Pool table math is a common feature of a lot of geometry textbooks. Billiards hit a cushion and leave it at about the same angle. We have a real-world application! But as we’ll see in this week’s WWIB installment, not all treatments of that application are equal. In fact, commenters found them all wanting in various ways. I invite you to click through to this week’s three contestants:

1. Discovering Geometry
2. CME Project
3. College Preparatory Mathematics

What You Said

In the preview post, commenters called out the following turn-offs in different versions.

• “It jumps to the math notation too quickly.”
• “There is a ton of language in these problems.”
• “Two of the books just state that the angle of incidence and angle of reflection are the same and the other just expects students to know that.”
• “I feel like if I sat down and solved the problem that follows their explanation, I’d be copying their steps rather than really thinking it out for myself in a way that would make sense of it.”

On Twitter, Rose Roberts urges us to be careful here as, “Problems involving pool and mini-golf were the reason I decided I hated geometry in 8th grade. The sole reason.”

I’ll try to summarize the critiques using language that’s common to this blog without putting too many words in my commenters mouths. These textbook treatments rush to a formal level of abstraction too quickly. They don’t do a sufficient job developing the question for which “angle of incidence = angle of reflection” is the answer, or helping students develop an intuition about that answer.

In Discovering Geometry, for example, the formal equivalence statement is given and then the text asks students to apply it with their protractor.

A number of my commenters offer variations on, “Just take ’em to the pool hall!” This idea sounds great and will scan to many as suitably progressive, inquiry-based, student-centered, etc. But I’m unsatisfied. Mr. Bishop took us to the pool hall when I was a high school student and let us watch a local pro knock down a rack. I think he let us shoot a bit ourselves. I remember enjoying myself. I don’t remember learning more math than I did in his classroom lesson.

Pro pool players don’t use protractors.

For one reason, they’ve internalized that mathematics through practice. For another, the player can’t measure the angle of the ball in real time. The ball moves too quickly and the pool player’s eye-level view of the pool table is unlike the bird’s-eye view that would allow her to measure that angle.

This is a problem.

What I Need

Here is the resource I need. I’d like students to experience mathematical analysis as power, rather than punishment.

So let’s start with a tool that comes easily to students: their intuition. Let’s invite them to use their intuition in the context of a pool table. And let’s establish the context so that their intuition fails them, or at most earns a C-.

Then, let’s help students learn how to analyze the path of the pool ball mathematically. We’ll repeat the previous exercise and point at the end to the superior results that accrue when students analyze the pool table mmathematically instead of intuitively. (If superior results don’t accrue, we should either re-design the context to better highlight math’s power on a pool table or admit to ourselves we were wrong about math’s power.)

John Golden gets us close to that resource, inviting teachers to pull out still frames from this video of billiard shots for student analysis. But that analysis is much more complex than the level of the textbooks we’re critiquing today. Billiards ricochet off of other billiards in that video.

The resource I need doesn’t seem to exist yet, so I’ll try to build it. I’ll start with this game. Stay tuned.

Every week this summer I’m posting three versions of the same real-world task. Please tell me: who wore it best?

• In what ways are they different?
• What do their differences say about their authors’ beliefs about students, learning, and math?
• Would you make changes? Which and why?

Every secondary teacher and secondary textbook author knows that parabolas are #realworld because they describe the path of projectiles subject to gravity. Forgive me. “Projectiles” are not #realworld. “Baseballs” are #realworld.

But let’s not relax simply because we’ve drawn a line between the math inside the classroom and the student’s world outside the classroom. Three different textbooks will treat that application three different ways.

Click each image for a larger version.

Version #1

Version #2

Version #3

Chris Hunter claims, “The similarities here overwhelm any differences.” That’s probably true. So let’s talk about some of those similarities and what we can do about them.

My Least Favorite Phrase in Any Math Textbook

They each include the phrase “is modeled by,” which is perhaps my least favorite phrase in any math textbook. Whenever you see that phrase, you know it is preceded by some kind of real world phenomenon and proceeded by some kind of algebraic representation of that phenomenon, a representation that’s often incomprehensible and likely a lie. eg. The quartic equation that models snowboarding participation. No.

Chris Hunter notes that the equations “come from nowhere” and seem like “magic.” True.

@dmcimato and John Rowe point out that what normal people wonder about baseball and what these curriculum authors wonder about baseball are not at all the same thing.

That isn’t necessarily a problem. Maybe we think we should ask the authors’ questions anyway. As John Mason wrote in a comment on this very blog on the day that I now refer to around the house as John Mason Wrote a Comment on My Blog Day:

Schools as institutions are responsible for bringing students into contact with ideas, ways of thinking, perceiving etc. that they might not encounter if left to their own devices.

But these questions are really strange and feel exploitative. If we’re going to use, rather than exploit, baseball as a context for parabolic motion, let’s ask a question like: “Will the ball go over the fence?”

And let’s acknowledge that during the game no baseball player will perform any of those calculations. This is not job-world math. So the pitch I’d like to make to students (heh) is that, yes, your intuition will serve you pretty well when it comes to answering both of those questions above, but calculations will serve you even better.

Ethan Weker suggests using a video, or some other visual. I think this is wise, not because “kids like YouTubes,” but because it’s easier to access our intuition when we see a ball sailing through the air than when we see an equation describing the same motion.

Here’s what I mean. Guess which of these baseballs clears the fence:

Now guess which of these baseballs clears the fence:

They’re different representations of the same baseballs — equations and visuals — but your intuition is more accessible with the visuals.

We can ask students to solve by graphing or, if we’d like them to use the equations, we can crop out the fence. If we’d like students to work with time instead of position, we can add an outfielder and ask, “Will the outfielder catch the ball before it hits the ground?”

This has turned into more of a Makeover Monday than a Who Wore It Best Wednesday and I shall try in the future to select examples of problems that differ in more significant ways than these. Regardless, I love how our existing curricula offer us so many interesting insights into mathematics, learning, and curriculum design.

Featured Comment

Karim Ani:

I’ll throw ours into the ring: In which MLB park is it hardest to hit a home run?

If you tell me you’re a fan of real-world math, I know almost nothing about what goes on in your classroom. That’s because there’s enormous variation within real-world tasks. Almost as much as there is between real and “fake” tasks. (I’ve written about this before.)

This summer we’ll interrogate that thesis. Every week I’ll post three versions of the same real-world task.

Please tell me: who wore it best?

• In what ways are they different?
• What do their differences say about their authors’ beliefs about students, learning, and math?
• Would you make changes? Which and why?

We’ll begin with Barbie Bungee, a lesson which is as old as math teaching itself. (The earliest reference I found in an exhaustive #lazyweb search on Twitter was this 1993 Mathematics Teacher article. Thanks, Norma.) If you’ve never heard of it, here is a video summary from Teacher Channel.

Click each image for a larger image. Or click through for the PDF.

Version #1

Version #2

Version #3

I previewed these problems on Twitter a week ago.

A number of people noticed that Version 3 asks for a lot of literacy in addition to numeracy. “Example 3 is too wordy for me and the students that I work most closely with,” said Bridget Dunbar.

I’m sympathetic. I was initially repelled by the dense text, but several educators I respect came along and noted that Version 3 leaves a lot of room for the teacher to develop the question along with students. Andrew Morrison said that “the structure of the activity is a lot more open ended than I expected based on the amount of text I initially saw.” Paul Jorgens used some of my favorite advice to support Version 3: “You can’t subtract but you can always add,” continuing to say that “the third one seems the easiest to start thin and add as necessary.”

“A thin start.” Great description.

A number of participants in the discussion said variations on, “It depends on the student.” That seems hard to falsify. Even if it’s true with these three worksheets, though, I don’t think that advice extends to any version of this task. Some are probably just bad.

For my part, I look at each version and try to imagine the verbs, the mental work students do. In each version of the task, the work becomes formal and operational very quickly. Version 1 has students measuring precisely in its first step. Version 2 has students graphing precisely in its first step. That’s important work but once the task has been formalized like that, it’s very difficult to ask students to do informal, imprecise work, which is just as important and often more interesting.

Like wondering what kinds of questions a bungee jump operator would wonder.

Like estimating how many rubber bands would be ideal for a given bungee jumping scenario. (Bridget Dunbar with the eagle eyes: “Version 2 misses estimation while Version 1 asks for it, but at the end of all of the directions.”)

Like abstracting the world of bungee jumping into a few manageable pieces of data which we can measure and track. (eg. The temperature outside probably doesn’t matter. The number of rubber bands probably does.)

Like sketching the relationship between rubber bands and fall height before graphing it.

It’s difficult to load all of those tasks onto the same piece or pieces of paper. Perhaps impossible, as later tasks will provide the answer to earlier tasks. My ideal Barbie Bungee task (and modeling task, in general) requires a dialog between teacher and students, with the teacher adding context, questions, and help, as the situation and students require it.

Watch Twitter for next week’s preview. You should find three versions of a task and play along at home.

Featured Comments

Chris Hunter:

I had a similar reaction to Version #3: all that text. Marc often asks “Who’s doing the math?” This, like “You can always add. You can’t subtract,” rattles around my head when designing or evaluating tasks. Version #3, for all that text, probably wore it best; in both Versions 1&2, the answer to Marc’s question is “The author.”

Elizabeth Raskin:

I wonder if the “step by step” worksheets that educators can be so fond of stems from the fear that students wouldn’t know what to do to solve the problem or from the idea that teachers have a vision for what the students’ output should look like. In versions 1 and 2, students will most likely have much cleaner products than those of version 3. I assume it’s some combination of both.

Michael Paul Goldenberg:

I wouldn’t say that it’s so much the case that “it depends on the student” as that it depends on the classroom culture. Teachers that have cultivated a culture of risk-taking, serious inquiry, and other habits of mind and practices that draw on the natural curiosity and need to know and understand that we all have (before it’s schooled out of us) are going to have enormous flexibility in designing or adapting tasks like this one so as to not wind up stifling individual thinking and productive struggle.