The Ceiling at the Top of the Class

Liketo

A hole scooped in the dirt. A fistful of small stones. Liketo is a Basotho girls’ game. You toss one stone into the air, and in the moment before it lands you have to move the others. You move them out of the hole, back into the hole, in counts that go higher with every round. Think about what a player is actually doing: arithmetic under pressure, tracking a stone in flight, splitting a set into parts, all while her friends lean in, hoping she slips so the turn passes. Nobody calls this mathematics. It’s a Tuesday afternoon.

The boys have morabaraba, a cattle-herding game played on lines scratched into stone or drawn in the dust, each player steering their likhomo (cows) into rows of three. It is similar to chess but played faster and louder. It is also a branch of mathematics called combinatorial game theory, though nobody playing it needs the word, the same way nobody singing needs the word “melisma”.

The word came later, in a classroom, and something strange came with it. The moment mathematics was named, it stopped belonging to us. It became something imported. Something with a gate.

Chalk

I taught at a high school a few years ago. One of my students, a brilliant girl who sat in the second row, always shot her hand up when I posed questions in class. She got most of the answers right. And that was the trouble.

I had been teaching and tutoring for a few years by then. High school mathematics, later first-year calculus. I had enough experience to know the sound of a right answer arriving with nobody home behind it. On this one occasion, she had run the quadratic formula the way you run a tap. Numbers in, roots out. When I asked her what the two answers meant, where the curve she had never drawn crossed a line she had never pictured, her face did the thing where it closed. Politely and apologetically.

At this point, I need to emphasise that she was not struggling. She was the strongest student in that room. First to finish, rarely wrong, the one everyone else copied from. That is exactly what unsettled me. The formula I had handed her was a good one, carrying centuries of human effort folded into one tidy package. It had carried her to the top of the class and also become her ceiling. She had mastered the tap so completely that nobody had thought to mention the water.

I want to be gentle here, because the formula is not the villain of this essay. But that afternoon stayed with me the way a splinter stays. Small, and always there when I pressed on anything.

Two Papers

For a while I owned two qualifications that refused to speak to each other. A BSc in Mathematical Statistics, earned in South Africa. An MFA in Creative Writing, earned in the United States, an ocean and most of a hemisphere away from Lesotho. At gatherings this combination was treated as a charming contradiction, like a butcher who’s vegan. People wanted to know which one was the real me. I never had an answer, because the question assumed a border I could no longer find.

Here is what a proof is: a sequence of statements, each one earning the next, building toward something the reader could not have accepted at the start. Here is what an essay is: a sequence of statements, each one earning the next, building toward something the reader could not have accepted at the start. I have just written the same sentence twice.

Both begin with something you suspect is true but cannot yet justify. Both go through drafts. Both fail the same way, through leaps the reader cannot follow, and both succeed the same way, through a rightness you feel in your body before you can explain it. Mathematicians call a good proof elegant. They don’t mean correct. Correct is just the entry fee. Elegant means the thing was said the way it wanted to be said.

There is something else, which took me longer to admit. The degree I earned near home taught me mathematics, but it was the degree I earned far from home that taught me to see home. Distance does that. In writing workshops in the US the questions came again and again. Where are you from? What do you know that we don’t? I found myself describing pieces of home, like patterned walls, to people who had never heard of Lesotho. In my own descriptions, I started to hear it: the mathematics I had supposedly left behind, alive in everything I was homesick for. I had to go that far away to notice what had been on the walls the whole time. Only after coming back did the two papers become one. I had spent years believing I had two educations. I had one. It had simply been taught to me in two buildings, on two continents and it was only ever going to make sense at home.

Litema

After the harvest, when the walls of a mokhoro (hut) have been freshly plastered with earth and dung, a woman kneels and begins to draw. She works with her fingers, sometimes a fork, sometimes a comb, pressing furrows into the wet surface before it dries. The furrows run in small square fields. Each field repeats its neighbour, or mirrors it, or turns it a quarter of the way around, until the whole wall becomes one pattern breathing in four directions at once. The patterns on the Seanamarena, the Basotho blanket, are drawn from this same tradition.

There is a name for what the woman at the wall is doing, and a branch of mathematics built around it. A pattern that repeats across a flat surface can be built in only a fixed number of ways. Not many ways. Not infinite ways. Seventeen. Every tiled floor in every building on earth, every printed fabric, every wall in every village, belongs to one of seventeen families, and there has never been an eighteenth. Mathematicians call them the wallpaper groups. A group, in their sense, is simply a set of moves you can combine and undo: turn the square, mirror it, slide it, turn it back. Group theory is the study of what such moves can do, and the proof that there are exactly seventeen of these families was not completed until 1891. A Mosotho woman with a comb and a wet wall had been working inside that classification for generations, choosing between its symmetries by eye, teaching her daughters which choices sat well beside which. She did not have the theorem. She had the thing the theorem is about.

The word for this is litema. It comes from ho lema, to plant, and planting was traditionally the work of women. The word for the art and the word for their labour are the same. A Mosotho woman did not decorate her home instead of working. The decorating was the last movement of the work itself.

Consider how the labour was divided, because it is a diagram of mathematics itself.

The men raised the structure. The foundation, the walls, the thatched roof and the door. This is procedure. It is governed by rules, it carries weight, and it is unforgiving of error. A crooked wall will not hold, and there is no creative interpretation of a roof. Nothing about this work is lesser. Get it wrong and the whole house falls on your head.

Nobody lives in bare walls. The building became a home on the day the women knelt at the walls and began to draw, when the sound structure received a pattern nobody required of it. A design debated in the dust first, by a council of women, the most skilled among them sketching her intention on the ground while the others weighed in, until everyone agreed and the work began. The structure made the art possible. The art was what the structure was for.

This is what school gets backwards. School mathematics stops at the raised walls and calls the house finished: twelve years of straight lines and rules that carry weight, then a certificate, as if anyone had ever been meant to live there. Advanced mathematics is litema. It begins where correctness ends, on the day the walls are solid enough to hold beauty, and someone hands you a comb instead of a level and asks what you see. You need the walls. You cannot draw on air. But the walls were never the point.

Notice who holds which task in this picture. In the story my schooling told, mathematics at its highest was men at blackboards. In the story my heritage told, the men built what was necessary. The women made it sing.

One more thing about these walls. The murals were prayers, appeals to the ancestors for rain. If the ancestors were satisfied, the rain came and washed the murals away, and in the next dry season the women knelt and began again with new designs. The art asks for the one thing guaranteed to erase it. Imagine holding your best work that lightly. Imagine a whole tradition resting on the understanding that the answer washes away the question, and that this is not a tragedy. It is weather.

Chalk

I have been hard on the quadratic formula, and I owe it an apology.

A formula is a memory. It is the polished stone of other people’s thinking, centuries of trial and error compressed into something a sixteen-year-old can hold in one hand. The woman drawing litema did not invent her pattern either. She received it, and the receiving is part of its beauty. Inheritance is how any craft survives.

So the trouble was never the formula. The trouble was handing over the pattern with the meaning removed, teaching the furrow without the seed. The woman at her wall knows what the furrows are for. The girl in my second row was given the furrows alone, and then we wondered why she could not imagine a harvest.

None of this needs a villain. Teachers teach the way they were taught, inside syllabi that reward the tap and never ask about the water. I did it for years, kindly, with good marks to show for it. And the correction, it turns out, is small. Before the formula, the question. Before the method, the itch the method was invented to scratch. A formula offered after curiosity is a gift. Offered before it, it is a gate.

Two Papers

When I finally let my two degrees speak to each other, here is what they said: mathematics is a language, and I had been teaching it as spelling.

Nobody falls in love with a language through its grammar. You fall in love because you have something to say, and suddenly there are words for it. The child playing liketo has something to say. I can hold five things in the air of my mind at once. And the equation is simply the sentence she was never invited to write.

So now I teach the sentence first. I teach it to the students who already have the grammar. The ones with the distinctions, the ones who finish early. Those are the students most at risk of never discovering what the subject actually is, because their marks keep telling them they have already arrived. They have raised perfect walls. No one has yet invited them to kneel and draw.

Together we read a situation the way you would read a paragraph. What do we know? What is missing? What shape might the missing thing be? Only then do we reach for symbols, the way a writer reaches for a semicolon, because the thought requires it. And when a top student meets a problem no formula fits, and stays in it, and begins to compose, that is the moment the ceiling lifts. I have watched it happen. It looks like relief.

An IB student I was tutoring a while back came to me troubled by differentiation. He wanted the methods, all of them, quickly, because the methods were what the exam would ask for. I did not give him the methods. I gave him rates of change instead, and I started where he was already strong. A straight line: how steep is it? He knew that one. Rise over run, a gradient, done. Then a curve, and two points on it: how fast is it changing between them? He drew the line joining the two points and found its gradient, and he was pleased with himself, because that was still a method he could reach for. Then I gave him one point. Just the one. How fast is it changing here? And there was nothing in his toolbox for that, because a gradient needs two points and he only had one. That was the moment. Not the answer, which came later. The moment he understood what the question was, and why anyone had needed to invent something new to answer it. He had been differentiating for months. That was the afternoon he found out what it was for.

The Tap

So. The girl from my second row.

She would have been superb at litema. She could hold a whole design in her head while working one square at a time. That was always her gift. What she needed from me was not another formula, and not some vague blessing to be creative. What she needed was for someone to walk her back along the pipe.

What I would teach her now is this. First, simply, water comes out of a tap. Not answers. Water. A living thing, with pressure and temperature and weight. Then, the tap is hers to play with. She can open it full blast and let a problem roar. She can bring it down to a slow, measured thread when the work is delicate and every drop counts. Rhythm is a mathematical decision. So is restraint, knowing what not to compute, what to save, which approaches would drain a whole afternoon for nothing. Finally, furthest back and most important, that the water did not begin at the tap. It fell as rain on the mountains. It gathered in rivers. It was held in dams, built because someone understood that water must be stored where it wants to run. In my country we know this in our bones because Lesotho’s mountains hold water for a thirsty region. The tap is only the last centimetre of a pattern the size of a watershed.

That is what I mean by enrichment. Not more taps, opened faster. The whole system, handed over: source, storage, pressure, play.

Basotho women knew. All of this was the practice long before anyone had a tap. They pressed their patterns into the walls and asked for rain, and the rain came and took the patterns, and the rivers rose, and somewhere far downstream, generations later, a tap filled a cup. Then the plaster dried, and they knelt, and began again.

Lerato Molisana