Using multiple solutions to mathematical problems to develop pedagogical and mathematical thinking: A case study in a teacher education program

ABSTRACT

Mathematics educators agree that linking mathematical ideas by using multiple approaches for solving problems (or proving statements) is essential for the development of mathematical reasoning. In this sense, geometry provides a goldmine of multiple-solution tasks, where a myriad of different methods can be employed: either from the geometry topic under discussion or from other mathematical areas—analytic geometry, trigonometry, vectors, complex number, etc. Employing multiple proofs fosters better comprehension and increased creativity in mathematics for the student/learner, enriching teachers’ pedagogical accomplishments and promoting lively class discussion. Given the important role of multiple-solution problems within and between mathematical topics, the evidence is astonishing that classroom teachers rarely introduce their students to multiple-solution tasks. Hence, one can conjecture that this gap between theory and practice could turn connecting tasks with the employment of technological tools into a powerful environment for the development of pre- and in-service mathematics teachers’ knowledge. For this reason the authors believe that exposing and providing mathematics teachers with an arsenal of specific tasks with a variety of solutions from different mathematical areas is essential. Based on a conducted case study, both teacher trainees and lecturers clearly indicated that solving problems in multiple ways is valuable in developing thinking ability for both students and teachers, encouraging creativity and increasing the quality of teaching—hence this technique should be included in the secondary school curriculum as well as in teacher training programs.

KEYWORDS:

• Dynamic geometry environments (DGE)
• dynamic geometry software (DGS)
• mathematical proof
• multiple solutions and proofs
• pedagogical methods
• problem solving

Acknowledgments

We would like to acknowledge the contribution of The National Teachers’ Center for Mathematics in High School, located at Haifa University, Israel, in preparing the clips using the GeoGebra software application, with special thanks to academic head Dr. Varda Talmon and staff members Geula Sever and Maya Katz.