This paper reports on data from investigations on learners' understanding of inclusion relations of quadrilaterals, building on the ideas from our earlier study (Fujita & Jones, 2007). By synthesising past and current theories in the teaching of geometry (van Hiele's model, figural concepts, prototype phenomenon, etc.), we propose a theoretical…

The theoretical and experimental solutions for vibrations of a vertical-oriented, prismatic, thin cantilever beam are studied. The beam orientation is "downwards", i.e. the clamped end is above the free end, and it is subjected to a transverse movement at a selected frequency. Both the behaviour of the device driver and the beam's weak-damping…

Infinite series is a challenging topic in the undergraduate mathematics curriculum for many students. In fact, there is a vast literature in mathematics education research on convergence issues. One of the most important types of infinite series is the geometric series. Their beauty lies in the fact that they can be evaluated explicitly and that…

Based on Steiner's fascinating theorem for trapezium, seven geometrical constructions using straight-edge alone are described. These constructions provide an excellent base for teaching theorems and the properties of geometrical shapes, as well as challenging thought and inspiring deeper insight into the world of geometry. In particular, this…

This case study describes and investigates the instructional practices of Grades 1 to 5 teachers and the levels of geometry thinking of the learners, according to the Van Hiele model, with a view to determining whether there is a match between the instructional practice and the learners' level of thinking. The instructional practices of the…

This study describes the mathematical construction of a real-life model by means of parametric equations, as well as the two- and three-dimensional visualization of the model using the software GeoGebra. The model was initially considered as "determining the parametric equation of the curve formed on a plane by the point of a pen, positioned…

A study was conducted on the involvement of visuospatial working memory (VSWM) in intuitive geometry and in school performance in geometry at secondary school. A total of 166 pupils were administered: (1) six VSWM tasks, comprising simple storage and complex span tasks; and (2) the intuitive geometry task devised by Dehaene, Izard, Pica, and…

This article discusses a writing project that offers students the opportunity to solve one of the most famous geometric problems of Greek antiquity; namely, the impossibility of trisecting the angle [pi]/3. Along the way, students study the history of Greek geometry problems as well as the life and achievements of Carl Friedrich Gauss. Included is…

"Carnegie Learning Curricula and Cognitive Tutor"®, published by Carnegie Learning, is a secondary math curricula that offers textbooks and interactive software to provide individualized, self-paced instruction based on student needs. The program includes pre-Algebra, Algebra I, Algebra II, and Geometry, as well as a three-course series…

This paper describes Singapore and Australian Grade 6 students' (n=1,187) performance on a symmetry task in a recently developed Mathematics Processing Instrument (MPI). The MPI comprised tasks sourced from Australia and Singapore's national assessments, NAPLAN and PSLE. Only half of the cohort solved the item successfully. It is possible that…

This paper looks at the role of visualisation in the proving process. It considers the different functions of proof and then describes student responses when engaged in the process of discovering Viviani's Theorem. The findings show that learners can attain high levels of conviction when working in a dynamic geometry environment. In particular,…

Much of the mathematical reasoning employed in the typical introductory physics course can be traced to Pythagorean roots planted over two thousand years ago. Besides obvious examples involving the Pythagorean theorem, I draw attention to standard physics problems and derivations which often unknowingly rely upon the Pythagoreans' work on…

One possible way of evaluating set curricula is to examine the consistency of study programmes with students' psycho-cognitive development. Three theories were used to evaluate matching between developmental theories and content proposed in the mathematics programmes (geometry section) for primary and the beginning of secondary education. These…

Opportunities for teacher candidates to investigate and enact the work of teaching in settings of reduced complexity--what are called "approximations of practice"--offer a promising path toward preparing more ambitious and equitable mathematics teachers. However, these approaches face the risk of not preparing individuals to continue…

Students often find geometrical concepts abstract and difficult to understand. This results in poor performance, which contributes in the declining interest in geometry. The aim of this study was to examine the impact of using the free educational software program, "GeoGebra" on 9th grade student's mathematics achievement in learning…