Manipulatives are concrete or virtual objects (e.g., blocks and chips) often used in elementary grades to illustrate abstract mathematical concepts. We conducted a systematic review to examine the effects of interventions delivered with manipulatives on the learning of children with mathematics learning disabilities (MLD). The outcomes observed in the sample (N = 38) were learning, maintenance, and transfer in a variety of mathematical domains. Interventions using manipulatives were reported to be effective for a range of learning objectives (e.g., conceptual understanding and computational fluency), but several methodological weaknesses were observed. Analyses also highlighted considerable heterogeneity in the studies reviewed in terms of participant characteristics, intervention approaches, and methodology. We discuss overall effects of interventions with manipulatives in the MLD population, the methodological quality across the sample, and implications for practice.

In the present research, we provide empirical evidence for the process of symbolic integration of number associations, focusing on the development of simple addition (e.g., 5 + 3 = 8), subtraction (e.g., 5 - 3 = 2), and multiplication (e.g., 5 × 3 = 15). Canadian children were assessed twice, in Grade 2 and Grade 3 (N = 244; 55% girls). All families were English-speaking, and parent education levels ranged from high school to postgraduate, with a median of community college. In Grade 2, children completed general cognitive tasks (i.e., receptive vocabulary, working memory, nonverbal reasoning, and inhibitory control). In both grades, children completed single-digit addition and complementary subtraction problems. In Grade 3, they completed single-digit multiplication problems and measures of applied mathematics, specifically, word-problem solving, algebra, and measurement. We found that addition and subtraction were reciprocally related (controlling for cognitive skills). Subtraction fluency predicted multiplication in Grade 3, whereas addition fluency did not. In Grade 3, both subtraction and multiplication fluency were predictors of applied mathematics, with multiplication partially mediating the relation between subtraction and applied mathematics performance. These findings support the view that learning arithmetic associations is a hierarchical process. As students practice each new skill, individual differences reflect the integration of the novel component into the developing associative network. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

In the current study, we used a multi-method approach to understand the quality of math homework-helping interactions between parents and their children and how parents’ and children’s own math achievement and math anxiety relate to the quality of the interaction. Forty Canadian parents and their children (ages 10–12 years; grades 5 to 7) completed self-report measures of math and general anxiety. Parents and children completed standardized assessments of math achievement and were then recorded as they engaged in a simulated math homework interaction. Coders assessed parent–child interaction quality during the interaction. Parent–child dyads generally performed well on the simulated math homework task. Nevertheless, task performance was correlated with the quality of the interaction, with high-quality interactions associated with high accuracy on the math task. Furthermore, the variability in the quality of the interaction was associated with parents’ and children’s math achievement and with the math anxiety of the children, but not the parents. Identifying the elements that influence parent–child interactions in math-related situations is essential to developing effective interventions to scaffold children’s math learning and attitudes.

The home learning environment includes parental activities, attitudes, affect, knowledge, and resources devoted to supporting children’s development, including literacy and mathematics skills. These factors are related to the academic performance of preschool children (aged 3 to 6 years), before formal schooling and possibly beyond. In the present research, we examined the home learning environment of Canadian families as reported by either the mother (n = 51) or father (n = 30) of their Grade 3 child (n = 81; M_age = 8.7 years; range 8 to 9 years of age). Importantly, mothers’ and fathers’ reports of the home learning environment for school children were similar. For literacy, parents’ knowledge of children’s books and attitudes toward literacy were related to children’s vocabulary skills; home literacy was not related to word reading skills. For mathematics, parents’ reports of the frequency of activities such as practicing arithmetic facts and their attitudes toward mathematics were related to children’s arithmetic fluency. Other aspects of the home learning environment (time spent helping with homework, parents’ math anxiety) were not related to children’s performance. These results suggest some continuity between home learning environments and academic skills after children’s transition to school.

Concrete objects used to illustrate mathematical ideas are commonly known as manipulatives. Manipulatives are ubiquitous in North American elementary classrooms in the early years, and although they can be beneficial, they do not guarantee learning. In the present study, the authors examined two factors hypothesized to impact second-graders’ learning of place value and regrouping with manipulatives: (a) the sequencing of concrete (base-ten blocks) and abstract (written symbols) representations of the standard addition algorithm; and (b) the level of instructional guidance on the structural relations between the representations. Results from a classroom experiment with second-grade students (N = 87) indicated that place value knowledge increased from pre-test to post-test when the base-ten blocks were presented before the symbols, but only when no instructional guidance was offered. When guidance was given, only students in the symbols-first condition improved their place value knowledge. Students who received instruction increased their understanding of regrouping, irrespective of representational sequence. No effects were found for iterative sequencing of concrete and abstract representations. Practical implications for teaching mathematics with manipulatives are considered.