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Existing research has identified pre-service teachers' (PTs') difficulties with measuring the area of rectangles and explaining their approaches to area measurement. Yet, little is known about elementary PTs' understandings of area measurement. Existing research has focused on children's understanding of area measurement in the context of rectangles and has identified mental processes they use to construct area measurement. The study described in this report contributes to and extends this literature by exploring elementary PTs' understandings of area measurement in the context of rectangles. Clinical interview methodology was used to investigate four PTs' mental processes used in area measurement tasks. Study participants engaged in a 45-minute explanatory interview during which they solved area measurement tasks, including finding areas of rectangles with whole and non-whole number side lengths. Retrospective analysis of interview transcripts was conducted using Battista's framework to identify mental processes used by PTs. While PTs correctly measured the area of rectangles with whole number side lengths, their approaches included identifying side lengths by counting units of area rather than units of length. This approach caused significant difficulties for PTs as they tried to find the area of rectangles with non-whole number side lengths (e.g. length of 0.1 cm). Findings include descriptions of the mental processes used by PTs to find the area of rectangles and are used to describe understanding needed to address measurement of area of rectangles with whole and non-whole number side lengths. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]

This study uses a new psychometric model (mixture item response theory-MIMIC model) that simultaneously estimates differential item functioning (DIF) across manifest groups and latent classes. Current DIF detection methods investigate DIF from only one side, either across manifest groups (e.g., gender, ethnicity, etc.), or across latent classes (e.g., solution strategies, speededness, etc.) leading to incomplete results. Alternatively, they consider one aspect as the real source of DIF and the other aspect as a proxy for the same source. This can only be true when manifest and latent classifications provide perfect or very high overlap. A combination of a Rasch type model for manifest group-DIF (G-DIF) and a mixture Rasch model for latent class-DIF (C-DIF) detection is applied as the mixture IRT-MIMIC model (MixIRT-MIMIC). A Markov chain Monte Carlo method called Gibbs sampler is applied for Bayesian estimation of parameters for MixIRT-MIMIC model as well as the Rasch model, and the mixture Rasch model.This study shows that in detection of DIF, when the group-class overlap is between 50% and 70%; manifest group approaches and latent class approaches can provide biased DIF, and item difficulty estimates for some test items that show G-DIF and C-DIF, simultaneously. However, for the same conditions MixIRT-MIMIC provides less biased estimates for latent class-DIF (C-DIF) and item difficulty parameters, while the confounding is reflected as bias in G-DIF parameter estimates. Main factors of importance are group-class overlap and the overlap between DIF items. MixIRT-MIMIC contributes by; (1) estimating the unbiased magnitudes of G-DIF and C-DIF, (2) estimating the unbiased estimates of item difficulties, (3) determining the overlap ratio (confounding) between groups and classes which is unknown a priori (4) determining the true source(s) of DIF. Researchers, test developers, and state testing programs that are interested in detecting true sources of differences (e.g. cognitive, gender, ethnic) across individuals are potential users of MixIRT-MIMIC. It is important to note that this study is an initial step to detect both types of DIF simultaneously, and is limited to binary data and a special case of 2 groups by 2 classes, which can be applied to most DIF detection purposes. Its performance and extensions will be investigated for other possible situation. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]