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…

Cross-lagged panel analysis was used to determine the causal relationship between arithmetic progress and two factors (concrete operational development and visuo-spatial figurative ability) in 28 learning disabled boys (9-12 years old). Results of seven operational tasks, five figurative tests, and a diagnostic arithmetic test administered early…

The relationship between the development of reversible thought and performance in arithmetic equations among children was investigated. Subjects were 86 second grade boys and girls. Two reversibility tasks and 20 addition and subtraction equations were administered. Results indicated significant correlations between the two variables only among…

Described in the context of a computer simulation are highlights of a program of research focusing on the storage of mathematics problem solving information in young children's memory and the development of such knowledge structures in older children. Specifically discussed are the problem size effect, the network nature of the memory…

Equivalent fractions are usually introduced in fourth grade and reviewed repeatedly in the subsequent grades as the four arithmetical operations are taught. In spite of this repeated instruction, the results are disappointing. This paper reviews some data from previous research documenting the difficulty of equivalent fractions, explains this…

In this paper four programs are described in which children learn multidigit number concepts and operations with understanding: (1) the Supporting Ten-Structured Thinking projects, (2) the Conceptually Based Instruction project, (3) Cognitively Guided Instruction projects, and (4) the Problem Centered Mathematics Project. The diversity in these…

If we consider the gap between mathematics at elementary and secondary levels, and the logical nature of the higher level, it is important that aspects of children's logical development in the latter grades in elementary school be clarified. We focused on 5th graders' learning "division with decimals" as it is known to be difficult to…

A longitudinal study of students' developing understanding of decimal notation has been conducted by testing over 3000 students in Grades 4 to 10 up to 7 times. A pencil-and-paper test based on a carefully designed set of decimal comparison items enabled students' responses to be classified into 11 codes and tracked over time. The paper reports on…

This paper considers whether or not curriculum initiatives within England are in fact leading to qualitatively improved levels of thinking among children identified as lower achievers in arithmetic. Revisiting data from two earlier studies and introducing current data, the paper draws comparisons in the strategies that 8-year-old children use to…

This investigation used a clinical interview technique to identify the difficulties of kindergarten children who are unable to develop models or representations of simple arithmetic story problems. It is hypothesized that effective human problem solvers first generate some type of "physical model" and use this model to create a…

This experiment tested the hypothesis that combining operational strategies with free verbalization facilitates development of competencies, percepts of self-efficacy, and interest in arithmetic activities. Children lacking division skills received treatments in which they either verbalized division strategies, verbalized freely, did both, or did…

Students in grades 1, 3, 4, 5, 6, and college were timed as they produced the answers to simple addition problems or verified a given problem as true or false. First graders clearly relied on a counting process for their performance, as advanced by the Groen and Parkman "min" (for minimum addend) model. Third grade appears to be a…

During the last decades, a shift from studying the product to studies of the processes has become more and more obvious. The major aim of this study was to investigate the mental performance process and to examine the changes the variables describing this process undergo with time. In this investigation, a mental arithmetic test was used. The test…

An influential proposal about aquiring mathematical knowledge is that it entails linking instruction-based concepts to intuitions derived from informal activities. In the case of non-positive numbers, informal knowledge is unlikely to emanate from observing physical objects, because non-positive objects or sets of objects do not exist. However, it…

Although many common problem-solving errors within the realm of school mathematics have been previously identified, a compilation of such errors is not readily available within learning disabilities textbooks, mathematics education texts, or teacher's manuals for school mathematics texts. Using data on error frequencies drawn from both the Fourth…