# What is a good pacing for a Calculus 1 undergraduate course?

I am going to teach a Calculus 1 course next semester, and I have 15 weeks for the course material. The class meets MWF for 50 minutes each. I have taught this class before using the same syllabus, but my colleague shared concerns that my pacing is too fast:

Week 1: Review of Functions

Week 2: Limits and Continuity; Infinite Limits

Week 3: Derivative (Limit Definition); Differentiation Rules; Transcendental Functions

Week 4: Implicit Differentiation; Related Rates; Linear Approximation and Differentials

Week 5: Extrema; Curve Sketching; L'Hopital's Rule

Week 6: Optimization; Newton's Method; Antiderivatives

Week 7: Integrals; Fundamental Theorem of Calculus

Week 8: Applications of Integrals: Work; Areas; Volumes of Solids

Week 9: Integration by Parts; Partial Fraction Decomposition

Week 10: Trig Substitution; Approximate Integration

Week 11: Arc Length; Surface Area

Week 12: First-Order Differential Equations

Week 13: Parametric Equations and Polar Coordinates

Week 14: Introduction to Sequences and Series

Week 15: Review for Final

This is the syllabus I used last Spring, and I didn't have any problems with running out of time. I get straight to the point with my lectures, and my student grades have been above average compared to other instructors. However, the exams I use from my department only cover up to the fundamental theorem. So my students end up with a lot of excess information, since I cover up towards the end of a traditional Calculus 2 course. This means they're more than prepared for Calculus 2. However, I don't feel like going slower if I do not have to. I mean, if I were to get rid of the first week review, I could theoretically cover all of Calculus 1 and 2 in one course. Not sure why my colleagues take so long in lecturing. I sat in on a class and it took my colleague the entire 50 minutes to teach about power rule and product rule, when in the same time frame I can cover all differentiation rules plus transcendentals. Student evaluations seemed to be good. In a class of 33 students, 26 got "A's" and 5 got "B's" and 2 got "C's". No one failed.

• Assuming this is in the U.S., or at least the students and course level are similar to that of a typical first semester calculus course in the U.S., I find it difficult to believe you can cover all this, which is essentially most everything in the first two semesters of calculus. Instead of sitting in on someone else's class (good to do, but you've done this), it would have helped to have someone sit in on yours. And I'm surprised your students didn't complain and rise-up-and-revolt at having to cover so much more than the other classes! Nov 10, 2022 at 9:53
• Do you squeeze in time for chapter or midterm exams, or do you just have the final exam in week 15? In the US, a MWF class meeting 50 minutes would be a 3 credit course (semester system). The vast majority of Calc 1 courses in the US are 4 or 5 credits, for a good reason, and typically cover up to the Fundamental Theorem of Calculus. Nov 10, 2022 at 12:34
• 1. Are your departmental exams graded by you, or by a departmental grader? 2. Have you received any feedback from the students you taught last Spring on how they have done with differentiation and its applications in other classes this semester? It seems like a possible downside of this schedule is insufficient student mastery of differentiation, hence the questions about the in-course and post-course measures of success. Nov 10, 2022 at 14:54
• I don't think the pace is too fast for lectures alone, but I do think it is pretty fast for material to "soak in". I usually teach high school calculus, so college may have a faster pickup, but it usually takes longer for students to actually internalize the material in my experience. I normally would at least split week 3 into three weeks - what is a derivative (orient to topic), what are basic rules (practice), and what are the more extended rules and compositions of rules (deeper). Another question, though, is whether or not your students had high school calculus. Nov 10, 2022 at 16:06
• Your question reminds me of this story: mathoverflow.net/a/53238/297 Nov 10, 2022 at 16:54

As fedja says, if your students are doing as well as you say, and you believe the students are similar this year, there isn't a reason to change.

It seems hard to believe, though. I've taught calculus at U Michigan and UC Berkeley, which are generally considered to be good schools, and in 14 weeks with 4 hours a week, we generally get to your week 7. To give very rough numbers, about 25% of our students skip over first semester calculus and take a more advanced class, and about 50% never take it; so this course is targeting the 50th-75th percentile of math preparation. A bit more than half of them have had some high school calculus. Of course, we also have honors courses that cover much more; I'm talking about the non-honors section.

The reason is that the students who take these classes don't have a strong understanding about how to think about functions, graphing and formulae. So that each concept has to be presented many ways before they can use it. Some reasons you might be seeing different results:

1. You are an amazing instructor.

2. You are at an extremely selective college. (But then why do your colleagues not get the same results?)

3. For some reason, you have unusually strong students in your section. Do your students come from a different major than the norm, or are a large number of them transfer students from a different educational system?

4. You are pushing out the students who would fail before they get to the final exam. How many students drop the course, or change out of your section, before they get to the final?

5. Your exams are way too easy.

6. There is a large amount of cheating on your exams. (Were they administered online?)

If 1,2 or 3 applies and will apply next term then what you are doing is working so there is no reason to change. If 4 applies, then I believe you should work to keep and help those students, although it gets complicated. Obviously, if 5 or 6 applies you should fix your exams!

And, if 1 applies, then congratulations!

• The exams are straight from the department, and I do not grade them. Median exam score for my section was 86% last time I taught this course. The problem is since the department exams are based on a traditional Calculus 1 pace. So the midterm covered up to derivative rules and applications, while my students were learning about applications of integrals by the midterm. Nov 10, 2022 at 18:31
• Colleagues generally have lower section scores, but I never revealed to them about my pacing until recently, hence the concern one of them raised. Nov 10, 2022 at 18:40
• Exams are in-person. Midterms and Final by the department. I give out weekly quizzes. Homework is assigned for extra credit up to 2% of the final grade. Besides that, both exams make up 60% of the grade, quizzes make up 20%, and a small quiz/test I call "quests" during Week 4 and 10 that add up to 20%. Nov 10, 2022 at 18:42
• I am not at a selective college. This is a small liberal arts school in rural Midwest. Nov 10, 2022 at 18:43
• Okay, central exams take out 5 and probably 6, you've ruled out 2, and presumably you would have mentioned 4. So I'm down to 1 or 3. Maybe you are just really good! Nov 11, 2022 at 5:27

Just do what works for you and your students. If they get a good grasp of the material (which you should check regularly) and do not look terribly overworked, I see nothing wrong with going fast. However, if you see that a noticeable portion of the class is falling behind, slow down and allow them more time to digest everything. Don't get surprised if the situation changes from year to year and even from topic to topic. The only danger arises when you fool yourself into believing that the students understand the lectures when they actually don't. Then the things can go astray really fast and get really ugly in the end.

26 A's is an amazing result for an undergraduate calculus course (when I have 30 students, I usually give not more than 10, though I never curve, just set the bar for the passing score in the beginning and stick to it). Make sure that you don't demand too little (another standard way of fooling oneself into believing that everything works just fine).

So, if everything is as you presented it, just don't change anything unless you see yourself that you have to. I have no way to check your claims against reality, so I leave it to you and your peers. But if you are not deceiving yourself, then my congratulations: you seem to have really good students and, probably, found an efficient way to teach them. Just make sure it is not a mirage. I agree with other commenters that it looks somewhat unbelievable. If not a secret, where are you teaching?

• I don't grade the exams. They are created and graded by the department. The grading scheme is as follows (set by the department): 85% is an A; 75% is a B; 60% is a C; 55% is a D Nov 10, 2022 at 18:38