The solution of problems and the provision of proofs have always played a crucial part in mathematics. In fact, they are the heart and soul of this discipline. Moreover, the use of different techniques and methods of proof in the same mathematical field, or by combining fields, for the same specific problem, can show the interrelations between the…

The butterfly theorem is proved by assigning point masses to the four vertices of the wings and using the distributive property of the mass centre of a mechanical system.

The nine-point circle theorem is one of the most beautiful and surprising theorems in Euclidean geometry. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. This paper describes a set of instructional activities that can help students discover the nine-point circle theorem through…

Translating an informal mathematical argument into a proof which conforms to the norms of the mathematical community in which it is situated is a non-trivial task. Here we discuss several types of products, other than the initial informal argument and its direct formalisation, which we observed students generating in a master's level analysis…

Our article aims to show how illuminating mathematical work as a concept from didactics of mathematics is useful in understanding issues relating to proving and learning of proof, with or without technology. After posing our hypotheses on the relationship of proof with mathematical work, the pedagogical intent of historical elements of geometry…

The purpose of this study was to investigate pre-service teachers' views about the history of mathematics course in which GeoGebra was used. The qualitative research design was used in this study. The participants of the study consisted of 23 pre-service mathematics teachers studying at a state university in Turkey. An open-ended questionnaire was…

In this article, we revisit the concept of strong differentiability of real functions of one variable, underlying the concept of differentiability. Our discussion is guided by the Straddle Lemma, which plays a key role in this context. The proofs of the results presented are designed to meet a young audience in mathematics, typical of students in…

Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs…

The high school curriculum sometimes seems like a disconnected collection of topics and techniques. Theorems like the factor theorem and the remainder theorem can play an important role as a conceptual "glue" that holds the curriculum together. These two theorems establish the connection between the factors of a polynomial, the solutions…

A part of high school geometry is devoted to the study of parallelograms in the context of proving some of their properties using congruent triangles (CCSSI 2010). The typical high school geometry book's chapter on quadrilaterals focuses on parallelograms (e.g., their properties, proving that a given quadrilateral is a parallelogram, and special…

This article deals with a short historical introduction to determinants with applications to the theory of equations, geometry, multiple integrals, differential equations and linear algebra. Included are some properties of determinants with proofs, eigenvalues, eigenvectors and characteristic equations with examples of applications to simple…

In this paper I describe how I have used the classic buried treasure problem with prospective and practicing mathematics teachers to enhance their problem solving abilities and disposition to integrate interactive geometry software (IGS) into the learning environment. I illustrate how IGS may be used as a strategic tool to gain insight into the…

In this article, a simple exercise for finding groups isomorphic to the symmetry group of the real line is presented. A mechanism for producing metrics on the real line is used to construct the group elements, and these transformations, by construction, are isometries with respect to the generating metric.

Archimedes calculated the centre of gravity of the cone but the proof of this theorem is not extant in his works. Knorr made a reconstruction of this proof utilizing geometrical arguments. This paper proves this theorem by means of a physical demonstration utilizing the law of the lever, and by adapting from Archimedes the method of mechanical…

In this paper, we present an analytic framework for investigating expert mathematical learning as the process of building a "network of mathematical resources" by establishing relationships between different components and properties of mathematical ideas. We then use this framework to analyze the reasoning of ten mathematicians and mathematics…