**Principal Investigators**Andrew Izsak,

*University of Georgia*(PI)

Joanne Lobato,

*San Diego State University*(Co-PI)

Chandra Orrill,

*UMass Dartmouth*(Co-PI)

Allan Cohen,

*University of Georgia*(Co-PI)

Jonathan Templin,

*University of Georgia*(Co-PI)

__Funding__National Science Foundation DR-K12 Program

2008 - 2012

$944,163

Award: DRL 0822064

__Abstract__Diagnosing Teachers' Multiplicative Reasoning (DTMR) is an exploratory project that addresses the assessment component of the DR-K12 Contextual Challenges strand. Investigating knowledge that teachers need to enable students' learning and developing assessments of that knowledge are central challenges for mathematics education. One approach, driven by accountability, emphasizes correlations between amounts of teachers' knowledge and students' achievement. Another, grounded in research on mathematical thinking, often uses case studies to investigate teachers' capacities for identifying and building upon opportunities in students' problem-solving strategies. Tensions exist between these approaches because instruments convenient for assessing large numbers of teachers are insensitive to capacities for reasoning, while case study methods used to investigate teachers' reasoning are not practical with large samples. The DTMR project will build a demonstration instrument both suitable for use with large samples of teachers and informative about their capacities to reason about content in ways that support students' thinking.

In particular, the DTMR project will develop and evaluate a test form that diagnoses teachers' capacities in two closely connected cases of reasoning about multiplicative relations among quantities. The first has to with measurement that often relies on multiplicatively nested levels of units when partitioning a given quantity. The second has to do with covariation that often relies on multiplicative relations between distinct quantities. The project will focus on aspects of such reasoning that are interconnected and fundamental to addition and subtraction of quantities, multiplication of quantities, quotative and partitive division of quantities, and ratios of quantities. We will consider fractions, decimals, and ratios. A main goal of the project is to address content and construct validity of the demonstration form in sufficient depth so that larger scale work and predictive validity studies may follow.

We will develop our instruments using a new class of psychometric models called cognitive diagnosis models (CDMs). Using CDMs involves specifying components of reasoning in a particular domain and then constructing test questions (typically multiple-choice) systematically so that each choice corresponds to reasoning with a different combination of those components. We will draw from the research on students' and teachers' multiplicative reasoning to identify useful components for building and validating one test forms of 30 to 40 items. CDM simulation studies of estimation and equating methods are also an important component of the project.

The proposed project is designed to answer the following research question in the sub-area of reasoning about multiplicative relations among quantities:

- How can instruments be designed to diagnose teachers' multiplicative reasoning?

__Intellectual Merit__

By coordinating underlying components of multiplicative reasoning about quantities with new statistical models, we will narrow the gap between evidence for reasoning garnered through case studies and large-scale studies. A strong research base exists regarding CDMs, but researchers have yet to develop instruments for these models from the ground up within the CDM framework. As a result, the project promises to interest both psychometricians and mathematics educators.

__Broader Impact__

The process by which we construct assessments using CDMs-where mathematics educators, mathematicians, and psychometricians work collaboratively to specify fundamental components of reasoning and develop validated instruments in a given domain could serve as a model for developing further instruments that diagnose reasoning in other areas of multiplicative reasoning and reasoning in other STEM content areas.