Abstract
Recent research on the phenomenon of improper proportional reasoning focused on students’ understanding of elementary functions and their external representations. So far, the role of basic function properties in students’ concept images of functions remained unclear. We add to this research line by investigating how accurate students are in connecting functions to their corresponding properties and how this accuracy depends on function types and representations. A large group of 10th graders evaluated for different function types, represented in either a graphical, a formulaic, or a tabular mode, the correctness of statements about their general properties and behavior. Results show that students succeeded rather well in making the right connections between properties and functions. Errors depended not only on the type of function for which the properties were evaluated but also on the kind of representation in which the function was presented. These results highlight the importance of function properties in students’ concept images of functions and suggest positive effects of making these properties explicit to students.
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Notes
In the Anglo-Saxon literature, an “y = ax + b” model is denoted as a “linear” model (etymologically derived from the Latin word linea which stands for (straight) line, the graphical representation of a linear function). In this paper, we follow this arrangement, but we differentiate between cases in which b = 0 and b ≠ 0, respectively, denoted by the terms “proportional” and “affine.” Affine models with a = 1 are denoted as “additive.” In line with these definitions, the “overuse of proportionality” (i.e. the overuse of an “y = ax” model) is a specific utterance the “overuse of linearity” (i.e. the overuse of an “y = ax + b” model).
In the rest of this paper, we will use the term “representation” to denote “external representation.”
The increasing affine and proportional function could not be involved in these comparisons because almost all students assigned statement MD2 correctly to these two function types.
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De Bock, D., Neyens, D. & Van Dooren, W. Students’ Ability to Connect Function Properties to Different Types of Elementary Functions: An Empirical Study on the Role of External Representations. Int J of Sci and Math Educ 15, 939–955 (2017). https://doi.org/10.1007/s10763-016-9724-z
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DOI: https://doi.org/10.1007/s10763-016-9724-z