I agree that this is a great idea. It's common knowledge that students tend to have difficulty in courses due to gaps and misconceptions from previous courses. I teach for the Math for Elementary teachers sequence. Once in a while, I get a calc. student in there. They become stronger students as a result of taking a deeper, hard and conceptual look at the "basics."
Too often, we reserve a course like the one you're describing to our developmental students. But then those courses become so full that we still have trouble taking the time on number sense and other important conceptual ideas leading up to or involving algebra.
At my school, I co-created a course called Introduction to College Mathematics as an accelerated model for students placing into our 2-course dev. ed. sequence. It's a 6 credit hour course giving us 2 hours and 50 minutes twice a week to (in a perfect world) take our time to explore number sense, the why's of algebra, etc.
I would love for their to be an optional course like the one you describe for students at all levels. Perhaps conversations between calc. student and students in development students about some of the big misconceptions and number sense would be fruitful. I would design it similar to the first math for elem. teachers course.
Of course, getting buy in from students and finding faculty to teach it in a way that isn't just a checklist of skills would involve some creativity.
Thanks for starting this thread.
Chris Sabino
Harold Washington College