Abstract
The paper promotes the notion of computational experiment supported by a multi-tool digital environment as a means of the development of new mathematical knowledge in the context of education. The main study of the paper deals with the issues of teaching this knowledge to secondary teacher candidates within a graduate capstone mathematics education course. The interplay of mathematics and education is considered through the lens of using technology to enhance one’s mathematical background by advancing ideas from mostly known to genuinely unknown. In this paper, the knowns consist of Fibonacci numbers, Pascal’s triangle, and continued fractions; among the unknowns are Fibonacci-like polynomials and generalized golden ratios in the form of cycles of various lengths. The paper discusses the interplay of pragmatic and epistemic uses of digital tools by the learners of mathematics. The data for the study were collected over the years through solicited comments by teacher candidates enrolled in the capstone course. The main results indicate the candidates’ appreciation of the need for deep mathematical knowledge as an instrument of the modern-day pedagogy aimed at making high schoolers interested in the subject matter.
Highlights
Whereas a spreadsheet is not the only digital tool discussed in this paper, it played an initial role in the development of mathematical ideas that the author, a mathematics educator of teacher candidates, intends to share
One of the earliest uses of a spreadsheet in tertiary mathematical education was in the context of numerical modeling of Fibonacci numbers [3], motivated, perhaps, by the ideas about a laboratory course in calculus and linear algebra in which “one student found experimentally that the limit of the ratio of terms in a Fibonacci sequence is independent of the two given initial values” [4] (p. 293)
Some 150 years before Dewey, Euler, as mentioned in [33], saw observations as a foundation for the discovery of new properties of numbers in mathematics. Such historically notable similarity of mathematical and educational perspectives on observations set a perfect context for writing a paper on the interplay of mathematics and education
Summary
Two types of materials have been used by the author when working on this paper as a reflection on teaching the capstone course titled “Topics and Research in Mathematics Education”. The university where the author works is located in upstate New York in close proximity to Canada, and many of the author’s students have been Canadians pursuing their master’s degrees in education This diversity of students suggests the importance of aligning mathematics education courses with multiple international perspectives on teaching mathematics to future teachers [27]. In accord with those perspectives, activities described and analyzed in the paper are aimed at developing deep conceptual understanding of mathematics in teacher candidates and may be characterized as action learning [28,29]. In the lab, during experimentation with mathematical assignments requiring the use of digital tools, didactic intervention in the form of a dialogue was mostly between the author and a pair of participants and it could evolve into the whole class discussion of an issue important for the attention of all participants
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