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Lower division undergraduate mathematics course structure

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This page gives information on the course structure for lower division undergraduate mathematics courses, including:

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

See also college course dependencies

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.

Recommendations

Pre-college recommendations

Your choices earlier on can affect the options you have to get started:

  • Make sure to take, and prepare well for, the Advanced Placement BC Calculus examination in high school, if available to you: The placement options this opens up constitute an added reason to take the test. Take a look at our single-variable calculus learning recommendations for suggestions on how to prepare well.
  • Make sure to review calculus quickly in the month or two prior to joining college: This is particularly important for people who take AP Calculus in their junior year, or take a gap year after school. But even for people who took AP Calculus (or equivalent) in their senior year a few months ago, a lot can be forgotten over a few months. A quick refresher can be helpful both for doing well on on-campus placement tests and for getting off to a good start in college mathematics.
  • In the run-up to beginning college, give more emphasis to reviewing precalculus and single-variable calculus than to reviewing multivariable calculus and linear algebra: Colleges generally grant credit and placement for single-variable calculus, but are less likely to grant credit or give placement based on multivariable calculus and linear algebra (there are some exceptions, such as MIT). Moreover, precalculus and single-variable calculus have wider applicability, so making sure you are thorough on these prepares you for a wider range of starting points.

Entry point decisions

  • Make sure to take, and prepare for, any general online or on-campus placement test offered by your college, even if you don't think it will improve your placement: Each college is different, and reading the online course descriptions may not give you a clear idea of the prerequisites for entrance. Taking the placement test and knowing how well you did on that can be an added input, even if you ultimately decide not to avail of improved placement. Generally, optional placement tests cannot be used to lower your placement, but you might want to check that with the specific university department.
  • Understand the amount of flexibility that colleges offer with respect to placement. For instance, Stanford is highly flexible about placement levels -- many courses have open enrollment, and almost all courses allow you to enroll with instructor consent. The University of Chicago has prerequisites that are rigid on paper (people are required to have placement credit to begin specific courses) but students can request to try out a course one level higher, and can officially have their placement changed if they do well in the first few weeks of the course. Some aspects of this flexibility may not be evident from online materials, so talking to people can help.
  • To the extent that there is flexibility, use it to figure out your best option, rather than simply accepting the college recommendation (but give strong weight to the college recommendation as an informed piece of advice). However, if you are doing this, you need to put in some prior effort in understanding the options. You should also have more willingness to change course if something unusual you tried out isn't working.
  • Understand the extent to which ordinary (non-honors) courses cover material well: At top universities, the regular (non-honors) sequences can be fairly challenging in and of themselves, and it is not necessary to take the honors sequence to challenge oneself mathematically. In addition, non-honors courses cover most mathematical topics relevant outside of mathematics, and students can generally enjoy the benefits of reasonably good peer groups. At mid-tier universities, the honors class might be the only place that provides adequate mathematical challenge and an interested and motivated peer group. Note that, regardless of your university, you should still consider honors options if you are interested in a math-based discipline or if it is the recommended placement you receive.

Once you get started

  • Consider variation between instructors within a course level: Some universities offer more flexibility to instructors with respect to the syllabus and course materials, and others offer less. In general, the greater the flexibility offered, the higher the variation between instructors, and the more important it is to select a good one. In some cases, getting a good instructor in a non-honors sequence may be better than getting a bad instructor in an honors sequence.
  • Take a look at our calculus learning recommendations, multivariable calculus learning recommendations, and linear algebra learning recommendations for suggestions on resources to use for learning to supplement your college learning.

Some example university course structures

Examples of some universities are below. This list is intended only to give a general idea of how different mathematics departments structure their lower division mathematics requirements. It is not intended as a substitute to gathering information about a specific university's policy. Information here is not kept up to date.

University Quarter or semester system? Regular single-variable sequence Regular sequence for multivariable calculus and linear algebra Honors route
Princeton University Semester 103-104 (optionally take 102 instead of the sequence) 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.
Massachusetts Institute of Technology (MIT) Semester 18.01 18.02 (multivariable calculus), 18.03 (differential equations), 18.04 (linear algebra) 18.024, the "Calculus with Theory" course
University of California, Berkeley (here, here) Semester 1A and 1B 53 and 54 Honors versions of the courses 1B onward. However, these are offered sporadically, and there is no predefined guideline on how these differ from the regular sequence.
University of Michigan (here, here, and here) Semester 115 and 116 215, 217, 316 185, 186 (there are also other options)