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Mathematics is a fundamental part of life, yet every one of us has a unique relationship with learning and understanding the subject. Working with numbers may inspire confidence in our abilities or provoke anxiety and trepidation. Stanford researcher, mathematics education professor, and the leading expert on math learning Dr. Jo Boaler argues that our differences are the key to unlocking our greatest mathematics potential. In Math-ish, Boaler shares new neuroscientific research on how embracing the concept of \"math-ish\"--a theory of mathematics as it exists in the real world--changes the way we think about mathematics, data, and ourselves. When we can see the value of diversity among people and multi-faceted approaches to learning math, we are free to truly flourish. --Provided by publisher.

Abstract The current meta-analysis quantifies the average effect of worked examples on mathematics performance from elementary grades to postsecondary settings and to assess what moderates this effect. Though thousands of worked examples studies have been conducted to date, a corresponding meta-analysis has yet to be published. Exclusionary coding was conducted on 8033 abstracts from published and grey literature to yield a sample of high quality experimental and quasi-experimental work. This search yielded 43 articles reporting on 55 studies and 181 effect sizes. Using robust variance estimation (RVE) to account for clustered effect sizes, the average effect size of worked examples on mathematics performance outcomes was medium with g = 0.48 and p = 0.01. Moderators assessed included example type (correct vs. incorrect examples alone or in combination with correct examples), pairing with self-explanation prompts, and timing of administration (i.e., practice vs. skill acquisition). The inclusion of self-explanation prompts significantly moderated the effect of examples yielding a negative effect in comparison to worked examples conditions that did not include self-explanation prompts. Worked examples studies that used correct examples alone yielded larger effect sizes than those that used incorrect examples alone or correct examples in combination with incorrect examples. The worked examples effect yields a medium effect on mathematics outcomes whether used for practice or initial skill acquisition. Correct examples are particularly beneficial for learning overall, and pairing examples with self-explanation prompts may not be a fruitful design modification. Theoretical and practical implications are discussed.

The purpose of this meta-analysis was to examine the relation between mathematics anxiety (MA) and mathematics performance among school-aged students, and to identify potential moderators and underlying mechanisms of such relation, including grade level, temporal relations, difficulty of mathematical tasks, dimensions of MA measures, effects on student grades, and working memory. A meta-analysis of 131 studies with 478 effect sizes was conducted. The results indicated that a significant negative correlation exist between MA and mathematics performance, r = -.34. Moderation analyses indicated that dimensions of MA, difficulty of mathematical tasks, and effects on student grades differentially affected the relation between MA and mathematics performance. MA assessed with both cognitive and affective dimensions showed a stronger negative correlation with mathematics performance compared to MA assessed with either an affective dimension only or mixed/unspecified dimensions. Advanced mathematical task that require multistep processes showed a stronger negative correlation to MA compared to foundational mathematical tasks. Mathematics measures that affected/reflected student grades (e.g., final exam, students 'course grade, GPA) had a stronger negative correlation to MA than did other measures of mathematics performance that did not affect student grades (e.g., mathematics measures administered as part of research). Theoretical and practical implications of the findings are discussed.

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.

Mathematical difficulties have been distinguished as mathematics learning disability (MLD) and persistent low achievement (LA). Based on 1,880 Finnish children who were followed from kindergarten (age 6) to fourth grade, this study examined the early risk factors for MLD and LA. Distinct groups of MLD (6.0% of the sample) and LA (25.7%) children were identified on the basis of their mathematics performance between first and fourth grades with latent class growth modeling. Impairment in the same set of cognitive skills, including language, spatial, and counting skills, was found to underlie MLD and LA. The finding highlights the importance of monitoring mathematical development across the early grades and identifying early cognitive precursors of MLD and LA for screening and intervention efforts.

We investigated the cognitive demand of the exercise problems that middle school students from Nepal are expected to complete. The middle school (grades 6, 7, and 8) mathematics textbooks examined in this study were published and distributed by the government of Nepal. Our data set consisted of textbooks that are currently in use in public schools of Nepal and used by majority of the middle school students in Nepal. Using the mathematical tasks framework to analyze the data, we found more than 92% tasks are of lower cognitive demands, and most tasks are at the level of procedures without connections. We also observed that the majority (more than 75%) of exercise problems are supplemented with similar worked examples. We discuss implications and suggest future studies.

This study draws upon social cognitive career theory and higher education literature to test a conceptual framework for understanding the entrance into science, technology, engineering, and mathematics (STEM) majors by recent high school graduates attending 4-year institutions. Results suggest that choosing a STEM major is directly influenced by intent to major in STEM, high school math achievement, and initial postsecondary experiences, such as academic interaction and financial aid receipt. Exerting the largest impact on STEM entrance, intent to major in STEM is directly affected by 12th-grade math achievement, exposure to math and science courses, and math self-efficacy beliefs—all three subject to the influence of early achievement in and attitudes toward math. Multiple-group structural equation modeling analyses indicated heterogeneous effects of math achievement and exposure to math and science across racial groups, with their positive impact on STEM intent accruing most to White students and least to under-represented minority students.

Identifying the types of mathematics content knowledge that are most predictive of students' long-term learning is essential for improving both theories of mathematical development and mathematics education. To identify these types of knowledge, we examined long-term predictors of high school students' knowledge of algebra and overall mathematics achievement. Analyses of large, nationally representative, longitudinal data sets from the United States and the United Kingdom revealed that elementary school students' knowledge of fractions and of division uniquely predicts those students' knowledge of algebra and overall mathematics achievement in high school, 5 or 6 years later, even after statistically controlling for other types of mathematical knowledge, general intellectual ability, working memory, and family income and education. Implications of these findings for understanding and improving mathematics learning are discussed.

Recent research in cognitive and developmental neuroscience is providing a new approach to the understanding of dyscalculia that emphasizes a core deficit in understanding sets and their numerosities, which is fundamental to all aspects of elementary school mathematics. The neural bases of numerosity processing have been investigated in structural and functional neuroimaging studies of adults and children, and neural markers of its impairment in dyscalculia have been identified. New interventions to strengthen numerosity processing, including adaptive software, promise effective evidence-based education for dyscalculic learners.