Betweenness and non-betweenness: absolute value inequalities and Patrick Callahan

I felt a little nervous about having Patrick Callahan come to observe my classroom yesterday, but in the end, it was fun. I had asked one of our security guards, to bring him down to my room when he arrived at our school. He walked in as he always does, all mathematical open-mindedness and pedagogical curiosity.

And we got started.

I felt anxious about having him observe my conceptual lessons about betweenness and non-betweenness. I have never seen anything even close to how I understand and talk about absolute value and inequalities. I talk about boundary points and betweenness and I have students hold up their fists and point their thumbs to show me their understanding. “Is this a situation of betweenness — or NON-betweenness?” I demonstrate with my own fists, swinging my thumbs inward or outward. “Your fists are the boundary points and your thumbs are how you shade your graph on the number line. So is this a situation of betweenness... or NON-betweenness?”

If it is a situation of "betweenness," then students point their thumbs inward towards each other, touching the tips together. If it is NON-betweenness, then they point their thumbs outward in either direction, like a group of indecisive hitchhikers. And once we have done this analysis, then we can do whatever calculations we may need to find our boundary points.

So much of advanced algebra and precalculus depends on having this kind of deep conceptual understanding and thinking. Am I looking for quantities that are GREATER than...? or LESS than? Is this quantity going to be positive? or negative?

For me, the whole thing is intimately hooked together with the real number line. And with number sense. 

When we started last week, we began with an inquiry into “more than” and “less than” and widened our thinking outward from there.We connected more than and less than to number line thinking. I always emphasize Number-Line-Order and Number-Line-Thinking in my Algebra 1 classes. If they think about the number line, then they can anchor their thoughts in their bodies. LHS (or Left-Hand Side) and RHS (Right-Hand-Side) are fundamental ways of thinking in algebra. These ideas are eternal and unchanging. The number line is the foundation of everything. It gives you the “true north” of the real number system.

So we always ground our thinking in our bodies. I ask, “Left Hand Side or Right Hand Side?” “Is this a situation of betweenness or non-betweenness?” “OK, now that we know that, now what?”

I also anchor this unit in what they know about logical reasoning. They have an intuitive sense of how many possible cases a situation may present. I've been a huge Yogi Berra philosophy fan all my life, so I believe that when you come to a fork in the road, you should take it. When you come to a fork in the road, you can go left or you can go right. Or you can stay right where you are. Three possible cases. Over and over I ask them, “What’s going on here? How do you know?”

Absolute value inequalities are either situations of betweenness or situations of non-betweenness. Figure that out and then everything else will run smoothly. Then all you have to do is to use what you already know.

Once students have gotten that figured out, it’s just one more small step to combining their new knowledge with their existing knowledge. Follow the order of operations and common sense. Plus everything you know about the real number line and multiple representations. Then things can naturally unfold the right way.

But I always come back to number sense to what we know about the real number line. Numbers are the ground, the foundation.

So when Patrick walked in yesterday — this world-class mathematician and math education expert — what he encountered was my bootcamp in algebraic thinking. “Hold up your fists! Is this a situtation of betweenness or non-betweenness?”  "How do you know?" And then my waiting until everybody’s thumbs are pointing in the same direction.

It is Logic 101 and numbers and anchoring our thoughts about numbers in our bodies. Like the ancient Greeks and Babylonians and Egyptians before us.

Our next step is to solidify our thinking through what How People Learn calls “deliberate practice with metacognitive awareness.” We are going to do two days of Speed Dating. Now I have to make up Speed Dating cards and a test to use on Thursday. 

And then to document my thinking.

When the class ended, Patrick came up to my tech podium and was excited. He grabbled a whiteboard marker and started sketching and pouring out ideas.

For me, that was the best possible review I could have gotten on this lesson. A five-unicorn review. A direct hit. :)