< Previous | Contents | Next >
Sebastian Wirthgen, Kathrin Gläser
Ostfalia University of Applied Sciences, Germany
It is common knowledge at universities that understanding of basic concepts in mathematics are needed to succeed in all STEM-disciplines. Concerning conceptual understanding education research has shown that students’ deficits in learning such concepts can often be traced back to characteristic difficulties within the subject matter. Quite often students develop alternative conception (so called misconceptions) instead of the scientific concept. The topic “Functions” is one of these fundamental mathematical concepts in STEM courses. Students often have difficulties with the concept of functions and develop misconceptions. In order to help students to overcome their difficulties, instructors need more information about the students’ deficient conceptual understanding.
Based on the idea of the “Force Concept Inventory” in physics (Hestenes, Halloun, Wells and Swackhamer, 1992) we are currently developing a “Function Concepts Inventory” to measure students’ misconceptions and to understand their alternative conceptions. The development of this inventory is based on the APOS theory which is a constructivist theory about the way mathematics is learned and understood. The inventory is constructed on the instructors’ reports about misconceptions, interviews with students and on literature. In the development process we are analyzing open questions to gain valid distractors for multiple choice tasks. So far those have been found and tested for the “Function Concepts Inventory” with more than 250 students.
We will take a closer look at the composition of functions which is one of the characteristic difficulties students struggle with. Based on the results of the APOS theory (Arnon I. et.al., 2014), there are three different types of problems concerning a composition of functions H=F○G. While these are mathematically speaking identical they require three different cognitive levels. The first level is to obtain H from given functions F and G. For the second level the functions G and H are given and F is to be determined. In the third and last level there are given functions F and H and the missing function is G. To understand and measure the students’ understanding of these different aspects of compositions we have specified two tasks of the “Function Concepts Inventory”. Both tasks are concerned with the three different levels, but using different representations.
We will present the tasks, their development and our findings obtained in courses in mathematics for software engineering. The data we have collected during the last three years suggest that the students already have difficulties with the first two levels regardless of the functions’ representations. Our results show that these tasks are an appropriate and valid indicator for students’ understanding of and their difficulties with the composition of functions.