kevin p. siu musings on technology, politics, and the world

Mathematics in Ontario High Schools: A Step Backwards

Some time in 2005, the Ministry of Education of Ontario decided they would evaluate the mathematics portion of Ontario's secondary school curriculum. Their first study led them to announce that they would remove the calculus course from the curriculum, replacing "Advanced Functions and Introductory Calculus" to "Advanced Functions", along with some slight adjustments to both the infamously difficult "Geometry and Discreet Math" course and the grade 11 "Functions and Relations" course.

All of these changes were designed to simplify the curriculum, or perhaps, in their mind, to make the curriculum more 'relevant' to students. The ministry cited increasing failure rates in mathematics, and the low enrollment numbers into the Calculus and Geometry courses. There was some noise made by students and parents, as well as the Ontario Society of Professional Engineers. To this end, I wrote the ministry a letter, which I subsequently posted on my blog (click to read). [For the record, after they replied with their initial acknowledgment of my letter, I never heard back from them.] After hearing the complaints, the ministry decided to postpone the proposed changes for a year, while creating a special task force to investigate the changes to the curriculum.

I, for one, was hoping the ministry would scrap the changes completely, and realize the mistake they had made. Unfortunately for me, that didn't happen. While I occupied myself with the various activities at university, I forgot about this issue during the past school year. So when I visited my high school again, I was shocked to hear that new changes were to be implemented next year which would have a dramatic effect on current secondary school math students.

The changes made were:

• Grade 11 Functions and Relations became Grade 11 Functions
• Grade 12 Advanced Functions and Introductory Calculus became Grade 12 Advanced Functions
• Grade 12 Geometry and Discrete Math was basically removed, making room for a new course, called Grade 12 Calculus and Vectors
• The third senior math course (and generally recognized as the easiest), Grade 12 Data Management, was essentially untouched

Now, it would appear that Calculus remained in the curriculum, and that the only real casualty was the Geometry and Discrete portion of mathematics. Upon closer inspection, in fact, both calculus and geometry+discrete were dumbed down and/or removed. Vectors, which made up about one-third of the old (and difficult) Geometry course, was added onto the already packed Calculus course, while some of the Grade 11 Functions and Relations content was moved to the new Grade 12 Advanced Functions course.

What did all this accomplish?

• Grade 11 students who don't intend to take calculus now have a much easier course;
• Grade 12 students who wished to take calculus, but not the more difficult geometry course must now take an extra course to get their credit;
• Stronger grade 12 math students who wished for a challenging course to stimulate them are now simply out of luck.

Not to mention, of course, the problems that they caused for all the students going into this 'transition' year. (Namely, all the students who took Grade 11 Functions and Relations last year will have to take Grade 12 Advanced Functions, which repeats some amount of material, before they can take Calculus)

Now, I am sure the Ontario Ministry of Education had the students' best interest at heart when they implemented these changes, but they have gone about it in a completely backwards way. To put it bluntly, in order to curb failure rates of senior mathematics students, the ministry has decided to dumb down the curriculum. Simple, right?

Well, as I wrote previously in my letter to the ministry, the failure rates, in fact, are representative of a larger problem, and that is the growing incompetence of our educators and the use of particularly bad learning material (anyone who remembers the Quest 2000 series of textbooks introduced by Harris will understand what I mean). Kids are no longer learning the fundamentals properly - of course they're having trouble in upper years.

I'd love to get into how the education system is flawed, but that's a topic for another day. Most of it, of course, has to do with most educators teaching all the wrong things, and the mostly forceful rote memorization and inherent boring-ness of the assigned work. If we start teaching people to ask why and motivate instead of lecture, we might actually see some positive results.

In any case, the fundamental flaw behind this new series of courses is that instead of fixing the fundamentals from the ground up, they have decided to build another obstacle and hoping for the best. Seriously, will requiring an extra math course before calculus really improve students' understanding of the concepts? Surely, those who were having trouble with calculus aren't going to suddenly get better at it just because they've been given more math.

A second fatal flaw in the new courses lies in the lack of a true mathematics course. Anyone who has taken high school calculus knows that for the most part, this is a course about memorizing techniques and, well, methods of differentiation. For the most part, students taking calculus don't realize its significance, nor do they expect to use it in any facet of their life after the course. Most merely enroll in the course to get the prerequisite for their university program. The only real mathematics course for senior students, Geometry and Discrete Mathematics, has been all but destroyed.

Sure, the ministry did note that the "Discrete" course was getting low enrollment numbers - but for good reason. It was a course designed, and in that respect, designed very well, for students who were genuinely interested in mathematics or were at least skilled in the subject. I say without hesitation that content-wise, it was definitely the most challenging course I took in high school, but it was also very enlightening, from a mathematics standpoint.

Geometry and Discrete brought everything we learned in mathematics together, from basic number theory to algebra to geometry. It required connection between concepts, and a deep level of understanding of what mathematics is. If nothing else, it was an unbelievable learning experience. The sheer elegance of mathematics was brought out in the course, and for those so inclined, it was even enjoyable.

Beyond just the learning experience, Discrete provided a solid basis upon which science, math, and engineering students could build during their postsecondary education. Without taking the Discrete course (which was not a requirement for my engineering program), I would surely have done far worse in both my Vector Algebra and Linear Algebra courses. Ironically, the only reason I had a vector algebra course at all in my first year was because it was removed as a prerequisite for entry into the program last year. The class average in that course was very high this year - and not because the material is easy, but because most of the students there had already learned the material.

Now, engineering at the University of Toronto is a rather diverse group. I would venture to say (although I do not have solid statistics here at the moment) that somewhere between a third and a half of the students in my program was from out-of-province, and yet, most still had the background knowledge for that course. This proves only one thing - the rest of the world is at least on par with the Ontario education system. Now, with the removal of the more challenging course, Ontario has surely fallen behind.

As an aside, but perhaps not so off-topic, AP Calculus in Ontario has all but hit an end. The flagship course of the American Advanced Placement program is a course which teaches university-level calculus to high school students who wish to get a head start, or wish for a truly challenging learning environment. With the addition of a second prerequisite to calculus, it essentially forces all secondary schools to semester their grade 12 math programs, and run calculus in the second semester. Thus, if AP Calculus classes were to be run, it would have to be in the second semester as well. Typically, schools in Ontario have their second semester from February to June. Unfortunately, the Advanced Placement exams run by the college board in the States is held in early May. That leaves only three months for AP Calculus to teach students all they need to know for the AP Exams in May, on top of all the other curriculum-required material. This little logistical problem has rendered AP Calculus virtually useless in Ontario. Now, it will only be a matter of time before that is ultimately canceled, leaving Ontario students further behind their counterparts from the rest of Canada as well as the United States. Oh, and not to mention the more impressive European education systems, and the ever-competitive Chinese students. So much for having a good and competitive education system.

Canada was recently ranked one of the worst in terms of innovation among modern industrialized countries around the world. Now, with Canada's largest province deciding to dumb down the education system, Canada will fall even further behind in innovation. Without solid mathematical foundations in secondary school, the postsecondary institutions will now have to shoulder the burden of teaching students these subjects, in an environment that is generally harsh for learning (for one, if you fail in university, you'll now have to pay for it - is that any incentive to take a challenging course?). With our mathematics lagging behind, it is no wonder that we're not innovating - how long will it be until our economy begins to suffer because our students have been denied the opportunity to excel in a global context?