Functions are a core concept of mathematics as well as of disciplines using mathematics. Difficulties related to the concept of function are well documented and analyzed in mathematics education research[1][2] [3] and other disciplines (e.g. computer science[4], physics[5]).
Core findings from mathematics education research
Students conceptual difficulties with respect to functions have been investigated at least since the 1980s. These findings have led to the development of APOS Theory[3] which is a general constructivist theory explaining how mathematical understanding works and develops. It models what might happen in a person's mind when one tries to learn a mathematical concept. It postulates that a person's understanding of a mathematical topic develops as one constructs or reconstructs certain mental structures and organizes them into schemas. The mental structures proposed by APOS theory are actions, processes, objects and schemata; hence the acronym APOS.
In case of functions in mathemtics, different mental structures are required depending on the developmental stage of understanding. At the action stage, a function is seen merely as an instruction that specifies how the function value (output) is to be calculated. In the process stage, a function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule/algorithm. At the object stage, the function becomes an object, which in turn can itself be affected or manipulated by processes (e.g. differentiation). Going even further, functions can be organized as objects with other mathematical processes or objects to form larger schematas (e.g. function spaces).
Note that there are two core transitions in a person's conceptualization of function: from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of treshold concepts both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.
Decoding work done
In their book Overcoming Student Learning Bottlenecks Joan Middendorf and Leah Shopkow give excerpts of two Decoding Interviews dealing with the concept of function.[6] There, however, the focus is on posing suitable interview questions.
In terms of APOS-theory the identified bottleneck is related to the conceptual transition from the action to the process stage.
Description of bottleneck
Students don't grasp the concept of functions in a calculus class.
Description of mental moves needed to overcome the bottleneck
The two interview excerpts revealed the following mental moves:
- Trying not to think of a function in terms of an algorithmic relationship between input and output. This can be achieved, for instance, via visualing functions as graphs.
- Having available for oneself examples of input-output relationsships which can be described as functions but not algorithmically (e.g. random processes in the stock market).
Available resources
Researchers have developed teaching materials to help students overcome conceptual difficulties with functions, see in particular the text books by Arnon et. al.[3] and Dubinsky et al.[7]
References
- ↑ Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational studies in mathematics, 22(1), 1-36.
- ↑ Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational studies in mathematics, 23(3), 247-285.
- ↑ 3.0 3.1 3.2 lana Arnon, Jim Cottrill, Ed Dubinsky, Asuman Oktaç, Solange Roa Fuentes, Maria Trigueros, Kirk Weller: APOS theory: a framework for research and curriculum development in mathematics education. Springer, New York Heidelberg 2014, ISBN 978-1-4614-7965-9.
- ↑ Philipp Marwan, Peter Riegler (2011): Entwicklung des Funktionenkonzepts bei Studierenden der Informatik, Wismarer Frege–Reihe, vol. 02/2011.
- ↑ Combining mathematics and physics beyond the introductory level: the case of partial differential equations, https://research.kuleuven.be/portal/en/project/3E211102
- ↑ Middendorf, J., & Shopkow, L. (2017). Overcoming student learning bottlenecks: Decode the critical thinking of your discipline. Stylus Publishing, LLC, pp. 55-57.
- ↑ Dubinsky, E., Schwingendorf, K. E., & Mathews, D. M.: Calculus concepts and computers (2nd ed.). McGraw-Hill College 1994, ISBN 978-0-314-01129-9
