My dissertation involved, among other things, the East Asian way of teaching and learning mathematics. (See, for example, Leung (2001).) I was particularly interested in the Kumon method. Although I found a nice paper in a refereed journal that describes it (Ukai, 1994), and some informal studies comparing it with other ways of teaching mathematics, I was unable to find any experimental studies (those involving random sampling or assignment).
What papers in refereed journals have presented experimental studies comparing the Kumon method (or something similar to it) with other ways of teaching and learning mathematics?
Leung, F. K. S. (2001). In search of an East Asian identity in mathematics education. Educational Studies in Mathematics, 47, 35-51.
Ukai, N. (1994). The Kumon approach to teaching and learning. Journal of Japanese Studies, 20, 87-113.
Here is additional information as requested in a comment. (I'm quoting from my dissertation.)
Students in East Asian countries outperform their counterparts in other countries in large-scale international studies of mathematics achievement (Leung, 2006). Leung (2006) asserted that this is because their teachers have good conceptual understanding and procedural skills, their instruction is teacher-dominated, and their culture strongly emphasizes the importance of education, which creates high expectations for them and their teachers. It also seems that East Asian teachers and students acquire their mathematics competence from instruction that focuses on procedures and repeated practice with variation (Leung, 2006).
How does this kind of education result in the high mathematics achievement observed? Leung (2006, p. 43) proposed this explanation:
The process of learning often starts with gaining competence in the procedure, and then through repeated practice, students gain understanding. Much of the mathematics in the school curriculum may need to be practiced without thorough understanding first. With a set of practicing exercises that vary systematically, repeated practice may become an important "route to understanding" [...].
The view that repeated practice of procedures leads to conceptual understanding is especially apparent in the program known as Kumon Math.
Kumon Math, published by Kumon North America, Inc., is a supplemental mathematics curriculum for students in preschool through secondary school. The curriculum is not structured by age or grade level, but by a student's own pace, and is composed of hundreds of short assignments that progress through increasingly difficult mathematics skills. Students complete one assignment every weekday and attend a Kumon center for two sessions per week. Every assignment is timed and graded. Students master skills through repetition and cannot progress to a new skill before completing an assignment within a set amount of time and with close to 100% accuracy. New skills and exercises build on previously mastered concepts. (WWC, 2009)
Ukai (1994) gives a detailed description of the Kumon method and the philosophy behind it. In Kumon Math, "[c]oncepts are not explicitly taught. Rather, through repetition, learners experience insight. For example, the means of solving such problems as $48\div 6$ or $132\div 12$ will be patently clear, without verbal or written explanation, to a child who has completed several hundred multiplication problems and to whom $6\times 8$ and $11\times 12$ are operations that have been 'overlearned' " (pp. 92-93).