Geometric Construction Problems Research Articles

Problems and challenges of using randomized automatically evaluating geometric construction problems in Moodle LMS

<abstract><p>The objective of this paper was to examine the present state of geometric constructions in both international and national mathematics assessments. Additionally, it aimed to shed light on the noticeable lack of geometric construction items in these assessments and propose a solution to incorporate such items by utilizing a web-based mathematics assessment tool. After providing a brief summary of the unique benefits associated with geometric construction problems, the subsequent paper explored the development of a web-based mathematics assessment tool, the procedure for generating automatic evaluation GeoGebra applets for geometric construction problems, and their integration into Moodle learning management system (LMS). The purpose of this assessment tool was to enhance the evaluation of a substantially larger group of students within the realm of geometric constructions. The final section of the paper examined the difficulties encountered in the creation of the automatic evaluation applets and their incorporation into Moodle LMS. Through the development of such a web-based mathematics assessment tool, we enabled the evaluation of a significantly larger number of students, thus aiding not only national and international mathematics examinations but also assisting mathematics teachers in tracking students' scores and progress.</p></abstract>

The effect of problem-based learning on cognitive skills in solving geometric construction problems: a case study in Kazakhstan

IntroductionThis study aims to investigate the impact of a Problem-Based Learning (PBL) course on cognitive skills (i.e., Critical Thinking, Problem-Solving, Logical Reasoning, Creativity, and Decision-Making) in the context of solving geometric construction problems.MethodsThe research utilized a quasi-experimental design involving a control group and an experimental group to assess the effects of the PBL intervention. Cognitive skills were measured using a custom-designed questionnaire. Additionally, Structural Equation Modeling (SEM) was employed in a subsequent phase to scrutinize the causal interrelationships among these cognitive skills.ResultsIn the initial phase, the findings revealed that the PBL intervention had a statistically significant positive impact on problem-solving and creativity skills. However, the effects on critical thinking, logical reasoning, and decision-making skills did not reach statistical significance. In the subsequent phase employing SEM, the analysis demonstrated significant positive relationships, particularly between critical thinking and problem-solving, critical thinking and logical reasoning, logical reasoning and problem-solving, and logical reasoning and creativity. Notably, creativity also exhibited a significant positive effect on problem-solving.DiscussionThis study underscores the nuanced impact of PBL on different cognitive skills, with clear enhancements observed in problem-solving and creativity. However, the study suggests that the effects may not be uniform across all cognitive skills. These findings offer valuable insights for educators and curriculum designers, emphasizing the need for tailored approaches when integrating PBL to foster cognitive skill development.

The Intuitive Way to Solve Construction Problems in the Dynamic Geometry Environment

The ability to solve geometric construction problems is justly regarded as an essential component of mathematical culture. The dynamic, general nature of objects provided by dynamic geometry systems allows the development of intuitive methods for solving construction problems. The core mathematical concept underlying this approach is the loci method, in which the main element of the required object forms as the intersection of two loci, each one obtained by purposely ignoring part of the conditions of the problem. With GeoGebra, the relevant locus may be visualized and detected in trace mode. Using dynamic geometry this way for problem solving replaces the single object by the infinite locus and is similar to the introduction of variables for solving equations in school algebra. This paper presents some examples of how this approach may be realized along with the results of a small-scale experiment with pre-service mathematics teachers.

Improving the Teaching of Geometric Construction Problems in the Process of Training Future Mathematics Teachers

The article examines the problem of improving the teaching of geometric construction problems using information and communication technology (ICT) tools in the process of training future mathematics teachers at pedagogical universities. During the clarifying experiment, methodological problems in teaching geometric construction problems were identified, and as one of the ways to solve them, the need for methodological training of future mathematics teachers to use applied information and computer resources in teaching geometric construction problems was based. Analysis of information and computer technologies and programs intended for use in teaching geometry courses was performed, and the effectiveness of using the “Live Geometry” program in creating geometric construction problems was shown. Learners are more interested in geometry and construction problems, and can save all work in electronic format for future use. While studying the topic, the tools in the program will help to draw figures and use these figures in solving problems, formulate the properties of geometric figures, and use them to model geometric bodies. In the given article, the authors analyzed the state programs and concepts, the works of domestic and foreign scientists, and the results of an online survey and interviews with students. 54 students of the mathematics education program were involved in a pedagogical experiment, and the results were shown in detail. As the results of the experiment show, after using the “Living Geometry” program in the lesson, students were recognized from a certain personal subjective point of view, which regulates their cognitive and practical activities, and an increase in theoretical knowledge and practical skills was determined. It has been proven by mathematical and statistical methods that the use of the “Living Geometry” program in the teaching of structural problems allows effective and efficient organization of education and increases the methodological training of future mathematics teachers.

Formation of students' mental activity by improving the applied focus of learning a geometric

Prerequisites for the study the problems of implementing the applied direction of teaching the topic of geometric construction in the geometry course of secondary school are most relevant, since there are no clear basic methods for the formation of students' thinking activities in this direction. This article discusses the results of a scientific study aimed at obtaining theoretical knowledge in any section of school geometry, which aims to give the student the skills to use the means of drawing the properties of geometric shapes and relationships. For this purpose, students were given a review of the literature on the implementation of the applied direction of geometry, focusing on mathematical statements in order to improve the applied direction of geometry. In the framework of the topic, the analysis was carried out in the course of improving this topic, due to the fact that mathematics teachers did not pay enough attention to the problems of construction in the geometry course of secondary school due to lack of time, students did not develop skills in solving problems of the applied direction of geometry. As a result of research, geometric construction problems are an important element of the mathematical preparation of schoolchildren. Using the problems of geometric construction, its application points were determined. The ways of implementing the applied direction in teaching geometry to students are described. Also, when teaching the geometry course, problems of the applied direction were collected in the school's geometry textbooks, ways to solve them were considered, and plot geometric tasks were prepared in order to increase the creativity of students. The stages of solving the practice-oriented geometric construction problem and the principles of implementing the applied direction of teaching mathematics, defined as a special type of plot problem, reflecting the use of geometric means in practical situations, are also studied. In the future, it can be used as a methodological tool for mathematics teachers in teaching construction problems.

Болашақ математика мұғалімдерін геометриялық салу есептеріне даярлауда «деңгейлеп оқыту» технологиясын қолдану

The article examines the issue of methodological preparation of future mathematics teachers for teaching geometric construction problems, analyzing the educational program "Mathematics Teacher Training" prepared by the University from the subject and meta-disciplinary point of view. Based on the use of "level learning technology" in teaching future mathematics teachers to solve construction problems, students' creativity and research abilities, independent work, abstract thinking, etc. problems of improving methodological training by developing skills are studied. Mastering new technologies of teaching in the field of modern education helps the teacher to develop himself and effectively organize the educational process. The use of new technological methods of teaching increases the level of education of students in accordance with modern requirements. In the given article, the authors analyzed the state programs and concepts, the works of domestic and foreign scientists, the results of an online survey, and interviews conducted with students. 68 studentsofthe mathematics education program were involved in a pedagogical experiment, and the results were shown. As the analysis of the survey shows, after using the "level learning technology" in the lesson, students are recognized from a certain personal subjective point of view, which regulates their cognitive and practical activities. It has been shown that the use of "Level-level teaching technology" in the teaching of structural calculations allows effective and efficient organization of education, and increases the methodological training of future mathematics teachers.

Болашақ математика мұғалімдерін геометриялық салу есептеріне даярлауда «деңгейлеп оқыту» технологиясын қолдану

The article examines the issue of methodological preparation of future mathematics teachers for teaching geometric construction problems, analyzing the educational programs “Mathematics Teacher Training” prepared by the University from the subject and metadisciplinary point of view. Based on the use of “level learning technology” in teaching future mathematics teachers to solve construction problems, students’ creativity and research abilities, independent work, abstract thinking, etc. problems of improving methodological training by developing skills are studied. Mastering new technologies of teaching in the field of modern education helps the teacher to develop himself and effectively organize the educational process. The use of new technological methods of teaching increases the level of education of students in accordance with modern requirements. In the given article, the authors analyzed the state programs and concepts, the works of domestic and foreign scientists, the results of an online survey and interviews conducted with students. 68 students of the mathematics education program were involved in a pedagogical experiment, and the results were shown. As the analysis of the survey shows, after using the “level learning technology” in the lesson, students are recognized from a certain personal subjective point of view, which regulates their cognitive and practical activities. It has been shown that the use of “Level-level teaching technology” in the teaching of structural calculations allows effective and efficient organization of education, and increases the methodological training of future mathematics teachers.

Doubling the Cube and Constructability in Higher Dimensions

Summary It is known that the three classical geometric construction problems introduced by the ancient Greeks: trisecting an angle, squaring a circle, and doubling a cube, cannot be solved using the Euclidean tools. However, ancient Greek mathematicians solved these three problems using other means. We present solutions to the doubling-the-cube problem using ideas that go beyond the Euclidean tools, and we consider generalizations to higher dimensions.

ДИАГНОСТИКА РЕГУЛЯТИВНЫХ УНИВЕРСАЛЬНЫХ УЧЕБНЫХ ДЕЙСТВИЙ ОБУЧАЮЩИХСЯ 8–9-х КЛАССОВ В ПРОЦЕССЕ РЕШЕНИЯ ГЕОМЕТРИЧЕСКИХ ЗАДАЧ НА ПОСТРОЕНИЕ

Statement of the problem. The article analyzes the studies of foreign and Russian authors in the field of diagnostics of regulatory universal learning skills (RULS) of students. It is concluded that there is a shortage of adequate tools for assessing RULS in the process of teaching geometry. The present research is aimed at solving the following tasks: finding a connection between the process of formation of regulatory skills and the formation of subject skills in teaching geometry to schoolchildren; assessment of the educational results of students, taking into account the characteristics of the taught subject. The purpose of the article is to scientifically substantiate and describe the RULS diagnostic tools for students in the process of solving geometric construction problems. The methodology of the study is based on the use of system-activity and personality-oriented approaches to learning; analysis of modern research in the field of diagnostics of personal and meta-subject results of students. Research results. The operational detailing of the students’ RULS is determined and justified; the principles of development of the structural and content RULS map are highlighted. The article describes the relationship between the stages of solving construction problems and RULS, manifested by students in the process of their implementation. Priority regulatory actions are highlighted at each stage. The results of RULS diagnostics among 8-9 grade students, participants of the Regional Olympiad in Geometry named after Prof. S.A. Anishchenko. Conclusion. The structural and content map of students’ RULS proposed in the article corresponds to the principles of completeness of structural elements; takes into account the systemic nature of the formation of RULS, the adequacy of the list of structural elements for the purposes of diagnosis and the choice of diagnostic tools. This map determined the direction of the search for links between the stages of solving construction problems and the operations manifested in the process of their implementation. The article presents a methodology for assessing students’ RULS in the process of solving construction problems on the basis of expert assessment and self-assessment cards. The results obtained can be extended to other groups of students.

AL-FARABI'S MATHEMATICAL LEGACY AND ALGORITHMIC APPROACH TO RESOLVING PROBLEMS REGARDING GEOMETRICAL CONSTRUCTIONS IN GEOGEBRA ENVIRONMENT

Al-Farabi had left a rich scientific heritage for the succeeding generations including studies in various scientific areas. So far, his proceedings are still valuable being of great interest in the scientific society. The actuality of analyzing the problems with regard to geometric constructions accomplished by Al-Farabi for modern mathematical and engineering education lies in the unique character of Al-Farabi’s efforts involving the use of the algorithmic approach for resolving mathematical problems and application-oriented researches. These algorithmic approaches enable the creation of didactic teaching techniques based on the use of information processing technology. One of the main Al-Farabi’s principles is to study and consider the mathematics in terms of natural phenomena and processes as well as all types of its practical implementation. This article pursues studying AlFarabi’s mathematical heritage concerning geometrical construction problems as well as the development of their solution methods using modern information technology, in particular, using the GeoGebra dynamic geometry system which is a problem-solving environment designed for studying, conversion and visualization of geometric object models. This scientific article shows examples of resolving geometric construction problems using only a mathematical compass and a ruler by the Al-Farabi’s procedures as well as problems for dividing squares and spheres and giving Al-Farabi’s visual evidence of proving Ptolemy’s theory from trigonometry area. The authors are interested in a targeted audience to familiarize themselves with algorithms developed by Al-Farabi and this software implementation; for this purpose, a scientific-education portal has been designed for the concerned researches and pedagogues may find out a lot of useful and interesting information.

Portfolio theorem proving and prover runtime prediction for geometry

In recent years, portfolio problem solving found many applications in automated reasoning, primarily in SAT solving and in automated and interactive theorem proving. Portfolio problem solving is an approach in which for an individual instance of a specific problem, one particular, hopefully most appropriate, solving technique is automatically selected among several available ones and used. The selection usually employs machine learning methods. To our knowledge, this approach has not been used in automated theorem proving in geometry so far and it poses a number of new challenges. In this paper we propose a set of features which characterize a specific geometric theorem, so that machine learning techniques can be used in geometry. Relying on these features and using different machine learning techniques, we constructed several portfolios for theorem proving in geometry and also runtime prediction models for provers involved. The evaluation was performed on two corpora of geometric theorems: one coming from geometric construction problems and one from a benchmark set of the GeoGebra tool. The obtained results show that machine learning techniques can be useful in automated theorem proving in geometry, while there is still room for further progress.

Using jointly geometry and algebra to determine RC-constructibility

In most cases in geometry, applying analytic or algebraic tools on coordinates helps to solve some difficult problems. For instance, proving that a geometrical construction problem is solvable using ruler and compass is often impossible within a synthetic geometry framework. But in an analytic geometry framework, it is a direct application of Galois theory after performing triangularizations. However, these algebraic tools lead to a large amount of computation. Their implementation in modern Computer Algebra Systems (CAS) are still too time consuming to provide an answer in a reasonable time. In addition, they require a lot of memory space which can grow exponentially with the size of the problem. Fortunately, some geometrical properties can be used to setup the algebraic systems so that they can be more efficiently computed. These properties turn polynomials into new ones so as to reduce both the degrees and the number of monomials. The present paper promotes this approach by considering two corpora of geometric construction problems, namely Wernick's and Connely's lists. These lists contain about 280 problems. The purpose is to determine their status i.e. whether they are constructible or not with ruler and compass. Some of these problems had unknown status that will be settled in this paper. More generally, the status of all problems of these corpora are fully automatically given by an approach combining geometry and algebra.

Automatic Constructibility Checking of a Corpus of Geometric Construction Problems

Straightedge and compass constructions play a special role in geometry. First, for a very long time, they were used in practice by land surveyors or architects in order to solve concrete problems. Second, they are an inexhaustible source of exercises used to learn concepts in geometry. And finally, some famous impossible problems related to straightedge and compass constructions waited centuries before being solved using algebra. In addition, not knowing if a well-constrained problem is constructible with straightedge and compass or not, make that kind of problems more difficult to address: should we synthesize a program or on the contrary find a counterexample has to be found and treated using algebra or reduction on it? In this paper, we perform a systematic checking of a whole corpus of problems proposed by William Wernick. We expose our methodology and the algorithm we used. For each problem, its constructibility status is computed and either an algebraic argument or a geometric construction is given.

Метод координат в геометрических задачах на построение

Метод координат в геометрических задачах на построение

Why was Wantzel overlooked for a century? The changing importance of an impossibility result

The duplication of a cube and the trisection of an angle are two of the most famous geometric construction problems formulated in ancient Greece. In 1837 Pierre Wantzel (1814–1848) proved that the problems cannot be constructed by ruler and compass. Today he is credited for this contribution in all general treatises of the history of mathematics. However, his proof was hardly noticed by his contemporaries and during the following century his name was almost completely forgotten. In this paper I shall analyze the reasons for this neglect and argue that it was primarily due to the lack of importance attributed to such impossibility results at the time.

Dragging as a conceptual tool in dynamic geometry environments

Some ‘drag-to-fit’ solutions given by student teachers to three geometric construction problems in a dynamic geometry environment (DGE) are analysed. The responses of a group of experienced mathematics teachers to the question whether or not such solutions can be considered ‘legitimate’ are then discussed. This raises fundamental questions concerning the concept of legitimacy, the relationship between DGEs and Formal Axiomatic Euclidean Geometry, the nature of ‘conceptual tools’ in different geometric environments, and the functions of dragging in DGEs. The authors argue that, if dragging is viewed as a conceptual tool, then certain drag-to-fit solutions, although soft constructions, may still be considered as conceptually legitimate and therefore valid. Finally, some important questions are raised concerning the impact that teachers’ different attitudes towards legitimacy might have on students’ learning through DGEs.

Using the history of mathematics as a starting point for investigations: some examples on approximations

This paper discusses an investigative approach to problems concerned with approximations. Two problems are analysed in detail and both are inspired by their historical origins. The first examines an ancient Chinese formula for calculating the area of a segment. The second looks at Dürer's construction for the trisection of an angle. It is argued that historical problems can provide a rich source for such an investigative approach. Discussion of the second problem also raises fundamental issues about the nature of ‘conceptual tools’ and ‘conceptually exact solutions’ to geometric construction problems, particularly in the light of using dynamic geometry environments such as Sketchpad.

Construction with Obstructions Involving Arcs

Geometric-construction problems progress to innovative problems when key parts of the figure are made inaccessible or an obstruction is inserted. Such problems encourage creative thinking and challenge the student while improving problemsolving skills and reviewing concepts of Euclidean geometry. The student is asked to (1) show the construction, (2) describe the solution, and (3) verify the correctness of the solution. Further, the student can develop original problems, thereby opening yet another avenue of creativity.

A problem with touching circles

Those who enjoy geometric construction problems will find much pleasure in drawing combinations of touching circles.