This review of the introductory instructional substance of functions and graphs analyzes research on the interpretation and construction tasks associated with functions and some of their representations: algebraic, tabular, and graphical. The review also analyzes the nature of learning in terms of intuitions and misconceptions, and the plausible approaches to teaching through sequences, explanations, and examples. The topic is significant because of (a) the increased recognition of the organizing power of the concept of functions from middle school mathematics through more advanced topics in high school and college, and (b) the symbolic connections that represent potentials for increased understanding between graphical and algebraic worlds. This is a review of a specific part of the mathematics subject mailer and how it is learned and may be taught; this specificity reflects the issues raised by recent theoretical research concerning how specific context and content contribute to learning and meaning.
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