Category: fake-world math

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Can Sports Save Math?

A Sports Illustrated editor emailed me last week:

I’d like to write a column re: how sports could be an effective tool to teach probability/fractions/ even behavioral economics to kids. Wonder if you have thoughts here….

My response, which will hopefully serve to illustrate my last post:

I tend to side with Daniel Willingham, a cognitive psychologist who wrote in his book Why Students Don’t Like School, “Trying to make the material relevant to students’ interests doesn’t work.” That’s because, with math, there are contexts like sports or shopping but then there’s the work students do in those contexts. The boredom of the work often overwhelms the interest of the context.

To give you an example, I could have my students take the NBA’s efficiency formula and calculate it for their five favorite players. But calculating – putting numbers into a formula and then working out the arithmetic – is boring work. Important but boring. The interesting work is in coming up with the formula, in asking ourselves, “If you had to take all the available stats out there, what would your formula use? Points? Steals? Turnovers? Playing time? Shoe size? How will you assemble those in a formula?” Realizing you need to subtract turnovers from points instead of adding them is the interesting work. Actually doing the subtraction isn’t all that interesting.

So using sports as a context for math could surely increase student interest in math but only if the work they’re doing in that context is interesting also.

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Marcia Weinhold:

After my AP stats exam, I had my students come up with their own project to program into their TI-83 calculators. The only one I remember is the student who did what you suggest — some kind of sports formula for ranking. I remember it because he was so into it, and his classmates got into it, too, but I hardly knew what they were talking about.

He had good enough explanations for everything he put into the formula, and he ranked some well known players by his formula and everyone agreed with it. But it was building the formula that hooked him, and then he had his calculator crank out the numbers.

Real Work v. Real World

“Make the problem about mobile phones. Kids love mobile phones.”

I’ve heard dozens of variations on that recommendation in my task design workshops. I heard it at Twitter Math Camp this summer. That statement measures tasks along one axis only: the realness of the world of the problem.

140822_1lo

But teachers report time and again that these tasks don’t measurably move the needle on student engagement in challenging mathematics. They’re real world, so students are disarmed of their usual question, “When will I ever use this?” But the questions are still boring.

That’s because there is a second axis we focus on less. That axis looks at work. It looks at what students do.

That work can be real or fake also. The fake work is narrowly focused on precise, abstract, formal calculation. It’s necessary but it interests students less. It interests the world less also. Real work – interesting work, the sort of work students might like to do later in life – involves problem formulation and question development.

That plane looks like this:

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We overrate student interest in doing fake work in the real world. We underrate student interest in doing real work in the fake world. There is so much gold in that top-left quadrant. There is much less gold than we think in the bottom-right.

BTW. I really dislike the term “real,” which is subjective beyond all belief. (eg. What’s “real” to a thirty-year-old white male math teacher and what’s real to his students often don’t correlate at all.) Feel free to swap in “concrete” and “abstract” in place of “real” and “fake.”

Related. Culture Beats Curriculum.

This is a series about “developing the question” in math class.

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Bob Lochel:

I would add that tasks in the bottom-right quadrant, those designed with a “SIMS world” premise, provide less transfer to the abstract than teachers hope during the lesson design process. This becomes counter-productive when a seemingly “progressive” lesson doesn’t produce the intended result on tests, then we go back not only to square 1, but square -5.

Fred Thomas:

I love this distinction between real world and real work, but I wonder about methods for incorporating feedback into real work problems. In my experience, students continue to look at most problems as “fake” so long as they depend on the teacher (or an answer key or even other students) to let them know which answers are better than others. We like to use tasks such as “Write algebraic functions for the percent intensity of red and green light, r=f(t) and g=f(t), to make the on-screen color box change smoothly from black to bright yellow in 10 seconds.” Adding the direct, immediate feedback of watching the colors change makes the task much more real and motivating.

[Fake World] Real-World Math Proves Tough To Pin Down

tl;dr – “Real world” is tougher to measure than “interest” and less important overall. So rather than asking which of these three different versions of a word problem is more “real world” I asked a couple hundred people which is more interesting. Only the geometry treatment was significantly better than a coin flip at generating questions.

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Here are some closing words about “real world” math, mostly distilled from your comments on the last post. As with previous investigations, I am indebted to the folks who stop by this blog to comment and make me smarter.

Real-World Math Is Hard To Define

What other conclusion can we draw from the dozen-or-so definitions of “real-world math” I found here and on Twitter?

  • It depends on whether we’re talking about the world of procedures, concepts, or applications. [Karim Ani]
  • A problem simulating a project / job / task being performed by someone performing their normal job duties, such as the example of the contractor building a pool to meet municipal code. [Ben Rimes]
  • A problem involving objects or tasks that would be considered an experience students are likely to have in the “real world.” [Ben Rimes]
  • If it came from a math teacher it’s basically not real-world, unless it was a math teacher doing something in the outside world leading to an interesting problem in mathematics. [Bowen Kerins]
  • An observation / question that would be interesting to humans outside a math class. [Kate Nowak]
  • Something is “real” to a student if it’s concrete, attainable, comprehensible. [Michael Pershan]
  • Something is “real” to a student if it has a non-mathematical purpose. [Michael Pershan]

Real-World Math Doesn’t Guarantee Interest

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David Taub argued the whole question confused “interest” with “real world.” M Ruppel listed other criteria for judging the value of a question, one of which was “kids want to solve it.”

Their arguments may seem obvious to you. They aren’t obvious to the three people who emailed me here or these presenters at the California STEM Symposium or Conrad Wolfram or the New York Times editorial board.

As Liz said, “Real world is just a means to an end. The goal is interest.” We should reject simple explanations of student interest.

So Which Version Is More Interesting?

Asking which of these three versions (candy, geometry, text) of the same math problem is more “real-world” was a pointless task since basically everyone has a different definition of “real-world” and it doesn’t guarantee interest anyway

So let’s ask about interest instead.

I used Mechanical Turk (and Evan Weinberg’s invaluable Internet skills) to show a random version of the problem to 99 people. I asked them if they had a question or not.

Of the three treatments, only the geometry treatment was statistically better than a coin flip at generating questions. (Here’s the experiment and the data.)

Then I showed another 80 people the same three treatments and asked how interested they were in the equal area question as measured on a Likert scale from -3 to 3, including 0. (This measured interest another way. Perhaps a question didn’t occur to you impulsively but once you heard it you were interested in it.)

Here again only the geometry treatment had an interest rating that was significantly different from “neutral.” (Experiment and data.)

Why the geometry treatment? I don’t know. It’s more abstract than the candy treatment, which features objects from outside the math classroom. 88% of the people I surveyed in the first experiment answered “within the last year” to the question “When did you last use math to solve a problem in life, work, or school?” That’s a math-friendly crowd. It’s possible that a class of elementary schoolers would find the candy treatment more interesting and that a coffee klatch of research mathematicians would tend towards the text treatment.

I don’t know. I’m just speculating here that real world is a pretty porous category. And for the sake of interesting your students in mathematics, it’s more important to know their world.

2014 Mar 26. Fawn Nguyen asked her eighth grade geometry students which version they preferred.

Dear Mathalicious: Which Of These Questions Is “Real World”?

An ongoing question in this “fake world” series has been, “What is real anyway, man?”

Are hexagons less real-world to an eighth-grader than health insurance, for example? Certainly most eighth graders have spent more time thinking about hexagons than they have about health insurance. On the other hand, you’re more likely to encounter health insurance outside the walls of a classroom than inside them. Does that make health insurance more real?

I don’t know of anyone more qualified to answer these questions than our colleagues at Mathalicious who produce “real-world lessons” that are loved by educators I love.

I’m sure they can help me here. Here are three versions of the same question.

Version A

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle.

Find P such that the area of the square and circle are equal.

Version B

medium

When do the circle and the square have equal area?

Version C

low

Where do the circle and square have the same number of candies?

Version D

[suggested by commenter Jeff P]

You and your friend will get candy but only if you find the spot where there’s the same number of candies in the square and the circle. Where should you cut the line?

Version E

[suggested by commenter Mr. Ixta]

Imagine you were a contractor and your building swimming pools for a hotel. Given that you only have a certain amount of area to work with, your client has asked you to build one square and one round swimming pool and in order to make the pools as large as possible (without violating certain municipal codes or whatever), you need to determine how these two swimming pools can have the same area.

Version F

[suggested by commenter Emily]

Farmer John has AB length of fencing and wants to create two pens for his animals, but is unsure if he wants to make them circular or square. To test relative dimensions (and have his farmhands compare the benefits of each), he cuts the fencing at point P such that the area of the circle = area of the square, and AP is the perimeter of the square pen and PB is the circumference of the circular pen.

Dear Mathalicious

Which of these is a “real world” math problem? Or is none of them a real-world math problem?

If anybody else has a strong conviction either way, you’re welcome to chip in also, of course.

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David Taub:

These are the two ideas that seem to be confused here are just “real world” and “interesting”. There seems to be an inherent assumption that they are somehow related when I doubt they even should be. Sometimes they will overlap, and sometimes not, but it is a bit random and quite personal in my opinion.

Kevin Polke:

I caught the maintenance staff at my high school using “real-world math.”

Ben Rimes:

The problem probably lies in the many ways that one could define “real world”.

What qualifies as “real world”?

  • A problem simulating a project/job/task being performed by someone performing their normal job duties, such as the example of the contractor building a pool to meet municipal code?
  • Is a problem involving objects or tasks that would be considered an experience students are likely to have “real world”(dividing up the M&Ms to be equal size)?
  • Actual evidence of math in the “real world “(video or otherwise) being applied as a part of someone’s job (the maintenance staff example above from @Kevin Polke) could qualify, or perhaps the application of math to a problem that is foreseeable in that person’s job duties.

I think it’s best to take the more pluralistic viewpoint on this one, as it would be quite a task to attempt to statically define the exact nature of “real world” math, as there are always countless more examples lining up to disprove whatever narrow definition you choose.

Liz:

“Real world” is just a means to an end. The goal is interest.

M Ruppel:

What matters for a ‘good task’ is not whether it’s real, it’s whether
1) its meaning is clear right away
2) kids want to solve it
3) have the mathematical tools to solve it (even if not very sophisticated tools)

Bowen Kerins:

I’ll say again though: I don’t really care if a problem is real-world. There are so many great problems that aren’t, and so many terrible problems that are. I don’t think it carries huge added value. Everyone decides what’s “real” to them (as Jeff P said). Right now for my kid, 7-5 and 5-7 being related to one another is plenty real, even though there is no connection yet to physical objects or money or any of that.

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2014 Mar 26. Fawn Nguyen asked her eighth grade geometry students which version they preferred.

[Fake World] Culture Beats Curriculum

a/k/a Worshipping The Real World

Here are three e-mails I received from three different people over the last three months. Spot the common theme.

November:

My co-teacher and I were puzzling over what kind of problem would create an intellectual need for systems. Do you have anything you could send, by chance?

December:

We are planning to launch a unit on systems of equations in early January (after December break) and wanted to try out your approach to create an intellectual “need.”

January:

Showing two straight lines on a piece of graph paper and finding points of intersection has very little significance to most people. I’m looking for a real-world problem that has an answer that is not self-evident, but which requires a little thinking and finding the intersections and is infinitely more productive and satisfying and will stay with them for the rest of their lives. That is what I am looking for.

I receive these questions on Twitter also. I find them almost impossible to answer because I don’t know what your class worships.

Here’s what I’m talking about:

Class #1

You start class by asking your students to write down two numbers that add to ten. They do. Most likely a bunch of positive integers result.

Then you ask them to write down two numbers that subtract and get ten. They do.

Then you ask them to write down two numbers that do both at the exact. same. time. “Is that even possible?” you ask.

Many of them think that’s totally impossible. You can’t take the same two numbers and get the same output with two operations that are natural enemies of each other. They’d maybe never phrase it that way but the whole setup seems totally screwy and counterintuitive.

Then someone finds the pair and it seems obvious in hindsight to most students. We’ve been puzzled and now unpuzzled. Then you ask, “Is that the only pair that works?” knowing full well it is, and the class is puzzled again.

You define systems of equations as “finding numbers that make statements true” and you spend the next week on statements that have only one solution, that have infinite solutions, and the disagreeable sort that don’t have any solutions.

Students learn to identify the kind of scenario they’re looking at and how to find its solutions quickly (if any exist) using strong new tools you offer them over the unit.

Class #2

The same lesson plays out but this time, after we’ve determined the pair of numbers that solve the system, a student pipes up and asks, “When will we ever use this in the real world.”

Worshipping the Real World

David Foster Wallace wrote about worship – the secular kind, the kind that applies to everybody, not just the devout, the kind that applies especially to us teachers in here:

If you worship money and things — if they are where you tap real meaning in life — then you will never have enough. Never feel you have enough.

If your students worship grades, they won’t complete assignments without knowing how many points it’s worth. If they worship stickers and candy, they won’t work without the promise of those prizes.

If you say a prayer to the “real world” every time you sit down to plan your math lessons, you and your students will never have enough real world, never feel you have enough connection to jobs and solar panels and trains leaving Chicago and things made of stuff.

If you instead say a prayer to the electric sensation of being puzzled and the catharsis that comes from being unpuzzled, you will never get enough of being puzzled and unpuzzled.

The first prayer limits me. The first prayer means my students will only be interested in something like The Slow Forty – a real world application of systems. The second prayer means my students will be interested in The Slow Forty (because it’s puzzling) but also the puzzling moments that arise when we throw numbers, symbols, and shapes against each other in interesting ways.

The second prayer expands me. Interested people grow more interested. Silvia writes, “Interest is self-propelling. It motivates people to learn thereby giving them the knowledge needed to be interested” (2008, p. 59). Once you give your students the experience of becoming puzzled and unpuzzled by numbers, shapes, and variables, they’re more likely to be puzzled by numbers, shapes, and variables later. That’s fortunate! Because some territories in mathematics are populated exclusively by numbers, shapes, and variables, in which cases your first prayer will be in vain.

That’s why I can’t tell you what to teach on Monday. Your classroom culture will beat any curriculum I can recommend. I need to know what you and your students worship first.

BTW

References

Silvia, PJ. (2008). Interest — the curious emotion. Current Directions in Psychological Science, 17(1), 57–60. doi:10.1111/j.1467-8721.2008.00548.x