In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P[subscript k](n) and Q[subscript k](n), such that P[subscript k](n) = Q[subscript k](n) = f[subscript k](n) for n = 1, 2,… , k, where f[subscript k](1), f[subscript k](2),… , f[subscript k](k) are k arbitrarily chosen…

We use the hyperbolic cotangent function to deduce another proof of Euler's formula for ?(2n).

In 1859, on the occasion of being elected as a corresponding member of the Berlin Academy, Bernard Riemann (1826-66), a student of Carl Friedrich Gauss (1777-1855), presenteda lecture in which he presented a mathematics formula, derived from complex integration, which gave a precise count of the primes on the understanding that one of the terms in…

Script writing by learners has been used as a valuable pedagogical strategy and a research tool in several contexts. We adopted this strategy in the context of a mathematics course for prospective teachers. Participants were presented with opposing viewpoints with respect to a mathematical claim, and were asked to write a dialogue in which the…

In this note, we introduce sequence factorial and use this to study generalized M-bonomial coefficients. For the sequence of natural numbers, the twin concepts of sequence factorial and generalized M-bonomial coefficients, respectively, extend the corresponding concepts of factorial of an integer and binomial coefficients. Some latent properties…

What does "to be a mathematician" mean? What is implied, and what image is created of "a mathematician"? Are "mathematicians" members of an exclusive club? Are mathematicians different to "other people"? Are mathematicians different because they are able to mathematize? These might not be the most oft asked questions, but are they questions to…

Algebraic notation has been bedazzling learners for generations, yet despite research, learned papers, comparative studies, and a great deal of hard work on the part of classroom practitioners the situation remains a stubborn constant. In this article some small-scale research has highlighted some issues that might find a resonance in many…

The purpose of this article is to examine one possible extension of greatest common divisor (or highest common factor) from elementary number properties. The article may be of interest to teachers and students of the "Australian Curriculum: Mathematics," beginning with Years 7 and 8, as described in the content descriptions for Number…

The 43rd meeting of Canadian Mathematics Education Study Group (CMESG) was held at St. Francis Xavier University in Antigonish, Nova Scotia (May 31-June 4, 2019). This meeting marked only the third time CMESG/GCEDM (Groupe Canadien d'Étude en Didactique des Mathématiques) had been held in Nova Scotia (1996, 2003), and the first time it had been…

At grade 7, Common Core's content standards call for the use of long division to find the decimal representation of a rational number. With an eye to reconciling this requirement with Common Core's call for "a balanced combination of procedure and understanding," a more transparent form of long division is developed. This leads to the…

For at least 2000 years people have been trying to calculate the value of [pi], the ratio of the circumference to the diameter of a circle. People know that [pi] is an irrational number; its decimal representation goes on forever. Early methods were geometric, involving the use of inscribed and circumscribed polygons of a circle. However, real…

The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the key ideas about infinitesimals via a proceptual analysis of the meaning of the ellipsis "..." in the real…

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…

The general purpose of this article is to shed some light on the understanding of real numbers, particularly with regard to two issues: the real number as the limit of a sequence of rational numbers and the development of irrational numbers as a continued fraction. By generalizing the expression of the golden ratio in the form of the limit of two…

This article explores the notion of collateral learning in the context of classic ideas about the summation of powers of the first "n" counting numbers. Proceeding from the well-known legend about young Gauss, this article demonstrates the value of reflection under the guidance of "the more knowledgeable other" as a pedagogical method of making…