Introduction to the Grade 12 Curriculum Project
My purpose here is to provide a set of good problems for the two new grade 12 university-bound courses, which could serve not only as an investigative supplement but as the curriculum itself. That’s a tough job, since a problem is “good” for me, only if it passes fairly high artistic standards. But I don’t feel I have to cover everything. My idea is that if the students study these problems, understanding some parts of them, “learning” other parts, and master the technical skills that are a crucial component of each problem, then by the end of the course they will be ready to move forward. The fact that they might have left a topic or two out is of little consequence. I promise you that a student who has some level mastery of most of this material will do well in any math, science or engineering program at Queen’s.
The following notes, prepared for a panel discussion on "reinventing the teacher,” will give you some idea of the philosophical background for this work.
Our job is to ready our students to learn, to teach them how to learn, to set them free. Too often all the “stuff” we feel we have to teach them gets in the way. And too often “stuff” is the word for it.
What is important, even crucial, about the content is not what it is but how it is, not its quantity but its quality. The classroom should be an artist’s studio and what the teacher, as artist, brings to the lives of the students are the works of art that are particularly wonderful to her, those works which she knows and loves, and always struggles to understand. And if she gets that right, the class will unfold naturally.
More than any previous generation, today’s students are in need of discipline, focus and allegiance. But traditional sources of authority won’t work because those particular emperors seem to have very few clothes. Discipline follows from respect, respect for the teacher, respect for what she gives them. Nothing else will work.
Today’s students will have to reinvent the world—even a casual reading of the newspapers will tell you that. And to prepare for that, the teacher will have to be reinvented too. My argument is that the teacher as artist is the only model that will work. As such, her task is to render the world for her students with imagination and light. And make them thirst for the technical skills that will allow them to be artists too. Given time and persistence, this will give us back the world and will give us one another as well.
An important caveat is that the teacher cannot do this alone. She needs the support, and the respect of society, and right now she has too little of that.
What is a good problem?
In putting this set of problems together I am guided by a sense of what mathematics is to me, and what it was that reached out to me when I was a student.
Where do I start? Perhaps with power. Once in grade 12 physics in Almonte high school we were using a convex mirror to focus the sun’s rays at a single point and I asked what the shape of the mirror was and Mr. Suter told me it was a part of a sphere. That night at home I played with the geometry of that, circular arcs and lines, and I produced an argument that he was wrong, that it couldn’t be spherical. I didn’t discover what the true shape was (a circular paraboloid), but I had proved that it couldn’t be a piece of a sphere. I was quite excited at what I had done, and I borrowed my father’s typewriter, and, late into the night, typed the argument out, drawing careful diagrams on the side. The next morning I presented the paper to Mr. Suter. I don’t recall his reaction, but I do remember sitting in the cafeteria afterwards eating the hot casserole that Mrs. Hughes always brought in for those who could spend 25 cents, and being a little awed at what I had done. I had given the paper to Mr. Suter without the slightest doubt that he was wrong and I was right. What did that mean? That the universe followed simple rules and that mathematics had the power to describe them and analyze them with absolute certainty. And that power was accessible to me––if I wanted, I could own it.
When I started doing university math, I began to have a new kind of experience. I’d be trying to solve a problem and it would come down to some brutal calculation, and I’d plow away at that and the mess on my page would get more and more horrible, and for that very reason I’d decide I must have made a mistake. That’s not because it was a problem assigned by one of my profs and could therefore I could count on it working out. These were more often problems that came out of nowhere, problems I’d invent myself, and I had no reason to think that the calculations would work, but somehow they usually did, and so when they got ugly, I’d stop and go back and look for a mistake, and I’d almost always find one. And that was another striking revelation for me, that aesthetics was a reliable guide to truth. I couldn’t see any reason for that; I couldn’t understand why that ought to be the case. As one of these seemed so subjective and the other so objective, it made no sense that they should be so connected. But they were. And I understood Keat’s affirmation that beauty was truth and truth beauty.
As an addendum, I move from Keats to Einstein, here in a 1929 letter:
“My latest results are so beautiful that I have every confidence in having found the natural field equations.” (p 35)
A Sense of the Mysterious Alan Lightman [In fact in that particular instance, Einstein was wrong.]
For some 25 years now I’ve been teaching, along with a colleague in the English Dept, a course called Mathematics and Poetry. One thing that struck me at the very beginning was that students of literature study works which are much more sophisticated than those that mathematics students are given to work with. The same thing seems to be true in the schools. In many ways, the examples that make their way into the school mathematics class are far less sophisticated than those that appear in the English classroom just down the hall. And so my objective in the Math and Poetry course was to produce math problems that were of equal sophistication to the poems my colleague was working with. Sophistication, by the way, is not really about difficulty; it’s about depth and meaning, there are different levels of meaning. So that’s my objective here too, to provide problems that have some sophistication, that yield but not completely, that always have another layer, another richness. Of course the student does somewhat different things with a sophisticated example. Understandings will be partial, but with the promise of greater richness ahead.
Here’s a reaction I often get. Your problems are great Peter, but they’re too hard for my students. Well they are and they aren’t. It’s important to emphasize this. They are (often) too hard for our students to fully understand, or to be able to reproduce on their own. Of course that routinely happens in the humanities and that is certainly what gives the subject its richness. In the humanities, it’s amazing what a student can catch on to and relate to in his or her own way. The point is that this is (or ought to be!) also the case in mathematics. We just have to be careful what we ask them to do, particularly on tests and exams, but also on assignments. Our requirements of them (for marks) have to be clear and within their compass. For the most part I give my students clear understandable models of what I want them to do, typically I ask them to ring a slight variation on a solution I have written for them. In so doing, they gain an understanding of the configuration at some level and they get some good practice at a needed technical skill.
In my experience understanding emerges from this process, not immediately but slowly over time, in fact over years. And if the example is beautiful and powerful, love and respect develop also.
When you read the policy document for MHF4U you will notice that some of my examples go beyond the specification of the curriculum, and, conversely, some of the detailed expectations don’t seem to be covered, at least not explicitly. In fact most of them will arise in one form or the other in exploring the examples but some of them will certainly be missed. That’s the price you pay for a sophisticated investigative curriculum, but it’s well worth the cost. And over the years of the student’s education, the important stuff will all get done.
Shortly after I had put out a very preliminary first draft of these problems, the Head of Guidance at a Toronto school wrote to me pointing our that I had too much physics and not enough business-oriented models. Didn’t I know that bright young folks today are much more interested in business than in physics?
I was not sure what to say in response to that. Certainly I have noticed the enthusiasm for business. And it is in fact a fine field to study. But I do feel that physics is a better place to start, that its basic principles are so fundamental to all science that everyone should grapple with them, certainly the bright young folks”. Besides, there’s a simplicity and a precision in physics that few other areas of science can claim. So for example, the hugely important process of exponential growth and decay is seen in physics in a way that allows one to understand why it is there, partly because one is able to descend to a lower level of detail and actually track the process. A good example of this is found in the punctured tire problem. By going down to the level of the motion of molecules, one can actually prove that the air pressure equation will be one of exponential decay. And, even better, we can actually make measurements and verify the theory. It doesn’t get better than that.
Having said that, I have in fact constructed some simple financial games and applications: maximum profit, optimal allocation, etc.
Mechanism. The comment about molecules brings me to the idea of mechanism. Kids need to cultivate at an early age the habit of asking “how does it work?” This is the case, not only in physics and engineering (and math) but in all walks of life. An industrial manager once told me that the students he hired for the summer were of two types, those who were always looking to see what was really happening and why, and those who weren’t. The former were naturally oriented towards seeing how things really worked, and usually have ideas for change and improvement. They were also ones who were more able to respond well when something unexpected happened. They were the ones he always tried to hire.
Anyway, if these ideas and ideals strike some chords with you and you want to examine these units more closely, drop me an email. I’d be happy to have you along.