On Experimental Mathematics and Mathematics Education

On the Interplay of Mathematics and Education: Advancing Computational Discovery from Recognition to Observation

“…Despite or perhaps due to their variety, the tools have to be used appropriately to ensure both pragmatic and epistemic development of learners. The importance of the interplay between the pragmatic and the epistemic in the context of using computational experiments as one of the main instruments of the 21st century mathematics education "is rooted in the fact that nowadays the very act of mathematical thinking is enhanced through the power of technology" [14] (p. 841). With these pedagogic ideas in mind, mathematical teaching of the "strange and remote" begins with constructing one-variable polynomials, the coefficients of which are the numbers in the staggered rows of the rearranged Pascal's triangle (Figure 3).…”

“…Entering the ratio of any two consecutive terms of sequence (1) into the Graphing Calculator [10], a program capable of graphing a relation from any two-variable equation or inequality, while keeping a and b as parameters, yields an approximation to the Golden Ratio, and the higher the ranks of the terms, the better the approximation. These discoveries, representing computational experiments made possible by the modern-day educational software tools which are transforming the mathematical practices of learners and teachers [11][12][13][14], open a window to students' learning methods of mathematical machinery, including trigonometry and complex numbers, as a way of using theory to confirm the results of the experiments. Among those students are prospective elementary teachers [15], impelling mathematics educators "to construct experiences which lead to the growth of subject matter knowledge in learners" [16] (p. 331).…”