This paper presents the early investigations about the nature of sound of the Pythagoreans, and how they started a tradition that remains valid up to present times--the use of numbers in representing natural reality. It will touch on the Pythagorean notion of musical harmony, which was extended to the notion of universal harmony. How the…

Two identities for the Bernoulli and for the Euler numbers are derived. These identities involve two special cases of central combinatorial numbers. The approach is based on a set of differential identities for the powers of the secant. Generalizations of the Mittag-Leffler series for the secant are introduced and used to obtain closed-form…

Based on an empirical study, we explore children's primary and secondary perceptions on infinity. When discussing infinity, children seem to highlight three categories of primary perceptions: processional, topological, and spiritual. Based on their processional perception, children see the set of natural numbers as being infinite and endow Q with…

This article explores prospective teachers' understandings of one million to gain insights into the development of adult understanding of large numbers. Themes in the prospective teachers' work included number associated with a quantity of objects, number as an abstraction, and additive and multiplicative approaches. The authors suggest that the…

A student noticed an interesting fact about the base two numerals for perfect numbers. Mathematical explanations for some questions are given. (MNS)

A brief survey of the elementary number systems is provided. The natural numbers, integers, rational numbers, real numbers, and complex numbers are discussed; numerals and the use of numbers in measuring are also covered. (DT)

The method used by Cantor to demonstrate the uncountability of the real numbers is applied to a proof showing that the set of natural numbers is uncountable; the error in the argument is discussed. (DT)

Some procedures are developed for testing divisibility by prime numbers composed of two or more digits. Accelerating the tests is also considered. (Contains 2 tables.)

This article describes the instructional process of helping students visualize irrational numbers. Students learn to create a spiral, called "the wheel of Theodorus," which demonstrates irrational and rational lengths. Examples of student work help the reader appreciate the delightful possibilities of this project. (Contains 4 figures.)

Calculation fluency weaknesses are a key characteristic of children with mathematics difficulties. The major aim of this dissertation was to uncover early predictors of calculation fluency weaknesses in second graders. Children's performance on number sense tasks in kindergarten along with general cognitive abilities, early literacy skills, and…

We investigated the relation between children's numerical-magnitude representations and their memory for numbers. Results of three experiments indicated that the more linear children's magnitude representations were, the more closely their memory of the numbers approximated the numbers presented. This relation was present for preschoolers and…

Textbooks, like many other resources teachers have at hand, are meant to be an aid for instruction; however there is little research with textbooks or on their potential to develop metacognitive knowledge. Metacognitive knowledge has received substantial attention in the literature, in particular for its relationship with problem-solving in…

To date, a number of studies have demonstrated the existence of mismatches between children's "implicit" and "explicit" knowledge at certain points in development that become manifest by their gestures and gaze orientation in different problem solving contexts. Stimulated by this research, we used eye movement measurement to…