Month: November 2013

Total 8 Posts

[Fake World] It’s “Real” If They Can Argue About It

Hans Freudenthal changed the conversation from “real world” to “realistic world“:

The fantasy world of fairy tales and even the formal world of mathematics can provide suitable contexts for a problem, as long as they are real in the student’s mind.

This complicates our task. It’s easy to create real world tasks that aren’t real in the student’s mind. It’s harder to create realistic tasks.

Here’s one way to test if the context is “real in the student’s mind”:

Can they construct an argument about it?

From Jennifer Branch’s presentation handout at CMC-South [pdf], I’ve pulled a series of questions she calls “Eliminate It!”

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None of these are “real” in the sense that most of us mean the word. But each of these groups is “real” to different students. Triangles are real. Pentagons are real. Diameters are real. We know they’re real because those students can construct an argument about which one doesn’t belong. That ability to argue proves their realness.

(Of course, the value of the task is that different arguments can be made for each member of the group.)

On the other hand, consider:

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These elements are definitely “real.” They’re metals. But are they realistic? Are they real in your mind? Can you construct an argument about their substance?

If not, how is it in our best interests to promote a definition of “real” that admits “magnesium” but denies “pentagons”?

2013 Nov 26. Similarly, it’s “real” if they can sort it meaningfully.

[Fake World] Two Different Definitions Of Relevance

Wolfram uses the word “relevant” to define the word “relevant.” This highlights (again) the contested and circular nature of terms like “real” and “relevant,” terms which everyone says and everyone else nods along to, but which are in reality so soft they fall apart to the touch.

Meanwhile, here’s Ben Blum-Smith, three years ago on this blog, offering a much more useful definition of “relevance”:

The real test of whether a math problem is “relevant” is not “do you use this in ‘real life’,” whatever that means, but “do you want to solve it?” It’s not that you want to solve it because it’s relevant; wanting to solve it is what it means to be relevant.

These two different definitions of “relevance” and “real world” lead to two very design questions for your math class.

One: “How do I make [some difficult math concept] relevant and real world?”

The terms here are contested and circular and for many math concepts the question is impossible to answer.

Two: “How do I make [some difficult math concept] something students want to solve?”

This question is often very hard to answer and the answers are often highly contingent on the culture you’ve developed in your classroom.

But it has the advantage of not being impossible.

[Fake World] Math Needs A Better Product, Not More Commercials

There’s a contest called Math-O-Vision, which your students should enter. Here’s the premise:

The Neukom Institute for Computational Science, at Dartmouth College, is offering prizes for high school students who create 4-minute movies that show the world of equations we live in. In 240 seconds, using animation, story-telling, humor, or anything you can think of, show us what you see: the patterns, the abstractions, the patterns within the abstractions.

The contest doesn’t articulate its goal – why should we ask students to participate? – but we can fill in that blank with plenty of good ones:

  • It promotes creative expression.
  • It marries math and the arts.
  • It allows students who like math to express what they like about math.
  • It has a $4,000 grand prize.

What the contest won’t do is convince people who don’t like math to like math. Here’s why.

I could divide my students into three rough categories. Here’s how each one would react to last year’s grand-prize winner.

The students who like math in every one of its flavors – pure, applied, whatever.

They’ll enjoy making these videos and they’ll enjoy watching these videos. But these students aren’t any concern of ours right here.

The students who dislike math but wish they liked math.

They’ll find themselves enticed by these videos. Seeing shots like this will re-activate their sense that math has some powerful and exhilarating role to play in the world around them.

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But then they’ll return to the classroom where they’re assigned real-world tasks like this:

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And the difference couldn’t be more stark. The promise of “real-world math” has gone unrealized. What seemed powerful and exhilarating and full of potential in the video is in reality debilitating and tedious

The students who dislike math, who think math dislikes them, and who want nothing to do with math.

These videos will be ineffective. Even counter-productive. Hearing the grand prize-winner say, “Mathematics. It’s everywhere,” won’t entice or engage these students. It’ll terrify them. Like telling someone who hates clowns, “Clowns. They’re everywhere.” Or showing videos of skydiving and bungee jumping to someone who’s terrified of heights.

More than they need “real-world” mathematical experiences (like the debilitating and tedious volleyball task above) these students need math to make them feel powerful and exhilarated and full of potential.

These videos are effectively commercials for math. Commercials are useful if someone is in the market for the product. They’re useless if someone already has the product and has found it defective, which describes how many of our students feel about math right now.

The pursuit of real world math can lead to lots of positive outcome but one outcome it leads to is effective commercials for a defective product. We need fewer commercials. We need a better product.

[Fake World] Conjectures

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“Real” isn’t a guarantee.

When I work with publishers, I find authors hesitant to tinker with the mechanics of a task and eager instead to spread the “real world” over the task like a thin coat of varnish.

Why are students bored by quartic equations in the problem I excerpted above? We are less eager to examine the structure of the task (which has been passed down and assumed constant for generations) and eager instead to just add stock photography of a snowboarder.

Students aren’t so easily fooled.

Similarly, when I find interesting real-world tasks that result in engaged, interested learners, the real world-ness of the task is often its least essential element.

“Real” is relative.

What’s “real” to you isn’t necessarily “real” to me. Our lived experiences are different.

An example on the extremes: one of my favorite proof tasks is Bucky the Badger. Lots of teachers and students enjoy it. The context and the math are often interesting. But I took it to Australia two years ago and interest was quite a bit muted for reasons I should already have anticipated: they don’t do football like we do football.

This obstacle to effective curriculum design isn’t insurmountable. Lots of great art has been made about contexts we’ve never experienced, for instance. But we assume this obstacle doesn’t exist at the expense of our students who haven’t been privileged with our life experiences.

It’s self-limiting to try to draw lines between “real” and “fake.”

Take hexagons, health insurance, hydraulic engineering, hydrogen gas, and heptominoes. Is anybody here confident they can tell me which of these is “real” to a sixth grader? Which is “fake”?

This is a self-limiting question. They can all be made real.

We should be very, very hesitant to write off entire worlds as foreign and unknowable, especially the world of pure mathematics, which a student can always conjure up given nothing more than a mind, a pencil, and a piece of paper.

The point of math class.

The point of math class isn’t to build a student’s capacity to answer questions about the world outside the math classroom. It isn’t to prepare her to get a job either. Both of those are happy outcomes of a different goal that’s broader and more interesting to me.

The point of math class is to build a student’s capacity to puzzle and unpuzzle herself – no matter what form those puzzles take.

Find those puzzles in the real world, the fake world, the job world, or any other world – it doesn’t matter.

Real-World Math That Isn’t Real To Students

Seen on a dessert menu at a fancy restaurant I crashed this weekend:

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“Well you guys said you wanted to know when you’d ever use this stuff.”

I’ll be dedicating this blog to a certain line of inquiry for the next few days or weeks or for however long it takes me to come to some kind of internal consensus. I’d appreciate your help with that.