Children's early accuracy on place value (PV) tasks longitudinally predicts their later multidigit calculation skills. However, another window into children's emerging base-ten concepts is the pattern of errors -- "smart errors" -- they exhibit on these measures. Past research has speculated that these smart errors -- similar to invented…

The results of this study revealed students' solution strategies progressed from inefficient strategies e.g., counting on fingers to more efficient strategies with mathematical connections. These findings should enable educators to gain more insight to use Number Talks to help student develop computational fluency with numbers.

This study used hierarchical linear modeling to examine the relationship between an internet-based mathematics formative assessment and data from a mathematics summative assessment for primary grade learners (ages 5-7). Results showed a positive relationship between formative assessment data related to the concepts of counting and decomposing…

There is an ongoing debate over the psychophysical functions that best fit human data from numerical estimation tasks. To test whether one psychophysical function could account for data across diverse tasks, we examined 40 kindergartners, 38 first graders, 40 second graders and 40 adults' estimates using two fully crossed 2 × 2 designs, crossing…

The Common Core State Standards for Mathematics (CCSSI 2010) expect second grade students to "fluently add and subtract within 20 using mental strategies" (2.OA.B.2). Most children begin with number word sequences and counting approximations and then develop greater skill with counting. But do all teachers really understand how this…

This study explored factors that are associated with calculation development among 176 children at-risk for learning difficulties. Data were collected on working memory, language, non-verbal reasoning, processing speed, decoding, numerical competence, incoming calculation, SES, and gender at the beginning of 1st grade and on calculations over 4…

Children's estimation skills on a bounded and unbounded number line task were assessed in the light of their familiarity with numbers. Kindergartners, first graders, and second graders (N = 120) estimated the position of numbers on a 1--100 number line, marked with either two reference points (i.e., 1 and 10: unbounded condition) or three…

This paper compares the counting and whole number knowledge and skills of primary school children in Australia and Germany at the end of Grade 1 and Grade 2. Children's learning was assessed using the Early Numeracy Interview and associated Growth Point Framework. The findings highlight substantial differences between the two groups that vary for…

One crucial issue in mathematics development is how children come to spontaneously apply arithmetical principles (e.g. commutativity). According to expertise research, well-integrated conceptual and procedural knowledge is required. Here, we report a method composed of two independent tasks that assessed in an unobtrusive manner the spontaneous…

The purpose of this study was to examine child-level pathways in development of prealgebraic knowledge versus word-problem solving, while evaluating the contribution of calculation accuracy and fluency as mediators of foundational skills/processes. Children (n = 962; mean 7.60 years) were assessed on general cognitive processes and early…

The purpose of this study was to examine child-level pathways in development of prealgebraic knowledge versus word-problem solving, while evaluating the contribution of calculation accuracy and fluency as mediators of foundational skills/processes. Children (n = 962; mean 7.60 years) were assessed on general cognitive processes and early…

The purpose of this study was to compare subgroups of students with various forms of learning difficulties (< 25th percentile) on cognitive and mathematics characteristics. Students with mathematics difficulty (MD, n = 105), reading difficulty (RD, n = 65), both (MDRD, n = 87), or neither (NoLD, n = 403) were evaluated on an array of cognitive…

We examined a web-based mathematics progress monitoring tool for second graders. The tool monitors the learning progress of two competences, number sense and computation. A total of 414 students from 19 classrooms in Germany were checked every 3 weeks from fall to spring. Correlational analyses indicate that alternate-form reliability was adequate…

Part of a larger initiation mixed methods study (Greene, Caracelli, & Graham, 1989), this paper discusses the changes in young children's learning performance and efficiency (one element of the quantitative portion of the larger study) during clinical interviews in which each child interacted with a variety of virtual manipulative…

Understanding the linear relationship of numbers is essential for doing practical and abstract mathematics throughout education and everyday life. There is evidence that number line activities increase learners' number sense, improving the linearity of mental number line representations (Siegler & Ramani, 2009). Mental representations of…