ONE PROBLEM, MULTIPLE SOLUTIONS: HOW MULTIPLE PROOFS CAN CONNECT SEVERAL AREAS OF MATHEMATICS
Problem solving and proofs have always played a major role in mathematics. They are, in fact, the heart and soul of the discipline. Even more so, using a number of different proof techniques - from within and between several domains of mathematics - for one specific problem can demonstrate the connections between these domains, as well as the wealth, beauty, and elegance of mathematics. We present one specific, interesting geometry problem, and present nine different proofs for it, using methods from a wide variety of different mathematical domains, including using dynamic geometry software (DGS) application. Encouraging high school as well as college students to derive results in such various ways will enhance their appreciation of mathematics and give them incentive to derive even more elegant solutions on their own. After conduction of a case study that involved a course on this topic as a part of a pre-service mathematics teacher education program (including student feedback via questionnaire and interviews), it was concluded that mathematics educators should be encouraged to introduce many authentic multiple-proof problems into their teaching program. As a bonus, the problem we used in this report provided a pleasant surprise in that it revealed a special triangle with interesting properties.
mathematical proof, problem solving, multiple solutions and proofs, DGS - dynamic geometry software, teacher education.