Spirograph® is a classic drawing toy that has intriguing connections with mathematics. Inspired by earlier articles in "Mathematics Teacher" on the classic drawing toy that targeted higher grade levels, the authors adopt and create activities that make Spirograph mathematics accessible to middle-grade learners.

Mathematics is a human creation, which has been developing for more than four thousand years. It emerged as a response to different social and economic needs of civilizations. Historical development of mathematics stresses that mathematics as a science has always been connected to economic and social context and development of society. There is…

This paper reports about professional learning related to mathematics instruction in student-teaching triads. The study focuses on how the joint work of triads offers opportunities for the professional learning of student teachers (interns), mentor teachers, and supervisors. Two elementary student-teaching triads reported different areas of and…

This text aims to present new research perspectives on teacher training. In particular, it is dedicated to analyzing the training of teachers who teach mathematics from a historical perspective. To this end, it uses the concepts of mathematics to teach and mathematics for teaching, indicating research results, characterizing the professional…

A simple in-class demonstration of integral Calculus for first-time students is described for straightforward whole number area magnitudes, for ease of understanding. Following the Second Fundamental Theorem of the Calculus, macroscopic differences in ordinal values of several integrals, [delta]"F"(x), are compared to the regions of area…

In today's digital society, people encounter a lot of information on a daily basis. This information should be understood by citizens in order for them to be able to properly engage in political issues, something the population of a healthy democracy does. Much of such information involves mathematics and fully comprehending it requires so-called…

What is the connection between advanced mathematics and teaching secondary mathematics? I address this question, drawing on insights from the Special Issue on "Personal Mathematical Knowledge in the Work of Teaching" ("CJSMTE" 13(2), 2013) and reflecting on my personal teaching experience.

In this article, Mark Colyvan argues that formal mathematical notation is more than just the language of mathematics. He gives examples of where notation plays a role in extending mathematical theory and in delivering explanations. He concludes that good mathematical notation, in both mathematics and empirical domains, helps us to think,…

In this paper, Richard Ollerton presents two alternative approaches to proving the six standard trigonometric angle sum and difference identities. They are suitable for students who have an understanding of rotation matrices.

Creativity plays an important role in mathematics, but it can be overlooked in undergraduate courses. This article describes three ways an existing first-year math seminar was modified to increase its focus on creativity: introducing readings and discussions related to creativity, drawing connections to creativity in other disciplines, and adding…

Currently, studies focusing on mathematics education pay more attention to the study of the lecturer's knowledge, aiming to understand this knowledge, its development, and how it is reflected in the practice of teaching. Following this line of research, the present study is based in the "Mathematics Teacher's Specialized Knowledge…

To design and improve instruction in mathematical proof, mathematics educators require an adequate definition of proof that is faithful to mathematical practice and relevant to pedagogical situations. In both mathematics education and the philosophy of mathematics, mathematical proof is typically defined as a type of justification that satisfies a…

We investigated how 53 elementary teachers interpreted the impact of the contexts in which they work on their mathematics instruction, and what those interpretations reveal about the agency individual teachers were able to achieve. Latent class analysis revealed two distinct classes, with teachers in one class perceiving that their contexts had a…

Researchers, who have studied the understanding of the density property in the set of decimal numbers, have shown that the student uses the property of the discrete of natural numbers to solve tasks related to density. So, a restructuring of concepts is necessary, that is, a conceptual change from "the discrete" to "the dense".…

The application of computer science is fast-growing and things change day by day; with this there is a necessity to adapt to the changes as software applications are powerful tools that herald significant development in diversified fields including education. Adoption of the computer systems in various operations of schools (primary, secondary and…