Fraction arithmetic is a foundational aspect of mathematics education, yet many students, particularly those with mathematics difficulties (MD), struggle with this skill. The present study examined the types of errors made by 251 sixth-grade students with MD when solving simple fraction addition, subtraction, and multiplication problems. Ten distinct error categories emerged from the coding process. The errors indicated that many sixth-grade students with MD rely on flawed procedural strategies rather than drawing on conceptual understanding of fractions or operations. Targeted interventions are suggested that strengthen foundational fraction concepts and operational knowledge, especially with familiar fractions, are needed to support students with MD in developing a deeper, more flexible understanding of fractions, enabling them to reason confidently rather than depend on flawed procedural approaches.

In this study, our team observed sixth-grade teachers as they taught fractions to students in their mathematics intervention classes to see whether they were including motivational-supportive messages within the framework of Situated Expectancy-Value Theory. Messages in the intervention lessons and teacher transcripts were explored and analyzed, comparing teachers taking part in the experimental fractions intervention and teachers teaching business-as-usual. The fraction lessons and motivational messages were coded via teachers' audio recordings over the span of 15 weeks (24 lessons). Findings revealed that teachers in the experimental group used significantly more motivationally supportive messages than the teachers in the control group, and they made deeper connections between content and students' lives. These results imply that the use of an intervention including motivational messages can further promote teachers' use of motivational strategies while teaching mathematical content.

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.

Mathematics difficulties (MD) are widespread. Higher mathematics achievement is associated with college success and greater job opportunities. In this chapter, we address the intersection between cognitive research on the science of learning and the education of students with MD. First, a conceptual framework for mathematics learning is provided, including consideration of individual differences in domain general and domain specific learning processes. Next, learning principles that promote successful learning in mathematics are described, along with examples of how the principles can be translated into educational practice for students with MD. We argue that in addition to providing developmentally appropriate instruction in content knowledge, such as whole number and fraction knowledge, application of more general learning principles validated by research will lead to deeper and more durable learning for struggling students. These learning principles include studying and comparing correct as well as incorrect worked out problem-solutions; using integrated visual and verbal models to reduce splitting attention; interleaving or varying practice with problems of different types; providing frequent cumulative practice or review that is spaced out over time; connecting and integrating concrete and symbolic representations; presenting arithmetic problems in different formats; using physical movements and gestures to promote learning; and incorporating activities with number lines to increase learning of whole numbers and fractions.

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.

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. (PsycINFO Database Record (c) 2020 APA, all rights reserved)

This chapter provides an overview of a kindergarten (ie, children 5–6 years of age) number sense intervention that was developed and tested over a 4-year period. The intervention targeted core understandings of number, number relations, and number operations in high-risk learners. Over multiple trials, we obtained moderate to strong effect sizes, relative to controls, on a measure closely aligned to the intervention topics (ie, a number sense brief screener) as well as on a test of mathematics calculation achievement. Many of the gains were still present 2 months after the intervention ended. Catching number sense weaknesses early positions children for later success in formal mathematics.

The goal of the present article is to synthesize findings to date from the Delaware Longitudinal Study of Fraction Learning. The study followed a large cohort of children ( N = 536) between Grades 3 and 6. The findings showed that many students, especially those with diagnosed learning disabilities, made minimal growth in fraction knowledge and that some showed only a basic grasp of the meaning of a fraction even after several years of instruction. Children with low growth in fraction knowledge during the intermediate grades were much more likely to fail to meet state standards on a broad mathematics measure at the end of Grade 6. Although a range of general and mathematics-specific competencies predicted fraction outcomes, the ability to estimate numerical magnitudes on a number line was a uniquely important marker of fraction success. Many children with mathematics difficulties have deep-seated problems related to whole number magnitude representations that are complicated by the introduction of fractions into the curriculum. Implications for helping students with mathematics difficulties are discussed.

The efficacy of a research-based number-sense intervention for low-achieving kindergartners was examined. Children (N= 126) were randomly assigned to 1 of 3 conditions: a number-sense intervention followed by a number-fact practice session, an identical number-sense intervention followed by a number-list practice session, or a business-as-usual control group. The interventions were delivered in a small-group setting over 24 half-hour lessons. Both intervention groups performed better than controls on measures of number sense, arithmetic fluency, and general mathematics calculation achievement at immediate posttest. However, the number-fact practice condition gave children an additional advantage over the number-list practice condition on the outcomes at delayed posttest 8 weeks later. The number-fact practice condition was especially effective for producing gains in English learners.

The present study involved examining whether a storybook reading intervention targeting mathematics vocabulary, such as “equal,” “more,” and “less,” and associated number concepts would increase at-risk children’s vocabulary knowledge and number competencies. Children with early numeracy difficulties (N = 124) were recruited from kindergarten classes in four schools. Participants were randomly assigned to one of three groups: a storybook number competencies (SNC) intervention, a number sense intervention, or a business-as-usual control. Interventions were carried out in groups of four children over 8 weeks (24 30-minute sessions). Findings demonstrated that the SNC intervention group outperformed the other groups on measures of mathematics vocabulary, both in terms of words that were closely aligned to the intervention and those that were not. There was no effect of the SNC intervention, however, on general mathematics measures, suggesting a need to provide the mathematics vocabulary work along with more intensive instruction in number concepts.

The present study involved examining whether a storybook reading intervention targeting mathematics vocabulary, such as "equal," "more," and "less," and associated number concepts would increase at-risk children's vocabulary knowledge and number competencies. Children with early numeracy difficulties (N = 124) were recruited from kindergarten classes in four schools. Participants were randomly assigned to one of three groups: a storybook number competencies (SNC) intervention, a number sense intervention, or a business-as-usual control. Interventions were carried out in groups of four children over 8 weeks (24 30-minute sessions). Findings demonstrated that the SNC intervention group outperformed the other groups on measures of mathematics vocabulary, both in terms of words that were closely aligned to the intervention and those that were not. There was no effect of the SNC intervention, however, on general mathematics measures, suggesting a need to provide the mathematics vocabulary work along with more intensive instruction in number concepts.

Positioned for future success: Evidence-based number sense interventions (NSI) can help kindergartners link their nonverbal understanding of quantities to the symbolic representations of number, number relations, and number operations.

Abstract Early number competencies predict later mathematical learning. Weaknesses in number, number relations, and number operations can be reliably identified before school entry in first grade. Income status, associated early home and preschool opportunities, and general cognitive capacity all influence children’s level of numerical knowledge. Interventions based on a developmental progression and targeted to specific areas of number, such as the ability to count and sequence numbers, compare numerical quantities, and add and subtract small quantities, have shown positive, meaningful, and lasting effects on children’s achievement. Guided practice is effective when configured to support efficient counting strategies, frequent correct responding, and meta-cognitive behaviour and when contextualized with a strong focus on number knowledge tutoring.

Domain-general skills that mediate the relation between kindergarten number sense and first-grade mathematics skills were investigated. Participants were 107 children who displayed low number sense in the fall of kindergarten. Controlling for background variables, multiple regression analyses showed that both attention problems and executive functioning were unique predictors of mathematics outcomes. Attention problems were more important for predicting first-grade calculation performance, whereas executive functioning was more important for predicting first-grade performance on applied problems. Moreover, both executive functioning and attention problems were unique partial mediators of the relationship between kindergarten and first-grade mathematics skills. The results provide empirical support for developing interventions that target executive functioning and attention problems in addition to instruction in number skills for kindergartners with initial low number sense.

This study examined prospective teachers’ (PSTs) ability to recognize evidence of children’s conceptual understanding of mathematics in three content areas before and after an instructional intervention designed to support this ability. It also investigates the role PSTs’ content knowledge plays in their ability to recognize children’s mathematical understanding. Results of content knowledge assessments administered at the beginning of the study revealed that content knowledge did seem to support PSTs’ analyses of children’s understanding when the child’s response demonstrated understanding or demonstrated a misconception. Content knowledge did not seem to support PSTs’ analyses of children’s procedural responses, as many PSTs with good content knowledge initially characterized procedural solutions as evidence of conceptual understanding. Similarly, content knowledge did not seem to support PSTs’ analyses of children’s responses with features commonly associated with understanding but not evidence of understanding. After the instructional intervention consisting of three multifaceted lessons in which PSTs examined many examples of student thinking, they showed improved ability to analyze responses with conceptual features and no evidence of conceptual understanding and responses demonstrating procedural knowledge. Results suggest that content knowledge is not sufficient for supporting PSTs’ analysis of children’s thinking, and that building activities such as the intervention into content courses may help develop this ability. Implications for teacher education programs and future research are considered.

Early number sense is a strong predictor of later success in school mathematics. A disproportionate number of children from low-income families come to first grade with weak number competencies, leaving them at risk for a cycle of failure. The present study examined the effects of an 8-week number sense intervention to develop number competencies of low-income kindergartners ( N = 121). The intervention purposefully targeted whole number concepts related to counting, comparing, and manipulating sets. Children were randomly assigned to either a number sense intervention or a business as usual contrast group. The intervention was carried out in small-group, 30-min sessions, 3 days per week, for a total of 24 sessions. Controlling for number sense at pretest, the intervention group made meaningful gains relative to the control group at immediate as well delayed posttest on a measure of early numeracy. Intervention children also performed better than controls on a standardized test of mathematics calculation at immediate posttest.