Here's the foundational thing that irritates me the most when teaching college algebra.
Up through the secondary level, I think that instructors and students are trained to understand subtraction and division in terms of the inverse operation. Focusing on division here, if one asked "Why is $6/2 = 3$?", then one would most likely say it's because:
$$3 \times 2 = 6$$
But in every college-level algebra book I've seen, a different definition is given (and this goes for any texts in remedial elementary algebra, intermediate algebra, college algebra, etc.). Specifically, such books begin with the accepted "properties of real numbers", which are basically a restatement of the axioms for a field. In particular, one of the basic axioms is the existence of inverses: e.g., for multiplication, for any $a \ne 0$, there exists a value $1/a$ such that $a \times 1/a = 1$. (This is already problematic because students at this level are not yet familiar with statements involving existential quantifiers.) Thereafter, division is defined this way: $a/b$ means $a \times 1/b$. Of course, that's exactly what we see for a definition in most abstract algebra texts. But then technically this commits us to justifying "Why is $6/2 = 3$?" with something like the following chain of reasoning from the axioms:
$$6/2 = 6 \times 1/2 = (3 \times 2) \times 1/2 = 3 \times (2 \times 1/2) = 3 \times 1 = 3$$
Which I'm pretty sure no one actually ever does. Rather, they continue to use the secondary-school justification, even though this is technically out-of-synch (although, obviously, provably consistent with) our starting textbook axiom-properties.
Furthermore: When radicals are defined in the college algebra text, then the definition will once again look like the understanding of inverses from secondary-school subtraction and division (so it is additionally irritating to have these definitions and justifications out-of-synch with each other).
In summary: Advantages of the secondary-school definition: (1) it's what students are familiar with, (2) it provides shorter justifications, (3) it better lays the groundwork for the definition of radicals. Advantages of the standard college-algebra definition: (1) it complies with any standard textbook, and (2) it synchronizes with standard abstract algebra definitions.
So I go back-and-forth about this proud nail every semester. It seems like there would be more advantages to redefining subtraction and division as per the customary secondary-school rules, and thus smooth the way for student entry and understanding of the course; but the labor of going off-book and rewriting everything always deters me.
What is the best resolution to this problem?