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Mathematical Analysis and Its Discontents, or: WTFCYDWT?!?

05.25.2011

It is impossible to escape the impression that people commonly use false standards of measurement — that they seek power, success and wealth for themselves and admire them in others, and that they underestimate what is of true value in life.

              -Sigmund Freud, Civilization and Its Discontents

I have opinions, and an internet connection!

Disclaimer: You don’t have to listen to what I say. In fact, I’m actually weirded out about how little push back there is on these blogs. Especially mine, since I spend more time floundering and being a shitty teacher than most. So, maybe not so much backslapping or grabassing or whatever.

Second disclaimer: I AM NOT DISCOURAGING THE USE OF WCYDWT.

Third disclaimer: I am sure that those cited in this post would totally agree with me, and have probably actually said the same thing at some point in the past.

Here are some fun facts:

  • 50% of all people looking for roommates on Craigslist will be heroin junkies.
  • 100% of all girls you date that you met at Daniel Johnston concerts will recently have been convicted of a crime.
  • 50% of visits to Columbus, OH will result in being punched in the face by a stranger (and for no reason!).
  • 33.3% of all visits to Six Flags will result in being taped as part of a Candid Camera show.

Of course, these are ridiculous. But true (!), if I limit the data to only my life experiences.

I have a real worry about unbridled WCYDWT (or #anyqs). I mean, I guess I have a real worry about bastardized math. Not just because it can be wrong, but dangerous. Look at the shit that Harold Camping just did: pulled together a bunch of random ass numbers with arbitrary significance and convinced a decent number that the rapture would occur.

The Washington Post even verified his calculations!

I’m not saying that any of you guys are Harold Camping, or that using this stuff in your class makes for bad math. Just the opposite is true. But we have students who are going to experience bad journalism, or politicians invoking some think tank that is politically backed and uses biased data to say whatever they want.  And so when people make these lessons that come out all neatly tied up with a ribbon around it and everything works out perfectly, kids begin to make bad assumptions about the way the world works, and the way math works. Like, that it ALWAYS WORKS.

We’ve all seen the magic that Dan Meyer pulls. We’ve also got the heir apparent, Dan Anderson, who keeps finding some really great ideas and is running full speed ahead with the “3 Act” stuff. I love how you are immediately drawn in. I love the story aspect. I love the engagement aspect. I love how well everything is done.

I’m not sure I always love the conclusions that are drawn.

THE SONG OF ROBERT WADLOW

I totally goofed by not framing this lesson in the 3 Act form, but you totes could. Start with the picture of Wadlow. They will freak out and be totally into it.

I was doing a unit on regressions, and so I had everyone collect data. They had to get the shoe size and height of 10 other people in the classroom. They plotted the points and found the line of best fit for height vs shoe size. Because everyone took different data, everyone had different resulting equations. I had them figure out how tall I should be given my shoe size of 12 (which, FUN FACT: is the same size I wore in 7th grade). I seem to have slightly bigger feet than I should, because most people went over my height by a couple inches. But everyone was pretty close, and thank god no one had me at a gangly height of like 6’6″.

Anyways, then I put up this picture. I hadn’t had this great of a reaction in a while. Of course, the first question they all asked was, HOLY SHIT, HOW TALL IS THAT GUY?

I mean, HOLY SHIT. So, I put up this picture too.

Could we predict his height if I gave them the shoe size? He wore a size 37.

They had answers all over the place. 7′ to 12′ was the range, if I remember right. They were really confused. Wait, what did we do wrong?

Nothing.

How can we all be this far off?

How can we make our answers better?

Uh.. is it because we all started with different data points? What if we combined all of our data? What if we took more data? I just kept shrugging my shoulders.

What followed was a really good discussion on the dangers of extrapolation, really led by them. They talked about predictions that they had heard about fossil fuels or the price of milk or world population and how it would be really hard to predict what these things would be like way down the road, like when they were 60 years old.

I followed the next day with some canned data (sorry, everybody) that modeled growth rates of sharks over time (I wanted something that they had a little less knowledge of). Plugged it all in, they did a linear regression, and BAM! We could predict the length of a shark given how old it was! The problem was, when we looked at old sharks, they should be something like 70 feet long. And the data just never really seemed to fit right anywhere. So we had to add another tool. Logarithmic regressions to the rescue!

I kept giving them data and kept just letting them assume stuff. We ended up with coffee that cooled to absolute zero, or the population of Brooklyn being that of India in the future. We kept having to refine our tools and refine our methods. (Editor’s note: You know, like logarithmic or exponential regressions. Because not everything is a line, dammit).

We are all afraid of being messy or things not working out, but those are the most valuable lessons. I’ve said before that people who observe my class loved how my classes didn’t get mad when we chased things that were totally wrong. We would just try a new approach when we saw that our answer was incorrect or unreasonable. I think that that is a much more important skill.

Editor’s note: Oh yeah, maybe I should emphasize that I did use the regression to make a prediction that was pretty close, when they found my height. Then, whenever we used a new tool, like a logarithmic regression, we would see the validity of our work. I’m not saying throw the baby out with the bathwater. Just, I guess, throw out a lot of bathwater.

19 Comments leave one →
  1. 05.25.2011 9:53 pm

    Great post! I did the same thing with prediction, but used our 7 foot 5th grade math teacher and a couple of 1st graders. We estimated foot size for both and it was WILDLY off. I did this on purpose. After working for two days on our line of best fit, I wanted to show them why our data didn’t always work (we only took data of 6th grade students in our school, they had different data points, different number of data,…). What I didn’t do however, was your second step. I didn’t throw them larger sets of data to play with. Great idea. I mess up in class all of the time, on purpose (mostly). It’s good for them to see that nothing and nobody’s perfect – especially in math. And, if you mess up it ok. The sky is not going to fall. You are just going to try again.

  2. 05.25.2011 10:04 pm

    Terrific article — one of my biggest challenges in teaching high school physics is finding ways to allow students to discover the power and dangers of extrapolation and statistics. This sounds like a great lab activity for the first rainy day of the school year. Especially if I can start some groups with a set of baby shoes / height data, other groups with a more random scattering of shoe/height sizes, and maybe even a few groups with just a point or two of data. Should make for some fun post-analysis discussions!

    • 05.26.2011 6:45 am

      That’s a pretty great idea.

      And crazy, I was just looking at your website for Physics review stuff. Looks like you just saved my ass. That should be really helpful for the kids who are struggling with some of the topics.

    • Jim Doherty permalink
      10.07.2011 6:59 am

      Dan,

      Another great extrapolation warning activity is to look at world record times for running the mile. We just did this in our AP Stat class. The current record is 12 years old, so you can predict how poor extrapolation would be in this case.

  3. LSquared permalink
    05.25.2011 10:11 pm

    Amazing. I love it.

    It also brings to mind a paper I read a while back that said that most students believe that all functions are lines (from which I infer the misconception that all data should be fit to a best fit line). Which brings to mind the realization that understanding geometry is crucial. It’s geometric reasoning that gets you to why light intensity varies inversely as the square of the distance (which you can readily get data to back up). It also has to be geometry that got Newton to infer that gravity varies as the square of the distance between the objects (something that’s a lot harder to verify with data–but that just goes to show that Newton was smarter than the rest of us).

  4. Rhett permalink
    05.25.2011 10:23 pm

    I am pretty sure that tall guy in your photo is Dan Meyer.

  5. 05.25.2011 10:29 pm

    How about this action?! http://commons.bcit.ca/physics/rjw/pers/womenrun.htm

    What about presenting data that is not even possibly related like number of american deaths in Iraq vs. number of iPods sold?

    • 05.26.2011 6:47 am

      That is one of the best things I’ve ever read. Looks like I’ve got a great intro to the statistics unit for next year.

  6. Kevin permalink
    05.26.2011 4:14 am

    I think the spirit of WCYDWT is exactly what you did with your lesson, at least the way I plan on using it. Its about engaging students, developing models, testing models and thinking critically about models. But then I don’t read every blog/tweet/burp about WCYDWT. The point is however that your students are not ONLY filling in worksheets but are actually DOING math (which is NOT filling in worksheets). That being said, kudos to you for dissenting.

    • 06.08.2011 4:14 pm

      Most people *are* using it well that are blogging about it. I had been getting on the Dans about some pretty hefty assumptions that they seemed to make in some regressions, such as a major extrapolation or assumption that it was a line. I don’t doubt that they had good discussions on it in class to reason out why, but it wasn’t totally clear. I’m afraid some of the n00bs may just run with something they put up without having those discussions. Heed my warnings!

  7. 05.26.2011 7:59 am

    Saw this on twitter yesterday:

    From @sheaphysics “I let kids do lab “wrong” sometimes because they learn more than if they did it “right”.”

    Love it.

    And so when people make these lessons that come out all neatly tied up with a ribbon around it and everything works out perfectly, kids begin to make bad assumptions about the way the world works, and the way math works. Like, that it ALWAYS WORKS.

    Spot on, Mr. Greg.
    You raise an excellent point about how perfect lessons lead to not so perfect learning. “Everything works out nicely in the math classroom” is a terribly dangerous idea to plant in the students. In every math textbook I’ve seen, there is one and only one perfect answer to every question with no noise provided. In some ways, the #wcydwt problems can turn into this fallacy if you aren’t careful about when and how to give the students information and strategies. A helicopter teacher will ruin #wcydwt because they’ll stop students as soon as they stray from the “right” path, and no real learning will happen.

    I will keep your words in mind when making (and teaching) these lessons. Let the students see that the world doesn’t always work out nicely, even in the math classroom. These problems are far away from the perfect problems delivered in the textbook, but they may need to stray further in order to enrich the learning atmosphere.

  8. 05.26.2011 8:53 am

    Gosh Greg, I can’t believe you would discourage the use of WCYDWT like that.

  9. cheesemonkeysf permalink
    05.26.2011 5:03 pm

    Apropos of absolutely nothing (or at best, very little), this is the kind of Fun Fact that keeps me coming back to your site like a heroin junkie to the Craigslist Rooms & Shares list:

    50% of visits to Columbus, OH will result in being punched in the face by a stranger (and for no reason!).

    I like how you mixed it up, with some actual, familiar real-world data and then LOTS of other kinds of data like shark growth rates that they are unlikely to have much experience with. This seems like a good way to give students the (truthful) insight that we are working with imperfect data AND with imperfect tools, but that doesn’t preclude our becoming skillful enough with them that we can do useful work in the real world regardless of their imperfection.

    This is a hard thing to get scientists and engineers to understand about business modeling when you are pitching and growing a start-up. There are a lot of assumptions and rules of thumb that make the whole model look like one giant steaming pile of wild-ass guesses. And yet… by using high-quality quantitative methods, we can come up with some extremely accurate predictive models.

    How fast DO sharks grow, BTW?

    - Elizabeth (aka @cheesemonkeysf on Twitter)

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  1. Messing with students and regression | Algebra 2 Blog

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