# Conceptual Understanding of Linear Relationships Across Various Mathematics Courses

May 2024

Dissertation

## Degree Name

Doctor of Philosophy

Urban Education

DeAnn Huinker

## Committee Members

Anne Marie Marshall, Kevin McLeod, Bo Zhang

## Keywords

conceptual, cross-sectional, linear, mathematics, relationships, understanding

## Abstract

This cross-sectional study investigated the conceptual understanding of linear relationships for 195 students enrolled in a single school in a large, urban district across five mathematics courses: Grade 7 Math (n = 24), Grade 8 Math (n = 52), Geometry (n = 43), Algebra 1 (n = 31), and Algebra 2 (n = 45). The following questions guided this study: (1) What differences exist in students’ conceptual understanding of linear relationships across mathematics courses? (2) What are common strengths and weaknesses in students’ conceptual understanding of linear relationships?

An assessment was created to assess three constructs of conceptual understanding of linear relationships: (1) Identifying unit rates in proportional relationships, (2) Moving fluidly between representations of the same linear relationships, and (3) Interpreting different representations of linear relationships. The assessment contained eight multiple-choice items and seven free-response items, with five items assessing each construct. Each student completed the assessment in their regular mathematics course in a single class period. The assessment was scored and analyzed by mathematics course.

The overall test score and the score on each construct were analyzed using an ANOVA with Tukey and Games-Howell post hoc analyses. A significant difference was found in overall understanding of linear relationships across courses. The difference existed between students enrolled in the follow mathematics courses: Grade 7 Math and Algebra 2, Grade 7 Math and Geometry, and Grade 8 Math and Algebra 2. Results also highlighted a significant difference between Grade 7 Math and Algebra 2 students in all three linear relationship constructs, with Algebra 2 students scoring significantly higher than Grade 7 Math students. Overall, students did not demonstrate a strong understanding of linear relationships, although Algebra 2 students were the most successful. The area in which students in all mathematics courses showed the greatest understanding was calculating unit rates in familiar contexts (e.g. speed, units per hour). Areas of weakness included interpreting linear relationships from any representation (e.g. table, graph, equation, verbal description) and moving fluidly between representations of linear relationships.

The results of this study suggested that students need to be given more opportunities to engage in learning experiences where they are interpreting multiple representations of the same linear relationship across all mathematics courses. Students should be asked to translate between tables, graphs, equations, and verbal representations of linear relationships in all directions (e.g., table to equation, verbal to equation, equation to graph). Curricular materials and learning experiences that only ask students to translate in select familiar directions between representations (e.g., table to graph, equation to graph) may be contributing to inequitable mathematics learning experiences and outcomes. Engaging in learning experiences in all directions and with all representations of linear relationships should help increase students’ conceptual understanding of linear relationships. Effectively implementing conceptual-based curricular materials along with research-based best teaching practices will help provide a more equitable mathematics experience for all students.

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