Part IIFunctional Thinking

Part II examines the function concept in school mathematics as it has evolved from Klein’s time.

Katja Krüger describes the goals of the Prussian reform movement at the beginning of the twentieth century in the Meraner Lehrplan, particularly their call for functional thinking as a foundational principle. Krüger describes the impact of this movement on German mathematics education, using examples from textbooks and writings for teachers, which shows a focus on a dynamic conception of function. She also describes the work of mathematics teachers influenced by Klein and the wider group of educators and mathematicians of which he as a member.

Patrick W. Thompson and Fabio Milner describe the results of a comparative study of secondary teachers from South Korea and the USA. They examine the meanings that teachers attached to functions and function notation by looking at results from the instrument Mathematical Meanings for Teaching secondary mathematics developed by the first author and collaborators. They find significant differences between the mathematical meanings of South Korean and US teachers and conclude with the observation that in the US Klein’s double discontinuity is in fact a continuity of flawed meanings that prospective teachers carry throughout their university careers and bring back to the school classroom.

Hyman Bass studies an important component of the function concept in school mathematics, the notion of a continuous number line that provides the domain for most functions students encounter. There is evidence that many students in the US lack a robust understanding of this continuum. Bass compares the construction narrative of the number line, common in US curricula, with the occupation narrative proposed by V. Davydov, in which numbers are discovered rather than built. The article describes activities from Davydov’s work and concludes with a description of some advantages of the approach.

Finally, William McCallum examines the image of the function concept in school mathematics, using internet image search. Searches in different languages produce collections with a greater or lesser degree of mathematical coherence and mathematical fidelity. The results vary along multiple dimensions: the presence of visible connections between ways of presenting functions; the density of meaningful annotation; the degree to which components are semantic rather than pictorial; and the extent to which extraneous features of an image violate mathematics properties. This also reveals a tension between the dynamic and static conceptions of function. These variations suggest directions of growth in professional discernment for in communities of educators.