Cows, Cubes and Connection
Regardless of how long one’s teaching career is in mathematics, there sometimes is just an element of “plain ol’ luck” that is involved in who/what crosses your lifelong learner journey.
This person shapes/sculpts and ultimately helps define your understanding/philosophy of teaching mathematics. My intervention — cosmic, I believe today — came very early in my career. It was 1998, I was nursing a rather fresh wound of being declared surplus at my first high school that I taught, and being bumped to a new school, Riverdale Collegiate(new teachers have low seniority and are quite often bumped around in the beginning of their careers if a school’s enrollment falls).
During the first day of school, I was in the mailbox area — having all the comfortable space of a submarine — when I bumped into a person named Peter Harrison. I didn’t know it at the time, but Peter had also suffered a similar kind of administrative upheaval(if memory serves me correctly, he came to my school later in the year). Except, in his case, due to Board reconstruction, his math coordinator position was shuffled back into the bureaucratic deck. He was not only a well-known math leader in Toronto, he was a well-known leader in the whole Province of Ontario — for quite a while. Our conversation was short — as we sporadically dodged other teachers retrieving their mail — but it was poignant, and above all, warm, hinting at our now lifelong friendship. I was ranting and raving about math education. Peter listened attentively and his parting words in that narrow space was “we’ll talk”.
We have been talking now for close to 20 years.
When I taught in Switzerland, at the International School of Lausanne, he came to visit me — in my classroom:) He just sat at the back and watched.
For 4 years, Peter Harrison became my teaching mentor. I not only learned invaluable content from him, but I learned how to colour outside the lines, and always do what is best for the student — first. In my book, Pi of Life: The Hidden Happiness of Mathematics, I talk about one of my favorite things I learned from him — that math discussion/discourse should go beyond the classroom. Literally(ME stands for Mathematical Expectation)
In 2017, teachers have available to them some of best resources in mathematics. However, when it comes to bridging arithmetic to algebra, there has been none better than Peter Harrison’s Cows and Cubes, that are now well over 30 years old.
I will save the delight of exploring this gold deposit of mathematical exploration/inquiry to you. Keep in mind, that the resources for “the Cows stuff” is just the tip of the iceberg. He has reams and reams of handouts from all the presentation he as done. If you email him, he will be more than glad to discuss mathematics with you and share any resources he might have still lying around(email@example.com).
The Cows is one of the most brilliant ideas I have come across. Even the simple cartoon sketch of a disarming cow invites students/teachers to explore problems with curiosity and persistence. The hidden beauty about the Cows is that it goes from simple addition/subtraction problems to high school algebra and beyond — to eigen vectors! All this on the same template of cows, bridges and fields.
The Cubes stuff is fascinating as almost any current attempts at modeling quadratic functions falls terribly short of the content here. What I like about the activities is that each have checkpoints of exhaustion. What I mean by that is that a particular method will only be so good until it hits a wall of practicality. In the beginning, you could keep drawing the growing structures and count the cubes. Nothing wrong with that. But eventually, it will become irritating. After that, once you have figured out the relationship of first, second, etc. differences, you could carry on this procedure for a good long time. Unless, you were asked to find the number of cubes in the 879th model — or “worse”, the nth model.
Welcome to The Bridge for Algebra:)