A quilt of mathematics

Our lives can be viewed as quilts, each piece representing a story in a larger narrative. Embedded in the quilt is the story of our educational experience: a great or bad teacher, a positive or negative experience with a subject, an insight or a setback. Mathematics is a part of all of our stories to a very tiny or very large degree, or more often somewhere in between.

We each have a mathematical story. Here is an early one of mine.

Trigonometry

I didn’t much like arithmetic in elementary school. Rote memorization of times tables was an endlessly boring exercise. My love of mathematics awakened in high school, with the abstraction found in basic algebra. In grades ten and eleven, we learned trigonometry. Older students had warned us this was sink or swim time with mathematics. My view of trigonometry was not that different from arithmetic: did it really matter to the world the value of sin(45) versus cos(45)?

Then I met trigonometric identities and that everything changed.

Trigonometric identities are equations involving the trigonometric functions like sine, cosine, tangent, etc. More complex identities build up from the simpler ones, using symbolic manipulation of equations and substitution. Here is a fundamental one, called the Pythagorean identity:

A more complicated identity, for example, is following:

Our homework contained a range of identities, from easy ones to the tougher ones. The very tough, super challenging ones were marked in our text with a star *. The * acted like a kind of “beware” sign to students.

Being the fearless teenager that I was, I tried the final starred exercise in the chapter on trig identities. If I could master that, then I could do any of them. I don’t remember the specific identity now. In any case, I toiled away, spending at least an hour or two playing with the identity. I tried everything I knew, and then brought in some new tricks. It was like exploring a cave. I would travel down a corridor and find myself at a dead-end. Other times, my exploration would open up to a whole network of passages and hidden paths.

Then I found it: the solution: multiply by 1 as fraction of trigonometric terms. That allowed the appropriate simplification.  I checked my work and it looked correct. I wrote it up carefully, and then went to bed.

Our Grade 11 math teacher, Mrs. Ives, asked us to present our solutions on the board for the class. I always do the same now in the mathematics courses I teach. Experiential learning gets students engaged quickly.

When Mrs. Ives asked volunteers to present their work, I raised my hand and began writing out the problem I was presenting. She said “Oh, that one.” I learned early the importance of coherently presenting your work: state what is given, then your steps (justifying each one), then write a concluding sentence. My teacher was floored with my solution and the other students were pissed. I had raised the bar, conquering the toughest trig identity in the book.

Onwards and upwards

From there, as the story goes, the rest is history. Over the coming years, I became increasingly in tune with mathematical abstraction, and fell in love with set theory, geometry, algebra and analysis in my undergraduate degree. Logic came after that, and now discrete mathematics is my thing.

What is your math story? Did you love math in high school? Hate it? Were you forced unwillingly to take statistics or Calculus in university? Are you a math teacher with a cool approach to teaching the subject? A fellow mathematician who had an epiphany that this would be your life’s work?

Quanta’s Pencils Down series

Quanta Magazine is a on-line magazine focusing on science and mathematics. I highly recommend it to people at all levels interested in mathematics. A recent article is part of their Pencils Down series, exploring through four pieces on the teaching of mathematics and science.  The latest article in the series Do You Love or Hate Math and Science?, which to me, is one of the most intriguing parts. Quanta is asking all of you to share your math stories, just as I did above.  The result is an unscientific snapshot of current thoughts and trends in mathematics education, all woven into a lovely hexagonal tessellation (or tiling) of the plane.

I’d encourage everyone reading to add their mathematics stories. Pieces of our own quilt will combine with others to tell a much broader story of our relationship to mathematics.

Anthony Bonato