The thrust of peer apprenticeship learning is in how an individual's personal beliefs, disposition, and epistemology can be transformed through an apprenticeship-like learning process. This paper describes the peer apprenticeship learning situation between two students, Dom (age: 13) and Ming (age: 14), in distributed computer-mediated co-construction of mathematical meanings. Ming was initially procedural or rule-based in his problem solving methods, encountering numerous difficulties with his approaches. Dom, however, was constantly engaged in playing with ideas through conjecturing and perceiving mathematical relationships. In the initial stages of co-construction efforts, Dom and Ming were solving the problems rather independently as each appropriated a radically different epistemology of mathematics – Ming was rule-based whereas Dom was idea-based. In the cause of increasing difficulties, Ming soon recognized that his methods were inadequate and, after a considerable struggle, positioned himself in an apprentice-like manner in order to follow Dom's conceptualizations. Through monitoring of Dom's conceptualizations and with personal experimentations to concretize his understanding, Ming was gradually able to assimilate Dom's mathematical meaning perspectives. We depicted such a learning situation as peer apprenticeship learning. As a result of assimilating the disposition towards playing with ideas, Dom and Ming were able to engage in meaningful idea-based social constructivism.