I posted earlier about how I was surprised that a typical Calculus 1 course that meets 3-4 hours each week for 15 weeks only barely manages to reach the fundamental theorem by the end of the course. If we consider James Stewart's Calculus: Early Transcendentals, then the fundamental theorem of calculus is towards the end of Chapter 5 (5.3-5.5). As a reference, Chapter 1 is a review of functions, Chapter 2 is limits and the notion of a derivative, Chapter 3 is methods of differentiation and basic applications (related rates; differentials), Chapter 4 is more applications of derivatives (optimization; curve sketching), and Chapter 5 is integrals (culminating in the fundamental theorem).
There are 5 chapters with a total of 38 sections. And given 3 lectures per week per 15 weeks, there are 45 class meetings in a semester. Taking into account days for exams and other holidays, that leaves still about 40 classes, which is two more lectures than I need. In other words, the rate is 0.3 chapters per week.
I mentioned in my previous post that I am able to, for the most part, end my Calculus 1 course at around Chapter 11.3, which covers the integral test for infinite sequences and series. This is usually the last chapter of a Calculus 2 course (Chapters 6 - 11; although some instructors skip Ch. 9). This ends up being an average of 0.73 chapters per week.
I have not had any negative effects due to teaching at such a fast pace, but I decided to draft a syllabus that goes at the common pace. In this syllabus, I basically ended up having one 50-minute lecture for every single section of the text, with room to spare for quizzes on certain days. This rate allows me to finish at the fundamental theorem of calculus. For example, in this new syllabus, I am dedicating an entire 50 minutes to Chapter 3.2 (Product and Quotient Rules).
In my "fast" version of the lectures, I typically cover all rules of differentiation and trigonometric derivatives all in about 50 minutes. As such, I do not know how to drag out only the product and quotient rule for 50 minutes. I do not see any point in providing the amateur proof of these rules, since I am not teaching for math majors.
If any of you are familiar with Professor Leonard's YouTube series, his lectures about the product and quotient rules is one hour long! Is his style the same way most instructors teach? The comments from students generally say positive things about his content, but my students also say positive comments about my teaching in the student evaluations. Most of the comments point out that I am quick to the point with the material and examples.
My students have had success, but the only issue I see is that since the exams are departmental, the material being tested is usually topics we covered weeks ago. But because calculus is cumulative, it ends up not being too much of a problem. For example by the time they are taking the final exam, my students have had a lot of experience taking derivatives when doing integration by parts and other integration methods. Similarly, they breeze through the exam questions that are about basic integration, since they have experience with more advanced methods of integration. They can also compute limits much faster with knowledge of L'Hopital's Rule and the problems they did with limits in the improper integral sections. I have even had a handful of students test out of Calculus 2 after taking my class, since I essentially covered 90% of the material of Calculus 2.
I am not at a selective college. It is a small liberal arts college in the midwest. My students are not in an honors section. Exams are departmental. I do weekly quizzes, and two semi-exams in the semester. Homework is assigned but not graded for credit. The entire grading scale is based on assessments only, there are no cushion points.
I will be teaching Calculus 1 again next semester, and I will also be teaching Calculus 3. One problem with this is that I have already created lectures for Calculus 3 (covering Chapters 12 - 16 of Stewart's). My current playlist has 21 lectures each that is 50-55 minutes long. That means that if I follow the pace of may already recorded lectures, I will end up covering all of Calculus 3 by Week 6, and that still leaves me with 9 more weeks of class with nothing to do. Should I just start them on differential equations after? I guess I am not sure how much the students will grasp the material, but even so, I think Calculus 3 is easier on students than Calculus 1 (I won't have to review functions or worry about the students understanding the actual notion of derivatives and integrals). I asked my colleague for a copy of his calendar for Calculus 3, and he spends about 3-4 lectures per a single section (about an entire week on just double integrals over general regions, whereas in my 50-minute lecture, I covered double integrals in rectangular and general regions, and in polar coordinates.)
All advice is appreciated. I am always seeking to be a better instructor.