- Invite students to try a task that is intuitive, but inefficient or inaccurate.
- Help them understand some math.
- Invite them to
*re-try*the task and see that with math it’s more efficient and accurate.

That’s an instructional design pattern meant to help students see that the math they learn is power rather than punishment. Most instructional resources do a great job at #2, which they decorate with images of *other people* using that math in their lives. *Some* resources invite students to use the math themselves in #3. But without experiencing #1 the advantage of math may be unclear. “Why do I need to learn this stuff?” they may ask. “I could have done this by guesswork just as easily.”

We should show them the limits of guesswork.

Last week’s installment of Who Wore It Best looked at three textbooks each trying to exploit billiards as a context in geometry. None of the textbooks applied all three steps. I needed a resource that didn’t exist and I spent two days building it. Here is how it works.

**Inefficient & Inaccurate**

Play this video. Maybe twice.

Ask students to write down their estimates for all eight shots on this handout.

For instance:

@ddmeyer CBCAABAB Great discourse between hubby and I. I want a paper version so I can sketch and protractor. Probably all wrong.

— Adrianne Burns (@a_schindy) July 19, 2016

**Some Math**

Several of the textbooks simply *assert* the principle that the incoming angle of the pool ball is congruent to the outgoing angle. Based on Schwartz & Martin’s work on contrasting cases, I’ll offer students this page as preparation for future instruction.

What do you notice about the reals that isn’t true about the fakes?

**Efficient & Accurate**

Now that they have an introduction to the principle that the incoming angle and the outgoing angle are congruent, ask them to apply it, now with analysis instead of intuition. Have them record those calculations next to their estimates.

Then show them the answer video.

Have the students tally up the difference between their correct calculations and their correct estimates. If that isn’t a positive number, we’re in trouble, and essentially forced to admit that the math we asked them to learn isn’t actually powerful.

I’ll wager your class average is positive, though, and on the last three shots, which bank off of multiple cushions, *very* positive.

Because math is power, not punishment.

[Download the goods.]

**2016 Jul 26**. I have changed a pretty significant aspect of the problem setup after receiving feedback from Scott Farrar and Riley Eynon-Lynch. Thanks, team.

**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 m*mathematically* 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.

]]>Here are eight different yet interacting moving parts that I believe has to go into any reform aimed at creating a high-achieving school using technology to prepare children and youth to enter a career or complete college (or both).

Notably, none of them are explicitly about technology.

]]>“These are the dimensions of the rectangle that has the largest area given a fixed perimeter.”

“WHAT IS A SQUARE!” I yell out while my competitors are still thinking quietly. I have disqualified myself and ruined the round, but I don’t care. I start high-kicking around the set while security tries to wrangle me away and I *still* don’t care because I finally found some use for this fact that takes up a *significant chunk* of my brain’s random access memory.

It’s a question you’ll find in every quadratics unit, every textbook, everywhere. I could have selected this week’s Who Wore It Best contestants from any print textbook, but instead I’d like to compare *digital* curricula. I have included links and attachments below to versions of the same task from GeoGebra, Desmos, and Texas Instruments, three thoughtful companies all doing interesting work in math edtech. (Disclosure: I work for Desmos, but don’t let that fact sweeten your remarks about the Desmos version or sour your remarks about the others. Just be thoughtful.)

So: who wore it best?

Click each image for the full version.

**Version #1 – GeoGebra**

**Version #2 – Desmos**

**Version #3 – Texas Instruments**

Steve Phelps suspects I stacked the deck in favor of Desmos here, taking full advantage of our platform while taking only *partial* advantage of GeoGebra and the Nspire. John Golden concurs, hypothesizing that “there would be a worksheet to go with the GeoGebra sketch.”

So a note on sampling: the GeoGebra example is the most viewed lesson on the subject I could find at their Materials site. The Texas Instruments lesson is the *only* lesson on the subject I could find at their Activities site. I told Steve, and I’ll tell you, that if anybody can come up with a better lesson on either platform. I’ll be happy to feature it. This isn’t much fun for me (or useful to Desmos) if I stack the deck.

Both Lisa Bejarano and John Golden call out the Desmos lesson as “too helpful” – they know how to make it sting – in the transition from screen 5 (“Collecting data!”) to screen 6 (“Here! We’ll represent the data as a graph for you.”).

I’l grant that it seems *abrupt*. I don’t think this kind of help is *necessarily* counterproductive, but it doesn’t seem as though we’ve developed the question well enough that the answer – “graph the data!” – is sensible. The Texas Instruments version has a solution to that problem I’ll attend to in a moment.

My concern with the GeoGebra applet is that the person who made the applet has done the most interesting mathematical thinking. I *love* creating Geogebra applets. I generally *don’t* have a good story for what students *do* with those applets, though. In this example, I suspect the student will drag the slider backwards and forwards, watching for when the numbers go from small to big and then small again, and then notice that the rectangle at that point is a square. The person who made the applet did much more interesting work.

Let me close with one item I prefer about the Desmos treatment and one item I prefer about the Texas Instruments treatment.

First, my understanding of Lisa Kasmer’s research into estimation and Paul Silvia’s research into interest led me to create this screen where I ask students “Which of these three fields has the biggest perimeter?” knowing full well they all have the *same* perimeter:

Still later, I ask students to estimate a rectangle they think will have the *greatest* area. That kind of informal cognitive work is largely absent from the TI version, which starts much more formally by comparison.

TI does have a technological advantage when they allow students to sample lots of rectangles and quickly capture data about those rectangles in a table.

Desmos is working on its own solution there, but for now, we punt and include prefabricated data, which I think both companies would agree is less interesting, less useful, and more abrupt, as I mentioned above.

That’s my analysis of these three computer-based approaches to the same problem. What’s your analysis? And it’s also worth asking, “Would a *non-*computer-based approach be even better?” Is the technology just getting in the way of student learning?

You can also pitch your thoughts in on next week’s installment: Pool Table Math.

**2016 Jul 8**. Steve Phelps has created a different GeoGebra applet, as has Scott Farrar.

**2016 Jul 9**. Harry O’Malley uploads another GeoGebra interpretation, one that strikes a very interesting balance between print and digital media.

- 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**

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

In the report, “Equations and Inequalities: Making Mathematics Accessible to All,” published on June 20, 2016, researchers looked at math instruction in 64 countries and regions around the world, and found that the difference between the math scores of 15-year-old students who were the most exposed to pure math tasks and those who were least exposed was the equivalent of almost two years of education.

The people you’d imagine would crow about these findings are, indeed, crowing about them. If I were the sort of person inclined to ignore differences between correlation and causation, I might take from this study that “applied math is bad for children.” A less partisan reading would notice that OECD didn’t attempt to control the pure math group for *exposure to applied math*. We’d expect students who have had exposure to *both* to have a better shot at transferring their skills to new problems on PISA. Students who have only learned skills in one concrete context often don’t recognize when new concrete contexts ask for those exact same skills.

If you wanted to conclude that “applied math is bad for children” you’d need a study where participants were assigned to groups where they *only* received those kinds of instruction. That isn’t the study we have.

The OECD’s own interpretations are much more modest and will surprise very few onlookers:

- “This suggests that simply including some references to the real-world in mathematics instruction does not automatically transform a routine task into a good problem” (p. 14).
- “Grounding mathematics using concrete contexts can thus potentially limit its applicability to similar situations in which just the surface details are changed, particularly for low-performers” (p. 58).

**BTW**. I was asked about the report on Twitter, probably because I’m seen as someone who is super enthusiastic about applied math. I *am* that, but I’m also super enthusiastic about *pure* math, and I responded that I don’t tend to find categories like “pure” and “applied” math all that helpful. I try to wonder instead, what kind of cognitive and social work are students *doing* in those contexts?

**BTW**. Awhile back I wrote that, “At a time when everybody seems to have an opinion or a comment [about mathematics education], it’s really hard for me to locate NCTM’s opinion or comment.” So credit where it’s due: it was nice to see NCTM Past President Diane Briars pop up in the article for an extended response.

**Featured Comment**:

]]>What is often overlooked in these kind of studies is the students who are enrolled in the various courses. The correlation between pure math courses and higher level math exists because higher achieving students are placed in the pure math classes, while lower performing students are placed in applied math.

Same thing is true for studies that claim that students who take calculus are the most likely to succeed in college. No duh! That is because those who are most likely to succeed in college take calculus.

The course work does not cause the discrepancy, the discrepancy determines the course work.

In spite of following 150 creative math teachers on Twitter and subscribing to 750 creative math teacher blogs (including one blog that’s dedicated *exclusively* to creative math), I’m only *now* learning about Gordon Hamilton’s Unsolved K-12 Project. It’s creative. It’s math. It’s almost *three years old*!

Better late than never.

See, Hamilton convened a bunch of creative math types in Banff in November 2013 to a) select unsolved math problems and b) adapt them for use at every grade in K-12. Not a simple task, and I’m enormously impressed by their results. You can watch videos introducing the problems at this page or read about them in these slides.

Here are two of my favorites. (Click for larger.)

**Grade 3: Graceful Tree Conjecture**

**Grade 10: Imbedded Square**

These two problems have the capacity to develop fluency just as well as any worksheet or worksite. In working out their solutions, students will perform the same operation *dozens* of times – subtracting whole numbers in the third grade task and calculating slope and distance in the Cartesian plane in the tenth grade task. But these problems ask students to think strategically and systematically *in addition to* practicing efficiently and accurately. That’s no easy feat, but Hamilton and his team pulled it off thirteen times in a row.

**Related**:

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.

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**

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.”

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.

]]>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.

Ann Shannon asks teachers to avoid “GPS-ing” their students:

When I talk about GPSing students in a mathematics class I am describing our tendency to tell students—step-by-step—how to arrive at the answer to a mathematics problem, just as a GPS device in a car tells us – step-by-step – how to arrive at some destination.

Shannon writes that when she used her GPS, “I usually arrived at my destination having learned little about my journey and with no overview of my entire route.”

True to the contested nature of education, we will now turn to someone who advocates exactly the opposite. Greg Ashman recommends novices learn new ideas and skills through explicit instruction, one facet of which is step-by-step worked examples. Ashman took up the GPS metaphor recently. He used his satellite navigation system in new environs and found himself able to re-create his route later without difficulty.

What can we do here? Shannon argues from intuition. Ashman’s study lacks a certain rigor. Luckily, researchers have actually studied what people learn and don’t learn when they use their GPS!

In a 2006 study, researchers compared two kinds of navigation. One set of participants used traditional, step-by-step GPS navigation to travel between two points in a zoo. Another group had to construct their route between those points using a map and then travel segments of that route from memory.

Afterwards, the researchers assessed the route knowledge and survey knowledge of their participants. Route knowledge helps people navigate between landmarks directly. Survey knowledge helps people understand spatial relationships *between* those landmarks and plan new routes. At the end of the study, the researchers found that map users had better survey knowledge than GPS users, which you might have expected, but map users outperformed the GPS users on measures of *route* knowledge as well.

So your GPS does an excellent job transporting you efficiently from one point to another, but a poor job helping you acquire the survey knowledge to understand the terrain and adapt to changes.

Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions.

I’ll take that trade with my GPS, especially on a dull route that I travel infrequently, but that isn’t a good trade in the classroom.

The researchers explain their results from the perspective of active learning, arguing that travelers need to do something *effortful* and *difficult* while they learn in order to remember both route and survey knowledge. Designing learning for the right kind of effort and difficulty is one of the most interesting tasks in curriculum design. Too *much* effort and difficulty and you’ll see our travelers try to navigate a route without a GPS *or* a map. While blindfolded. But the GPS offers too *little* difficulty, with negative consequences for drivers and even worse ones for students.

**2016 Jun 17**. The two most common critiques of this post have been, one, that I have undervalued step-by-step instructions in math, and two, that this GPS study offers very few insights into math education. I respond to both critiques in this comment.

A poker face? A bit of malice? Nitsa Movshovits-Hadar argues [pdf] that it requires only the ability to trick yourself into forgetting that you know every triangle has the *same* interior angle sum. “Suppose we do not know it,” she writes, which is easier said than done.

The premise of her article is that “… all school theorems, except possibly a very small number of them, possess a built-in surprise, and that by exploiting this surprise potential their learning can become an exciting experience of intellectual enterprise to the students.”

This is such a delightful paper – extremely readable and eminently practical. Without knowing me, Movshovits-Hadar took several lessons that I love, but which seemed to me totally disparate, and showed me how they connect, and how to replicate them. I’m pretty sure I was grinning like an idiot the whole way through this piece.

[via Danny Brown]

**Featured Tweets**

@rawrdimus i.e. think less like a math teacher who knows how to write a circle equation

— Dan Anderson (@dandersod) June 10, 2016

Not easy for math teachers to do!

@ddmeyer I did a similar thing with my Year 9 students and the trig ratios!!! Heaps of fun and surprise!

— David Ross Lang (@Davidinho_78) June 12, 2016

What if you asked two questions: which triangle has the longest perimeter and which triangle has the largest angle sum? It might clarify what can change in a triangle and what cannot. Also it hides the surprise better. If you teach via surprise consistently, kids start looking for the punchline.

**Featured Comments**

Jo:

Elementary may actually have an advantage here! We play these games all the time. Some favorites:

Draw me a two-sided quadrilateral

Draw me a triangle with three right angles (or three obtuse angles)

(We have a manipulative that consist of little plastic sticks that snap together to build things)–Build me a triangle with the red stick (6″), the purple stick (1″) and the green stick (2″ )Once the whole class is convinced they can’t you can get at why and then writing a rule for it. There is nothing an 8 year old likes better than proving the teacher wrong.

]]>Theorems and formulae in textbooks should be marked with a “spoiler alert”.