Developing proficiency in algebra is the focus of instruction in high school mathematics courses and is a minimum expectation for high school completion for all students including those with learning difficulties. However, the foundation for success is laid in grades 4-8 (aged 9-14). In this paper, we assert that students' development of algebraic…

Just as athletes stretch their muscles before every game and musicians play scales to keep their technique in tune, mathematical thinkers and problem solvers can benefit from daily warm-up exercises. Jessica Shumway has developed a series of routines designed to help young students internalize and deepen their facility with numbers. The daily use…

Children are often intrigued by number patterns and games and so it makes sense for teachers to include them in their mathematics lessons. Puzzles encourage the use of critical thinking skills and provide practice in important skills areas. The use of games fosters mathematical learning and encourages the mathematical processes that children use.…

The aim of this paper is to investigate the ways in which the number line can function in solving mathematical tasks by first graders (6 year olds). The main research question was whether the number line functioned as an auxiliary means or as an obstacle for these students. Through analysis of the 32 students' answers it appears that the number…

Many students encounter difficulty in their transition to advanced mathematical thinking. Such difficulty may be explained by a lack of understanding of many concepts taught in early school years, especially multiplicative reasoning. Advanced mathematical thinking depends on the development of multiplicative reasoning. The purpose of this study…

Although increasing emphasis is being placed on mathematical justification in elementary school classrooms, many teachers find it challenging to engage their students in such activities. In part, this may be because the teachers themselves have not had an opportunity to learn what it means to justify solutions or prove elementary school concepts…

Infinity and infinitely small numbers pique the curiosity of middle school students. From a very young age, many students are intrigued, interested, and even fascinated by extremely large or extremely small numbers or quantities. This article describes an activity that takes this curiosity about infinity into the domain of adding the numbers of an…

The goal of this study was to investigate understanding of inservice elementary school teachers in Taiwan about number sense, teaching strategies of number sense and the development of number sense of students; and the profile of integrating number sense into mathematical instruction , and teaching practice. Data was gathered through interviews of…

This book provides a landscape of learning that helps teachers recognize, support, and celebrate their students' capacity to structure their worlds algebraically. It identifies the models, contexts, and landmarks that facilitate algebraic thinking in young students and provides insightful and practical methods for teachers, math supervisors, and…

"The power of two" is a Royal Institution (Ri) mathematics "master-class". It is a two-and-a half-hour interactive learning session, which, with varying degree of coverage and depth, has been run with students from Year 5 to Year 11, and for teachers. The master class focuses on an historical episode--the Josephus…

How do composing and decomposing numbers connect with the properties of addition? Focus on the ideas that you need to thoroughly understand in order to teach with confidence. The mathematical content of this book focuses on essential knowledge for teachers about numbers and number systems. It is organized around one big idea and supported by…

We present a general formula for a triple product involving four real numbers. As a particular case, we get the sum of a triple product of four odd integers. Some interesting results are recovered. We derive a general formula for more than four odd numbers.

This study investigates two sixth grade students' dilemmas regarding the parity of zero. Both students originally claimed that zero was neither even nor odd. Interviews revealed a conflict between students' formal definitions of even numbers and their concept images of even numbers, zero, and division. These images were supported by practically…

Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…

This classroom note includes the following sections: Introduction; The General Case; and Related Formulas.