The Development of the Concept of Rates of Change and its Impact on Students' Understanding of Functions

A Thesis Submitted in Partial Fulfillment of the Requirements for Degree of MASTER OF SCIENCE in The Graduate Mathematics Program (Curriculum Content Option) from Texas A&M University-Corpus Christi.The focus of the study was to identify similarities and differences of grade seven through grade twelve students' understanding of rates of change and functions while keeping in mind that upperclassmen study participants will have more mature mathematical thinking in comparison to the participants from middle school. In addition, the study explored how students' understandings and difficulties relate to their teachers' expectations for prerequisite knowledge, and expected outcomes for students at each grade level. In contrast to the prior studies, the current study provides a comprehensive picture of progression in students' understanding of rates of change (or slope) by the use of the study instrument that included focused problems on mostly all unique conceptualizations of slope. A cross-sectional study was conducted in a public high school in South-Texas with 187 Grades 7- 12 students enrolled in Math-7 to Calculus (AB), and 8 teachers. All student participants completed the Diagnostic Test on various representations of Rates of Change (DTRC) in physical and functional situations. Fourteen students and six teachers were interviewed. To analyze student performance by grade, and by current math course, Welch's Test for Analysis of Variance including Games-Howell Simultaneous Tests for Difference of Means (One-way ANOVA) was employed besides computing descriptive statistics. The student performance on DTRC increased very little in both cases, across grades and by math course. Results from DTRC data and students' interviews showed that students had difficulty computing unit rate in contextual problems, and when required to compare and extract information from multiple representations within the same problem. A large proportion of study participants including Grades 11, and Grades 12 had difficulty in estimating average rate of change from a data table or when presented with a graph of a non-linear function. There was a lack of evidence in the students' work to explore the rate of change of a dependent variable with respect to an independent variable. The data from the interview points towards a need for enhancing teacher horizon knowledge on how slope is connected to the ideas of unit rate, average rate of change, and to the instantaneous rate of change in functional situations. The same instrument and procedures may be used with a greater cross-section of the general population to have more variability in participant's socio-economic status to allow enhancement in interpretation and generalization of the findings of the current study.Mathematics and StatisticsCollege of Science and Engineerin