PERSONAL STORIES OF DISCOVERING MATHEMATICS

Do you have a specific first memory of discovering mathematics, that is, a piece of mathematics that was original to you, that you felt you owned, and, as such, gave you a rush of joy? Did you recognize it as mathematics?

I certainly have had such a formative childhood experience, one that I today regard as the key experience that turned me to mathematics. But I did not at all regard the enterprise as mathematics at the time — nor could I since my childhood definition of math was dictated by the Australian-Anglophilic “just memorize and do” education culture of the 70s. Thank goodness for my bedroom ceiling!

I grew up in Australia (so let’s replace “math” with “maths” throughout this essay) in an old Victorian house. Each room of the house had a patterned press-tin ceiling and the one in my bedroom displayed the simple geometric design of a five-by-five grid of squares. Each and every night throughout my childhood I fell asleep staring at that grid. And I made up puzzles for myself. Lots of different puzzles.

Of course, I counted how many squares there were — twenty-five — and I remember the sudden realization that I could count two-by-two squares too (sixteen of those), three-by-three squares (nine of those), four-by-four squares (four of those), and of course the one five-by-five square. This act of counting I recognized as school mathematics, but asking Why is the count of squares of a particular size in a square grid always a square number? — which I did wonder — was not a school math question in my mind!

I tried counting rectangles, too, and eventually convinced myself that there were 225 of them. (Another square number. Whoa!)

I certainly saw that there were five rows of five squares in the grid, five columns of five squares, and nine diagonals with 1, 2, 3, 4, 5, 4, 3, 2, and 1 squares in turn. Since the diagonals account for all twenty-five squares, I realized that, without a lick of arithmetic, the sum 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 must equal 25. (So if my bedroom ceiling had a 1000-by-1000 grid of squares for its design, would I have discovered that the sum of all the numbers from 1 up to 1000 and back down is a million?)

The diagonals of an N-by-N square reveal the sum 1+2+3+…+(N-1)+N+(N-1)+…+3+2+1.

Here is the puzzle I made up for myself that I truly believe made me a mathematician. Starting at the top left corner of the grid it is clear one can walk a path of horizontal and vertical steps that visits each and every cell exactly once. (There are many ways to do this but going across and down, row-by-row, might be the most obvious route.) One can also walk such a path starting at the center cell. (A spiral path works.) One can also start such a journey from the center cell of the bottom row, or from a cell one place diagonally in from the bottom-left corner. And so on. Finding paths that work from these starting positions usually take but a moment.

From which cells is it possible to start a journey of vertical and horizontal steps that visits each and every cell exactly once?

But there are other possible starting cells for which finding a path that visits each cell of the grid, with just vertical and horizontal steps, seem far from straightforward. Try finding such a path starting from the second cell on the top row. Try finding one starting one place to the left of the center cell. I couldn’t!

And I tried for many a night.

I tried so many times that I was convinced that paths starting from 12 certain cells were impossible. But I wanted something more, something that would defend against the worry that my searches may have stopped just short of catching working paths. Should I keep searching? (When would the searching end?) I began seeking iron-clad logical reasoning that would convince me — or anyone — that paths starting from these troublesome cells simply do not exist. And that is the task that stuck with me — for years! Many years! Six years, in fact.

I carried this puzzle in my mind for so very long, actively returning to it at odd moments, every now and then trying to trace out paths segments in my mind’s eye on the bedroom ceiling, trying to devise some universal “theory of corner stuckness.”

And then, out of the blue, while walking to school in grade 10, I had the most astounding epiphany. I wasn’t even thinking about the problem at the time when suddenly a simple image flashed into my mind: it was the five-by-five grid colored like a checkerboard with 13 black cells and 12 white cells. And it dawned on me right then and there how to explain, in one astounding elegant fell swoop, why no path starting from one of the white cells could possibly exist. (Can you figure out my logical argument?)

I had never experienced such euphoria! It was a high. But I couldn’t call it a math high as this work, this six-year journey, had absolutely no connection to the mathematics I knew from school.

I found my mathematical schooling by-and-large joyless, but I did continue with the subject in university. It wasn’t until I took advanced courses that I realized that playing with my self-created puzzle and its solution was mathematics! I was asking a why question, I had discovered parity and used it to devise convincing reasoning, that is, to devise a proof. I was a young lad operating as a mathematician. I had discovered, on my own, true mathematics and its true joy.

How unusual is my experience? I don’t know.

In his article Who Knows Two? [1] mathematician Jim Propp describes day-dreaming during his Latin class. He recalls having to recite two-column tables of noun and adjective declinations, and they had to be recited by column. He began to wonder if there might ever be a table that when recited by row instead would sound the same? There is!

A five-row table of two columns that reads the same row-by-row as it does by column.

Actually, one can create a “boring” two-column table with this reading duality with any desired number of rows: simply have all entries, save the first and the last, be the same. But the interesting question is: For which counts of rows in a two-column table do only these “boring” examples exist?

Mathematics author Cathy O’Neil writes in her piece How Dominoes Helped Make me a Mathematician [2] about her parents, both mathematicians, introducing her to classic domino tiling problems on mutilated checkerboards and its influence on her mathematical development. And mathematician Tatiana Shubin once described to me her fascination as a child with the grid lines on the graph paper issued to her at school. A formative moment for her was her discovery that taking a steps to the right and b steps up, and taking a steps up and b steps left is sure to produce two line segments orthogonal to one another, no matter the values chosen.

The vectors <a,b> and <-b,a> are orthogonal.

Professor Susan Crook as a child developed a passion for palindromes and with her father tried to figure how many times a day a digital time piece will display a palindrome.

Each example presented here occurred outside of the regular mathematics classroom, or at least, outside the stated activities being studied in the classroom. But it is possible for classroom mathematics itself to provide formative mathematics moments for our students. Master teacher Michael Cole described to me his early experience of being utterly bored with an arithmetic worksheet that required students to practice long multiplication by computing the squares of two-digits numbers. But then young Michael noticed that 24² and 25² differ by 24 + 25 = 49, and conjectured that 25² and 26² will differ by 25 + 26 = 51. (They do!) He checked that the pattern seemed to hold for other consecutive squares, but, like me, wanted a convincing argument that the pattern had to be true. He eventually decided to take an abstract approach, calling one square x² and the next (x+1)². He muddled his way through working out an alternative expression for (x+1)² and saw, alas, that it differs from x² by 2x+1 , not at all the expression he was expecting … until he realized that 2x+1 can be interpreted as x + (x+1)!

He used his discovery to breeze through the worksheet, simply adding odd numbers to previous answers, euphoric too about the power of his work. The next day he told his teacher of his discovery and explained why it was true. Unfortunately, his teacher was unimpressed and accused him of taking the idea from a later section in the text that introduces algebra. “I was deflated,” Michael told me, “and [I] never got excited about math again until many years later. But for a short time there I was really proud!”

We each have our stories. We each have moments from our pasts, some particularly formative moments perhaps, that jolted us with the unbridled joy of a math “aha.” These are extraordinarily potent moments that come from just noticing, wondering, asking, playing, and exploring. They invite powerful mathematics.

So let’s share our stories with our students. Let’s welcome their stories too and encourage them to look for many moments of wonder and sheer delight in pattern, structure, and reason. (And might I suggest taping a five-by-five grid of squares on your classroom ceiling? If you do, your job is to say nothing!)

Mathematics is an uplifting and intensely human enterprise. Let’s honor joyous mathematics with our human stories.

References:

[1] “Who knows two?,” Jim Propp, April 2018, Mathematical Enchantments. https://mathenchant.wordpress.com/2018/04/16/who-knows-two/

[2] “How Dominoes helped make me a mathematician” Cathy O’Neil, July 2018, Bloomberg. https://www.bloomberg.com/view/articles/2018-07-21/how-dominoes-helped-make-me-a-mathematician