# Functional Thinking in the Mathematics Curriculum

**By David Bressoud**

**You can now follow him on Twitter @dbressoud**

The International Congress on Mathematics Education (ICME) meets every four years and produces a series of monographs based on the contributed papers. One important monograph from ICME-13, held in 2016 in Hamburg, Germany, has just been released: *The Legacy of Felix Klein*, published Open Access by Springer. Part II of this volume is comprised of four important articles on one of Klein’s central tenets: the need to place functional thinking at the heart of the entire mathematical curriculum. This month’s column will focus on two of them: Katja Krüger’s “Functional thinking: the history of a didactical principle” and “Teacher’s meanings for function and function notation in South Korea and the United States” by Pat Thompson and Fabio Milner.

Klein was frustrated by what he called the “double discontinuity,” first the fact that university mathematics made little or no attempt to connect with the mathematics that students had learned in earlier years, second that those returning to teach in school drew little or nothing from the mathematics they had learned at university. The solution from the German National Teaching Commission for Mathematics that he led was to bring elements of university mathematics into secondary school mathematics, in particular analytic geometry and concepts from differential and integral calculus (in a secondary curriculum that extends to grade 13), and to place “functional thinking” at the center of instruction for grades 5 through 13.

Krüger explains the meaning of functional thinking through quotes and examples. It has nothing to do with the formal, static definition employing ordered pairs. Rather, in the 1909 words of Heinrich Shotten, a member of this commission, “It is about making students aware of the variability of quantities in arithmetic or geometric contexts and of their shared dependence and mutual relationship.” In modern language, functional thinking involves understanding co-variational relationships. As Klein would elaborate in 1933, “It [function] should not, of course, be introduced by means of abstract definitions, but should be transmitted to the student as a living possession, by means of elementary examples, such as one finds in large number in Euler.”

Functional thinking lies at the heart of the research and curricular reforms with which Thompson and Milner have engaged. In their article, they explore the *meanings* that high school teachers associate with aspects of function. They come to the depressing conclusion that, rather than Klein’s double discontinuity, too many U.S. teachers exhibit a *continuity* from inappropriate meanings learned in high school that are preserved during their university education and reappear in their understandings as teachers.

In this article, they consider three aspects of function notation, *f*(*x*). As I reported in my May 2017 column, *Re-imagining the Calculus Curriculum, I*, many students see *f*(*x*)* *as simply a long way of expressing the dependent variable. Figure 3 shows a question given to 253 U.S. high school mathematics teachers. Of those who had taught calculus at least once, only 43.2% correctly inserted *v* into all four spaces, while 33.8% retained the variable names *s* and *t* from the function definitions. Of those high school teachers who had never taught calculus, 29.6% inserted *v*, and 41.3% retained the *s* and *t*. For comparison, this question was also presented to 366 South Korean high school and middle school teachers. Of the high school teachers, 76.9% put in *v*; only 5.3% retained *s* and *t*. Even South Korean middle school teachers did considerably better than teachers in the U.S., with 63.7% inserting *v* and only 5.9% retaining *s* and *t*. The point is that students often view *w*(*t*) and *q*(*s*) as the names of the functions, a misconception that is rampant in the United States and carries forward from one generation of teachers to the next.

There were similarly dispiriting results on a question that explored teacher understanding of the role of the domain in the definition of a function.

The third exploration went directly to the question of whether teachers could use functional thinking. Showed a circle with a dot at the center, teachers were presented with the following problem:

Hari dropped a rock into a pond creating a circular ripple that spread outward. The ripple’s radius increases at a non-constant speed with the number of seconds since Hari dropped the rock. Use function notation to express the area inside the ripple as a function of elapsed time.

What one would hope to see is something like *A*(*t*) = π (*r*(*t*))², with functional notation employed on both sides of the equality. What appeared included *A* = π (*r*(*t*))², *A*(*t*) = π*r*², and *A* = π*r*². The point of the exercise was to see whether teachers would recognize that they could use *r*(*t*) or similar functional notation as a model for this unknown function of time. Answers were categorized by whether teachers employed functional notation on the left, right, both sides, or neither.

These results should be a wake-up call to those who prepare our future teachers. Preparing teachers for their role in the mathematical preparation of the next generations of students is about more than filling them with mathematical knowledge. It also must consciously address the misunderstandings with which they enter university and work to correct them.

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**References**

Krüger, K. (2019). Functional thinking: The history of a didactical principle. Pages 35-53 in *The Legacy of Felix Klein*, Weigand, McCallum, Menghini, Neubrand, and Schubring, editors. Cham, Switzerland: Springer Nature. https://www.springer.com/us/book/9783319993850

Lorey, W. (1938). *Der Deutsche Verein zur Förderung des mathematischen und naturwissenschaftlichen Unterrichts*. Frankfurt: Otto Salle.

Thompson, P. and Milner, F. (2019). Teachers meanings for function and function notation in South Korea and the United States. Pages 54–66 in *The Legacy of Felix Klein*, Weigand, McCallum, Menghini, Neubrand, and Schubring, editors. Cham, Switzerland: Springer Nature. https://www.springer.com/us/book/9783319993850