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

The Removal of Calculus from Ontario High Schools

(Some quick comments: At the time of the writing, the Ontario Ministry of Education has deferred the decision to change the curriculum until 2007, but the facts remain and still hold true.)

For those who are unfamiliar with Calculus, it is a small strand of mathematics that essentially studies the rates of change of variables, but is often extended to include broader concepts such as functions and relations and equations on the Cartesian plane. Its ideas have been around since the ancient Greeks, but were not formalized in the 17th century by Newton and Leibniz. It has been used since the 17th century by various fields of science and mathematics to solve difficult problems and develop new concepts.

Early in the 2005-2006 school year, the Ontario Ministry of Education held a review of the secondary school curriculum in Ontario, particularly in the mathematics area. Among the most notable changes was the removal, or indeed, substantial reduction, of calculus from the grade 12 curriculum. The primary reasons given for the sweeping changes in the grade 12 math program were that dropout rates were increasing, enrollment into certain mathematics courses (particularly the infamous Geometry and Discrete Mathematics) were decreasing, and rates of failure in grade 12 Calculus were increasing. The proposals also noted that there was 'not enough time' to teach Calculus in a 4-year high school system, and that the subject was only necessary for careers in engineering, science, political science or physics. These proposals not only do not make any sense for the advancement of education, but will also hinder the future of Ontario's workforce.

First, the allegation that Calculus is too difficult a course for high school students is ridiculous. Calculus has been around at the high school level for several decades. Ask your parents, and they have probably sat through a Calculus class, or at the very least, had the option of doing so during their high school years. Calculus is also offered worldwide in high school, and in almost every other province in Canada. So what has changed in the last few years in Ontario that has caused an increase in the failure rate of Calculus students? Well, to answer that question, we will not look at the Calculus course, but at the foundations of mathematical education in Ontario. In Ontario, mathematics is often unfairly treated as an extraneous subject that is 'feared' by students. This attitude of education has caused a dramatic lag on the teaching of mathematical concepts in early education. Simple concepts of counting, addition, and subtraction are often left out in kindergarten and are sometimes taught as late as the second grade, usually out of fear from the teachers that students will find the subject "too difficult" for them. In turn, there is a cascading effect on the rest of the student's life in school. By the time students reach the sixth grade, they have learned only the bare basics of mathematics, but have developed a culture of fear around the subject. Back to Calculus, the subject itself requires a few mathematical concepts to have been learned by the student. They include an understanding of equations, functions, the Cartesian plane, and the ability to manipulate algebraic expressions. These skills are not very difficult to master, and generally do not require much conceptualization on the part of the student once it has been taught. Therefore, it is not much of a stretch to say that these concepts should have been taught from grades 9 through 11. Unfortunately, because of the culture of fear of mathematics and the delay of its teaching, students are still learning about fractions and division in grades 9 and 10. Thus, it is easy to see how students are going into Calculus without the necessary skills, and will naturally find the subject difficult. Calculus teachers are left with no other choice but to reteach these concepts and skills, thus leaving no time for the actual "calculus" part of the course. The fault in the system here is that mathematics is being dragged along the curriculum too slowly at the elementary level, leaving students wholly unprepared for the real world of math by the time they reach high school. Therein lies the biggest problem in the approach to this problem by the Ministry of Education - instead of starting at the root of the problem, they are only attacking the symptom, and the fatal flaw remains. Indeed, a change from the bottom up would take some time and considerable effort, but with the advancement of the modern world, it is inevitable that changes will happen, and if the current trend continues, Ontario's education system will lag far behind the rest of the world, and schools will not be prepared for the future.

Recently, Calculus has been used as a convenient excuse for students dropping out of school, due to its irrelevance to the students and its level of difficulty. This is no more convincing than the student's popular "I left my homework at home" excuse. It may be true that dropout rates are increasing in high school, but this fact is completely independent of students taking the Calculus course. In Ontario, Calculus is only offered as a '12U' course, meaning it is geared towards more hard-working, intelligent students aiming towards post-secondary education in university. In any graduating class, the number of students heading towards university and college are about the same (although the trend is leaning towards the university side of things due to parental and educational pressure). The system is designed in such a way (like it or not) that only the more academically inclined half of students from grades 9 and 10 move on to grade 11 and 12 'U' level courses. Generally, students who drop out of high school are failing courses, and by the same token, if a student has bad grades during their first two years of high school, they are strongly encouraged against or even prevented from taking 'U' level courses in the senior years. Thus, although it is more difficult to prove statically due to the lack of information at this level, it is probable that of the students who drop out, only a very low percentage have taken Calculus, and were not influenced by the subject. The reasons behind the dropout of students vary, but they will not be discussed in the space of this text.

The extent of use of Calculus in various fields has been debated by the Ministry of Education and their study, and they have come up with only a short list of careers that require the study of the strand of math. Those include engineering, science, political science, and physics. As a Calculus student, I find that list highly selective and is a gross misinterpretation of the flexibility of Calculus. In fact, many concepts and skills learned in Calculus are essential to business, medicine, and even certain arts, like architecture. Those who believe otherwise do not understand the essential concepts behind Calculus and do not realize its power. It is inconceivable that students will enter university programs without learning any calculus, even though it is an integral part of modern science. Where I will concede to changes in the curriculum is in its focus, but I will propose an alternate method of changing grade 12 math. Currently, Calculus is the focus of the grade 12 mathematics curriculum, and the main mathematics courses in grades 9 through 11 all gear students toward grade 12 Calculus. Although Calculus is very powerful and can be used for a great many purposes, it should still remain an optional course reserved for students who are interested in the subject, or have a need for it in the career path they choose (particularly for computing, engineering, physics, and mathematical studies). In place of Calculus, a fourth grade 12 university level mathematics course should be implemented. This course would include in it the concepts of algebra, equations, graphical analysis, and more real-world applications of mathematics, which could mean a movement of the current grade 11 finance unit into this new grade 12 course. Other changes to grade 12 math could potentially include the removal of the last unit of Geometry and Discrete, as it overlaps with the Data Management course and is often not taught, and the movement of Conic Sections into the grade 12 Geometry & Discrete course. These changes would allow teachers in grade 11 to focus on the development of vital mathematical skills required for all of the grade 12 math courses, and allow more time in earlier grades to cover geometric and algebraic concepts that have been neglected in the current curriculum (these include Euclidean Geometry and trigonometry). The choice of four different grade 12 mathematics courses for university-bound students with the new focus on the algebra skills course would make much more sense than cramming many unrelated concepts into the Advanced Functions and Introductory Calculus course as it is being done currently, and would leave Calculus as the third or fourth math course behind the new proposed Algebra course and the current Data Management course, the latter of which has had a reputation of being the 'easiest' of the three current senior math courses, and is often taken by students in grade 11. The Calculus course and the Geometry & Discrete course would thus be left only for stronger math students and those who are motivated for a challenge. This solution allows more flexibility and choices for the average student, while both reducing the focus on Calculus as well as keeping the course available for students who require it.

Moving forward, it is important for Ontario's Ministry of Education to realize that removing Calculus from the curriculum will not solve the problems they have identified themselves. The problem should be attacked at its root, that is, the culture of fear that has developed around mathematics, even with its simplest concepts and skills, especially in early education. Without a solid foundation of the concepts behind math, it is inevitable that students will find Calculus and its related subjects hard to understand. The focus of Calculus in the grade 12 curriculum should be reduced, and deferred to a new course which will encompass and bring to a closure all the algebraic skills that will be useful in practically every career path. Calculus should be kept as a third or fourth senior mathematics course, for students who still need it for their post-secondary education or have strong interest in the subject.