# Formats for Calculus instruction at different colleges and universities

In the comments under another question, a couple of people expressed interest in how Calculus is taught at the University of Michigan. I'm not convinced a question that narrow is appropriate for this site, so I am opening the topic up to a broader discussion. I suggest that this question be made Community Wiki to encourage a broad range of different responses.

The question:

What format does Calculus instruction take at different post-secondary institutions, both in the United States and elsewhere in the world?

"Format", here, includes such issues as:

• Whether classes meet in a large lecture, and if so how many students are typically enrolled in each lecture
• Whether classes meet in smaller sections, and if so how large those sections are, and what they are called ("discussions", "practica", "problem sessions", "labs"...)
• Whether students meet in one-on-one tutorials, or small group tutorials, and if so who leads those tutorials (graduate students, lecturers, advanced undergraduates...)
• Whether classes are held in a combination of the above (e.g. "large 300-student lecture that meets twice per week, supplemented by optional 20-student problem sessions twice per week")
• How many times per week, how long each lecture or small session lasts, how many weeks per semester (or "term", or equivalent)
• Whether instructors have broad autonomy to teach however they want, or whether instruction is coordinated across sections to ensure some kind of consistency
• and anything else that seems relevant for a specific institution.

I am hoping to gather a wide range of responses, including from small liberal arts colleges, large research institutions, community colleges, and others.

I don't particularly think "textbook" is one of the most important variables here, but feel free to include that information if you think it is salient.

Feel free to include information about how things vary from, say, "Precalculus" to "Calculus 1" to "Calculus 2" to "Calculus 3" (or whatever they are called at your institution).

I teach in the Universidad Politécnica de Madrid, which is a fairly large public engineering school with research objectives. The students are comparable to those I have taught in engineering degrees at places like Georgia Tech or the University of Washington, although they enter the univesity with better preparation. The teaching of calculus in the UPM varies somewhat from school to school (the UPM is divided into many different schools) but is in broad strokes similar to what is standard in Spanish universities, which are more uniform in their practices than are their counterparts in the US (although of course there are lots of variations in details). It is strongly conditioned by the structure of Spanish degree programs and by the comparativaley poor financing of Spanish universities (when compared with other countries having a comparable ecomonic level professors and administrative staff are few and poorly paid).

Classes typically meet in large lectures, somewhere in the 50-150 students range. There is plenty of willingness to teach in smaller groups, but there are usually not enough professors available to do so. A typical class meets 4 or 5 hours a week, perhaps 1 or 2 of which are formally or nominally dedicated to working problems (this depends on local practices), although this would often be done in the usual classroom, by the usual instructor. For example, in my school we have 4 hours of instruction in a large lecture hall, and 1 hour in a room with movable tables so students can work problems in small groups or in the computer lab to use Matlab, R, or whatnot (with the one professor circulating to help them - I have only about 70 students so this is viable). Sometimes these problem sessions, computer lab session are broken into smaller groups but not for educaitonal reasons, simply because they don't fit in the rooms used (this requires assigning additional professors). Semesters are 15 weeks by law.

There are no teaching assistants or graders in the US sense, so it is generally not viable to assign homework in calculus classes. Often one gives some sort of quiz or weekly in class assignment or computer exercise, that might be evaluated automatically via some cell phone app, or something of this sort, or might be done in groups, so as to make grading viable, but this is often left up to individual professors, and is not universal. The main components of grading typically are one or two tests and a final exam.

Whether instructors have broad autonomy to teach however they want, or whether instruction is coordinated across sections to ensure some kind of consistency and anything else that seems relevant for a specific institution.

Such autonomy is basically inconceivable in the Spanish system. Each degree program has an official plan of study approved by a national accreditation agency. Modifying the study plan requires substantial bureaucratic movement at the level of a dean's office, so principally occurs in response to flagrant problems, or at regular 5 year intervals when these plans are obligatorialy revised. The study plan stipulates course contens, and for something like calculus it sometimes essentially establishes the curriculum. In engineering degrees the contents of even calculus courses are in any case basically obligated by law and professional accreditation requirements.

Every year the department responsible for teaching a course approves a "study guide" for the course (later approved by the large faculty - this all occurs many months before the semester starts). This is written by the official course coordinator in conjunction with the other professors assigned to teach the class. It is a document of some pages that stipulates course objectives and learning goals, and details the syllabus (in my school to the point of indicating what contents will be covered in what lecture). It describes the evaluation scheme to be used (and the description is supposed to be obligatory for the professor). It contains recommended books and other information about learning resources. The professor is expected to adhere to it more or less rigidly. Of course there are professors that don't, and not much can be done about this, but the general spirit is to do what has been stipulated in advance.

This sort of planning guarantees considerable uniformity, at least where such uniformity is desired. I teach one class where the study guide intentionally leaves 20% of the grading, as well as the structuring of problem sessions, at the discretion of the individual professor (there are 9 professors). In other classes all grading is done in common. Certainly there is no grading on a curve, or a posteriori changing of grading schemes, etc. Such practices, common in the US, are generally viewed as unprofessional, unfair to the student, and potentially unethical.

Generally speaking books are only recommended, not assigned. It is economically impossible to do anything else. Many families can't afford to shell out euros for books and no one thinks it reasonable to force them to do so. In any case everything can be downloaded from the web, and everyone knows this. Books are formally recommended, but it is essentially unheard of to assign problems from a book. Typically professors prepare weekly problem sheets (voluntary) and provide lecture notes that constitute unedited books. When several professors teach together this effort is often coordinated. There might be an official course problem list accompanied by official lecture notes, often distributed through the university's internal publication service.

At the university level there is no precalculus as this is not considered (essentially in a legal sense) a university level subject and could not count towards an official university degree. The high school courses required to enter the university in a degree program that involves calculus include calculus and linear algebra. Typical first year calculus courses assume knowledge of basis differentiation and integration, and typical first year linear algebra courses assume ability to analyze and solve systems of linear equations. A typical first year program in mathematics includes calculus, one variable and multivariable, linear algebra, and often gets to more advanced topics such as Laplace and Fourier transforms, ODEs, a bit of numerical methods.

In the old days (a decade ago, before the Bologna treaty), pass rates were in the 10-30 percent range, at least in engineering programs. Now they are typically more in the 30-70 percent range. (I'm guessing a bit at the numbers, but the upwards shift has been undoubtedly pronounced, a response to changing administrative exigencies).

Professors are obligated to hold office hours, in my institution 6 hours a week, and in my experience this obligation is taken more seriously than it is in the US, although students come to office hours just as infrequently as they do in the US.

The bureaucratic overhead associated with teaching a calculus course is substantial in Spain, in my experience more onerous than in the US and with substantially less administrative support and fewer instructors per student. This involves things like the annual preparation of the aforementioned study guide, various offical reports on class outcomes, etc., in addition to the day to day handling of matters related to enrolment and the like. Teaching a course of 150 in the US I would have 4 TAs and several graders at my disposition, also some departamental administrative personel to attend to purely administrative matters. In Spain there is one administrative person serving three departments with a total of 50 professors, and there are no TAs or graders.

Teaching style is generally a professor's prerogative. When teaching calculus I mainly use a blackboard (when I teach programming this is done entirely in a computer lab). Some professors teach entirely using a computer/projector. Most instruction is classic lecture format, but there are people trying other approaches and this is encouraged administratively.

• What do you attribute the upward jump from 10-30% passing to 30-70%? Better prepared incoming students, more supports, lowered standards/easier exams, something else? – Daniel R. Collins Dec 1 '18 at 7:42
• Retroactively establishing a grading scheme to fit student performance does not,to me, seem unfair -- it seems like a recognition that creating exams of consistent difficulty is basically impossible; the only way to know how hard an exam is, is to give it to people. If two groups of 1000 students, in consecutive semesters, each take a (different) final exam, and one group has a median score of 80% and the other group has a median score of 70%, which seems more likely: that one group of students is collectively stronger than the other, or that one exam was harder than the other? (1/ ) – mweiss Dec 2 '18 at 0:32
• I was trying to describe what I experience as the prevailing sentiment among professors in Spain regarding grading. My answer does not intend to advocate for one thing or another (an interesting discussion). My sense is that different approaches can work, and that what works depends in part on cultural and administrative context. When I came to Spain I was convinced of the good sense of a posteriori adjustments to accomodate actual performance; after some years in Spain I am much less convinced that such grading on a curve is plainly reasonable, and I doubt very much it would be accepted here. – Dan Fox Dec 2 '18 at 11:28
• @mweiss: There's copious real-world evidence that's not the case. Haertel (2013) has an analysis that only 10% of variation in exam scores is attributable to the instructor. I've taught two sections of a course identically in the same semester and had a 23% pass rate in one and 60% in the other. The SAT, NAEP, and TIMSS tests all have very large sample sizes and vary from year to year (not randomly; conclusions for trends in different ethnicities, cohorts, socioeconomic groups are taken from these tests and considered valid). – Daniel R. Collins Dec 2 '18 at 17:39
• @DanielR.Collins I think this conversation would best be moved to another question, rather than stay here in the comments under this one. – mweiss Dec 2 '18 at 18:36

At the University of Michigan, all instruction at the level of Precalculus, Calculus 1, and Calculus 2 is conducted in small "Recitation" sections -- no lectures!. Traditionally (since the mid-1990s), these sections were capped at a maximum enrollment of 30 students; since 2016, the University has invested in lowering the class sizes to a ceiling of 20 students, which has resulted in many more sections and instructors.

In a typical Fall semester we have upwards of 1600 students in Calculus 1, distributed across 80-90 sections, taught by 60+ instructors; these numbers are roughly halved in the Winter semester. A tremendous amount of effort goes into trying to ensure that the instruction is inquiry-based and centered around group work, and that we deliver a consistent experience to all students in every section.

A recent article (https://www.tandfonline.com/doi/abs/10.1080/10511970.2017.1315474) summarized the systems that Michigan has put in place to manage a system with this many moving parts, and some of the challenges that remain in consistently delivering quality instruction. Among the highlights:

All sections are taught by an instructor who has full responsibility for running each class, and most of the instructors are graduate students or postdoctoral assistant professors. All of the sections exist in an extensive structural framework: the day-by-day course schedule, all deadlines, homework assignments, and exams are set by a course coordinator, who also runs weekly meetings with all instructors to address conceptual and logistical questions. Individual instructors teach three days a week, with 80 minutes per class-session, and are expected to have an engaged classroom in which students are working, in groups, on meaningful problems for about half of each class period.

Student assessment in the program has five primary components: two midterm exams and a final; individual, on-line web homework; team homework worked on by groups of four students, with each team submitting a solution paper that requires extensive written work; “gateway” (skills) tests, as noted below; and quizzes written and graded by each instructor. Exams are written by the course coordinator, are highly conceptual, and require significant problem- solving skills. They are written with the expectation that students will be using (graphing) calculators, and they have few “standard” problems that reappear semester to semester (an example exam is shown in Appendix A). Basic skills are developed and assessed with “gateway tests,” which are run in a proctored lab environment in which calculators are not allowed...

Ideally, a Michigan calculus classroom is noisy, with students sitting at tables of four happily engaged in inquiry-focused problem solving. While the students collaborate on solving the problem(s) at-hand, the instructor moves from table to table, ensuring students stay on track and gently guides them when necessary. In particular, instructors are encouraged to spend less than half of each class meeting lecturing.

Although the article quoted above describes Calc 1 (Math 115), much the same description applies to our Precalculus course, Math 105 (which is not actually called "Precalculus"; we call it "Data, Functions, and Graphs", and it is the lowest-level mathematics course offered at Michigan), and to Calc 2 (Math 116).

The first course taught in a lecture format is Calc 3 (Math 215, Multivariable Calculus), and which is usually capped at 110 students per section. Concurrent with their Calc 3 lectures, students also take a computer-based lab section, run by a graduate student, in which they use various software packages (currently MatLab; in the past, both Mathematica and Maple have been used) to visualize and explore surfaces, contour plots, and other representations of multivariable functions that are difficult to produce and modify by hand.

It is also worth mentioning that, completely separate from all of the above, Michigan also has multiple independent "honors" options. These are:

• A three semester applied honors sequence, Math 156-285-286. From the website: Applied Honors Calculus II (MATH 156) is designed for engineering and science majors who received a score of 4 or 5 on the AP exam (AB or BC). MATH 156 is an alternative to MATH 116 with more emphasis on science applications and theory. The sequence continues with courses in multivariable calculus and differential equations (MATH 285-286), which are alternatives to MATH 215-216.
• A four semester honors seminar mathematics sequence, Math 175-176-285-286. From the website: MATH 175 (Introduction to Cryptography) and MATH 176 (Explorations in Topology and Analysis) are taught in the Inquiry-Based Learning (IBL) style. The IBL method emphasizes discovery, analysis, and investigation to deepen understanding. These courses assume a knowledge of calculus roughly equivalent to MATH 115; they cover a substantial amount of basic number theory (MATH 175) and provide a good, high level understanding of calculus (MATH 176). The sequence concludes with Honors versions of multivariable calculus (MATH 285) and differential equations (MATH 286).
• A four semester honors calculus sequence, Math 185-186-285-286. From the website: Honors Calculus I and II (MATH 185 and 186) rigorously develop the concepts of calculus. These courses are intended for students who desire a complete understanding of the theoretical underpinnings of calculus, and they lay a solid foundation for future Mathematics courses. The sequence concludes with Honors versions of multivariable calculus (MATH 285) and differential equations (MATH 286). Most students who take MATH 185 have taken a high school calculus course, but it is not required.
• And a four-semester honors mathematics sequence, Math 295-296-395-396. From the website: The Honors Mathematics Sequence provides a rigorous introduction to theoretical mathematics. These courses require an extremely high level of interest and commitment and provide excellent preparation for mathematics at the advanced undergraduate and beginning graduate level. Most students electing MATH 295 have completed a thorough high school calculus course.

The honors courses are also taught in small "Recitation" sections, typically 20 students or less, and meet either four days per week for 50 minute sessions, or three days per week for 80 minute sessions, depending on the course. Because there are only a few sections of each honors course, my sense is that instructors typically have much more latitude in how they decide to run things. Also, as both the description and the course numbers suggest, the $$x9x$$ sequence is taught at a much higher level of abstraction than the other honors courses; in many ways it really is the equivalent of an upper-division, proof-based analysis course.

All of these courses -- 105 (Data, Functions & Graphs), 115 (Calc 1), 116 (Calc 2), 215 (Calc 3), and the various honors alternatives -- are 4 credit courses.

• I like the small class sizes because it mimics how I learned in high school and the uncollege. I don't think research grade professors are needed or even desirable for calculus. You want "teachers". Not crazy about the group work and inquiry stuff though. Just feel like conventional approach is best. But I admit this is a prejudice with limited experience of the converse. – guest Nov 30 '18 at 4:42
• They changed the numbering from 44 years ago, when I was there. It was 195-196-295-296. But I bet it's the same sequence. I got upper division credit for all the courses. But I left U of Michigan thinking I didn't like math. (And regained my love at a much less prestigious school next door, EMU.) – Sue VanHattum Dec 2 '18 at 2:58
• Yeah, it was 195-etc when I was there in 1989-1993. They changed it sometime in the late 1990s, I think. – mweiss Dec 2 '18 at 3:03
• "Ideally, a Michigan calculus classroom is noisy, with students sitting at tables of four happily engaged in inquiry-focused problem solving." - I can imagine that a noisy environment is not necessarily productive for all students. – J W Dec 3 '18 at 11:27

I'm studying physics at Eötvös Loránd University, Hungary, so my answer will not be complete, but I will try my best.

In the first semester, we can choose between "advanced" and "normal" level calculus (there aren't many proofs on calculus for us). Both of them last for a $$2/3$$ semester, with $$3$$hr lecture + $$3$$hr practice/week. "In my times", there were $$1$$ lecture for both levels and $$1$$ practice group for the advanced class (with about $$20$$ students), and $$3$$ practice groups for the normal class (with about $$20$$ students/group). In the remaining part of the semester, we have a "differential equations in physics 1" class, with only $$1$$ difficulty level, but $$4$$ practice group + $$1$$ lecture group (and the same $$3+3$$ hr/week). Also, we are having a "vector calculus" class, "advanced" and "normal" level, $$4+4$$ (lecture + practice) hr/week, and every occasion was $$2$$hr long. About $$2/3$$ part of this class is about linear algebra, and $$1/3$$ is about differentiating and integrating in higher dimensions (It's a rather practical class).

In the second semester, there's the "differential equations in physics 2" class, with $$1+1$$ hr/week, with $$1$$ lecture and $$4$$ practice groups. There's also a "mathematical methods in physics" class, with only $$1$$ group and $$4$$ hr/week, all at once. It's a practice and lecture at the same time. This class contains $$4$$ parts: Fourier analysis, Complex analysis, Calculus of variations and Curvilinear coordinates. It's also a practical class. This class will be reformed this year, there'll be $$2$$ levels here as well, etc.

There are also "Analysis 1" and "Analysis 2" classes on the theoretical physics specialization, with only lectures (there were an "Analysis 3" before as well, but it's on the MSc now). I don't know much about them, because Analysis 1 is on the 4th semester, but I know that they are differentiating in normed spaces. On the Analysis 2, there's some complex analysis and measure theory, and on Analysis 3, there's some functional analysis.

I'm taking the "Analysis 1" class at the moment (which is a first semester subject for mathematicians), so I can talk about it a little bit as well. Basically, they can choose between $$2$$ paths: They can either take the "Analysis 1" and "Analysis 2", or the "Calculus 1", "Calculus 2" and the "Introduction to analysis" classes. I don't know much about the calculus path, but I was told that it's much easier, so I think they are going slower, and proving fewer things. Back to the analysis, there is only $$1$$ group for the lecture, and they take $$2+1$$ hr/week. But the practices are a bit different: There is an intensive group (this is the hardest) with $$24$$ student, and $$4$$ advanced groups, with $$18$$ student/group. And all of the groups are having $$2+2$$ hr/week of practices.

So usually the lectures are collective, while there are smaller groups on the practices, with different difficulty levels.

Remark 1: This kind of separation is quite common for us, but there are some classes without it, like the mathematical methods in physics was.
Remark 2: In our university, the "Analysis" and "Calculus" are different, as you could see. But I didn't see this kind of distinction between them elsewhere in Hungary. For example the engineers at different universities are having only analysis, which is, of course, not as deep as the analysis here. (Or at least this is how I saw it).