**ERIC Number:**EJ1044653

**Record Type:**Journal

**Publication Date:**2014-Sep

**Pages:**9

**Abstractor:**ERIC

**Reference Count:**15

**ISBN:**N/A

**ISSN:**ISSN-0025-5769

Pose and Solve Varignon Converse Problems

Contreras, José N.

Mathematics Teacher, v108 n2 p98-106 Sep 2014

The activity of posing and solving problems can enrich learners' mathematical experiences because it fosters a spirit of inquisitiveness, cultivates their mathematical curiosity, and deepens their views of what it means to do mathematics. To achieve these goals, a mathematical problem needs to be at the appropriate level of difficulty, challenging but not insurmountable, and rich enough to be a fruitful source of new and interesting problems. As learners embark on problem-posing and problem-solving adventures, they need to be equipped with powerful tools of inquiry that will support their efforts in exploring uncharted territory. The first tool that learners need is a systematic problem-posing strategy. Posing converse problems is a strategy that is applicable to a wide range of problems. As Movshovits- Hadar (1988) suggests, most nontrivial mathematical problems are an endless source of surprise. Varignon converse problems, as shown here, are no exception. The second tool that learners need to continue their investigations is a technological device such as GeoGebra or The Geometer's Sketchpad® that can facilitate the discovery of patterns or regularities to formulate a plausible conjecture (Pólya 1945). Finally, learners need a mathematical and pedagogical tool to transform their conjecture into a theorem. Such a tool is a proof. Constructing a mathematical proof allows learners not only to verify the validity of a theorem but also, and perhaps more important, to gain insight about why it is true. A proof allows the learner to connect the underlying mathematical concepts.

Descriptors: Problem Solving, Questioning Techniques, Educational Practices, Educational Strategies, Teaching Methods, Mathematical Applications, Mathematics Achievement, Mathematics Instruction, Mathematics Skills, Investigations, Geometry, College Mathematics, Secondary School Teachers, Inservice Teacher Education, Mathematical Logic, Mathematical Concepts, Technology Uses in Education

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**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**Higher Education; Postsecondary Education; Secondary Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A