Although the theory of functions, when compared to other mathematical domains such as algebra, is a relatively new field in mathematics, it is notable that functions represent a valuable piece of mathematics and mathematics education (Lucus, 2005). The concept of function is classified mainly into two subtopics: composition of functions and the inverse function. While composition of function is mostly computational, the concept of inverse function is considered to be a class of functions that undo each other. This notion of inverting functions and creating new functions (the inverse functions) is particularly difficult students, including pre-service teachers, because inverting functions involves understanding other similarly complex concepts, like the one-to-one property, functions, domain and range, and multiple representations. A review of the literature revealed there are few research studies that have focused on pre-service teachers’ content knowledge of the concept of the inverse function.
My goal for this study was therefore to gain insight into the pre-service teachers’ content knowledge of the inverse function based on the following research questions:
- How do prospective teachers conceive of and understand the concept of inverse function?
- What level of proficiency do prospective teachers have in linking inverse functions using different representations?
I collected the data for this study in the months of October and November 2010. This study involved 10 pre-service teachers (6 females, 4 males) who were enrolled in a teacher preparation program at a medium-sized research university in the northeastern part of the United States. Four of the participants were enrolled in a master’s program while six participants were enrolled in an undergraduate program. Participant selection for this study was on a voluntary basis. I used five questionnaire tasks in which I asked participants to interpret the inverse functional situations that are based on the grade 8-12 mathematics curriculum such as defining inverse functions, composing functions, classifying relations as having inverse functions or not, and working with logarithmic, linear, and quadratic functions. From among the 10 participants, I selected eight pre-service teachers for the semi-structured interviews. I audio-taped a total of approximately 510 minutes from the eight interviews after which I transcribed the audio talk into Word documents for analysis. I used the theoretical framework proposed by Even (1990) to categorize the participants’ knowledge of the inverse function. Further, I drew on Hiebert’s (1986) framing of conceptual and procedural knowledge, and Shulman’s (1986) notion of teachers’ content knowledge in analyzing the data for this study.
Two major findings emerged from this study. First, these pre-service teachers had limited understanding of the inverse function based on the fact that: (1) although most pre-service teachers were successful in defining the inverse function using composition of function, their definitions of the inverse function were not based on the one-to-one property; and (2) most of these pre-service teachers believed that the vertical line test was used to determine if a function was one-to-one, while the horizontal line test determined if a function was onto. Second, the findings showed that most of these pre-service teachers could not link functions and their inverse functions using different representations. This may have been caused by these participants’ limited understanding of the symbolic and the verbal representations of the inverse function, restriction of the domain, and the lack of knowledge on the non-existent of the inverse function for constant functions. However, these pre-service teachers possessed strong procedural skills of composition of function. Overall, these pre-service teachers showed disjointed conceptual and procedural knowledge, and in particular, a weak conceptual understanding of the inverse function.
Even, R. (1989). Prospective secondary mathematics teachers’ knowledge and understanding about mathematical functions. Unpublished Doctoral dissertation, Michigan State University, East Lansing, MI.
Even, R. (1990). Subject matter knowledge for teaching and the case of function. Educational Studies in Mathematics, 21(6), 521-544.
Even, R., & Tirosh, D. (1995). Subject matter knowledge and knowledge about students as sources of teacher representations of the subject matter. Educational Studies in Mathematics, 29, 1-20.
Hiebert, J. (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Erlbaum.
Lucus, C. A. (2005). Composition of functions and inverse function of a function: Main ideas, as perceived by teachers and pre-service teachers. Unpublished doctoral dissertation, Simon Fraser University, Burnaby, British Colombia, Canada. Retrieved from: http://ir.lib.sfu.ca/retrieve/728/etd1606.pdf
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
This task-based, qualitative study reported the findings of ten preservice secondary school teachers’ knowledge of inverse functions. Findings from data analysis using Even’s framework showed that these teachers had strong procedural skills, profound misconceptions, and weak conceptual understanding of the concept.
Session Type: Poster Session