Jungeun Park

Is the derivative a function? If so, how do students talk about it?

This study explores how calculus students talk about the derivative as a function based on the concepts of function at a point and function on an interval. Twelve Calculus I students took a survey and explained their solution process during interviews. The analysis of students’ discourse with the commognitive lens focusing on their use of words and visuals showed that most students did not appreciate the derivative at a point as a number and the derivative function as a function. Moreover, one of the common descriptions of the derivative as a tangent line implies that they considered the ‘derivative’ as a point-specific object but also a (linear) function defined on an interval. This description was closely related to their use of the word, ‘derivative’ for both ‘the derivative function’ and ‘the derivative at a point.’

Fermat's point from five perspectives

The Fermat point of a triangle is the point such that minimizes the sum of the distances from that point to the three vertices. Five approaches to study the Fermat point of a triangle are presented in this article. First, students use a mechanical device using masses, strings and pulleys to study the Fermat point as the one that minimizes the potential energy of the system. Second, students use soap films between parallel planes connecting three pegs. The tension on the film will be minimal when the sum of distances is minimal. Third, students use an empirical approach, measuring distances in an interactive GeoGebra page. Fourth, students use Euclidean geometry arguments for two proofs based on the Torricelli configuration, and one using Vivianis Theorem. And fifth, the kinematic method is used to gain additional insight on the size of the angles between the segments joining the Fermat point with the vertices.

Interactive Technology to Foster Creativity in Future Mathematics Teachers

This chapter discusses ways in which the use of interactive technology in a problem-based course that integrates mathematics, science, and technology fosters creativity among future secondary mathematics teachers in their first year in college. The course was built on research-based principles to learn mathematics for understanding. We found that creativity is fostered naturally by teaching mathematics based on those principles. Creativity is fostered, promoted and developed when (a) learners themselves grapple with concepts and make concepts explicit; (b) learners actively build new understanding on previous knowledge; (c) learners engage with mathematics as a social process; (d) learners use multiple representations and connections to enhance their understanding; (e) learners pose and solve problems; and (f) learners exercise multiple modes of learning—when they read, talk, write, draw, analyze, apply, present, and reflect. We discuss the use of technology and issues related to future teachers’ creativity as they solve problems; design experiments and collect, represent, and analyze data; develop mathematical models for phenomena in the physical, biological, and social sciences; and build and program their own robot.

Soap films and GeoGebra in the study of Fermat and Steiner points

ABSTRACT We discuss how mathematics and secondary mathematics education majors developed an understanding of Fermat points for the triangle as well as Steiner points for the square and regular pentagon, and also of soap film configurations between parallel plates where forces are in equilibrium. The activities included the use of soap films and the interactive geometry program GeoGebra. Students worked in small groups using these tools to investigate the properties of Fermat and Steiner points and then justified the results of their investigations using geometrical arguments. These activities are specific approaches of how to encourage prospective teachers to use physical experiments to support students’ development of mathematical curiosity and mathematical justifications.

Use of words and visuals in modelling context of annual plant

ABSTRACT This study looks at the various verbal and non-verbal representations used in a process of modelling the number of annual plants over time. Analysis focuses on how various representations such as words, diagrams, letters and mathematical equations evolve in the mathematization process of the modelling context. Our results show that (1) visual representations such as flowcharts are used not only in the process to symbolization, but also used in the justification of symbols, (2) some of the visual representations serve as a bridge between the words in the problem context and the symbols that represent the mathematical equations of the number of annual plants and (3) words and context help to introduce visual representations and symbols. Also, once students come up with the visual representations and symbols, they show better understanding about words used in the problem context. These observations imply that the modelling and mathematization process is not just one-directional and linear from words describing real-life situations to the symbols in mathematical equations and expressions. Rather, the mathematization can be promoted through using other visuals that help make this transition smooth by organizing the given information in a way that can be used towards mathematization.