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More than 35 years have passed since Shulman introduced the term "pedagogical content knowledge" [PCK] to the field of teacher education. As is the case with any new construct in a field, Shulman's advocacy for PCK in the mid-1980s was almost certainly a reaction to the priorities of the field at that time. A primary goal of this paper is to refresh our collective memories of the origins of and development of PCK as a research domain, with a particular focus on what may have prompted scholars such as Shulman to propose the construct of PCK in 1986. With this historical lens as background, I also seek to consider what might be fruitful future directions in the field's thinking about teaching and teacher knowledge. [ABSTRACT FROM AUTHOR]

Growth in fully online mathematics education has been steady for decades. This is where essentially all interactions associated with the teaching and learning process are mediated via the online medium. Initially spurred on by technological innovation and a variety of socio-cultural forces, growth has more recently rapidly accelerated due to pandemic restrictions. Many suggest we are on the precipice of profound transformation in mathematics education, the outlines of which remain unknown. In leveraging online education possibilities and advancing online mathematics education, this paper considers aesthetic experiences associated with learning mathematics fully online. First, three assumptions are made to highlight the importance of these experiences. Second, three challenges are identified regarding the current state of fully online mathematics education. Finally, specific instructional design and online teaching strategies are outlined that potentiate experiences of wonder and beauty often associated with mathematical discovery. As its central theme, the paper seeks to answer how we may take advantage of this technology in delivering quality mathematics education without allowing it to diminish our human essence. [ABSTRACT FROM AUTHOR]

Lent's (2003) Engineering Fields Questionnaire assesses major components of Social Cognitive Career Theory (SCCT) and includes measures of self-efficacy, coping efficacy, outcome expectations, technical interests, and educational goals. These measures have demonstrated good internal consistency in previous research, but validation information is more limited for use with African American STEM students. This paper discusses a validation study of an adapted version of Lent's Engineering Fields Questionnaire, entitled the STEM Fields Questionnaire, with a sample of African American undergraduate STEM students (n = 526). Validating the STEM Fields Questionnaire for African Americans is particularly important given the role of cultural values and certain experiences in career development among this population. Seven factors resulted from an exploratory factor analysis conducted in the present study: engineering/technology interests, outcome expectations, self-efficacy, STEM coping, goals, bio-chemical sciences interest/self-efficacy, and mathematics interest/self-efficacy, with four of six original subscales represented. Implications for research and practice were discussed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate how the contents related to the arithmetic of fractions, as described in the official curriculum issued by the Japanese Ministry of Education, Culture, Sports, Science, and Technology (MEXT), are implemented in actual teaching at the so-called Japanese supplementary school in Denmark. Additionally, at a methodological level, this case study demonstrates how so-called praxeological reference models can be developed at different levels of detail to describe classroom teaching at a particular grade level and the overall curriculum of primary school in a connected way. This study is based on the notions of praxeological reference models and moments of didactic processes from the Anthropological Theory of the Didactic. The findings reveal that (1) not much time is allocated to each moment and (2) the flow of gradually building up techniques, as intended by the textbook authors is to some extent disrupted in the lesson. [ABSTRACT FROM AUTHOR]

Valuing in mathematics is one of the areas in mathematics education that has attracted the attention of mathematics education researchers in recent times. While efforts have been made to research the attributes students value in their mathematics learning, the same cannot be said about attributes teachers and their students value in mathematics learning, and value alignment in mathematics learning between teachers and students. This paper reports on a study that sought to explore what teachers and their students value and the alignment between attributes teachers and their students value in learners' mathematics learning. A survey of what students and their teachers value in learners' mathematics learning was conducted with 690 senior high school students and 66 teachers from nine schools, using adapted What I Find Important in my mathematics learning questionnaires. A multistage sampling procedure involving the use of stratified random sampling and purposive sampling procedures was used to select the research participants. The data obtained from the study were then analysed, using frequency counts, mean and standard deviation and principal component analysis. The results from the study revealed that the senior high school mathematics teachers valued Authority, Engagement, Strategy and Communication in students' mathematics learning, while the students valued Authority, Relevance, Understanding, Engagement, Strategies, ICT and Achievement in their own mathematics learning; an indication of a gap in valuing among the two groups of research participants. Implications of the findings for mathematics teaching and learning as well as in-service and pre-service education programmes are provided. [ABSTRACT FROM AUTHOR]

This paper reports on the findings of a baseline study that fed on to a broader investigation exploring ICT integration in the teaching of functions. The baseline study was premised on the observation that the notion of function, despite being a fundamental and central idea in the mathematics school curriculum, presents pedagogical challenges to both teachers and learners. The study used the qualitative case study design to investigate Grade 11 mathematics learners' errors and misconceptions under hyperbolic functions at a rural high school in the Limpopo province of South Africa. Informed by the constructivist APOS theory conception levels, learners' errors were detected from their written responses in a diagnostic test. The learners were then individually interviewed during task-based interviews to infer on the misconceptions underlying their errors. Qualitative analysis of the data revealed the following categories of misconceptions: algebraic, conceptual, asymptotic, graphical, and notational misconceptions. The study established that errors, if diagnosed and corrected, are springboards and milestones to the formulation and construction of proper mathematical ideas. To resolve learners' misconceptions, this study recommends that teachers should use explorative and discovery pedagogical strategies to enable learners realize their own misconceptions and self-correct. Based on the findings of this study, the integration of Information and Communication Technologies (ICTs) in teaching functions as an intervention strategy will be explored. [ABSTRACT FROM AUTHOR]

Procedural flexibility is a crucial element of deep procedural knowledge involving the selection of "the most appropriate strategy." However, there exists unavoidable nuance when one attempts to define "the most appropriate" strategy, which leads to two types of procedural flexibility: (a) structure-informed flexibility (Struct-Flex), which refers to a preference for situationally appropriate strategies; (b) algorithm-oriented flexibility (Algo-Flex), which refers to a preference for standard algorithms. The current study investigated the distinction between these two flexibility types and tested potential predictors for both types of flexibility. The study data were collected from 412 Grade 9 through 12 students from 19 math classrooms in the southeastern region of the U.S. Chi-squared tests and regression models revealed that: (a) problem type was strongly correlated to and predictive of a preference for using algorithms in algebra; (b) students' current high school math course grade was associated with preference for standard algorithms; (c) students' current high school math course level and their classroom assignment were related to preference for situationally appropriate strategies. The results suggest that, in this sample, Algo-Flex was as prevalent as Struct-Flex, and the two flexibility types had different predictors. This paper can advance our practical knowledge of U.S. students' strategy choices. The identified predictors of Algo-Flex and Struct-Flex can also inform the design of relevant educational interventions. Future studies may explore longitudinal datasets to untangle the relationship between course level, age, and flexibility, as well as examine flexibility in Asia in comparison to Europe and North America. [ABSTRACT FROM AUTHOR]

In this paper, we present a detailed action plan under continuous quality improvement (CQI) exercise of outcomes based education framework related to Mathematics concerns highlighted by the Electrical Engineering (EE) Department. Failing and low performed student learning outcomes (SOs) and course learning outcomes (COs) in range of core EE courses were observed to be linked with weak Mathematics basis of students. This feedback was achieved through an automated CQI system in EvalTools under Faculty Class Assessment Report (FCAR) tab, where each instructor is required to make reflections about failing as well as low performed SOs and COs. Feedback was reviewed comprehensively, and mathematical concepts which significantly affect the performance of students in the core EE courses were derived. Moreover, a high level of Mathematics pre-requisite covering required mathematical concepts is identified for the range of core EE courses. Consequently, well-defined tasks were assigned to continuously evaluate and monitor improvement in required Mathematics concepts to ensure SOs and COs achieved in core EE courses. Results are given to exhibit the effectiveness of continuous improvement management system (CIMS). [ABSTRACT FROM AUTHOR]

Teaching requires attention to individual student needs by providing both adequate challenge and sufficient support to help students successfully gain academic skills (Shurr et al., 2019). The learning stages framework divides typical learning into four distinct stages: acquisition, fluency, maintenance, and generalization (Collins, 2012; Haring & Eaton, 1978). Thinking in terms of the learning progression can help teachers assess student performance and determine how they can best be supported to progress. This article will lead readers through the process of using the four stages of learning as a framework for assessment (i.e., understanding where students are currently performing), goal setting (i.e., setting the instructional aim), and instruction (i.e., planning for and delivering instruction aligned to individual student needs) within the context of mathematics for students with a variety of disabilities and support needs. [ABSTRACT FROM AUTHOR]

Students with disabilities experience differential levels of achievement in mathematics when compared with their nondisabled peers. Identifying and implementing evidence-based practices (EBPs) is essential to increase their mathematics achievement. However, an argument is re-emerging that calls into question the effectiveness of well-known EBPs, such as explicit/direct instruction. This argument is based on opinion and conjecture, rather than student outcome measures. In this commentary, we use research on explicit and direct instruction with mathematics to (a) highlight the confusion over theoretical implications, (b) call for researchers to reduce personal bias within research, and (c) emphasize the need for improving outcomes of students with disabilities that affect mathematics learning. [ABSTRACT FROM AUTHOR]

As students enter the upper elementary grades, word problems become a main component of mathematics instruction, increasing in complexity as students advance through the curriculum. For students identified as emergent bilinguals with mathematics difficulty (MD), the linguistic complexity inherent in word problems may serve as a barrier to word‐problem proficiency. The current study investigated the potential relation between academic English proficiency and word‐problem outcomes for emergent bilinguals with MD. After analyzing data from 241 third‐grade students, results indicated students who participated in an evidence‐based word‐problem intervention outperformed students who did not receive the intervention. Moreover, students' academic English‐language proficiency scores in the domains of reading and writing positively correlated with higher scores on a measure of word‐problem solving. [ABSTRACT FROM AUTHOR]

Using PISA 2012 data, we conducted multilevel modeling analyses to explore how student–teacher relationships (STRs) affected mathematics motivation across EU countries. We compared this relationship across high and low achievers and explored how sex, economic, social, and cultural status (ESCS), ability grouping, class size, and teacher intentions moderated this relationship. Results show that positive STR was associated with higher motivation. High-ability students, boys, and high ESCS students had higher motivation, but we found no statistically significant interactions between STR and low or high ability. Class size, ability grouping, and teacher intentions had no statistically significant effects on mathematics motivation. Overall, our findings suggest that positive perceptions of STR can positively impact student motivation in mathematics and that this effect is consistent across different demographic groups. Thus, general trends around STR and motivation also apply to high-achieving populations. [ABSTRACT FROM AUTHOR]

Disrupting forms of exclusion in higher education requires culturally sustaining practices. However, in some areas of higher education, the notion of culturally sustaining practice remains unexplored. The present study addresses this disparity by exploring the concept of culturally sustaining supervision in the specific case of Pasifika research students in mathematics education. In particular, this case study examines what a group of 22 Pasifika research students in mathematics education valued most about their respective supervisors. Using talanoa (respectful and free-flowing conversation and discussion) to gather data, three characteristics of supervisors that the participants valued most were identified via thematic analysis: (a) being available and accessible; (b) developing the student's confidence; and (c) providing support beyond the thesis. The characteristics are discussed with respect to the Pacific concept of vā (relationships), which refers to the intricate relational and spatial realm in which connections among people exist and evolve. [ABSTRACT FROM AUTHOR]

Students in any grade level vary widely in their mathematics achievement, with the typical classroom including four to seven grade levels of mathematics proficiency. Due to this large range of mathematics learning needs, some schools offer certain courses in earlier grades than is typical. In this study, we analyzed multiple, large, national datasets to understand which schools provide access to seventh-grade algebra and eighth-grade geometry. Next, we explored what factors predicted school-level access to those courses and the proportion of eligible students enrolled in either course. We found that access varies widely across, and even within, states and districts, with factors such as parental education and within-district economic segregation remaining predictive even after controlling for average school achievement. In general, predictors of the proportion of a school identified as gifted were weaker, but schools in more-segregated districts and those with larger proportions of White students enrolled did tend to enroll fewer students in either math course. [ABSTRACT FROM AUTHOR]

We presented a design-based study within the context of a four-session Scratch programming activity among 23 fourth-grade students in Hong Kong. Inspired by the computational thinking (CT) strategies and the 5E instructional model, we investigated students' mathematical learning of fractions in a Scratch (block-based programming) environment. Students developed CT concepts, practices, and perspectives by building a "fraction magic calculator" in groups. This study analyzed the lesson design, students' drawings, interviews, and work expressing their mathematical understanding of fractions in Scratch applications. The learning tasks were designed to support the students' fraction learning and utilized computational abstractions in the form of variables, functions, and iterations to formulate mathematical models in a programming context. Students' artifacts and feedback showed they were interested in learning fractions in a programming learning context, contributing to exercising and improving their fraction concepts and CT. Ultimately, we emphasized the benefits of CT integrated into mathematics education, promoting students' understanding of fraction concepts, a set of CT abilities (concepts, practices, perspectives), and learning motivation. Moreover, we suggested a set of non-cognitive skills (e.g., socializing, expressing) to enrich the CT perspectives in the framework and show the importance of developing coding communities to co-create digital artifacts among learners. Overall, we highlighted that mathematics teachers should apply and create learning tasks that promote computational thinking to forge mathematical ideas and thinking. [ABSTRACT FROM AUTHOR]

Empirical research has demonstrated that mathematical errors originate from various causes that may have different implications for the learning process. Mathematical errors are an international phenomenon as all teachers from different cultural background must handle them every day during teaching-and-learning processes. When errors arise, teachers must determine whether and how to deal with them and whether the error should be corrected. Although errors and how they are addressed by teachers are of high pedagogical importance, teachers' approaches to handling student errors remain underexplored. The study reported herein examines the ways in which mathematics teachers handle student errors in various situations, and whether any connection between the error type and the manner in which it is corrected may be reconstructed. The study participants comprised 13 secondary mathematics teachers in a large city in northern Germany. Each teacher was video-recorded over two 90-min-lessons, and the data were subsequently evaluated using qualitative content analysis. The results indicate that teachers' approaches to correcting mathematical student errors diverged considerably depending on the phase of the lesson during which the error occurred. In particular, during the class discussion phase, a high proportion of the corrections were performed by the students' classmates, whereas corrections executed during the group work, partner work, or individual work phases were primarily performed by students who had made the errors and the teachers. Given that the teacher's approach to handling errors exerts a significant influence on teaching-and-learning processes, further research is required to identify the most prolific approaches to handling errors in order to foster students' mathematical understanding. [ABSTRACT FROM AUTHOR]

We present results from a meta-analysis of 37 contemporary experimental and quasi-experimental studies of summer programs in mathematics for children in grades pre-K–12, examining what resources and characteristics predict stronger student achievement. Children who participated in summer programs that included mathematics activities experienced significantly better mathematics achievement outcomes compared to their control group counterparts. We find an average weighted impact estimate of +0.10 standard deviations on mathematics achievement outcomes. We find similar effects for programs conducted in higher- and lower-poverty settings. We undertook a secondary analysis exploring the effect of summer programs on noncognitive outcomes and found positive mean impacts. The results indicate that summer programs are a promising tool to strengthen children's mathematical proficiency outside of school time. [ABSTRACT FROM AUTHOR]

Schema instruction improves word-problem solving for students with mathematics difficulty (MD; e.g., Powell et al., 2015); however, prior research suggests that students need word-problem-specific language comprehension in addition to schema knowledge to build word-problem models (Fuchs et al., 2015 , 2018). Students have general word knowledge (e.g., more, less) but require word-problem-specific word knowledge to solve word problems successfully (e.g., more than, less than ; Kintsch & Greeno, 1985). Prior word-problem schema interventions have not provided explicit instruction in word-problem vocabulary to support students' development of word-problem models. We aimed to determine if word-problem vocabulary instruction combined with schema instruction provided an advantage to word-problem solving for third-grade students with MD. We randomly assigned 75 third graders with MD to three conditions: Word-problem intervention with explicit vocabulary instruction (WP + V), word-problem intervention without explicit vocabulary instruction (WP-only), and business-as-usual intervention (BAU). Students in the treatment conditions received 22, 50-min lessons provided virtually in small groups. On a word-problem vocabulary measure, students in the WP + V condition significantly outperformed students in the WP-only (ES = 0.43) and BAU (ES = 0.83) conditions. On a word-problem measure, students in the WP + V condition outperformed students in the WP-only (ES = 0.08) and BAU conditions (ES = 0.43), although these were not statistically significant. These findings provide preliminary support for the benefits of incorporating explicit vocabulary instruction in a word-problem schema intervention; future research is needed to fully understand the effect of vocabulary instruction on word-problem outcomes. [ABSTRACT FROM AUTHOR]

This small-scale comparative study investigated Spanish and Turkish middle school students' mastery of four core cognitive skills required for proportional reasoning using the log-linear cognitive diagnosis model. We compared students in the two samples for their proportional reasoning and provided a diagnostic assessment of their strengths and weaknesses. The study included 314 Spanish eighth grade and 282 Turkish seventh grade students. The students were administered a proportional reasoning test with 22 multiple-choice items. We found significant differences between the two samples in two of the cognitive skills. While the Spanish sample were better at understanding nonproportional relationships, the Turkish sample were better at understanding the concept of ratio. Furthermore, while the Spanish sample had a more dispersed distribution of students among different attribute profiles, the Turkish sample mostly consisted of either low-performing or high-performing students. To better understand the similarities and differences between the two samples, we compared students' performance on some specific test items. The study shows how cognitive diagnostic models, which are not widely used in small-scale comparative studies, can be very useful in revealing similarities and differences between two samples that cannot be detected by traditional psychometric models. [ABSTRACT FROM AUTHOR]

Online learning is a relatively new but emerging approach in Philippine education. The approach's rise in popularity warrants an emphasis on self-regulated learning (SRL). Hence, the current study aimed to propose and assess an intervention using a digital note-taking application as an SRL tool for Business Math. Data collection included the administration of a pre-test and a posttest and weekly ratings of digital notebook usage. Two-way ANCOVA results suggest that skillful self-regulators consistently achieved more than naïve self-regulators. Hierarchal regression analysis results imply that learners with better digital notebooks are likelier to achieve more in the course. These results demonstrate the effectiveness of the SRL intervention used in the study. Furthermore, repeated measures of ANOVA results suggest that different groups of learners use their SRL tools differently. Despite the effect of SRL tools on achievement, higher achievers are not certainly better self-regulators than lower achievers. [ABSTRACT FROM AUTHOR]

Both educational policies and current mathematics education research highlight that all students should be given the opportunity to be creative in mathematics class. However, if creativity is of such significance to mathematics learning, how can we describe the mathematical creativity of school children? Conducting an integrative literature review, this article aims to answer this question by synthesizing conceptual aspects and research results on the mathematical creativity of young school children working on mathematical open tasks. Here, concepts of creativity in mathematics education based on a divergent production approach are emphasized in detail. This leads to the introduction of the term "individual mathematical creativity" and to the suggestion of a framework to qualitatively characterize the individual mathematical creativity of school children working on open tasks. Its relevance and application as a scientific and/or educational tool to analyze and foster the creativity of students are explained exemplarily using the creative task processing of the 6-yearold student Jessica. [ABSTRACT FROM AUTHOR]

Success in mathematics has been shown to increase career opportunities; thus, improving student learning in mathematics is vital for educational policymakers. This study investigated the effects of five parental involvement variables on young children's mathematical achievements in 10 countries, including seven Arab countries, and found significant effects for all of them. Students who attended pre-primary education achieved higher mathematics scores compared to those who did not. Students who had been helped with or had their homework reviewed fewer times by their parents achieved significantly higher scores than those who had been helped or checked on more frequently. Students who had not attended any extra lessons to excel over or keep up with classmates achieved significantly higher scores. Students whose parents had university degrees attained significantly better results. Finally, students whose parents had higher expectations achieved significantly better results. These results have implications to enhance students' mathematical achievements in Arab countries. [ABSTRACT FROM AUTHOR]

Using a dataset that includes over 17 million students from across all 50 states, we estimate the causal impact of making structural transitions into middle school (in grades 4, 5, 6, or 7) on student math and reading achievement trajectories. This dataset provides an ideal opportunity to engage in the valuable scientific practice of conducting replication studies. Prior research on the impacts of middle school transitions is of high quality and rests on a strong causal warrant, but the study settings vary greatly and use data from a prior decade. We conduct a replication (i.e., using the same methods on different data) using larger, broader, and more recent data. We extend prior analyses in ways that may further strengthen the causal warrant. Finally, we explore heterogeneity of effects across subgroups and states, which may help reconcile differences in the magnitude of estimated effects across studies. [ABSTRACT FROM AUTHOR]

Since 1975, researchers have conducted interventions to improve the word-problem performance of elementary school students facing mathematics difficulties. The current study reports a meta-analysis of 52 studies that examined the effect of these interventions. We estimated multivariate, random-effects models (REM) with robust variance estimation (RVE) with and without outliers. Results showed a large, positive, and significant mean weighted effect size (g = 1.01 for the model with outliers; g = 0.81 for the model without outliers). Findings of meta-regression analyses showed several moderators, such as sample composition, group size, intervention dosage, group assignment approach, interventionist, year of publication, and dependent measure type, significantly explained heterogeneity in effects across studies. A sensitivity analysis showed these results were generally robust to outliers. We offer possible explanations for the findings and discuss study limitations. Finally, we propose recommendations for future research and classroom practice. [ABSTRACT FROM AUTHOR]

Proportional reasoning is the cornerstone of many higher-level mathematics topics. We evaluated the sustainability effects of schema-based instruction (SBI) on proportional reasoning and the impact of professional development (PD) on the instructional practices of two cohorts of seventh-grade teachers (i.e., SBI-experienced-teacher implementers, who taught SBI in a previous study; SBI-novice-teacher implementers, who taught SBI for the first time with PD) and mathematics outcomes of students with mathematics learning difficulties. SBI was implemented for 6 weeks by 49 teachers in two distal states. Despite the cessation of PD provided to SBI experienced-teacher implementers, teachers in both cohorts demonstrated similar levels of fidelity. Results indicated no SBI experience effect on the immediate and 11-week retention tests of proportional problem solving, on a general measure of problem solving, and on an end-of-year mathematics achievement test, suggesting that the effectiveness of SBI generalizes over time to different cohorts of teachers and students. [ABSTRACT FROM AUTHOR]

Background/Context: Across the United States, secondary school curricula are intensifying as a growing proportion of students enroll in high-level academic math courses. In many districts, this intensification process occurs as early as eighth grade, where schools are effectively constraining their mathematics curricula by restricting course offerings and placing more students into Algebra I. This paper provides a quantitative single-case research study of policy-driven curricular intensification in one California school district. Research Questions: (1a) What effect did 8th eighth grade curricular intensification have on mathematics course enrollment patterns in Towering Pines Unified schools? (2b) How did the distribution of prior achievement in Towering Pines math classrooms change as the district constrained the curriculum by universalizing 8th eighth grade Algebra? (3c) Did 8th eighth grade curricular intensification improve students' mathematics achievement? Setting: Towering Pines is an immigrant enclave in the inner-ring suburbs of a major metropolitan area. The district's 10 middle schools together enroll approximately 4,000 eighth graders each year. The districts' students are ethnically diverse and largely economically disadvantaged. The study draws upon administrative data describing 8th eighth graders in the district in the 2004-20-05 through 2007-20-08 school years. Intervention/Program/Practice: During the study period, Towering Pines dramatically intensified middle school students' math curricula: In the 2004-20-05 school year 32 % of the district's 8th eighth graders enrolled in Algebra or a higher- level mathematics course; by the 2007-20-08 school year that proportion had increased to 84%. Research Design: We use an interrupted time-series design, comparing students' 8th eighth grade math course enrollments, 10th grade math course enrollments, and 10th grade math test scores across the four cohorts, controlling for demographics and prior achievement. Findings/Results: We find that students' odds of taking higher level mathematics courses increased as this district implemented the state's Algebra mandate. However, even as the district implemented a constrained curriculum strategy, mathematics achievement growth between 6th sixth and 10th grade slowed and the achievement advantages associated with 8th eighth grade Algebra declined. Conclusions/Recommendations: Our analyses suggest that curricular intensification increased the inclusiveness and decreased the selectivity of the mathematics tracking regime in Towering Pines middle schools. However, the findings suggest that this constrained curriculum strategy may have may have unintended negative consequences for student achievement. [ABSTRACT FROM AUTHOR]

Word‐problem features such as text complexity, charts and graphs, position of the unknown, calculation complexity, irrelevant information, and schemas impact word‐problem performance. We compared the word‐problem performance of typically achieving (TA) students and students with mathematics difficulty (MD). First, we measured the word‐problem performance of all students for schemas and position of the unknown, followed by the performance of students with MD for schemas, position of the unknown, irrelevant information, and charts or graphs. Across schemas, while TA students outperformed students with MD, all students typically scored higher on Change and Difference problems than on Total problems. For position of the unknown, students often scored highest on problems with the final position unknown. Students with MD also demonstrated higher scores on problems with irrelevant information than charts and graphs. Although patterns emerged, not all problems followed the same trends, suggesting the need for further research to investigate the impact of word‐problem features on word‐problem accuracy. [ABSTRACT FROM AUTHOR]

Manipulative-based instructional sequences—including both concrete and virtual manipulative instructional sequences—are evidence-based or research-based mathematical interventions for students with disabilities. However, as options for manipulative-based instructional sequences increase, educators need support in deciding the best approach. In this manuscript, we provide practical tips for educators to make decisions about selecting and implementing manipulative-based instructional sequences for students with disabilities. We present and discuss the following steps educators should take when preparing to use a manipulative-based instructional sequence: (a) determine resources, (b) know the student, (c) identify the targeted mathematics, (d) evaluate time constraints, and (c) decide upon the ultimate goal of learning. [ABSTRACT FROM AUTHOR]

Researchers and practitioners alike often look to meta-analyses to identify effective practices to use with students with disabilities. The number of meta-analyses in special education has also expanded in recent years. The purpose of this systematic review is to evaluate the quality of reporting in meta-analyses focused on mathematics interventions for students with or at risk of disabilities. We applied 53 quality indicators (QIs) across eight categories based on recommendations from Talbott et al. to 22 mathematics intervention meta-analyses published between 2000 and 2020. Overall, the meta-analyses met 61% of QIs and results indicated that meta-analyses most frequently met QIs related to providing a clear purpose (95%) and data analysis plan (77%), whereas meta-analyses typically met fewer QIs related to describing participants (39%) and explaining the abstract screening process (48%). We discuss the variation in quality indicator scores within and across the quality categories and provide recommendations for future researchers. [ABSTRACT FROM AUTHOR]

Previous work has noted that math anxiety may have a profound effect on math performance; however extant research has relied on measures that explicitly assess math anxiety. This study examined the effects of implicit math anxiety on the performance of a math achievement task. We hypothesized that combined measure of implicit anxiety and explicit anxiety would better predict math achievement than measures of explicit math anxiety alone. In addition, we hypothesized that an individuals' measured implicit anxiety and measured explicit anxiety would share only a modest correlation. To test these hypotheses, 175 participants completed measures of explicit anxiety, an implicit associations test designed to measure implicit anxiety, and a measure of math achievement. As expected, math achievement was better predicted when implicit anxiety was combined with explicit anxiety. Furthermore, scores on the implicit and explicit anxiety measures were not significantly correlated. These results suggest that implicit and explicit math anxiety are two distinct constructs, thus traditional methods for helping students deal with math anxiety may not be entirely successful if the implicit component is ignored. [ABSTRACT FROM AUTHOR]

In this systematic review, we explored mathematics interventions for middle school (Grades 6, 7, and 8) implemented with students who experienced difficulty in the area of mathematics, including students with an identified learning disability in mathematics. A total of 72 single‐subject and group comparison studies met inclusion criteria, with 59 studies demonstrating positive effects on student‐level mathematics outcomes. The majority of mathematics interventions focused on foundational, prealgebraic skills (e.g., operations and problem solving) related to algebraic reasoning. To understand the landscape of effective mathematics interventions and inform instruction within mathematics intervention, we identified six instructional components used with regularity within the effective studies. These components included explicit instruction, multiple representations, problem‐solving instruction, mathematical language, mnemonics, and graphic organizers. [ABSTRACT FROM AUTHOR]

This study examined the classification accuracy for subskill mastery measures administered in mathematics for students in kindergarten and Grades 1, 3, 5, and 7 at fall (n = 564) and winter (n = 602) screening. In addition, response to classwide math intervention was examined as another layer of screening for students in kindergarten and Grades 1, 3, and 5 (n = 387). Criterion measures were the year-end accountability measure for math in Grades 3, 5, and 7. For kindergarten and Grade 1 a researcher-constructed composite measure was used. For all measures (screening and criterion), 20th percentile performance was the risk criterion. Authors hypothesized that classwide intervention response would provide more accurate identification of risk than fall or winter screening. Receiver operating characteristic (ROC)–generated sensitivities and specificities supported this hypothesis. Data indicated that intervention dosage was adequate and did reduce the base rate of risk in each screening group. Posttest probabilities indicated that subskill mastery measures and response to classwide intervention could be used to accurately indicate math risk and the need for more intensive intervention for students in a multitier system of supports (MTSS) model. [ABSTRACT FROM AUTHOR]

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Using the test scores of more than 2,250,000 students who participated in the American Mathematics Competitions (AMC) from 2009 to 2019, this study examined the direction and magnitude of the trends in gender disparity in participation and high achievement in self-select mathematics exams. The results of this study indicated that the male to female ratio among the AMC participants increased significantly from 2009 to 2019. The findings also showed an established disparity that favored males for each year and for all competitions in both the top 1% and 5% levels, while the trend in the male to female ratios over a decade was stable, except for the top 1% of the population in the AMC 8, in which there was a significant increasing trend that favored males. The possible impacts of these findings within the context of the underrepresentation of women in STEM (science, technology, engineering, and mathematics) fields are discussed. [ABSTRACT FROM AUTHOR]

Using data from the Programme for International Student Assessment (PISA), this study implements two statistical analyses to investigate the effects of student and teacher characteristics on students' mathematical achievement. First, the authors conduct an exploratory factor analysis to explore the factor structure for the various student and teacher variables of interest in this study. Second, they perform hierarchical linear modeling to analyze students' and teachers' multilevel structure in a school. The results suggest that student characteristics such as mathematics interest, instrument motivation, mathematics self-efficacy, mathematics anxiety, mathematics self-concept, and out-of-school study time predicted 39.9% of mathematical achievement variance. The results also suggest that mathematics self-efficacy had the largest effect on mathematical achievement. Teacher characteristics such as teacher-directed instruction, cognitive activation, teacher support, classroom management, and student–teacher relations predicted 34.9% of mathematical achievement variance. This study's results have implications for educators in fostering a positive learning environment to increase students' mathematics interest and self-efficacy, and focus on specific teacher characteristics to increase students' mathematical achievement. [ABSTRACT FROM AUTHOR]

While mathematics education is key to the post-school outcomes of students with disabilities, it has received less attention in research and practice compared with other aspects of educating this population. Skill maintenance is particularly crucial in mathematics because students build upon prior knowledge across grade levels. They also need to be able to apply mathematical skills in everyday life. Hence, this study utilized a multiple probe across-participants single-case experimental design to evaluate the effectiveness of the virtual–representational–abstract (VRA) with overlearning instructional sequence in teaching multiplication and supporting its maintenance among three students with disabilities. For each student, a functional relation existed between the VRA with overlearning instructional sequence and accuracy of solving multiplication problems. Students also maintained the skill up to 8 weeks after the intervention. [ABSTRACT FROM AUTHOR]

This single‐case meta‐analysis is the first to provide a quantitative synthesis of the published literature on mathematics word problem‐solving intervention studies for English learners with learning disabilities and mathematics difficulties. A total of ten single‐subject studies were included for analysis. The current study investigated the magnitude of the effect of mathematics interventions targeting the improvement of students' mathematics word problem‐solving performance, and analyzed variables that moderated this effect. Results indicate a moderate overall effect size of 0.81 (95% CI [0.71, 0.90]) on word problem‐solving performance for English learners with learning disabilities and mathematics difficulties. Moderator variables such as the implementer of the interventions, instructional focus, and word problem‐solving content significantly moderated the mathematics intervention effects. [ABSTRACT FROM AUTHOR]

Students with mathematics learning disability (MLD) are entitled to a free appropriate public education (FAPE) as outlined by their Individualized Education Program (IEP). The IEP is a roadmap to enhance in‐school and postschool outcomes for students with a disability. We reviewed IEPs (n = 89) for secondary students with MLD enrolled in 15 rural independent school districts in the southeast part of the United States. Results suggested secondary students with MLD have needs in (a) calculation; (b) fractions, decimals, and percentages; (c) functional mathematics; (d) reading; and (e) social and behavioral skills based on annual goals written in their IEPs. Yet 70% of the IEPs reviewed included only instruction in the general education setting, without specialized services. IEPs included a variety of accommodations, but few goals facilitated access to grade‐level content or special education services to support specialized instruction to meet goals. Implications for practice, limitations, and future research are discussed. [ABSTRACT FROM AUTHOR]

It is with increasing importance that mathematics education research considers the role of noncognitive motivation variables alongside elements of the classroom context in investigations of student mathematics performance. This study uses a hierarchical linear modeling framework to predict mathematics achievement from three classroom variables, project-based learning, group collaboration, and student-driven curriculum, and two noncognitive factors, mathematics anxiety and mathematics self-concept, utilizing data from the 2012 Programme for International Student Assessment (PISA) United States sample. Findings suggest that mathematics classroom contexts that are student-driven and integrate project-based learning positively impact mathematics achievement, and that both mathematics anxiety and mathematics self-concept contribute significantly towards explaining variation in mathematics achievement after accounting for gender, race, socioeconomic status, truancy, and school-level poverty. [ABSTRACT FROM AUTHOR]

Secondary students with persistent mathematics difficulties need the most intensive intervention in order to improve their mathematics outcomes. One approach to intensifying and individualizing intervention is through data‐based individualization (DBI). The present study used a single‐subject, multiple‐baseline‐across‐participants, replicated‐across‐skills design to investigate the effectiveness of DBI process on mathematics computation and mathematics concept and application performance of high school students who had intense needs in mathematics. Results suggest the interventions guided by the DBI process improved mathematics computation and mathematics concept and application performance of the participants. Implications of these findings and indications for future research are discussed. [ABSTRACT FROM AUTHOR]

Background. Middle school students' math anxiety and low engagement have been major issues in math education. In order to reduce their anxiety and support their math learning, game-based learning (GBL) has been implemented. GBL research has underscored the role of social dynamics to facilitate a qualitative understanding of students' knowledge. Whereas students' peer interactions have been deemed a social dynamic, the relationships among peer interaction, task efficiency, and learning engagement were not well understood in previous empirical studies. Method. This mixed-method reseacrh implemented E-Rebuild, which is a 3D architecture game designed to promote students' math problem-solving skills. We collected a total of 102 50-minutes gameplay sessions performed by 32 middle school students. Using video-captured and screen-recorded gameplaying sessions, we implemented behavior observations to measure students' peer interaction efficiency, task efficiency, and learning engagement. We used association analyses, sequential analysis, and thematic analysis to explain how peer interaction promoted students' task efficiency and learning engagement. Results. Students' peer interactions were negatively related to task efficiency and learning engagement. There were also different gameplay patterns by students' learning/task-relevant peer-interaction efficiency (PIE) level. Students in the low PIE group tended to progress through game tasks more efficiently than those in the high PIE group. The results of qualitative thematic analysis suggested that the students in the low PIE group showed more reflections on game-based mathematical problem solving, whereas those with high PIE experienced distractions during gameplay. Discussion. This study confirmed that students' peer interactions without purposeful and knowledge-constructive collaborations led to their low task efficiency, as well as low learning engagement. The study finding shows further design implications: (1) providing in-game prompts to stimulate students' math-related discussions and (2) developing collaboration contexts that legitimize students' interpersonal knowledge exchanges with peers. [ABSTRACT FROM AUTHOR]

Manipulatives are widely considered an effective practice and have been recommended as an evidence-based practice for students identified with a learning disability when used within the concrete–representational–abstract instructional framework. The aim of the current study was to evaluate single-case experimental designs that implemented a mathematics intervention using manipulatives on the mathematical outcomes of students at risk or identified with a disability. A total of 53 studies were included in the review. The Tau- U effect size (ES) across studies ranged from 0.34 to 1.00, with an omnibus ES of 0.91 (CI95 = [0.87, 0.95]). The between-case standardized mean difference for individual studies ranged from 0.03 to 18.58. Moderator analyses revealed that out of nine variables analyzed (i.e., study quality, design, age, interventionist, manipulative type, perceptual richness, math concept, dependent variable, and disability category), only disability category served as a moderator. Implications for research and practice are discussed. [ABSTRACT FROM AUTHOR]

This manuscript presents the results from a study to investigate the technical characteristics of two versions of a number line assessment (NLA 0–20 and NLA 0–100). The sample consisted of 60 kindergarten and 46 first grade students. Both number line versions had sufficient alternate form and test–retest reliability. The NLA 0–20 had low and the NLA 0–100 had low to moderate correlations with math achievement. Results indicated that the NLA 0–100 explained a small, but unique portion of the variance in first grade mathematics performance when controlling for performance on the Assessing Student Proficiency in Early Number Sense (ASPENS) a set of early numeracy screening measures. We discuss study results related to the utility of adding number line assessment tasks to mathematics screening batteries and propose additional areas of research. [ABSTRACT FROM AUTHOR]

National assessments suggest that many students do not display proficiency with fractions. Considering that time and materials are precious resources, many educators are looking to use technology to provide supplemental instruction with low demand on the instructor. The current study utilized a multiple‐baseline design to examine the effects of the Motion Math: Bounce application on the fraction quantity comparison and number line placement skills of three fifth‐grade students. Results indicate that Motion Math: Bounce improved the quantity comparison skills of all three participants and the number line skills of one participant. Additionally, all three participants made improvements from pre‐ to posttest on a measure of general fraction knowledge. Implications for future research and intervention implementation are discussed. [ABSTRACT FROM AUTHOR]

Using an ecological approach to trauma-informed care and radical healing, this case study explores how one Title I public middle school mathematics classroom offered students opportunities to engage in healing practices through the use of Social Justice Mathematics. Findings indicate that students identified their emotions, engaged in structural analyses of local social issues, and expressed plans to take action. This study suggests the possibility of using a Healing-Informed Social Justice Mathematics approach to support development of students' sociopolitical consciousness, mathematics learning, and well-being. [ABSTRACT FROM AUTHOR]

Several studies show that using technology as an intervention tool in mathematics influences student outcomes, motivation to learn, and attitude about learning. However, no meta-analysis exists that examines all three of these characteristics combined along with several facets of the interventions. Understanding how motivation and attitude changes and influences student learning when technology is involved is essential in effectively using technology to enhance mathematical achievement. The current study uses a systematic review process to determine the effects of technology use on student achievement, motivation, and attitude. Different aspects of the intervention are examined (type of intervention, type of treatment, duration of the intervention, mathematical content area, and context of the learning environment). Results from 24 articles (4,522 subjects) indicate a significant overall impact of technology on student achievement, motivation, and attitudes; however, results vary based on the different aspect of the intervention examined. [ABSTRACT FROM AUTHOR]

As classrooms begin to adopt a greater number of digital technologies such as computers and tablets, it is important for educators to understand how effective such tools can be in aiding in the delivery of instruction to students who struggle in mathematics, such as those identified with a learning disability in mathematics. One digital‐based instructional strategy with a limited research base for students with a learning disability is video modeling. Through a single subject alternating treatments design, this study compared the use of video modeling to face‐to‐face explicit instruction for teaching geometry word problems to three secondary students with a learning disability in mathematics. Across 10 sessions of intervention, all three students demonstrated improved performance on all dependent variables with both interventions, while the explicit instruction condition produced slightly greater accuracy scores for two of the three students. The results and their implications for the field of mathematics are discussed. [ABSTRACT FROM AUTHOR]

Based on the control-value theory of achievement emotions, this longitudinal study examined students’ control-value appraisals as antecedents of their enjoyment and boredom in mathematics. Self-report data for appraisals and emotions were collected from 579 students in their final year of primary schooling over three waves. Data were analyzed using latent interaction structural equation modeling. Control-value appraisals predicted emotions interactively depending on which specific subjective value was paired with perceived control. Achievement value amplified the positive relation between perceived control and enjoyment, and intrinsic value reduced the negative relation between perceived control and boredom. These longitudinal findings demonstrate that control and value appraisals, and their interaction, are critically important for the development of students’ enjoyment and boredom over time. [ABSTRACT FROM AUTHOR]