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A computer science camp for pre-collegiate students was operated during the summers of 2022 and 2023. The effect the camp had on attitudes was quantitatively assessed using a survey instrument. However, enrollment at the summer camp was small, which meant the well-known Pearson's Chi-Squared to measure the significance of results was not applied. Thus, a quantitative analysis method using a multinomial probability distribution as a model of a multilevel Likert scale survey was used. Exact calculations of a multinomial probability model with likelihood ratio were performed to quantitatively analyze the results of questionnaires administered to participants in two cohort groups (combined N=17). Probabilities per Likert categories were determined from the data itself using Bayes theorem with a Dirichlet prior. Each cohort functioned as part of a homogenous sample, thus allowing cohorts to be pooled. Post-test revealed significant changes in participants' attitudes after camp completion. Using this technique has implications for studies with small sample sizes. Using exact calculation of the multinomial probability model with the use of likelihood ratio as a statistical test of evidence has advantages: a) it is an exact value that can be used on any size sample, although it offers a quantitative analysis option for small sample size studies; b) depends only on what was observed during a study; c) does not require advanced calculation; d) modern spreadsheet and statistical package programs can calculate the analysis; and e) likelihood ratio employed in Bayes theorem can update prior beliefs according to evidence. Utilizing small sample size quantitative analysis can strengthen insights into data trends and showcase the importance of this quantitative technique.

Suicide, which is defined as an individual's intentionally attempting to end his or her life, is considered an important public health problem. In this study, it was aimed to examine the relationship between cognitive distortions and suicide probability in university students who are in age groups at risk for suicide, and to examine the mediating roles of the attachment dimensions of anxiety and avoidance, in this relationship. The study group consisted of 441 university students. In the study, Suicide Probability Scale, Experiences in Close Relationships Inventory, Cognitive Distortions in Relationships Scale and a personal information form were used as data collection tools. In accordance with the purpose of the study, a correlation analysis between variables and regression analyses were applied to examine the mediating roles of avoidance and anxiety in the relationship between cognitive distortions and suicide probability. The findings showed that there was a moderate positive correlational relationship between suicide probability scores and avoidance scores and there was a moderate positive correlational relationship between suicide probability scores and cognitive distortion scores. It was shown that there were low and moderate positive correlations between cognitive distortion scores, and avoidance and anxiety scores. The results of the mediation analyses showed that anxiety and avoidance had a partial mediating role in the relationship between cognitive distortion and suicide probability. The results were discussed in light of the literature.

Grade repetition is an issue with important educational implications. This study analyzes the effect of cognitive variables, socio-economic status, as well as the interaction of socioeconomic status with ethnic origin, on the probability of not repeating. Data were collected to calculate socioeconomic status using the Hollingshead Socioeconomic Index, and the following tests were administered to collect information on cognitive variables: the Primary Mental Aptitude Test (PMA), the Raven's Progressive Matrices Test, and the Wechsler Intelligence for Children Information Scale (WISC IV). The sample consisted of 664 students of Compulsory Secondary Education (ESO), Baccalaureate and intermediate and higher vocational training. The results confirm the predictive power of socioeconomic status. With regard to cognitive variables, significant results were only obtained when the WISC-IV Information Scale was taken as a predictor, this measure being related to crystallized intelligence. The influence of the origin of the students did not yield significant results. There is a clear need for a debate and a rethinking of the issue leading to the identification of new strategies to reduce grade repetition and its negative effects.

This research aims to determine the thinking activity types dominated by a mental process in producing answers characterized by automatic, unconscious, and subjective-empirical processes (system 1) in solving problems so that the default-interventionist interaction occurs. This research novelty is the formulation of the contents and thinking activity arrangement adapted to students' thinking when solving problems. The problem used in this research is a mathematical problem that triggers students to produce answers quickly with confidence that the answers are correct at a high level. Another problem is about probability because the mode of occurrence of students' learning difficulties at the secondary school level occurs when learning the concept of probability. This is qualitative research with a case study approach. The research subjects were students of Mathematics Education in semester 1. The results showed that thinking activity one could condition the occurrence of type 1 default-interventionist interaction. Thinking activity two could condition the occurrence of type 2 default-interventionist interaction. Thinking activity three could condition the occurrence of type 3 default-interventionist interaction. This research concluded that the default-interventionist interaction occurred because the content and arrangement of the thinking activity conditioned the subjects to pay attention to information gradually and change the subjects' beliefs. Lecturers were recommended to produce, develop, and research thinking activities on topics other than probability at various levels of education. The default-interventionist interaction was essential to be conditioned when system one dominated students' thinking, causing difficulties.

This study aims to determine students learning obstacles in probability material based on their probabilistic thinking (PT) level using the theory of didactical situation (TDS) perspective. This is qualitative research with the case study method. The subject consisted of 23 grade 9 students in junior high school who had studied the material and had taken the test. The test results were used to classify students depending on their PT level using the framework developed by Graha A. Jones. Furthermore, interviews were performed with three representative students from each PT level. The interviews indicated that students in each PT level continue to face learning obstacles, which include instrumental ontogenic and epistemological obstacles. The lowest PT level demonstrated a more complex and comprehensive learning obstacle in all constructs of PT. From the TDS perspective, all students who are at various levels of PT have reached an action situation. Only 33% of the students at the subjective level and 83% of the students at the transition level reached the formulation situation. Still, students at that level needed help to reach the other two situations. Unlike the two previous levels, students at the level of quantitative informal thinking can achieve situations of validation and institutionalization.

Education plays a very important role in improving the quality of human resources. In higher education, students need learning that is innovative and not monotonous so that learning goals can be achieved. In this research, what will be discussed further is the development of learning methods by combining 2 methods, namely Cooperative Based Learning (CBL) with Problem-Based Learning (PBL) in the Advanced Financial Management course. This research method uses Pretest - Posttest Design. The data population is 711 students collected from 17 classes taking Advanced Financial Management courses at Putra Indonesia University YPTK Padang. The sampling technique uses probability sampling. The data that will be used as a sample are observations of 117 students collected from 3 classes. The CPBL method is implemented by grouping students into groups of 4-5 people to discuss each case in the material. Data will be analyzed based on the completeness of learning outcomes and calculating N-Gain. From the data processing, the results obtained were that by combining these 2 methods, it was successful in improving student learning outcomes and the students' completeness test scores were also proven to be very successful in being upgraded. The effectiveness of the collaboration between these 2 methods reaches the high category of the N-Gain test. This is a new variation in making learning calculations that use many formulas more challenging, increasing students' enthusiasm for learning and providing satisfactory learning results. This method is worth utilizing and can be used in all courses that involve calculations and case questions.

Mathematical symbols, such as those embodying quantum concepts, are indispensable for conveying complex ideas and relationships in academic writing. However, some education researchers and students keep a distance from anything mathematical: algebraic equations, geometrical reasoning, or statistical symbols. How to lower the access threshold for this type of mathematical narrative and reveal the meanings of a range of quantum conceptions to modern educators thus becomes a real problem. Using the pendulum motion equation as a reference point, I argue in this article for the advantages of academic English or French writing genres that fuse a range of mathematical symbols of quantum concepts and conceptual change. Such writings help demonstrate how incorporating the idea of probability (a) refines the debate among conceptual, verbal, and mathematical academic writing; (b) allows new conceptions that draw on the insights from quantum cognition-supported theories; (c) helps explain students' understanding of mathematical symbols; and (d) offers a new taxonomy for categorizing academic writings.

Teaching and learning undergraduate statistics has been a most challenging task for undergraduate psychology majors (Salkind, 2017). A seasoned statistics instructor consulted with a seasoned instructional designer on a method to improve a particularly demanding course using a performance improvement approach to address learning difficulties she had noted in previous semesters. The Decoding the Disciplines methodology identified the most challenging concepts and provided a methodology to improve student learning performance. The methodology focused on five core concepts in psychological statistics: probability, variability, central limit theorem, independent/dependent variables, and degrees of freedom. The Decoding the Disciplines curriculum was used for three semesters. In these three semesters, performance was compared pre and post on these concepts within the semester. Repeated measures t-tests found a significant change in the percentage of correct answers between the pretest and the final exam on the five core concepts.

Background/purpose: Awareness of the mathematical skills and knowledge children possess in their early years is widely accepted. This includes various common positive aspects, not only for educators but also for researchers and policymakers. This study presents a systematic review conducted to meticulously identify empirical studies published in the Scopus-Index Journal database about the mathematical skills children aged 3 to 8 years old have mastered. Materials/methods: This review followed the PRISMA guidelines and the research database comprised of Scopus-indexed journals. The technique followed used "keywords" and Boolean operators. The screening processes included reviewing abstracts, scanning complete texts of published articles, and rejecting those not meeting preset inclusion criteria. Moreover, systematic reviews, meta-analyses, and papers not written in English were also excluded. Of the 801 studies initially identified, a total of 15 empirical studies were included in the systematic review. Results: Children master various math skills from a very young age, mainly in "numbers and operations", but face difficulties in skills related to "algebra" as well as "geometry and measurement". Additionally, several preschoolers' characteristics help to explain the acquisition of these skills, with "age" being the primary factor. Researchers use various research instruments and mainly conduct individual semi-structured interviews. Children's geometry skills and knowledge appear to have been studied to a small extent. The areas of "measurement" and "data analysis and probability" were found to be under active investigation. Conclusion: It is worth noting that not only does a noticeable research gap exist for the math domains of "measurement," "geometry," and "data analysis and probability". Factors that seem to affect young children's math skills, such as "gender", "parents' educational level", and "attendance to a preparatory preschool" need further investigation. The implications of the current study's results extend beyond academia, providing valuable insights that educators and policymakers can leverage to enhance the quality of mathematics education during the early years period.

Faced the vast amount of information, choosing the appropriate materials is a prerequisite for effective self-directed learning. The recommendation algorithm is a kind of intelligent technology that can accurately locate the required information which the users care about most. However, many recommendation techniques experience can not be trained adequately in scenarios with small sample data and extremely sparse ratings. Moreover, DLRAs (Deep learning based Recommendation Algorithms) require high hardware support. The probabilistic graph (PG) can effectively represent the implicit complex relations among nodes, but it still has the problem of sparse data sensitivity. Therefore, we propose a Matrix-Factorization-based Probabilistic Graph Model for Recommendation Algorithm (MF-PGMRA): By matrix-factorizing the sparse rating matrix, the users and items are mapped to the user/item spaces, respectively; We employ the inner product to data-enhance and overcome the problems of sparse data and cold start; Then, we build Probabilistic Graph to construct the "user-item" latent spaces and estimate the probability distribution based on expectation maximization (EM), so as to predict the ratings; Finally, we built a library management system with the recommendation module to highlight the benefits of MF-PGMRA for students' subject learning. According to a questionnaire, we confirmed that the students are satisfied with the system from four aspects of speed, accuracy, usability and convenience, which can confirm that the library management system based on MF-PGMRA can efficiently and accurately recommend suitable materials for students from the huge amount of learning materials to improve students' self-directed learning efficiency.

The purpose of this study was to investigate the effects of the anonymity of the internet and gender differences in online trolling and cybervictimization. A sample of 151 college students attending a southeastern university completed a survey to assess their internet activities and online trolling and cybervictimization. Multivariate analyses of logistic regression and ordinary least squares regression were used to analyze online trolling and cybervictimization. The results indicated that the anonymity measure was not a significant predictor of online trolling and cybervictimization. Female students were less likely than male students to engage in online trolling, but there was no gender difference in cybervictimization. In addition, the total hours spent on the internet increased the likelihood of the decision of college students to participate in online trolling, but not cybervictimization. Further implications for research related to online trolling and risk factors are discussed.

During the last two years, the COVID-19 pandemic had a secondary effect of increased media content loaded with mathematical, often probabilistic information (and misinformation). Our exploratory study investigates the probabilistic intuitions, misconceptions, biases, and fallacies in conditional probability reasoning of mathematics teacher candidates in the context of the pandemic. The pre-service mathematics teachers who participated in our study were given a questionnaire with five contextual conditional probability problems, all formulated similarly to media statements often encountered when discussing the COVID-19 pandemic. Our findings confirm the previous findings on biases and fallacies related to conditional probability problems with a social context. They were also indicative of several types of errors (both numerical and logical) as more common than expected. Our results also reveal that pre-service mathematics teachers apparently separate the content learned in the classroom from the application of the knowledge in critical examination of the information to which they are daily exposed by the media.

This paper reports on an exploratory study about probabilistic intuition in learning mathematics for decision-making. The analysis was carried out on a group of high school students in relation to their probabilistic intuition in problem-solving, after performing playful learning activities on a simulation platform specifically designed for this study. The analysis used a mixed qualitative-quantitative method. The results showed that the simulation activities facilitated the students' probabilistic intuition and boosted their performance in solving problems, particularly avoiding decisions based on previous results (when unnecessary) and identifying situations, where binomial distribution can be used.

Problem-based learning (PBL) has been widely applied as an alternative to improve learning outcomes, but it is still little studied in the context of the probability theory course. This study described how implementing the PBL model improves students' problem-solving and critical thinking skills in probability theory course and evaluates its quality. This design research involved 58 undergraduate students and two probability theory course lecturers from two universities in Indonesia as participants. Data collection used observation to describe PBL implementation, then questionnaire and pretest-posttest to evaluate the quality of the model. This study produced a PBL model for the probability theory course which is implemented through five steps: i) Orienting students on problems; ii) Organizing students to study; iii) Assisting individual and group investigations; iv) Developing and presenting work or solutions; and v) Analyzing and evaluating problem solving processes. Lecturers and students consider that the implementation of PBL is practical. PBL implementation can also improve students' problem-solving skills and critical thinking in the probability theory course. Thus, implementing PBL can be used as a solution to optimize learning outcomes in the probability theory course.

The aim of this study is to classify the subjects and skills of middle school mathematics course in the context of MATH Taxonomy and to determine their relations. For this purpose, the questions and answers related to the mathematics subtest of a national exam were analyzed over the answers of 20154 students. The study continued with the analysis of the data set containing the answer patterns of the students according to the rules of association of data mining methods by using RapidMiner program and FP-Growth algorithm. The results show that students with any level of skill in MATH taxonomy related to mathematical skills could perform skills in subcategories at a high rate. Students who have attainment in the learning areas provided later in the mathematics program had a high rate of knowledge in the previously provided learning areas. However, this result is not valid for Probability of Simple Events.

Mixed Membership Models (MMMs) are a popular family of latent structure models for complex multivariate data. Instead of forcing each subject to belong to a single cluster, MMMs incorporate a vector of subject-specific weights characterizing partial membership across clusters. With this flexibility come challenges in uniquely identifying, estimating, and interpreting the parameters. In this article, we propose a new class of "Dimension-Grouped" MMMs (Gro-M[superscript 3]s) for multivariate categorical data, which improve parsimony and interpretability. In Gro-M[superscript 3]s, observed variables are partitioned into groups such that the latent membership is constant for variables within a group but can differ across groups. Traditional latent class models are obtained when all variables are in one group, while traditional MMMs are obtained when each variable is in its own group. The new model corresponds to a novel decomposition of probability tensors. Theoretically, we derive transparent identifiability conditions for both the unknown grouping structure and model parameters in general settings. Methodologically, we propose a Bayesian approach for Dirichlet Gro-M[superscript 3]s to inferring the variable grouping structure and estimating model parameters. Simulation results demonstrate good computational performance and empirically confirm the identifiability results. We illustrate the new methodology through applications to a functional disability survey dataset and a personality test dataset.

Clear and rigorous quantum reasoning is needed to explain quantum physical phenomena. As pillars of true quantum physical explanations, we suggest specific quantum reasoning derived from quantum physical key ideas. An experiment is suggested to support such a quantum reasoning, in which a quantized radiation field interacts with an optical beam splitter, leading to experimental results conflicting with classical physical predictions. The results, however, can be explained consistently with a quantum reasoning based on the key ideas of probability, superposition, and interference (PSI). In this quantum optical key experiment the optical beam splitter prepares a superposition of single photon states and a Michelson interferometer is used to detect the superposition via controlled propagation phases. Although different single photon experimental setups (aimed at helping students to gain access to foundational issues in quantum physics) have been discussed in the past, the wave-particle dualism bound to classical physics maintains its predominance as an explanation pattern for the interpretation of these experiments. The study presented here investigates the effect of the quantum optical key experiment on the ability of students to use quantum reasoning based on the key ideas of PSI to overcome the naive wave-particle dualism. The current state of relevant studies that test student access to quantum physics can roughly be divided into two distinct areas: one tests how mathematical abilities help them to understand quantum physics and one tests how nonmathematical representations of a set of specific quantum theoretical traits ("Wesenszüge") lead to a deeper understanding of quantum physics. There is a lack of questionnaires that focus on the idea of developing quantum reasoning based on superposition, probability, and interference of quantum states combined with a real experiment using true quantum light. In the first part of the article, we describe the physical modeling and present the development of the questionnaire. The set of items has been constructed from newly developed items and combined with well-tested ones. The validation of the set addresses qualitative and quantitative methods. In the second part, we give a pre- and poststudy examination of the impact of the quantum optical key experiment on students' quantum reasoning. A significant increase in the number of students using quantum arguments is based on PSI reasoning for the explanation of an interference, such as the behavior of single photon states. Though the increase is significant, we found only minor changes in a particular issue to the students' reasoning when approaching quantum physics as illustrated by a sample of answers given in the second part of the article. The concept of quantum states and the principle of superposition still appear particularly difficult.

Using software to teach statistical inference in introductory courses opens the door for methods and practices that are more conceptually appealing to students. With an increasing number of fields requiring competency in statistics including data science, natural and social sciences, public health and more, it is crucial that we as instructors deliver the basic concepts of statistics effectively. In line with guidelines presented in the GAISE College Report, this article demonstrates intuitive approaches to teaching proportion and mean inference that take advantage of statistical software and emphasize conceptual understanding. The article recommends putting aside asymptotic-based methods for proportion inference and using the exact binomial method. Regarding mean inference, we propose a more contextualized and simplified process that uses the distribution of the sample mean directly and avoids standardized statistics such as z or t. In both the proportion and mean inference contexts, we discuss the benefits of the proposed approaches and provide detailed examples that demonstrate the methods using the Rguroo statistical software.

Effective undergraduate statistical education requires training using real-world data. Textbook datasets seldom match the complexities and messiness of real-world data and finding these datasets can be challenging for educators. Consulting and industrial datasets often have nondisclosure agreements. Academic datasets often require subject area expertise beyond those of a general education or lack connections to real-world applications. Many governments, including the United States, now require the release of data from projects they directly complete or fund though grants and contracts. We show how statistical educators may find datasets and incorporate them into courses. Specifically, we use two examples from the U.S. Geological Survey (USGS) and one example from the ecology literature. We demonstrate the use of these datasets in an upper-level analysis of variance (ANOVA) class. In addition to describing how we found the datasets, we describe how to include them into course work and the course's student assessments. We have used these datasets over multiple semesters and included student feedback from these courses. Although our examples focus on an ANOVA class, the general methods for finding data shared here could be used for statistical classes ranging from high school to graduate education. Supplementary materials for this article are available online.

In a range of settings, human operators make decisions with the assistance of automation, the reliability of which can vary depending upon context. Currently, the processes by which humans track the level of reliability of automation are unclear. In the current study, we test cognitive models of learning that could potentially explain how humans track automation reliability. We fitted several alternative cognitive models to a series of participants' judgements of automation reliability observed in a maritime classification task in which participants were provided with automated advice. We examined three experiments including eight between-subjects conditions and 240 participants in total. Our results favoured a two-kernel delta-rule model of learning, which specifies that humans learn by prediction error, and respond according to a learning rate that is sensitive to environmental volatility. However, we found substantial heterogeneity in learning processes across participants. These outcomes speak to the learning processes underlying how humans estimate automation reliability and thus have implications for practice.

Alternative grading methods, such as standards-based grading, provide students multiple opportunities to demonstrate their understanding of the learning outcomes in a course. These grading methods allow for more flexibility and help promote a growth mindset by embracing constructive failure for students. Implementation of these alternative grading methods requires developing specific, transparent, and assessable standards. Moving away from traditional methods also requires a mindset shift for how both students and instructors approach assessment. While providing multiple opportunities is important for learning in any course, these methods are particularly relevant to an upper level mathematical statistics course where topics covered often provide an additional challenge for students as they lie at the intersection of both theory and application. By providing multiple opportunities, students have the space for constructive failure as they tackle learning both a conceptual understanding of statistics and the supporting mathematical theory. In this article we share our experiences--including both challenges and benefits for students and instructors--in implementing standards-based grading in the first semester of a mathematical statistics course (i.e., focus primarily on probability). Supplementary materials for this article are available online.

Sequence learning is fundamental to a wide range of cognitive functions. Explaining how sequences--and the relations between the elements they comprise--are learned is a fundamental challenge to cognitive science. However, although hundreds of articles addressing this question are published each year, the actual learning mechanisms involved in the learning of sequences are rarely investigated. We present three experiments that seek to examine these mechanisms during a typing task. Experiments 1 and 2 tested learning during typing single letters on each trial. Experiment 3 tested for "chunking" of these letters into "words." The results of these experiments were used to examine the mechanisms that could best account for them, with a focus on two particular proposals: statistical transitional probability learning and discriminative error-driven learning. Experiments 1 and 2 showed that error-driven learning was a better predictor of response latencies than either n-gram frequencies or transitional probabilities. No evidence for chunking was found in Experiment 3, probably due to interspersing visual cues with the motor response. In addition, learning occurred across a greater distance in Experiment 1 than Experiment 2, suggesting that the greater predictability that comes with increased structure leads to greater learnability. These results shed new light on the mechanism responsible for sequence learning. Despite the widely held assumption that transitional probability learning is essential to this process, the present results suggest instead that the sequences are learned through a process of discriminative learning, involving prediction and feedback from prediction error.

In any visual search task in the lab or in the world, observers will make errors. Those errors can be categorized as "deterministic": If you miss this target in this display once, you will definitely miss it again. Alternatively, errors can be "stochastic", occurring randomly with some probability from trial to trial. Researchers and practitioners have sought to reduce errors in visual search, but different types of errors might require different techniques for mitigation. To empirically categorize errors in a simple search task, our observers searched for the letter "T" among "L" distractors, with each display presented twice. When the letters were clearly visible (white letters on a gray background), the errors were almost completely stochastic (Exp 1). An error made on the first appearance of a display did not predict that an error would be made on the second appearance. When the visibility of the letters was manipulated (letters of different gray levels on a noisy background), the errors became a mix of stochastic and deterministic. Unsurprisingly, lower contrast targets produced more deterministic errors. (Exp 2). Using the stimuli of Exp 2, we tested whether errors could be reduced using cues that guided attention around the display but knew nothing about the content of that display (Exp3a, b). This had no effect, but cueing all item locations did succeed in reducing deterministic errors (Exp3c).

Logistic regression (LoR) is a foundational supervised machine learning algorithm and yet, unlike linear regression, appears rarely taught early on, where analogy and proximity to linear regression would be an advantage. A random sample of 50 syllabi from undergraduate business statistics courses shows only two percent of the courses included LoR. Conceivable reasons for this dearth of LoR content is likely related to topic complexity, time constraints, and varying degrees of tool ease of use and support. We propose that these constraints can be countered by: [1] introducing logistic regression early, [2] informed tool selection prioritizing ease of use with comprehensive output, and [3] using/developing innovative, accessible, and easy to understand concept learning aids. This approach would leverage the proximity to linear regression and probability readily embed distributed practice for student understanding of a foundational technique.

When constructed-response items are administered for a second time, it is necessary to evaluate whether the current Time B administration's raters have drifted from the scoring of the original administration at Time A. To study this, Time A papers are sampled and rescored by Time B scorers. Commonly the scores are compared using the proportion of exact agreement across times and/or t-statistics comparing Time A means to Time B means. It is common to treat these rescores with procedures that assume a multinomial sampling model, which is incorrect. The correct, product-multinomial model reflects the stratification of Time A scores. Using direct computation, the research report demonstrates that both proportion of exact agreement and the t-statistic can deviate substantially from expected behavior, providing misleading results. Reweighting the rescore table gives each statistic the correct expected value but does not guarantee that the usual sampling distributions hold. It is also noted that the results apply to a wider class of situations in which a set of papers is scored by one group of raters or scoring engine and then a sample is selected to be evaluated by a different group of raters or scoring engine.

Many challenges are identified when students solve mathematic problems especially in solving probability word problems. This study was conducted to identify the major challenges faced by matriculation college students while solving the probability of an event word problem. Seven matriculation college students were the sample for this case study. Clinical interviews were employed as data collection. This data collection technique was selected based on the researcher's observation on the participants as they answered the task. The probability word problem task was given during the interview session. Semi-structured interviews are used to obtain in-depth information. The think-aloud analysis involves observations leading to individual behaviours in the oral or nonverbal form of participants and the researcher's field notes. Participants were found to have difficulty interpreting probabilities. Three categories of difficulties were identified, unfamiliar with the meaning of the word, not well versed in the nature of the probability, and unable to identify the goal of the probability word problem. The implication of this study can contribute ideas to instructors in developing the pedagogical techniques practiced in probability word problems.

We tested two accounts of the cognitive process underlying the N400 event-related potential component: one that it reflects meaning-based processing and one that it reflects the processing of specific words. The experimental design utilized separable Persian phrasal verbs, which form a strongly probabilistic, long-distance dependency, ideal for the study of probabilistic processing. In sentences strongly constraining for a particular continuation, we show evidence that between two low-probability words, only the word that changed the expected meaning of the sentence increased N400 amplitude, while a synonym of the most probable target word did not. The findings support an account of the N400 in which its underlying process is driven by the processing of meaning rather than of specific words.

The reliability of a test score is usually underestimated and the deflation may be profound, 0.40 - 0.60 units of reliability or 46 - 71%. Eight root sources of the deflation are discussed and quantified by a simulation with 1,440 real-world datasets: (1) errors in the measurement modelling, (2) inefficiency in the estimator of reliability within the selected measurement model, (3) inefficiency in forming of the score variable (X) as the manifestation of the latent trait [theta], (4) non-optimal characteristics of the items (g[subscript i]) in relation to the estimator, and (5) inefficient weight factor, that is, coefficient correlation (wi) that links [theta] with the observed values of the test item (x[subscript i]), (6) a small sample size, (7) extreme test difficulty, and (8) a narrow scale in the score. If willing to maximize the probability that the estimate of reliability would be as close as possible the true, population value, these sources should be avoided, or their effect should be corrected by using deflation-corrected estimators of reliability.

High school calculus has become indispensable for students seeking a college degree in a STEM field. However, in the present study, we argue that the mathematics opportunities that students seize (when afforded) in middle grades are the key to earning calculus credit in high school. To take calculus in high school, students usually need to take advanced mathematics in middle school to take the prerequisite courses. We analyzed the probability of earning credit in calculus based on a sample of (n = 17,765) students and their eighth-grade mathematics courses. Using descriptive statistics and odds ratio effect sizes we found that taking advanced (i.e., algebra in eighth grade) mathematics courses greatly increased a student's chances of earning calculus credit in high school. However, the results indicated that taking double advanced (i.e., geometry in eighth grade) only statistically significantly improved the odds of Asian students earning calculus credit in high school. While participating in double advanced mathematics statistically significantly lowered the odds of earning calculus credit compared to the advanced track for Black and Latinx students. These findings have major implications for providing equitable math instruction for all students, especially in light of the many math students who are tracked into courses with limited ways to earn the calculus credit necessary for STEM-related college and career success.

The Philippines has undergone a fundamental overhaul of its educational system to highlight basic education and overcome its deficiencies. The educational reform prompts prospective teachers to connect the concepts taught to instill a more profound understanding. As such, the researchers explored the mathematical connections made by prospective secondary mathematics teachers while completing investigative tasks. The study employed a concurrent triangulation mixed-methods design. Data were gathered from 39 prospective secondary mathematics teachers enrolled in the state universities' flagship campuses in Northeastern Philippines. Data were gathered using mathematical connections evaluation, think-aloud strategy, and interviews. Frequency counts, percentages, mean percent scores, Pearson product-moment correlation coefficient, and analysis of the interviews' transcriptions were employed in the study. Results showed that the prospective teachers performed best in making instruction-oriented connections but have difficulty in making implications connection. Also, the prospective teachers' ability to successfully make the mathematical connection is directly linked to their knowledge of the concept, the algorithm, and the part-whole relationship. As such, reform efforts should be made to enhance mathematical connections, emphasizing statistical thinking, and reasoning. Moreover, higher education institutions should incorporate connecting as one of the intended learning outcomes for prospective mathematics teachers.

Although educational research and evaluation generally occur in multilevel settings, many analyses ignore cluster effects. Neglecting the nature of data from educational settings, especially in non-randomized experiments, can result in biased estimates with long-term consequences. Our manuscript improves the availability and understanding of artificial neural networks, an underutilized method trending in other disciplines. This method also shows promise for dealing with challenges faced by educational researchers, such as analyzing clustered data. Therefore, we simulated data to generalize the potential benefits of artificial neural networks to different data types. We also compared artificial neural networks to more familiar methods and investigated the time it demanded to perform each technique. Hence, readers can decide when it may be more appropriate to use one method instead of another.

Simulation modeling is an effective tool for solving problems that cannot be explained analytically or when data cannot be collected. This is done by simulating the observed behavior of a problem under study using a computer program. In math education, this can develop knowledge and fundamental competencies of simulation modeling at a higher level and foster its applications in everyday life. This study created learning activities for secondary school students to simulate the probabilistic behavior of random events using Python. 28 grade 12 students took part in these activities using appropriate scaffolding strategies and a powerful mathematical tool, Python. After completing the activities, student competency in simulating the probabilistic behavior of random events with Python was evaluated using rubrics and the factors of student enjoyment, perceived value, interest, and self-efficacy were determined through a Likert-scale questionnaire. The assessment results showed that the activities had a positive effect on student competencies and emotions. The outcomes of the study can serve guidelines for teachers who are interested in expanding the results for further student development.

This study developed a teaching module that incorporated the Monty Hall problem to introduce conditional probability in a college introductory statistics course. This teaching module integrated a guess -- experiment -- discussion approach with game-based instruction. The researchers piloted this module and made modifications. The work of 20 non-mathematics major undergraduate students was examined for changes in their perceptions about conditional probability through a pre-and-post survey design. The Structure of the Observed Learning Outcome (SOLO) taxonomy was used for data analysis to show the quality of the student work. Findings suggest that most students' perceptions were at higher levels after the teaching module. [Note: The page range (93-109) shown on the PDF is incorrect. The correct page range is 93-108.]

This study examines the development of learning designs based on Realistic Mathematics Education (RME) on the probability topic at the eighth-grade junior high school level. Probability abounds in everyday life, and the RME approach is believed to develop students' mathematical communication skills. This learning design development used the Plomp model combined with Gravemeijer and Cobb model. The research phase includes preliminary research, prototyping phase, and assessment phase. The products of this research are a valid, practice, and effective Local Instructional Theory (LIT) and teacher and student books. The results showed that the students who participated in learning using RME-based designs could determine the concept of probability and effectively improve students' mathematical communication skills.

This research aimed to generate a learning trajectory in an introduction to early mathematics, precisely to measure learning using educational games and Realistic Mathematics Education (RME) and to describe young children's curiosity in learning early mathematics. Children need to have an understanding to take a measurement and use a learning trajectory to learn how to differentiate measurable and countable quantities and to tell the longer and the greater in number. The design of this research involves three phases, namely, preliminary design, experimentation, and retrospective analysis. A Hypothetical Learning Trajectory (HLT) using educational games was developed by collecting data from documentation, observations, and interviews. The results obtained from the implementation in the classroom showed that the educational games were able to help young children get to know early mathematics, particularly on measurements of lengths and volumes. There is an association with several factors in the selection of realistic problems. Communications during the game could stimulate an introduction to measurement. The use of objects in students' vicinity also contributed to their experiments relating to measurement. The results of this research inform policymaking for teachers in designing learning in the classroom in introducing early mathematics based on educational games and Realistic Mathematics Education (RME).

Reform movements in mathematics education advocate that mathematical argumentation play a central role in all classrooms. However, research shows that mathematics teachers at all grade level find it challenging to support argumentation in mathematics classrooms. This study examines the role of teachers' mathematical knowledge in teachers' support of argumentation in mathematics classroom. The study addresses a documented need for a better understanding of the relationship between mathematical knowledge for teaching and instruction by focusing on how the knowledge influences teachers' support of argumentation. The results provide insights into particular aspects of teachers' mathematical knowledge that influence teachers' support of students' development of valid mathematical arguments in mathematics classrooms and suggest implications for research and practice.

A range of pedagogical learning theories has been proposed to guide science teachers' classroom teaching. This study presents the results of the development and use of an 18-item Arabic language rating scale survey to assess Omani teachers' (N = 400) views towards the application of selected pedagogical learning theories of potential use in their science classrooms. The current study was a quantitative approach following the survey design. Rasch analysis was used to guide the analysis of the instrument. The results suggested that one dimension was defined by survey items and item independence. Person and item reliability were acceptable. Analysis of category probability curves showed acceptable measurements from the rating scale. The Wright Map matched theory and so provided guidance for the future teaching of Omani science teachers. This study adds to the literature reporting the results of science education data collected from science teachers who teach for international use in general, and for Oman and Arab countries in particular. Perspectives and contributions of the study findings on improving science teaching and learning are discussed.