Katedra matematiky - Katolícka univerzita v Ružomberku

Catholic University in RužomberokScientific Issues, Teaching Mathematics II: Innovation, New Trends, Research, Ružomberok 2010CONJECTURING: AN INVESTIGATIONINVOLVING POSITIVE INTEGERSJan Kopka a , Leonard Frobisher b , George Feissner ca Přírodovědecká fakulta UJEP v Ústí nad LabemČeské mládeže 8, 400 96 Ústí nad Labem, Czech Republice-mail: kopkaj@sci.ujep.czb 58 Buckstone Avenue, LeedsLS 17 5HP, Englande-mail: ljfrobisher@btinternet.comc State University of NY, College at CortlandCortland, New York, 13045, USAe-mail: gffeisner@hotmail.comAbstract. Here we demonstrate investigative approach and creation of conjectureswith the help of a problem. A conjecture is a statement which appearsreasonable, but whose truth has not yet been established. Conjecturing forms thebackbone of mathematical thinking and reasoning.We present an approach to investigate a "mathematical situation." Wechoose not to define "mathematical situation" but will illustrate it withseveral examples. Since it is the first step on what can be a long and interestingjourney, we feel it is best to allow it to be as general as possible.A typical method of investigating a mathematical situation follows thissequence of processes:Mathematical situation - experimentation - conjecture -test - proof - mathematical theorem.The central activity during the investigation of many mathematicalproblem situations is that of conjecturing. A conjecture is a statementwhich appears reasonable, but whose truth has not yet been established.Conjecturing forms the backbone of mathematical thinking and reasoning.We look at a number of famous and interesting conjectures associated withpositive integers which can be explored at different levels in elementary andsecondary schools.