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Archive for the 'dissent' Category

Christopher Danielson:

Your quest for the perfect image that will get 100% of viewers on board with the same mathematical question may be a bit quixotic …

Avery Pickford:

In [my ideal] world, I imagine spending a greater amount of time talking about the aesthetics of what makes for an interesting math problem and much less time cajoling students to ask the “right” question.

John Scammell

It’s unfortunate that we are so curriculum driven that we have to trick them into inventing the question we want them to come up with.

Here's the thing: nobody watches Jaws and feels cajoled into wondering the question, "Won't anybody stop that shark?!" No one watches Star Wars and feels tricked into wondering, "Will the rebels defeat the Galactic Empire?!" Those questions are irresistible, not on account of any deception on the part of the cast or crew, but because the cast and crew evoked the central conflict of their story skillfully.

This isn't to say those questions are irresistible to everybody. Some people lack the cultural prerequisites to care about Star Wars. Some people possess the prerequisites and simply don't care. Not everyone is interested in every movie, however skillfully it creates a narrative.

The point of the #anyqs challenge is to evoke a perplexing situation so skillfully that the majority of your students will wonder the same question (whatever that is) and the rest of the class won't find that question unnatural or uninteresting, even if it wasn't the first question that struck them.

H. Wells Wulsin:

I recognize that I am probably not going to persuade you (or most of your readers) on this point. But these kinds of strategies have never been tried before in a math software package, and if they do work, then the developers stand to make a lot of money, and it could help a lot of students. I can’t be sure how effective these strategies would be until they’re tried, but I have a lot of reasons (which I tried to explain in the article) to think that they have the potential to make a big difference. That’s why I’d like to see a publisher or software company invest a few million dollars to produce a really high-quality software product.

I respond in the comments.

Dissent Of The Day

Sean Geraghty:

The one nagging question I continue to have about WCYDWT is…what exactly does it accomplish?

Yes, it appeals tremendously to our intuition. Students are looking at the world and designing questions about curiosities. This certainly appears to be the kind of skill we want to teach. But research is at best divided about the kinds of gains similar projects have had in the past.

Dan has pointed out that his students out-performed his entire department, that they showed real and true gains according to every measure we currently use.

But still I wonder. How much of those gains were attributable to the actual WCYDWT lessons? How much of it was attributable to his skill and highly developed craftsmanship as a manager, questioner, evaluator? WCYDWT doesn't strike me as terribly efficient. And I know that's not the point, but where is the research showing that these kinds of problems are effective?

I think Dan also mentioned that he would do these kind of lessons bi-weekly (about 1 in 10 days). To which I say, fair enough. He was efficient and skilled enough in those other nine days to experiment with something like this.

I don't know, though. These problems appear to make students conceptually flexible, which is brilliant. To make them procedurally flexible- arguably more important and more difficult to teach- is probably what Dan was doing the other 90% of the time.

I know the focus of this blog is WCYDWT and debunking conventional textbook wisdom. But when there's no coke or sprite, escalators, three-pointers, or cheese, how do you do the other stuff?

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."

Mike:

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.)

Mike:

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.