Category: fake-world math

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[Fake World] Teaching The “Boring” Bits

tl;dr

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).

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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:

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.

[Fake World] The “Real World” Guarantees You Nothing

There were two particularly useful comments in response to this problem:

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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.

[Fake World] The New York Times Goes All-In On “Real World” Math

In case this whole series seemed to you like a bit of a straw man (it did to Kate Nowak and Michael Pershan) here’s the New York Times Editorial Board:

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:

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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.

Featured Comments

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 …

[Fake World] Limited Theories of Engagement

Let’s just call them “theories of engagement” for now. Every teacher has them, these generalized ideas about what engages students in challenging mathematics. Here’s the theory of engagement I’m trying to pick on in this series:

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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:

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

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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:

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.

[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.