It always bugged me the way formulas are often written in ways that hide the relationships between the math. like using lxw, s^2 and 1/2bh – calling it a length, width, side, base and height is so strange to me and doesn’t really make sense from a teaching and learning perspective.

Having students discover pi by measuring circumference & diameter of different sized circles (quarter, orange, record, frisbee, wall clock, can of beans, tupperware lid, etc) was huge step for me learning to teach. It was the 1st time I was able to successfully transfer my own newly learned sense of wonder in math to my students. I had learned C=(Pi)d but who could remember when you used that formula and when you used Pi(r)^2? Not me, until I understood, exactly as you say, that pi is a relationship, not just a weird number. It conveys all kinds of things to students about who they are and who they can be in math.

Thanks for this thoughtful post!