摘要：Concrete objects used to illustrate mathematical ideas are commonly known as manipulatives. Manipulatives are ubiquitous in North American elementary classrooms in the early years, and although they can be beneficial, they do not guarantee learning. In the present study, the authors examined two factors hypothesized to impact second-graders’ learning of place value and regrouping with manipulatives: (a) the sequencing of concrete (base-ten blocks) and abstract (written symbols) representations of the standard addition algorithm; and (b) the level of instructional guidance on the structural relations between the representations. Results from a classroom experiment with second-grade students (N = 87) indicated that place value knowledge increased from pre-test to post-test when the base-ten blocks were presented before the symbols, but only when no instructional guidance was offered. When guidance was given, only students in the symbols-first condition improved their place value knowledge. Students who received instruction increased their understanding of regrouping, irrespective of representational sequence. No effects were found for iterative sequencing of concrete and abstract representations. Practical implications for teaching mathematics with manipulatives are considered.
