How do children develop associations among number symbols? For Grade 1 children (n = 66, M = 78 months), sequence knowledge (i.e., identify missing numbers) and number comparison (i.e., choose larger number) predicted addition, both concurrently and indirectly at the end of Grade 1. Number ordering (i.e., touch numbers in order) did not predict…

Although basing instruction on a learning trajectory (LT) is often recommended, there is little direct evidence to support the premise of a "LT approach"--that to be maximally meaningful, engaging, and effective, instruction is best presented one LT level beyond a child's present level of thinking. The present report serves to address…

The understandings prospective mathematics teachers develop by focusing on quantities and quantitative relationships within real numbers have the potential for enhancing their future students' understanding of real numbers. In this article, we propose an instructional sequence that addresses quantitative relationships for the construction of real…

Although basing instruction on a learning trajectory (LT) is often recommended, there is little direct evidence to support the premise of a "LT approach"--that to be maximally meaningful, engaging, and effective, instruction is best presented one LT level beyond a child's present level of thinking. The present report serves to address…

Recent intervention research has demonstrated success in raising the achievement level of students who are struggling with mathematics. This practice guide, developed by the What Works Clearinghouse™ (WWC) in conjunction with an expert panel, distills this contemporary mathematics intervention research into six easily comprehensible and practical…

The procedural deficit hypothesis claims that impaired procedural learning is a causal risk factor for developmental dyslexia and developmental language disorder. We investigated the relationships between measures of basic cognitive processes (declarative learning, procedural learning and attention) and measures of attainment (reading, grammar and…

Students must leave elementary school with a firm understanding of fractions and decimals in order to make satisfactory progress in advanced mathematics. Firm understanding includes manipulation of rational numbers using different models of representation such as area and length. Research has demonstrated that a graduated sequence such as…

Are there differential benefits of training sequential number knowledge versus spatial skills for children's numerical and spatial performance? Three- to five-year-old children (N = 84) participated in 1 session of either sequential training (e.g., what comes before and after the number 5?) or non-numerical spatial training (i.e., decomposition of…

During the semester break, 36 second-grade students accessed a set of resources and completed a series of online math activities focused on the application of the model method for arithmetic in two contexts 1) addition/subtraction and 2) multiplication/division. The learning environment first modeled and then supported the use of a scripted series…

This article describes a comprehensive and novel approach to assessment in early numeracy. Topics include the significance of early numeracy, developing a nomenclature for early numeracy, and describing the context for the development of this approach to assessment. The largely unrealised importance of numerals and numeral sequences in early…

Visuospatial working memory and its involvement in arithmetic were examined in two groups of 7- to 11-year-olds: one comprising children described by teachers as displaying symptoms of nonverbal learning difficulties (N = 21), the other a control group without learning disabilities (N = 21). The two groups were matched for verbal abilities, age,…

Teachers can help students with learning problems in solving arithmetic word problems by following the principles commonly used in the design of Direct Instruction curricula: teach rules and strategies rather than rote memorization; sequence examples and skills carefully; and introduce and review skills cumulatively. (CB)

The way in which two teachers, Elizabeth and Carolyn, posed problems to develop their own conceptual understanding of division of fractions in terms that would also be meaningful for their students is described. Carolyn and Elizabeth's approach is to pose several problems of various degrees of difficulty and complexity for each aspect of the…

Studies of young children's sequence of development from counting to the beginnings of formal arithmetic are reviewed. Four essential basic skills are identified: use of counting words, enumeration, the cardinality rule, and quantitative comparison. The contribution of counting to the development of arithmetical proficiency is stressed. (JDD)

Presented is the task analysis for "Developing Mathematical Processes, Arithmetic, Book 1: Comparing and Equalizing Objects and Sets." This task analysis consists of identifying the behavioral objectives necessary for comparing and equalizing objects and sets and arranging them in a logical sequence for instruction. The 52 items in this task…