A method and apparatus are provided for teaching rigorous mathematicalthinking to a learner. The method includes the steps of mediating the learnertoappropriate a set of cognitive tasks as general physiological tools based upontheirstructure-function relationship, mediating the learner to perform the set ofcognitive tasks through the use of the physiological tools to construct higher-ordercognitive processes, mediating the learner to systemically build basicessentialconcepts needed in mathematics from everyday experiences and language,mediating the learner to discover and formulate the mathematical patterns andrelationships in the cognitive processes, mediating the learner to appropriatemathematically specific psychological tools based upon their uniquestructure--function relationships and mediating the learner to practice theuse of eachmathematically specific psychological tool to organize and orchestrate the useofcognitive functions and to construct mathematical conceptual understanding.
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