#### Graduation Year

2015

#### Document Type

Dissertation

#### Degree

Ph.D.

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Degree Granting Department

Teaching and Learning

#### Major Professor

Denisse R. Thompson, Ph.D.

#### Committee Member

Yi-Hsin Chen, Ph.D.

#### Committee Member

Gladis Kersaint, Ph.D.

#### Committee Member

Janet Richards, Ph.D.

#### Keywords

graphing calculators, HLM, mathematics education, path analysis, teaching functions, UCSMP

#### Abstract

The concept of function is one of the essential topics in the teaching and learning of secondary mathematics because of the central and unifying role it plays within secondary and college level mathematics. Organizations, such as the National Council of Teachers of Mathematics, suggest students should be able to make connections across multiple representations of mathematical functions by the time they complete high school. Despite the prominent role functions play in secondary mathematics curriculum, students continue to struggle with the complex notion of functions and especially have difficulty using the different representations that are inherent to functions (algebraic, graphical and tabular).

Technology is often considered an effective tool in raising student achievement, especially in learning functions where the different representations of a graphing calculator are analogous to the different representations of a function. Opportunity to learn is another important consideration when examining achievement and is generally considered one of, if not the most important, factor in student achievement. Opportunity to learn, or the measure of to what extent students have had an opportunity to learn or review a concept, is often measured with self-reports of content coverage.

This study examined the relationship between opportunity to learn, students'; use of graphing calculators, and achievement within a curriculum that supports integrated use of technology and focuses on conceptual understanding of mathematical concepts. The research questions focused on what opportunities students had to learn functions from the enacted curriculum, what calculator strategies students used when solving function problems, how both opportunity to learn and calculator strategies influenced student achievement, and what relationships exist between opportunity to learn, use of calculator strategies, and student achievement.

This study is an in-depth secondary analysis of a portion of data collected as part of the evaluation study of Precalculus and Discrete Mathematics (Third Edition, Field-Trial Version) developed by the University of Chicago School Mathematics Project. Participants in this study (n = 271) came from six schools, seven teachers, and 14 classes. Instruments in this study include two pretests (one with technology and one without) and three posttests (two with technology and one without) and a calculator usage survey for one posttest. In addition to five student assessments, teachers completed opportunity-to-learn surveys for the posttests and chapter evaluations forms on which they indicated the lessons taught and the homework problems assigned from the textbook. Some students (n = 151) had access to graphing calculators equipped with computer algebra systems (CAS) while others (n = 120) had access to graphing calculators.

Students had multiple opportunities to learn functions as measured by lessons taught, homework assigned, and posttest items teachers reported as having taught or reviewed the content necessary for students to correctly answer the items. Overall, students showed a positive increase in achievement between the pretests and posttests. In general, achievement was positively correlated to OTL Lessons, negatively correlated to OTL Homework, and had no correlation to OTL Posttests when controlling for prior knowledge. Results indicate students appear to be, for the most part, making wise choices about when and how to use graphing calculators to solve function items. Students prefer the graphical representation and are rarely using CAS features or tables, even when they are the best choices for solving a problem.

Results from hierarchical linear models (HLM) show use of strategies (beta = 0.96), access to CAS (beta = 5.12), and OTL lessons (beta = 0.75) all had significant and positive impacts on student achievement for one of the posttests, when controlling for prior knowledge. Results from path analyses also indicated use of strategies had a direct and positive effect (beta =0 .14) on student achievement but showed access to CAS had a negative indirect effect (beta = -0.64) on student achievement for the same posttest mitigated through OTL Lessons (beta = 0.30).

The results of this study have implications for both researchers and mathematics educators who seek to understand ways in which teachers can increase students'; understanding of functions and student achievement. The relationship between the use of technology and student achievement in relation to opportunity to learn is complex, but use of calculator strategies appears to have a positive effect on students' opportunity to learn functions and student achievement when used in a curriculum that focuses on conceptual understanding and integrates technology.

#### Scholar Commons Citation

Hauser, Laura A., "Precalculus Students' Achievement When Learning Functions: Influences of Opportunity to Learn and Technology from a University of Chicago School Mathematics Project Study" (2015). *Graduate Theses and Dissertations.*

https://digitalcommons.usf.edu/etd/5497