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Prior research highlights a positive effect of incorrect worked examples on mathematics learning. Yet the mechanisms underlying these benefits are unclear. To investigate potential mechanisms of the benefits of various worked example types, we examined process data from a previously published classroom-based experiment. More specifically, we analyzed students' explanations made while explaining worked examples in three varying example conditions as well as students' problem-solving errors made when solving problems. These data operationalize two potential mechanisms: a reduction of misconceptions (i.e., fewer targeted conceptual errors), and an increase in principled algebra knowledge (i.e., explanations focusing on principles underlying procedures). Mediation analyses revealed both as important mechanisms of varying effects. A reduction of misconceptions explained greater benefits of all three worked example conditions, compared to a problem-solving control, on an algebra concepts posttest. More principled explanations of procedures explained the benefits of incorrect worked examples on problem-solving at posttest compared to the two other example conditions. These findings help explain differential findings in prior work by example type and may elucidate potential avenues for errorful instruction. [ABSTRACT FROM AUTHOR]

Background: Providing students with worked out problem solutions is a beneficial instructional technique in STEM disciplines, and studying examples that have been worked out incorrectly may be especially helpful for reducing misconceptions in students with low prior content knowledge. However, past results are inconclusive and the effects of incorrect worked examples alone or in combination with correct examples remains unclear. Objectives: We aim to address whether studying incorrect examples alone or in combination with correct examples can support the reduction of students' fraction misconceptions, operationalized as errors made with high confidence. Methods: After incorrectly solving a sampling problem, 130 students in 4th through 11th grade in the U.S. were randomly assigned to a condition in an online problem set focused on fraction equivalence. Students studied either single‐type worked examples (i.e., correct or incorrect; n = 49) or combination‐type worked examples (correct and incorrect; n = 41) or engaged in a problem‐solving control (n = 50). Results: Studying a combination of correct and incorrect worked examples was as effective as the problem‐solving control with feedback at improving fraction equivalence knowledge and reducing the rate of high‐confidence errors. Students in both the combination condition and the problem‐solving with feedback condition outperformed those who studied either correct or incorrect worked examples alone. Conclusions: Results support the inclusion of a combination of correct and incorrect worked examples when teaching students with low prior content knowledge. Studying a combination of example types within an online tutor helps to reduce misconceptions about fractions, a topic students commonly struggle with. A problem‐solving task with corrective feedback worked equally well. Lay Description: What is already known about this topic: Studying worked examples of mathematical solutions improves student learning.Incorrect examples, either alone or in combination with correct examples, are particularly effective for improving learning for students with low prior knowledge in the target content.Comparing correct and incorrect examples is also effective at improving problem‐solving but this effect may be specific to those with high prior knowledge.Still unknown are the impacts of these different combinations of examples on misconceptions in particular. What this paper adds: We show that studying a combination of correct and incorrect examples reduces math misconceptions.Problem‐solving with corrective feedback within an online tutor also has this effect. Implications for practice and/or policy: Using a combination of example types will help reduce students' misconceptions about fractions. [ABSTRACT FROM AUTHOR]

• Parents align into 4 profiles based on variations in expectations and activities. • Parent race, nationality, education and income are related to profile membership. • Profiles were differentially related to kindergarteners' reading and math outcomes. • High parental expectations and home activities promoted children's academic skills. The kindergarten transition has implications for children's short- and long-term academic trajectories. In this study, we assess the role that parent expectations about kindergarten readiness and parent–child home activities have on children's early reading and mathematics skills. We conducted latent profile analyses (LPA) using the Early Childhood Longitudinal Study Kindergarten (ECLS-K) Class of 2010–2011 parent interviews (n = 12,670). Results revealed four distinct profiles: (1) high expectations, fewest home activities; (2) high expectations, high activities; (3) very high expectations, moderate home activities; (4) very high expectations, most home activities. We examined both predictors and distal outcomes of displaying these four profiles. When examining children's academic skills at kindergarten entry, children with parents in the very high expectations, most home activities profile (Profile 4) have the most advanced academic reading and mathematics skills. Therefore, this profile was deemed most optimal. Children with parents in the very high expectations, moderate activities profile (Profile 3) had higher reading and math scores than children in Profiles 1 and 2. Findings of this descriptive study reveal the heterogeneity of parental involvement during the kindergarten transition and the need for more nuanced research investigating how families assist children during this critical developmental period. [ABSTRACT FROM AUTHOR]

The effectiveness of an experimental middle school fraction intervention was evaluated. The intervention was centered on the number line and incorporated key principles from the science of learning. Sixth graders (N = 51) who struggled with fraction concepts were randomly assigned at the student level to the experimental intervention (n = 28) or to a business-as-usual control who received their school's intervention (n = 23). The experimental intervention occurred over 6 weeks (27 lessons). Fraction number line estimation, magnitude comparisons, concepts, and arithmetic were assessed at pretest, posttest, and delayed posttest. The experimental group demonstrated significantly more learning than the control group from pretest to posttest, with meaningful effect sizes on measures of fraction concepts (g = 1.09), number line estimation as measured by percent absolute error (g = −.85), and magnitude comparisons (g =.82). These improvements held at delayed posttest 7 weeks later. Exploratory analyses showed a significant interaction between classroom attentive behavior and intervention group on fraction concepts at posttest, suggesting a buffering effect of the experimental intervention on the normally negative impact of low attentive behavior on learning. A number line–centered approach to teaching fractions that also incorporates research-based learning strategies helps struggling learners to make durable gains in their conceptual understanding of fractions. Educational Impact and Implications Statement: A mathematics intervention that used a number line–centered approach and validated learning principles to teach fraction concepts helped struggling sixth graders improve their fraction understanding. After participating in the intervention, students performed better on assessments of fraction concepts, number line estimation, and magnitude comparisons than a group of students who received their school's regular intervention, and these improvements held seven weeks later. Findings suggest that students who are struggling with fractions, even after receiving several years of formal fraction instruction in school, can still make large gains in their understanding, preparing them for more advanced mathematics and for success in STEM related fields. [ABSTRACT FROM AUTHOR]

The current study examined the predictive utility of sense of belonging to math relative to a range of other beliefs previously established as important for achievement (i.e., self-concept, interest, importance, and entity view). A racially and ethnically diverse sample of U.S. public middle-school students (N = 206) completed surveys and an algebra assessment before and after a classroom unit. Of the motivation and belief measures taken, sense of belonging to mathematics was the only significant predictor of learning. Underrepresented racial and ethnic minority students (URM) reported a markedly lower sense of belonging than non-URM students which partially explained their lower posttest scores, even when accounting for socioeconomic status. URM students did not differ from non-URM students on any other motivation or belief measures. We provide the necessary foundational knowledge of one understudied but potentially malleable factor for consideration in improving URM students' experience and opportunities in mathematics. • Sense of belonging to mathematics is a significant predictor of algebra learning. • Self-concept, entity view, interest, and importance were not significant factors. • Underrepresented minority students (URM) had lower sense of belonging than non-URMs. • URMs did not differ from non-URMs on other belief measures or prior knowledge. • URMs' lower belonging partially explained learning differences, accounting for SES. [ABSTRACT FROM AUTHOR]

Reasoning about numerical magnitudes is a key aspect of mathematics learning. Most research examining the relation of magnitude understanding to general mathematics achievement has focused on whole number and fraction magnitudes. The present longitudinal study (N = 435) used a 3-step latent class analysis to examine reasoning about magnitudes on a decimal comparison task in 4th grade, before systematic decimals instruction. Three classes of response patterns were identified, indicating empirically distinct levels of decimal magnitude understanding. Class 1 students consistently gave correct responses, suggesting that they understood decimal properties even before systematic decimal instruction. Class 2 students were accurate when a 0 immediately followed the decimal, but were inaccurate when a zero was added to the end of the decimal string, suggesting a partial understanding of place value; their performance was also negatively influenced by a whole number bias. Class 3 students showed misunderstanding of both place value and a whole number bias. Class membership accurately predicted 6th grade mathematics achievement, after controlling for whole number and fraction magnitude understanding as well as demographic and cognitive factors. Taken together, the findings suggest students may benefit from instruction that emphasizes decimal properties earlier in school. [ABSTRACT FROM AUTHOR]

Prior research has documented differences in both performance and motivation between students with learning disabilities (LD) and non-learning disabled (non-LD) students. However, few studies have conducted a finer grained analysis comparing students with LD with nondisabled students of varying achievement levels. The present study examines differences between LD, low-achieving, average-achieving, and high-achieving adolescents on algebra performance and readiness, motivational constructs (competence expectancy, interest, and goal orientation in mathematics), and the discrepancy between students' competence and their perceptions of their own competence. Results indicate that while students with LD may demonstrate lower algebra readiness and algebra achievement and more inaccurate judgments of their own competence compared with the whole non-LD sample, critical differences in performance and motivation were most evident between high-achieving and low-achieving students, not students with learning disabilities. [ABSTRACT FROM AUTHOR]

The efficacy of a research-based fraction sense intervention for sixth graders with or at risk for mathematics difficulties (N = 52) was examined. The intervention aimed to build understanding of fraction magnitudes on the number line. Key concepts were taught with a narrow range of denominators to develop deep understanding. The intervention was centered on a visual number line in the meaningful context of a color run race. Students were randomly assigned to the fraction sense intervention (n = 25) or a business-as-usual control group (n = 27). Students in the intervention condition received 21 lessons in small groups (45 min each) during their regular mathematics intervention period. Students in the intervention group performed significantly better than those in the control group on a measure of fraction number line estimation and a more general measure of fraction concepts, both at immediate posttest and delayed posttest, with large effect sizes; lesser effects were shown for fraction arithmetic. [ABSTRACT FROM AUTHOR]