On the interplay of cognitive, tactile, and computational approaches to number theory in teacher education

Abstract

This paper focuses on topics from elementary number theory used by the author in the mathematical preparation of K-12 teacher candidates by juxtaposing concrete materials, digital technology, and formal reasoning. The topics include triangular numbers, their connection to trapezoidal numbers and extension to other figurate numbers. The paper shows how problem solving may be based on the integration of the modern-day approaches to mathematics supported by the creation of images, their numeric interpretation followed by algebraic generalization and computational verification of general statements in a symbolic form. Digital tools used in the paper include spreadsheets, Wolfram Alpha, and Maple. Solicited comments by teacher candidates about their use of the digital tools are shared.