Learning to compute with fractions is a major challenge for many students and especially for students with disabilities (SWD). Phase 1 of this study employed a randomized pretest–posttest comparison design to test the effects of two versions of formative assessment combined with an instructional program called Fractions at Work. In one condition, teachers used technology-assisted prompts to assess student performance and remediate errors. In the comparison condition, teachers gave students the same items for assessing progress but used their own methods of reteaching. Results indicated no difference between the two methods. However, pretest-to-posttest gain scores were significantly higher on all three measures regardless of type of formative assessment, and students maintained much of what they had learned. Phase 2 examined issues related to instructional dosage. Students who received additional weeks of instruction scored significantly higher than students who went back to their business-as-usual curriculum.

A multiple probe across participants design was used to examine the effectiveness of a treatment package to teach students with moderate intellectual disability how to solve simple linear equations. The investigator read realistic scenarios of a problem, used actual items as manipulatives, presented a visual aid of the equation, and used a system of least prompts procedure to teach the students to solve the problems. The results revealed the treatment package was effective in teaching students with moderate intellectual disabilities to solve linear equations. Participants generalized the ability to solve algebraic equations to problems in the natural environment.

A topic model is a statistical model for extracting latent clusters or themes from the text in a collection of documents. The purpose of this study was to apply a topic model to two educational assessments. In the first study, the model was applied to students’ written responses to an extended response item on an English Language Arts (ELA) test. In the second study, a topic model was applied to the errors students’ made on a fractions computation test. The results for the first study showed five distinct writing patterns were detected in students’ writing on the ELA test. Two of the patterns were related to low scores, two patterns were associated with high scores and one pattern was unrelated to the score on the test. In the second study, five error patterns (i.e., latent topics) were detected on the pre-test and six error patterns were detected on the post-test for the fractions computation test. The results for Study 2 also yielded evidence of instructional effects on students’ fractions computation ability. Following instruction, more students in the experimental instruction condition made fewer errors than students in the business-as-usual condition.

This article presents a multilevel longitudinal nested logit model for analyzing correct response and error types in multilevel longitudinal intervention data collected under a pretest–posttest, cluster randomized trial design. The use of the model is illustrated with a real data analysis, including a model comparison study regarding model complexity and cluster bias. Two substantive research questions regarding the intervention effect on correct response probability and error patterns are investigated using the proposed model. The recovery of item parameters for the proposed model using two sample size conditions is examined via a simulation study. The accuracy of the parameter estimates is comparable with those found in previous studies for the same family of models, except for the intercept parameters of correct responses. Finally, the impact of ignoring cluster membership in the model on the parameter estimation is also studied by fitting a single-level model to multilevel data. Ignoring cluster membership in the model adversely affects the estimation of intercept parameters in correct and error responses.

In this article, we describe results of a reanalysis of two randomized studies that tested the effects of enhanced anchored instruction (EAI) on the fractions computation performance of students in special education resource rooms and inclusive mathematics classrooms. Latent class analysis and latent transition analysis classified students according to error subtypes and tracked their performance patterns. Results indicated that EAI was more effective than business as usual in reducing combining errors (e.g., adding denominators) and denominator errors (e.g., not finding common denominator) of students with disabilities (SWD) and students without disabilities in both settings. SWD in inclusive classrooms scored higher on the pretest than SWD in resource rooms, but EAI reduced the disparity on the posttest. An important additional finding revealed that the SWD who received more support from special education teachers in inclusive classrooms scored higher and made fewer errors than the SWD who were provided only limited support.

The purpose of this study was to apply a random item mixture nominal item response model (RIM-MixNRM) for investigating instruction effects. The host study design was a pre-test-and-post-test, school-based cluster randomized trial. A RIM-MixNRM was used to identify students’ error patterns in mathematics at the pre-test and the post-test. Instruction effects were investigated in terms of students’ transitioning in error patterns. That is, we compared students’ error patterns in the Enhanced Anchored Instruction (EAI) condition with students’ error patterns in a business-as-usual (BAU) instructional condition following each instruction. We also compared error patterns of students with math disabilities and students without math disabilities following the two types of instruction.

In a pre-test-post-test cluster randomized trial, one of the methods commonly used to detect an intervention effect involves controlling pre-test scores and other related covariates while estimating an intervention effect at post-test. In many applications in education, the total post-test and pre-test scores, ignoring measurement error, are used as response variable and covariate, respectively, to estimate the intervention effect. However, these test scores are frequently subject to measurement error, and statistical inferences based on the model ignoring measurement error can yield a biased estimate of the intervention effect. When multiple domains exist in test data, it is sometimes more informative to detect the intervention effect for each domain than for the entire test. This paper presents applications of the multilevel multidimensional item response model with measurement error adjustments in a response variable and a covariate to estimate the intervention effect for each domain.

The Common Core State Standards for Mathematics will place more pressure on special education and math teachers to raise the skill levels of all students, especially those with disabilities in math (MD). The purpose of this study was to assess the effects of enhanced anchored instruction (EAI) on students with and without MD in co-taught general education classrooms. Results showed that students in the EAI condition improved their performance on math skills contained in several of the standards. Effect sizes were especially large for students with MD when the special education teacher more actively participated in the instructional activities with the math teacher. Classroom observations provided examples of how teachers can work together to benefit students in inclusive math settings.

Latent transition analysis (LTA) was initially developed to provide a means of measuring change in dynamic latent variables. In this article, we illustrate the use of a cognitive diagnostic model, the DINA model, as the measurement model in a LTA, thereby demonstrating a means of analyzing change in cognitive skills over time. An example is presented of an instructional treatment on a sample of seventh-grade students in several classrooms in a Midwestern school district. In the example, it is demonstrated how hypotheses could be framed and then tested regarding the form of the change in different groups within the population. Both manifest and latent groups also are defined and used to test additional hypotheses about change specific to particular subpopulations. Results suggest that the use of a DINA measurement model expands the utility of LTA to practical problems in educational measurement research.

Multilevel modeling (MLM) is frequently used to detect group differences, such as an intervention effect in a pre-test-post-test cluster-randomized design. Group differences on the post-test scores are detected by controlling for pre-test scores as a proxy variable for unobserved factors that predict future attributes. The pre-test and post-test scores that are most often used in MLM are summed item responses (or total scores). In prior research, there have been concerns regarding measurement error in the use of total scores in using MLM. To correct for measurement error in the covariate and outcome, a theoretical justification for the use of multilevel structural equation modeling (MSEM) has been established. However, MSEM for binary responses has not been widely applied to detect intervention effects (group differences) in intervention studies. In this article, the use of MSEM for intervention studies is demonstrated and the performance of MSEM is evaluated via a simulation study. Furthermore, the consequences of using MLM instead of MSEM are shown in detecting group differences. Results of the simulation study showed that MSEM performed adequately as the number of clusters, cluster size, and intraclass correlation increased and outperformed MLM for the detection of group differences.

This article describes a follow-up analysis of findings from a randomized study that tested the efficacy of a blended version of Enhanced Anchored Instruction (EAI) designed to improve both the computation and problem-solving performances of middle school students with disabilities. The goals of the secondary analysis were to track overall error patterns of students in computing with fractions and to compare the effects of EAI and Business As Usual (BAU) on making these errors. Results showed that students taught with EAI reduced their errors compared to students in BAU classrooms and that reducing the one common error led to improved performance. Error pattern analysis provided clues about how to modify instructional materials for improving computation with fractions.

A pretest-posttest cluster-randomized trial involving 31 middle schools and 335 students with disabilities tested the effects of combining explicit and anchored instruction on fraction computation and problem solving. Results of standardized and researcher-developed tests showed that students who were taught with the blended units outscored students in Business As Usual classes. Students made the largest gains in computing with fractions and on problems related to ratios, proportions, and geometry. The findings suggest important implications for the way curriculum is designed for middle school students with disabilities who exhibit low performance in math.

This study compared how students with learning difficulties in math (MLD) who were randomly assigned to two instructional conditions answered items on problem solving tests aligned to the Common Core State Standards Initiative for Mathematics. Posttest scores showed improvement in the math performance of students receiving Enhanced Anchored Instruction (EAI) and typical instruction but the improvement of students in the EAI condition was greater. Much of the total variance in latent ability scores was explained at the teacher level. As we have observed in previous studies, teachers' use of interactive technology tools combined with engaging hands-on applications can make important differences in what students with MLD achieve in mathematics, particularly in problem solving.

A multilevel latent transition analysis (LTA) with a mixture IRT measurement model (MixIRTM) is described for investigating the effectiveness of an intervention. The addition of a MixIRTM to the multilevel LTA permits consideration of both potential heterogeneity in students’ response to instructional intervention as well as a methodology for assessing stage sequential change over time at both student and teacher levels. Results from an LTA–MixIRTM and multilevel LTA–MixIRTM were compared in the context of an educational intervention study. Both models were able to describe homogeneities in problem solving and transition patterns. However, ignoring a multilevel structure in LTA–MixIRTM led to different results in group membership assignment in empirical results. Results for the multilevel LTA–MixIRTM indicated that there were distinct individual differences in the different transition patterns. The students receiving the intervention treatment outscored their business as usual (i.e., control group) counterparts on the curriculum-based Fractions Computation test. In addition, 27.4 % of the students in the sample moved from the low ability student-level latent class to the high ability student-level latent class. Students were characterized differently depending on the teacher-level latent class.

Current methods for detecting growth of students' problem-solving skills in math focus mainly on analyzing changes in test scores. Score-level analysis, however, may fail to reflect subtle changes that might be evident at the item level. This article demonstrates a method for studying item-level changes using data from a multiwave experiment with a teaching method called enhanced anchored instruction (EAI). The analysis combines a mixture Rasch model for detecting individual differences within latent groups with a latent transition analysis model for tracking changes in latent group membership over the course of EAI. The analysis clearly indicates the effects of EAI and how they differ for members of each latent class. Comparisons are provided with a standard analysis of changes in test scores. Implications of the new approach are discussed for detecting subtle transformations in the math performance of students across a range of ability levels.

Middle school students with learning disabilities in math (MLD) used two versions of Enhanced Anchored Instruction (EAI). In one condition, students learned how to compute with fractions on an as-needed basis while they worked to solve the EAI problems. In the other condition, teachers used a computer-based instructional module in place of one of the EAI problems to deliver formal fraction instruction. The results indicated that students in both instructional formats improved their fraction computational skills and that formal instruction provided an added benefit. Both instructional conditions improved students' problem-solving skills by about the same amount. The findings suggest that combining formal fraction instruction with EAI is a viable way to improve the problem-solving and computational skills of students with MLD.

A latent transition analysis (LTA) model was described with a mixture Rasch model (MRM) as the measurement model. Unlike the LTA, which was developed with a latent class measurement model, the LTA-MRM permits within-class variability on the latent variable, making it more useful for measuring treatment effects within latent classes. A simulation study indicated that model recovery using the LTA-MRM was good except for small sample size-short test conditions. A real data application of a mathematics intervention with middle school students indicated that the LTA-MRM clearly detected the intervention effect and also provided a means of helping to better understand the effects compared to a standard multiwave analysis of variance.

Students with learning disabilities in math (denoted MD) display difficulties in developing conceptual understanding of number, in computation, and in formulating correct strategies for solving problems. Often, these students have concomitant reading difficulty, which severely limits their understanding of text-based problems. While explicit instruction of basic computation skills remains important, a greater emphasis is placed on the ability to solve problems, especially as a growing number of students with MD are included in general education classrooms. This article summarizes a small sample and brief descriptions of instructional interventions that hold promise for educating students with MD.

While curriculum specialists and committees often decide how mathematics is taught, it is ultimately principals who influence the extent to which these initiatives are carried out. The overall goal of this article is to provide school leaders with classroom-based research that describes one way of improving the math skills of middle school students. The study employed a randomized pretest-posttest comparison group design to examine the effects of two versions of Enhanced Anchored Instruction (EAI) and a Business as Usual (BAU) condition on the math skills of middle school students in technology education classrooms. Results showed that both EAI conditions were effective at improving the math skills of students over those of students in the BAU classes. The findings suggest that technology education teachers can make important contributions in helping students develop their computation and problem-solving skills.

Building a skateboard ramp motivates students to practice and apply fraction operations and measurement while developing problem-solving skills.