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Here is a very valuable conjecture:

The spelling of every whole number shares at least one letter with the spelling of the next whole number.

Which is to say that:

  • “one” and “two” both share an “o”
  • “two” and “three” both share a “t”
  • etc.

Could that possibly be true for every whole number?

If I were starting a course on geometry or a unit on proof or an activity on deductive logic, I would introduce this conjecture very early in the process. Let me explain what I find so very valuable about this conjecture.

Deduction is hard. It’s an abstract mental act that adults find difficult. (See: the van Hiele’s and their levels.) Too often we rush students to that abstract act, rushing them past the lower van Hiele levels, and we ask them to argue deductively about objects that, to them, are also abstract.

I suspect that, to many students, those proof prompts read something like this:

Given that the base bangles are twice the tonnage of the circumwhoozle and the diagonalized matrox is invertible, prove that all altimeters cross the equation at Quito, Ecuador.

The word “prove” is weird. And, unfortunately, so is every other word in the sentence.

So I cherish opportunities to help students argue deductively with concrete objects, which is what we’re working with here, with the spelling of whole numbers. This conjecture also gives students several different angles on the proof act.

You can ask students to find a counterexample, for example, a useful strategy when first interrogating a conjecture.

Once students have tried several different numbers they may satisfy themselves that the conjecture is true. This is one of the naive proof schemes Harel & Sowder observed in the students they studied. When this proof scheme surfaces in conjectures about geometric shapes, it’s challenging to summon up one new shape after another to challenge the student’s proof by example. It’s trivial, by comparison, to summon up one new number after another and ask the student to check her hypothesis again.

At a certain point in this process, likely after you give several numbers in the millions, your students may transform in two ways:

  1. They’ll get tired of trying example after example. “Proof by examples means you have to try all the examples,” you can say, giving you both a moment to reflect on the need for a more rigorous proof scheme, like deductive reasoning.
  2. They’ll notice that every number in the millions shares an “n” with every other number in the millions. And same for the billions. And same for the trillions. And … same for the hundreds. And so on.

And suddenly we’re on our way to a proof by exhaustion, which is much more rigorous than a proof by example. Nice.

This conjecture also leaves ample room for you and your students to pose follow-up conjectures. Like, “Does it work for all integers, or just whole numbers?”

I saw the conjecture and saw its value immediately. This is a very valuable kind of conjecture, I thought. But I don’t have many of them. Do you have another you can trade?

[via Futility Closet.]

BTW. You’re worse off in at least one way now than before you knew the conjecture was true. Now, when you ask your students, “Could that possibly be true?” you’re going to have to pretend.

Featured Comments:

Max:

What I like about game strategies is you go from “what seems to work,” to “will this always work,” to “here’s why this does/doesn’t work” pretty seamlessly.

Paul Hartzer:

Is it true for any other language with an alphabet? It fails for German (5/6 using umlauts, 7/8 otherwise), French (2/3), and Spanish (7/8).

Michael Serra:

What letter(s) of the alphabet do(es) not appear in the spelling of the first 999 whole numbers? Prove it.

Lynn CP:

List all the factors of every number from 1-100. What do you notice? Which numbers have an even number of factors? An odd number? What do you notice about numbers with an odd number of factors? Can you prove which numbers beyond 100 will have an odd number of factors?

[3ACTS] Pool Bounce

There are three steps:

  1. Invite students to try a task that is intuitive, but inefficient or inaccurate.
  2. Help them understand some math.
  3. 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.

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For instance:

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.

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

2016 Jul 31. Edited to add this literature review, which elaborates the positive effect of contrasting cases (and building explanations on student solutions) in more detail.

2016 Jul 31. Also, in the spirit of “you can always add, you can’t subtract,” I’m sure that before I showed all four contrasting cases and the labels “real” and “fake,” I’d show the individual cases without those labels. Students can make predictions without the labels.

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.

2016 Jul 26. I’ll be changing the name of this activity shortly, on request from a Chicago educator who thinks his students will read violence into the title. That makes sense to me.

2016 Jul 28. Changed to “Pool Bounce.” I am amazing at titles.

Featured Comment

Julie Wright:

I love this partly because the fake ones look fake, and students have to think about why and are given materials to test their hypothesis. You’re making students refine their intuition to include mathematical precision, which they can then use to solve the rest. I feel like this honors and builds on the knowledge they already have in a way that’s far more motivational than throwing out some big-words statement about angles of incidence and reflection.

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.

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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.)

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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.

Larry Cuban has spent the last year observing and documenting the practices of schools that are known for successful technology implementation.

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.

I have a recurring happy dream that I’m on Jeopardy. It’s the final round. The Trebekbot 2000 reads the final clue:

“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

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Version #2 – Desmos

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Version #3 – Texas Instruments

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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.”).

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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:

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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.

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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.

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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.
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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.

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