What do you do in order to drag out lectures?

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.

• Assuming you're truthful about your students success, apparently you don't need the following, but for what it's worth I note that you are almost entirely focused on lecture time. What about student questions about homework problems, having students present (at least some) homework problems at the board, in-class working together in small groups, student presentations of various sorts (usually this for upper level classes only, but this is not all that unusual even in first year calculus courses at small liberal arts colleges from my personal experience), etc.? Nov 14, 2022 at 10:52
• If your Calc I lectures are recorded like your Calc 3 material, they can easily be benchmarked against Youtube comps to more concretely understand where your time savings is coming from. For instance, the Prof Leonard product and quotient rule video is about 1 hr long which includes about 10 minutes of stating and motivating the rules, 45 minutes spread across 7 examples, and 5 minutes of overhead. I have not found a course that purports to cover "all differentiation rules" in fewer than 3 lectures, so under rough assumptions your pace is at least 3x faster than a Prof Leonard video.... Nov 14, 2022 at 11:25
• "I do not see any point in providing the amateur proof of these rules, since I am not teaching for math majors." How do you know? I didn't switch my major to math until I had already completed Calc 1-3 and linear algebra. Besides, plenty of things at this level can be proved (put quotes around it if you prefer) for any audience. A car driving down the road is still my preferred "proof" of the FTC, for example. Nov 14, 2022 at 14:43
• The title of your question is disrespectful to the ways many of us help our students learn. I do not "drag out" lectures. In fact, lecture is a small part of my classtime. Nov 14, 2022 at 15:57
• @mrwillparker: Could you clarify what you mean by "amateur proof" (as opposed to "non-amateur proof") of the product rule? In any case, I'm not sure why you think that no proof (or at least heuristic argument) for the product rule should be given to non-math majors. The argument is pretty simple and, more importantly, one can actually give a very clear heuristic explanation of the rule by a simple picture. In fact, I've first seen such a picture in a basic book on technical thermodynamics. Apparently, the author thought it would benefit the readers to understand why the product rule is true. Nov 15, 2022 at 8:10

8 Answers

What you are describing is so far outside of my, and I suspect most educators, experience that it appears to be literally incredible.

The very strongest Universities in the country, with some of the best prepared students and very well designed Calculus courses (such as the University of Michigan), still struggle to fit all of the material of Calculus 1 into a single semester while having the majority of students achieve competence. Doing both semesters of Calculus in a single semester via a "straight lecture" approach and having students excel is an extreme anomaly.

My first suspicion would be either rampant cheating or tests which vary so little from year to year that students can easily memorize their way to a passing grade.

Do you have one on one conversations with your students? Do you find that they are able to solve problems equally well during office hours as during the exam?

The number of basic misconceptions even bright students bring with them from high school is staggering in most cases.

Here is an example (which has research to back it up): Describe a car driving along a N/S road with an intersection. Give a graph of a function $$f$$, together with the interpretation that $$f(t)$$ represents the displacement of the car northward, in feet, relative to the intersection $$t$$ seconds after passing the intersection.

Many students will believe that the graph of $$f$$ is a sketch of the motion of the car in two dimensions. They are not able to correctly interpret that the car is moving only in one dimension, and that the horizontal axis represents time, not a spatial dimension. If you give them a toy car and ask them how the car is moving, they will move it along the graph, not N/S in a line.

A lot of the time I use to "drag out" my lectures is spent on correcting such fundamental misunderstandings by giving students a wide variety of conceptual problems using multiple representations and having students work (often in groups) on engaging with these problems meaningfully, and bringing the mathematical techniques we are learning to bear on these problems.

• How are the axes labeled on that graph? Nov 16, 2022 at 2:51
• Hopefully something like "Displacement (feet)" and "Time (seconds)" Nov 16, 2022 at 13:06
• That should be more than enough information. Then there is something fundamentally wrong with education system. Nov 18, 2022 at 1:11
• @PeterMortensen I agree! When students mostly have lecture based instruction, and when the questions posed to them are almost entirely of the form "Find the equation of this line in slope intercept form" and other mindless procedure following, students really do not even understand that they can attempt to make sense of mathematics or to use it to model the real world. Nov 18, 2022 at 11:10
• Information from lectures can only be analyzed as long as it's in working memory, or some other form of memory (and without a fundamental understanding to build on, it's harder to remember the blocks). Even a very fast typist can have trouble keeping up with what a person is saying, taking notes (especially if it's complicated). If someone doesn't understand a problem, it's difficult to determine what's relevant to write down or not, if they have to pick something. Nov 25, 2022 at 0:54

What do I do? First, explain the product rule. Let's say 3 minutes on that. Then write a random product of 2 functions and differentiate it. 2 more minutes. Then write a product of 2 functions and ask students to differentiate that until you get at least 5 answers ("I got the same" is a legitimate answer after the first one) and go over that example together. 5 more minutes, say. Then you play the same game again. 15 minutes into the class. Then you do two examples with students suggesting the products to differentiate (25 minutes in). Now you give a product of 3 functions and let the students to figure out how to do it in two steps. Let's say, 5 more minutes with the game repeated if everything moves too fast. You can also write it for arbitrary many functions in the product times the logarithmic derivative form (I find it useful to compare with the quotient rule later, which is much easier to remember that way). Once you have 15 minutes left, give a 5 problem quiz. If someone submits early, he or she is free to go and enjoy a longer break until the next class but tell them that there will be no partial credit to scare them a bit into checking their work). Then you collect the quizzes (with no partial credit the grading is done in under 45 seconds per paper, so you aren't too hard on yourself) and announce the topic for the next class.

And if you have a young Gauss among your students, ask him to find the value $$\frac d{dx}[(x-1)(x-3)(x-5)(x-7)\dots(x-99)]$$ at $$x=50$$.

• And that is if you do not want to include any proofs. If you do, you can stretch much more! :lol: Nov 14, 2022 at 23:20
• This is actually a very good idea. Although students already do a lot of problems in their separate discussion sessions. My question now is why should I slow down if going fast has been workinh so far? Nov 15, 2022 at 0:51
• @mrwillparker "My question now is why should I slow down if going fast has been working so far?" I didn't say you should (moreover, in my previous answer to your other post, I said that you shouldn't unless you see some indications that things may go astray yourself). I just tried to show you how to slow down if you find it necessary :-) Nov 15, 2022 at 17:55

When I was in school, what we did was to fill the class time by doing every exercise in the book on board. Naturally some exercises were kept as homework.

But I personally disliked this approach. I think it would be better if there was more of a student focused approach. Maybe some problems the students can do on board, and others the teacher can. Maybe make it competitive?

I think there is large appeal to doing problems require only basic calculus knowledge to solve (see problem solving channel like Michael Penn, Presh Talwalker, Blackpenred pen). Perhaps you could rip some problems from them and show in Class.

I was tasked with teaching calculus to some begineers at times and I often noticed that the perception we have of them understanding what we say, the perception they have of understanding what we say and what their actual understanding in practice is, are all at a disconnect.

There were many scenarios were I thought the person understood it, the person thought they understood it but when I gave them a practical problem, they were totally lost on how to even begin it.

You may feel you're able to communicate well, but sometimes what ends up happening with clear lectures which are "too clear" is that the student ends up in a sort of dunning krueger state where they over-estimate their own competance.

Just my two cents.

• begineers = beginner + engineers :) Nov 14, 2022 at 12:17

Going at such a pace with lectures can work fine, but all that lecture time should be supplemented with time for students to ask questions and work on problems. In my physics degree (in Europe) we covered all of single and multivariable calculus in the first semester, which is 16 weeks if you exlude exam weeks, with around 6 hours of lectures and 4 hours of exercise classes per week. In terms of lecture time this seems similar to your "fast" pace.

Later I was a TA for this class a couple of times and noticed that often people would think that they had a fairly good idea of the theory, just from the lectures. They could easily solve exercises that looked like the examples presented by the lecturer or as found in the textbook, but anything a little beyond that was too difficult. With students often barely knowing how to start. Having a TA or lecturer then guide you with hints or a further explanation is a lot more valueable than students studying the answer and simply remembering the solution for a test. Which is what I believe happens a lot.

I think that when you calculate lecture time you should take into account time for exercise classes, or something similar, rather than just time spent talking and explaining the theory. If you don't have that much time to spend per week, it is probably a good idea to sacrifice some of the pace for a higher quality learning experience.

• Which country in Europe? I am always under the impression that syllabuses in Spain are particularly overextended, for instance my Calculus I course includes ODEs. Nov 17, 2022 at 13:06
1. Slow down your actual rate of speech.

2. Explain the same thing three different ways.

3. Give five separate concrete examples for every abstract idea. Make them physical demonstrations if needed.

4. Ask questions.

5. Give physical demonstrations of the principles.

6. Solve problems using TPS. (Think, Pair, Share: do it individually; do it with a partner or two; do it as a group. Add or skip parts of this process as time requires.)

7. Incorporate more breaks, but use the class time more intensively. (Not 85 min + 10 min + 85 min but 50 min + 10 min + 60 min + 10 min + 50 min, ensuring that the non-break time is dense and rich.)

8. Run competitive games.

9. Run work periods during which you talk with (and orally evaluate the understanding of) individual students or small groups.

• Although the "competitive games" will undeniably be popular with a certain segment of the population, they'll have a negative effect with other segments. Math does not need to be a competitive endeavor... etc. Nov 16, 2022 at 22:34
• @paulgarrett just ideas. Not many work with everyone. The purpose is not to compete (not many subjects are competitive endeavours), but to provide another avenue of engagement. I find teaching works best when throwing a lot of ideas at the wall and seeing what sticks, which is different with each group. Nov 16, 2022 at 22:46
• The original post brought up Professor Leonard's YouTube videos in his post, and I think what makes Professor Leonard so effective is that he does pretty much everything on this list (except 7, 9, 10). But he definitely does 1-4 extremely well. He doesn't talk too fast, he uses tons of examples, he stops and checks for student understanding by asking the class questions and he repeats the main ideas three different ways.throughout the examples. Nov 17, 2022 at 2:15

IF you want to fill more time (and that is what the question asks), the clear answer is to spend more time on in-class drill. You are verging back to debating if you should spend more time, in responding to answers. But that's not this question (which is "how", not "if", as your other question was).

FWIW, if you're really crushing it like you say, I would say the answer to the "if" question is no. [And this is coming from someone who usually pushes in class drill...but if you're getting results, you're getting results.]

You could just cover 1st semester calc and give students half the periods back. Or stick with your original scheme and teach both semesters of calculus (at least there is some drill of first semester and of algebra buried within the work of second semester) in a normal semester. I mean...it worked, right. Just keep your head low and ignore colleagues (no bragging and just go your way on the sly).

Either the centralized tests do not properly assess your students' understanding, or your students are being inspired by your fast-paced style to spend more of their own time on reading and doing exercises - things that the lecturer or a TA would usually be doing in the settings I have been part of, as a student long ago and as an educator for the past few decades. It is unlikely (not impossible, but improbable) that you have found a magic recipe for accelerated learning of the same skills, at the same level, that take twice as long to master for other groups of students. Comparisons between US and any other country can sometimes be misleading, because of the often huge differences in math curriculum and the level of skill retained from lower grades, but I assume you are comparing apples and apples here. Some students can learn at an amazing speed. However, calculus is hard for most people, and they require time for repetition and rote practice of problem solving. You might possibly be giving your students a false sense of accomplishment by preparing them to pass the tests but not for applying math to actual problem solving. Note that I say "possibly". Only you and the people near you can know.

There have been some nice responses, and your experience goes counter to mine (I am at a "fancy" school, and I have a daily struggle to get through the material...). But, I want to give a different framing:

The question I ask when thinking about lectures is: what can the students get out of class time with me that they couldn't get on their own? While your exposition may be good and efficient, your students have excellent exposition in their textbook and on YouTube.

One thing they cannot get outside of the classroom is immediate feedback from an expert.

I design my classes so that I can get real-time information from students about what their misconceptions are and then work to address those misconceptions on the spot.

For example, many misconceptions come out when students try and explain their reasoning. I pose a question to the class and give students time to work on it (~5 minutes). Many students are able to "solve" the question; I then ask for 3 or 4 student answers and transcribe them to the board verbatim (or have the students write them on the board). I then have students discuss for each answer whether it is totally correct, mostly correct, or incorrect, and how they would fix any response that wasn't totally correct. Activities like this take quite a bit of time (~20 minutes depending on how involved their answers are), but the time spent emphasizes to the students that explaining an answer is just as important as having an answer. It also teaches me a lot about the students! (For example, today I asked my students what the derivative of sin(t degrees) was. I gave them time to work it out. The results were illuminating...)

Some notes on teaching this way:

• Coming up with questions that promote good discussion and reasoning is hard! If the explanation is "I took the derivative and set it to zero", that's not interesting to discuss. There are many places where you can find pre-made questions. For calculus, you can look at Cornell's Good Questions Project. If you are teaching Linear Algebra, I wrote a textbook with in-class questions built into the book.
• Creating a classroom environment where students are willing to share is hard! There's always a few good students willing to answer any question, but when I teach, I want to target the "middle 60%". That means, I need to make a comfortable environment for those students to share their (often wrong) ideas.
• For more details about classroom techniques, I suggest looking at the MAA Instructional Practices Guide. It is meant to be "flipped through", so you can start reading from any section that catches your attention.
• I looked through some of the questions in the Good Questions Project, and I'd caution anyone wanting to use them to give them a careful read. There seem to be some serious errors in the answers they suggest and the reasoning advanced.
– KDN
Nov 25, 2022 at 13:06