The short answer to your question is "of course there is". I think that other responses will probably adequately take care of the many exceptions to this in terms of how questionable university rankings are. But there is absolutely no question that even in a course like calculus which (in the United States) has a fairly well-defined set of topics (see the MAA calculus report) there is considerably difference in how rigorously things are done, how difficult of applications are considered, etc. (An example might be whether one barely talks about implicit differentiation, whether it's connected to parametric curves, whether one then does logarithmic differentiation, or whether one then introduces the implicit function theorem.)

However, I want to focus on a different problem with this, which is that at many universities (most?) there may be different versions of the "same" course taught.

• In graduate school we had three different varieties of calculus to teach, for instance, ranging from fairly standard to using Spivak.
• Some institutions will have separate linear algebra (often not in a course by that name) for engineers, for math majors, for computer scientists, and for others - all depending on the application.
• Even (US) upper-division courses like number theory or geometry might come in several varieties; for instance, particularly if there is a large teacher preparation program, you could have two courses covering non-Euclidean geometry, one focusing on the axiomatic viewpoint and one on the differentiable viewpoint.

In all of these cases the content might be extremely different, certainly with different texts, but in general the "hard" version at University C probably would fall higher under your hierarchy than the "easiest" version at University A. If there were just one calculus or linear algebra (or whatever) at each university it might be easier to make such comparisons, but there isn't.

The short answer to your question is "of course there is". I think that other responses will probably adequately take care of the many exceptions to this in terms of how questionable university rankings are. But there is absolutely no question that even in a course like calculus which (in the United States) has a fairly well-defined set of topics (see the MAA calculus report) there is considerable difference in how rigorously things are done, how difficult of applications are considered, etc. (An example might be whether one barely talks about implicit differentiation, whether it's connected to parametric curves, whether one then does logarithmic differentiation, or whether one then introduces the implicit function theorem.)

However, I want to focus on a different problem with this, which is that at many universities (most?) there may be different versions of the "same" course taught.

• In graduate school we had three different varieties of calculus to teach, for instance, ranging from fairly standard to using Spivak.
• Some institutions will have separate linear algebra (often not in a course by that name) for engineers, for math majors, for computer scientists, and for others - all depending on the application.
• Even (US) upper-division courses like number theory or geometry might come in several varieties; for instance, particularly if there is a large teacher preparation program, you could have two courses covering non-Euclidean geometry, one focusing on the axiomatic viewpoint and one on the differentiable viewpoint.

In all of these cases the content might be extremely different, certainly with different texts, but in general the "hard" version at University C probably would fall higher under your hierarchy than the "easiest" version at University A. If there were just one calculus or linear algebra (or whatever) at each university it might be easier to make such comparisons, but there isn't.

The short answer to your question is "of course there is". I think that other responses will probably adequately take care of the many exceptions to this in terms of how questionable university rankings are. But there is absolutely no question that even in a course like calculus which (in the United States) has a fairly well-defined set of topics (see the MAA calculus reportenter link description here) there is considerably difference in how rigorously things are done, how difficult of applications are considered, etc. (An example might be whether one barely talks about implicit differentiation, whether it's connected to parametric curves, whether one then does logarithmic differentiation, or whether one then introduces the implicit function theorem.)

However, I want to focus on a different problem with this, which is that at many universities (most?) there may be different versions of the "same" course taught.

• In graduate school we had three different varieties of calculus to teach, for instance, ranging from fairly standard to using Spivak.
• Some institutions will have separate linear algebra (often not in a course by that name) for engineers, for math majors, for computer scientists, and for others - all depending on the application.
• Even (US) upper-division courses like number theory or geometry might come in several varieties; for instance, particularly if there is a large teacher preparation program, you could have two courses covering non-Euclidean geometry, one focusing on the axiomatic viewpoint and one on the differentiable viewpoint.

In all of these cases the content might be extremely different, certainly with different texts, but in general the "hard" version at University C probably would fall higher under your hierarchy than the "easiest" version at University A. If there were just one calculus or linear algebra (or whatever) at each university it might be easier to make such comparisons, but there isn't.

The short answer to your question is "of course there is". I think that other responses will probably adequately take care of the many exceptions to this in terms of how questionable university rankings are. But there is absolutely no question that even in a course like calculus which (in the United States) has a fairly well-defined set of topics (see the MAA calculus report) there is considerably difference in how rigorously things are done, how difficult of applications are considered, etc. (An example might be whether one barely talks about implicit differentiation, whether it's connected to parametric curves, whether one then does logarithmic differentiation, or whether one then introduces the implicit function theorem.)

However, I want to focus on a different problem with this, which is that at many universities (most?) there may be different versions of the "same" course taught.

• In graduate school we had three different varieties of calculus to teach, for instance, ranging from fairly standard to using Spivak.
• Some institutions will have separate linear algebra (often not in a course by that name) for engineers, for math majors, for computer scientists, and for others - all depending on the application.
• Even (US) upper-division courses like number theory or geometry might come in several varieties; for instance, particularly if there is a large teacher preparation program, you could have two courses covering non-Euclidean geometry, one focusing on the axiomatic viewpoint and one on the differentiable viewpoint.

In all of these cases the content might be extremely different, certainly with different texts, but in general the "hard" version at University C probably would fall higher under your hierarchy than the "easiest" version at University A. If there were just one calculus or linear algebra (or whatever) at each university it might be easier to make such comparisons, but there isn't.