Date of Award:


Document Type:


Degree Name:

Doctor of Philosophy (PhD)


School of Teacher Education and Leadership

Committee Chair(s)

Patricia S. Moyer-Packenham


Patricia S. Moyer-Packenham


Beth L. MacDonald


Mariana Levin


Marla Robertson


Colby Tofel-Grehl


Research shows that fractions concepts play an essential role in the learning of later mathematics. However, fractions are notoriously difficult to learn and difficult to teach. Division with fractions is a frequent subject in mathematics education research because division is the most conceptually difficult of the four basic arithmetic operations and rational numbers are the most conceptually difficult numbers in K-12 mathematics curricula.

In the U.S., teachers are generally proficient with mathematical procedures, but often have difficulty explaining the concepts underlying the procedures. Research indicates a positive association between student learning and teachers’ depth of conceptual understanding of mathematics. Thus, it is important to ensure that future and practicing teachers are competent with fractions operations at a deep, conceptual level.

In order to gain a better understanding of teachers’ conceptions of division with fractions, this study engaged teachers in a 4-hour professional development program designed to deepen the teachers’ understanding of fractions and their ability to represent fraction operations through the construction of rectangular area models. Eight teachers were given one-to-one professional development. Analysis of these videos showed that teachers constructed idiosyncratic conceptions yet faced some common challenges. One common challenge was that a central part of the division concept was readily visible to the teachers in some contexts but not in other contexts. Another common difficulty teachers experienced was conceptually explaining why the quotient to a fraction division problem should be based on a whole unit of the divisor.

Additionally, teachers constructed different modules of division arising from the structure of the situation in which the division was conceptualized. Models of partitive division with one apparent referent were easier to conceptualize and represent. Models of partitive division with two apparent referents were more difficult to conceptualize and represent. Two-referent models of partitive division with fractions are fundamental to rate and intensive quantity, and directly relate to other topics in mathematics, such as proportion and derivatives. Results of this study shed light on potentially common conceptual difficulties as well as suggest ways that learners can facilitate a conceptual understanding and representational fluency with fractions division.



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