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?
Comments (11)
Well, so what are you going to do?
Are there any politicians out there who are at least a little bit enthusiastic about math? Hmm.. no, by the looks of it.
Teachers cannot do much (so they say) because it’s all about the politicians. Obviously, as you’ve mentioned, students couldn’t do much.
What are we going to do next? How come Russian students know math inside and out? (No, they’re not all geniuses) Math is just drilled into them back in that country. In any case, until at least some powerful politician decides to completely change the mathematical education in Ontario, thing will be the same.
Personally, I think that teachers who do not have a mathematics degree should not teach math. But hey, most people say that it’s impossible to find these teachers. Moreover, how do you make sure that teachers actually like teaching and teach well?
How do you make sure that poorer schools in T.O. get great teachers? (At the moment, you can’t)
Anyway. I totally agree, but what are we going to do?
And: Discrete was really not that enlightening. The only great chapters were the vector chapters. The teacher was really biased, as well (my personal opinion… he was lovely toward other people). And the course, as ‘hard’ as it is, really teaches not much of ‘real’ math, aka university-taught math =) (that’s as real as we get right now anyway)
Discrete was as close to ‘math’ as we really got in high school (as unfortunate as that is). The rest of the ‘math’ was, well, mostly boring memorization of rules.
In any case, there are things that can be done, but whether *we* can do that is another question entirely. The politicians and beaurocrats don’t seem interested in real ‘improvements’. So I guess we can only keep bugging them until they budge. Spread the word, get people thinking, and hopefully it’ll reach the right ears eventually.
For example, the OSPE (Ontario Society of Professonal Engineers) did have a good say in the process, and they helped delay the process enough for them to reconsider the calculus situation. There are other organizations, or perhaps even those in postsecondary institutions, who could have a similar say. The Fields Institute did the initial report on math, and that’s affiliated with the Math Faculty at UofT.
Is it possible to find teachers who can teach math? Of course it is. And it isn’t necessary for the teacher to have a math background to teach math. The joys of self-learning are great. All you need to do is to motivate students.
How do we get good teachers to schools? Well this may sound a bit offbeat, but I think you don’t need to get good teachers at *every* school. Here’s what you do: get a select few talented teachers, and have them go around to different schools. Have them go to, say 4 or 5 schools regularly, or maybe more. Kind of like the way motivational speakers do, but on a more regular basis, and teaching real material.
And here’s what we need to change in the curriculum: stop dumbing it down! Move material in the curriculum down the grades, not up. Eliminate more useless parts of subjects (what is with the ‘keep a journal of your math discoveries’ a la grade 6, or some really dumb writing assignments which end up being only useful for satisfying the EQAO preps?). Start connecting different subjects earlier, not later. Invest in libraries, give students something to be excited and interested about.
How do you make sure teachers are doing their job? Talk to the damn students! How many times have we seen those staged teacher evaluations? The teachers (who aren’t supposed to) tell us they have a teacher evaluation, get us on our best behaviour, and they teach an uncannily ‘boring’ lesson. The only people who know how good teachers are doing are students. Get them to fill out surveys, or get some admins/staff to talk to students. It’s not THAT hard.
Those are just off the top of my head. I’m sure people can think more about it, and get more ideas. Things can be better.
Discrete was stupid and so is every course in the ontario curriculum.
It was just a course that crushed together everything that should’ve been taught back in elemntary school. It’s challenging because it rushes through the material fast and doesn’t teach the stuff properly. Honestly, wtf does euclidean geometry, combinatorics, linear algebra and a tiny bit of number theory have in common? nothing except they were all ignored by the ontario curriculum.
AP calc is pretty noob too, go look at IB HL math exam and come back tell me AP calc is challenging. AP calc is not nearly as challenging at other schools, RHHS AP calc is challenging cause brar doesn’t teach anything and expects u to find everything on ur own.
@Anonymous:
Discrete was as close as real mathematics as you’ll ever get from an Ontario high school. They were teaching pure mathematics, and nothing else. Whether the course had a theme or not is a different matter.
Is it challenging because it’s fast? Well, sure – but if you were an interested math student, then the material speed would not have been ‘fast’, but just ‘adequate’. The course was designed for interested and intelligent students – not just for everybody. Perhaps you may not have enjoyed it; or maybe you didn’t belong in the class.
AP Calculus, of the BC variety (as opposed to the AB taught in most AP programs in Ontario) is in fact at about the same level as the IB exam. It is also on par with the British A-level math programs.
“RHHS AP calc is challenging cause brar doesn’t teach anything and expects u to find everything on ur own.”
-That’s the only sentence in that paragraph that I agree with.
IB maths only gives you the basic scope of calculus. it doesnt even challenge you. most of the questions are like “integrate this”, which requires no thinking. And about that brar comment, its true he doesnt really teach much, but thats teh beauty of learning- you shouldnt have someone always feeding you all the answers, you ahve to look for the answers yourself. especially in more technical subjects such as math and science.
Dear Anonymous number 1,
You are obviously a person that has yet to come and experience the beauty of ‘university learning.’ In university is where I think you will get to experience the true meaning of “rushing through the material too fast without teaching the stuff properly” from professionals of the subject area lightyears ahead of your knowledge in the subject and who barely even care if you even learn the material at all.
However, in addressing your main point. I agree with Kevin that Discrete it is probably the closest thing you’ll ever come to real math in high school, but I also agree with you that the course does is fairly convoluted with some random bits here and there and some sections that are given as methods without any mathematical basis (esp. the linear algebra portion with the determinants, Cramers Rule, etc.) All I have to say is don’t worry, if you want, you can get the full explanation a real university linear algebra course =).
-ray
Oh, I thought Brar actually taught us some stuff…Maybe it’s just me.
@Anonymous(1)
It bothers me when I hear people say something like every course in the Ontario curriculum “sucks” or is “stupid”. I agree with Kevin that overall our quality of education is degrading, but that doesn’t mean one can overgeneralize it for “every course”.
I’m going to assume you haven’t taken “every course” in the Ontario curriculum to actually make a viable statement like that. Or maybe you did, then my bad.
Now I’m not defending all courses of the Ontario curriculum, but believe it or not a course or two has actually changed my life. But that was because I actually chose courses in my electives that were useful to myself. And the courses only changed my life… because I let it.
You’ll be institutionalized if you let your institution “define” your leaning. Learning has nothing to do with what you’re taught.
(Sorry that was completely off topic.)
Do you know what a characteristic polynomial is ? or the harmonic division of a line segment is?
Math is not just about how in depth you learn calculus. IB Math learns a much greater variety of subjects in Math that AP can’t compare with.
I’m not saying IB Math is the best and I do admit they stress breadth rather than depth, but it is better than AP, IMO. And I mistyped when I said “every other course in the Ontario curriculum”. I meant “every other Math course…”. I also have to say that how “good” a course is often in Ontario determined more by which teacher is teaching it rather than the material that is in the course (This explains the sometimes ridiculous discrepancies in marks, how one school’s 95% is another school’s 80%).
And how many have seen a Chinese University Entrance Math Exam Paper? I’d be surprised if more than 10% of the Ontario students going into Math and Engineering programs could pass it.
I’ve seen a friend of mine use his Grade 10/11 Algebra book from Hong Kong for our first year EngSci course on Linear Algebra (normally a second year course for most programs in Ontario).
Just more evidence that Ontario’s system is quickly and surely falling behind the rest of the world.
Outside of their regular studies, students are still free to ‘learn’ anything they choose. Therefore, I suspect that any intelligent student who ‘really wants to know’ will end up studying geometry and discrete mathematics on his or her own.