A challenge encountered when monitoring mathematics teaching and learning at high school is that taxonomies such as Bloom's, and variations of this work, are not entirely adequate for providing meaningful feedback to teachers beyond very general cognitive categories that are difficult to interpret. Challenges of this nature are also encountered in…

To support teachers in their quest to incorporate reasoning as a mathematical proficiency as espoused in the Australian Curriculum: Mathematics, a professional learning research project using demonstration lessons was carried out. This paper reports on the impact of demonstration lessons on one participating teacher's pedagogical knowledge about…

This paper describes the use of the free software, to teach graphs of logarithmic functions at an Ethiopian College of Teacher Education. Data comprised two video-recorded lessons and interview data provided by a mathematics teacher educator, and three primary school mathematics pre-service teachers who were in the class of learners. Pre-service…

The Isoperimetric Quotient, or IQ, introduced by G. Polya, characterizes the degree of sphericity of a convex solid. This paper obtains closed form expressions for the surface area and volume of any Archimedean polyhedron in terms of the integers specifying the type and number of regular polygons occurring around each vertex. Similar results are…

The Fibonacci series has been studied since it was first described by Leonardo of Pisa--Fibonacci--in 1202. It begins with the sequence 1, 1, 2, 3, 5, 8... Each succeeding number is the sum of the previous two. In number theory courses, students are introduced to the concept of modulo arithmetic, sometimes called "clock" arithmetic. In modulo…

This article presents an applied calculus exercise that can be easily shared with students. One of Kepler's greatest discoveries was the fact that the planets move in elliptic orbits with the sun at one focus. Astronomers characterize the orbits of particular planets by their minimum and maximum distances to the sun, known respectively as the…

In this article we explore the consequences of modifying the common definition of a parabola by considering the locus of all points equidistant from a focus and (not necessarily linear) directrix. The resulting derived curves, which we call "generalized parabolas," are often quite beautiful and possess many interesting properties. We show that…

Modern conceptions of critique express preoccupations with obtaining certainty and truth, with establishing foundations, and with linking knowledge and rationality. Modern conceptions of critique have been challenged, and critique has taken new formats. In particular critique has developed without demands of providing foundations for knowledge and…

The comments below are meant to show that considerations of public and private realms and the tension between these realms arise in a natural and persistent way in discussions connected with semiotics. In particular, they arise out of the themes of body and sociocultural mathematical meaning-making, which are recurring themes of the papers in this…

Relatively little is known about how students come to reason coherently about the formal definition of limit. While some have conjectured how students might think about limits formally, there is insufficient empirical evidence of students making sense of the conventional [superscript epsilon]-[delta] definition. This paper provides a detailed…

We explore conditions for productive synthesis between formal reasoning and intuitive representations through analysis of college students' understanding of the limit concept in the definition of the derivative. In particular, we compare and contrast cognitive processes that accompany different manifestations of persistence of intuitions and tacit…

The main scope of this article is to prove that if a continuous function attains its minimum before maximum on every compact interval, than it is nondecreasing. Finally, other similar results are established.

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.

Extending a recent paper by Derek Holton, we show how to represent the algorithm for the Frog Problem diagrammatically. This diagrammatic representation suggests a simpler proof of the symmetrical case (equal numbers of frogs of each colour) by allowing the even and odd cases to be treated together. It also provides a proof in the asymmetrical…

The aim of this article is to discuss some results about the converse mean value theorem stated by Tong and Braza [J. Tong and P. Braza, "A converse of the mean value theorem", Amer. Math. Monthly 104(10), (1997), pp. 939-942] and Almeida [R. Almeida, "An elementary proof of a converse mean-value theorem", Internat. J. Math. Ed. Sci. Tech. 39(8)…