The intersection of a plane and a cone is a conic section and rotating the plane leads to a family of conics. What happens to the foci of these conics as the plane rotates? A classical result gives the locus of the foci as an oblique strophoid when the plane rotates about a tangent to the cone. The analogous curve when the plane intersects a…

The geometrical shapes in Fig. 1 recently came into the Greenslade Collection. The semi-triangular pieces of wood looked like slices from a pie, each 22.5 degrees in angular width. They were tied together with a thin leather band tacked around their curved surfaces. What is this apparatus?

Following Husserl's speculations on how geometry originated, we suggest that spatial perception is "seduced" by language as a result of human attempts to capture, signify and share its concepts. And this language traps geometry and humans themselves in to the forms that have guided and regulated past practices, thereby obscuring possibilities for…

The international debate about experimental approaches to the teaching and learning mathematics is very current. While number theory lends itself naturally to such approaches, elementary geometry can also provide interesting starting points for creative work in class. This article shows how simple considerations about right triangles and the…

We will first recall useful formulas in integration that simplify the calculation of certain definite integrals with the quadratic function. A main formula relies only on the coefficients of the function. We will then explore a geometric proof of one of these formulas. Finally, we will extend the formulas to more general cases. (Contains 3…

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.

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in…

The purpose of this research is to develop a hands-on spatial geometry learning system to facilitate the learning of geometry. The development of this system was based on Duval's four critical elements of geometric learning: perceptual apprehension, sequential apprehension, operative apprehension, and discursive apprehension. The system offers…

In this article, by way of an analysis of a case of mathematics teacher training, I explore the general idea of pedagogic expectation of an alignment of pedagogic identities and specific realisations of mathematics in pedagogic contexts. The particular case analysed has a constructivist orientation, but the analytic resources brought to bear in…

In order to address concerns with the underrepresentation of spatial thinking in K-12 curricula, (National Research Council, 2006), Valentine and Kopcha (in press) designed and implemented a learning environment integrating cases as alternative perspective (Jonassen, 2011) in the context of eighth-grade mathematics. The design aimed to provide…

We investigate the relationship between teacher licensure test scores and student test achievement and high school course-taking. We focus on three subject/grade combinations-- middle school math, ninth-grade algebra and geometry, and ninth-grade biology--and find evidence that a teacher's basic skills test scores are modestly predictive of…

This study investigated the learning process of preservice elementary teachers (PSTs) through lesson study in a mathematics teaching methods course. Lesson study meeting data of kindergarten PSTs were analyzed to examine the shifting focus on (1) student collaboration in classroom (vision), (2) teaching practices (making content accessible), and…

Children can learn mathematics and sciences effectively even before being exposed to formal school curriculum if basic Mathematics and Sciences concepts are communicated to them early using activity oriented (Hands-on) method of teaching. Mathematics and Science are practical and activity oriented and can best be learnt through inquiry (Okebukola…

It remains difficult to foster problem-solving and mathematical-reasoning capabilities in classrooms where students and teachers are accustomed to the more traditional forms of education. Several studies suggest that this difficulty might be related to the kind of knowledge students acquire in such environments, which could be fragmented and…

Group work has been a main activity recommended by mathematics education reform. We aim at describing the patterns of interaction between teachers and students during group work. We ask: How do teachers scaffold group work during a problem-based lesson? We use data from a problem-based lesson taught in six geometry class periods by two teachers…