#### Date of Award

Spring 1992

#### Project Type

Dissertation

#### Program or Major

Mathematics Education

#### Degree Name

Doctor of Philosophy

#### First Advisor

William Geeslin

#### Abstract

The purpose of this study was to investigate the apparent effects of students' beliefs about mathematics and autonomy on their learning of mathematics. The study utilized a multiple-case study design with analysis by and across cases. The cases represented six high school students enrolled in either Algebra II or Algebra II/Trigonometry. Data was collected in three phases: (a) classroom observations and assessment of the teacher's perception of her role in the learning process, (b) an assessment of students' beliefs about mathematics and autonomy, and (c) an assessment of students' newly formed mathematical constructs on functions.

The beliefs' assessment included observing and questioning students while they solved mathematics problems. Follow-up probes explored the students' rationale for their strategies and their dependency on rules and algorithms when solving problems, particularly problem-solving situations. To further corroborate the beliefs expressed by the students or those inferred from their solutions, the students also marked and discussed a mathematics topics ranking grid and vocabulary lists, graded a sample algebra test, and responded to scenarios on student's/teacher's roles. From the data, a detailed portrait of each individual's beliefs about mathematics was developed. These portraits were compared with the students' understanding of functions and the classroom expectations.

The results from this research investigation suggest three hypotheses concerning students' beliefs about mathematics, autonomy, and mathematical knowledge. First, students' beliefs about mathematics rather than being dichotomous form a continuum from strongly conceptual in outlook to strongly procedural. Second, students' autonomy augments their beliefs about mathematics and often meditates them. Third, students' beliefs and autonomy appear to concur with their problem-solving strategies and with their knowledge of mathematics. Collectively these hypotheses suggest that students' beliefs and autonomy are an integral component of students' conception of mathematics and influence both how problems are approached and how mathematics is learned. Further study needs to be done on how and when these beliefs are formed and under what conditions these beliefs are modified and changed. Finally, the interplay among beliefs, autonomy, and learning needs to be investigated in the actual classroom context.

#### Recommended Citation

Rector, Judy Ann, "Beliefs, autonomy, and mathematical knowledge" (1992). *Doctoral Dissertations*. 1687.

https://scholars.unh.edu/dissertation/1687