>It is certainly how I was taught about slope, and I went on to graduate from MIT, so whatever nuance this definition allegedly didn't catch did me no harm.

In that case, this debate really isn't about you. You were probably an exceptional student, who saw the connections between mathematical concepts easily, regardless of instruction. You probably found yourself predicting the next concept a teacher would introduce, because it just "makes sense." Not all students are that way. Most are not.

The two people involved here are fighting over two different ideas. Sal is being pedantic, but is right, slope is defined as ∆x/∆y. What the other guy was saying is that slope represents rate of change, which is a much more important concept to early algebra, and the underpinning of why you actually care about slope in physics and calculus. You probably made the connection effortlessly. I assure you, many students do not.

I teach high school mathematics to both honors and special needs students, and it's important to keep in mind that the instruction is very different between the two populations.