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Several studies indicate that exploring mathematical ideas by using more than one approach to prove the same statement is an important matter in mathematics education. In this work, we have collected a few different methods of proving the multinomial theorem. The goal is to help further the understanding of this theorem for those who may not be familiar with it. These proofs can also be used by undergraduate college instructors in a calculus, a discrete mathematics or a probability course.

This study aims to establish a knowledge base on how to support students in learning. We develop an initial hypothetical learning trajectory by formulating learning activities and predicting the development of students' thinking and understanding. The methodological framework employed in this study is design research, which seeks to generate actionable knowledge for achieving various educational goals through design. The design process incorporates the Hypothetical Learning Trajectory (HLT) thought experiment and the teaching experiment based on the six Realistic Mathematics Education (RME) principles. By integrating individual learning activities with the context and principles of the RME approach, students can construct their knowledge and rediscover the quadratic equation formula. Through comparing the HLT with observed learning outcomes, we redesign the process, revise our HLT, and provide answers to research questions regarding the attainment of specific learning objectives. Similar to our retrospective data analysis, we create a revised HLT to reinforce the concept of the completing the square and reinvent the quadratic formula.

This study investigates the problem-solving strategies employed by ninth-grade students when addressing symbolic and real-world contextual problems involving trigonometric ratios. Conducted with 46 ninth-grade students from a Turkish public high school, this research employed a worksheet consisting of six problems aligned with the Turkish ninth-grade mathematics curriculum. Three of these problems were based on real-world contexts, while the other three were conventional symbolic problems. The findings indicate that students exhibited proficiency in identifying similarity ratios using side length ratios. Additionally, the results revealed that students were more adept at solving real-world mathematical scenarios compared to purely symbolic tasks. This study offers significant insights into the problem-solving strategies of ninth-grade students when confronted with trigonometric ratio problems. It underscores crucial implications for mathematics curricula and pedagogy, highlighting the importance of preparing ninth-grade students for success in their future academic and professional endeavors. The study emphasizes the necessity for a balanced approach in teaching, integrating both real-world and symbolic problem-solving tasks to enhance students' mathematical understanding and application. By identifying the strengths and areas for improvement in students' problem-solving strategies, this research contributes to the development of more effective educational practices that address the diverse needs of learners.

Multiple representations are widely recognized for their significant role in concept learning. This study aimed to investigate the multiple representation translation skills of high school students at different grade levels about the concept of one-dimensional motion. 239 9th, 10th, and 11th-grade students participated in the study using a developmental research model. The data collection tool consisted of questions that required translating figures, tables, graphs, verbal explanations, and algebraic representations into other representation types in a multiple-representation translation test focusing on one-dimensional motion. Data analysis involved evaluating the translation among representations for each category and analyzing the multiple representation translation skills across different grade levels using one-way analysis of variance (ANOVA). The results revealed that students successfully translated from figure, table, and graphical representations to other forms while encountering challenges in translation from verbal and algebraic representations. Furthermore, the ANOVA results indicated a significant difference between the 9th and 11th grades, favoring the 11th grade.

Post-instrument covariates are often included as controls in instrumental variable (IV) analyses to address a violation of the exclusion restriction. However, we show that such analyses are subject to biases unless strong assumptions hold. Using linear constant-effects models, we present asymptotic bias formulas for three estimators (with and without measurement error): IV with post-instrument covariates, IV without post-instrument covariates, and ordinary least squares. In large samples and when the model provides a reasonable approximation, these formulas sometimes allow the analyst to bracket the parameter of interest with two estimators and allow the analyst to choose the estimator with the least asymptotic bias. We illustrate these points with a discussion of the settler mortality IV used by Acemoglu, Johnson, and Robinson.

Obtaining reliable and timely estimates of migration flows is critical for advancing the migration theory and guiding policy decisions, but it remains a challenge. Digital data provide granular information on time and space, but do not draw from representative samples of the population, leading to biased estimates. We propose a method for combining digital data and official statistics by using the official statistics to model the spatial and temporal dependence structure of the biases of digital data. We use simulations to demonstrate the validity of the model, then empirically illustrate our approach by combining geo-located Twitter data with data from the American Community Survey (ACS) to estimate state-level out-migration probabilities in the United States. We show that our model, which combines unbiased and biased data, produces predictions that are more accurate than predictions based solely on unbiased data. Our approach demonstrates how digital data can be used to complement, rather than replace, official statistics.

Over the last decades, a multitude of results in educational and psychological research have shown that the implementation of multiple external representations (MERs) in educational contexts represents a valuable tool for fostering learning and problem-solving skills. The context of science, technology, engineering, and mathematics (STEM) education has received great attention because it necessitates using various symbolic (e.g., text and formula) and graphical representations (e.g., pictures and graphs) to convey subject content. Research has mainly explored effects of combining two representations, but the potential benefits of integrating more than two representations on students' learning remain underexplored. This gap limits our understanding of promising educational practices and restricts the development of effective teaching strategies catering to students' cognitive needs. To close this gap, we conducted a systematic review of 46 studies and a meta-analysis that included 132 effect sizes to evaluate the effectiveness of using more than two representations in STEM education and to identify moderating factors influencing learning and problem-solving. A network diagram analysis revealed that the advantages of learning and problem-solving with MERs are also applicable to more than two representations. A subsequent meta-analysis revealed that the learning with more than two representations in STEM can have advantageous effects on students cognitive load (Hedges'g = 0.324, p < 0.001, 95% CI [0.164, 0.484]) and performance (Hedges' g = 0.118, p < 0.001, 95% CI [0.050, 0.185]) compared to learning with two representations without notable differences in learning time. The analysis of moderating factors revealed that benefits of learning with more than two representations primarily depend on the provision of appropriate support.

Grasping mathematical objects as related to processes is often considered critical for mathematics understanding. Yet, the ontology of mathematical objects remains under debate. In this paper, we theoretically oppose internalist approaches that claim mental entities as the endpoints of process-object transitions and externalist approaches that stress mathematical artifacts--such as physical manipulatives and formulas--as constituting mathematical objects. We search for a view on process-object duality that overcomes the dualism of mind and body. One such approach is commognition that describes mathematical objects as discursive entities. This paper expands the nature of mathematical objects beyond discourse and highlights the role of learners' interaction with the environment by adopting ecological onto-epistemology. We develop a functional dynamic systems perspective on process-object duality in mathematics learning emphasizing embodied actions and the re-invention of artifacts' affordances. As a main result, we reconsider process?-object duality as a reification of repetitive actions into a cultural artifact that consists of two steps: (1) forming a new sensory-motor coordination that brings new perception to the fore and (2) crystallizing a new artifact in a mathematical environment that captures this new perception. An empirical example from research on embodied action-based design for trigonometry illustrates our theoretical ideas.

A central topic in Bioelectricity is the generation of the extracellular potential that results from the propagation of a transmembrane action potential along the muscle fiber. However, the way in which the extracellular potential is determined by the propagating action potential is difficult to describe, conceptualize, and visualize. Moreover, traditional quantitative approaches aimed at modeling extracellular potentials involve complex mathematical formulations, which do not allow students to visualize how the extracellular potential is generated around the active fiber. The present study is aimed at presenting a novel pedagogical approach to teaching the generation of extracellular potentials produced by muscle fibers based on the convolution operation. The effectiveness of this convolutional model was tested using a written exam and a satisfaction survey. Most students reported that a great advantage of this model was that it simplifies the problem by dividing it into three distinct components: 1) the input signal (associated with the action potential), 2) the impulse response (linked to the system formed by the fiber and the recording electrode), and 3) the output signal (the extracellular potential). Another key aspect of the present approach was that the input signal was represented by a sequence of electric dipoles, which allowed students to visualize the individual contribution of each dipole to the resulting extracellular potential. The results of the survey indicate that the combination of basic principles of electrical fields and intuitive graphical representations largely improves students' understanding of Bioelectricity concepts and enhances their motivation to complete their studies of biomedical engineering.

Perceptual learning theory suggests that perceptual grouping in mathematical expressions can direct students' attention toward specific parts of problems, thus impacting their mathematical reasoning. Using in-lab eye tracking and a sample of 85 undergraduates from a STEM-focused university, we investigated how higher-order operator position (HOO; i.e., multiplication/division operators and the presence of superfluous brackets impacted students' time to first fixation to the HOO, response time, and percent of correct responses). Students solved order-of-operations problems presented in six ways (3 HOO positions × presence of brackets). We found that HOO position and presence of superfluous brackets had separate and combined impacts on calculating arithmetic expressions. Superfluous brackets most influenced undergraduates' performance when higher-order operators were located in the center of mathematical expressions. Implications for learning and future directions are discussed about observing eye movements and gaining insights into students' processes when solving arithmetic expressions.

This article proposes an in-depth mathematics analysis of simple and compound interest through an exploration of the mathematical structures underlying these concepts. Financial numeracy concepts (simple and compound interest, interest rates, interest period, present and future value, etc.) have been added to the mathematics curriculum of the Canadian province of Quebec in 2016. Yet, little is known with regards to mathematics teachers' knowledge of financial concepts. We mobilized the relational paradigm framework in the context of simple and compound interest to construct an assessment instrument of teacher knowledge. Based on 36 teachers' responses to the questionnaire, we observed that the majority of teachers did not use mathematical structures in problem solving; they were not able to make sense of simple and compound interest in ways that do not involve standard formulas. Such results indicate opportunities for professional development in regard to secondary mathematics teachers' knowledge of mathematics in the context of simple and compound interest situations.

Functions and formulas in spreadsheets provide an instructional opportunity to help students build their skills in identifying errors in their own work and identifying a path to go about correcting the errors. This paper provides examples of functions and equations used to create two approaches to calculating the solution to one problem, such as a weighted average calculation or a loan payment calculation. Solving the problem two ways gives students the opportunity to see why both approaches work and to reconcile them if errors cause the results to differ. Using a guided errors pedagogy, instructors can also create an error in one computational approach so that students then work to reconcile the two. This method helps build problem-solving skills and confidence in a process for persisting to resolve errors. Learning from failure and correcting errors in one's work can contribute to perceptions of self-efficacy.

The COVID-19 pandemic has forced the rapid adoption of remote teaching modalities including "hyflex" where students attend some class sessions in person and some online. Managing the hyflex course requires faculty to quickly generate several reports and to update these reports rapidly when the authorities adjust the rules, students add/drop, or the number of course sessions is changed. The creation of these reports is tedious and error-prone, so they need to be automated. However, the nature of the task precludes traditional Excel programming approaches. We use Excel's new "dynamic array" capabilities (which are available only in Excel 365) to automate the creation and updating of the reports needed to manage the hyflex course. We show how to program the reports, and we discuss the importance of taking an iterative approach to creating effective, error-free cell formulas. To help the reader acquire genuine access to the dynamic array functions, we provide a practical tutorial on the principles and new concepts of dynamic arrays, explain how they relate to legacy array functions, and present selected dynamic array functions including SEQUENCE, FILTER, and XLOOKUP.

In this pedagogical action research study, we, as post-secondary mathematics teacher educators, built on an existing effort to improve pre-service teachers' mathematical vocabulary understandings by intentionally addressing their struggles related to polygonal area formulas. Utilizing cognitive load theory and Bruner's levels of developmental learning, we adapted and refined an existing "Area of Polygons" lesson to eliminate extraneous elements and scaffold the introduction of essential elements in the context of a cognitively engaging activity. Comparing our resulting lesson components to existing literature on polygonal area, we found two main approaches towards exploring area of polygons. Both approaches emphasized conservation of polygonal area with one focused on the details of attributes and square units and the other focused on comparisons of areas of figures. We discuss the implications of these approaches and the use of cognitive load theory in tandem with Bruner's levels for future curriculum redesign.

The benefits of technological and theoretical scaffolding were observed when pre-service teachers aiming to teach upper elementary grades were given three learning-based geometrical inquiry tasks involving inscribed circles. They were asked to collaboratively examine the accompanying geometrical illustration and data for some new or interesting feature and then propose a hypothesis resulting from their observations and prove them. Due to the difficulty generally involved in proposing and proving geometrical hypotheses, two forms of scaffolding were provided: theoretical scaffolding based on revising previous learning or specific attributes of the given data and technological scaffolding in the form of specifically designed GeoGebra applets that allowed dynamic observation of the attributes of the geometrical shapes and the changes they underwent during modification. We found that the two forms of scaffolding led to relatively pre-service teachers' high levels of success. They exhibited high levels of interest and participation, were engaged in the tasks, and underwent high-quality learning processes. In follow-up interviews, they confirmed that the exercise improved their inquiry skills, and developed their pedagogical and technological knowledge.

Studies that address preservice teachers' knowledge of area measurement emphasize their lack of knowledge and their tendency towards the use of formulas, without offering a body of knowledge that helps to address such difficulties. This study offers an approximation of the mathematical knowledge necessary for preservice teachers to solve area tasks. For this, the Mathematics Teacher's Specialized Knowledge model is used with emphasis on the subdomain of Knowledge of Topics and Knowledge of the Structure of Mathematics. Preservice teachers' resolutions and written justifications are analyzed using qualitative and quantitative tools. The results indicate that those resolutions that manage to mobilize mathematical knowledge are associated with the joint mobilization of different procedures, properties, and geometric principles. Results also indicate that the strategic coordination between different registers of representation allows Preservice Teachers to mobilize categories of specialized knowledge and establishing connections with other mathematical contents.

The study of loops and spaces in mathematics has been the subject of much interest among researchers. In Part 1 of "The Theory on Loops and Spaces," published in the "Mathematics Teaching Research Journal," introduced the concept and the basic underlying idea of this theory. This article continues the exploration of this topic and aims to advance the understanding of the theory through observation and analysis of patterns. A systematic examination of intersection points and their numerical Sum is conducted, and the effect of the order of numbering on the analysis is analyzed. Furthermore, the physical implications of the theory are discussed, and the validity of the theory in the third dimension is confirmed through analysis. This article provides a solid foundation for the understanding of the elementary principles of Graph Theory and paves the way for the development of more advanced theorems in the field. Additionally, the article demonstrates how patterns in nature can be analyzed and expressed mathematically, offering a unique perspective on the interplay between mathematics and the natural world. The work is inspired by the video posted by mathematician Dr. James Tanton on his YouTube channel on September 27th, 2021. [For "The Theory on Loops and Spaces. Part 1," see EJ1350656.]

Proportional reasoning has been greatly influencing the development of students' mathematical abilities. Along with the area conservation ability, it helps elementary students comprehend area measurement. This exploratory study aimed to produce qualitative-descriptive data on elementary students' proportional reasoning in solving the conservation of plane figures. The study used guided-unguided area conservation problems using a proportional reasoning level as the analysis framework. Data were collected from 4 primary school students in Sidoarjo, Indonesia, who were in fifth-grade class. The students' strategies used were identified to analyze the students' proportional reasoning in solving area conservation. Results show that the level of proportional reasoning varies from zero to two. Regarding the students' proportional reasoning levels, most of the students' strategies use visual clues and cute paste strategies. Only one student can reach the level of quantitative reasoning by using a formula to compare both area measurements. Interestingly, the problem of the conservation of plane figures failed to reveal students' formal proportional reasoning due to their insufficient knowledge of fractions, division, multiplication, and decimals. Some implications regarding students' proportional reasoning and interventions in the area conservation problem are discussed.

The purpose of the study was to develop mathematical instructional card game titled "algebrameter" and determine its efficacy in remediating junior secondary school (JSS) students' learning difficulties in algebra and geometry. The study employed research and development design. A sample of 120 JSS II students drawn from a population of 4800 JSS II students in Nsukka Education Zone, using multi-stage sampling procedure participated in the study. Algebra and geometry achievement test (AGAT) comprised 30 multiple choice questions developed by the researchers was used for data collection. The internal consistence of AGAT was determined using Kuder-Richardson formula 20 (KR-20) formula, which yielded a value of 0.82. Research questions were answered using mean and standard deviation while the hypothesis was tested using analysis of covariance. The results of this study revealed that students who were exposed to "algebrameter" significantly performed better than students who were not exposed to "algebrameter" package.

Various features have been found hidden in the Pascal triangle. In this paper, some very well-known properties of the Pascal triangle will be presented, as well as the properties related to different extensions of the triangle, namely the Pascal pyramid. Given that in the textbooks of the tenth grade, respectively in the school, where we realised the research but also in general in other schools, the importance of the Pascal triangle is not at the right level, then in this paper it has been shown very well that many different exercises in mathematics. The purpose of this paper is to look at the difference and importance of explaining mathematical units against units that students do not have knowledge of, namely the explanation of Pascal's triangle in the efficiency of solutions to various mathematical exercises. This research is mainly based on descriptive and quantitative method, while research instruments are two tests. From the study of Pascal's triangle, many solutions of problems in mathematics emerged through this triangle, starting from the binomial formula, the extension of the binomial formula, and the combinatorics as well as the probability. So the students realized that Pascal's triangle enables the solution of all these exercises in an easier and more understandable way and this also came from the results of two tests. Also, the students were open and motivated for the idea of ??using Pascal's triangle for other exercises but what now remains for them to find other possible solutions to the exercises through Pascal's triangle.

Although positive effects of lesson study on teachers learning are reported, only some studies have investigated teacher noticing as an analytical tool for supporting teachers with an explicit focus, as in LS, and more empirical evidence is needed. This qualitative interpretive case study design aims to investigate the noticing processes of a group of mathematics teachers conducting a lesson study cycle focused on teaching algebraic expressions using manipulatives in middle school. Data is collected through the audio recordings of the participants' lesson study meetings. Participants were six elementary mathematics teachers who attended a graduate course selected based on voluntariness. This study aims to incorporate Lee and Choy's (2019) teacher noticing framework with Sfard's (2008) commognitive theory, which views learning "as changes in discourse" and noticing as a "discourse structure" covering observation, interpretation, and reasoning processes (van Es, 2011). Results showed that teachers focused more on aspects of students' learning than issues of their instructional practice. However, their noticing was mostly related to future decisions and actions regarding issues of teaching methods and sequencing of the lesson, whereas teachers' dominant noticing form related to students learning was interpretive. Results illustrate the applicability of these noticing frameworks as an analytic tool where noticing is conceptualized as a pedagogical discourse for the analysis of a lesson study review discussion by a group of mathematics teachers who focus on teaching algebraic expressions.

An onto-semiotic analysis of the mathematical connections established by one in-service mathematics teachers and university students when solving a problem about launching a projectile using the derivative was carried out. Theoretically, this research was based on the articulation between the Extended Theory of Mathematical Connections and the Onto-semiotic Approach. The methodology was qualitative-descriptive where data was collected through interviews based on a task. Subsequently, following the joint analysis method of both theories, the mathematical activity of the participants when they solved the task was analyzed. The results show that, teacher and students established a system of connections of feature type, different representations, meanings, part-whole, procedural and implications in terms of practices, processes, objects, and semiotic functions that relate them. However, some students presented difficulties caused by some incorrect mathematical connection such as stating that the maximum height of the projectile is the time obtained with the critical number, errors in performing arithmetic calculations when evaluating the function, graphically representing the quadratic function as a straight line and use the general formula in an inappropriate way that prevents the procedural connection from being made.

The present study investigates the effect of the separator lines on the learning of 8th grade students in simplifying algebraic expressions with parenthesis. An experimental study was designed to achieve this goal involving 60 girl students in 8th Grade (13 and 14 years old) randomly selected and assigned to two experimental and control groups. After taking the pre-test, both groups were taught by one teacher. The control group was led as usual and the experimental group was taught by using the separator lines. As a result of the covid-19 disease, students were taught online using WhatsApp. In the end, a post-test was carried out for both groups. Data was also collected using WhatsApp. For the analysis of the data, a Covariance test was conducted. The results showed positive effects of separator lines on the student's performance, as well as reducing their mistakes when working with parenthesis.

This study used the RBC+C model to reveal the abstractions of the 6th-grade students in the process of transition to the parallelogram area formula. Also, constructing parallelogram area information was employed as a teaching experiment based on the basic interpretive approach, one of the qualitative research methods. The study participants comprised 9 volunteer sixth-grade students with high, medium and low mathematics success levels in a public school in Istanbul in the 2021-2022 academic year. Four activities prepared by the researcher on triangular, rectangular and parallelograms were used as data collection tools. In the study, activities were recorded on video and then transcribed. As a result of the research, it is seen that students with a high and medium success level in the recognizing and building phases did not have difficulty finding the quadrilateral and triangular areas and guided the students with a low success level. Moreover, it was found that students constructed parallelogram area information. Still, when asked about the area formula, they had difficulty expressing their operations even though they correctly found the area. This is because students process by rote and cannot explain mathematical information logically. In line with these results, it is considered necessary and recommended to organize teaching activities that will allow students to learn meaningfully.

Subjects with initially extreme observations upon remeasurement are found closer to the population mean. This tendency of observations toward the mean is called regression to the mean (RTM) and can make natural variation in repeated data look like real change. Studies, where subjects are selected on a baseline criterion, should be guarded against the RTM effect to avoid erroneous conclusions. In an intervention study, the difference between pre-post variables is the combined effect of intervention/treatment and RTM. Thus, accounting for RTM is essential to accurately estimate the intervention effect. Many real-life examples are better modeled by a bivariate binomial model with varying probability of success. In this article, a bivariate beta-binomial distribution is used that allows the probability of success to vary from subject to subject. Expressions for the total, RTM, and treatment effect are derived, and their behavior is demonstrated graphically. Maximum likelihood estimators of RTM are derived, and their statistical properties are studied via Monte Carlo simulation. The proposed techniques are employed to estimate the RTM effect by utilizing data related to the Countway WM-class circulation.

One expected outcome of physics instruction is for students to be capable of relating physical concepts to multiple mathematical representations. In quantum mechanics (QM), students are asked to work across multiple symbolic notations, including some they have not previously encountered. To investigate student understanding of the relationships between expressions used in these various notations, a survey was developed and distributed to students at six different institutions. All of the courses studied were structured as "spins-first," in which the course begins with spin-1/2 systems and Dirac notation before transitioning to include continuous systems and wave function notation. Network analysis techniques such as community detection methods were used to investigate conceptual connections between commonly used expressions in upper-division QM courses. Our findings suggest that, for spins-first students, Dirac bras and kets share a stronger identity with vectorlike concepts than are associated with quantum state or wave function concepts. This work represents a novel way of using well-developed network analysis techniques and suggests such techniques could be used for other purposes as well.

In the context of college physical education curriculum reform, fostering students' interest and promoting lifelong physical exercise have become crucial. Aerobics, an integral component of physical education, plays a key role in achieving these objectives. However, existing data flow analysis technologies lack integration, limiting their ability to leverage information from various sources. To address this issue, this paper proposes an aerobics teaching model utilizing few-shot learning technology for data flow analysis. The model incorporates a label feature network based on metric learning, enhancing its ability to analyze multi-scale features and label features within classes. Comparative analysis demonstrates an 8.12% improvement in accuracy compared to traditional image feature combined classifier models.

While research on the opportunity to learn about mathematics concepts provided by textbooks at the secondary level is well documented, there is still a paucity of similar research at the undergraduate level. Contributing towards addressing this knowledge gap, the present study examined opportunities to engage in quantitative and covariational reasoning, in the context of the derivative concept, provided by two textbooks commonly used in the teaching of calculus in the United States-a regular calculus textbook and an applied calculus textbook. Analysis of expository sections, examples, and exercises related to the derivative provided in these textbooks revealed three main results. First, the applied calculus textbook provides plenty of opportunities to engage in quantitative reasoning. Similar opportunities are limited in the regular calculus textbook. Second, opportunities to engage in covariational reasoning are not only minimal in both textbooks, but also limited to low levels of covariational reasoning, namely coordination, direction, and quantitative coordination. Third, the applied calculus textbook consistently defines ordinary and partial derivatives as rates of change while the regular calculus textbook consistently defines these concepts as limits of a difference quotient. Findings of this research have implications for several stakeholders, including calculus textbook authors and calculus instructors.

The present study seeks to explore the mathematical connections that 13-14-year-old secondary school students establish when solving area tasks. Emphasis is placed on different mathematical objects, and the connections between them, that allow students to successfully solve the tasks. The study follows a mixed methodology using qualitative and quantitative method of analysis. The results show that representations play a key role in solving area tasks, as they condition the use of alternative procedures to the use of formulas. Likewise, the properties involved in area measurement processes may condition the use of geometric procedures, such as surface decomposition and reorganisation. Finally, results show that to accurately carry out area measurement processes, it is necessary to bring different mathematical objects into play simultaneously. If these connections between mathematical objects do not occur, there is a risk of using the formulas in a mechanical way.

The data we encounter in real life often contain missing values. In sampling methods, missing value imputation is done with different methods. This article proposes novel logarithmic type imputation methods for estimating the population mean in the presence of missing data under ranked set sampling (RSS). According to the determined theoretical results, the proposed imputation methods are found to be the most efficient in comparison to popularly known imputation methods like mean imputation, Al-Omari and Bouza (2014) imputation methods, Sohail et al. (2018) imputation methods, and Bhushan and Pandey (2016) type imputation methods utilizing RSS. Apart from this, a simulation study has been accomplished utilizing artificially drawn symmetric and asymmetric populations. The outcomes are encountered to be rather satisfactory, showing improvement over all existing imputation methods.

This article introduces an innovative sampling scheme, the median sampling (MS), utilizing individual observations over time to efficiently estimate the mean of a process characterized by a symmetric (non-uniform) probability distribution. The mean estimator based on MS is not only unbiased but also boasts enhanced precision compared to its simple random sampling-based counterpart. Moreover, a new EWMA chart based on the mean estimator within the MS scheme is proposed. The performance of the EWMA charts, derived from both simple random sampling (SRS) and MS schemes, is evaluated using the metrics of steady-state average run-length and average number of items-to-signal. The findings underscore the superiority of the EWMA-MS chart over the EWMA-SRS chart. Additionally, as the magnitude of ranking errors escalates, the behavior of the EWMA-MS chart converges toward that of the EWMA-SRS chart. The practical implementation of the newly introduced EWMA chart is demonstrated through an illustrative example.

Multisite cluster randomized trials (MCRTs), in which, the intermediate-level clusters (e.g., classrooms) are randomly assigned to the treatment or control condition within each site (e.g., school), are among the most commonly used experimental designs across a broad range of disciplines. MCRTs often align with the theory that programs are delivered at a cluster-level (e.g., teacher professional development) and provide opportunities to explore treatment effect heterogeneity across sites. In designing experimental studies, a critical step is the statistical power analysis and sample size determination. However, the statistical tools for power analysis of moderator effects in three-level MCRTs are not available. In this study, we derived formulas for calculating the statistical power and the minimum detectable effect size difference (MDESD) with confidence intervals for investigating the effects of various moderators in three-level MCRTs. We considered the levels of the moderators (level-1, -2, and -3), the scales of the moderators (binary and continuous), and random and nonrandomly varying slopes of the (moderated) treatment effects. We validated our formulas through Monte Carlo simulations. Finally, we conclude with directions for future work.

QR decomposition is widely used for solving the least squares problem. However, existing materials about it may be too abstract for non-mathematicians, especially STEM students, and/or require serious background in linear algebra. The paper describes theoretical background and examples of GNU Octave compatible MATLAB scripts that give relatively simple but complete explanations about how to use QR decomposition for the least squares problem solution. Only basic knowledge of linear algebra and calculus are required. Both Givens rotations and Householder reflections usage for the linear least squares problem were considered. It was shown that the algorithm based on Givens rotations is even easier to program than explicit formation of the normal equations with subsequent usage of Gaussian elimination.

The most famous equation in physics, E = mc[superscript 2], is rarely introduced in middle school physics curricula. Recent research has shown that teaching Einsteinian concepts at the middle school level is feasible and beneficial. This paper analyses an Einsteinian energy teaching module for Year 8 students (13-14 years old), which encompasses the two fundamental energy formulas in modern physics, E = mc[superscript 2] and E = hf. In the context of activity-based learning, the Einsteinian energy module relates to all the forms of energy in traditional school curricula. This study uses a design-based research approach within the Model of Educational Reconstruction framework. Modern experiments, historical events, and educational research helped us identify relevant Einsteinian energy concepts, activities, and assessments. The study included 22 students who participated in nine in-class Einsteinian energy lessons. Analysing results in the post-test showed a 31% mean increase from the pre-test, a clear and significant positive change in students' conceptual understanding. The results demonstrated students' ability to deal with very large and small constants of proportionality and physical concepts involved in the module.

Comprehensive evaluation of treatment effects is aided by considerations for moderated effects. In educational research, the combination of natural hierarchical structures and prevalence of group-administered or shared facilitator treatments often produces three-level partially nested data structures. Literature details planning strategies for a variety of experimental designs when moderation effects are of interest but has yet to establish power formulas for detecting moderation effects in three-level partially nested designs. To address this gap, we derive and assess the accuracy of power formulas for detecting the different types of moderation effects possible in these designs. Using Monte Carlo simulation studies, we probe power rates and adequate sample sizes for detecting the different moderation effects while varying common influential factors including variance in the outcome explained by covariates, magnitude of the moderation effect, and sample sizes. The power formulas developed improve the planning of experimental studies with partial nesting and encourage the inclusion of moderator variables to capture for whom and under what conditions a treatment is effective. Educational researchers also have some initial guidance regarding adequate sample sizes and the factors that influence detecting moderation effects in three-level partially nested designs.

This study builds on the simulation framework of a recent paper by Stanley and Doucouliagos ("Research Synthesis Methods" 2023;14;515--519). S&D use simulations to make the argument that meta-analyses using partial correlation coefficients (PCCs) should employ a "suboptimal" estimator of the PCC standard error when constructing weights for fixed effect and random effects estimation. We address concerns that their simulations and subsequent recommendation may give meta-analysts a misleading impression. While the estimator they promote dominates the "correct" formula in their Monte Carlo framework, there are other estimators that perform even better. We conclude that more research is needed before best practice recommendations can be made for meta-analyses with PCCs.

In this article, the authors share their favorite "Construct It!" activity, which focuses on rate of change and functions. The initial approach to instruction was procedural in nature and focused on making use of formulas. Specifically, after modeling how to find the slope of the line given two points and use it to solve for the y-intercept, the authors gave students time to practice. They recognized that their approach lacked an opportunity for students to engage in reasoning and sense making. According to Lithner (2008), imitative reasoning (i.e., a focus on memorizing) is ubiquitous in mathematics classrooms in the United States, whereas creative reasoning (i.e., reasoning that results in knowledge creation) is rare. To help students generalize the process of identifying a linear relationship that holds for two ordered pairs, explain the connection to the slope formula, and engage in creative reasoning, the authors designed the activity they call Turn A Into B. This activity focuses on reasoning about the relationship between two quantities and reflecting on how this is related to rate of change and functions.

This study explores the usefulness of covariates on equating test scores from nonequivalent test groups. The covariates are captured by an estimated propensity score, which is used as a proxy for latent ability to balance the test groups. The objective is to assess the sensitivity of the equated scores to various misspecifications in the propensity score model. The study assumes a parametric form of the propensity score and evaluates the effects of various misspecification scenarios on equating error. The results, based on both simulated and real testing data, show that (1) omitting an important covariate leads to biased estimates of the equated scores, (2) misspecifying a nonlinear relationship between the covariates and test scores increases the equating standard error in the tails of the score distributions, and (3) the equating estimators are robust against omitting a second-order term as well as using an incorrect link function in the propensity score estimation model. The findings demonstrate that auxiliary information is beneficial for test score equating in complex settings. However, it also sheds light on the challenge of making fair comparisons between nonequivalent test groups in the absence of common items. The study identifies scenarios, where equating performance is acceptable and problematic, provides practical guidelines, and identifies areas for further investigation.

Cognitive diagnosis models (CDMs) are the statistical framework for cognitive diagnostic assessment in education and psychology. They generally assume that subjects' latent attributes are dichotomous--mastery or nonmastery, which seems quite deterministic. As an alternative to dichotomous attribute mastery, attention is drawn to the use of a continuous attribute mastery format in recent literature. To obtain subjects' finer-grained attribute mastery for more precise diagnosis and guidance, an equivalent but more explicit form of the partial-mastery-deterministic inputs, noisy "and" gate (DINA) model (termed continuous attribute profile [CAP]-DINA form) is proposed in this article. Its parameters estimation algorithm based on this form using Bayesian techniques with Markov chain Monte Carlo algorithm is also presented. Two simulation studies are conducted then to explore its parameter recovery and model misspecification, and the results demonstrate that the CAP-DINA form performs robustly with satisfactory efficiency in these two aspects. A real data study of the English test also indicates it has a better model fit than DINA.

Background: The ability to translate concrete manipulatives into abstract mathematical formulas can aid in the solving of mathematical word problems among students, and metacognitive prompts play a significant role in enhancing this process. Aims: Based on the concept of semantic congruence, we explored the effects of metacognitive prompts and numerical ordinality on information searching and cognitive processing, throughout the process of solving mathematical word problems among primary school students in China. Sample: Participants included 73 primary school students (38 boys and 35 girls) with normal or corrected visual acuity. Methods: This study was based on a 2 (prompt information: no-prompt, metacognitive-prompt) × 2 (number attribute: cardinal number, ordinal number) mixed experimental design. We analysed multiple eye-movement indices, such as fixation duration, saccadic amplitude, and pupil size, since they pertained to the areas of interest. Results: When solving both types of problems, pupil sizes were significantly smaller under the metacognitive-prompt condition compared with the no-prompt condition, and shorter dwell time for specific sentences, conditional on metacognitive prompts, indicated the optimization of the presented algorithm. Additionally, the levels of fixation durations and saccadic amplitudes were significantly higher when solving ordinal number word problems compared with solving ordinal number problems, indicating that primary school students were less efficient in reading and faced increased levels of difficulty when solving ordinal number problems. Conclusions: The results indicate that for Chinese upper-grade primary school students, cognitive load was lower in the metacognitive prompting condition and when solving cardinal problems, and higher when solving ordinal problems.

Permutations and combinations are generally taught by requiring students to memorize formulas and solve problems using the appropriate formula. Students who learn these topics may succeed in gaining high scores on end-of-chapter exams in textbooks, while lacking the conceptual understanding required to deal with problems in the real world. Therefore, this study aimed to examine in-service mathematics teachers' pedagogical content knowledge (PCK) to determine students' mistakes in solving permutations and combinations problem and their teaching strategies to eliminate these errors. Data were collected by distributing vignettes, CoRe, and PaP-eRs to thirteen mathematics teachers from ten provinces in Indonesia after they finished an online professional teacher education program to determine their PCK in teaching permutations and combinations. The data collected were analyzed qualitatively using a content analysis approach to obtain categories inductively. The result showed that PCK of in-service mathematics in teaching permutations and combinations was observed by identifying student mistakes conceptually and procedurally, even though some could not determine their mistakes in permutations. On the other hand, the knowledge of instructional strategies can engage all students in active learning, but most of them only give general answers. Furthermore, an in-depth understanding of permutations and combinations topic is needed to support the development of teachers' pedagogic competencies sustainably. The contribution of this research will be of interest to curriculum development and mathematics educators.

The MY MAMOVONO technique was created as an innovative way to improve pre-university students' ability to master the mole concept. The Malaysian pre-university students mentioned that they were confused by the formulas they needed to remember. The technique required them to build a card that contained the four (4) formulas of the mole concept using their creativity. Consequently, the students performed better postintervention than in the pre-intervention phase. The paired sample t-test analysis revealed that the mean difference (24.44) between the students' post-intervention mean (82.78; SD = 10.46) and pre-intervention mean (58.33; SD = 7.48) is significant with p < 0.001. Moreover, the target students' post-intervention performance was significantly better than the minimum distinction mark -- the difference between the post-intervention (82.78) and minimum distinction mark (70) is 12.78 with p < 0.001. The MY MAMOVONO technique positively impacted the students' level of understanding.