Math to the People

(this post is a fleshed-out version of the 5 minute talk I gave yesterday at the First Year Math and Stats in Canada conference in Hamilton, Ontario)

Every single precalculus textbook I have used introduces the material in roughly the same order. First, functions and some of their general properties (domain, range, etc.) are introduced. Then, the algebraic operations on functions, function compositions, and inverses are introduced. Polynomial and rational functions then tend to be the subject of the next few chapters, followed by exponential functions and trigonometric functions. Usually, following the textbook order results in students being first exposed to logarithms around the midpoint of the course, and trigonometry occupies the final third of the course.

I believe that this order does a disservice to our precalculus students. I believe that the all transcendental functions should be introduced before the midpoint of any college-level precalculus course, and that the exponential, sine and cosine functions should be introduced before the first midterm if there are two midterms. What this looks like in practice is of course up to the individual instructor, but I believe that the exponential, sine, and cosine functions are such central examples used in calculus that they should be treated as examples in precalculus rather than subjects unto themselves. For example, when composition of functions is first introduced, a composition of an exponential and a sine function like e^(sin x) should be one of the examples shown.

Some may wonder what we will be doing in the second half of a precalculus course if all the important functions are introduced before the midpoint of the course. I believe that, just as the first half of precalculus should be spent on defining and learning to evaluate the basic functions, the second half should be devoted to learning what we can do with these functions. Topics such as learning the algebraic laws of logarithms and using trigonometric identities to solve equations can take up a fair portion of the second half of the course. Applications of both trigonometric and exponential functions can also be saved for the second half of the course.

There are five main advantages that this treatment of precalculus will have over the traditional textbook order. The first is that introducing all the basic transcendental functions early makes it easier for instructors to create assignments that allow for interleaved practice. Interleaved practice is a technique where students alternate between problem types when practicing instead of completing all problems of type A before moving on to type B. Interleaved practice – while students tend not to view it favourably – has been show to produce better outcomes in studies such as Interleaved Practice Enhances Memory and Problem-Solving Ability in Undergraduate Physics. By introducing the basic functions early, it is easier to give students an algebraic example followed by a trigonometric example followed by an exponential example, forcing students to undertake interleaved practice.

The second advantage of the Early Transcendentals approach in precalculus is that it allows for spaced recall to improve retention of knowledge among students. If a fundamental piece of knowledge is tested at multiple times throughout a course, it has been shown (in studies such as Spaced Recall Reduces Forgetting of Fundamental Mathematical Concepts in a Post High School Precalculus Course) that students are much more likely to remember this piece of knowledge in a future course. The basics of exponential, logarithmic, and trigonometric functions are so fundamental in Calculus that it is necessary to ensure that they are tested at multiple times throughout a course. If problems involving those functions appear on two midterms and on the final exam, students are much more likely to remember their definitions when they get to Calculus.

The third advantage of this approach is that it provides a treatment (but not a cure) to what I refer to as end-of-semester syndrome. At many institutions where I have taught, attendance has dropped dramatically in the last three weeks of a semester, largely due to term projects and lab exams in other courses. These last three weeks of the semester are the wrong time to be introducing something as fundamental as unit circle trigonometry. I believe that these last three weeks would be better used to extend concepts that were already introduced earlier in the course.

The fourth advantage of Early Transcendentals is that it makes it easier to establish connections between ideas. For example, if trigonometry has already been introduce when rational functions are first introduced, we can immediately use the method taught to find the vertical asymptotes of a rational function and apply it to the secant or cosecant function to find its asymptotes. If logarithms and inverse trigonometric functions are introduced in consecutive classes, the abstract concept of an inverse function will be more clear. Similarly, if power functions and exponential functions are introduced in quick succession, it is easier to contrast them with each other to prevent students from confusing them in calculus.

The fifth and last advantage of the Early Transcendentals approach that I have identified is that, when entering calculus, students need to be able to be prepared to solve problems that put different “types” of functions together. For example, in calculus, students will need to be able to differentiate composite functions such as sin(x^2) or x^3 e^x . If the exponential, sine and cosine functions are not introduced until later in the course, there is less time to expose students to these sort of examples, which may cause difficulty for students when they get to calculus. The more that students are exposed to ways in which the different basic types of functions can be used together, the more prepared they will be for calculus.

While I do believe that this approach to precalculus will result in students perceiving the course as being a ‘harder’ course, I believe that introducing the exponential, sine, and cosine functions in the first third of the course will result in better outcomes for students. I believe that the interleaved practice will give students a greater ability to score higher on their final exam in Precalculus and that the effects of spaced recall will ensure that students will remember these topics when they get to Calculus. I believe that more time to compare and contrast functions and mix them together will result in students being more ready for their calculus course when they get to it.