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.