No time to write now, but here's the abstract:
THE EFFECTS OF CUMULATIVE PRACTICE ON MATHEMATICS PROBLEM SOLVING (pdf file)
KRISTIN H. MAYFIELD AND PHILIP N. CHASE
JOURNAL OF APPLIED BEHAVIOR ANALYSIS
2002, 35, 105–123
NUMBER 2 (SUMMER 2002)
This study compared three different methods of teaching five basic algebra rules to college students. All methods used the same procedures to teach the rules and included four 50-question review sessions interspersed among the training of the individual rules. The differences among methods involved the kinds of practice provided during the four review sessions. Participants who received cumulative practice answered 50 questions covering a mix of the rules learned prior to each review session. Participants who received a simple review answered 50 questions on one previously trained rule. Participants who received extra practice answered 50 extra questions on the rule they had just learned. Tests administered after each review included new questions for applying each rule (application items) and problems that required novel combinations of the rules (problem-solving items). On the final test, the cumulative group outscored the other groups on application and problem-solving items. In addition, the cumulative group solved the problem-solving items significantly faster than the other groups. These results suggest that cumulative practice of component skills is an effective method of training problem solving.
Note: the effects of cumulative practice on problem solving.
Not "procedural fluency" or "automaticity" or "mastery" etc.
The path to problem solving goes through a particular form of practice - cumulative practice - not through "do the problem 3 ways" (Trailblazers) or "explain how you got your answer."