Competence with rational numbers is essential for mathematics proficiency in secondary mathematics. However, many students struggle with rational number concepts, and students with mathematics difficulties struggle even more. The purpose of this study was to examine the effects of an intervention that incorporated the use of explicit instruction…

A pivotal information about the structure of students' comprehension underlying from which the knowledge is gained can be identified through the mental model. This study aimed at describing the six levels of students' mental model in comprehending the concept of the integer. The subject of this research was 40 students consisting of 20 students…

In this article, we report the findings of a study conducted with 6 Grade 8 students in the United States. The students posed stories for open number sentences involving addition and subtraction of integers. We analysed the stories posed by the students to build models that describe the conceptual structures behind these posed stories -- the…

The number line and the neutralization model have been used very extensively in teaching integer additions and subtractions for decades. Despite their advantages, issues concerning subtractions on these models are still debatable. Therefore, the neutralization on an empty number line (NNL) model is proposed in this research to help students better…

The author wants her students to see any new mathematics--fractions, negative numbers, algebra--as logical extensions of what they already know. This article describes two students' efforts to make sense of their conflicting interpretations of 1/2 × -6, both of which were compelling and logical to them. It describes how discussion, constructing…

In this proposal, I introduce a method for modeling the dynamics of a sixth-grade student's accommodation of his fractions scheme to include a disembedding operation (Steffe & Olive, 2010). I will describe a three-part approach consisting of a constructivist teaching experiment, retrospective analysis, and stochastic modeling of the student's…

The intention of the present study is to know how the pupils can learn to make a group of ten to understand the idea of unitizing. The pupils were given a contextual problem "Counting the Beads" in order to promote their understanding about the idea of unitizing. The process of designing the problem was based on the 5 tenets of…

This article reviews a particular computational modeling approach to the study of psychological development--that of constructive neural networks. This approach is applied to a variety of developmental domains and issues, including Piagetian tasks, shift learning, language acquisition, number comparison, habituation of visual attention, concept…

Lamon ("Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers," 2nd edn. Lawrence Erlbaum Associates, Mahwah, 2005) claimed that the development of proportional reasoning relies on various kinds of understanding and thinking processes. The critical components suggested were…

A solid understanding of equivalent fractions is considered a steppingstone towards a better understanding of operations with fractions. In this article, 55 rural Australian students' conceptions of equivalent fractions are presented. Data collected included students' responses to a short written test and follow-up interviews with three students…

In this study children between the ages of 3 and 6 years were each presented with six number problems in which length and density were varied according to the proposed composition rules. (Authors)

Several researchers have noted how children's whole number schemes can interfere with their efforts to learn fractions. An Australian study found that children who were successful with the solution of rational number tasks exhibited greater whole number knowledge and more flexible solution strategies. Behr and Post (1988) indicated that children…

The research on conceptual change has so far mainly dealt with cognitive outcomes, but especially during the last few years there has been a growing interest in and discussion about the processes of conceptual change. The purpose of the article is to contribute to this discussion and to present a theoretical model of the dynamics among the…

Examines bar models as graphical representations of rational numbers and presents related real life problems. Concludes that, through pairing the fraction bars with ratio tables and other ways of teaching numbers, numeric strategies become connected with visual strategies that allow students with diverse ways of thinking to share their…

Studies concerned with the synchronous emergence prediction of Piaget's structures-of-the-whole principle are discussed in conjunction with three groups of concrete-operational skills: (1) transitivity/conservation/class inclusion; (2) double classification/double seriation; and (3) ordinal, cardinal, and natural number concepts. Findings show…