Reflections on Symmetry and Proof

Autor/in	Merrotsy, Peter
Titel	Reflections on Symmetry and Proof
Quelle	In: Australian Senior Mathematics Journal, 22 (2008) 1, S.38-49 (12 Seiten) PDF als Volltext (1); PDF als Volltext kostenfreie Datei (2) Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0819-4564
Schlagwörter	Stellungnahme; Logical Thinking; Mathematics Instruction; Cognitive Ability; Mathematical Logic; Mathematical Concepts; Problem Solving; Mathematics Skills; Generalization; Methods; Spatial Ability; Developmental Stages; Geometry; Geometric Concepts; Validity; Secondary School Mathematics + Suchen Sie Ihr Suchwort? Mathematics lessons; Mathematikunterricht; Denkfähigkeit; Mathematical logics; Mathematische Logik; Problemlösen; Mathmatics achievement; Mathematics ability; Mathematische Kompetenz; Method; Methode; Räumliches Vorstellungsvermögen; Geometrie; Elementare Geometrie; Gültigkeit
Abstract	The concept of symmetry is fundamental to mathematics. Arguments and proofs based on symmetry are often aesthetically pleasing because they are subtle and succinct and non-standard. This article uses notions of symmetry to approach the solutions to a broad range of mathematical problems. It responds to Krutetskii's criteria for mathematical ability as well as the outcomes which guide the Extensions 1 & 2 Mathematics courses of the Board of Studies NSW. For Krutetskii (1976, pp. 84-88), mathematical ability is seen in terms of a student's ability (1) to formalise; (2) to symbolise; (3) to generalise; (4) to carry out sequential deductive logic; (5) to syncopate or to curtail logic or argument; (6) to reverse logical thinking or find the converse; (7) to be flexible in mathematical methods used; (8) to conceptualise spatially; and (9) to develop before puberty a "mathematical mind." Students with high ability or high potential in mathematics enjoy and express these abilities in a way which is markedly and qualitatively differentiated from the ability of typical age peers, and which is measurable in their ability to solve problems. Here, the author presents mathematical problems and their solutions which are intended to illustrate "Krutetskii's abilities." The solutions to the problems present arguments using various aspects of the fundamental concept of symmetry: rotational symmetry, reflection (in several senses of the word), the symmetry of proportion, symmetry in an abstract sense (for example, in the conics problem), and asymmetry. The cognitive abilities called on in these arguments involve each of Krutetskii's "mathematical abilities." (Contains 11 figures.) (ERIC).
Anmerkungen	Australian Association of Mathematics Teachers (AAMT). GPO Box 1729, Adelaide 5001, South Australia. Tel: +61-8-8363-0288; Fax: +61-8-8362-9288; e-mail: office@aamt.edu.au; Web site: http://www.aamt.edu.au
Erfasst von	ERIC (Education Resources Information Center), Washington, DC
Update	2017/4/10