Few of us succeed in truly bringing a beginner’s mind to our experience of mathematics. By the time we reach middle school, it is already impossible. We have experiences, beliefs, and opinions about what it takes to do math that have been formed by other people’s experiences, beliefs, and opinions. Students show up in my classes believing — as early as age eleven — that they are good at math or bad at math, that they love math or hate it, prefer decimals over fractions, use a calculator as a defense mechanism and a talisman, and even those who love books believe that word problems are evil. These attitudes get locked in early, and they continue to shape our experiences with math for the rest of our lives.
When I was in first grade, Mrs. Williams gave us each a little muslin drawstring bag of colored wooden blocks and rods. There were little ivory cubes and stubby red rods that were about the same scale as the houses and hotels in a Monopoly set. The apple green rods were just a little bit longer. One, two, three. Each rod got one unit longer, but not wider, than its predecessor. All the way from one to ten. We each kept our little bag in our desk all year long.
I did not know this then, but these are called manipulatives. Mathematical manipulatives. Cuisenaire rods, more specifically. They were invented in the 1920s by Georges Cuisinaire, an imaginative Belgian teacher, and were popularized in the 1950s by the great mathematical educator Caleb Gattegno. As a teacher and as a learner, I have always had a crush on Gattegno. His math pedagogy mirrors the inner development teachings of my root teacher, Fred Orr. Gattegno said, “Only awareness is educable.” Fred said, “Noticing shifts the energy.” When I first encountered those Cuisenaire rods, I could not have known that these two currents of awareness would flow together into a stream that would subtend every aspect of my life.
Those little rods became my friends. I loved the way they felt between my fingers. I loved the smell of the muslin bag and the way they clicked together inside the bag when I took them out during arithmetic lessons or tucked them back into into the darkness of my desk’s inner compartment.
I loved the sound they made when they clattered onto the desk in preparation for our daily lessons. We used them for counting, equivalence, and for modeling mathematical operations. I also used them for personal puppet shows. I traced their shapes on paper with a fat yellow pencil, and I colored in my outlines according to their shapes — white, red, green, blue. Those little blocks cast a long, cool shadow in my mathematical memory. They kept me connected to mathematical wonder even when I felt certain I had no clue about anything mathematical.
Years later, I realized you could buy them on Amazon, and I bought the traditional wooden set. They no longer come with the muslin bag. I cleared off the old oak dining table so I could hear that long-ago sound of the blocks on my desk. It was still exactly the same. I picked up one block or rod at a time and turned it between my fingers. I inspected them. I smelled them. I lined them up: white, red, green. It was one of my madeleine moments.
Before my parents moved away from that old town, I borrowed my mother’s Jeep and drove back to my old school. It was exactly as I had remembered it, only a little bit uglier and smaller. It was after school, and the door was open. I followed the corridor around, past my kindergarten classroom at the first corner that had long ago become the school library. I walked down the linoleum-floored, painted cinder block corridor to the room that had once been Mrs. Williams’ classroom and my own. It was unchanged, except that the blond spinet piano was no longer in the corner. That spot now housed a small cluster of computers.
I wanted to lift the lid of one of the desks to see if there was a drawstring bag of Cuisenaire rods inside.
The rods make me think about beginner’s mind in mathematics. Our minds and emotions get so tangled up and cluttered, it’s not easy to approach math without preconceptions. My first Zen teacher was Keido Les Kaye of Kannon Do. He was the thirteenth monk ordained in America by Shunryu Suzuki, known by most American dharma practitioners as Suzuki Roshi. Suzuki Roshi said, “In the beginner’s mind, there are many possibilities, while in the expert’s mind, there are few.”
This is as true in school mathematics as it is in the study of mindfulness.
As learners, we quickly become experts at either embracing or avoiding math. Our defense mechanisms against shame, humiliation, and confusion become fierce. They harden our hearts. Those of us who suffer from trauma and anxiety around math come by it honestly. In response to years of competition and sorting and embarrassment, our unconscious minds develop powerful and uniquely personal sets of defense mechanisms. These arise as protective functions in the Self, emerging to shield us against intolerable feelings of panic and confusion and shame.
But there is a different kind of relationship with mathematics that is possible, and it is not only possible for anyone to enter it — it is also essential. It begins with setting aside what we think we know about math, and allowing ourselves to experience it through beginner’s mind — through a mind that is open and self-aware and unselfconscious. It is about setting aside our adult preconceptions for a few moments and allowing ourselves to sink down into the flow of curiosity — the kind of curiosity I remember feeling when I sat on the curb and watched a millipede shuffling along in the ninety-degree New Jersey heat and in the ninety-degree angle between the curb and the gutter, its black legs fluttering along like little windmills, flipping over and over in an articulated lumber, like a deck of cards being shuffled before the magic begins.