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Whole number arithmetic is the foundation of higher mathematics and a core part of elementary mathematics. Awareness of pattern and underlying problem structure promote the learning of whole number arithmetic. A growing consensus has emerged on the necessity to provide students with the opportunity to engage in algebraic reasoning earlier in their education. In fact, the U.S. National Council of Teachers...
This paper makes a proposal, from the perspective of a research mathematician interested in mathematics education, for broadening and deepening whole number arithmetic instruction, to make it more relevant for the twenty-first century, in particular, to enable students to deal with large numbers, arguably an essential skill for modern citizenship. It suggests that, rather than being content with accurate...
Whole-number arithmetic is a core content area of primary mathematics, which lays the foundation for children’s later conceptual development. This paper focuses on teaching whole-number multiplication (WNM) to build a stepping stone for children’s proportional reasoning. Our intention in writing this paper is to obtain a practice-based perspective on the teaching of WNM, given the growing attention...
In this paper we analyze the outcomes of a design experiment that sought to improve the multiplicative reasoning of 12–13 year-old learners across two schools in a South African context of low performance. Using a hybrid theoretical base in Realistic Mathematics Education, variation theory and analogical reasoning, a short-term intervention consisting of four lessons, designed with attunement to classroom...
A central issue in the mathematics curriculum is that we want students to make connections. This issue has been analysed in a series of curricula and instruction design and analysis studies. Moving towards mathematics connections—and away from treating mathematics as a body of isolated concepts and procedures—is an important goal of mathematics education. Although many studies identify “whole number...
The idea of using fingers as a key component in arithmetic development has received a great deal of support, much of which is based on neuroscientific evidence. However, this body of work pays limited attention to how fingers are used and possible different outcomes in arithmetic problem solving. The aim of our paper, based on an analysis of 126 observations of 4–5-year-olds solving a simple subtraction...
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 understanding and teaching number and place value. The research...
This study investigates the teaching and learning of single-digit whole number multiplication in China. Analysis of data from documents, classroom teaching, and semi-structured interviews revealed three salient characteristics of emphasizing oral calculation, calculation speed, and understanding across standards, textbook and classroom practices. It also showed how mathematics teachers enact these...
This paper is centred around a framework for studying teaching—Mediating Primary Mathematics (MPM)—developed in the context of the teaching of Whole Number Arithmetic (WNA) in South Africa. Findings from the analysis of four WNA lessons are used to illustrate how the application of the MPM framework can measure nuanced differences in the meditational means—both artefact- and human-based—employed by...
The goal of this paper is to present a succinct consolidation of the why, what and how of the ‘Model’ method as there is a lack of such documentation for researchers, practitioners and policy makers thus far. The ‘Model’ method is a tool for representing and visualising relationships when solving whole number arithmetic (WNA) word problems. The paper traces the history of the method in the Singapore...
Teachers and mathematicians hold different perspectives on the teaching and learning of whole number arithmetic. Though these perspectives may be complementary, sharing them across communities is challenging. An unusual professional development course for primary school teachers, initiated and taught by research mathematicians, provided a setting for productive sharing of both mathematical and pedagogical...
The paper presents some reflections and activities developed by researchers and teachers involved in teacher education programs on cultural transposition. The construct of cultural transposition is presented as a condition for decentralizing the didactic practice of a specific cultural context through contact with other didactic practices of different cultural contexts. We discuss the background theoretical...
Numeration units (ones, tens, hundreds, etc.) are an epistemological foundation for place value. Here, we assume that they are a didactic support in countries where spoken numbers are incongruent with written numbers. In France, teaching practices neglect the relations between units, and numeration units are typically used only to designate the positions of digits. Our research, which adopts the didactical...
Research has shown that word problem difficulty is influenced by various linguistic, numerical and general variables, but results in this area are inconsistent. This paper focuses on the question of how the order of numerical data, the context, the position of the unknown transformation, and the length of the statement in an additive word problem, influence the achievement and reasoning of primary...
Whole Number Arithmetic (WNA) appears as the very first topic in school mathematics and establishes the foundation for later mathematical content. Without solid mastery of WNA, students may experience difficulties in learning fractions, ratio and proportion, and algebra. The challenge of students’ learning and mastery of fractions, decimals, ratio and proportion, and algebra is well documented. Most...
Solving word problems is challenging in elementary schools, both for the teacher in teaching students to solve word problems and for the student in learning to solve them. This paper examines how the ideas of learning trajectory and variation pedagogy could be integrated as an instructional principle for teaching this content in the context of solving additive comparison problems. Based on research...
This paper addresses teachers’ corporeality and its influence on the structuring of mathematics classroom interaction. The present contribution explores the way a teacher’s bodily activity is implicated in the (re)production of social inequality, and challenges the assumption that the teaching and learning of mathematics is primarily a linguistic and semiotic accomplishment. I argue that the teacher’s...
The growing number of interpretive lenses used in mathematics education research are often seen in terms of either/or, such as the individual or the social, the discursive or the bodily, the classroom interactive or the neurological events, the humanist or the post-humanist. We propose stratigraphy as a research (meta-)method that is conjunctive rather than disjunctive, resisting the temptation to...
In this paper, we examine historically the onto-epistemological framework of mathematics education that generates cultural theses about modes of life. We pursue the idea of able bodies to think about schooling as producing kinds of people as well as their differences and divisions despite efforts for inclusion. Two historical moments in United States mathematics education are examined: pre-post WWII...
This paper responds to the frequently occurring phenomena of ‘sad affects’ experienced by student teachers as they confront the logic of ‘proper mathematics’ with young children of differently abled bodies. The notion of ‘proper’ entails a normalizing fixity of ‘what counts’ as mathematics for certain children and fails to recognize mathematics as a sensual encounter amongst bodies. The study draws...
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