Graduation Term

Spring 2026

Degree Name

Doctor of Philosophy (PhD)

Department

Department of Mathematics: Mathematics Education

Committee Chair

Amanda L. Cullen

Committee Member

Neet Priya Bajwa

Committee Member

David Barker

Committee Member

Cynthia Langrall

Committee Member

Mariana Levin

Abstract

In this dissertation study, I sought to identify the initial ideas that middle school students had when approaching Multiple Batches and Variable Parts proportion word problems, as well as the connections they made among those initial ideas. I conducted semi-structured individual and paired interviews with 13 sixth- and seventh-grade students at the end of their school year. I then used the Knowledge-in-Pieces theoretical perspective to identify knowledge resources that the participants accessed when approaching the tasks, prior to any intervention on my part.

Through my analysis, I identified 18 initial ideas that the middle school students employed when solving the seven different proportion tasks. I categorized the initial ideas as Operational or Reasoning initial ideas. Of the 10 Operational initial ideas, I classified three as Additive initial ideas—Difference Across, Difference of Same, and Total Number of Items—and seven as Multiplicative initial ideas, including Divide Across, Divide Same, Equal Groups: Number of Groups, Equal Groups: Find 1 Part, Factors and Multiples, Given Values, and Derive Multiple Parts from 1 Part. Of the eight Reasoning initial ideas, I identified five as Rational Number initial ideas (i.e., Fraction, Ratio: Other, Ratio: Scale Up, Ratio: Simplify, and Unit Rate) and three as Algebraic initial ideas (i.e., Equivalence, Patterns, and Variables). I also noted that participants only accessed the Find 1 Part initial idea when approaching the Variable Parts tasks, whereas they accessed the Number of Groups initial idea when approaching Multiple Batches tasks or re-interpreting Variable Parts tasks into Multiple Batches tasks. Finally, I determined that the participants connected initial ideas to form a procedure, to explain a procedure, or to confirm or reject other initial ideas. My findings suggest that Variable Parts proportion tasks are a necessary addition to middle school students’ proportion curriculum since they provide a unique opportunity to tap into a specific initial idea.

Access Type

Dissertation-Open Access

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