This article describes students learning to build their own numbering system by recognizing and identifying patterns with interlocking cubes in different place values. The students used the Egyptian hieroglyphic numeral system in conjunction with this activity to connect learning in other subjects. Students used prior knowledge of place value to…

Psychological studies of early numerical development fill a void in mathematics education research. However, conflations between magnitude awareness and number, and over-attributions of researcher conceptions to children, have led to psychological models that are at odds with findings from mathematics educators on later numerical development. In…

Prospective elementary school teachers (PTs) come to their mathematics courses fluent in using procedures for adding and subtracting multidigit whole numbers, but many are unaware of the essential features inherent in understanding the base-ten place-value system (i.e., grouping, place value, base). Understanding these features is crucial to…

The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the…

A growing body of evidence suggests that non-symbolic representations of number, which humans share with nonhuman animals, are functionally related to uniquely human mathematical thought. Other research suggesting that numerical and non-numerical magnitudes not only share analog format but also form part of a general magnitude system raises…

Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…

This paper describes a design research study with ten second-grade students who are part of the Measure Up (MU) research and development project underway at the University of Hawai'i. Students were asked how they counted in multiple bases, specifically how they knew when to go to a new place value and why it was necessary to do so. All ten…

Discusses mastery goals and developmental levels of students aged 3-7 for learning numbers in informal situations. (KHR)

A cross-sectional study of 132 Australian rural children from grades K-6 assessed children's understanding of the number system. Task-based interview data exhibited lack of understanding of the base ten system, with little progress made during Grades 5 and 6. Few Grade 6 children used holistic strategies or generalised the structure of the number…

The goal of this study was to describe characteristic stages in the development of children's understanding of place value. Sixty children (15 in each grade, 2 through 5) at 5 elementary schools in Butte County, California, were individually administered 18 tasks designed to reveal their thinking about place value and part-whole relations. On six…

Presents an insightful approach to algebra that enhances both personal and classroom communication through the process of solving a linear equation. (KHR)

Presents a covariation approach to learning exponential and logarithmic functions based on a primitive multiplicative operation labeled splitting that is not repeated addition. Suggests that students need the opportunity to build a number system from splitting structures and their geometric forms. (30 references) (MKR)

Two studies indicate that Oksapmin children progress from premediational to mediational phases in their use of body parts to compare and reproduce number and that this change generally occurs prior to the development of concepts of number conservation. A third study shows that this general change is manifested in culturally specific ways.…

The question is raised: What comes first: rules of calculation or the meaning of concepts? The pressures on the teacher to teach and simplify knowledge to algorithms are discussed. The relation between conceptual and procedural knowledge in school mathematics and consequences for the teacher's professional knowledge are considered. (DC)