Exploring the impact of a fraction sense intervention in authentic school environments: An initial investigation.

A solid understanding of fractions is the cornerstone for acquiring proficiency with rational numbers and paves the way for learning advanced mathematical concepts such as algebra. Fraction difficulties limit not only students' educational and vocational opportunities but also their ability to solve everyday problems. Students who exit sixth grade with inadequate understanding of fractions may experience far-reaching repercussions that lead to lifelong avoidance of mathematics. This article presents the results of a randomized controlled trial focusing on the first two cohorts of a larger efficacy investigation aimed at building fraction sense in students with mathematics difficulties. Teachers implemented an evidence-informed fraction sense intervention (FSI) within their sixth-grade intervention classrooms. The lessons draw from research in cognitive science as well as mathematics education research. Employing random assignment at the classroom level, multilevel modeling revealed a significant effect of the intervention on posttest fractions scores after controlling for pretest fractions scores, working memory, vocabulary, proportional reasoning, and classroom attentive behavior. Students in the FSI group outperformed their counterparts in the control group, with noteworthy effect sizes on most fraction measures. Challenges associated with carrying out school-based intervention research are addressed.

The development of number line estimation in children at risk of mathematics learning difficulties: A longitudinal study.

Children with mathematics learning difficulties (MLD) show poorer performance on the number line task, but how performance on this task relates to other mathematical skills is unclear. This study examined the association between performance on the number line task and mathematical skills during the first 2 years of school for children at risk of MLD. Children (N = 100; Mage = 83.63 months) were assessed on four occasions on the number line task and other mathematical skills (math fluency, numerical operations, and mathematical reasoning). Estimation patterns were analyzed based on the representational shift and proportional judgment accounts separately. More consistent longitudinal trends and stronger evidence for differences in mathematical skills based on estimation patterns were found within the representational shift account. Latent growth curve models showed accuracy on the number line task as a predictor of growth in some mathematical skills assessed. We discuss impacts of methodological limitations on the study of estimation patterns.

Does Conceptual Transparency in Manipulatives Afford Place-Value Understanding in Children at Risk for Mathematics Learning Disabilities?

We investigated the effect of conceptual transparency in the physical structure of manipulatives on place-value understanding in typically developing children and those at risk for mathematics learning disabilities. Second graders were randomly assigned to one of three manipulatives conditions: (a) attachable beads that did not make the denominations or ones in the denominations transparent, (b) pipe cleaners that made only the denominations transparent, and (c) string beads that made both the denominations and the ones in the denominations transparent. Participants used the manipulatives to represent double- and triple-digit numerals. Statistical analyses indicated that the transparency of the denominations, but not the transparency of the ones in the denominations, is responsible for children's number representation and place-value understanding. Descriptive analyses of their responses revealed that the at-risk children were at a greater disadvantage than their typically developing peers with the attachable beads, failing to use place-value concepts to interpret their representations.

Embodied mathematical pedagogy to liberate racialized and multilingual bodies.

When language is defined narrowly in mathematics classrooms, racially and linguistically minoritized students in classrooms could be systematically positioned as "learners of deficiency." Recent scholarship calls for expanding the notion of language to emphasize embodied expression of mathematical ideas. Taking a critical perspective to understand racialized experiences of using languages in disciplinary learning spaces, this article proposes the reconceptualization of embodiment as a language for racialized multilingual learners. This study was conducted in a Grade 1 classroom in a linguistically and racially diverse school in Canada. Through a series of professional development sessions, we worked with an experienced teacher to redesign the normalized and institutionalized pedagogy toward greater mobility of racialized multilingual learners' bodies, which was intertwined with their intellectual liberation. Focusing on the spatiality of pedagogy, the previously restrictive areas in the school were transformed into a place that augments embodied expression of mathematical ideas and agentive participation of minoritized learners. The analysis focused on the embodied discourse that participating racialized multilingual students used to actively engage in mathematical discussion. Our findings show that the designed pedagogy, characterized by the spatial and temporal expansion of the learning environment, offered more spaces for uncertainty and spontaneity with the decreased control of the teacher as an explicator. Our article furthers anti-colonial approaches to understand the intersection of racialized bodies and language in mathematics education.

Parents' experiences of mathematics learning at home during the COVID-19 pandemic: a typology of parental engagement in mathematics education.

The COVID pandemic disrupted the schooling of students worldwide resulting in many having had a period of at-home learning. Many parents found themselves assuming responsibility for supporting their children's at-home learning. Parents often find it difficult to support their children's mathematics learning compared with other curriculum areas. There has been limited research exploring parental engagement in mathematics education generally, and little into parental engagement in mathematics education during the COVID pandemic. This paper examines how parents supported their child's mathematics education during the school closures and identifies the factors that impacted this engagement. The Ecologies of Parental Engagement (EPE) model was used to help describe the engagement of different parents in mathematics education during the school closures and to examine the way the home space and available capital shaped parental engagement. Eight parents were selected from a larger Australian study that explored the impact of the pandemic-induced period of at-home schooling on primary school mathematics and science. One-on-one narrative interviews were conducted online with participants. Analysis identified three categories of parental engagement: monitors, facilitators, and enhancers. Parents in each category responded to their role in at-home learning differently, and accessed and activated different capital to support their child's at-home learning in mathematics during the pandemic. Results highlight the value of emotional capital, as well as knowledge of mathematics and mathematics education, with implications for schools hoping to engage parents in mathematics learning. The study offers a typology to be explored in future research concerning parental engagement in mathematics education.

The effectiveness of web-based Mathematics instruction (WBMI) on K-16 students' mathematics learning: a meta-analytic research.

Given the increasing prevalence of web technology, web-based mathematics environments have been increasingly widely used in mathematics education for the past two decades. The COVID-19 pandemic has led to an urgent transition from traditional mathematics instruction (TMI) to web-based mathematics instruction (WBMI) at all levels of mathematics education. At this point, it is crucial to scrutinize the effects of WBMI on K-16 students' mathematics learning comprehensively. This meta-analysis research contained a total of 63 studies with 115 effect sizes, which aimed to investigate the effectiveness of WBMI on K-16 students' mathematics learning by incorporating potential moderators, namely mathematics topics, mathematical content standards, feedback status, type of instructional features, age (i.e., grade level), and assessment methods. Based on findings, WBMI has a significantly strong effect on K-16 students' mathematics learning (g = 1.10, p = 0.01, 95% CI [0.95, 1.27]). Moderator analyses reveal that the effect sizes of WBMI on K-16 students' mathematics learning varied significantly depending on all these potential moderators. Additionally, higher-level mathematical concepts, statistics and probability, WBMI with providing feedback, tutorial systems, undergraduate students, and traditional paper-pencil assessment are the strongest moderators in their context. The most notable results of this research are that WBMI is significantly more effective on students' mathematics learning than TMI, while even in the context of WBMI, traditional paper-pencil assessment is significantly more effective than online assessment. This meta-analytic research provides a comprehensive and up-to-date perspective on the effectiveness of WBMI on K-16 students' mathematics learning.

Looking for "us": power reimagined in mathematics learning for Black communities in the pandemic.

In this reflective essay, a BlackCrit lens is used to explore new and evolving possibilities for Black teachers, families, leaders, and students in ways that highlight and honor parents' agency, expand notions of digital equity in mathematics, and preview new and re-prioritized approaches which aid liberatory mathematics, teaching, and learning spaces that resurfaced in the pandemic. Several actions reimagine the work of mathematics as building blocks for engaging the flourishing for Black communities: (1) expanding and amplifying direct networks for Black parents to share, communicate, and advocate for their own needs and spaces around mathematics; (2) making visible and amplifying our advocacy for racial justice in the content creation and representation found in current digital platforms for meeting the needs of Black communities; and the need to (3) invest in, prioritize usage of, and illuminate mathematics commercial and academic entities focused solely on creating content and centering Black (and other people's) knowledge and experiences in mathematics for Black families.

Predicting Age of Becoming a Cardinal Principle Knower.

Children's first mathematics concept is their understanding of the quantities represented by number words (cardinal value), and the age at which they achieve this insight predicts their readiness for mathematics learning in school. We provide the first exploration of the factors that influence the age of becoming a cardinal principle knower (CPK), with a longitudinal study of 197 (94 boys) children from the beginning to the end of two years of preschool. Core symbolic and non-symbolic quantitative competencies at the beginning of preschool, as well as measures of intelligence, executive function, preliteracy skills, and parental education were used to predict timing of CPK status. Children who achieved early CPK status had higher IQ scores, knew more count words and numerals, and had a better intuitive understanding of relative quantity than their peers. Children who were delayed CPKs, in contrast, had deficits in executive function and poor preliteracy skills. The results add to our understanding of children's conceptual development in mathematics and have implications for the identification of at-risk children and design of interventions for them.

Differential effects of the classroom on African American and non-African American's mathematics achievement.

We examined whether African American students differentially responded to dimensions of the observed classroom-learning environment compared with non-African American students. Further, we examined whether these dimensions of the classroom mediated treatment effects of a preschool mathematics intervention targeted at students from low-income families. Three observed dimensions of the classroom (teacher expectations and developmental appropriateness; teacher confidence and enthusiasm; and support for mathematical discourse) were evaluated in a sample of 1,238 preschool students in 101 classrooms. Using multigroup multilevel mediation where African American students were compared to non-African American students, we found that teachers in the intervention condition had higher ratings on the observed dimensions of the classroom compared with teachers in the control condition. Further, ratings on teacher expectations and developmental appropriateness had larger associations with the achievement of African American students than for non-African Americans. Findings suggest that students within the same classroom may react differently to that learning environment and that classroom learning environments could be structured in ways that are beneficial for students who need the most support.

The benefits of computer-generated feedback for mathematics problem solving.

The goal of the current research was to better understand when and why feedback has positive effects on learning and to identify features of feedback that may improve its efficacy. In a randomized experiment, second-grade children received instruction on a correct problem-solving strategy and then solved a set of relevant problems. Children were assigned to receive no feedback, immediate feedback, or summative feedback from the computer. On a posttest the following day, feedback resulted in higher scores relative to no feedback for children who started with low prior knowledge. Immediate feedback was particularly effective, facilitating mastery of the material for children with both low and high prior knowledge. Results suggest that minimal computer-generated feedback can be a powerful form of guidance during problem solving.

Transcranial direct current stimulation of the posterior parietal cortex modulates arithmetic learning.

The successful acquisition of arithmetic skills is an essential step in the development of mathematical competencies and has been associated with neural activity in the left posterior parietal cortex (PPC). It is unclear, however, whether this brain region plays a causal role in arithmetic skill acquisition and whether arithmetic learning can be modulated by means of non-invasive brain stimulation of this key region. In the present study we addressed these questions by applying transcranial direct current stimulation (tDCS) over the left PPC during a short-term training that simulates the typical path of arithmetic skill acquisition (specifically the transition from effortful procedural to memory-based problem-solving strategies). Sixty participants received either anodal, cathodal or sham tDCS while practising complex multiplication and subtraction problems. The stability of the stimulation-induced learning effects was assessed in a follow-up test 24 h after the training. Learning progress was modulated by tDCS. Cathodal tDCS (compared with sham) decreased learning rates during training and resulted in poorer performance which lasted over 24 h after stimulation. Anodal tDCS showed an operation-specific improvement for subtraction learning. Our findings extend previous studies by demonstrating that the left PPC is causally involved in arithmetic learning (and not only in arithmetic performance) and that even a short-term tDCS application can modulate the success of arithmetic knowledge acquisition. Moreover, our finding of operation-specific anodal stimulation effects suggests that the enhancing effects of tDCS on learning can selectively affect just one of several cognitive processes mediated by the stimulated area.

The contribution of general cognitive abilities and approximate number system to early mathematics.

BACKGROUND: Math learning is a complex process that entails a wide range of cognitive abilities to be fulfilled. There is sufficient evidence that both general and specific cognitive skills assume a fundamental role, despite the absence of shared consensus about the relative extent of their involvement. Moreover, regarding general abilities, there is no agreement about the recruitment of the different memory components or of intelligence. In relation to specific factors, great debate subsists regarding the role of the approximate number system (ANS). AIMS: Starting from these considerations, we wanted to conduct a wide assessment of memory components and ANS, by controlling for the effects associated with intelligence and also exploring possible relationships between all precursors. SAMPLE AND METHOD: To achieve this purpose, a sample of 157 children was tested at both beginning and end of their Grade 1. Both general (memory and intelligence) and specific (ANS) precursors were evaluated by a wide battery of tests and put in relation to concurrent and subsequent math skills. Memory was explored in passive and active aspects involving both verbal and visuo-spatial components. RESULTS: Path analysis results demonstrated that memory, and especially the more active processes, and intelligence were the strongest precursors in both assessment times. ANS had a milder role which lost significance by the end of the school year. Memory and ANS seemed to influence early mathematics almost independently. CONCLUSION: Both general and specific precursors seemed to have a crucial role in early math competences, despite the lower involvement of ANS.

The impact of reading metacognitive strategies on mathematics learning efficiency and performance: An analysis using PISA 2018 data in China.

The current study employed latent profile analysis to examine the application patterns of students' reading metacognitive strategies using PISA 2018 data in China. Subsequently, it explored the differences in students' mathematics learning efficiency and performance. The results revealed that: (1) Six types of reading metacognitive strategies application patterns were identified: "Novice - indifferent," "Veteran - average," "Novice - low evaluating," "Veteran - skilled," "Novice - mixed," and "Novice - arbitrary." (2) The primary factors that affect the classification of reading metacognitive strategies application patterns were gender, and family economic, social, and cultural statuses (ESCS). (3) Mathematics learning time could positively predict performance overall, but the mathematics learning time of "Veteran - skilled" and "Novice - mixed" students had no significant correlation with their mathematics performance. The findings suggests that educators should not blindly increase students' mathematics learning time but instead provide appropriate guidance based on their mastery patterns of reading metacognitive strategies to enhance mathematics learning efficiency and performance.

A reciprocal association between interpersonal relationship quality and student's positive affect: A three-wave random intercept cross-lagged study in a Chinese primary mathematics learning context.

AIM: This study adopted a three-wave random intercept cross-lagged panel model to explore the longitudinal reciprocal relationships between (a) the teacher-student relationship (TSR) quality and (b) the parent-child relationship (PCR) quality and positive affect among Chinese primary school students. SAMPLES: Two primary school student samples, including 3505 and 2505 students, were tracked with their perceived relationship quality with math teachers and parents and their positive affect levels in mathematics learning over three academic years. RESULTS: The results demonstrated that more closeness with parents could significantly predict students' subsequent higher level of positive affect in mathematics learning. However, more closeness with their math teachers did not show significant prediction. Meanwhile, more conflict with math teachers and parents could significantly predict their subsequent lower degree of perceived positive affect in mathematics learning. That is, a reciprocal association lines in the PCR quality and positive affect, whereas only a unidirectional association exists between the TSR and positive affect. The predictions of the experienced positive affect on their perceived interpersonal relationships with math teachers and parents were stronger than those in the reverse association. CONCLUSIONS: This study identifies that while the effects of closeness with math teachers and parents on positive affect in students' math learning differ, conflict with math teachers and parents indeed harms students' experienced positive affect in math learning. More attention should also be paid to fostering positive affect in math learning.

The Relationship Between Resilience, Interactive Distance, and College Students' Online Mathematics Learning Engagement: A Longitudinal Study.

Introduction: Resilience, a pivotal construct in positive psychology, remains incompletely understood in its facilitation of learners' online engagement. This study aims to investigate the relationship between resilience, transactional distance, and Online Mathematics Learning Engagement (OMLE) among first-year university students. Methods: Utilizing a cross-lagged path analysis approach, the study surveyed 612 first-year students. Multiple models were constructed and compared to explore the mutual predictive relationships between resilience, transactional distance, and OMLE. Results: Among the compared models, Model 4 demonstrated the best fit. The model revealed that: (1) resilience at Time 1 and Time 2 positively predicted transactional distance at Time 2 and Time 3; (2) transactional distance at Time 1 and Time 2 positively predicted OMLE at Time 2 and Time 3; (3) resilience at Time 1 significantly predicted OMLE at Time 3; and (4) transactional distance at Time 2 fully mediated the relationship between resilience at Time 1 and OMLE at Time 3. Furthermore, mediational model analysis confirmed that transactional distance played a mediating role in the longitudinal relationship between resilience and OMLE. Using a cross-lagged mediational model with 5000 bootstrap samples, the indirect effect of transactional distance on the relationship between resilience at Time 1 and OMLE at Time 3 was significant and remained stable over time. Discussion: The findings suggest that resilience, as a positive psychological resource, stimulates students to seek and utilize protective resources in online environments, leading to more active participation in interpersonal communication and classroom interactions. Additionally, resilience helps students overcome emotional and practical difficulties encountered in online learning, thereby enhancing their OMLE. These insights offer valuable implications for educators, highlighting the potential to improve students' online learning engagement by fostering their psychological resilience.

Perceived parental involvement influences students' academic buoyancy and adaptability: the mediating roles of goal orientations.

Academic buoyancy and adaptability (i.e., student capacities to deal with difficulties and challenges in daily school lives and to make appropriate cognitive, behavioral as well as affective adjustments in interacting with new, uncertain, and/or changing situations, circumstances, and conditions) can help learners regulate and protect themselves in the failure-prone learning environment. This study examined how students' perceptions of parental academic involvement and their goal orientations were related to their academic buoyancy and adaptability in mathematics learning. We recruited a sample of 1,164 Chinese junior high school students. Using structural equation modeling, the results indicated that after controlling for family socioeconomic status and gender, perceived parental involvement was positively related to the students' academic buoyancy and adaptability. Furthermore, parental involvement was significantly associated with students' mastery and performance-approach goal orientations, which further partially mediated the relationship between parental involvement and academic buoyancy and adaptability. However, the mediating role of a performance-avoidance goal orientation in this relationship was not significant. Findings highlight the important roles that parenting practices and individual achievement motivation play in the development of academic buoyancy and adaptability in the Chinese context. Future research directions and implications are discussed.

Chinese students' filial piety beliefs and procrastination in mathematics learning: The mediating role of academic emotions.

This study examined the relations between Chinese students' filial piety beliefs and mathematics procrastination and the mediating role of academic emotions in the relations. Analysis of data on 1,476 primary school students in China with structural equation modeling revealed that students' reciprocal and authoritarian filial piety beliefs were positively related to academic enjoyment and anxiety, respectively. Students' procrastination in mathematics learning was positively related to anxiety and authoritarian filial piety beliefs and had negative associations with enjoyment and reciprocal filial piety beliefs. The bootstrap analysis results confirmed the mediating role of anxiety in the relation between authoritarian filial piety beliefs and procrastination. Reciprocal filial piety beliefs had negative indirect relationship with procrastination via enjoyment. The results were explained from a socio-cultural perspective. The theoretical contributions and practical implications are discussed.

Some Correct Strategies Are Better Than Others: Individual Differences in Strategy Evaluations Are Related to Strategy Adoption.

Why do people shift their strategies for solving problems? Past work has focused on the roles of contextual and individual factors in explaining whether people adopt new strategies when they are exposed to them. In this study, we examined a factor not considered in prior work: people's evaluations of the strategies themselves. We presented undergraduate participants from a moderately selective university (N = 252; 64.8% women, 65.6% White, 67.6% who had taken calculus) with two strategies for solving algebraic word problems and asked them to rate these strategies and their own strategy on a variety of dimensions. Participants' ratings loaded onto two factors, which we label quality and difficulty. Participants' initial evaluations of the quality of the strategies were associated with whether they used the strategies at posttest, and this effect held even when controlling for individual and contextual factors. However, people's evaluations of the difficulty of the strategies were not consistently associated with their later adoption of those strategies. We also examined individual and contextual predictors of strategy ratings and strategy adoption. Participants' need for cognition and their spatial visualization ability were associated with their strategy evaluations, and the framing of the story problems was associated with their strategy adoption. The findings highlight that strategy adoption depends on multiple interacting factors, and that to understand strategy change, it is critical to examine how people evaluate strategies.

Secondary school students' use and perceptions of textbooks in mathematics learning: A large-scale investigation in China.

Students' use of textbooks is the key link of students engaged and learned curriculum and has received much attention recently. However, existing studies were mainly case studies or small-scale investigations and few addressed the context of China. Hence, this study provided a general overview of mathematics textbook use by Chinese secondary students through a large-scale investigation. Using a mixed-method approach, we collected the quantitative data from 2,145 students in eight provinces through a questionnaire survey and the qualitative data from 20 students and 8 teachers by the interviews. The results revealed that (1) Chinese students relied heavily on mathematics textbooks and pointedly used a portion of components in textbooks, mainly kernels, examples, and exercises; (2) Chinese students used mathematics textbooks for various but typical reasons, particularly to understand basic knowledge and skills, and showed self-regulation and teacher-mediation behind their use; and (3) Chinese students had a positive view about textbook use in mathematics learning, especially in developing mathematical knowledge, skills, and abilities. Furthermore, there were significant differences in mathematics textbook use among different students in terms of school regions, grade levels, and teachers' demographic variables. Finally, explanations and implications of the results were discussed.

Ordinal models to analyze strategy sophistication: Evidence from a learning trajectory efficacy study.

Investigators often rely on the proportion of correct responses in an assessment when describing the impact of early mathematics interventions on child outcomes. Here, we propose a shift in focus to the relative sophistication of problem-solving strategies and offer methodological guidance to researchers interested in working with strategies. We leverage data from a randomized teaching experiment with a kindergarten sample whose details are outlined in Clements et al. (2020). First, we describe our problem-solving strategy data, including how strategies were coded in ways that are amenable to analysis. Second, we explore what kinds of ordinal statistical models best fit the nature of arithmetic strategies, describe what each model implies about problem-solving behavior, and how to interpret model parameters. Third, we discuss the effect of "treatment", operationalized as instruction aligned with an arithmetic Learning Trajectory (LT). We show that arithmetic strategy development is best described as a sequential stepwise process and that children who receive LT instruction use more sophisticated strategies at post-assessment, relative to their peers in a teach-to-target skill condition. We introduce latent strategy sophistication as an analogous metric to traditional Rasch factor scores and demonstrate a moderate correlation them (r = 0.58). Our work suggests strategy sophistication carries information that is unique from, but complimentary to traditional correctness-based Rasch scores, motivating its expanded use in intervention studies.