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.

Featured Email

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.

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.

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:

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.

Featured Tweet

@ddmeyer @mpershan my kids working with http://t.co/ig98dLeSGS are doing real work, fake world, and loving it.

— Pamela Rawson (@rawsonmath) September 17, 2014

Featured Comment

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.

————————–

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

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.

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

When do the circle and the square have equal area?

Version C

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.

Featured Mathalicians:

• Kate Nowak says “real world” means “interesting to humans outside a math class.”
• Ginny Stuckey asks if I’m trolling.
• Karim Ani is on vacay and promises a response when he gets back to business. [2013 Mar 22: He delivers.]
• Matt Lane asks, instead, if the task imbues people with joy for mathematics.
• Chris Lusto asks, “Is the question self-referential?”

Featured Comments

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.

Featured Tweets

@ddmeyer Intriguing matters to my students -Wonder if 'real world' imperative comes from assumption math is horrible http://t.co/WkgCG7lAmr

— Cathy Bruce (@drcathybruce) March 11, 2014

@ddmeyer a missing aspect in the "Real-world math debate? "Nothing ever becomes real 'til it is experienced." – John Keats

— Micah Hoyt (@MicahHoyt) March 11, 2014

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

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

• Review assignment: Which prescriptions from our earlier review of curiosity research are evident in Class #1 above?
• Michael Pershan draws a similar distinction between pedagogy and curriculum.

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

Provoking curiosity in our students about anything requires us to manage several tensions simultaneously. It requires keeping several lines tight — not slack — but not so tight they snap.

Read on for recommendations from some careful researchers.

Previously

Here is where the series stands: I’ve suggested that educators dramatically overvalue the real world as a motivator for students (one example) and that pinning down a definition of what is “real” to a child is no light assignment. I’ve suggested, instead, that the purpose of math class is to build a student’s capacity to puzzle and unpuzzle herself. And we shouldn’t limit the source of those puzzles. They can come from anywhere, including the world of pure mathematics. As an existence proof, I listed some abstract experiences that humans have enjoyed (everything from Sudoku to the Four Fours).

Currently

This is a bunch of question-begging, though, and my commenters have rightly called me out:

Okay, bud, how do you turn the “boring” bits of math into puzzles?

We struggle here. Across three conferences this fall, I’ve received different answers from respected educators about how we should handle the boring bits of math. (No one offers a definition of “boring,” by the way, but I take it to mean questions about mathematical abstractions — numbers, variables, etc. — instead of the material world.)

One educator suggested we “flip” those boring bits and send them home in a digital video. Another suggested we ask for sympathy, telling kids, “Hey, it can’t all be fun, okay? Just go with me here.” Another suggested we aim for empathy, that by accentuating our own enthusiasm for boring material our students might follow our lead. These answers all require you to believe there are mathematical concepts that are irredeemably boring, that there are aspects of the world about which we can’t possibly be curious. I don’t.

So I spent my holiday reading about curiosity, starting with some bread crumbs laid out by Annie Murphy Paul. These are some lessons I learned about teaching “the boring bits.”

Disclaimer

Interest and curiosity aren’t binary variables. They aren’t “on” or “off.” In his research on curiosity, Paul Silvia noted that “people differ in whether they find something interesting” and that “the same person will differ in interest over time” (2008, p. 58).

In 1994, George Loewenstein wrote a comprehensive review of the literature around curiosity [pdf] and even he despaired of locating some kind of universal theory of curiosity. (“Extremely ambitious” were his exact words; p. 93.)

We should temper our expectations accordingly. So my goal here is only to locate high-probability strategies for making students more interested more often. That’s the best we can hope for.

Lessons Learned

Provoking curiosity in our students about anything requires us to manage several tensions simultaneously. It requires keeping several lines tight — not slack — but not so tight they snap.

These are tensions between:

• The novel and the familiar. Stimuli that are too familiar are boring. Stimuli that are too novel are scary. (Silvia, 2008; Berlyne, 1954)
• The comprehensible and the confusing. Material that is too comprehensible is boring. Material that is too difficult is intimidating (Silvia, 2006; Silvia, 2008; Sadoski, 2001; Vygotsky, 1978).

It’s probably impossible to maintain this tension for each of your students but here are some recommendations that can help.

Start with a short, clear prompt that anyone can attempt.

Kashdan, Rose & Fincham (2004) claimed that curious people experience “clear, immediate goals … and feel a strong sense of personal control” (p. 292). Watch where we find that even in pure, abstract tasks:

• The Magic Octagon. “Pick where you think the arrow will be next.”
• Jinx Puzzle. “Just pick a number, any number.”
• Area v. Perimeter. “Just draw a rectangle, any rectangle.”

These could all fit in a tweet. In half a tweet. They’re light on disciplinary language, which keeps them comprehensible. In the final two tasks, students maintain a sense of personal control as they select individual starting points for each task.

Start an argument.

In 1981, Smith, Johnson & Johnson ran an experiment that resulted in one group of students skipping recess to learn new concepts and another group proceeding outside as usual, uninterested in learning those same concepts. The difference was controversy. The researchers engineered arguments between students in the first group, but not in the second.

Can you engineer arguments between students about the boring bits of mathematics?

• Is zero even or odd?
• Does multiplying numbers always make them bigger?
• Can you create a system of equations that has no solution?
• True or false: doubling the perimeter of a shape doubles its area.

At NCTM’s annual conference in Denver, Steve Leinwand said “the most important nine words of the Common Core State Standards are ‘construct viable arguments and critique the reasoning of others’.”

So not only can arguments stir your students’ curiosity but they’re an essential part of their math education. That’s winning twice.

Engineer a counterintuitive moment.

Hunt (1963, 1965) and Kagan (1972) popularized the “incongruity” account of curiosity. George Loewenstein summarizes: “People tend to be curious about events that are unexpected or that they cannot explain” (1994, p. 83). These events are difficult to engineer, of course, because they depend on the knowledge a student brings into your classroom. You have to know what your students expect in order to show them something they don’t expect.

Here are several existence proofs:

• The Magic Octagon. The arrow isn’t where the student thought it would be. “Wait, what?”
• Jinx Puzzle. We all wind up with the number 13. “Wait, what?”
• Area v. Perimeter. Swan asks, “Now where are the impossible points.” Impossible points? “Wait, what?”
• Ben Blum-Smith’s Pattern Breaking.

These moments are everywhere, though it’s an ongoing effort to train my eyes to find them. You can find them when the world becomes unexpectedly orderly or unexpectedly disorderly. When we all choose numbers that add to five and graph them, we get an unexpectedly orderly line. When we try to apply a proportional model to footage of a water tank emptying (“It took five minutes to empty halfway so it’ll take ten minutes to empty all the way.”) the world becomes unexpectedly disorderly. For younger students, the fact that 2 + 3 is the same as 3 + 2 may be a moment of curiosity and counterintuition. You know you landed the moment because the expression flashes across your student’s face: “Wait. What?”

You also create a counterintuitive moment when you …

Break their old tools.

Students bring functional tools into your classroom. They may know how to count sums on their fingers. They may know how to calculate the slope of a line by counting unit-squares and dividing the vertical squares by the horizontal squares. They may know how to write down and recall small numbers.

You create counterintuition when you take those old, functional tools and assign them to a task which initially seems appropriate but which then reveals itself to be much too difficult.

• “Great. Now go ahead and add 6 + 17.”
• “Great. Now go ahead and find the slope between (-5, 3) and (5, 10,003).”
• “Great. I’m going to show you the number 5,203,584,109,402,580 for ten seconds. Remember it as accurately as you can.”

These tasks seem easy given our current toolset but are actually quite hard, which can lead to curiosity about stronger counting strategies, a generalized slope formula, and scientific notation, respectively.

Create an open middle.

When you look at successful, engaging video games (even the fake-world games with no real-world application) they generally start with the same initial state and the same goal state, but how you get from one to the other is left to you. This gives the student the sense that her path is self-determined, rather than pre-determined, that she’s autonomous. (See: Deci; Csikszentmihalyi.)

• Sudoku. You start with a partially-completed game board and your goal is to complete it. You can wander down some dead ends as you accomplish the task. How you get there is up to you.
• Jinx Puzzle. You get to choose your number. It will be different from other people’s numbers. What you start with is up to you.
• Area v. Perimeter. You get to choose your rectangle. It will be different from other people’s rectangles. What you start with is up to you.

I’m not recommending “open problems” here because the language there is too flexible to be meaningful and too accommodating of a lot of debilitating student frustration. I’m not recommending you throw a video on the wall and let students take it wherever they want. I’m recommending that you’re exceptionally clear about where your students are and where they’re going but that you leave some of the important trip-planning to them.

Give students exactly the right kind of feedback in the right amount at the right time.

Easy, right? Feedback has been well-studied from the perspective of student learning but feedback’s effect on student interest is complicated. Some of you have recommended “immediate” feedback in the comments, but this may have the effect of prodding students down an electrified corridor where every deviation from a pre-determined path will register an alarm, creating a very closed middle. Students need to know if they’re on the right track while simultaneously preserving their ability to go momentarily off on the wrong track. This isn’t simple, but the best games and the best tasks maintain that balance.

• Four Fours. You arrange the fours into whatever configuration you want and then you check your answer. The feedback isn’t immediate. But you can check it yourself.
• Area v. Perimeter. You develop a theory about the impossible points then later test that theory out on different rectangles and coordinates.
• Sudoku. You don’t receive feedback immediately after you write a number in a box. But eventually you’re able to decide if the gameboard you’ve created matches the rules of the game.

Conclusion

You could very well say that these rules apply to “real world” tasks just as well as they apply to the world of pure mathematics. Exactly right! In the first post of this series, I said that “the real world-ness of [an engaging real-world] task is often its least essential element.” Real-world tasks are sometimes the best way to accomplish the pedagogy I’ve summarized here but it’s a mistake to assume that the “real world,” itself, is a pedagogy.

As I’ve tried to illustrate in this post with different existence proofs, it’s also a mistake to assume that pure math is hostile to student curiosity. The recommendations from these researchers can all be accomplished as easily with numbers and symbols and shapes as with two trains leaving Chicago traveling in opposite directions.

Often times, it’s even easier.

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

Loewenstein, G. (1994). The psychology of curiosity: a review and reinterpretation. Psychological Bulletin, 116(1), 75—98.

Berlyne, DE. (1954). A theory of human curiosity. British Journal of Psychology. General Section, 45(3), 180—191.

Turner, SA., Jr., & Silvia, PJ. (2006). Must interesting things be pleasant? A test of competing appraisal structures. Emotion, 6, 670—674.

Sadoski, M. (2001). Resolving the effects of concreteness on interest, comprehension, and learning important ideas from text. Educational Psychology Review, 13, 263—281.

Kashdan, TB., Rose, P., & Fincham, FD. (2004). Curiosity and exploration: facilitating positive subjective experiences and personal growth opportunities. Journal of Personality Assessment, 82(3), 291—305. doi:10.1207/s15327752jpa8203_05

Smith, K., Johnson, DW., & Johnson, RT. (1981). Can conflict be constructive? Controversy versus concurrence seeking in learning groups. Journal of Educational Psychology, 73(5), 651.

Hunt, JM. (1963). Motivation inherent in information processing and action. In O.J. Harvey (Ed.), Motivation and Social Interaction, 35—94. New York: Ronald Press

Hunt, JM. (1965). Intrinsic motivation and its role in psychological development. In D. Levine (Ed), Nebraska Symposium on Motivation, 13, 189—282. Lincoln: University of Nebraska Press.

Kagan, J. (1972). Motives and development. Journal of Personality and Social Psychology, 22, 51—66.

Vygotsky, L. (1978). Interaction between learning and development. From: Mind and Society, 79—91. Cambridge, MA: Harvard University Press.

Featured Comments

blink:

I wonder, though, if declaring some content “boring bits” gives up the game. (If they are *truly* and irredeemably boring, why teach them?) First, we need to understand these “bits” better and think about big-picture coherence.

Consider a jigsaw-puzzle. A missing piece is “interesting” quite apart from its shape or color (although they may be independent sources of interest). What matters is that it fits! Similarly, no one salivates over dates and locations in history. Yet we care about these facts because they allow us to tell stories, discover relationships, and find patterns. It is silly to complain that a particular fact is “boring” on its own. Here again, what matters it how it fits.

Evan Weinberg:

I like that these six methods have nothing to do with making math easy. This is often the identified goal that students and teachers (and countless online videos) work towards, particularly in the context of students that have failed math in the past. Having success is enough of a motivator for students to push through some material that is not particularly engaging, it does not have staying power.

Jason Dyer challenges the group to apply these principles to fifth-degree polynomial inequalities:

I don’t suppose we could take something extra-”boring” and try to do a makeover? Sort of like Dan’s makeover series except start from the hardest point possible?

I am having to currently teach working out polynomial inequalities like 2x^5 + 18x^4 + 40x^3 < 0. There's an intense amount of drudge and fiddly bits and the extra pain of having a lot of steps to do. Any ideas?

I give it a shot.

gasstationwithoutpumps:

The moment of inertia for rotating a I-beam about its long axis has no practical relevance in structural engineering. This is a fake-world problem, of no interest either mathematically or to engineers.

Even if this task did have practical interest for structural engineers, its presentation here will move the needle on student engagement only a fraction of a degree. The issue here isn’t the usefulness of the application to professionals but the tedious, pre-determined work students do.

Jane Taylor:

When I saw the two boards, I wanted to go get a board and try standing on it. How much weight could we put on the board in each position before it broke? That would be an engaging problem.

I don’t know. That might be an engaging problem.

There are 100 different directions that question can go in terms of the work students do in class and only a handful of them will actual leave kids mathematically powerful and capable.

Watch me ruin the problem:

The maximum load a board can hold before it snaps is given by the formula:

[formula involving cross-sectional area and mass]

Dan weighs 90 kilograms and the dimensions of the board are 2 inches by 4 inches by 70 inches. Will the board hold his weight?

I have no confidence this task will result in the sense of accomplishment and connection the editors of the NYT seem to think it will.

There are other ways to present this kind of task, though. Which is my point. The “real world”-ness or “job world”-ness of the task is one of its least important features.

A growing number of schools are helping students embrace STEM courses by linking them to potential employers and careers, taking math and science out of textbooks and into their lives. The high school in Brooklyn known as P-Tech, which President Obama recently visited, is a collaboration of the New York City public school system and the City University of New York with IBM. It prepares students for jobs like manufacturing technician and software specialist.

[..]

Though many of these efforts remain untested, they center around a practical and achievable goal: getting students excited about science and mathematics, the first step to improving their performance and helping them discover a career.

Pick any application of math to the job world and I promise you I can come up with 50 math problems about that application that students will hate. Get a little coffee in me and I’ll crank out 49 more. It’s that one problem, the one out of 100 that students might enjoy, that’s really tricky to create, and often times its “real world”-ness is its least important aspect.

Chris Hunter reminds me (via email) that the British Columbia Institute of Technology has made a similar bet on “real-world” math. Here’s an example:

Once again, we’re asking students to substitute given information for given variables and evaluate them in a given formula. Does anyone want to make the case that our unengaged students will find the nod to structural engineering persuasive?

The “real world” isn’t a guarantee of student engagement. Place your bet, instead, on cultivating a student’s capacity to puzzle and unpuzzle herself. Whether she ends up a poet or a software engineer (and who knows, really) she’ll be well-served by that capacity as an adult and engaged in its pursuit as a child.

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Chris Hartmann points out that these application of math to jobs often miss the math that’s most relevant to those jobs:

And, in the job world a lot of the mathematics isn’t done by human minds or hands anymore, with good reason. Faster, more accurate means are available using technology. What often remains is puzzling out the results.

Mr. K:

The telling thing is that the Times’s example of a real world problem that real world people can’t solve, that of calculating the cost of a carpet for a room, is pretty much a guaranteed loser for any math class that I have ever taught at any level.

On the other hand, yesterday I had a room full of third round algebra students engrossed in building rectangles with algebra tiles. That’s about as non real world as it gets.

gasstationwithoutpumps:

The moment of inertia for rotating a I-beam about its long axis has no practical relevance in structural engineering. This is a fake-world problem, of no interest either mathematically or to engineers.

There are real-world applications for moment of inertia problems, but this is not one of them.

nerdypoo:

This seems to be a perennial favorite. In 2011 the Times asked if we needed a new way to teach math, with this quote:

“A math curriculum that focused on real-life problems would still expose students to the abstract tools of mathematics, especially the manipulation of unknown quantities. ”

I’m certain I could find an example of such an article from every few years …

This theory says, “For math to be engaging, it needs to be real. The fake stuff isn’t engaging. The real stuff is.” This theory argues that the engagingness of the task is directly related to its realness.

This is a limited, incomplete theory of engagement. There are loads of “real” tasks that students find boring. (You can find them in your textbook under the heading “Applications.”) There are loads of “fake” tasks that students enjoy. For instance:

• Tetris.
• Tic Tac Toe.
• Sudoko.
• Shikaku.

No context whatsoever in any of them. Perhaps the relationship actually looks more like this:

I’m being a little glib here but not a lot. Seriously, none of those tasks are “real-world” in the sense that we commonly use the term and yet they captivate people of all ages all around the world. Why? According to this theory of engagement, that shouldn’t happen.

Here are fake-world math tasks that students enjoy:

• Ihor Charischak’s Jinx Puzzle.
• David Masunaga’s Magic Octagon.
• Malcolm Swan’s Area v. Perimeter.
• NRICH’s Factors and Multiples Puzzle. (Megan Schmidt: “… I have one student in particular who is not particularly motivated by much … when I bust out a puzzle, he’s all in.”)
• Matt Vaudrey’s Magical Triangle Theorem.
• Andrew Stadel’s Weekly Puzzle.
• The meaning of the sequence 3, 3, 5, 4, 4, 3, … , which drove kids bananas the day I wrote it on the board at the end of a test.
• The proof that 2 = 1.
• 2013 Dec 3. The four fours puzzle.
• 2013 Dec 5. Timon Piccini’s Broken Calculator.
• 2013 Dec 10. Patrick Vennebush sends along “How much is your name worth?”

My point is that your theory of engagement might be limiting you. It might be leading you towards boring real-world tasks and away from engaging fake-world tasks.

We need a stronger theory of engagement than “real = fun / fake = boring.”

Homework Time!

Choose one:

• Write about a fake-world math task you personally enjoy. What makes it enjoyable for you? What can we learn from it?
• Write about an element that seems common to those enjoyable fake-world tasks above.

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!”

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:

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.