Aesthetics in School Mathematics: A Potential Model and A Possible Lesson

Abstract

IntroductionEarlier studies on improving classroom practice in mathematics have suggested a closer attention to nurturing an aesthetic appreciation for mathematics in students' learning experiences (Krutetskii, 1976; Papert, 1980; Silver & Metzger, 1989; Smith, 1927; Sriraman, 2009). Recent evidence nonetheless reveals little indication of its presence (Dreyfus & Eisenberg, 1986; Tjoe, 2015). We discuss in this article how current considerations of aesthetics in school mathematics, if any, might have inadvertently emphasized perfunctory precision over creative process. Given its current state, we argue how aesthetics can evolve into a compelling case in school mathematics.We begin with a survey of the notions of mathematical aesthetics and its interpretations. We present a typical contemporary classroom episode of a first grade mathematics lesson in oneand two-digit addition. We explain how exposing students to such a lesson might overlook the opportunity to reveal and foster an aesthetic appreciation for mathematics. We then offer a potential model of the case for aesthetics in school mathematics. Central to this model is the harmonious hierarchy of necessity, existence, and uniqueness without any of which the case for aesthetics in student learning might be suboptimal, if not untenable. We exemplify our model with a possible lesson designed to engage students aesthetically in the learning of mathematics. Pedagogical implications are discussed to reflect and revisit an interpretation of learning mathematics through problem solving.Mathematical AestheticsAesthetics has been one of the driving forces behind the activities that gave life to the advancements in mathematics as a discipline (Davis & Hersh, 1981). Its subtlety creates guidelines that many research mathematicians follow as one of the foremost principles in their professions. It is in the search of mathematical beauty that research mathematicians often seek approvals that lead to the crowning achievement in their mathematical experience (Hardy, 1940).Sinclair (2004) analyzes the role of aesthetic values from several conceptual insights. She draws examples from existing empirical findings such as those by Dreyfus and Eisenberg (1986) and Silver and Metzger (1989). In one of her interpretations of their work, she suggests that mathematicians' aesthetic choices might be at least partially learned from their community as they interact with other mathematicians and seek their approval (Sinclair, 2004, p. 276). Furthermore, she indicates that mathematical beauty is only feasible in the process young mathematicians are having to join the community of professional mathematicians-and when aesthetic considerations are recognized (unlike at high school and undergraduate levels) (p. 276).Related to Sinclair's (2004) interpretations of mathematical aesthetics, Karp (2008) conducts a comparative study on the aesthetic aspect of mathematical problem solving. Karp's comparative study involved middle and high school mathematics teachers from the U.S. and Russia. In his study, teachers are asked to provide examples and explanations of beautiful mathematics problems and approaches in solving those problems. Karp's (2008) findings confirm that the curricular system of education has a tremendous impact on students' aesthetic preference in mathematics problem solving. Each group of teachers shows different perspectives on what count as mathematical beauty. In particular, these differences stand out from their selections of mathematics topics. American teachers put extra weight on mathematics topics as prescribed by the American curriculum, which is typically associated with real-life situations and applications. Russian teachers do likewise as recommended by Russian curriculum with its traditionally heavy emphasis on algebra, number theory, and geometry. Evidently, these Russian problems tend to require longer approaches and are more algebraically demanding than their American counterparts. …