Can you imagine the time before the discovery of zero? My husband and I got a glimpse of this when we witnessed the discovery of zero, not on the world-historical scale, but by our two-year-old daughter.
It was not an easy road. Counting had come uneventfully, but when numbers became numerals and the number 10 appeared on the page not with its own symbol, but with a 1 and 0, suddenly everything had changed. Till that moment, in her world it was still possible to have a system of enumeration like the one used by Ireneo Funes in Borges’ story, “Funes the Memorius.” Funes gives every number its own unique name. He has “an infinite vocabulary for the natural series of numbers” and no use for the concept of place value.
When our daughter saw that the numeral 10 comprised a 1 and a 0 she flung herself upon a chair and cried. We were taken aback, unprepared for the blow this dealt to her understanding of number and how she would struggle to make sense of it.
This was the struggle of an artist. The world had changed in a fundamental way. Something was lost that would never return. Why would there be a zero in the number ten? Till now the numerical representations were incidental to the concept of number, of quantity, of this thing that could go on forever …. and had now abruptly, jarringly, come back to zero.
Some time later, she confronted a blankness of another kind. In tears, she ran towards me holding a white crayon. It didn’t show up on the paper, she cried. “Therefore I am throwing away the white crayon,” she declared painfully. Her eyes brimming over pleaded for a way out of this harsh sentence. I drew something with the white crayon and painted with water color on top of it The water color surrounded the crayon image to reveal it. Saved! In its own way, the white crayon was a place holder.
We often hear people describe the joy and exultation of mathematics, but rarely the pain and suffering, arising not from inability, but rather from the wholehearted engagement with the ideas in all their beauty and tragedy. Immense was my gratitude when I came across a mathematician who wrote:
Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion— not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.
– Paul Lockhart, A Mathematician’s Lament
A few days later, our daughter explained to us that “in the number 10, the zero stands for 9 numbers. In the number 100, the two zeros stand for 99 numbers.”
So if you had to subtract, say 6 from 10, you could subtract it from 9 (instead of from 0) and then add the 1 to get the correct answer. Of course no one would ever do this because when you start doing single-digit subtraction you have your 10 fingers and no zeros are involved. But try it in 2 digits:
If you have to subtract 42 from 100, you could just subtract it from 99 (instead of from 00) and then add the 1. We didn’t do that either because by the time she started doing subtraction on paper, her resistance to zero was long forgotten and she learned to “borrow.” Funny word, but arithmetic can be funny that way. More importantly, it doesn’t matter that we didn’t use her method to subtract. What matters is that she thought about it and explained the solution to her problem.
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How often do we get a chance to appreciate the creative, expressive side of mathematics? Too often, people see it as a mechanical process and if at all they believe in the beauty and thrill of mathematics they perhaps feel it comes only to those who master its mechanics to a highly advanced level. Passionately opposing this approach to teaching mathematics, Paul Lockhart, who teaches in St. Ann’s School in New York, wrote an essay called “A Mathematician’s Lament.” First circulated privately, it was eventually published by the Mathematical Association of America, followed by a sequel, and expanded into a small book.
Lockhart says: “I want [students] to understand that there is a playground in their minds and that that is where mathematics happens.” He insists that mathematics is an art:
… if the world had to be divided into the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the latter category.
Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.
Lockhart wants children to have the opportunity to observe mathematicians at work, practicing mathematics, the way they see musicians sing or artists paint. He also wishes that it would be taught along with its social and historical context, so that we could explore such questions as: why did these mathematicians discover these things at these times in these places? Because the people, not only their formulae, matter.
In spite of mounting pressures to the contrary, teachers like Lockhart are doing what they can to bring humanity and adventure into the classroom. Lockhart urges teachers to play games such as chess, go, hex, to do puzzles, and be willing to get their hands messy and express their ideas through math:
Mathematics is an art, and art should be taught by working artists, or if not, at least by people who appreciate the art form and can recognize it when they see it.
Observing children learn gives us a second chance to appreciate mathematics as an art form. Even one such moment of struggle with the meaning of zero can inspire confidence in our capacity to probe the depths of ideas … if we are allowed that moment.
As artists have urged us to stop telling children what color to use or where to draw, we must also allow for creative expression in the discovery of numbers. We ought to refrain from intruding into the pre-numerate state with our preconceived notions and allow every child to investigate from scratch as if the world had not yet settled on a numbering system, adding system, dimensional system and so on. Would we want to skip the stage when little ones make up their own words and move expeditiously to standardized language? If we delight in the words and usages invented by our little ones, it is because we are so confident in our own language that we are free to tinker with it, to produce as well as consume.
A friend’s toddler started counting, “2, 2, 2, 2 …” At least, it sounded like counting. We may not always get it. But if we believe there is something there to get, we won’t rush to “correct” it. And if we believe in our children’s capacity to puzzle things out we won’t be tempted to give away the ending. Recently, our daughter struggled with the concept of negative exponents. A few days later she extended Cookie Monster’s song “One cookie and one cookie makes two cookies” up to 256 and used it to explain powers of 2. It goes for negative powers as well, “1 cookie and 2 people makes ½ cookie …” (Per person is understood.) When we share such stories we find that every parent has one – or many. Indeed, this kind of thing becomes commonplace when you recognize that math is everywhere.
Shapes and patterns in nature, ideas in our mind, games we play, doodling we do, rhythms we tap all lead us to mathematical discovery. As Lockhart says,
If everyone were exposed to mathematics in its natural state, with all the challenging fun and surprises that that entails, I think we would see a dramatic change both in the attitude of students toward mathematics, and in our conception of what it means to be “good at math.”
Practicing mathematics as an art form is nice work if you can get it. Vi Hart, who calls herself a mathemusician, shares her artwork through Khan Academy and her Youtube channels. Kjartan Poskitt’s series Murderous Maths contains stories full of historical and imaginary characters involved in various mishaps and misadventures, using plenty of math, which the author explains along the way. Popular internet sites like Numberphile air fascinating math puzzles and problems, with guest mathematicians from universities and research institutes around the world. What these artists have in common is that it is hard to watch or read their work without wanting to try it out yourself.
What if you never meet such artist-mathematicians? If I were to paraphrase Picasso, I might say that every child is an artist-mathematician. The problem is how to remain an artist-mathematician once we grow up.
Jorge Luis Borges, “Funes, The Memorius,” in Ficciones, translated by Emecé Editores. New York: Grove Press, 1962, pp. 107 ff. Accessed online at http://www4.ncsu.edu/~jjsakon/FunestheMemorious.pdf
Vi Hart, on Khan Academy, http://www.khanacademy.org/math/recreational-math/vi-hart
Paul Lockhart, A Mathematician’s Lament, 2002. Published by Mathematical Association of America in 2008. Accessed online at https://www.maa.org/external_archive/devlin/devlin_03_08.html | Selected Excerpts: One Real Teacher
Kjartan Poskitt, Murderous Maths published by Scholastic, Inc. Website at http://murderousmaths.co.uk/