This article presents a case study of two Grade 5 boys' argumentation concerning addition and subtraction of negative numbers while using an interactive tablet-based application simulating positive and negative tiles. We examine the properties of integers they conjectured, and the kinds of evidence and arguments they used to support their…

Adaptive teachers use student data to guide instruction. Learn about using an anecdotal record form to support adaptive teaching. In this article, the authors describe how teachers can use learning trajectories to make adaptive decisions to meet the needs of their students. They provide an example using the U .S. Math Recovery Council's learning…

Evidence in South Africa of poor performance on word problems points to particular challenges among learners relating to difficulties with, on the one hand, making sense of mathematical word problems and, on the other, moving beyond inefficient counting based calculation methods. While many studies in mathematics education in South Africa have…

The aim of this research is to determine the mental processing strategies used by 3rd grade primary school students in addition and subtraction using the empty number lines. The model of the research is a case study, one of the qualitative research methods. The participants consist of 143 students; 75 girls and 68 boys, attending the third grade…

This paper frames children's mathematics as mathematics. Specifically, it draws upon our knowledge of children's mathematics and applies it to understanding the prime number theorem. Elementary school arithmetic emphasizes two principal operations: addition and multiplication. Through their units coordination activity, children construct two…

We examine a hypothesis implied by Steffe's constructivist model of children's numerical reasoning: a child's "spontaneous" additive strategy may relate to a foundational form of multiplicative reasoning, termed multiplicative double counting (mDC). To this end, we mix quantitative and qualitative analyses of 31 fourth graders' responses…

We report on 25 Year 5-6 students' written responses to two items taken from an assessment of mental computation fluency with multiplication, alongside their reasoning of the strategy they had employed, which may or may not have made use of the associative property. Coding of this interview data revealed four distinct levels of conceptual…

In this study we investigated the development of 245 kindergarteners' mathematics problem solving (MPS) at two time points within 7 months using a cognitive diagnostic test, which measured three cognitive components (mathematical knowledge and skills, semantic understanding, and quantitative reasoning) with eleven cognitive attributes. The effects…

Assessing student learning traditionally involves determining whether students can solve a certain percentage of problems correctly, under the assumption that this achievement indicates they have the knowledge and understanding they need to progress to new topics. This article explores what teachers can learn from looking closely at student…

This paper examines a problem described as widespread and long-standing in mathematics education: supporting pupils into multiplicative reasoning, a form of reasoning that has been noted as central to large tracts of secondary mathematics and beyond. Also noted, however, is a persistent perception of multiplicative situations only in terms of…

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…

In a cross-sectional study, 160 students in Grades 2, 4, 7, and 11 were interviewed about their reasoning when solving integer addition and subtraction open-number-sentence problems. We applied our previously developed framework for 5 Ways of Reasoning (WoRs) to our data set to describe patterns within and across participant groups. Our analysis…

According to numerous studies (Barrouillet & Camos 2002; Brousseau 1988; Chevallard 1988; Riley et al. 1984; Schubauer-Leoni & Ntamakiliro, "Revue Des Sciences de L'éducation," 20(1): 87-113, 1994; Vergnaud 1982; Xin, "The Journal of Educational Research," 100(6):347-360, 2007), a combination of many factors, including…

Mathematical relationships are crucial elements to consider for learning mathematics. However, too often students pay more attention to the calculations to be done rather than the reasons for doing them. Relying on the relational paradigm to support elementary school students, we proposed two specially designed tasks to help students recognize and…

Many general and special education teachers teach mathematics word problems by defining problems as a single operation and linking key words to specific operations. Unfortunately, teaching students to approach word problems in these ways discourages mathematical reasoning and frequently produces incorrect answers. This article lists eight common…