Testing the Reorganization of the Equipartitioning Learning Trajectory Using Rasch Item Response Theory Modeling

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

Abstract

Objectives
This paper reports the results of a reanalysis of data previously subjected to Rasch IRT analysis to provide validation for a learning trajectory on equipartitioning (Confrey, Rupp, Maloney, & Nguyen, in review; Penuel, Confrey, Maloney, & Rupp, 2011). There are approximately 30,000 responses to 116 items administered to 4,800 students K-8. In the first analysis, item difficulty parameters for the K-8 data set were estimated as well as narrower grade bands including K-2, 3-5, and 6-8. Also conducted was an external one-way ANOVA comparing mean item difficulty estimates across the targeted 16 proficiency levels of the equipartitioning LT. Results showed that children in lower grades performed, on average, lower than children in higher grades, and the average difficulty of items monotonically increased, reflecting the ordering of proficiency levels. However, calibrations in the K-2 grade band and across the K-8 grade band (including K-2) provided improper reliability estimates, and across all grade bands, items were difficult for all students.

As we designed how the diagnostic assessments would be used in the classroom, we recognized the need to reconfigure the LT into fewer diagnostic instruments for effectiveness of use. We created the LPPSync System (Confrey et al., 2011) which organizes the LT into 7 e-packets to which the various proficiency levels are assigned. A sequential representation of the equipartitioning LT has been transformed into seven overlapping subsets of the original 16 LT levels, presenting an alternative framework for evaluating the validity of the LT in relation to the original assessment items. Our LT was organized around three primary cases of equipartioning: sharing a collection, sharing a whole, and sharing multiple wholes. Its target was the generalization that, in all cases, n objects shared among p people produces shares of size n/p. It was originally conceptualized as being divided among strategies, mathematical practices, emergent properties and relations, and broad and abstract generalizations. The reorganization preserves the order of the primary cases, weaves in the properties and relations but distributes the practices of naming, justifying and reassembling across the cases.

We are now analyzing the original field test data using the e-packet levels as the underlying framework, to determine whether reliability estimates of the grade bands are improved, and to compare whether mean item difficulty estimates across the 7 packets model the item response data. The paper will report on the results of these analyses using Rasch IRT analysis for all grade bands. We will also run double explanatory models to predict the variation of item difficulty per e-packet across grades.

Scientific and Scholarly Significance of the Work
The design of the LPPSync system embeds on-screen manipulatives in activities that more closely mimic the questioning and reasoning possible in clinical interviews and classroom interactions, and led to clustering overlapping subsets of LT proficiency levels into 7 packets. This non-linear configuration of proficiencies defined by the equipartitioning LT provides an alternate hypothesis of student cognitive development against which to map diagnostic items and then analyze item difficulties in relation to student competencies.

Authors

• Kenny Huy Nguyen, North Carolina State University

• Andre A. Rupp, University of Maryland

• Jere Confrey, North Carolina State University

• Alan Maloney, North Carolina State University