Aim of the study: This study examines numerical methods for solving the problems in gas dynamics, which are based on an exact or approximate solution to the problem of breakdown of an arbitrary discontinuity (the Riemann problem). Results: Comparative analysis of finite difference schemes for the Euler equations integration is conducted on the…

It is well documented in literature that rational number is an important area of understanding in mathematics. Therefore, it follows that teachers and students need to have an understanding of rational number and related concepts such as fraction multiplication and division. This practitioner reference paper examines models that are important to…

Many studies have shown the difficulties of learning and teaching the decimal number system for whole numbers. In the case of numbers bigger than one hundred, complexity is partly due to the multitude of possible relationships between units. This study was aimed to develop conditions of a resource which can help teachers to enhance their teaching…

The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1,701 primary…

A common theme of models of conceptual growth is to establish the hierarchical structures of abilities that can be interpreted along developmental lines. Integrating the literature on the development of mathematical concepts and skills in children, a comprehensive 6 level model for describing, explaining and predicting the development of key…

The paper describes the history of how learning trajectories (LTs) were associated with the Common Core State Standards for Mathematics (CCSS-M) and discusses the degree to which the two correspond faithfully. It reports on a website, www.turnonccmath.com, which organizes the K-8 standards into 18 LTs describing the development of big ideas over…

Although learning mathematics certainly depends upon accurate understanding of the facts of multiplication, it requires much more. This study examines the relationship between a meaningful understanding of arithmetic operations and the mastery of basic facts. The study began with a joke about a mistaken mathematical fact. The children appreciated…

In mathematics education, physical manipulatives such as algebra tiles, pattern blocks, and two-colour counters are commonly used to provide concrete models of abstract concepts. With these traditional manipulatives, people can communicate with the tools only in one another's presence. This limitation poses difficulties concerning assessment and…

Exploring and deriving proofs of closed-form expressions for series can be fun for students. However, for some students, a physical representation of such problems is more meaningful. Various approaches have been designed to help students visualize squares of sums and sums of squares; these approaches may be arithmetic-algebraic or combinatorial…

Teaching students how to multiply fractions is challenging, not so much from a computational point of view but from a conceptual one. The algorithm for multiplying fractions is much easier to learn than many other algorithms, such as subtraction with regrouping, long division, and certainly addition of fractions with unlike denominators. However,…

The effectiveness of different concrete and pictorial models on students' understanding of the part-whole construct for fractions was investigated. Using interview data from fourth and fifth grade students from three different districts that adopted the "Mathematics Trailblazers" series, authors identified strengths and limitations of models used.…

For over a decade, research studies of mathematics education in high-performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on the promise of common standards, the standards…

Reports on a study comparing the use of Multi-base arithmetic blocks (MAB) and an alternative, LAB (linear arithmetic blocks) on the basis of epistemic fidelity and accessibility for students. Use of LAB was associated with more active engagement by students and deeper discussion. (Author/MM)

The author argues that a knowledge of mathematical structure is only helpful to the high ability student. (MM)