What comes to mind when one thinks about building? One may envision constructions with blocks or engineering activities. Yet, constructing and building a number system requires the same sort of imagination, creativity, and perseverance as building a block city or engaging in engineering design. We know that children invent their own notation for…

This paper is concerned with developing a new class of generalized numbers. The main advantage of this class is that it generalizes the two classes of generalized Fibonacci numbers and generalized Pell numbers. Some new identities involving these generalized numbers are obtained. In addition, the two well-known identities of Sury and Marques which…

By the time they reach middle school, all students have been taught to add fractions. However, not all have "learned" to add fractions. The common mistake in adding fractions is to report that a/b + c/d is equal to (a + c)/(b + d). It is certainly necessary to correct this mistake when a student makes it. However, this occasion also…

This paper presents guided classroom activities that showcase two classic problems in which a finite limit exists and where there is a certain charm to engage liberal arts majors. The two scenarios build solely on students' existing knowledge of number systems and harness potential misconceptions about limits and infinity to guide their thinking.…

A facility with signed numbers forms the basis for effective problem solving throughout developmental mathematics. Most developmental mathematics textbooks explain signed number operations using absolute value, a method that involves considering the problem in several cases (same sign, opposite sign), and in the case of subtraction, rewriting the…

This article deals with a brief history of Fibonacci's life and career. It includes Fibonacci's major mathematical discoveries to establish that he was undoubtedly one of the most brilliant mathematicians of the Medieval Period. Special attention is given to the Fibonacci numbers, the golden number and the Lucas numbers and their fundamental…

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…

"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…

The method of proof by mathematical induction follows from Peano axiom 5. We give three properties which are often used in the proofs by mathematical induction. We show that these are equivalent. Supposing the well-ordering property we prove the validity of this method without using Peano axiom 5. Finally, we introduce the simplest form of…

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.

Different representations of rational numbers are considered and students are lead through activities that explore patterns in base ten and other bases. With this students are encouraged to solve problems and investigate situations designed to foster flexible thinking about rational numbers.

This article reports on the use of a unique number system to facilitate teachers' understanding of the concepts of place value. Teachers' mastery of base-ten may hinder their recognition of the difficulties students have with place value, so the authors created a number system that used five symbols to represent values. Using this system, teachers…

In this paper four programs are described in which children learn multidigit number concepts and operations with understanding: (1) the Supporting Ten-Structured Thinking projects, (2) the Conceptually Based Instruction project, (3) Cognitively Guided Instruction projects, and (4) the Problem Centered Mathematics Project. The diversity in these…

Illustrates the multiple strategies used by second graders to solve a problem with three addends and how their teacher tries to map their thinking into the system of mathematical notation. Describes the American Federation of Teachers' Thinking Mathematics program that the teacher uses. (MKR)