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This study examines why students have difficulty with inverse functions (inverse functions is the process of doing and undoing operations) and what we can do to support their learning. This was a quasi-experimental design in a math classroom in an urban comprehensive high school in California. After two weeks of instruction one group of students was taught the traditional way of inverse functions and another group was taught conceptually. About (N=80) mathematics students in the sampling were assessed before and after the study. Students were given a test to measure their learning of inverse functions and a questionnaire to measure their perspectives on the unit of study of inverse functions. Qualitative and quantitative methods were used to analyze the data. The results will be discussed hoping that in this study students taught conceptually would perform better than the controlled. Also, this study will be useful for teachers and educators to recognize that conceptual teaching yields better results than direct instruction of rote instruction

This paper develops a data structure based on preimage sets of functions on a finite set. This structure, called the sigma matrix, is shown to be particularly well-suited for exploring the structural characteristics of recursive functions relevant to investigations of complexity. The matrix is easy to compute by hand, defined for any finite function, reflects intrinsic properties of its generating function, and the map taking functions to sigma matrices admits a simple polynomial-time algorithm . Finally, we develop a flexible measure of preimage complexity using the aforementioned matrix. This measure naturally partitions all functions on a finite set by characteristics inherent in each function's preimage structure.

The main purpose of the study was to understand preservice secondary mathematics teachers&rsquo / pedagogical content knowledge of composite and inverse functions. The study was conducted with three preservice secondary mathematics teachers in Graduate School of Education at Bilkent University. The instruments of the study were qualitative in nature and in four different types of data forms: observations, interviews, documents, and audiovisual materials. Observation data came from fieldnotes by conducting an observation of lessons participants taught at Private Bilkent High School. Interview data came from the transcriptions of semi-structured interviews. Document data came from survey of function knowledge, journal writings, vignettes, and lesson plans. Audiovisual data came from the examination of the videotape of the lessons participants taught. The findings reveal that preservice secondary mathematics teachers&rsquo / knowledge levels in components of pedagogical content knowledge were not at the desired levels and also they experienced difficulty while integrating that knowledge. The results of the study indicate that teacher education should provide courses that cover the content relevant to students in order to assure both depth and breadth in subject matter knowledge of the preservice teachers. Moreover, the activities which mimics the classroom cases and assures the integration of knowledge components like vignettes would be used in teacher education programs. Results can inform educational practices, and reforms in Turkey, and provide a basis for further research, with increased pedagogical content knowledge as the ultimate goal.

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