This paper explores the specialized knowledge mobilized by a mathematics teacher in the selection and use of examples for teaching sequences. Taking an experimental case study approach, we analyse the examples deployed in a series of third-year secondary level lessons on sequences and identify the different knowledge subdomains activated according…

This study investigates the analogies used by a sample of secondary mathematics teachers when they described the concept of function. The study is concerned with understanding what analogies were used by the participant teachers and conceptions of functions positioned in those analogies. Using examples from a set of responses to five open-ended…

In this article special sequences involving the Butterfly theorem are defined. The Butterfly theorem states that if M is the midpoint of a chord PQ of a circle, then following some definite instructions, it is possible to get two other points X and Y on PQ, such that M is also the midpoint of the segment XY. The convergence investigation of those…

The article presents one of the possible ways of overcoming the misconception that is one of the myths about mathematics, which says that teaching mathematics is based only on the transfer of mathematical knowledge in the finished form -- often in the form of certain algorithms. The newly proposed method of access to mathematics teaching is…

The goal of this note is to obtain the area enclosed between a parabola and two tangent lines using visual arguments. The first step that we have to follow, in order to do that, is noticing that the area between a parabola and a tangent line only depends on the leading coefficient of the quadratic function of the parabola. Finally, the…

The surface area of a sphere as a function of radius, "r," is a derivative of the volume of the sphere, and a few other families of solids with this property are known. Sometimes, the derivative relationship can be reached with a tricky choice of size parameter "r." This paper presents an activity for students that discovers…

We propose a divisibility criterion for elements of a generic Unique Factorization Domain. As a consequence, we obtain a general divisibility criterion for polynomials over Unique Factorization Domains. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities for college and high school…

We propose some generalizations of the classical Division Algorithm for polynomials over coefficient rings (possibly non-commutative). These results provide a generalization of the Remainder Theorem that allows calculating the remainder without using the long division method, even if the divisor has degree greater than one. As a consequence we…

The rules of indices, e.g. a[superscript n] b[superscript n] = (ab)[superscript n], are a particularly important part of elementary algebra. This paper reports results from a textbook analysis which examined how the shift from integer to rational exponents in the rules of indices is discussed in school textbooks. The analysis also considered…

This paper reports the results of a research exploring the mathematical connections of pre-university students while they solving tasks which involving rates of change. We assume mathematical connections as a cognitive process through which a person finds real relationships between two or more ideas, concepts, definitions, theorems, procedures,…

Many students do not have a deep understanding of the integral concept. This article defines what a deep understanding of the integral is in respect to integration involving one independent variable; briefly discusses factors which may inhibit such an understanding; and then describes the design of a mathematical resource for introducing students…

Studies in mathematics education often point to the necessity for students to engage in more cognitively demanding activities than just solving tasks by applying given solution methods. Previous studies have shown that students that engage in creative mathematically founded reasoning to construct a solution method, perform significantly better in…

This paper compares sections on functions and linear functions from two Korean textbooks and an American standards-based textbook (University of Chicago School Mathematics Project [UCSMP] Algebra) to understand differences and similarities among these textbooks through horizontal and vertical analyses. We found that these textbooks provide…

Previous studies have produced several typologies of teacher questions in mathematics. Probing questions that ask students to explain are often included in the types of questions. However, only rare studies have created subtypes for probing questions or investigated how questioning differs depending on whether technology is used or not. The aims…

Similar to countries such as the Netherlands and the United Kingdom, secondary schools in Ireland can decide how to allocate instruction time between curriculum subjects. Although there are national guidelines available from the Department of Education and Skills (DES), the majority of schools make their own decisions about how much time they…