John Napier was born in 1550 in the Tower of Merchiston, near Edinburgh, Scotland. Napier's work on logarithms greatly influenced the work that was to be done in the future. The logarithm's ability to simplify calculations meant that Kepler and many others were able to find the relationships and formulas for motion of bodies. In turn, Kepler's…

Two studies were conducted to investigate, firstly, children's focusing on the aspect of numerosity in utilizing enumeration in action, and, secondly, whether children's Spontaneous Focusing on Numerosity (SFON) is related to their counting development. The longitudinal data of 39 children from the age of 3.5 to 6 years showed individual…

In Experiment 1, adults (n = 48) performed simple addition, multiplication, and parity (i.e., odd-even) comparisons on pairs of Arabic digits or English number words. For addition and comparison, but not multiplication, response time increased with the number of odd operands. For addition, but not comparison, this parity effect was greater for…

Learning trajectories are presented of 2 fifth-grade children, Jason and Laura, who participated in the teaching experiment, Children's Construction of the Rational Numbers of Arithmetic. 5 teaching episodes were held with the 2 children, October 15 and November 1, 8, 15, and 22. During the fourth grade, the 2 children demonstrated distinctly…

As with natural numbers, a greatest common divisor of two Gaussian (complex) integers "a" and "b" is a Gaussian integer "d" that is a common divisor of both "a" and "b". This article explores an algorithm for such gcds that is easy to do by hand.

Using the power series solution of a differential equation and the computation of a parametric integral, two elementary proofs are given for the power series expansion of (arcsin x)[squared], as well as some applications of this expansion.

The SNARC (spatial numerical associations of response codes) effect reflects the tendency to respond faster with the left hand to relatively small numbers and with the right hand to relatively large numbers (S. Dehaene, S. Bossini, & P. Giraux, 1993). Using computational modeling, the present article aims to provide a framework for conceptualizing…

This paper proposes a genetic development of the concept of limit of a sequence leading to a definition, through a succession of proofs rather than through a succession of sequences or a succession of epsilons. The major ideas on which it is based are historical and depend on Euclid, Archimedes, Fermat, Wallis and Newton. Proofs of equality by…

This paper investigates the ability of infants to attend to continuous stimulus variables and how this capacity relates to the representation of number. We examined the change in area needed by 6-month-old infants to detect a difference in the size of a single element (Elmo face). Infants successfully discriminated a 1:4, 1:3 and 1:2 change in the…

Using the modular ring Zeta[subscript 4], simple algebra is used to study diophantine equations of the form (x[cubed]-a=y[squared]). Fermat challenged his contemporaries to solve this equation when a = 2. They were unable to do so, although Fermat had devised a rather complicated proof himself. (Contains 2 tables.)

A triple (x,y,z) of natural numbers is called a Primitive Pythagorean Triple (PPT) if it satisfies two conditions: (1) x[squared] + y[squared] = z[squared]; and (2) x, y, and z have no common factor other than one. All the PPT's are given by the parametric equations: (1) x = m[squared] - n[squared]; (2) y = 2mn; and (3) z = m[squared] +…

Pascal's Triangle is, without question, the most well-known triangular array of numbers in all of mathematics. A well-known algorithm for constructing Pascal's Triangle is based on the following two observations. The outer edges of the triangle consist of all 1's. Each number not lying on the outer edges is the sum of the two numbers above it in…

This article describes the pivotal roles that Marin Mersenne played--as a recreational mathematician in search of prime number patterns and as a mentor to young mathematicians and scientists. His work is used as an example for today's mathematics teachers in encouraging students to work together and creating environments that foster success for…

A simple proof to some known results on the convergence of linear recursive sequences with nonnegative coefficients is given, using the technique of monotone convergence.

This document reports on a small-group teaching experiment whose goal was to understand how fourth- and sixth-grade children develop concepts of common and decimal fractions. Both the Grade 6 and the Grade 4/5 children were taught common and decimal fractions through discussion and by using manipulatives, beginning with basic concepts of common…