## Understanding in MathematicsThis book discusses the act of understanding in mathematics. The first chapter, "Understanding and Meaning," inquires into the various senses and uses of the word "understanding" in ordinary language and discusses the notion of meaning and relations between understanding and meaning. The second chapter, "Components and Conditions of an Act of Understanding," is central to the book and discusses the act of understanding. Chapter 3, "Processes of Understanding," looks at whole processes of understanding, and the roles therein of explanations and validations, examples, previous knowledge, figurative speech, and activity, both practical and intellectual. The question of evaluation of understanding is dealt with in chapter 4, "Good Understanding." One of the problems raised is concerned with the relativity of any evaluation of understanding. Two important determinants of this relativity, namely the developmental stage of the understanding subject and the culture, are the objects of chapter 5, "Developmental and Cultural Constraints of Understanding." Contains 227 references. (MKR) |

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### Contents

Understanding and Meaning | 1 |

Components and Conditions of an Act of Understanding | 27 |

Processes of Understanding | 72 |

Good Understanding | 112 |

Developmental and Cultural Contraints of Understanding | 138 |

170 | |

186 | |

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abstract act of understanding action activity Ajdukiewicz arithmetic Bachelard Bolzano child cognitive complex concept concrete constructed continuous continuous functions culture curve defined definition didactical domain ematics End of example epistemological obstacles epsilon equation experience explanation expressions fact formal function given Greeno hyperbola ibidem ical idea identified important inference integers kind knowledge language learning Leibniz limit linear algebra linear independence linear operators logical math mathematical proofs mathematicians mathematics education meaning mental mental models metaphor metonymies mind natural numbers notion object of understanding philosophers Piaget and Garcia problem process of understanding proof quantities question quotient structure real numbers reasoning relations representations role rules scientific sense sequence Sierpinska situation solving speak standing statement structure symbolic synthesis teacher teaching theory things thought tion understanding in mathematics understood variable vector space Vygotski word