## Contents of Vol. 26 No. 4 (2019)

### Classroom Connectivity Technology to Enhance the Social Construction of

Mathematical Knowledge

By Oscar Castrillón-Velandia and Omar Hernández-Rodríguez

This study focuses on how the combined use of graphic calculators with connectivity software promotes the social construction of mathematical knowledge. Particular emphasis is placed on the aspects related to design, development, implementation, and evaluation of didactic activities that integrate these technologies. We use Didactical Engineering (DE) and Realistic Mathematics Education (RME) as frames of reference. Twenty, eighth grade students from a private institution in the metropolitan area of San Juan, Puerto Rico participated. The topics discussed were those related to the conic sections as geometric loci. The results show that the careful planning of graphing calculator integrated with a system of connectivity in the classroom promotes the discussions and reflections of students. These reflections provide insight as to how students engage in the social construction of mathematical knowledge. In addition, students showed that they could reflect on powerful mathematical ideas and acquire meaningful knowledge.

### English Language Learners’ Cognitive Load and Conceptual Understanding of

Probability Distributions after Using an Animated Simulation Program

By Jase Moussa-Inaty and Mark Causapin

The majority of university students in the United Arab Emirates are English language learners. As a country that has only recently established its educational system based on an American model, it has adopted English as its language for teaching and learning. Challenges related to the use of a second language have been noted and simple interventions such as the use of Arabic translations and glossaries have not shown reasonable effectiveness, suggesting that limited English language proficiency in itself is not the sole cause of learning difficulties. The challenge to understand and find a solution to this problem led to considering Cognitive Load Theory, which suggests that certain approaches to teaching may hinder learning because of unnecessary burdens on working memory. This theory has been previously used to explain how the additional language burden negatively affect second language learners. Within this context, a quasiexperiment was conducted where students were taught the concept of probability distributions using an animated simulation of a coin tossing experiment. Animated simulation was hypothesized to create lower cognitive load and thus result in better learning and higher test scores. Performance and cognitive load were measured throughout the study. Although it was found that using animated simulation was not associated with better fact and procedural retention, students performed better in a test of conceptual understanding. As predicted by Cognitive Load Theory, the researchers found a negative relationship between test scores and cognitive load, albeit weak. Nonetheless, the cognitive load of students using the animated simulation was lower for most of the duration of the experiment. Results are further discussed from a cognitive load perspective and future research directions are proposed.

### A Formative Path in Tertiary Education through GeoGebra Supporting the

Students’ Learning Assessment and Awareness

By Francesca G. Alessio, Lucio Demeio and Agnese Ilaria Telloni

In this paper, we present a “route of activities” mainly addressed to first and second year engineering students attending the basic mathematics courses. These activities are intended to be a tool both for students and teachers. On the one hand, they should help the students to acquire a deeper understanding of the mathematical ideas and increase their awareness about the learning progress; on the other hand, they should be useful to the teachers in adjusting their educational strategies by synchronizing them with the students’ outcomes. The activities are based on working with coordination of multiple representations of sets in the plane; the motivation for this choice is given by the difficulties encountered by the students in the calculations of double integrals, either in the Calculus courses or in the Mechanics courses. We describe the design and the aims of the activities and analyze the tasks with regard to the mathematical competencies and the feedback given to the students. The tasks have been designed and implemented with GeoGebra and submitted to the students by using a Moodle platform.

### Surprising Geometrical Properties – Their Investigation, Proof and Generalization

By Moshe Stupel, Avi Sigler and Idan Tal

We perform dynamic investigation of two surprising geometrical properties, each of which involves additional properties. In the first task the property belongs to two regular polygons with the same number of sides and with one common vertex. It is found that all the straight lines that connect corresponding vertices of the two polygons intersect at a single point. This point the second point of intersection of the circumcircles of the polygons. In the second task the property concerns the situation in which there are two disjoint circles which are tangent to the sides of an angle, with an isosceles triangle between them which is tangent to the two circles, so that its base lies on one angle side and its head vertex lies on the second side – the sum of the radii of the circles equals the altitude of the triangle.

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