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IES Grant

Title: Creating Cross-Grade Assessments of the Development of Core Algebraic Constructs
Center: NCER Year: 2010
Principal Investigator: Bauer, Malcolm Awardee: Educational Testing Service (ETS)
Program: Science, Technology, Engineering, and Mathematics (STEM) Education      [Program Details]
Award Period: 4 years Award Amount: $1,572,975
Type: Measurement Award Number: R305A100518

Co-Principal Investigators: Caroline Wylie, Karen Harris

Purpose: There is a large body of research linking the consistent and systematic use of formative assessments to improved student learning. However, many of these programs fall short because either they do not provide specific questions for teachers to ask students, or they do not provide training or support materials to help teachers understand what to do next. The current study seeks to address this gap between the research on formative assessments and what it takes to implement these types of assessments in the classroom by developing and validating a set of formative assessments that are built around developmental models of student learning in middle school mathematics.

Project: The researchers will develop and validate assessments that middle school mathematics teachers can readily incorporate into their existing curricula in order to guide instructional decisions related to three key concepts in algebra—equality, notion of a variable, and multiplicative reasoning.

Products: Products include formative assessments that middle schools mathematics teachers can use to improve student learning and instruction and published reports.

Structured Abstract

Setting: The setting for this study includes middle schools in diverse regions and locations across the United States.

Population: The study sample will consist of a nationally representative sample of approximately 6,000 students in Grades 6 to 8. The students will be diverse in academic proficiency, socioeconomic status, and ethnicity.

Intervention: Two types of assessments will be developed and validated—periodic locator assessments to provide profiles of groups of students that require different instructional approaches, and incremental assessments to inform ongoing instructional decision-making. The incremental assessments will be small sets of items targeted on specific concepts that teachers could use with a group of students in the classroom as part of on-going formative assessment activities.

Middle school mathematics teachers can readily incorporate these assessments into their existing curricula in order to guide their instructional decisions related to three key concepts in algebra—equality, notion of a variable, and multiplicative reasoning. In addition to these three algebra constructs, the assessment framework includes four dimensions of competency (conceptual depth, conceptual breadth, procedural fluency/flexibility, and representational fluency). This two-dimensional assessment matrix—algebraic constructs by dimensions of competency—is further layered by a third dimension focusing on developmental models for describing how student learning develops over time.

Research Design and Methods: The project has three distinct phases of assessment development and validation activities. During Phase I, the researchers will develop the assessments and evaluate the items by seeking input and review from 12 middle school teachers and an advisory group of content experts. In Phase 2, the researchers will explore the validity of the items and assessments using two approaches. First, a content validity study using expert judgments will evaluate the degree to which student responses to items reflect the hierarchy of the developmental model and the dimensions of competency. Thirty experts, including middle grade mathematics teachers, teacher educators, and mathematics cognition researchers will be randomly assigned to one of three groups, with each group responsible for evaluating student responses focused on one of three algebraic constructs. Second, the researchers will carry out a field test of the items with groups of students in order to conduct statistical analyses to examine the validity of the items and sets of items. To field test the items, a sample of 4,000 to 6,000 middle school students will be assigned a set of items using a randomized-block design. The researchers expect that approximately 180-240 items will be tested. The goal will be to collect at least 1,000 student responses per item in order to have sufficient data for the exploratory and confirmatory factor analyses. Finally, in Phase 3, the researchers will examine the consequential validity evidence for the assessment. Twelve middle school teachers will use the assessments throughout the school year. Teachers will be asked to complete an online log each time they use either the periodic locator assessment or the incremental assessment. In addition, teachers will complete an assessment of pedagogical content knowledge in algebra, and periodic interviews will be conducted with the teachers.

Control Condition: There is no control condition.

Key Measures: The key measures for the study include students' responses on the assessments, student scores on standardized state assessments, online logs, teacher interviews, and teacher pedagogical content knowledge in algebra.

Data Analytic Strategy: Exploratory and confirmatory factor analysis will be used to investigate the dimensionality of the proposed matrix structure of the assessment. Differential item functioning analyses will also be performed. In addition, multi-dimensional item-response theory approaches will be used to construct score reports that provide information to teachers about groups of students with similar estimates of proficiency.


Journal article, monograph, or newsletter

Graf, E.A., and Arieli-Attali, M. (2015). Designing and Developing Assessments of Complex Thinking in Mathematics for the Middle Grades. Theory Into Practice, 54 (3): 195–202.