## [Fake World] Conjectures

"Real" isn't a guarantee.

When I work with publishers, I find authors hesitant to tinker with the mechanics of a task and eager instead to spread the "real world" over the task like a thin coat of varnish.

Why are students bored by quartic equations in the problem I excerpted above? We are less eager to examine the structure of the task (which has been passed down and assumed constant for generations) and eager instead to just add stock photography of a snowboarder.

Students aren't so easily fooled.

Similarly, when I find interesting real-world tasks that result in engaged, interested learners, the real world-ness of the task is often its least essential element.

"Real" is relative.

What's "real" to you isn't necessarily "real" to me. Our lived experiences are different.

An example on the extremes: one of my favorite proof tasks is Bucky the Badger. Lots of teachers and students enjoy it. The context and the math are often interesting. But I took it to Australia two years ago and interest was quite a bit muted for reasons I should already have anticipated: they don't do football like we do football.

This obstacle to effective curriculum design isn't insurmountable. Lots of great art has been made about contexts we've never experienced, for instance. But we assume this obstacle doesn't exist at the expense of our students who haven't been privileged with our life experiences.

It's self-limiting to try to draw lines between "real" and "fake."

Take hexagons, health insurance, hydraulic engineering, hydrogen gas, and heptominoes. Is anybody here confident they can tell me which of these is "real" to a sixth grader? Which is "fake"?

This is a self-limiting question. They can all be made real.

We should be very, very hesitant to write off entire worlds as foreign and unknowable, especially the world of pure mathematics, which a student can always conjure up given nothing more than a mind, a pencil, and a piece of paper.

The point of math class.

The point of math class isn't to build a student's capacity to answer questions about the world outside the math classroom. It isn't to prepare her to get a job either. Both of those are happy outcomes of a different goal that's broader and more interesting to me.

The point of math class is to build a student's capacity to puzzle and unpuzzle herself – no matter what form those puzzles take.

Find those puzzles in the real world, the fake world, the job world, or any other world – it doesn't matter.

## Real-World Math That Isn’t Real To Students

Seen on a dessert menu at a fancy restaurant I crashed this weekend:

"Well you guys said you wanted to know when you'd ever use this stuff."

I'll be dedicating this blog to a certain line of inquiry for the next few days or weeks or for however long it takes me to come to some kind of internal consensus. I'd appreciate your help with that.

## Great Classroom Action

Mitzi Hasegawa links up a clever game called Entrapment that helps student understand these reflections, translations, and rotations that (I'm told) now constitute the entirety of K-12 mathematics:

One table debated the transformation highlighted in today’s picture. Is the figure on the grid a reflection of the figure or a rotation?

Federico Chialvo asks his students "How safe is a Tesla?"

Jonathan Claydon poses the deceptively simple challenge to spin a measuring tape handle at exactly one mile per hour:

Most students experience miles per hour in a car. The challenge would imply "I should spin pretty slow." Yet, the handle isn't very long. And there's the curiosity. What does 1 mph look like at this small scale? So they spun.

Mary Bourassa repurposes a classic party game for the sake of learning features of quadratic equations:

They would ask "Is my h value positive?" but then either interpret the answer incorrectly or not be sure whether the person they had asked truly understood what a positive h value meant. They all figured out their equations and had fun doing so. And they want to play again next week when the whole class is there. I'll be happy to oblige.

## Geoff Krall’s Problem-Based Learning Starter Kit

You’ve seen the tasks. You’ve read the research. You’re basically bought in. But how do you begin?

Almost shockingly free of buzzwords, platitudes, soft descriptions, or anything close to moralizing. Bookmark it and send it around.

## Great Palm Springs Action

I'm just now back from CMC South in Palm Springs where attendance was about 1,000 people higher than the organizers expected. My already-pretty-high expectations for California math education conferences were also exceeded several times over. What follows are resources and takeaways from the sessions I attended and from one I didn't.

Robert Kaplinsky

Real-World Problem-Based Learning Using Perplexing Tasks

Robert showed us this image and asked us to figure out how much it cost.

I've seen his lesson plan before but it didn't prepare me for how interesting the math became.

We used the In-N-Out prices for a hamburger, a cheeseburger (a hamburger + cheese), and a Double-Double (a cheeseburger twice over) to figure out the cost of the 100×100 burger.

Our calculation was exactly right but we arrived at it differently. I used a system of three linear equations. My seatmates used a bit more conceptual creativity and got the same answer with a lot less computation. Robert highlighted all of these methods.

My takeaway: it's really, really hard to describe in a text-based lesson plan all the awesome, heady moments it may provoke. I knew the lesson would be fun and productive. I didn't know it would be this fun and productive. How do we create lesson plans that convey all those highs, rather than just the nuts and bolts of their implementation?

Breedeen Murray

Telling Stories, Teaching Math

Bree claimed that stories are a useful medium for student learning. She backed this up with lots of citations that I hope she'll share somewhere. [Update: She has.] She posed ideas for filtering our own lessons through the logic of stories – setting context, adding conflicts, etc. She closed by asking us try to create story-based lessons for exponentials and fraction division.

My thoughts went to St. Matthew Island, which I'll link without elaboration.

Allan Bellman

Manipulatives vs. Technology: Bring Your Bias and an Open Mind

Great premise for a session:

Pose two lesson objectives. For instance:

1. Students will be able to understand why the angles in a triangle always add to 180 degrees.
2. Students will be able to understand how to calculate the shortest distance from a point to a line and then back to another point.

Allan then brought any resource you'd want, from low-tech to high-tech, everything from tracing paper and scissors to a class set of TI-Nspire's. We used what we wanted to explore those objectives and then debated the merits of the analog and digital technologies.

The debate hit a high register pretty quickly. For the 180 degrees question, people tended to favor the diverse ways you could demonstrate it with paper. (Cutting, tracing, drawing, etc.) With the NSpire, you had one. You downloaded an applet from Allan and moved vertices around, watching the angle sum stay constant.

For the shortest path problem, most people preferred the calculator because you were able to set up the constraints and then drag a point around to see where the distance bottomed out. The only analog method we discussed was to draw the scenario and then use string to measure the different possibilities, slowly narrowing in on the answer.

For my part, I was bothered that we never discussed a) any other digital technology, aside from Texas Instruments calculators, or b) the cost of a class set of any kind of technology.

Granted, I probably make sport of Texas Instruments too much (and I'm hardly unbiased here) but truly I just find the user experience miserable. From the untouchable, low-DPI screens, to the time it takes me to find the right button out of the millions, to the mindless, button-pushing worksheets teachers have to pass out just to make the devices comprehensible, to the lurching way the cursor moves across the screen in Cabri, I find the whole experience pretty painful.

It would have been great to see the same problems approached with Geogebra or Desmos, for instance. Or even an Excel spreadsheet.

Then there's the cost. All other things held equal, the high-tech solution will still cost thousands of dollars more per classroom. So we shouldn't be talking about which solution comes out barely ahead of the other. Technology should shoulder the greater burden of proof here.

Michael Serra

Polygon Potpourri

Five interesting investigations with polygons [pdf]. Michael spent ten minutes prefacing the set, then let us investigate them for twenty minutes, and then asked a volunteer to debrief each one at the end.

If nothing else, it was a nice morning moment to talk about math with Internet friends. That was enough. But I've been struck also by how hard it is to make a given math concept more challenging for students and more interesting at the same time. We use bigger numbers. We mix in fractions and decimals. We lengthen the problem set. We time it.

For instance, once students understand how to find the sum of the interior angles of a polygon, it's like, what do you do to make this more challenging and more interesting?

Michael introduced donut polygons:

Finding the interior angle sum of a donut polygon makes the original task more challenging and more interesting at the same time. In particular, it has a great stinger at the end when you find out whether or not a triangle inside of a pentagon has the same angle sum as a pentagon inside of a triangle.

Michael had two questions at the end that asked, basically, "Do your conclusions hold if there's a dent in the polygon?" Then, "What about two dents in the polygon?" This messed me up a little bit, because, no, it shouldn't matter, but then why would Serra include the two questions? Basically, Serra had your correspondent feeling briefly but completely off balance.

Featured Comment

The absolute value was a pleasant surprise. Now I have ammunition when a student says, “When am I ever going to use absolute value?” “Well, when calculating the interior angles of a n-sided star polygon, of course.” ;)

Avery Pickford

Proof Doesn't Begin With Geometry

Avery wins the prize for Best Session Description by sneaking in the totally droll line, "All hail CCSSM MP3."

He briefly dinged two-column proofs, the Disaster Island where we usually sequester conjecture and argument. I'm all for broading and deepening the definition of proof but I think Avery stretched it too far to include skills like estimation and asking us to justify our answers to the Locker Problem. Is "justify your answer" any different than "prove your conjecture"?

Later, he got into territory I found challenging and really well-thought-out. He showed how simple puzzles like Shikaku could be used to introduce the difference between axioms and theorems. He showed a Shikaku puzzle and its answer (below) and asked us, "What are the rules here?"

"The numbers define the area of a rectangle" and "the side lengths of those rectangles are integers" are axioms, without which the game wouldn't make any sense. Theorems are the consequences of the axioms, like "Prime-numbered areas result in long, skinny rectangles with side-length 1."

He also used a variation on the old game Mastermind to create a class-wide context for conjecturing, contradicting, and proving. Great stuff. Wish you were here.

2013 Nov 4. Avery elaborates on all of this in a blog post.

2013 Nov 6. Avery continues his self-recap.

Brent Ferguson

Geometry, Numeracy, & Common Core: A Vigorous Hands-On Task

I didn't actually attend Brent's session, but he explained it to me in the coffee shop and I wished I had. Basically, you give your kids a number line with 0 and 1 marked off. What other numbers can they construct with a compass and a straightedge?

The other integers fall pretty quickly. A lot of irrational numbers fall when students learn the Pythagorean theorem. Trisecting an angle is impossible but trisecting a line is possible using similarity and the properties of parallel lines and the constructions are varied enough to be interesting.

This seemed like a task that could extend through a good stretch of a Geometry course, drawing in lots of conjecture and proof along with some good review of numbers and operations.

The odd thing: I found Brent's personal description totally engrossing. But this is a really tough idea to sell in a 500-character workshop description.

Takeaway: I probably skip over loads of really interesting, but hard-to-describe sessions at conferences.

Mine

I'll be giving this talk a few more times and then posting video here. I cited these resources. People were real nice to stop by even though it was the last session on a very pretty day. Thanks for the hospitality, everybody.