ERIC Number: EJ1231134
Record Type: Journal
Publication Date: 2018
Pages: 16
Abstractor: ERIC
ISBN: N/A
ISSN: ISSN-0819-4564
EISSN: N/A
Integrating Various Fields of Mathematics in the Process of Developing Multiple Solutions to the Same Problems in Geometry
Stupel, Moshe; Oxman, Victor
Australian Senior Mathematics Journal, v32 n1 p26-41 2018
The solution of problems and the provision of proofs have always played a crucial part in mathematics. In fact, they are the heart and soul of this discipline. Moreover, the use of different techniques and methods of proof in the same mathematical field, or by combining fields, for the same specific problem, can show the interrelations between the fields, as well as the richness, beauty and elegance of mathematics. In addition to the specific roles of proof in mathematics, the authors suggest that attempts to also prove a certain result (or solve a problem) using methods from several other different areas of mathematics (geometry, trigonometry, analytic geometry, vectors, complex numbers, etc.) are very important in developing deeper mathematical understanding, creativity, and appreciation of the value of argumentation and proof in learning different topics of mathematics. Their approach, that of presenting multiple proofs to the same problem, as a device for constructing mathematical connections is supported by (Polya,1973,1981; Schoenfeld, 1988; NCTM, 2000; Ersoz, 2009; Levav-Waynberg & Leikin, 2009). Multiple solution tasks (MSTs) contain an explicit requirement for proving a statement in multiple ways. The differences between the proofs are based on using: (1) different representations of a mathematical concept; (2) different properties (definitions or theorems) of mathematical concepts from a particular mathematical topic; (3) different mathematics tools and theorems from different branches of mathematics; (4) different tools and theorems from different subjects (not necessarily mathematics); and (5) different strategies of problem solving. In this case, the authors apply the third type of differences between the proofs. In a student activity, they present various solutions to the problems using the tools and theorems of Euclidean geometry, analytic geometry, trigonometry, and vectors.
Descriptors: Mathematics Instruction, Geometry, Problem Solving, Mathematical Logic, Validity, Mathematics, Mathematical Concepts, Equations (Mathematics)
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
Publication Type: Journal Articles; Reports - Descriptive
Education Level: N/A
Audience: N/A
Language: English
Sponsor: N/A
Authoring Institution: N/A
Grant or Contract Numbers: N/A