We left Portland this morning and headed south. After a brief stop at Silver Falls State Park so we could hike behin a waterfall, we headed on to Bend for a couple nights. In case you’ve ever wandered what the back side of a waterfall looks like, here’s your answer:

This was our first significant driving opportunity in a few days and so my first time to ponder some things. One of the books my department is reading this summer is *The Falconer* by Grant Lichtman. I’d never read it but suggested it based on the recommendation of those I respect and on how I thought it might speak to some of the things we’ll be thinking about in our curriculum rewrite. I’m three or four chapters into the book and am really enjoying it. Something I read recently has been bothering me, and I want to try to play out an idea here.

In the fifth chapter, “Step 1: The Art of Questioning,” Lichtman describes a fictional classroom in which a teacher challenges his students to write the questions for an upcoming test on evaporation. The students pose a number of questions, all of which have factual answers that can be found in a textbook. The teacher points this out and challenges the students to ask questions that might not be so easy to answer, suggesting they consider questions that start with, “What if…?” A student comes up with the question, “What if water didn’t evaporate?” which inspires other students to ask similarly-challenging questions. In the story, the teacher points out that these questions are great because (a) there’s more than one way to answer them, (b) answering them requires a deep understanding of the topic, and (c) they naturally lead to other questions and lines of inquiry.

This idea of asking “What if?” isn’t new to me, but I like the idea of using it to frame assessments. In fact, I think that using “What if?” is a good way to develop essential questions to frame a topic or investigation. The problem I’m having is how to pose the right kind of “What if?” questions in a math class.

I can pose a “What if?” question in physics and we can develop experiments to test our hypothesized answers to the question. While it’s not my field of expertise, I would imagine that a “What if?” question in a history class can be played with by looking at other things that were happening and how they might change. With math, though, this doesn’t seem quite as easy.

At one end of the spectrum of questions is something like “What if we make the coefficients of these equations negative?” which is really just a factual question disguides as a “What if?” question. At the other end of the spectrum is something like, “What if a plane isn’t flat but is the surface a sphere?” which leads us to spherical geometry, a fascinating subject that’s incredibly complex. What I think we need in our integrated math curriculum is “What if?” questions that sit in between these two ends of the spectrum so that they’re interesting and challenging but can still be explored deeply.

What I’m wondering is, can we develop “What if?” questions that hit the right level but are theoretical? I can come up with good “What if?” questions that involve applications of math; in fact, I’ve come across a couple great examples recently. A couple months ago a video was posted on Numberphile th at asked the question, “What if baseball was played in a hyperbolic geometry?” Part of the answer is that you’d need about the same number of infielders but 10^94 outfielders (you can click here to see the full video). More recently, mathematician Alex Bellos asked the question, “What if pool was played on an elliptical table instad of a rectangular one?” You can see the answer to this question here.

These are great questions that ask students to understand something deeply, but they’re also questions that challenge us to apply the math in a physical context. Are there similar questions that exist entirely in the theoretical realm? It turns out that this question is actually less interesting than others that it implies, which I think would make Lichtman happy, at least. I’ll try to address them in my next post.