I have heard contradicting views that both parties say are research based. Specifically, I am looking for research on which teaching methods (direct instruction or student centered approaches) are most effective in high school mathematics (grades 9-12). I went to a workshop about a year ago that said direct instruction is proving to be more effective, but every curriculum and instruction department in academia focuses on teaching pre-service math teachers student-centered approaches. What does the research say? Searching google scholar does not seem to yield any useful results.
Abstract from Alfieri, Brooks, Aldrich, "Does Discovery-Based Instruction Enhance Learning?", Journal of Educational Psychology, 2011:
Discovery learning approaches to education have recently come under scrutiny (Tobias & Duffy, 2009), with many studies indicating limitations to discovery learning practices. Therefore, 2 meta-analyses were conducted using a sample of 164 studies: The 1st examined the effects of unassisted discovery learning versus explicit instruction, and the 2nd examined the effects of enhanced and/or assisted discovery versus other types of instruction (e.g., explicit, unassisted discovery). Random effects analyses of 580 comparisons revealed that outcomes were favorable for explicit instruction when compared with unassisted discovery under most conditions (d = –0.38, 95% CI [–.44, –.31]). In contrast, analyses of 360 comparisons revealed that outcomes were favorable for enhanced discovery when compared with other forms of instruction (d = 0.30, 95% CI [.23, .36]). The findings suggest that unassisted discovery does not benefit learners, whereas feedback, worked examples, scaffolding, and elicited explanations do.
Clark, Kirschner, Sweller, "Putting Students on the Path to Learning: The Case for Fully Guided Instruction", American Educator, Spring 2012, also make the following points: (1) less-skilled students learn more from guided instruction, more-skilled from self-guided discovery (although if allowed to choose, each group will pick as a preference the type they learn less from), and (2) per Richard Mayer, the cycle of self-guided instruction has been disabused, and then resurrected under a different name every decade or so since the 1950's (and each generation of proponents seems unaware/disinterested in the findings that came before).
In the recent MAA publication "Insights and Recommendations from the MAA National Study of College Calculus" active learning is one of the 7 characteristics of successful Calculus programs. I would probably start with Chapters 3 and 8 and the references therein. For example, on page 97 you can find:
A meta-analysis of 39 studies in multiple STEM fields found that small-group learning had a positive impact on achievement, persistence, and student attitudes (Springer, Stanne, & Donovan, 1999)
These studies have consistently found that students from [calculus] reform courses developed stronger conceptual understanding and were more likely to persist in STEM fields while showing little or no negative impact on procedural fluency (Chappell & Killpatrick, 2003; Hurly, Koehn, & Ganter, 1999; Joiner, Malone, & Haimes, 2002).
I think you need to consider what is really under each of those terms (student-based versus direct instruction). For example one of the first answers here assumes that student based equates to discovery method. If I self-study a textbook (reading the chapter, doing homework problems, grading myself, etc.), is that student based? (yes). But is it discovery method (no).
There are more than two simple choices for how to teach a subject. To even have a good hypothesis to test, a little more effort is needed on definition of the alternatives.
I do like that you have restricted the subject (HS math), but I would also consider that different methods may work better with different populations, especially student ability.