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On the psychology of mathematical problem solving by gifted students
289-301Views:116This paper examines the nature of mathematical problem solving from a psychological viewpoint as a sequence of mental steps. The scope is limited to solution processes for well defined problems, for instance, which occur at International Mathematical Olympiads. First the meta-mathematical background is outlined in order to present problem solving as a well defined search problem and hence as a discovery process. Solving problems is described as a sequence of elementary steps of the so called "relationship-vision" introduced here. Finally, non-procedural aspects of the psychology of problem solving are summarized, such as the role of persistence, teacher-pupil relationship, the amount of experience needed, self-confidence and inspiration at competitions. -
Levels of students' understanding on infinity
317-337Views:188Here 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. -
Compositions of dilations and isometries in calculator-based dynamic geometry
257-266Views:99In an exploratory study pre-service elementary school teachers constructed dilations and isometries for figures drawn and transformed using dynamic geometry on calculators. Observational and self assessments of the constructed images showed that the future teachers developed high levels of confidence in their abilities to construct compositions of the geometric transformations. Scores on follow-up assessment items indicated that the prospective teachers' levels of expertise corresponded to their levels of confidence. Conclusions indicated that dynamic geometry on the calculator was an appropriate technology, but one that required careful planning, to develop these future teachers' expertise with the compositions. -
Sage and scribe – asymmetrical pair work that can easily fit into any mathematics lesson, yet still have cooperative benefits
133-164Views:519This article uses a case study experiment to learn the characteristics of a pair work, called the sage and scribe method (Kagan, 2008). We also wished to explore the positive and negative effects of the systematic application of this single cooperative element without any other structural changes during the lessons. In the case study experiment, we asked two teachers, accustomed to traditional frontal teaching methods, to substitute individual work tasks in their standard lesson plans with the sage and scribe method. Our experiments indicate that this method wastes insignificant time, requires little extra effort on the part of the teacher, yet has many of the positive effects of cooperative methods: in our experiments, students received immediate feedback, corrected each other’s mistakes, learned from each other in meaningful discussions and engaged in collaborative reasoning to address emerging problems.
Subject Classification: 97D40
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Process or object? Ways of solving mathematical problems using CAS
117-132Views:75Graphing 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 to be well-connected?" An example for instructional process planning with Problem Graphs
145-155Views:195Teachers’ design capacity at work is in the focus of didactical research worldwide, and fostering this capacity is unarguably a possible turning point in the conveyance of mathematical knowledge. In Hungary, the tradition hallmarked by Tamás Varga is particularly demanding towards teachers as they are supposed to be able to plan their long-term processes very carefully. In this contribution, an extensive teaching material designed in the spirit of this tradition will be presented from the field of Geometry. For exposing its inner structure, a representational tool, the Problem Graph is introduced. The paper aims to demonstrate that this tool has potential for analyzing existing resources, helping teachers to reflect on their own preparatory and classroom work, and supporting the creation of new designs.
Subject Classification: 97D40, 97D50, 97D80, 97G10, 97U30
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Experiences in the education of mathematics during the digital curriculum from the perspective of high school students
111-128Views:278Due to the COVID-19 epidemic, Hungarian schools had to switch to a digital curriculum for an extended period between 2019 and 2021. In this article, we report on the experiences regarding the education of mathematics during the digital curriculum in the light of the reinstated on-site education, all through the eyes of high school students. Distance education brought pedagogical renewal to the lives of many groups. Students were asked about the positives and negatives of this situation.
Subject Classification: 97C90
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Heuristic arguments and rigorous proofs in secondary school education
167-184Views:135In 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. -
Mapping students’ motivation in a problem oriented mathematics classroom
111-121Views:187This research focuses on mapping students’ motivation by implementing problem-solving activities, namely how the problem-oriented approach affects the students’ commitment, motivation, and attitude to learning. As a practicing teacher, the author faced difficulties with motivation and sought to improve her practice in the form of action research as described in this paper. Based on the literature, the author describes sources of motivation as task interest, social environment, opportunity to discover, knowing why, using objects, and helping others. The author discusses the effect of problem-oriented teaching on the motivation of 7th-grade students. In this paper, the results of two lessons are presented.
Subject Classification: 97C20, 97D40, 97D50, 97D60
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Research studies in didactics of mathematics supported by the Operant Motive Test
153-173Views:124The present paper reports a case-study which took place within an EUsupported international program organized for research and development of multi-grade schools (NEMED, [16] [26]). One of the main goals of the research was to develop the connection between disadvantageous social situations and the efficiency (success or failure) in learning mathematics especially from the point of view of average and above-average (talented) students: Why does the talent of children with socially disadvantageous background remain undiscovered? How can we make school mathematics more aware of hidden talents?
The author was looking for a didactical solution that compensated for social disadvantages without restricting the development of "average" students by using sociological, educational, psychological and mathematical (experimental and theoretical) studies in interaction with a series of experimental (hypothesis testing and exploratory) investigations.
We constructed tools and methods for exploration and experimental teaching, adapted to Hungarian conditions (Curriculum Development, teacher training, materials, interviews, Kuhl's motivation test, Malara's "researchers and practicing teachers in cooperation" method, etc., see [18], [20]).
The teaching materials and methodological guidelines are based on Bruner's representation theory (see [5]). The empirical research took place in 16 multi-grade schools located in different parts of the country. The author co-operated with nearly 250 students and 25 teachers for 3 years. In this paper we try to demonstrate how an Operant Motive Test can be involved in this research (see [18]). -
Constructing the disk method formula for the volume obtained by revolving a curve around an axis with the help of CAS
363-376Views:122Calculus concepts should have been taught in a carefully designed learning environment, because these concepts constitute a very important base for almost all applied sciences. The integral, one of the fundamental concepts of Calculus, has a wide application area. This paper focuses on constructing the disk method formula for the volume obtained by revolving a curve around an axis with the help of a CAS.
In this study, a semi-structured interview was carried out. In this interview, we tried to construct the disk method formula.
The levels of constructing the disk method formula in this study are:
• Introducing the concept: evaluating the volume of an Egyptian pyramid.
• Evaluating the volume of a cone obtained by revolution (using Maple worksheet).
• Designing their own ring and evaluating its price (using Maplet).
In this study, the interview has been presented as a dialog between teacher and students. When we look at feedback from students, we see that such a teaching method effects students in a positive way and causes them to gain conceptual understanding directed towards the concepts of approximation and volume. -
Some Remarks on History of Mathematical Problem Solving
51-64Views:123In this contribution, it is our goal is to look on history of mathematics as a resource for a long-term study of mathematical problem solving processes and heuristics. In this way we intend to get additional information, e. g., about heuristics which proved to be extremely successful to create new mathematics. "Changing representation" and "false position" are examples of such strategies, which are illustrated by concrete examples to demonstrate the use for classroom teaching and teacher education. Our methods are based on hermeneutic principles. -
Teaching probability theory by using a web based assessment system together with computer algebra
81-95Views:123In the course of Maths Basics 2, the Faculty of Economic Science students of Kaposvár University learn the classical chapters of Probability Theory, namely random variables and the well-known probability distributions. Our teaching experiences show that students' achievement is weaker in case of problems concerning continuous random variables. From school year 2012/13 we have had an opportunity to take Maple TA, the web-based test- and assessment system, into the course of education. It is sufficient for the users of Maple TA to have a browser. Maple computer algebra system, which runs on the server, assesses students' answers in an intelligent way, and compares them with the answers that are considered correct by the teacher. In our presentation we introduce some elements of Maple TA system, the didactic considerations the test sheets were made by, as well as our research results concerning the use of Maple TA. -
Experiences using CAS and multimedia int teaching vectorcalculus
363-382Views:76The development of informatics brings new opportunities that need reevaluating of the teaching concepts. For this reason we have performed a comprehensive educational development for engineering students. Our main goals were to work out a new educational strategy, to develop the needed package of the subject material, to introduce the strategy in the practice, to analyze and evaluate the experiences. In the developed and adapted teaching-learning strategy the teacher is the organizer, designer and the manager of the process. In this paper we summarize the concepts, the results and experiences of the 3-years-long development.
