Assessing Mathematical Skills, Understanding, and Thinking
Jan de Lange
Many mathematicians, among them Ahlfors, Bers, Birkhoff, Courant, Coxeter, Kline, Morse, Pollak, and Polya, signed a 1962 statement from which the following quotes were taken ( Ahlfors, 1962):
To know mathematics means to be able to do mathematics: i.e., to use mathematical language with some fluency, to do problems, to criticize arguments, to find proofs, and, what may be the most important activity, to recognize a mathematical concept in, or to extract it from, a given concrete situation. Therefore, to introduce new concepts without sufficient background of concrete facts, to introduce unifying concepts where there is no experience to unify, or to harp on the introduced concepts without concrete applications which would challenge the students, is worse than useless. Premature formalization may lead to sterility; premature introduction of abstractions meets resistance especially from critical minds who, before accepting an abstraction, wish to know why it is relevant and how it could be used...Extracting the appropriate concept from a concrete situation, generalizing from observed cases, inductive arguments, arguments by analogy, and intuitive grounds for an emerging conjecture, are mathematical modes of thinking...The best way to guide the mental development of individuals is to let them retrace the mental development of the race (the genetic principle). (p. 8.)
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Publication information: Book title: Assessment of Authentic Performance in School Mathematics. Contributors: Richard Lesh - Editor, Susan J. Lamon - Editor. Publisher: AAAS Press. Place of publication: Washington, DC. Publication year: 1992. Page number: 195.
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