Date of Award

December 2023

Degree Type


Degree Name

Doctor of Philosophy


Urban Education

First Advisor

DeAnn Huinker

Committee Members

Henry Kepner, Kevin McLeod, Nancy Rice, Christopher Lawson


Benchmarks, Comparison, Decimals, Fractions, Magnitude, Number Line





Elizabeth Cutter-Lin

The University of Wisconsin-Milwaukee, 2023Under the Supervision of Professor DeAnn Huinker

This study investigated how three fifth-grade students’ understanding of fraction and decimal magnitude evolved over the course of a five-week teaching experiment. Students participated in teaching and learning sessions focused on developing concepts of fraction and decimal magnitude. The following questions guided this study: (1) How do fifth grade students reason about the magnitude of fractions and decimals? (2) What are the shifts in mathematical thinking that occur with students’ evolving understanding as they progress towards generalization of fraction and decimal magnitude? (3) What are the characteristics of instructional experiences that lead to shifts in students’ mathematical understanding of fraction and decimal magnitude? A teaching experiment methodology was used to investigate these questions. In keeping with the essential elements of a teaching experiment, the study included several iterations of teaching, student thinking, reflection, and analysis, and then teaching again. Data was analyzed both during the teaching experiment and retrospectively following the conclusion of the teaching and learning sessions. Primary data included recordings and transcripts from the teaching sessions as well as students’ written work. Retrospective analysis generated three major themes as most prominent in the students’ reasoning. First, experiences physically partitioning fraction strips and number lines emerged as critical to students’ developing reasoning. The students drew upon the foundations they established when engaging in their own partitioning work as they worked with questions of size, equivalency, and density. Second, critical relationships such as the relationship between the numerator and denominator and between fractions to other fractions and decimals were essential, but also complex and challenging for the students. In particular, students appeared to anchor much of their work in relationships of fractions and decimals to the benchmarks of 1/2 and one whole. Distance reasoning (i.e., considering the distance of a fraction to a benchmark) appeared to become a preferred reasoning strategy for the students. Lastly, the number line appeared more complicated and challenging than initially anticipated but powerful in illuminating misconceptions. Pushing the students to place both decimals and fractions on the same number line appeared especially critical in supporting the students in integrating their developing understanding of both of these notations and expanding and deepening their mental number lines.