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.