Through methods developed by the Association of Mathematical Instruction (AMI) in Japan, children can be helped toacquire mathematical knowledge. Problem solving that enhances acquisition of mathematical knowledge is divided intowo types, semantic problem solving (SPS) and pragmaticoblem solving (PPS). The focus in SPS is a systematic organization of the target knowledge and the schematic representations through which children acquire it. PPS emphasizes thefunctional composition of the situation and the use of constraints in accordance with a child’s ability. SPS and PPS cover not only the beginning but the whole process of learning. support of this assertion, the process of becoming an adaptiveproblem solver beyond the initial learning situation is illustrated and examined from the viewpoint of interiorization and generalization. SPS and PPS are also helpful for restructuring learning in mathematics through clarifying referents and demonstrating the significance of a new concept. Finally, the authenticity of learning in relation to effective PPS-based teaching is discussed.

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