This brief investigation exemplifies such considerations by relating concepts from number theory, set theory, probability, logic, and calculus. Satisfying the call for students to acquire skills in estimation, the following technique allows one to "immediately estimate" whether or not a number is prime. (MM)

Many assertions about the occurrence of the golden ratio phi in art, architecture, and nature have been shown to be false, unsupported, or misleading. For instance, we show that the spirals found in sea shells, in particular the "Nautilus pompilius," are not in the shape of the golden ratio, as is often claimed. Some of the most interesting…

Three experiments examined the effects of target-distractor (T-D) similarity and old age on the efficiency of searching for single targets and enumerating multiple targets. Experiment 1 showed that increasing T-D similarity selectively reduced the efficiency of enumerating small (less than 4) numerosities (subitizing) but had little effect on…

Everyone knows what makes a 3-4-5 triangle special, but how many know what makes a 4-5-6 triangle special? It is an integer-sided triangle in which one angle is twice another. It is enjoyable to search for these things, but for those who are impatient, this article derives explicit polynomial formulas that generate all of the basic examples of…

Do preschool children appreciate numerical value as an abstract property of a set of objects? We tested the influence of stimulus features such as size, shape, and color on preschool children's developing nonverbal numerical abilities. Children between 3 and 5 years of age were tested on their ability to estimate number when the sizes, shapes, and…

Vision was for a long time considered to be essential in the elaboration of the semantic numerical representation. However, early visual deprivation does not seem to preclude the development of a spatial continuum oriented from left to right to represent numbers (J. Castronovo & X. Seron, 2007; D. Szucs & V. Csepe, 2005). The authors investigated…

We applied overlapping waves theory and microgenetic methods to examine how children improve their estimation proficiency, and in particular how they shift from reliance on immature to mature representations of numerical magnitude. We also tested the theoretical prediction that feedback on problems on which the discrepancy between two…

In this article, we show how "Laplace Transform" may be used to evaluate variety of nontrivial improper integrals, including "Probability" and "Fresnel" integrals. The algorithm we have developed here to evaluate "Probability, Fresnel" and other similar integrals seems to be new. This method transforms the evaluation of certain improper integrals…

Number processing is characterized by the distance and the size effect, but symbolic numbers exhibit smaller effects than non-symbolic numerosities. The difference between symbolic and non-symbolic processing can either be explained by a different kind of underlying representation or by parametric differences within the same type of underlying…

In this study, the author examined the relationship of probability misconceptions to algebra, geometry, and rational number misconceptions and investigated the potential of probability instruction as an intervention to address misconceptions in all 4 content areas. Through a review of literature, 5 fundamental concepts were identified that, if…

This is unit three of a fifteen-unit secondary mathematics textbook. This unit contains two chapters. The first chapter discusses integers and the second chapter discusses rational numbers. Operations with both types of numbers as well as the structure of the systems are discussed. (MK)

This is one in a series of SMSG supplementary and enrichment pamphlets for high school students. This series is designed to make material for the study of topics of special interest to students readily accessible in classroom quantity. Topics covered include natural numbers, positive integers, sets, well ordering, lower bound, upper bound,…

Virtual organisms animated by a computational theory of selection by consequences responded on symmetrical and asymmetrical concurrent schedules of reinforcement. The theory instantiated Darwinian principles of selection, reproduction, and mutation such that a population of potential behaviors evolved under the selection pressure exerted by…

We describe the performance of an aphasic individual who showed a selective impairment affecting his comprehension of auditorily presented number words and not other word categories. His difficulty in number word comprehension was restricted to the auditory modality, given that with visual stimuli (written words, Arabic numerals and pictures) his…

The manipulation of numbers required during calculation is known to rely on working memory (WM) resources. Here, we investigated the respective contributions of verbal and/or spatial WM manipulation brain networks during the addition of four numbers performed by adults, using functional magnetic resonance imaging (fMRI). Both manipulation and…