I think there may be a basic computer programming course that could be given before such a course that introduces various basic ideas of modern programming, e.g. what is an IDE, what is an algorithm, what are basic ways to measure complexities, etc. where there are 2 choices given to the student: Use what the course suggests for a language, or pick your own from this list that TA's and others would know enough about to determine if the solution is acceptable or not. One issue to consider is whether or not various exotic languages like Logo, Lisp, Modula-3 or Fortran would be OK or not. The idea of understanding how to use various tools would be the focus more so than using a specific programming language that may not be used outside of academia.
In some respects the final exam for such a course would act as a way to determine whether or not one would be able to skip such a course. The main idea is to have some basics within a specific realm pinned down, not unlike how in elementary school an Arabic number system in base 10 with the operations of add, subtract, multiply and divide and place value, like tens, hundreds, etc. is the basics of Mathematics as opposed to having learned any of these as one's introduction to Mathematics, which are all possible starting points:
Trigonometry using a hexadecimal number base defining functions such as sine, secant, tangent as well as the co- of each and the inverse of all of those and the graphs of such functions.
Graph theory consisting of paths, breadth first search, depth first search, and minimum weight spanning trees.
Derivatives, integrals, and partial differential equations over the Complex Number system or some higher dimensional spaces like N x N matrices of real numbers.
Probability and Statistical theory including arithmetic and geometric mean, median, mode as well as the idea of least squares and linear regression with just enough Algebra so it all makes sense.
Finite Mathematics such as permutations, combinations, enumeration and asymptotes.
Modular arithmetic and floating point number systems.
Geometry of various shapes examining types of angles and parallel sides as ways to categorize various shapes. Also in here are the formulas for perimeter, area, surface area and volume for various shapes.
Sequences and series with the idea of limits and infinity as something special to bring to the class along with capital sigma for sums and pi for products.
Boolean algebra involving various logic gates such as AND, OR, XOR, and NOT in the form of truth tables.
Linear Algebra using only Matrices and linear transformations.
Logic problems similar to the board game Clue.
Pure Mathematics involving proofs of various Theorems and using a syntax full of abbreviations including the following: <=> as if and only if, => implies, backwards E for existence, backwards E followed by an exclamation point for there exists and is unique, upside down A as a "For all" or "For each" qualifier, three dots in one arrangement for "Therefore" and another for "Since," as well as ideas of necessary conditions compared to sufficient conditions as well as proof by Induction.
Algebraic concepts like groups, rings and fields. Here would also have the notion of inverses, associative and commutative operations.
Picture if one had a classroom of 30 students, 10 of which have a basic North American education and the other 20 each learned one of the 12 above parts of Mathematics but not necessarily the others. Could terms like "graph" be cases where more than one branch use the term with a very different meaning, e.g. graph of a function in the (x,y) plane versus a set of vertices and edges as a graph or net as in the 2-D representation of how to build various 3-D shapes? That would be a similar situation I believe.