#### Date of Award

Winter 2005

#### Project Type

Dissertation

#### Program or Major

Mathematics Education

#### Degree Name

Doctor of Philosophy

#### First Advisor

Karen Graham

#### Abstract

At the end of the standard first calculus course, the student is expected to learn the Fundamental Theorem of Calculus, and to be able to use the integral to produce new functions, or numbers which, they are told, represent the "area under the curve". At the beginning of the standard second calculus course, students are expected to generalize their knowledge, and use the integration process to generate solids of revolution, surface areas, arc length and work, among other applications. Looking at students' success or failure in these endeavors, it was detected that there are marked differences in an aspect that, in this study, is called mathematical fluency.

The concept of mathematical fluency was developed, and the four parameters used in foreign language learning: reading comprehension, writing, speaking and listening comprehension were employed to measure mathematical fluency as defined in the present study. The types of mistakes made by the fluent and nonfluent students can be related to the types of mistakes made by the native or fluent speaker of a natural language, and those made by one who has not reached---or might never reach---that stage. The classification of local fluencies, with mathematical fluency as a global amalgam of these, was developed. Theoretical constructs such as Knisley's four stage model of mathematical learning, Tall and Gray's procept classification, Brousseau's cognitive obstacles as well as analysis of schemas, mental models and metaphors provided the language and concepts with which mathematical fluency, as detected by the four parameters, was described.

The study was realized through interviews, action research and observations of students in a second calculus course. In depth analysis using the four parameters of foreign language learning offers a methodology for studying student learning and understanding, that can be generalized to other mathematical areas, and adapted to quantitative as well as qualitative methods.

#### Recommended Citation

Montiel, Mariana, "The process of integration and the concept of integral: How does success with applications and comprehension of underlying notions such as accumulation relate to students' mathematical fluency" (2005). *Doctoral Dissertations*. 307.

https://scholars.unh.edu/dissertation/307