The purpose of this note is to introduce two challenging problems in geometry that were inspired by some well-known problems in high school geometry. These problems can be explored in the dynamic geometry environments of computer construction programmes.

The use of the dynamic mathematics software GeoGebra to solve geometric problems on conics and loci from an 18th century textbook will be presented. In particular, examples will be shown of how the use of this program helped the authors to understand the method that our predecessors used to deal with conic sections together with solving loci…

In this article, we present investigative tasks that concern loci, which integrate the use of dynamic geometry software (DGS) with mathematics for proving the obtained figures. Additional conditions were added to the loci: ellipse, parabola and circle, which result in the emergence of new loci, similar in form to the original loci. The…

This paper concerns teaching Euclidean geometry at the university level. It is based on the authors' personal experience. It describes a sequence of learning activities that combine geometrical constructions with explorations, observations, and explanations of facts related to the geometry of triangle. Within this approach, a discussion of the…

The representation of figures in mongean projection (double system planned orthographic projection used in the studies of Descriptive Geometry), specially when placed in a particular situation in relation to the projection plans, possesses the quality that, through them, the actual dimensions of represented spatial objects can be found directly…

Learning geometry emphasizes the importance of exploring different representations such as virtual manipulatives, written math formulas, and verbal explanations, which help students build math concepts and develop critical thinking. Besides helping individuals construct math knowledge, peer interaction also plays a crucial role in promoting an…

Students in a graduate geometry class used items such as paper, ribbon, plastic spheres, cardboard tubes, and markers to carry out investigations in spherical geometry. The hands-on activities helped students develop a new appreciation of geometry as a study of shape and space; however, the difficulty of subduing wayward elastics and drawing lines…

An online interactive geometry item was developed to explore students' abilities to create prototypical and "tilted" rectangles out of line segments. The item was administered to 1,002 students. The responses to the item were hand-coded as correct, incorrect, or incorrect with possible evidence of a misconception. A variation of the nearest…

The present study is concerned with assessing Logo programming experiences among seventh grade students. A formal multiple-choice test and five performance tasks were used to collect data. The results provided that students' performance was better than the expected score by the probabilistic laws, and a very low correlation between their Logo…

During a two-week summer professional development workshop, teams of intermediate-level school teachers and college methods instructors crafted mathematics learning modules--activities, lesson plans, work sheets, and technology-oriented tasks--with the primary goal of strengthening students' understanding of various geometric concepts. They recast…

We explore how curvature and torsion determine the shape of a curve via the Frenet-Serret formulas. The connection is made explicit using the existence of solutions to ordinary differential equations. We use a paperclip as a concrete, visual example and generate its graph in 3-space using a CAS. We also show how certain physical deformations to…

The performances of 4- and 5-year-olds and rhesus monkeys were compared using a computerized task for quantity assessment. Participants first learned two quantity anchor values and then responded to intermediate values by classifying them as similar to either the large anchor or the small anchor. Of primary interest was an assessment of where the…

Using the film "Journey to the Centre of a Triangle" with a group of 15-year-old pupils, the author describes how they had done some work on constructions such as angle bisector and perpendicular bisector. The pupils were given A3 papers, rulers, compasses and pencils and they were asked to recreate their favourite scence from the film. The film…

Mathematics can be perceived as being a difficult subject to learn due to the conceptual leaps required to understand particular mathematical topics. In some areas of mathematics, part of the difficulty may be associated with applying sufficient imagination to visualize a particular mathematical concept, and applying sufficient visio-spatial…

Technological tools have the potential to offer students the possibility to represent information and relationships embedded in problems and concepts in ways that involve numerical, algebraic, geometric, and visual approaches. In this paper, the authors present and discuss an example in which an initial representation of a mathematical object…