Lower division undergraduate mathematics course structure
The information on this page has been constructed based on an in-depth knowledge of the course structure of a few undergraduate programs and a broad survey of many others. The program surveyed have been based in the United States. The structure of undergraduate programs differs from country to country and even varies widely within country, so please do not treat this as a substitute for information about the course structure at a specific educational institution.
Basics
Parallel versus sequential courses
Large colleges generally offer a diverse range of calculus courses. It's worth distinguishing between two different types of relationships between courses:
- Sequential relationships: Here, one course comes after the other. For instance, Calculus 2 comes after Calculus 1.
- Parallel relationships: Here, the courses substitute for one another, but target slightly different niches. For instance, honors versions of calculus are parallel tracks to take in place of regular versions of calculus.
Placement and credit
Due to significant differences in the level of mathematical skill and preparation of entering undergraduates, colleges typically rely on some mechanisms to place students in the most suitable starting course. The mechanism include a combination of:
- Using students' school performance, such as SAT/ACT scores, high school performance, and Advanced Placement credit.
- Using a college-specific placement test, typically administered at move-in time or online (a few weeks before).
The goal of placement is to determine:
- The appropriate starting point among many parallel options (for instance, whether students are ready to start with honors courses).
- The appropriate starting point within a sequence (for instance, people who have seen the material typically covered in Calculus 1 may be ready to start with Calculus 2).
It's worth noting that college policies for placement and credit are not uniform. For instance, the University of Chicago treats an Advanced Placement AB score of 5 as a substitute for one quarter of calculus, even though the content of AB Calculus corresponds to two quarters of the University of Chicago calculus sequence, possibly reflecting their view that it's better for students to repeat some material rather than begin their studies with completely new material.
Honors Calculus
Most top universities offer Honors Calculus sequences distinct from their regular offerings, intended for people who satisfy some combination of prior calculus background and general mathematical ability.
Some example university course structures
Examples are below:
| University | Quarter or semester system?Regular single-variable sequence | Regular sequence for multivariable calculus and linear algebra | Honors route | |
|---|---|---|---|---|
| Princeton University | 103-104 (optionally take 102 instead of the sequence) | Semester | 201-202 (multivariable calculus and linear algebra for scientists) or 175 (multivariable calculus and linear algebra for social scientists) | No honors substitute for 103-104 (but it can be placed out of) 203-204 is an "honors" replacement of 201-202, targeting at physicists and applied math people; 214-215 is the recommended option for people interested in theoretical mathematics (See here) |
| University of Chicago (here, here) | Quarter | 151, 152, 153 | 195, 196 (social science majors) 200, 201 (physical science majors) |
Single-variable calculus: 161-163 (Honors Calculus) (placement through on-campus placement test) Multivariable calculus and linear algebra: take a bridge course (Math 199), then the analysis sequence. |
| Stanford University (here and here) | Quarter | 19-21, 41-42 | 51, 52, 53 (encouraged starting point for people with AP AB 5 or BC 4 or 5 credit) | Honors version 51H-53H available for 51-53 sequence (placement into honors version requires instructor consent or AP BC 5) |
| Harvard University | Semester | 1a and 1b | 18 (social sciences) 19a and 19b (life sciences) 21a and 21b (physical sciences) |
No specifically demarcated honors versions, though mathematics majors generally take 21a and 21b. |