Suppose, for instance, that a political science professor was tasked to design a curriculum for his quantitative methodology class. Generally, a quantitative methodology class covers the application of mathematical techniques in analyzing data. Can a professor, then, given his academic discretion, design a curriculum that focuses on the philosophy of quantitative analysis and not on the actual math?
Answering for the R1 schools with which I am familiar:
Of course a professor has this discretion. There is no universal standards board that decides what must be taught in what course -- nor would we want such a thing. Otherwise we might end up teaching intelligent design alongside evolutionary biology and the great flood alongside plate tectonics.
The professor might be better off choosing a different course title, but there are a host of bureaucratic reasons why this doesn't always happen. So long as the syllabus accurately reflects the course material, I see no problem whatsoever. (And even if it doesn't, the professor still has the ultimate right to determine the course curriculum).
In addition to the question of what a professor is permitted to do, there is a second question about what is a reasonable standard of conduct for the professor. This too offers quite a lot of latitude, particularly in higher level classes.
Caveats: For certain large introductory classes or multi-selection courses, the academic department or a committee within the department may select a set of topics to be covered in order to standardize what the students learn -- but in this case the professor would not be tasked with developing the curriculum himself or herself. As we move from introductory classes toward advanced classes. Similarly there could potentially be a problem if a professor taught oceanography in an English literature class, or vice versa -- but decisions about where to fall on the philosophy / calculation spectrum within a given topic area are entirely up to the professor.
Addendum This whole thing may not seem like a big deal to those of us in STEM fields (with the possible exception of those fields that sometimes contradict big oil, big tobacco, or a narrow reading of the Book of Genesis). But faculty discretion over the contents of the curriculum is extremely important the humanities and social sciences where political agendas are more immediate and where the choice of a text may itself be an overtly political act.
I think the answer is "yes or no, depending on extent". When a regular course is first proposed, there is usually a procedure for describing the content of the course, and that proposal generally has to be approved by the department (thought it might by by a single person with a relevant administrative function, such as "director of graduate studies"). The proposal is scrutinized and approved or rejected by some number of higher layers of bureaucracy, and ultimately approved by the trustees, regents or whatever they are called. That, in principle, puts a limit on individual discretion.
To take an obvious extreme, if a course is described as "Introductory Calculus", then making the content actually be about GLBT theories of Dowland lute music, that would be a serious deviation from the official content of the course. Especially since other departments have an interest in the content of such a course which may be a requirement for some major, you can expect that such deviation would be nipped in the bud. On the other hand, if the official description names a particular calculus textbook that was predominant in the 40's, changing texts would not be considered a deviation from the described content.
Whether or not the specific case you describe would be considered a sufficiently egregious deviation from described content would depend on the politics of the university and the role of the course in the overall structure of university offerings. Deviations would be dealt with in a post-hoc manner, such as by the chair having a talk with the offending faculty member, and possibly re-assigning the person to a different course.
In addition to the other answers given here, the answer also depends somewhat on the locale. For example, in Germany, a university professor has the right to set the curriculum for his course as she sees fit. The department may give him some hassles if the proposed curriculum too closely duplicates or overlaps with an existing course, but the right to teach as she chooses in principle remains.
Sometimes it may also depend on if the course is part of an "accredited" degree program. For example, the instructor of an introductory thermodynamics course in an ABET-accredited engineering program probably couldn't get away with not teaching the Second Law of Thermodynamics.
This varies tremendously from country to country. In the US, the professor has a lot of latitude.
However, in the UK and Australia, this is not the case. At Oxford, you are presented with the examination statutes when you arrive and these contain very clear statements of the topics that will be covered in each exam. Plus the examiners are not the lecturer so if the lecturer gets creative and covers different things, everyone is going to be very unhappy.
At Melbourne, there is a handbook that lists the content of every subject and how you will be assessed. The professor has to apply to change the contents well in advance, and students can successfully complain if the handbook is not followed.
There are many, many reasons why colleges and universities do not simply allow their faculty to teach any random stuff they choose as part of a course. They have to do this for accreditation. They have to do this for their students who want to transfer coursework to other schools. They have to do this so that students who take prerequisite courses are prepared for later work, which may be in other departments. Departments want to make sure that, e.g., if a student gets a degree in French, the student has actually learned certain things about French phonetics or literature.
For these reasons, every accredited college or university in the US has some kind of formal curriculum process. At my school, for example, we have a curriculum committee and a computer database in which curriculum information is stored. Courses are required to be revised every 6 years, and can be revised more often if the department faculty wish. When a course is revised, it goes through multiple rounds of commentary and editing. For a class in physics, for example, the following would all get a chance to comment: the faculty member originating the new version; all other physics faculty; administrators; and a faculty committee including faculty from other departments.
An example of the kind of thing that is hashed out in this process would be issues about class sizes. Professors would like to have small class sizes, but administrators have to balance this against fiscal concerns. So for example if I want my course to have a section size of 25, I need to demonstrate some need for that, e.g., because the class is a writing-intensive one.
Mercifully, in the U.S., apparently unlike the U.K. in recent years, there is no central body whose purpose is determining or checking-up-on either the catalogue description of, or the specific syllabi of, or the actual on-the-ground-content-of mathematics courses.
Sure, for lower-division courses, and for the "core" upper-division courses, there are compatibility issues with other math courses and other science courses, and the vast majority of math faculty do not do anything to wreck those relationships. (Even for the very few faculty who do not seem to be able to understand that part of their job is to meet the needs of other people, it seems easier to administratively "work around" them rather than rush to confrontation, ... in part, I think, because "thinking outside the box" is a good thing, although there will be mistakes and infelicities. That is, the principle should not be to chastise people for erring while trying to think deeply about what would be good for students (which is the almost universal story even in cases where the conclusion is dubious or naive).)
Indeed, for upper-division undergrad math courses, and certainly for graduate-level math courses, it is my opinion/conceit that my (R1...) university should exactly expect (if not "demand") that I'd be regularly/endlessly updating the content of the courses I teach to reflect contemporary state-of-the-art. In math, it's not really that old things become "wrong", but that there are new things that may have higher priority, in the finite time-and-space of a course.
Over the years, I have indeed been involved with official revision of the course catalogue and such. For graduate courses, especially, even if my colleagues and I have a rough consensus on what a year-long core course should include, it is obviously inconceivable that "144 words or less, with minimal technical language" could describe what the dang course is about. In the 1990s we did another round of this, and the constraints and demands were so ridiculous that it cured me of belief in catalogue descriptions of courses.
Further, at my own Uni, despite the silliness of central administration and so on, the virtue of the situation is that, for example, "introduction to modular forms and L-functions" can be significantly different depending on which of several possible instructors take the course in a given year. Sure, there's a mock-up catalogue syllabus (which I made up 20+ years ago, because we ... had to), but no one worries about it, because they'll all exercise their own best professional judgement. Luckily!
Maybe the potential volatility in math is less than in other subjects...
How much discretion the professor has depends on the context. Suppose that course 1 is a prerequisite to one in which student are required to have learned certain methods taught only in course 1 and that is course 1's purpose according to official policy. Then omitting everything about those methods might reasonably be considered malpractice. However, suppose the professor reasonably thinks that students should be able to figure out the quantitative methods based on the conjunction of his material on "philosophy of quantitative analysis" and things they've learned in a math course that they also took. Then it might be reasonable. Things like this can depend on facts about the student's purposes and the course's purpose.