Middle Grades Teachers’ Partitioning Activity

• In Event: Harnessing Psychometric Models to Develop Next-Generation, Research-Based Assessments of Rational Number Knowledge

Abstract

Objectives
It is well-known that students can divide in half at an early age but struggle with more complex partitioning tasks. Less is known about teachers’ capacities to partition. As part of the Diagnosing Teachers’ Multiplicative Reasoning assessment development project, we used interviews to investigate whether in-service middle grades teachers experience difficulties partitioning drawn quantities (e.g., lengths and areas) when solving problems involving fractions. We used results from the interviews to develop test items for studying teachers in large samples.

Perspectives
In the past two decades, mathematics education researchers have been heavily influenced by Shulman’s (1986) discussion of subject-matter knowledge, pedagogical knowledge, and pedagogical-content knowledge. More recently, Ball and colleagues (e.g., Ball, Thames, & Phelps, 2008) have developed a construct termed mathematical knowledge for teaching (MKT) that refines Shulman’s categories and situates them in the discipline of mathematics. From these perspectives, teaching mathematics requires specialized knowledge of the subject. Understanding how drawn models (e.g., number lines and rectangular areas) can be used to develop general numeric methods for computing with fractions is specialized knowledge for teaching mathematics which depends, in turn, on a strong command of partitioning.

Methods and Data Sources
We conducted qualitative analyses of video taped interviews in which 14 in-service middle grades teachers solved a series of increasingly difficult partitioning tasks. The tasks began with talking halves of halves and progressed to partitioning using common numerators, as required by some fraction divisions tasks. The intent was to determine where partitioning got hard for teachers. We used the results of the interviews to develop a set of multiple-choice and constructed response test items. We included the items in a measure of MKT that we designed to assess teachers’ proficiency with core components of reasoning about fractions as drawn quantities (e.g., attention to referent units). We have begun administering the measure to a representative national sample of in-service middle grades teachers and will complete data collection this fall. We will use Diagnostic Classification Models (a family of psychometric models) to analyze the dataset. The models will generate profiles of teachers’ strengths and weaknesses with respect to partitioning and the other components of reasoning built into our measure of MKT.

Results
All of the interviewed teachers could readily take halves of halves but, as the tasks became more complex, we observed differences in the resources teachers used to guide their partitioning activity. Some teachers, but not others, used common multiples of numbers and prime factorizations as resources, but no teachers used common numerators to effectively guide their partitioning when trying to use drawn models to solve fraction division problems. Analysis of the national dataset will allow us to see how representative these results are and how teachers’ capacities to partition are related to other components of reasoning built into the full measure.

Scientific/Scholarly Significance
Recent standards documents have emphasized the use of drawn models for teaching fractions. Our results suggest that researchers have overlooked partitioning as an important area with which teachers can struggle when using drawn models.

Authors

• Andrew G. Izsak, University of Georgia

• Erik D. Jacobson, University of Georgia