## APOS Theory: A Framework for Research and Curriculum Development in Mathematics EducationIlana Arnon, Jim Cottrill, Ed Dubinsky, Asuman Oktaç, Solange Roa Fuentes, Maria Trigueros, Kirk Weller In spite of the fact that APOS Theory has been used extensively in numerous scholarly publications, in the design of textbooks, and in teaching practice, there is no single references that contains all the relevant information about its components, and provides guidance about its application. The goal of this book is to present the main elements of APOS theory. It should be useful for researchers who work with, or would like to learn more about, this theoretical approach, people who are interested in the way which mathematical conceptions are constructed according to this theory, Mathematics Education researchers, graduate students in Mathematics Education, and Mathematics instructors. |

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APOS THEORY

### Contents

1 | |

5 | |

Mental Structures and Mechanisms APOS Theory and the Construction of Mathematical Knowledge | 16 |

Genetic Decomposition | 27 |

The Teaching of Mathematics Using APOS Theory | 57 |

The APOS Paradigm for Research and Curriculum Development | 92 |

Schemas Their Development and Interaction | 109 |

Totality as a Possible New Stage and Levels in APOS Theory | 137 |

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### Common terms and phrases

abstract algebra ACE Teaching Cycle Action to Process activities analysis analyzed APOS Theory APOS-based instruction APOS-based research approach Arnon Asiala authors axiom based on APOS binary operation chain rule chapter cognitive considered coordination cosets Cottrill de-encapsulated decimal expansion decimal strings derivative described determine difficulty domain Dubinsky elementary elements encapsulation equivalence classes example framework func help students ideas individual individual’s infinity instructional treatment interiorization interview involved ISETL learner learning levels Line linear algebra linear transformation mathematical concepts mathematical induction Mathematics Education Mathematics Education Vol mental constructions mental mechanisms mental Object mental structures non-unit fractions notion Object conception Oktac perform Piaget preliminary genetic decomposition Process conception programming progression properties quotient groups reflective abstraction repeating decimal representations Research and Curriculum Sect sequence spanning set specific Springer Science+Business Media stage thematization theoretical thinking Totality Trigueros understanding vector space Weller