The starting point of this article is a search for pairs of quadratic polynomials x[superscript 2] + bx plus or minus c with the property that they both factor over the integers. The search leads quickly to some number theory in the form of primitive Pythagorean triples, and this paper develops the connection between these two topics.

A result known as the Borel-Cantelli lemma is about probabilities of sequences of events. This article presents an example in which it appears that the hypotheses of the lemma are satisfied but the conclusion is not. The explanation of why not combines elements of probability theory, number theory, and analysis.

This article discusses prime numbers, defined as integers greater than 1 that are divisible only by only themselves and the number 1. A positive integer greater than 1 that is not a prime is called composite. The number 1 itself is considered neither prime nor composite. As the name suggests, prime numbers are one of the most basic but important…

The term "random" is frequently used in discussion of the theory of evolution, even though the mathematical concept of randomness is problematic and of little relevance in the theory. Therefore, since the core concept of the theory of evolution is the non-random process of natural selection, the term random should not be used in teaching the…

After extensive laboratory testing of the famous memorist Rajan, Thompson, Cowan, and Frieman (1993) proposed that he was innately endowed with a superior memory capacity for digits and letters and thus violated the hypothesis that exceptional memory fully reflects acquired ''skilled memory.'' We successfully replicated the empirical phenomena…

This article features questions regarding logarithmic functions and hair growth. The first question is, "What is the underlying natural phenomenon that causes the natural log function to show up so frequently in scientific equations?" There are two reasons for this. The first is simply that the logarithm of a number is often used as a replacement…

This note explores ways in which the Fibonacci numbers can be used to introduce difference equations as a prelude to differential equations. The rationale is that the formal aspects of discrete mathematics can provide a concrete introduction to the mechanisms of solving difference and differential equations without the distractions of the analytic…

Mathematical historians place Heron in the first century. Right-angled triangles with integer sides and area had been determined before Heron, but he discovered such a "non" right-angled triangle, viz 13, 14, 15; 84. In view of this, triangles with integer sides and area are named "Heron triangles." The Indian mathematician Brahmagupta, born in…

The number Pi has a rich and colorful history. The origin of Pi dates back to when Greek mathematicians realized that the ratio of the circumference to the diameter is the same for all circles. One is most familiar with many of its applications to geometry, analysis, probability, and number theory. This paper demonstrates several examples of how…

This article illustrates the misconceptions that students have when using the equals sign and describes a lesson used to give students the foundation for an accurate conception of equivalency.

The Pre-Kindergarten Inventory of Demonstrated Skills (Pre-KIDS) is designed to be used before the school year begins so that teachers and staff can obtain information about incoming kindergarten students. This information is intended to give teachers insight about what skills a student may or may not have before entering their classrooms. This…

In this article, the author argues that coordinate geometry and all its trappings should be banned from key stage 2 in English schools. To explain why he makes such a strong statement, he discusses geometry problems tackled by the Ancient Greeks, showing how meaningful problem solving can occur without the use of coordinates and the corresponding…

This study examined whether singular/plural marking in a language helps children learn the meanings of the words "one," "two," and "three." First, CHILDES data in English, Russian (which marks singular/plural), and Japanese (which does not) were compared for frequency, variability, and contexts of number-word use.…

Several visuo-motor tasks can be used to demonstrate biases towards left hemispace in schizophrenic patients, suggesting a minor right hemineglect. Recent studies in neglect patients used a new number bisection task to highlight a lateralized defect in their visuo-spatial representation of numbers. To test a possible lateralized representational…

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities…