How could the Presentation of a Geometrical Task Influence Student Creativity?

This study aims to investigate high school students’ geometry learning by focusing on mathematical creativity and its relationship with visualisation and geometrical figure apprehension. The presentation of a geometrical task and its influence on students’ mathematical creativity is the main topic investigated. The authors combine theory and research in mathematical creativity, considering Roza Leikin’s research work on Multiple-Solution Tasks with theory and research in visualisation and geometrical figure apprehension, mainly considering Raymond Duval’s work. The relations between creativity, visualization and geometrical figure apprehension are examined through four Geometry Multiple-Solution Tasks given to high school students in Greece. The geometrical tasks are divided into two categories depending on whether their wording is accompanied by the relevant figure or not. The results of the study indicate a multidimensional character of relations among creativity, visualization and geometrical figure apprehension. Didactical implications and future research opportunities are discussed.

Enhancing the Skill of Geometric Prediction Using Dynamic Geometry

This study concerns geometric prediction, a process of anticipation that has been identified as key in mathematical reasoning, and its possible constructive relationship with explorations within a Dynamic Geometry Environment (DGE). We frame this case study within Fischbein’s Theory of Figural Concepts and, to gain insight into a solver’s conceptual control over a geometrical figure, we introduce a set of analytical tools that include: the identification of the solver’s geometric predictions, theoretical and phenomenological evidence that s/he may seek for, and the dragging modalities s/he makes use of in the DGE. We present fine-grained analysis of data collected during a clinical interview as a high school student reasons about a geometrical task, first on paper-and-pencil, and then in a DGE. The results suggest that, indeed, the DGE exploration has the potential of strengthening the solver’s conceptual control, promoting its evolution toward theoretical control.

On Hayekian Triangles

The triangle is an integral part of the history of economic thought. It has been used by writers such as Jevons (1871), Taussig (1896), Wicksell (1934, 1969) to illustrate and to help us understand capital theory. Since Hayek (1931) this geometrical figure has been used as a basic pedagogical device to explain the Austrian Business Cycle Theory (ABCT). The purpose of the present paper is to argue that the triangle is highly problematic, if not fatally flawed, and that if ABCT is to be made intelligible this tool of analysis must be either completely jettisoned, or heavily supplemented with a list (see below) of its shortcomings. Moreover in some ways the triangle has been responsible for the relative lack of development of ABCT for over a half century. Key words: Austrian economics, business cycle theory, praxeology, economic geometry, triangles. Clasificación JEL: E3, E32. Resumen: El triángulo es una parte integral de la historia del pensamiento económico. Ha sido utilizado por escritores como Jevons (1871), Taussig (1896), Wicksell (1934, 1969) para ilustrar y ayudarnos a comprender la teoría del capital. Desde Hayek (1931) esta figura geométrica se ha utilizado como un instrumento pedagógico básico para explicar la teoría austriaca del ciclo económico. El propósito de este trabajo es sostener que el triángulo es altamente problemático, sino fatalmente defectuoso, por lo que si deseamos que la teoría austriaca del ciclo económico sea comprendida debemos desecharlo comple-tamente, o complementarlo fuertemente con una lista de sus limitaciones. Además, en algunos casos el triángulo ha sido responsable de la relativa falta de desarrollo de la teoría austriaca del ciclo durante un periodo de medio siglo. Palabras clave: Economía austriaca, teoría del ciclo económico, praxeología, geometría económica, triángulos.

О видах калмыцких родовых тамг

The study gives a review of the Kalmyk clan sings called tamga. The article identifies the main types of these signs. The author points out that tamgas in their image have simple geometrical figures that can be the evidence of their ancient origin and the high probability of duplication in different nations. The most simple geometrical figure in the system of the clan signs has a form of a straight line and also an angle with sides of different length. One of the most popular images is a cross that in the folk culture is associated with cross piece of the smoke flap of the yurt. There are a lot of tamgas in the shape of a circle or, as the Kalmyks call it in the system of tamgas, iron ring. The variety of this type of tamgas is set up based on the number of rings and their position in relation to each other. There were popular tamgas in the shape of a semicircle that differed in the direction of the image and also in the shape of bident or trident. The authors’ materials point to the fact that there could also be sings in the form of Chinese characters. All things considered, there are several desiderata in the study of Kalmyk tamgas and their further study will allow to enlarge the list of tamga types.

ON THE ESSENCE, THE PLACE, THE PART AND CHARACTER OF THE TASKS WITH PARAMETERS IN THE GEOMETRY COURSES OF THE INSTITUTIONS OF GENERAL SECONDARY EDUCATION

The so-called tasks with parameters for a long time now have become an integral part as of the every to some extent profound course of algebra or of algebra and the beginnings of cultures at the institutions of general secondary education, as of the corresponding tasks of the State Final Attestation in Mathematics and the External independent assessment in mathematics. And it isn’t accidental because in the most often cases the solution of the task with a parameter turns for the student into a small investigation by his own. The realization of such investigation favors the formation of the creative practical-oriented personality. Simultaneously we must state that, despite of the existence of a lot of the high scientific and methodical level created corresponding training books, it is difficult just now to find in the methodical literature the clear answers to the natural questions of what is meant on the whole by the task with parameter (or with parameters) and its solution. At the same time, in the courses of geometry of the institutions of general secondary education to the tasks with parameters it is given next to nothing consideration. But in fact such tasks in the courses are present, their importance for the proper construction of the courses can be exaggerated. In the paper the problems of what must be understand by the task with the parameter or with the parameters and by its solution are analyzed. The essence, the part and the place of the tasks with parameters in the geometry courses of institutions of general secondary education are elucidated. Euclidean geometry as an axiomatic theory investigates the sets that in their overwhelming majority represent by themselves the mathematical abstructions of the spatial forms of the surrounding, some relations between such set and quantities that characterize such sets and relations. In the contrast to the courses of algebra, in the geometrical courses the part of parameters may be played by all of the three mentioned components. Geometrical figures can change by the size and by the form. Changing by the size bring us to the concept of the scalar quantity. Changing by the form are considered in the tasks of paving and, for example, in the tasks of finding the amount and the types of symmetries of geometrical figure in dependence of its form. The part of the parameter-relation can be played by different variants of mutual displacement of the given figures in Euclidean plane or in Euclidean space. According to their content, different geometrical tasks with parameters are considered in the work. The task of the existence of geometrical figures, the tasks, conserning the character of some geometrical places of points, the tasks of tracing with the help of a compass and a ruler are among them.

Cognitive Processes of High Mathematics Anxiety Student in Solving Area Conservation Problem

Area conservation is a concept to modify the shape or position of a geometrical figure without changing its area. Skipping the concept of area conservation in learning area measurement causes students’ difficulties in this topic. One of students’ activity in math class is problem solving which is a cognitive process of the brain to search a solution for a given problem. Cognitive process is an activity that consist of receiving, processing, and using information. Cognitive process may affected by mathematics anxiety. Mathematics anxiety is a condition in which students experience unexplained anxiety during learning mathematics. This study is qualitative descriptive research which aimed at describing the cognitive processes of high mathematics anxiety student in solving area conservation problem. The data were collected by using mathematics anxiety test, area conservation problem test, and interview. The result showed that at the stage of receiving information, high mathematics anxiety students read the given problem and observe the given figures. At the stage of processing the information, the students devising plan by linked the received information with their knowledge, solving the problem, and evaluating the obtained solution. Student with high mathematics anxiety use the concept of area conservation to modify some figures. Moreover, high mathematics anxiety student did a simple estimation to determine the area of the given figures. At the stage of using information, student with high mathematics anxiety re-explain the given problem, the idea to solve the problem, every single steps in solving problem, and the obtained solution.

SEMI-AUTOMATIC METHOD OF FAN SURFACE ASSESSMENT TO ACHIEVE GORGONIAN POPULATION STRUCTURE IN LE DANOIS BANK, CANTABRIAN SEA

<p><strong>Abstract.</strong> This study presents a semi-automatic method to estimate fan surface of a <i>Placogorgia</i> sp. octocoral assemblage using 3D point clouds in El Cachucho MPA at 550&amp;thinsp;m of depth. The presence of gorgonian forests and deep-sea sponge aggregations in Le Danois Bank was the cause of its declaration as ‘El Cachucho’ Marine Protected Area (MPA), being included in the Natura 2000 network. The <i>Placogorgia</i> sp. is a structuring species of the deep Cantabrian Sea; parameters such as population structure and morphology inform on the overall health of this vulnerable habitat, but the estimation of gorgonian metrics often requires destructive sampling. The use of non-invasive methodology, which does not cause damage or alterations on benthic communities, is particularly necessary in vulnerable ecosystem studies and Marine Protected Areas (MPA) monitoring. This study proposes a semi-automatic methodology to assess gorgonian morphometries fitting planes to colonies. Video transects acquired in Le Danois Bank, during the ECOMARG-2017 survey using the Politolana underwater towed vehicle were used. Using Pix4D Mapper Pro and Cloud Compare software, size and morphometry of fan-shaped gorgonians and forest population structure were assessed. RMS of fitting planes shows that the geometrical figure chosen is suitable to retain the morphometric characteristics of the specimens of this species. The adjustment of semi-automatic values with a sample of digitized surfaces manually is validated (R²=0.97). The results show that gorgonian population was mostly dominated by small colonies. The population structure distribution shows a high proportion (~22%) of recruits (&amp;lt;&amp;thinsp;0.05&amp;thinsp;m²) of fan surface.</p>