Abstract
In this paper we hypothesize that education, especially at the scale of curriculum, should be treated as a complex system composed of different ideas and concepts which are inherently connected. Therefore, the task of a good teacher lies in elucidating these connections and helping students make their own connections. Such a pedagogy allows students to personalize learning and strive to be ‘creative’ and make meaning out of old ideas. The novel contribution of this work lies in the mathematical approach we undertake to verify our hypothesis. We take the example of a precalculus course curriculum to make our case. We treat textbooks as exemplars of a specific pedagogy and map several texts into networks of isolated (nodes) and interconnected concepts (edges) thereby permitting computations of metrics which have much relevance to the education theorists, teachers and all others involved in the field of education. We contend that network metrics such as average path length, clustering coefficient and degree distribution provide valuable insights to teachers and students about the kind of pedagogy which encourages good teaching and learning.
Highlights
This project aims to utilize the mathematical theory of networks to understand the significance of connectivity in mathematics education
Throughout the effort to meaningfully analyze quantitative patterns exhibited through these network representations -oriented textbooks, it initially became visually evident via generation of node-oriented degree distributions relative to frequency that there is a natural power law exhibited through these connections
Upon analysis of these metrics, we find that there is reasonably no significant additional error induced when five percent (n = 80) of nodes are randomly removed; with a mean average path length of 5.7123, there is less than one additional topic one must introduce on average per selected topic that is required to introduce in order to meaningfully move throughout an extensive union course in precalculus, which indicates that relatively little information is lost through this level of error
Summary
This project aims to utilize the mathematical theory of networks to understand the significance of connectivity in mathematics education. The role of an effective educator lies in effectively highlighting these features and helping students make their own connections [1]. Such a pedagogy, based on a constructivist philosophy of learning, allows students to personalize learning and strive to be ‘creative’ which essentially amounts to ‘meaning making’ or generating new meaning out of old ideas [2,3,4]. While the ensuing discussion will surround the effectiveness of precalculus pedagogy, this article is primarily meant to be an exposition on the importance of a complexity theory-based approach towards curricular design. The current work is one ‘micro-scale’ such attempt in the endeavor to understand the complex structure of a curriculum and its implications throughout the entirety of education
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