How to “best” teach pupils has long been heatedly and unprofitably debated. On one hand, reformers have long been dissatisfied with the limited and even harmful effect of traditional direct instruction. Concerned that traditional direct instruction often fails to promote conceptual understanding, mathematical thinking (e.g., problem-solving and reasoning ability), and a positive disposition toward mathematics (e.g., autonomy, confidence, perseverance), constructivists and other reformers have recommended discovery learning.

Early in the 20th century, the progressive education movement pioneered by John Dewey advocated methods diametrically opposite to traditional methods (e.g., child-centered, instead of teacher-centered, instruction; unguided, rather than guided, instruction). In the 1960s and 1970s, the radical constructivist philosophy of some followers of Jean Piaget reinforced this laissez-faire “free play” or “free school” approach to instruction, which was ...