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Regula falsi in lower secondary school education
169-194Views:35The aim of this paper is to offer some possible ways of solving word problems in lower secondary school education. Many studies have shown that pupils in lower secondary school education (age 13-14) encounter difficulties with learning algebra. Therefore they mainly use arithmetical and numerical checking methods to solve word problems. By numerical checking methods we mean guess-and-check and trial-anderror. We will give a detailed presentation of the false position method. In our opinion this method is useful in the loweer secondary school educational processes, especially to reduce the great number of random trial-and-error problem solving attempts among the primary school pupils. We will also show the results of some problem solving activities among 19 grade 8 pupils at our school. We analysed their problem solving strategies and compared our findings with the results of other research works. -
Heuristic arguments and rigorous proofs in secondary school education
167-184Views:31In this paper we are going to discuss some possible applications of the mechanical method, especially the lever principle, in order to formulate heuristic conjectures related to the volume of three-dimensional solids. In the secondary school educational processes the heuristic arguments are no less important than the rigorous mathematical proofs. Between the ancient Greek mathematicians Archimedes was the first who made heuristic conjectures with the methods of Mechanics and proved them with the rigorous rules of Mathematics, in a period, when the methods of integration were not known. For a present day mathematician (or a secondary school mathematics teacher) the tools of the definite integral calculus are available in order to calculate the volume of three dimensional bodies, such as paraboloids, ellipsoids, segments of a sphere or segments of an ellipsoid. But in the secondary school educational process, it is also interesting to make heuristic conjectures by the use of the Archimedean method. It can be understood easily, but it is beyond the normal secondary school curriculum, so we recommend it only to the most talented students or to the secondary schools with advanced mathematical teaching programme. -
Regula falsi in lower secondary school education II
121-142Views:93The aim of this paper is to investigate the pupils' word problem solving strategies in lower secondary school education. Students prior experiences with solving word problems by arithmetic methods can create serious difficulties in the transition from arithmetic to algebra. The arithmetical methods are mainly based on manipulation with numbers. When pupils are faced with the methods of algebra they often have difficulty in formulating algebraic equations to represent the information given in word problems. Their troubles are manifested in the meaning they give to the unknown, their interpretation what an equation is, and the methods they choose to set up and solve equations. Therefore they mainly use arithmetical and numerical checking methods to solve word problems. In this situation it is necessary to introduce alternative methods which make the transition from arithmetic to algebra more smooth. In the following we will give a detailed presentation of the false position method. In our opinion this method is useful in the lower secondary school educational processes, especially to reduce the great number of random trial-and-error problem solving attempts among the lower secondary school pupils. We will also show the results of some problem solving activities among grade 6-8 pupils. We analysed their problem solving strategies and we compared our findings with the results of other research works.
Subject Classification: 97-03, 97-11, 97B10, 97B50, 97D40, 97F10, 97H10, 97H20, 97H30, 97N10, 97N20
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The theory of functional equations in high school education
345-360Views:40In this paper, we are going to discuss some possible applications of the theory of functional equations in high school education. We would like to line up some problems, the solution of which by functional equations are mostly not new results – they have also been treated in [1] and [2] –, although their demonstrations in high school can show a new way in teaching of talented students. The area of the rectangle, the calculating method of compound interest, binomial coefficients, Euler's formula, the scalar product and the vector product of vectors – we are looking for the reasons behind the well-known formulas. Finally, we are going to give a functional equation in connection with mean values. It can be understood easily, but its solution is beyond the high school curriculum, so we advise this part only to the most talented students. -
Connections between discovery learning through the Pósa Method and the secondary school leaving examination in three Hungarian mathematics classrooms
67-85Views:214The Pósa Method is a guided discovery learning method that has been used in Hungarian education in the form of extracurricular activities for "gifted" mathematics students. A four-year experiment implemented the method in three more "average" classrooms. This article reports on the relationship between the Pósa Method and the standardized secondary school leaving mathematics exam (Matura Exam in short) in Hungary. Data consists of students' survey responses, teacher interviews, and exam results from the three Hungarian classrooms who took part in the four-year experiment. We identify aspects of the Pósa Method that can benefit and hinder exam performance. In addition, we find that learning through the Pósa Method for the four years of high school has adequately prepared students for the exam.
Subject Classification: 97D44, 97D54, 97D64
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Process or object? Ways of solving mathematical problems using CAS
117-132Views:26Graphing and symbol manipulating calculators are now a part of mathematics education in many countries. In Norway symbol manipulating calculators have been used at various exams in upper secondary education. An important finding in mathematics education is the duality of mathematical entities – processes and objects. Building on the theoretical development by Anna Sfard and others, the students' solutions on exam problems in upper secondary education are discussed with reference to procedural and structural knowledge. -
How do secondary school students from the Kurdistan Region of Iraq understand the concept of function?
221-244Views:150The study investigates secondary school students' understanding of the concept of function. The paper focuses on three main aspects: students' ability to define the concept of function; students' ability to recognize different representations of function; and students' ability to convert between different representations. A test was developed to assess the three main constructs of the study and administered to 342 students in secondary schools in the Kurdistan Region of Iraq. According to the results, students have diffculties in recognizing different representations of function and conversion between them. Connections between different parts of the test may provide hints on educational challenges of how to appropriately teach functions.
Subject Classification: 26Bxx, 97D60
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Conventions of mathematical problems and their solutions in Hungarian secondary school leaving exams
137-146Views:8Collecting and analyzing the conventions indispensable for interpreting mathematical problems and their solutions correctly assist successful education and objective evaluation. Many professional and didactic questions arose while collecting and analyzing these conventions, which needed clarification, therefore the materials involved concisely in the conventions enrich both the theory and practice of mathematics teaching. In our research we concentrated mainly on the problems and solutions of the Hungarian school leaving examinations at secondary level in mathematics. -
Maximum and minimum problems in secondary school education
81-98Views:31The aim of this paper is to offer some possible ways of solving extreme value problems by elementary methods with which the generally available method of differential calculus can be avoided. We line up some problems which can be solved by the usage of these elementary methods in secondary school education. The importance of the extremum problems is ignored in the regular curriculum; however they are in the main stream of competition problems – therefore they are useful tools in the selection and development of talented students. The extremum problem-solving by elementary methods means the replacement of the methods of differential calculus (which are quite stereotyped) by the elementary methods collected from different fields of Mathematics, such as elementary inequalities between geometric, arithmetic and square means, the codomain of the quadratic and trigonometric functions, etc. In the first part we show some patterns that students can imitate in solving similar problems. These patterns could also provide some ideas for Hungarian teachers on how to introduce this topic in their practice. In the second part we discuss the results of a survey carried out in two secondary schools and we formulate our conclusion concerning the improvement of students' performance in solving these kind of problems. -
Guided Discovery in Hungarian Education Using Problem Threads: The Pósa Method in Secondary Mathematics Classrooms
51-67Views:116In Hungary, ‘guided discovery’ refers to instruction in which students learn mathematical concepts through task sequences that foster mathematical thinking. A prominent figure of guided discovery is Lajos Pósa, who developed his method to teach gifted students. Rather than teaching mathematics through thematic blocks, the Pósa Method employs webs of interconnected problem threads in which problems are built on each other, and different threads are presented simultaneously, so that students work on problems from multiple threads at the same time. It was found that this method has been successful as extracurricular training for gifted students since the 1980s; however since 2017, as part of an ongoing research, the method has been applied to mainstream curriculum in two public secondary school classrooms. The present paper examines the design and implementation processes of problem threads in this public secondary school context.
Subject Classification: 97D40
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Fehleranalyse beim Lösen von offenen Aufgaben Ergebnisse einer empirischen Studie in der Grundschule
83-113Views:12Open problems play a key role in mathematics education, also in primary school. However, children in primary school work in many relations in a different way from learner in secondary school. Therefore, the (possibly) first confrontation with an open task could be problematical. Within the framework of an international paper and pencil test it was examined how far children of primary school notice the openness of a task and which mistakes they do during working on that task. In particularly are meant by openness different interpretations of the task, which all lead to a set of numbers with more than one element as a result. For evaluation, a common classification system was adapted by slightly modification of the original system. -
The effects of chess education on mathematical problem solving performance
153-168Views:50We investigate the connection between the "queen of sciences" (mathematics) and the "royal game" (chess) with respect to the development of mathematical problem solving ability in primary school education (classes 1-8, age 7-15) where facultative chess education is present. The records of the 2014 year's entrance exam in mathematics – obligatory for the enrollment to secondary grammar schools in Hungary – are compared for the whole national database and for the results of a group containing chess-player students. The problems in the tests are classified with respect to the competencies needed to solve them. For the evaluation of the results we used standard mathematical statistical methods. -
Evaluating admission procedures for teacher education in Finland
231-243Views:34In Finland the number of applicants for elementary teacher education is many times greater than the number of accepted persons. In this article we focus on the significance of the entrance examination procedures at three Finnish universities. Our findings imply that the differing admission procedures at the institutions yielded different student profiles. The test component "mathematics-science" used on the entrance examination in Turku was found to be a significant separating factor, but also the applicants' mathematics achievement in upper secondary school seems to be an applicable criterion for developing admission procedures. -
Teaching student teachers: various components of a complex task
55-72Views:26In this paper we summarize various aspects of teacher training and teaching student teachers (mainly concerning teachers of upper secondary school and High school). We stress several hints and recommendations to better achieve the obviously important aim: they should learn doing, understanding and teaching mathematics!
Of course, our view is particularly influenced by European traditions, but we think most of them equally apply to teacher training and teaching student teachers elsewhere. Neither is the paper meant to give an all sided overview about the problem field of teacher education as a whole, nor does it contain provocative, completely new ideas. We just want to describe our view of some aspects, based primarily on our personal experience in the mentioned field. -
Levels of students' understanding on infinity
317-337Views:24Here we report some results of a two-year study for grades 5-6 and 7-8 (during the academic years 2001-03). The study included a quantitative survey for approximately 150 Finnish mathematics classes out of which 10 classes were selected to a longitudinal part of the study. Additionally, 40 students from these classes participated also a qualitative study. This paper will focus on students' understanding of infinity and the development of that understanding. The results show that most of the students did not have a proper view of infinity but that the share of able students grew, as the students got older. -
Comments on the remaining velocity project with reports of school-experiments
117-133Views:14The aim of this article is to introduce different possible solutions to the exercise referring to the calculation of "remaining velocity". We explain the possible approaches to the problem with the help of either using the tools of mathematics or other subjects. During the past few years, we have made Hungarian and Slovakian secondary school students solve the exercise, choosing from both children of average and of high abilities. The experince has shown that very few students were able to solve the problem by themselves, but with the help of their teachers, the exercise and the solution has been an eye-opener experience to all of them. A lot of students were even considering to drive more carefully in the future after getting their driving licenses. -
Report of Meeting Researches in Didactics of Mathematics and Computer Sciences: January 22-24, 2016 Bratislava, Slovakia
115-137Views:22The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Bratislava, Slovakia from the 22th to the 24th of January, 2016 at Comenius University in Bratislava. It was organized by the Doctoral School of Mathematical and Computational Sciences of University of Debrecen and the Faculty of Education of Comenius University.
The 60 participants – including 47 lecturers and 15 PhD students – came from 5 countries, 23 cities and represented 32 institutions of higher and secondary education. -
A retrospective look at discovery learning using the Pósa Method in three Hungarian secondary mathematics classrooms
183-202Views:186While the Pósa Method was originally created for mathematical talent management through extracurricular activities, three "average" public secondary school classrooms in Hungary have taken part in a four-year experiment to implement the Pósa Method, which is based on guided discovery learning of mathematics. In this paper, we examine the students' and teachers' reflections on the Pósa Method, and how student perspectives have changed between their first and last year of high school. Overall, teachers and students had a positive experience with the Pósa Method. Furthermore, our research indicated that this implementation has met several objectives of the Pósa Method, including enjoyment of mathematics and autonomous thinking.
Subject Classification: 97D40
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Analysis of a problem in plane geometry discussed in an 11th grade group study session
181-193Views:27The main aim of this paper is to show those strategies and proof methods we try to teach in secondary maths education through an interesting geometric problem: Find a relation for the sides of a triangle where an angle is the double of another angle. Is the converse also true? Is it possible to generalize the problem? We try to answer these questions while discussing the upcoming difficulties in detail and presenting more possible solutions. Hopefully the paper can be successfully used in study group sessions and problem solving seminars in secondary schools. -
What can we learn from Tamás Varga’s work regarding the arithmetic-algebra transition?
39-50Views:82Tamás Varga’s Complex Mathematics Education program plays an important role in Hungarian mathematics education. In this program, attention is given to the continuous “movement” between concrete and abstract levels. In the process of transition from arithmetic to algebra, the learner moves from a concrete level to a more abstract level. In our research, we aim to track the transition process from arithmetic to algebra by studying the 5-8-grader textbooks and teacher manuals edited under Tamás Varga's supervision. For this, we use the appearance of “working backward” and “use an equation” heuristic strategies in the examined textbooks and manuals, which play a central role in the mentioned process.
Subject Classification: 97-01, 97-03, 97D50
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Report of Meeting Researches in Didactics of Mathematics and Computer Sciences, April 1-3, 2022 Baja, Hungary
135-155Views:156The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Baja, Hungary, at Eötvös József College, from the 1st to the 3th of April, 2022. It was organized by the Doctoral School of Mathematical and Computational Sciences of the University of Debrecen and by Eötvös József College. The 62 participants - including 18 PhD students - came from 8 countries and represented 26 institutions of higher and secondary education. There were 3 plenary and 40 session talks in the program.
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Application of a color education software to improve color aptitude
267-285Views:13The ability to handle colors smoothly and consciously may be vital to professionals in various fields, including engineers specializing in architecture or design. Education in sciences related to colors and a developed color aptitude are essential. In our experience, many secondary school graduates in Hungary exhibit inadequate competence and need to be trained in both fields by college instructors, thereby laying the foundations for their future professional work. In our paper, we introduce a computer-based method to teach color theory using a self-developed interactive educational software. We also demonstrate the results of a test measuring the efficiency of the software. Our method was shown to be capable of familiarizing students with the basic fields of visual computing, e.g. graphics and image processing. -
Straight line or line segment? Students’ concepts and their thought processes
327-336Views:100The article focuses on students’ understanding of the concept of a straight line. Attention is paid to whether students of various ages work with only part of a straight line shown or if they are aware that it can be extended. The presented results were obtained by a qualitative analysis of tests given to nearly 1,500 Czech students. The paper introduces the statistics of students’ solutions, and discusses the students’ thought processes. The results show that most of the tested students, even after completing upper secondary school, are not aware that a straight line can be extended. Finally, we present some recommendations for fostering the appropriate concept of a straight line in mathematics teaching.
Subject Classification: 97C30, 97D70, 97G40
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Report of Meeting Researches in Didactics of Mathematics and Computer Sciences: January 27-29, 2017 Budapest, Hungary
109-128Views:12The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Budapest, Hungary from the 27th to the 29th of January, 2017 at Eötvös Lorand University. It was organized by the Doctoral School of Mathematical and Computational Sciences of University of Debrecen and the Department of Mathematics Teaching and Education Centre Institute of Mathematics.
The 62 participants – including 43 lecturers and 20 PhD students – came from 7 countries, 22 cities and represented 35 institutions of higher and secondary education. -
Zur Veränderung des Stellenwertesvon Beweisen im Mathematikunterricht - eine Analyse von ungarischen Abiturprüfungenzwischen 1981 und 2020
35-55Views:71Proofs are not just an essential, crucial part of mathematics as a science, they also have a long tradition in Hungarian mathematics classrooms. However, the school in general and, mathematics education in particular, have been changed in the last few decades enormously, including the final secondary school examinations in mathematics. The current paper's main goal is to answer the question, how has been changed the weight and the content of reasoning and especially proving tasks in the relevant examinations.
Subject Classification: 97E54, 97D64, 97U44