Engaging Students in a Visual and Conceptual Approach to Taylor Series

AbstractIn this reflection of teaching, we describe a series of activities that introduce the Taylor series through dynamic visual representations with explicit connections to students’ prior learning. Over the past several decades, educators have noted that curricular materials tend to present the Taylor series in a way that students often interpret as formulaic, which leads them to focus on procedural aspects. We share our instructional approach, which introduces students to graphs of polynomial functions that resemble transcendental functions. However, students are not told that they are approximating polynomials initially, and through carefully sequenced experiences, students gradually build an intuitive understanding of the Taylor series. With the goal of equipping instructors who wish to embark upon similar journeys, we describe the visual representations employed, the questions we pose to students, students’ typical responses, and important topics for discussion. The activities described engage students in reasoning and sense-making throughout the learning process, which is essential for developing a robust understanding of Taylor series as well as other calculus topics.KEYWORDS: Conceptually-basedvisual displayscalculusseriesintuitionsactive learningexploration ACKNOWLEDGMENTWe would like to acknowledge the innovative and supportive contributions of Stuart Price, a mathematics teacher in the United Kingdom, whose activities in teaching the Taylor series through exploration and use of dynamic visual representations inspired our work.DISCLOSURE STATEMENTNo potential conflict of interest was reported by the author(s).Notes1 The fact that f(x) is equal to T(x) is a consequence of the Lagrange Error formula.Additional informationNotes on contributorsElisa SoosloffElisa Soosloff is a mathematics teacher at Creston High School in Iowa where she is developing a calculus program. Elisa enjoys reading research about alternative teaching methods and producing new activities that can be beneficial to teachers.Maryann HueyMaryann Huey teaches mathematics and mathematics education courses at Drake University in Des Moines, Iowa. She is interested in how cognitive research and models of student thinking can inform and improve mathematics and statistics instruction.Daniel S. AlexanderDaniel Alexander is an emeritus professor of Mathematics at Drake University. His research areas involve the history of math, specifically the history of Complex Dynamics, as well as scholarship related to the teaching and learning of mathematics.