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College course dependencies
This page reviews some general background on the structure of dependencies between college courses. Details vary from college to college, and you should use the websites, plus talk to relevant people, to figure out what applies to a given college.
A course sequence refers to a collection of courses intended to be taken in a particular order. The courses typically have consecutive course numbers. For instance, at Stanford University, the regular sequence for multivariable calculus and linear algebra is numbered 51, 52, 53.
Note that, when inferring the time length of a course sequence from the number of courses in the sequence, it's important to remember the difference between course lengths in the quarter and semester system. A course in a semester system is about 1.5 times as long as in a quarter system.
Course sequences may differ in the extent to which the courses are linked.
- In some cases, later courses in the sequence list earlier courses as formal prerequisites, and in others, they don't. The extent to which the prerequisites are enforced varies by university and departments.
- In some cases, the courses are heavily linked: they have the same set of sections, with the instructor carrying over from one term to the next for a given section, and with students automatically enrolled in subsequent courses in the sequence. Although it is technically possible to switch sections between terms, it can be quite difficult given space limitations. In these cases, making a good initial choice of section can be important since the consequences of that choice affect the entire sequence. Further, in cases where the level of coordination between the sections is low, switching can be harder because instructors in other sections have covered, and expect knowledge of, somewhat different content.
A few other points related to course sequences:
- Course sequences may not begin every term. For instance, some course sequences begin only in the fall term (the first term of the academic year). If you miss taking the course in the fall term, you may end up having to delay the sequence by a year.
- Course sequences can also conflict with study abroad plans.
Aside from course sequences, there are also other dependencies between courses. Some courses may list other courses as formal prerequisites, while in other cases, the prerequisite listing may be only at the level of "recommended" rather than required. The prerequisite courses may be from other academic departments, and those might themselves be at the end of sequences or have their own prerequisites. Understanding the structure of dependencies can be crucial to making sure one is able to take one's desired courses on time.
For instance, an econometrics course offered by an economics department may list a linear algebra course and a statistics course as prerequisites. The linear algebra course may in turn list a single-variable calculus course sequence as prerequisite, while the statistics course may list a multivariable calculus course as prerequisite, which in turn lists a single-variable calculus course sequence as prerequisite. The upshot is that one needs to get the single-variable calculus course sequence finished with as soon as possible to then move on to multivariable calculus, linear algebra, and statistics, so that one can take the econometrics course.
In some cases, there are multiple parallel levels or tracks of a course. Examples could include "honors" versus "ordinary" tracks. For people who intend to major in the subject with honors, the honors track may be recommended. For people who are learning the material for use in other subjects, either track may suffice.
For instance, Stanford University has a standard regular sequence for multivariable calculus and linear algebra numbered 51, 52, and 53. There is an honors sequence numbered 51H, 52H, and 53H. The honors sequence covers similar material as the regular sequence, but is a more advanced version.
Similarly, the University of Chicago has a standard regular single-variable calculus sequence numbered 151, 152, 153, and an honors sequence numbered 161, 162, 163. The honors sequence is more proof-based but has a broadly similar syllabus as the regular version. Math majors are strongly encouraged to take the honors sequence, but others may take either sequence.
In the case of course sequences with parallel tracks, it is usually easy to drop down from a higher track to a lower track. For instance, at the University of Chicago, one can complete Math 161 and then drop down to Math 152 and 153. It is not, however, permissible to move up from a lower track to a higher track.
Courses that compress more information in less time
Some universities offer parallel tracks between a longer and shorter course sequence. The shorter course sequence may be intended for people who are starting off with more prior knowledge and can therefore afford a quicker refresher. It may also be the case that the shorter course sequence is intended for people who do not need to know the material in as much depth, so that many details can be skipped.
For instance, Princeton University offers Math 102 as a substitute for the sequence Math 103-104.
Credit and placement
Some university departments recognize Advanced Placement (AP) or International Baccalaureate (IB) course scores and use them to confer credit or placement. Some universities have their own online or in-person placement tests that can be used instead of or in addition to AP tests. Policies vary by university and department.
Placement within sequences
In some cases, the university offers placement within a sequence. For instance, the University of Chicago allows people to use AP credit or an on-campus placement test to place out of Math 151 (the first quarter of calculus) and directly into Math 152, or out of Math 151 and 152 and directly into Math 153, or out of all quarters (thereby not needing any calculus at all). Princeton University allows people to use their AP scores to place out of Math 103 and/or Math 104.
Placement within sequences is usually accompanied by the conferring of course credit. For instance, if somebody places out of Math 151 and directly into Math 152, that person gets credit for Math 151. This credit goes towards fulfilling the general education requirements (two quarters of calculus) at the University of Chicago, so the person need only complete Math 152 to finish the requirement. Similarly, if somebody places out of Math 151 and 152 and directly into Math 153, the person has two quarters of credit for mathematics and has therefore fulfilled the general education requirement for mathematics.
PENDING: Discussion for the implications of taking Advanced Placement or IB courses in high school.
Placement between parallel tracks
A different aspect of placement is that of placement between parallel tracks. This determines, for instance, whether somebody is eligible for the honors version of a course instead of the ordinary version. Placement in a higher parallel track does not confer credit -- it simply means one starts in a higher parallel course sequence.
In some cases, there is an either/or between placement within sequences and placement between parallel tracks. For instance, for University of Chicago mathematics, a person may get an honors recommendation, indicating that the person can start with the honors sequence (Math 161 on), and also get a quarter of credit (so the person can start with Math 152 instead of Math 151). But the person cannot combine the honors recommendation and the credit, i.e., the person cannot start with Math 162. Again, details vary by university and department, but generally, availing of honors options might conflict with taking advantage of course credit.