I do most things slowly and always have, starting all the way back with my birth – I was due on my mom's birthday, which also happened to be April Fools' Day, so of course I delayed for a week. Naturally then, in the first grade, attending a conservative, parochial school in Ann Arbor, Michigan, I was among those the teacher sent out into the hallway. We were to finish our worksheets while the other kids, having finished by working at an authorized pace, moved on to something else.

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We were a small group, following orders, and clutching our sheets containing addition and subtraction problems: 3 + 8 = ?** **or 9 – 4 **= ?**. I don't recall that the problems had anything in common apart from their tedium and lack of connection to our surroundings, which was a typical fear-inducing school hallway (they often made me afraid, at least). But I can confidently say this much: we were not, working through the problems, *doing math*, despite the way that common expression is applied to taking sums and differences. We were doing *abstract calculations*, an exercise not in understanding and subsequent skill-building, but in disenchantment. We plodded along, lying around on the floor, sharing more yawns than ideas. And I continued to go slowly: nothing in the shame of being set apart and excluded prodded me to go faster.

Why was I doing arithmetic problems at the pace of a ground sloth? Note something important here: I didn’t ask why I was *learning *arithmetic slowly, but *doing problems* slowly.

I suppose you can usually speed up an assembly line. Speeding up learning is another matter.

In second grade, I moved on to a different school in an entirely different midwestern state. Early in my third grade year, perhaps in the first week or so, the teacher placed a transparency on a projector showing a 3 x 3 array of black dots illustrating the elementary mathematical truth that 3 x 3 = 9. I was smitten. With that visual display, multiplication clicked for me.

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Even before third grade, I had (apparently) been approaching mathematics the way I worked jigsaw puzzles: take in the big picture on the box first, absorb its shades and shapes, then work to fit all the pieces together. I never did memorize multiplication tables: I vaguely recall refusing to do so, or maybe I didn’t have to. Asked to multiply 9 and 7, I drew an array - 7 rows of 9 dots or 9 rows of 7 dots - so I had a picture of the entire problem, and then I counted all the dots (if there were a lot of them). I started with what the operation of multiplication *means*, at least in much of everyday life*. *I always wanted to take each and every problem back to first principles and visualize at once how these principles combine in the final result. I didn’t get any faster at working through problems of any kind; in fact I probably got slower, but my understanding skyrocketed – and often came quickly.

When you think carefully and fully about what you are learning, sifting through many nuances for thorough understanding as well as unexpected connections, it takes longer. Even the fastest runners slog through thicket-filled, rocky canyon floors, especially if they check out any of the cave openings along the sides, or any of the intersecting branches from neighboring ravines. And when you want to start with the big picture before filling in the details, taking in the basic structure of the network of chasms as a whole first, well that takes time and resources.

My approach to calculation is visual. Today, when I mentally compute 3 x 7, I imagine the numbers three and seven as pieces of a jigsaw puzzle, and in my mind's eye I lay seven threes down end-to-end, or three sevens down end-to-end, and get a (linear) picture of the number twenty-one (one of my favorite numbers:). In my imagination, numbers are movable pieces of an indefinitely long, thin puzzle. They aren’t colored in any way; they aren’t even black and white: color doesn’t apply. But there is shape, and it’s long and linear, and flows off into a potential infinite as the numbers cascade, one on another.

. . .

So how do these past and present experiences of mine affect how I teach others today?

What they suggest most of all is the importance of talking less and listening.

Before I could direct my own learning and do so well, I had to listen to and observe myself to understand how I learned and what I needed and wanted to learn. Before I can facilitate or direct the learning of another, I must listen to them to understand how they learn and where they want to go.

That might seem obvious, but, as my experiences illustrate, teachers often do not take the time or make the effort to listen to their students (in part for practical reasons, given the large number of students they have). To my knowledge, it is not a significant part of their training (for most of them). It wasn't part of mine.

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The first step in teaching is to bond with the student, to earn their trust and respect. And one way to earn that trust and respect is to first offer them to the student – to open yourself up to where the student is on their educational escapade and where they want to go and then trust that their natural learning style (a unique blend of personality, temperament, natural interests and abilities, and past experience), appropriately nurtured, guided, and channeled, and at times disciplined, will take them there.

When I first discovered multiplication, no one was teaching me directly. And no one was hurrying me along. But the opening, the opportunity for me to learn, was created by a teacher, by a display she allowed me to explore and think about. By listening to students and responding to what I discern, one of my primary aims is to create these openings. I have learned the importance of being and remaining open to each student so that I can create opportunities for them to learn at their own pace what they find meaningful.

This is so promising to see that someone who managed to get a PhD in math had the same struggles and non-conformist (?) way of arriving at the answer as my children. Thanks so much for sharing, Glen. This truly demonstrates your unique vantage point and your deep relationship with math. I'd love to hear more on how you progressed!