**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 *any*where, 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 andÂ 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**

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.

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?