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The current study examined the efficacy and efficiency of incorporating instructive feedback within matrix training to teach children with autism spectrum disorder (ASD) to label common characters and cities. Experimenters taught one set of responses using a non-overlapping matrix, a second set of responses using an overlapping matrix, and a third set of responses using a non-overlapping matrix along with secondary targets to three individuals with ASD. The results demonstrated that all teaching methods were effective, and all trained and untrained responses were acquired. Matrix training with instructive feedback was equally as efficient as non-overlapping matrix training and overlapping matrix training, requiring about the same number of sessions for each participant to acquire the responses. The findings demonstrated that establishing recombinative generalization through matrix training and instructive feedback is equally and maybe even more effective and efficient than matrix training in isolation in some circumstances.

The aim of this study is to examine the ability of pre-service mathematics teachers to detect errors made in solving questions about matrices. The study particularly focused on revealing the internalization of the teachings such as the meanings and relational dimensions of concepts and operations about matrix. The study was conducted with 26 teacher candidates at a university in the Eastern Anatolia Region. They were given a written exam, and their responses were analyzed by two field experts. The results showed that the pre-service teachers did not fully understand the concepts and operations of matrices. They made a variety of errors, including misconceptions and incomplete understanding. They were also not very good at solving proof-based questions. However, they were more successful at solving problems that were based on plain logic or could be solved using rules.

We consider the statistical inference for noisy incomplete binary (or 1-bit) matrix. Despite the importance of uncertainty quantification to matrix completion, most of the categorical matrix completion literature focuses on point estimation and prediction. This paper moves one step further toward the statistical inference for binary matrix completion. Under a popular nonlinear factor analysis model, we obtain a point estimator and derive its asymptotic normality. Moreover, our analysis adopts a flexible missing-entry design that does not require a random sampling scheme as required by most of the existing asymptotic results for matrix completion. Under reasonable conditions, the proposed estimator is statistically efficient and optimal in the sense that the Cramer-Rao lower bound is achieved asymptotically for the model parameters. Two applications are considered, including (1) linking two forms of an educational test and (2) linking the roll call voting records from multiple years in the United States Senate. The first application enables the comparison between examinees who took different test forms, and the second application allows us to compare the liberal-conservativeness of senators who did not serve in the Senate at the same time.

Feedback is a powerful instructional tool for motivating learning. But effective feedback, requires that instructors have accurate information about their students' current knowledge status and their learning progress. In modern educational measurement, two major theoretical perspectives on student ability and proficiency can be distinguished. Latent trait models identify ability as a continuous uni- or multi-dimensional construct, with unidimensional item response theoretic (IRT) models presumably the most popular type of latent trait models. They report a single ability score that allows for locating examinees relative to their peers on the latent ability dimension targeted by the test. Latent trait models have been criticized for lacking diagnostic information on students' specific skills, their strengths and weaknesses in a knowledge domain. Cognitive diagnosis (CD) models, in contrast, describe ability as a combination of discrete skills (called "attributes") that constitute (partially) ordered latent classes of proficiency. The focus of CD is on collecting information about the learning progress for immediate feedback to students in terms of skills they have mastered and those needing study. CD has been underused in education; performance assessment still mostly relies on latent-trait-based methods. The motivation for the study reported here arose from the desire to conduct a side-by-side evaluation of the two seemingly disparate psychometric frameworks, CD and IRT. Data from a biochemistry end-of-term exam were used for illustration. They were fitted with multiple CD and IRT models, among them also HO-GDINA models that permit for a close approximation to several unidimensional IRT models.

Matrix decomposition structural equation modeling (MDSEM) is introduced as a novel approach in structural equation modeling, contrasting with traditional structural equation modeling (SEM). MDSEM approximates the data matrix using a model generated by the hypothetical model and addresses limitations faced by conventional SEM procedures by emphasizing factor analysis with L[subscript 2] penalization. Key advantages of MDSEM include preventing improper solutions, the ability to compute observation-wise residuals without post-hoc factor score estimation and ease in identifying equivalent models. These benefits are attributed to its matrix decomposition techniques, allowing for direct model fitting to the data matrix, unlike the covariance structure fitting in CS-SEM. An iterative algorithm for parameter estimation is proposed, guaranteeing a monotonically decreasing function value. Theoretical properties of MDSEM are examined, revealing its shared characteristics with existing factor analysis and SEM. Numerical simulations and real data examples validate that MDSEM produces results comparable to existing methods when adequately calibrated.

A two-level data set can be structured in either long format (LF) or wide format (WF), and both have corresponding SEM approaches for estimating multilevel models. Intuitively, one might expect these approaches to perform similarly. However, the two data formats yield data matrices with different numbers of columns and rows, and their "cols : rows" is related to the magnitude of eigenvalue bias in sample covariance matrices. Previous studies have shown similar performance for both approaches, but they were limited to settings where "cols << rows" in both data formats. We conducted a Monte Carlo study to investigate whether varying "cols : rows" result in differing performances. Specifically, we examined the p:N ("cols : rows") effect on convergence and estimation accuracy in multilevel settings. Our findings suggest that (1) the LF approach is more likely to achieve convergence, but for the models that converged in both; (2) the LF and WF approach yield similar estimation accuracy; which is related to (3) differential "cols : rows" effects in both approaches; and (4) smaller ICC values lead to less accurate between-group parameter estimates.

In order to promote the growth of contemporary music and the reform of music, this article designs an improved collaborative filtering (CF) algorithm to solve the problem of sparse matrix in traditional recommendation algorithms. The data matrix is dimensionally reduced to find the nearest neighbor, so as to realize personalized recommendation of music teaching resources in colleges and universities. The test results show that the accuracy of the proposed teaching resource recommendation algorithm is improved by 22.56% compared with the traditional CF algorithm. The improved CF algorithm can provide more accurate prediction, and the recommendation effect of the improved algorithm is better than the original algorithm, which can effectively avoid the sparse matrix problem faced by the CF algorithm, and provide technical support for the development of contemporary music discipline and the reform of music discipline.

Simple matrix sampling planned missing (SMS PD) design, introduce missing data patterns that lead to covariances between variables that are not jointly observed, and create difficulties for analyses other than mean and variance estimations. Based on prior research, we adopted a new multigroup confirmatory factor analysis (CFA) approach to handle missing data in such designs, in comparison to a regular CFA with full information maximum likelihood estimator. In Study 1, we tested the two approaches in 36 scenarios (4 sample sizes x 3 inter-item correlations x 3 numbers of x-set items) given a total of 20 items. We found that, the multigroup CFA approach performed with acceptable convergence rates, power to recover population values, acceptable standard errors and model fit in certain scenarios by larger sample size, higher bivariate correlation, and more items in the x-set. We found a few scenarios where regular CFA with FIML performed well. These findings suggested that the approaches can be implemented to handle the special missing data introduced by the SMS PM designs, and, thereby, enhance the utility of SMS PM data. In Study 2, we applied the multigroup CFA approach in real-world data to demonstrate the feasibility and analytic value of this approach.

The digitization of library and archive collections over the last two decades has enabled efforts to reform history education through the integration of primary sources. Currently the Library of Congress provides one of the largest digital collections of its kind. The authors' project, with support from the Library, provides social studies teachers with explicit guidance for teaching historical thinking. The authors, a group of university-based researchers and experienced social studies classroom teachers, conducted an iterative, action research study to identify learning trajectories in history education. In this article, the authors describe their process, introduce the learning trajectories matrix, and provide suggestions for integrating it into the classroom.

Managing and analysing large qualitative datasets pose a particular challenge for researchers seeking a consistent and rigorous approach to qualitative data analysis. This paper describes and demonstrates the development and adoption of a matrix tool to guide the qualitative data analysis of a large sample (N = 122) of interview data. The paper articulates the theoretical and conceptual underpinnings of the matrix analysis tool and how it was developed and applied in a longitudinal mixed methods out-of-home-care research study. Specific illustrations and examples of data integration and data analysis are provided to demonstrate the benefits and potentials of constructing matrix tools to guide research teams when working with large qualitative data sets alone or in combination with quantitative data sets.

Using the sawtooth map as the basis of a coupled map lattice enables simple analytic results to be obtained for the global Lyapunov spectra of a number of standard lattice networks. The results presented can be used to enrich a course on chaos or dynamical systems by providing tractable examples of higher dimensional maps and links to a number of standard results in matrices. A number of suggestions for classroom use are given.

Figural matrices tasks are one of the most prominent item formats used in intelligence tests, and their relevance for the assessment of cognitive abilities is unquestionable. However, despite endeavors of the open science movement to make scientific research accessible on all levels, there is a lack of royalty-free figural matrices tests. The Open Matrices Item Bank (OMIB) closes this gap by providing free and unlimited access (GPLv3 license) to a large set of empirically validated figural matrices items. We developed a set of 220 figural matrices based on well-established construction principles commonly used in matrices tests and administered them to a sample of N = 2572 applicants to medical schools. The results of item response models and reliability analyses demonstrate the excellent psychometric properties of the items. In the discussion, we elucidate how researchers can already use the OMIB to gain access to high-quality matrices tests for their studies. Furthermore, we provide perspectives for features that could additionally improve the utility of the OMIB.

Project-based learning (PjBL) is seeing increasing scholarly interest and pedagogical use in higher education, but instances of PjBL do not necessarily seek the same educational outcomes. Using the grounded theory method, the authors plot five courses in a PjBL program on a matrix of course design characteristics ranging from Fixed to Flexible and Individualistic to Cooperative. They describe four major variations of PjBL based on this matrix. Recognizing that PjBL courses vary in their use of student choice and student collaboration, the authors make recommendations for assessment researchers and for teachers wishing to develop new strategies that fit their institutional and disciplinary contexts.

All cognitive diagnostic models that evaluate educational test data require a Q-matrix that combines every item in a test with the required cognitive skills for each item to be answered correctly. Generally, the Q-matrix is constructed by education experts' judgment, leading to some uncertainty in its elements. Various statistical methods are suggested to validate misspecifications in the Q-matrix. This paper evaluates the performance of the Q-matrix validation methods, the sequential expectation-maximization-based [delta]-method (SEM [delta]-method), and the Q-matrix refinement (QMR) method using a study with real data and simulations. The simulation design results showed that the misspecification percentage and the length of the test had a small or no effect on the mean q-entry recovery rates (MRRs) of both methods, while the increase in sample size had an improving effect. The MRRs of both methods decreased when the number of attributes and guessing (g) - slip (s) parameters increased. According to simulation study results, the QMR method performed better than the SEM [delta]-method. For the q-matrix validation, it can be suggested that CDM practitioners prioritize the QMR method and use a sample size of 1,000. On the other hand, the real data results revealed that the MRRs of both methods were at the base rates. This result highlights the need for further research on method comparison, specifically for real-world data applications where the number of attributes is relatively large and the test duration is short.

This study designed a localized approach to working and writing action research suited for basic education. The researchers used multiphase mixed method research with three analysis stages focused on the identified problem. These stages include identifying the preferred approach in writing action research for basic education, designing a localized approach, and evaluating the developed action research approach. Five key informants participated in the preference assessment, while thirty respondents assessed the acceptability and significance levels of the designed localized approach. The key data analysis treatments employed qualitative data analysis, mean, and independent t-test. Preferred contextualized design for action research includes multiphase mixed-method research approaches. Grounded on the analyzed preference, the researchers designed the "Multi-Analysis Layer Nexus (MALN)" approach for basic education action research, featured with its focus approach matrix and a question-procedure-analysis (QPA) design alignment. The designed localized process received highly acceptable and highly significant ratings. No significant difference is derived from the two assessments. Recommendations included the conduct of MALN approach capacity building, skills and abilities profile, and data analysis treatments in the QPA matrix. The study is limited to the SALIKSIK (Strategic Action for Learning, Innovation, Knowledge Systems, and Instructional Keystones) Research Program of Lopez East and Lopez West Districts. Though, the DepEd Division of Quezon can use this for its localized action research approach. The developed MALN approach is an innovative, novel action research process specifically designed for basic education settings.

Before embarking on an individual participant data meta-analysis (IPDMA) project, researchers should consider the power of their planned IPDMA conditional on the studies promising their IPD and their characteristics. Such power estimates help inform whether the IPDMA project is worth the time and funding investment, before IPD are collected. Here, we suggest how to estimate the power of a planned IPDMA of randomised trials aiming to examine treatment-covariate interactions at the participant-level (i.e., treatment effect modifiers). We focus on a time-to-event (survival) outcome with a binary or continuous covariate, and propose an approximate analytic power calculation that conditions on the actual characteristics of trials, for example, in terms of sample sizes and covariate distributions. The proposed method has five steps: (i) extracting the following aggregate data for each group in each trial--the number of participants and events, the mean and SD for each continuous covariate, and the proportion of participants in each category for each binary covariate; (ii) specifying a minimally important interaction size; (iii) deriving an approximate estimate of Fisher's information matrix for each trial and the corresponding variance of the interaction estimate per trial, based on assuming an exponential survival distribution; (iv) deriving the estimated variance of the summary interaction estimate from the planned IPDMA, under a common-effect assumption, and (v) calculating the power of the IPDMA based on a two-sided Wald test. Stata and R code are provided and a real example provided for illustration. Further evaluation in real examples and simulations is needed.

Network meta-analysis has been an essential methodology of systematic reviews for comparative effectiveness research. The restricted maximum likelihood (REML) method is one of the current standard inference methods for multivariate, contrast-based meta-analysis models, but recent studies have revealed the resultant confidence intervals of average treatment effect parameters in random-effects models can seriously underestimate statistical errors; that is, the actual coverage probability of a true parameter cannot retain the nominal level (e.g., 95%). In this article, we provided improved inference methods for the network meta-analysis and meta-regression models using higher-order asymptotic approximations based on the approach of Kenward and Roger ("Biometrics" 1997;53:983-997). We provided two corrected covariance matrix estimators for the REML estimator and improved approximations for its sample distribution using a t-distribution with adequate degrees of freedom. All of the proposed procedures can be implemented using only simple matrix calculations. In simulation studies under various settings, the REML-based Wald-type confidence intervals seriously underestimated the statistical errors, especially in cases of small numbers of trials meta-analyzed. By contrast, the proposed Kenward-Roger--type inference methods consistently showed accurate coverage properties under all the settings considered in our experiments. We also illustrated the effectiveness of the proposed methods through applications to two real network meta-analysis datasets.

Purpose: Miniature linguistic systems (also known as matrix training) is a method of organizing learning targets to achieve generative learning or recombinative generalization. This systematic review is aimed at determining whether matrix training is effective for individuals with autism spectrum disorder (ASD) in terms of improving recombinative generalization for instruction-following, expressive language, play skills, and literacy skills. Method: A systematic review methodology was employed to limit bias in the various review stages. A multifaceted search was conducted. Potential primary studies were imported into Covidence, a systematic review software, and inclusion criteria were applied. Data were extracted regarding (a) participant characteristics, (b) matrix designs, (c) intervention methods, and (d) dependent variable. A quality appraisal using the What Works Clearinghouse (WWC) Single-Case Design Standards (Version 1.0, Pilot) was carried out. In addition to the visual analysis of the data, an effect size estimate, non-overlap of all pairs (NAP), was generated for each participant. Independent t tests and between-subjects analyses of variance were conducted to identify moderators of effectiveness. Results: Twenty-six studies including 65 participants met criteria for inclusion. All included studies were single-case experimental designs. Eighteen studies received a rating of "Meets Standards Without Reservations" or "Meets Standards With Reservations." The aggregated combined NAP scores for acquisition, recombinative generalization, and maintenance of a range of outcomes were in the high range. Conclusions: Findings suggested that matrix training is an effective teaching method for individuals with ASD for acquisition, recombinative generalization, and maintenance of a range of outcomes. Statistical analyses to identify moderators of effectiveness were insignificant. Based on the WWC Single-Case Design Standards matrix training meets criteria to be considered an evidence-based practice for individuals with ASD.

Word learning studies traditionally examine the narrow link between words and objects, indifferent to the rich contextual information surrounding objects. This research examined whether children attend to this contextual information and construct an associative matrix of the words, objects, people, and environmental context during word learning. In Experiment 1, preschool-aged children (age: 3;2-5;11 years) were presented with novel words and objects in an animated storybook. Results revealed that children constructed associations beyond words and objects. Specifically, children attended to and had the strongest associations for features of the environmental context but failed to learn word-object associations. Experiment 2 demonstrated that children (age: 3;0-5;8 years) leveraged strong associations for the person and environmental context to support word-object mapping. This work demonstrates that children are especially sensitive to the word learning context and use associative matrices to support word mapping. Indeed, this research suggests associative matrices of the environment may be foundational for children's vocabulary development.

This article utilizes a framework for classifying different scientific methods suggested by a philosopher of science (Brandon "Synthese," 99, 59-73, 1994) called Brandon's Matrix. It presents findings from teachers who took part in a funded project in England that looked at the nature of scientific methods in science investigations. Science investigations are an integral aspect of science education and, as such, are often included in high stakes examinations. Therefore, teachers need to have a good understanding of a range of scientific methods and their purposes in science investigations. The framework was used to ask teachers to classify science investigations based on how they teach them. It was also employed to devise assessments to measure students' understanding of scientific methods. The teachers were introduced to the new approaches and their perceptions were gathered to understand if they supported this as a framework for their classroom practice. Evidence from the study suggested that Brandon's Matrix appealed to teachers as a framework for practical science in schools, and they see potential benefits for its use in the teaching, learning, and assessment of science. Findings from the study showed it appealed to the teachers as a tool for classifying scientific methods, and how they also recognized the importance of assessing practical work and had an appreciation of the constraints and drivers in the current curriculum and assessment requirements in England. Implications for teachers' professional development are discussed.

This study investigated how and how often to present prompts to enhance students' source evaluation and multiple-text comprehension. Participants were 72 undergraduates who read a set of digital texts on a controversial topic of smartphone use and mental health, wrote a justification statement for their selection of trustworthy texts, and answered open-ended comprehension questions. To explore the optimal presentation conditions, this study varied the presentation format (matrix vs. question) and frequency (once vs. repeated) of prompts. The results showed that participants benefited more from the matrix prompt than the question prompt in source evaluation and multiple-text comprehension. An interaction effect occurred only in multiple-text comprehension, indicating that repeated prompting via matrix was an optimal approach to facilitate integration of text information. In addition, participants perceived less cognitive load when matrix was presented than when questions were presented. Taken together, these results have classroom implications for instructors to consider both the format and frequency of presenting prompts to facilitate source evaluation and comprehension of multiple conflicting-view articles.

Cognitive diagnostic assessment based on Bayesian networks (BN) is developed in this paper to evaluate student understanding of the physical concept of buoyancy. we propose a three-order granular-hierarchy BN model which accounts for both fine-grained attributes and high-level proficiencies. Conditional independence in the BN structure is tested and utilized to validate the proposed model. The proficiency relationships are verified and the initial Q-matrix is refined. Then, an optimized granular hierarchy model is constructed based on the updated Q-matrix. All variants of the constructed models are evaluated on the basis of the prediction accuracy and the goodness-of-fit test. The experimental results demonstrate that the optimized granular-hierarchy model has the best prediction and model-fitting performance. In general, the BN method not only can provide more flexible modeling approach, but also can help validate or refine the proficiency model and the Q-matrix and this method has its unique advantage in cognitive diagnosis.

Technological advances have significantly affected education, leading to the creation of online learning platforms such as virtual learning environments and massive open online courses. While these platforms offer a variety of features, none of them incorporates a module that accurately predicts students' academic performance and commitment. Consequently, it is crucial to design machine learning (ML) methods that predict student performance and identify at-risk students as early as possible. Graph representations of student data provide new insights into this area. This paper describes a simple but highly accurate technique for converting tabulated data into graphs. We employ distance measures (Euclidean and cosine) to calculate the similarities between students' data and construct a graph. We extract graph topological features (GF) to enhance our data. This allows us to capture structural correlations among the data and gain deeper insights than isolated data analysis. The initial dataset (DS) and GF can be used alone or jointly to improve the predictive power of the ML method. The proposed method is tested on an educational dataset and returns superior results. The use of DS alone is compared with the use of DS+GF in the classification of students into three classes: "failed", "at risk", and "good". The area under the receiver operating characteristic curve (AUC) reaches 0.948 using DS, compared with 0.964 for DS+GF. The accuracy in the case of DS+GF varies from 84.5 to 87.3%. Adding GF improves the performance by 2.019% in terms of AUC and 3.261% in terms of accuracy. Moreover, by incorporating graph topological features through a graph convolutional network (GCN), the prediction performance can be enhanced by 0.5% in terms of accuracy and 0.9% in terms of AUC under the cosine distance matrix. With the Euclidean distance matrix, adding the GCN improves the prediction accuracy by 3.7% and the AUC by 2.4%. By adding graph embedding features to ML models, at-risk students can be identified with 87.4% accuracy and 0.97 AUC. The proposed solution provides a tool for the early detection of at-risk students. This will benefit universities and enhance their prediction performance, improving both effectiveness and reputation.

In application-oriented mathematics, particularly in the context of nonlinear system analysis, phase plane analysis through SageMath offers a visual display of the qualitative behaviour of solutions to differential equations without inundating students with cumbersome calculations of the plane-phase. A variety of examples is usually given to illustrate phase-plane behaviour. We approach these problems by considering a problem containing a single real parameter that exemplifies the various situations clearly and simply. We developed two computer programs in SageMath: one program calculates aspects of the Jacobian matrix and displays the phase plane portraits, the other determines the centre manifold. Computations and images generated with computer codes are useful in understanding dynamic models in biology, physics and engineering that involve planar non-linear autonomous differential equations. This paper covers analytical and computational skills that are helpful for students and teachers.

This paper discusses findings from an ongoing study investigating mental mechanisms involved in the conceptualisation of linear transformations from the perspective of Action (A), Process (P), Object (O), and Schema (S) (APOS) theory. Data reported in this paper came from 44 first-year linear algebra students' responses on a task regarding the range of a linear transformation. Our analysis revealed parallels between Levels/Stages of the range concept and the use of representations of matrix multiplications. More importantly, these representations appeared to have been the enablers of transitions from lower to higher APOS Stages for the range. Conversely, mental mechanisms employing other means showed little to no progressions, some, furthermore, revealed faulty knowledge structures.

Raven's Standard Progressive Matrices (SPM) test and related matrix-based tests are widely applied measures of cognitive ability. Using Bayesian Item Response Theory (IRT) models, I reanalyzed data of an SPM short form proposed by Myszkowski and Storme (2018) and, at the same time, illustrate the application of these models. Results indicate that a three-parameter logistic (3PL) model is sufficient to describe participants dichotomous responses (correct vs. incorrect) while persons' ability parameters are quite robust across IRT models of varying complexity. These conclusions are in line with the original results of Myszkowski and Storme (2018). Using Bayesian as opposed to frequentist IRT models offered advantages in the estimation of more complex (i.e., 3-4PL) IRT models and provided more sensible and robust uncertainty estimates.

Network models of the WAIS-IV based on regularized partial correlation matrices have been reported to outperform latent variable models based on uncorrected correlation matrices. The present study sought to compare network and latent variable models using both partial and uncorrected correlation matrices with both types of models. The results show that a network model provided better fit to matrices of partial correlations but latent variable models provided better fit to matrices of full correlations. This result is due to the fact that the use of partial correlations removes most of the covariance common to WAIS-IV tests. Modeling should be based on uncorrected correlations since these represent the majority of shared variance between WAIS-IV test scores.

Teaching the noncommutativity of the product of matrices to high school or college level students is a difficult task when approached from a purely formal perspective. The aim of this paper is to present a simple experimental activity for teaching the noncommutativity of the matrix product, based on the Jones calculus, a mathematical formalism for describing polarised light by means of matrices and vectors. This activity can also be useful to introduce students to the use of matrices in physics, and to illustrate how abstract mathematics can become a powerful tool to help us explain and describe the real world.

Children with autism spectrum disorder often lack fundamental play skills, which can aid development with social, language, and imitation skills (Boutot et al. 2005). The purpose of this study was to extend the previous literature that successfully combined video modeling and matrix training. Matrix training is an efficient way of teaching that encourages generalization without direct teaching of some skills. In this study, play actions were selected from a 2D, 6 × 6 matrix to teach pretend play skills to 3-5-year-old children with a diagnosis of ASD. Play actions were made up of different toy kitchen foods and play actions within a play kitchen setting (e.g., rinse the carrot and cut the pear). Using a multiple probe design across behaviors, the play actions were taught using video modeling and other play actions from the matrix were later assessed for recombinative generalization. Overall, matrix training was effective for producing recombinative generalization, although additional training was required for 1 out of the 3 participants.

It is proposed a network approach for electric circuits simplification, that through a unified systematic procedure allows simplifying circuits of any complexity, and evaluation of the equivalent resistances, capacitors and inductors. Circuits to be simplified are characterized by their nodes, and by the elements of different types (resistances, capacitors and inductors) connecting each pair of nodes. Once that information organized in the matrix form, circuit simplification is straightforward based on the simplification conditions, which are key elements of the proposed approach. Simplification process evolves eliminating nodes from the original circuit, node by node, and evaluating the equivalent element types connecting the remaining nodes, up to the required simplification level. The key elements of the simplification procedure are the same for all the element types. For circuits with elements of different types simplification can be made in a segregated way, each time for one element type, or simultaneously for all the element types. The simplification procedure can be used with pencil and paper, or easily programmed for automatic and fast circuits simplification. Use of the proposed approach is illustrated through some appealing examples.

Intending to improve the teaching and learning of the notion of mathematical proof this study seeks to uncover the kinds of flaws in postgraduate mathematics education student teachers. Twenty-three student teachers responded to a proof task involving the concepts of transposition and multiplication of matrices. Analytic induction strategy that drew ideas from the literature on evaluating students' proof understanding and Yang and Lin's model of proof comprehension applied to informants' written responses to detect the kinds of flaws in postgraduates' proof attempts. The study revealed that the use of empirical verifications was dominant and in situations. Whereby participants attempted to argue using arbitrary mathematical objects, the cases considered did not represent the most general case. Flawed conceptualizations uncovered by this study can contribute to efforts directed towards fostering strong subject content command among school mathematics teachers.

Detection of students' ability levels is one of the common aims in educational studies. Cognitive Diagnosis Modeling approach has been used recently for the purpose of ability level detection by defined Q-matrices. To evaluate students' strengths and weaknesses, determine their mastery skills, and design instructions and interventions in learning process, Cognitive Diagnosis Modeling approach can be helpful. Cognitive Diagnosis Modeling is an alternative approach to Item Response Theory, and provides more information using multiple fine-grained skills in problem solving process rather than order students on a latent proficiency continuum This paper aims to use Cognitive Diagnosis Modeling (CDM) in order to investigate the definition of a Q-matrix across the cognitive skills of different years and countries in Trends in International Mathematics and Science Study (TIMSS). There is a subjective way in defining Q-matrices, an intuitive definition of Q-matrices, for this purpose, an application of building Q-matrices under specific Cognitive Diagnosis Models, from a set of expert proposed attributes is examined. The proposed attributes are used to build Q-matrices for TIMSS mathematics questions across its cycles, and across different nations.

There has been increased interest in assessing the quality and usefulness of short versions of the Raven's Progressive Matrices. A recent proposal, composed of the last twelve matrices of the Standard Progressive Matrices (SPM-LS), has been depicted as a valid measure of "g." Nonetheless, the results provided in the initial validation questioned the assumption of essential unidimensionality for SPM-LS scores. We tested this hypothesis through two different statistical techniques. Firstly, we applied exploratory graph analysis to assess SPM-LS dimensionality. Secondly, exploratory bi-factor modelling was employed to understand the extent that potential specific factors represent significant sources of variance after a general factor has been considered. Results evidenced that if modelled appropriately, SPM-LS scores are essentially unidimensional, and that constitute a reliable measure of "g." However, an additional specific factor was systematically identified for the last six items of the test. The implications of such findings for future work on the SPM-LS are discussed.

No matter how strong the theoretical infrastructure of a study is, if the measurement instruments do not have the necessary psychometric qualities, there will be a question of trust in interpreting the findings, and it will be inevitable to make wrong decisions with the results. One of the important steps in scale development/adaptation studies is to provide evidence of the experimental validity. In order to reveal evidence of construct validity of Likert type scales, to identify factor structures, to confirm previously predicted structures, factor analysis is used. The primary issue to be examined is the level of measurement of the variable and one of the leading decisions that must be taken is which relation matrix will be used. This descriptive research is based on the effects of using Pearson or polychoric correlation matrix in the factor analysis. It is determined that items showed different "item-total correlations", "loading values" and "correlation coefficients", different factor numbers emerged, different items were removed out of the scale, confirmation status of the structure has changed.

Recombinative generalization is the production of responses in the presence of novel combinations of known components. For example, after learning "red triangle" and "blue square," recombinative generalization is observed when a child can tact "red square" and "blue triangle." Recombinative generalization can emerge from a history of matrix training, which involves carefully selecting and arranging stimuli and responses along at least two axes and training a subset of responses. With three children with autism spectrum disorder, we compared recombinative generalization of object-action or feature-object tacts when the component stimuli were trained before combination stimuli, trained along with combination stimuli, or untrained (i.e., combination only). For two participants, training the components along with some combinations led to the most untrained targets acquired without direct teaching. For the other participant, training the combinations only led to the greatest proportion of untrained targets acquired without direct teaching. We discuss stimulus control promoted by each teaching arrangement and suggestions for future research on recombinative generalization.

Managing education and research during pandemics has increased in importance since the onset of epidemics such as avian flu, SARS and now COVID-19. Successful management in times of crisis ensures business continuity and institutional survival, making preparedness preceding an impending pandemic essential. Institutions of higher education (IHEs) must maintain balance between academic continuity and preventing morbidity during a pandemic crisis. To date, however, no general pandemic preparedness frameworks exist for IHEs. The aim of this paper is to report on the development of a Haddon matrix framework for IHE pandemic preparedness based on a scoping literature review of past IHE responses including pre-, during and post-pandemic phases. First, a review of previous global responses by IHEs during past pandemics was carried out. The review findings were then collated into a new IHE-centric Haddon matrix for pandemic preparedness. The content of the matrix is then illustrated through the documented responses of Malaysian universities during the early stages of the COVID-19 pandemic. The resulting IHE Haddon matrix can be used by universities as a general guide to identify preparedness gaps and intervention opportunities for business continuity during pandemics.

This study proposes transformation functions and matrices between coefficients in the original and reparameterized parameter spaces for an existing linear-linear piecewise model to derive the interpretable coefficients directly related to the underlying change pattern. Additionally, the study extends the existing model to allow individual measurement occasions and investigates predictors for individual differences in change patterns. We present the proposed methods with simulation studies and a real-world data analysis. Our simulation study demonstrates that the method can generally provide an unbiased and accurate point estimate and appropriate confidence interval coverage for each parameter. The empirical analysis shows that the model can estimate the growth factor coefficients and path coefficients directly related to the underlying developmental process, thereby providing meaningful interpretation.

Systematic reviews and meta-analyses are essential for drawing conclusions regarding etiologic associations between exposures or interventions and health outcomes. Observational studies comprise a substantive source of the evidence base. One major threat to their validity is residual confounding, which may occur when component studies adjust for different sets of confounders, fail to control for important confounders, or have classification errors resulting in only partial control of measured confounders. We present the confounder matrix--an approach for defining and summarizing adequate confounding control in systematic reviews of observational studies and incorporating this assessment into meta-analyses. First, an expert group reaches consensus regarding the core confounders that should be controlled and the best available method for their measurement. Second, a matrix graphically depicts how each component study accounted for each confounder. Third, the assessment of control adequacy informs quantitative synthesis. We illustrate the approach with studies of the association between short interpregnancy intervals and preterm birth. Our findings suggest that uncontrolled confounding, notably by reproductive history and sociodemographics, resulted in exaggerated estimates. Moreover, no studies adequately controlled for all core confounders, so we suspect residual confounding is present, even among studies with better control. The confounder matrix serves as an extension of previously published methodological guidance for observational research synthesis, enabling transparent reporting of confounding control and directly informing meta-analysis so that conclusions are drawn from the best available evidence. Widespread application could raise awareness about gaps across a body of work and allow for more valid inference with respect to confounder control.

Introduction: This paper provides a report of social validity and its usefulness in understanding study effects. Social validity data were drawn from a larger single-case study designed to investigate the effectiveness of the Matrix Approach in the practice of early intervention visual impairment professionals. Methods: Qualitative measures were used to assess social validity. According to Wolf (1978), validation of a study should encompass the significance of goals, appropriateness of procedures, and importance of outcomes. Therefore, data sources were created to provide evidence of social validity in these areas including pre- and post-intervention interviews of parents and professionals. Results: Three themes emerged from the social validity data. These are organized under the topics introduced by Wolf (1978) and identified as measures of quality by Horner et al. (2005). The themes are (a) Goals: role of the parent as learner and the professional as expert before using the Matrix Approach, (b) Procedures: helpfulness of coaching and the structure of the Matrix Approach, and (c) Outcomes: personal and professional growth as a result of using the Matrix Approach. Discussion: Evaluation of social validity in single-case research is an important component in a study's design and in interpreting the study's outcomes. Wolf's framework proved valuable in promoting a robust evaluation of study effectiveness, especially when incorporated into the study design. Intentionally planning to measure social validity held the researchers accountable to the practical needs of the participants. In fact, there is value in gathering social validity data at various points in a study. Further, hearing the perspective of stakeholders can provide valuable insights as researchers seek to understand the complexities of change evident in the outcomes of a study. Implications for Practitioners: Collaborative planning is an essential component of early intervention. The Matrix Approach shows promise as a mechanism to foster such collaboration.

This paper discusses a project for linear algebra instructors interested in a concrete, geometric application of matrix diagonalization. The project provides a theorem concerning a nested sequence of tetrahedrons and scaffolded questions for students to work through a proof. Along the way students learn content from three-dimensional geometry and apply eigenvectors, eigenvalues, and diagonalization to calculate a limit. Other concepts found within the project apply cross products and normal vectors. We describe the project's background, offer comments and variations for the given questions, and supply results from administering it ourselves.

This paper contains interesting facts regarding the powers of odd ordered special circulant magic squares along with their magic constants. It is shown that we always obtain circulant semi-magic square and special circulant magic square in the case of even and odd positive integer powers of these magic squares respectively. These magic squares will also take a particular form in the case of both even and odd positive integer powers. Moreover, they are singular under some conditions.

In cognitive diagnosis assessment (CDA), the impact of misspecified item-attribute relations (or "Q-matrix") designed by subject-matter experts has been a great challenge to real-world applications. This study examined parameter estimation of the CDA with the expert-designed Q-matrix and two refined Q-matrices for international large-scale data. Specifically, the G-DINA model was used to analyze TIMSS data for Grade 8 for five selected countries separately; and the need of a refined Q-matrix specific to the country was investigated. The results suggested that the two refined Q-matrices fitted the data better than the expert-designed Q-matrix, and the stepwise validation method performed better than the nonparametric classification method, resulting in a substantively different classification of students in attribute mastery patterns and different item parameter estimates. The results confirmed that the use of country-specific Q-matrices based on the G-DINA model led to a better fit compared to a universal expert-designed Q-matrix.