Contents of Vol. 26 No. 1 (2019)

 

The Comparative Effects of Concrete Manipulatives and Dynamic Software on the Geometry Achievement of Fifth-Grade Students

By Ozge Disbudak and Didem Akyuz

DOI: 10.1564/tme_v26.1.01

Abstract: Concrete manipulatives and dynamic geometry software (DGS) are both commonly used for geometry education in elementary and middle schools. This study sets out to understand which of the two approaches was better in improving the conceptual understanding of quadrilaterals for fifth graders. A pre-/post-test design was conducted in which the same topic was taught by the same teacher to three groups of students, with each group receiving a different type of instruction, namely concrete manipulative based, DGS based and paper-and-pencil based. The initial levels of the students were almost equal as determined by pre-tests. Quantitative analysis of post-test scores revealed that the DGS based approach contributed most to students’ geometrical understanding, followed by the concrete manipulative based approach. Interviews and qualitative analysis were also conducted to gain deeper insights into students’ reasoning and the effect of different instruction types. The interview results revealed that more students in the DGS group could overcome the misconceptions and difficulties than the students in other groups.

 

 Ariadne – A Digital Topology Learning Environment

By Moritz Summermann

DOI: 10.1564/tme_v26.1.02

Abstract: Ariadne is a touch-based program for the learning of homotopies of paths, without the use of formalism, by building mental models. Using Ariadne, the user can construct points, paths by dragging points and homotopies by dragging paths as well as compute winding numbers of paths, all on a variety of surfaces, through touch gestures. Ariadne provides surfaces in two and three dimensions and an optional number of punctures. This environment enables the user to tackle questions regarding the equivalence of points by paths or paths by homotopies, because it allows only mathematically valid operations, i.e., paths and homotopies cannot pass through punctures on the surface. Ariadne is designed to let students of all ages and prior states of knowledge approach problems ranging from the construction of a path connecting points to the classification of all paths up to homotopy on a punctured plane, effectively calculating the fundamental group of a sphere.

 

An Interesting Dynamic Investigation of a Sangaku Problem, and What Else Can Be Asked, as a Research Activity

By Moshe Stupel, Avi Sigler and Idan Tal

DOI: 10.1564/tme_v26.1.03

Abstract: An interesting Sangaku problem was chosen and a multidirectional study was performed, which included finding the locus of a circle’s center moving while being tangent to two circles. It also included finding the radius of that circle given the datum that it is also tangent to the side of an isosceles triangle. Two different proofs are presented of the conserved property of the problem. To exploit the value of the problem from the methodological point of view and to enrich the mathematical knowledge, the problem was expanded by adding a few questions in the style of what else can be asked about the problem? As part of the study of the problem, dynamic geometric software was used to enable hypothesizing and testing the hypotheses as a basis for writing formal proofs

 

Visualizing and Understanding Sampling Distributions Using Dynamic Software

By Sheldon Gordon and Florence Gordon

DOI: 10.1564/tme_v26.1.04

Abstract: This article illustrates ways that dynamic software using some sophisticated techniques in Excel can be used to demonstrate fundamental ideas related to sampling distributions and their properties to increase student understanding of the concepts and methods in elementary statistics courses. The article considers such sampling distributions as the distributions of sample means, sample medians, sample modes, sample midranges, sample proportions, and differences of sample means and examines how they depend on the underlying population and on the sample size of the repeated random samples drawn from the population.