Mike has also misquoted the definition of pseudocontext in small but crucial ways.
Mike’s: “It completely ignores reality.”
Mine: “Context that is flatly untrue.”
Mike’s: “Problems that ask for an irrelevant measurement.”
Mine: “Operations that have nothing to do with the given context.”
Personally, I find the Car Talk problem kind of boring and not very mathematically rich.
Once again we find that a problem’s basis in either pseudocontext or context has nothing to do with how much anyone enjoys or profits from it. (Seems only to fair to mention, though, that Alex’s class had the opposite reaction.)
Another word of caution: Mathematics is part utility and part artistry. By limiting mathematical study to problems related to genuine physical phenomena can only serve to retard the growth of mathematics.
It’s worth clarifying my total agreement here. My blog covers math applications pretty much exclusively not because I think those are the only problems worth studying but because those problems are the easiest to create and teach poorly.
Gilbert BernsteinDecember 15, 2010 - 1:39 am -
Kind of just guessing here, but I imagine the skill and showmanship of the teacher has a huge amount to do with whether the Car Talk problem goes over well with a group of students.
Gilbert BernsteinDecember 15, 2010 - 1:43 am -
eh, sorry that was kind of terse and not useful. I mean to say: the objection that the problem is boring seems like a presentation issue, not an issue with the problem… Also, solving the problem certainly requires a good amount of geometric reasoning, so I don’t buy the “not mathematically rich” objection.
AndrewDecember 15, 2010 - 7:00 am -
“In both of these cases, I deferred to the authority of the radio hosts. If either of Mike’s complaints were valid, why wouldn’t the hosts have echoed them?”
I think this a misreading of CarTalk as both an automotive and educational institution. CarTalk Puzzlers are often classic math problems lightly wrapped in automotive context, or flat out logic puzzles.
It’s a good show, but the place to look for authentic questions is in listener calls and questions, rather than the puzzler.
JerzyDecember 15, 2010 - 7:13 am -
“An arbitrary number maybe (why not one fifth of a tank?) but not irrelevant.”
Woo, there’s an extension to the problem: how little gas can still be left in the tank before the trucker really needs to start looking for a place to refuel?
Have the kids research or guesstimate how much gas such trucks can hold, how many miles per gallon these trucks get, and what the typical distance is between gas stations.
Try it with “typical” numbers and compare to “worst-case scenario” numbers (low MPG going uphill, and out in the boonies where gas stations are far apart).
Then maybe it’ll be clear that it’s safe to wait for refueling until you’re down to (for example) 1/5th of the tank, but (say) 1/8th would put you at risk of running out of gas before you reach the next station.
James McKeeDecember 15, 2010 - 7:16 am -
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.
James McKeeDecember 15, 2010 - 7:20 am -
As an aside, why might fuel gauges in this context be inaccurate? It should be plain to students after messing with this problem for a while why a typical, float-type fuel gauge might be inaccurate, and how it is inaccurate (for what readings is the fuel gauge close, for what readings is it “off,” and is it reading high or low?) How could we design a fuel gauge that is accurate?
Carl MalartreDecember 15, 2010 - 7:35 am -
Mike said: ” Personally, I find the Car Talk problem kind of boring (…)”
I’m with him on that. Depending on who you are, some things may or may not interest you. Gilbert’s explanation also sounds true. Depending on how the story is told and by who, you could become interested in the problem.
If he was a truck driver living before gauge were invented, maybe the story would sound more useful.
At the same time, solving stuff is just fun enough.
JenWDecember 15, 2010 - 7:42 am -
I guess this essentially echoes what James McKee said @5, but this question wasn’t a “puzzler” to begin with. Some trucker really called in and asked the question. Tom and Ray didn’t know the answer right away, though they spent a few minutes working on it, and I believe asked listeners to help out. It’s been a good 10 years since the original call, so I don’t remember for sure. Unfortunately, it’s also before the days of “PodCasts” and the like, so they may not have easily accessible archives of the show.
Also, fixing fuel gauges is ridiculously expensive, for something that’s essentially a convenience rather than an actual operating necessity. Often, by the time a fuel gauge breaks, the cost of replacing it is more than what the car is currently worth. I guess I don’t know that the same is necessarily true for trucks, but I’ll go ahead and assume it is. :)
Alex EckertDecember 15, 2010 - 1:37 pm -
I don’t feel that my presentation of the problem did anything to make it any more interesting for the students. I think what interested my students was them doing the following:
1) discovering on their own that it’s not 5 inches
2) each taking a guess at what the actual length was
3) understanding that THEY could figure out if their guess was correct or not
Neil StephensonDecember 15, 2010 - 1:58 pm -
#3) reminds me that the discipline (Social Studies for me) needs to set the benchmarks for success – not the teacher.
OlgaDecember 21, 2010 - 6:44 am -
A couple of things: how come there is nothing about the lunar eclipse yet?
Second, @ JenW, a great extension (into other disciplines as well) may be, if the fuel gauge is more of convenience, what gauges are a must? Which gauges would you have if you only could have 5 on your dashboard?
And finally, is anyone doing anything about delicious bookmarks? Are we just hoping that the service would just change hands and we would keep on using it?