Mathematics: The Year She Lived
The year was 2004. The month was May. My friend and colleague, Peter Harrison, were having dinner near the school that we both taught at from 1999 to 2004. He was retiring in a couple of months. Well, yes and no. His teaching career in Ontario, spanning four decades, was coming to an end. But, he was retiring in a fashion befitting his wanderlust nature— he was moving to Hong Kong to teach there for two years. Taking Peter’s cue for the adventure and the unknown, I too left the school — one of the best in Toronto — to go down the street to a school that was one of the most challenging socio-economic high schools.
To be blunt. That school had poor kids taking poor math courses. It was an emotionally exhausting four years. Teaching mathematics was secondary. Most of my duties and responsibilities were at the bottom of Maslow’s Hierarchy of Needs. Calculus and higher math ideas like that would go dormant. My focus was on being a compassionate caregiver in this academic hospice…(this is a story for another day). By the end of my tenure here, ironically, but not surprisingly, mathematics became a refuge for many of these kids.
It was almost like I didn’t want to be in the school that built all these amazing memories of teaching with the best mathematics teacher that I have ever come across. It has taken me over a decade to realize this, but essentially, my mathematical teaching career peaked with Peter.
Specifically, it peaked in 2004, when two wholly dissatisfied math educators decided to break the exam rules — and free mathematics.
In 2003, the Province of Ontario rolled out a new grade 12 math course, Geometry and Discrete Mathematics. It was the best math course that, in my opinion, had ever been offered. Sadly, it would be eliminated a few years later for political reasons.
That passage in the above section comes from the Cumulative Performance Problems section at the back of the book. Do you recognize those words? I mean have you ever seen these kinds of author words in a math textbook? I am guessing no.
There are 55 problems in this section. For this particular book, Peter Harrison was one of the main authors. So was Peter Taylor, Peter Harrison’s mentor. The course and the book were mathematical feasts of demanding proofs. The guiding words of the authors in the last section of the book would come full circle.
As we were eating dinner, Peter asked “So, who is writing the final exam?”. He didn’t ask it in an enthusiastic way. He asked it like he was dutifully checking off an inventory box. I think I responded back with a similar disinterest. In this collective ennui with traditional assessment, we found a disruptive nugget of gold.
Keep in mind, this is 2004. The times are not conducive to disruption. But, Peter Harrison was the Department Head, and was one of the most respected and well known math educators in Ontario. So, he was given a lot of rope to do things. I just followed his lead.
The students that Peter and I taught were among the brightest math students in the school. Almost all of them had proven their mathematical mettle over the last 8 months and all of them had already been accepted into the universities of their choice. As such, many had unofficially checked out. So, giving them a cliched exam of the year’s greatest hits in a 2 hr writing block seemed anticlimactic — and almost an insult to the students, to us, and to mathematics.
Peter suggested that we look at the 55 problems at the back of the book. We each pick 6, hand these 12 questions to the students, and tell them that 5 will be on the final exam in six weeks. It took me a fraction of a second to figuratively leap out of my seat to accept this wonderfully unconventional idea! So, over the weekend, we chose our problems, and put them all on one long piece of paper(double-sided).
We handed these out in class and told the students the format. The reaction was a mix of bewilderment and intrigue. We told them there would be class time to work on the problems together. The “worst case” scenario would be that the weakest student would memorize 12 detailed solutions of geometric proofs and problems.
Yeah. We were more than okay with that…
The days where our students were given class time to work on these potential exam questions had some of the best classroom energy I have experienced. Students working in groups, in pairs, and in isolation. Students moving around. Students scrawling their ideas on paper and the chalkboard. Students trying their best to run their ideas by us and see if they could break our poker faces.
Freeing the mathematics freed our students to think deeply, calmly, and passionately about the problems in front of them.
One of the 5 problems that appeared on the “final exam” was this one. Problem 33.
A square. Three small numbers. Find the side length(s). There is zero ambiguity of what is being asked. Also, there is an immediate attraction and entry for all students into this problem. Most will go to the well of Pythagorean Theorem. What almost none of would realize that a solution involving that is hidden away. But, was found by one remarkable student.
While there are multiple solutions to this problem, the simplest one is the most elusive. But, it is also the one most symbolic of thinking mathematically — and thinking “outside the box”. Literally.
My own personal solution looked aesthetically pleasing — using coordinate geometry and trying to find the intersection of three circles, using 3, 4, and 5 as radii. I will spare you the rough algebra of three equations in three unknowns that came out of this. But, my brain has always been wired to find algebraic solutions — almost stubbornly. Sigh. I suppose it’s just my own gift/curse.
The most elegant solution was offered by the student Yanna Kang. I could live a thousand years, and I wouldn’t have seen her solution. And, the only way even Yanna got to her solution was that she was given ample time to scrawl, scribble, and doodle over a leisurely period of time. Something which has always been afforded to mathematicians, but not K to 12 math students. That is one major reason why students do not fully grasp the creativity and persistence required in mathematics — and also its wonderful surprises and historic charm/beauty.
Yanna rotated the top triangle out of the square. Created a copy adjacent to one of the sides, and positioned the 90 degree angle in this new isosceles triangle. Now, the Pythagorean Theorem was applied. Some grade 10 trig of the triangles in this area, and eventually the side of the square was fleshed out. But, that was all the anticlimactic. That was all the denouement of the problem. The entire excitement/ingenuity was in the first move — that is what we should be living for as students and teachers of mathematics. To own the problem. To rethink the problem. To change the initial diagram. Not because you think it will work, but because you have been given creative license and mathematical understanding to do so. Ironically, in the same year, Lockhart would pen his highly influential “A Mathematician’s Lament”. The quote below sums up what happened in our Geometry and Discrete classes that Spring.
“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 break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.”
The final answer was inconsequential. So are most of the steps. The mathematical gold lies in the initial comprehension and moves.
The average final mark between the two classes was 91%. The solutions that we received from these 5 questions spoke to the achievement possibilities when both the freedom of mathematics and students are honored.
I would never see that kind of freedom again. Ten years later, I would quit teaching. I think a part of that tough decision was maybe that my mathematical soul died a little bit after that experience in 2004. I also realized how difficult it is to find honest mathematical experiences in the constraints of most K to 12 math curriculum.
The New Hampshire state motto is “Live Free or Die”. I can truthfully say that applies to mathematics. As when it is completely freed, it breathes and gives life to all those that are around it. Don’t ask me. Ask the 50 or so students who were awoken from their Spring slumber to do the hardest math problems that had ever faced.
Kids don’t hate because it is hard. Kids hate math because it is boring.
