An In-Depth Design of Introductory Calculus

Response paper for the session "Course Design"
Certificate in University Teaching Program - University of Waterloo

In this essay, I plan to investigate course design alternatives for introductory Calculus, often called "Calculus I" in various science and engineering disciplines. It is a course upon which virtually all university mathematics courses are built, therefore its thorough design for maximal effectiveness is of utmost importance. Mathematics is a primarily cumulative subject, thus out of consideration for the learner, the design of a math course must be carried out with great care. Despite the rational nature of the field of mathematics, it does not necessarily imply ease of reasonable course design. The "elementary" nature of the course for many professors, in itself can imply carelessness. In this essay, I will attempt to extensively detail some pitfalls of this course's design, as well as make constructive practical suggestions which I believe would improve in general the prevalent methodologies in use. It is a course I have been involved in teaching several times, and I feel confident in sharing my predispositions still under development.

I believe that the recurring issues with first-year Calculus courses spring from two main sources of pitfalls: (1) the "toolset" nature of the material, and (2) its elementary nature for the instructor. These two sources may branch out into an avalanche of consequences, that may fall upon the unlucky learner at any time, without them realizing in time, or even later, its adverse affects on their early mathematical foundations. Naturally the learner is mostly helpless, and must rely on the instructor's wisdom. I believe a wise instructor remains a student when s/he enters the classroom to teach, regardless of where s/he might be in terms of career. This is clearly difficult to do so, and is perhaps the main challenge of teaching, to see the material fresh and anew every time one teaches it. Then and only then, may one's teaching become fully didactic. This is imperative with such a foundational course, as a didactic structure transfers into the mind of the learner, in building their own inner picture of the material. Therefore via attempting to remain a "learner in progress", the instructor consciously must shred the "elementary feeling" inherent in the task of teaching Calculus I.

As far as the first pitfall stated above, it is somewhat more striking and therefore easier to identify. First let us confess that Calculus is indeed a toolset, but at the same time it is an interconnected living breathing theory, with a historical past and a future. In my opinion, a well-designed course incorporates a permeating veil of rhetoric in the material, which subconsciously brings the learner to these realizations, essentially showing the beauty and relevance of Calculus. This imperative factor in the design has its own consequences, not the least of which is student motivation. The mindset of all the students in a class can be taken on an ideal course, by an instructor who projects their own appreciation for the theory. These emotional ties may be coupled with rational reasoning for the theory's relevance, with the vast spectrum of theories and applications which hinge upon it. This double parallel subconscious reasoning should enter the learner's mind sufficiently, to induce a seductive motivational effect. Among all the distractions of a freshman's college life, such an attempt at seduction seems quite reasonable, the ends thereby justifying the means.

The above stated points are meant to assist the instructor in taking an ideal philosophical approach to teaching an introductory Calculus course. The actual course design however must be guided by the intended learning outcomes. As stated in the pre-workshop worksheet, by the end of my Calculus course the students should be able to...
• provide an overview of elementary Calculus.
• state the definition of a function, and derive its domain, range and inverse.
• calculate the limits of a wide variety of functions.
• explain and apply differentiation and integration techniques.
• elaborate on the conceptual meaning of the learned techniques.
• model and solve mathematical and real-world problems using Calculus.
I shall elaborate below on the often ineffective ways of attaining these ILO's, and suggest some alternative approaches, in light of Fink's taxonomy.

By "providing an overview" I mean the ability to rise to a global perspective above the material and seeing its inner multilevel connections. This is not a skill that comes easily, and it might even be one of the highest-level skills attainable, which could be why many students tend to lack it, and most instructors fail to encourage it. It ties in directly with the first pitfall above, as an overtly toolset-style teaching works against the development of a global perspective. Therefore, the instructor must consciously focus on this element, in order to counter-balance. It ties in with the idea of "conceptual understanding" under foundational knowledge in Fink's taxonomy. Indeed, the goal must be to develop a complete understanding of elementary calculus, as a foundation for future subjects. On the other hand, this ILO also has an "integration" aspect as per Fink, since connecting ideas is essential to being able to having an oversight.

Stating the definition of a function, and working with its domain, range, or inverse are basic elements of the toolset of Calculus. As such, they represent foundational knowledge that serves as skills for proper performance in cumulative topics building upon them. The pitfall here is that the instructor may forget to remain a student, and may rush through these imperative ideas, without the full comprehension of students. I have observed this multiple times as a student and as a teaching assistant. Clearly, this ILO may be addressed with a sufficiently thorough treatment by a conscientious instructor.

Calculating the limits of functions, can become a stale and mechanical topic, if presented without the intricacies giving its beauty. It can quite easily be reduced to a set of calculation rules and dogmatic methods. This reduction is certainly not the challenge of an educator, but quite the opposite. Without going into too much detail, one may build towards this intended learning outcome, by discussing the nature of the "limit" in the more rigorous sense. This would justify and establish its relevance, giving it sufficient weight in the mind of the learner. Otherwise, one may take this seemingly simple concept for granted. Such a somewhat more precise discussion, would lay the foundations for the eventual introduction of the concepts of continuity and differentiability. Thus making the effort to introduce the approach early on, shall pay off later to the learner's benefit. In Fink's taxonomy, this may be considered a way of motivation, by projecting the exciting implications into the future.

The next learning outcome is tied to the one following it for a good reason: to once again avoid the potential reduction of the material to mechanical calculations. Understanding the conceptual undercurrents of the topics of differentiation and integration, in and of itself is sufficient to motivating students towards exploring these theories created by scientists over two hundred years ago. The idea of time-traveling to walk alongside these pioneering explorers of the world of mathematics, may awaken the more romantic side of a learner, thereby serving as motivation for learning. I am speaking from personal experience, that such motivation can be deeply effective and long-lasting. It is precisely the "human dimension" of caring to develop feelings, interests, and values in Fink's taxonomy, which these thoughts of mine converge towards.

Reaching the level of ability to independently model and solve real-world problems, is clearly an expected goal for learners in applied disciplines. It is also clear, that the maturity and readiness for independence comes gradually, over the span of several courses, not only introductory Calculus. Yet, I do believe that the instructor must make steps towards and assist the development of learner maturity - again, this is one of the intricacies of teaching that often goes ignored, however essential. These steps could be gradually built up from smaller exercise-level problems, to the level of term projects in later terms. In Fink's model, it invokes the full spectrum of implications of concept "application", such as critical thinking and the ability to make project-directing decisions. It is primarily a way of "learning how to learn" - or even to research - independently via intriguing applicational problems. Inducing independence in students may thus be the most challenging yet most important aspect of teaching, matching the magnitude of its relevance.

In my essay I have reflected on my personal beliefs and strategies for designing an ideal introductory Calculus course. The elementary nature of the material is never to be confused with the significance of this contemplation. Indeed I have outlined two major pitfalls such confusion may lead to, as well as analyzed the intended learning outcomes which should guide the process of course design. I found this discussion all-in-all valuable and considerably self-reflective, hopefully increasing the effectiveness of my future endeavors.