In this article, we analyze a first grade classroom episode and individual interviews with students who participated in that classroom event to provide evidence of the variety of understandings about variable and variable notation held by first grade children approximately six years of age. Our findings illustrate that given the opportunity,…

Studies that consider the displays that students create to organize data are not common in the literature. This article compares fifth through eighth graders' difficulties with the creation of bar graphs using either raw data (Study 1, n = 155) or a provided table (Study 2, n = 152). Data in Study 1 showed statistical differences for the type…

Recent reform efforts in mathematics education have stimulated a focus on learning trajectories. At the same time, a global increase in high-stakes testing has influenced instructional practices. This study investigated how four fourth grade teachers within a school planned and enacted lessons to understand what mediated their planning and…

Mathematics curricula in the early years emphasize exploring patterns as an important stepping stone to algebraic thinking. Key expectations for young children are for them to recognize and describe visual patterns in the environment. When children are asked to identify patterns, what criteria do they apply to make a decision and on what do they…

Explanations are considered to be key aids to understanding the study of mathematics, science, and other complex disciplines. This paper discusses the role of students' explanations in making sense of data and learning to reason informally about statistical inference. We closely follow students' explanations in which they utilize their experiences…

People often have difficulties in understanding situations that involve non-linear processes. Also, the topic of non-linear functions is introduced relatively late in the curriculum. Previous research has nevertheless shown that already children aged 6 years and older are able to discriminate non-linear from linear processes. Within the present…

This critical ethnographic study of an after-school mathematics club for elementary-aged Latina/o youth focuses on connecting critical, community, and mathematical knowledge in the context of authentic, community-based investigations. We present cases of two extended projects to highlight tensions and dilemmas that emerged, particularly tensions…

Little research has focused on an informal understanding of subtractive negation (e.g., 3 - 3 = 0) and subtractive identity (e.g., 3 - 0 = 3). Previous research indicates that preschoolers may have a fragile (i.e., unreliable or localized) understanding of the addition-subtraction inverse principle (e.g., 2 + 1 - 1 = 2). Recognition of a small…

Subtraction problems of the type a - b = ? can be "flexibly" solved by various strategies, including the indirect addition strategy ("how much do I have to add to b to get at a?"). Little research has been done on the use of the indirect addition strategy with multi-digit numbers. The present literature review entails a summary of three recent and…

Two intervention studies are described. Both were designed to study the effects of teaching children about the inverse relation between addition and subtraction. The interventions were successful with 8-year-old children in Study 1 and to a limited extent with 5-year-old children in Study 2. In Study 1 teaching children about inversion increased…

This study analyzed the processes used by students when engaged in modeling activities and examined how students' abilities to solve modeling problems changed over time. Two student populations, one experimental and one control group, participated in the study. To examine students' modeling processes, the experimental group participated in an…

Learning trajectories are presented of 2 fifth-grade children, Jason and Laura, who participated in the teaching experiment, Children's Construction of the Rational Numbers of Arithmetic. 5 teaching episodes were held with the 2 children, October 15 and November 1, 8, 15, and 22. During the fourth grade, the 2 children demonstrated distinctly…

This article highlights the similarities between sixth-grade students' developing notions of angle and mathematicians' struggles to define this complex concept. Students learning from a reformed curriculum had the opportunity to voice their ideas concerning the definition of angle. These conversations were compared with past discussions and…