Development of the cardinality principle, an understanding that the last number-word recited in counting a collection of items specifies the number of items in that collection, is a critical milestone in developing a concept of number. Researchers in early number development have endeavored to theorize its development. Here we critique two widely…

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…

Background: Recent research showed that cross-notation magnitude knowledge of fractions and decimals was related to better performance in fraction arithmetic, but it remains unclear whether it made an independent contribution to fraction arithmetic longitudinally when other cognitive variables are considered. Aims: To examine the extent to which…

The purpose of this paper is to propose an effective learning environment for the initial stages of mathematical learning. Basic numerical skills and the objects and actions that trigger those skills are conceptualized as a mathematics-learning environment. We discuss numerical learning mechanism and the basic skills and environments we use to…

Understanding how young learners come to construct viable mathematical arguments about general claims is a critical objective in early algebra research. The qualitative study reported here characterizes empirically developed progressions in Grades K-1 students' thinking about parity arguments for sums of evens and odds, as well as underlying…

The integer is a basic concept in studying arithmetic and algebra. However, students still frequently experience misconceptions, especially in negative integer, count operations. Traditional games are activities that are often carried out by students in coastal areas so that they are relevant to be used as a tool to construct learning trajectories…

Although many U.S. children can count sets by 4 years, it is not until 5½--6 years that they understand how counting relates to number--that is, that adding 1 to a set necessitates counting up one number. This study examined two knowledge sources that 3½- to 6-year-olds (N = 136) may leverage to acquire this "successor function": (a)…

In this article we show how students' productive struggle on a mathematical task can lead to collective mathematical creativity. We use observable (co)actions and interactions from a video record that features three Grade 6 students in a problem-solving session to document the emergence of collective creativity leading to a solution. We discuss…

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…

We investigate how students make sense of irrational exponents. The data comprise 32 interviews with university students, which revolved around a task designed to examine students' sensemaking processes involved in predicting and subsequently interpreting the shape of the graph of f(x)=x[superscript square root of 2]. The task design and data…

This study aims to determine the effect of the cognitive conflict approach on the elimination of misconceptions in square root numbers. For this purpose, this study was conducted with 8th-grade students of a secondary school in a region of Turkey. A three tiered diagnostic test was used to determine the impact of the 5-step cognitive conflict…

In this study, a situation-based interactive e-book system was developed for improving children's learning performance of numerical operations. To evaluate the effectiveness of the system, an experiment was conducted in one kindergarten. The results showed that the proposed approach significantly improved the children's learning achievements.…

Place value is a foundational competency for primary school mathematics and for this reason we have sought to investigate what the recent and current academic conversations are around this important concept. In this paper we present a survey of literature presented in the Australasian, European and Southern African contexts through a review of…

Infants and toddlers experience the world in interaction with others. Likewise, social interactions are important in learning about math--concepts of number ("one," "two," "three"), space ("on top," "upside-down," "round"), and magnitude ("more," "big,"…

This is the first of a two-part article that presents a theory of unit construction and coordination that underlies radical constructivist empirical studies of student learning ranging from young students' counting strategies to high school students' algebraic reasoning. My explanation starts with the formation of arithmetical units, which presage…