Briefly: I am looking for research on the extent to which learning mathematics (let's say "college algebra" if we want to be specific) impacts problem solving skills, abstract reasoning, etc.
Less briefly: I teach a college algebra class in an american research university. Most of my students struggle with motivation, and want to know (for example) why they need to learn how to complete the square. My stock answer is to compare the classroom to a gym and to the instructor (me) as a personal trainer: the purpose of mathematics is then to strengthen their "analytic-reasoning muscles." The students accept this answer for the most part, but I feel a little guilty in that I don't actually know of any research that supports the argument that learning mathematics improves abstract reasoning or problem solving skills.
Possibly unnecessary: What do I mean by "abstract reasoning / problem solving skills"? That is a good question. I guess what I want to know is whether or not learning to complete the square is helps students learn how to solve problems beyond those that ask them to complete the square. I suspect this depends on what I am actually doing when I say that I am teaching them to complete the square. Am I giving them a recipe for them to follow? Or am I giving them something more to engage with?