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. :)

The concept of betweenness

I am coming to believe that, much like the concept of substitution, developing a deep understanding of the idea of betweenness is a huge part of the psychological and conceptual work of Algebra 1.

I have been dissatisfied for years now with the fact that we tell students about the geometric interpretation of absolute value, but we don't really get them to live it. And yet that idea of the "distance from zero" on the real number line is not something we give students time to really marinate in.

And you know what? That feels dumb to me.

So this week and part of the next, my students and I are really wallowing in that idea.

I've taken a page from my studies as a young piano student. In the study of the piano, there are certain studies of technique that really force you to slow down and take apart the finger movements. There are specific figures that you have to practice over and over and over so that they become part of your finger memory. How they feel in your fingers is how you come to relate to them.

This is not just about developing automaticity, although that is a side benefit. This is about learning to feel these foundational figures in your bones. In your body. They become so fundamental that as you learn and grow as a musician, you come to feel them when you see them coming up in a new score you are studying.

The technique does not replace musicianship. The technique supports the musicianship.

I've been noticing lately how my own experience of absolute value is about noticing boundary points at the periphery of my mathematical perception. I see them out of the corner of my mathematical mind's eye. And how an inequality is said to relate to them defines how I relate to those boundary points.

So I am taking the risk of sharing this mathematical experience with my students.

As with young piano students, we take this slowly. One figure at a time. Right now we are only dealing with the case of an absolute value being less than a nonnegative quantity. We are dealing with situations of betweenness, where an inequality presents us with a figural situation that is going to wind up with a quantity being between two boundary points.

That is all. And that is enough.

I see the effort in their faces and in their fingers as they rewrite, revise, calculate, solve, and sketch graphs. I see them noticing and wondering whether they need to use a closed dot or and open dot.

And I hear them developing the confidence that comes from experience in developing a relationship with these quantities.

They are not following rules. They are listening to their own deeper wisdom. Everybody knows something about the situation of being "between" other things. Betweenness is one of the most elemental human ideas.

They are making friends with mathematics.

The deeper wisdom of the body in math class

This post is for Malke.

As I was eavesdropping on a recent conversation between Lani Horn (@tchmathculture) and Malke Rosenfeld (@mathinyourfeet), I received a pointer to an article summarizing recent research that shows that kids with ADHD actually need to squirm in order to learn.

This makes sense to me. The deeper wisdom of the body is usually overlooked in thinking about teaching, learning, and assessment in mathematics. And yet, it can provide a vital link for our students in claiming their mathematics as well as their humanity.

I was thinking about this on Friday afternoon when I tweeted out the following:

I love the dulcet tones of compasses, rulers, & pencils during a Friday afternoon constructions quiz. #geomchat

Malke tweeted back:

I love that they are doing all that by hand. And that there are dulcet tones. :)

I responded:

Geo has affirmed my belief in the life made by hand. Huge benefits to Ss [students] from physically constructing their understanding.

I had another in a series of lightbulb moments this past month about what How People Learn says about externalizing our understanding.

And ever the good online learning partner, Malke tweeted back:

have you blogged about it?

So here I am.

In How People Learn, the authors talk about how we can use skits, presentations, and posters in group work to help students externalize their emerging understanding. This makes sense to me. In order to learn something, one first has to notice it, and that means developing a metacognitive self-awareness of the process and how it’s going.

Over the last few years, I have found that teaching students to use foldables, INBs (Interactive Notebooks), guided note-taking, and physical constructions is another extremely rich field of helping students to externalize their emerging understanding — only in these cases, they are externalizing their understanding through physical, kinesthetic processes — not just through talk, listening, and presentation processes.

The physical dimension is a good grounding for conceptual understanding. Teaching students how to literally use their tools can be a multidimensional process of making their learning both physical and tangible. Flipping open a flap or a page in a composition book is a physical manifestation of the process of retrieval or comparison or evaluation. Likewise, the process of using patty paper as a tracing medium to externalize the concept of superposition and projection of a figure to confirm congruence is a way of helping students to slow down their speeding monkey minds and to become present with the mathematics that are right in front of them.

When I tuned in to the clatter of compasses, rulers, and pencils on Friday, I really noticed how deeply engaged my students were with the geometry they were working on. Their body postures indicated how deeply immersed they were in the experience of flow: set your compass opening to an appropriate width and draw an arc across the angle you want to copy. Stab the endpoint of the segment where you want to create a new copy of your original angle and swipe the same arc there. Go back to the first angle. Refine your compass opening so that it now matches the width between the intersections of your arc and the original angle. Shift your paper, stab at the lower intersection of your copied angle-in-process and swipe an arc that will intersect with the arc you just drew there. Drop your compass; pick up your straight edge. Carefully draw a line to connect the endpoint of your target segment with the intersection of the arcs you have drawn, completing the terminal side/ ray you need to draw. Drop the straight edge; position the patty paper over your original angle and use your straight edge to trace it. Drop the straight edge and carefully slide your traced angle over your constructed angle. Does it match your figure perfectly?

The concentration etched on their brows matched the precision of their work on the page in front of them. Bisect an angle. Construct the perpendicular bisector of a segment. Construct a parallel line through an external point by using it to define an angle that you can copy.

Hopping from one stone to the next, you can cross an entire river. By placing one foot on the Earth after another in a pattern of glide reflections, you can complete a journey of a thousand miles or more.

That is one of the lessons of deductive and spatial reasoning at the heart of any good Geometry course. Noticing that it is happening in my classroom — really happening through physical, mental, and whole-hearted engagement — is one of the greatest blessings of being a teacher.