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On the legacy of G. Pólya: some new (old) aspects of mathematical problem solving and relations to teaching
169-189Views:36In this article are given some new aspects of mathematical problem solving. A framework is presented by three main resources: (1) Pólya's studies about mathematical heuristics are augmented by information drawn from a study of the history of mathematical problem solving. (2) Connections are presented between mathematical problem solving and mathematical beliefs. (3) Experience with a special program for mathematical talented students is sketched. On this background a new textbook-series has been developed and some teaching examples are taken from this context. An outlook is given on some new research on teaching of problem solving, including possible relations to modern brain research. -
Problem-solving in mathematics with the help of computers
405-422Views:33One of the most important tasks of the didactics of mathematics is the describing of the process of problem-solving activity and problem-solving thinking. The psychological theories concerning the problem-solving thinking leave the special demand of school subjects out of consideration, and search for connections of universal validity. In this article we attempt to connect an abstract theory of psychology concerning problem-solving thinking and a more practical conception of the problem-solving activity of mathematics, which is based on Polya's idea. In this way we can get a structure of problem-solving, which has scientific bases and at the same time it is useful in computer aided learning. Our result was developed and tested in Hungary so this is suitable especially for the Hungarian conditions of mathematics teaching. -
The tradition of problem-posing in Hungarian mathematics teaching
233-254Views:182Based on the literature, Pólya was influential in problem-posing research. The present paper draws attention to a book written with Pólya's collaboration, which has not yet received sufficient emphasis in the problem-posing literature. On the other hand, Pólya's impact on mathematics education in Hungary has been significant, including the problem-posing paradigm. Two works, published only in Hungarian, that rely heavily on problem-posing are highlighted. Furthermore, it is presented how problem-posing appeared in the Hungarian Complex Mathematics Teaching Experiment (1962-78) led by Tamás Varga.
Subject Classification: 97D50
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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. -
Problemorientierung im Mathematikunterricht – ein Gesichtspunkt der Qualitätssteigerung
251-291Views:7The aim of this article is to give a synopsis of problem orientation in mathematics education and to stimulate the discussion of the development and research about problem-orientated mathematics teaching. At the beginning we present historical viewpoints of problem orientation and their connection with recent theories of cognition (constructivism). Secondly we give characterizations of concepts that stand in the context of problem-orientation and discuss different forms of working with open problems in mathematics teaching. Arguments for more problem orientation in mathematics education will be discussed afterwards. Since experience shows that the implementation of open problems in classroom produces barriers, we then discuss mathematical beliefs and their role in mathematical learning and teaching. A list of literature at the end is not only for references but also can be used to further research.
Zusammenfassung. Ziel des Beitrags ist es, eine Synopsis in Bezug auf Problemorientierung im Mathematikunterricht zu geben und die Diskussion bezüglich Entwicklung und Forschung eines problemorientierten Mathematikunterrichts zu stimulieren. Als Erstes werden historische Gesichtspunkte von Problemorientierung und deren Verkn üpfung mit neueren Erkenntnistheorien (Konstruktivismus) vorgestellt. Zweitens werden Erläuterungen zu Begriffen, die im Kontext von Problemorientierung stehen, gegeben und verschiedene Ausprägungen der Behandlung offener Probleme im Mathematikunterricht diskutiert. Argumente für eine stärkere Berücksichtigung von Problemorientierung im Mathematikunterricht werden danach erörtert. Auf Barrieren bei der Implementierung von offenen Problemen im Unterricht, die durch mathematische Beliefs (Vorstellungen, Überzeugungen) geprägt sind, wird zum Schluss eingegangen. Die abschließend aufgeführte Literaturliste dient nicht nur dem Beleg der Zitate, sondern kann auch zu weiterer Vertiefung genutzt werden. -
Analyse von Lösungswegen und Erweiterungsmöglichkeiten eines Problems für die Klassen 7–11
231-249Views:31Making several solutions for a problem i.e. the generalization, or the extension of a problem is common in the Hungarian mathematics education.
But the analysis of a problem is unusual where the connection between the mathematical content of the task and of its different formulations is examined, solutions from different fields of mathematics are presented regarding the knowledge of different age groups, the problem is generalized in different directions, and several tools (traditional and electronic) for solutions and generalizations are presented.
This kind of problem analysis makes it viable that during the solution/elaboration several kinds of mathematical knowledge and activities are recalled and connected, facilitating their use inside and outside of mathematics.
However, an analysis like this is not unfamiliar to the traditions of the Hungarian problem solving education – because it also aims at elaborating a problem – but from several points of view.
In this study, a geometric task is analysed in such a way. -
The role of representations constructed by students in learning how to solve the transportation problem
129-148Views:107The purpose of the research presented in this paper was to study the role of concrete and table representations created by students in learning how to solve an optimization problem called the transportation problem. This topic was learned in collaborative groups using table representations suggested by teachers in 2021. In 2022, the researchers decided to enrich the students’ learning environment with concrete objects and urged the students to use them to present the problem to be solved. The students did it successfully and, to be able to record it in their notebooks, they constructed a table representation by themselves without any help from their teacher. After that, they managed to solve the problem by manipulating the objects. At the same time, each step in the solution was presented with changes in the table. The students were assessed before (pre-test) and after collaborative learning (test) in both academic years. The pre-test results were similar, but the test results were better in 2022. Therefore, it can be concluded that using concrete and table representations constructed by students in learning how to solve transportation problems makes collaborative learning more constructivist and more effective than when they use only table representations suggested by their teachers.
Subject Classification: 97M10, 97M40
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On the psychology of mathematical problem solving by gifted students
289-301Views:34This 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. -
Looking back on Pólya’s teaching of problem solving
207-217Views:229This article is a personal reflection on Pólya's work on problem solving, supported by a re-reading of some of his books and viewing his film Let Us Teach Guessing. Pólya's work has had lasting impact on the goals of school mathematics, especially in establishing solving problems (including non-routine problems) as a major goal and in establishing the elements of how to teach for problem solving. His work demonstrated the importance of choosing rich problems for students to explore, equipping them with some heuristic strategies and metacognitive awareness of the problem solving process, and promoting 'looking back' as a way of learning from the problem solving experience. The ideas are all still influential. What has changed most is the nature of classrooms, with the subsequent appreciation of a supporting yet challenging classroom where students work collaboratively and play an active role in classroom discussion.
Subject Classification: 97D50, 97A30
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Teaching of problem-solving strategies in mathematics in secondary schools
139-164Views:8In the Hungarian mathematics education there is no explicit teaching of problem-solving strategies. The best students can abstract the strategies from the solutions of concrete problems, but for the average students it is not enough. In our article we report about a developmental research. The topic of the research was the explicit teaching of two basic strategies (forward method, backward method). Based on our experiences we state that it is possible to increase the effectivity of students' problemsolving achievement by teaching the problem-solving strategies explicitly. -
Zbigniew Michalewicz - Matthew Michalewicz: Puzzle Based Learning: An introduction to critical thinking, mathematics, and problem solving. Hybrid Publishers Melbourne 2008 (Book review)
415-420Views:42Based on their experiences with engineering, mathematics, computer science, business students concerning the puzzle based learning in different countries the authors summarize their main problem solving teaching ideas. With help of interesting, motivating, nice problems they analyze the main mathematical principles and problem types. The review gives an overview about the main ideas, results of an interesting book. -
Dressed up problems - the danger of picking the inappropriate dress
77-94Views:15Modelling and dressed-up problems play an inevitably unavoidable role in mathematics education. In this study we would like to point out how dangerous is it to dress up mathematical problems. We go back to the principle of De Lange: The problem designer is not only dressing up the problem, but he is the solution designer, as well. We show three examples selected from Hungarian high school textbooks where the intended solution does not solve the problem, because the dressing changes the context and changes the problem itself. -
Computer cooking vs. problem solving
35-58Views:66Computer cooking is a task-related phenomenon where students (end-users) must blindly follow a long list of orders without any connection to the content of the problem, if there is any. Despite its low efficacy, this method is widely used and accepted in informatics both in the learning-teaching process and testing. The National Base Curriculum 2020 in Hungary is in complete accordance with the ‘Informatics Reference Framework for Schools’, but the course books hardly use the latest results of computer education research. The present paper provides examples of how the results of computer education research can be integrated into teaching-learning materials and classroom practices and discusses the effectiveness and consequences of the different solutions, where tool-centred approaches are compared to problem-focused solutions.
Subject Classification: 94-01
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Programming Theorems and Their Applications
213-241Views:117One of the effective methodological approaches in programming that supports the design and development of reliable software is analogy-based programming. Within this framework, the method of problem reduction plays a key role. Reducing a given problem to another one whose solving algorithm is already known can be made more efficient by the application of programming theorems. These represent proven, abstract solutions – in a general form – to some of the most common problems in programming. In this article, we present six fundamental programming theorems as well as pose five sample problems. In solving these problems, all six programming theorems will be applied. In the process of reduction, we will employ a concise specification language. Programming theorems and solutions to the problems will be given using the structogram form. However, we will use pseudocodes as descriptions of algorithms resembling their actual implementation in Python. A functional style solution to one of the problems will also be presented, which is to illustrate that for the implementation in Python, it is sufficient to give the specification of the problem for the design of the solution. The content of the article essentially corresponds to that of the introductory lectures of a course we offered to students enrolled in the Applied Mathematics specialization.
Subject Classification: D40
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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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"How to be well-connected?" An example for instructional process planning with Problem Graphs
145-155Views:96Teachers’ 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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Is it possible to develop some elements of metacognition in a Mathematics classroom environment?
123-132Views:89In an earlier exploratory survey, we investigated the metacognitive activities of 9th grade students, and found that they have only limited experience in the “looking back” phase of the problem solving process. This paper presents the results of a teaching experiment focusing on ninth-grade students’ metacognitive activities in the process of solving several open-ended geometry problems. We conclude that promoting students’ metacognitive abilities makes their problem solving process more effective.
Subject Classification: 97D50, 97G40
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Report on "Problem Solving in Mathematics Education": ProMath 6 Conference, 8–11 September, 2005, Debrecen, Hungary
313-319Views:33The sixth ProMath Conference was organized at the University of Debrecen (Hungary) in the year 2005. There were 12 presentations. After a short historical introduction we present the 12 abstracts written by the authors. -
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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Realizing the problem-solving phases of Pólya in classroom practice
219-232Views:124When teaching mathematical problem-solving is mentioned, the name of Pólya György inevitably comes to mind. Many problem-solving lessons are planned using Pólya's steps and helping questions, and teachers often rely on his heuristics even if their application happens unconsciously. In this article, we would like to examine how the two phases, Making a plan and Looking back, can be realized in a secondary school mathematics lesson. A case study was designed to observe and analyse a lesson delivered using cooperative work.
Subject Classification: 97B10, 97C70, 97D40, 97D50
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The transition problem in Hungary: curricular approach
1-16Views:120The curricular background of the transition problem from highschool to universty is analysed in Hungary. While students finish their mathematical studies successfully at highschool, pass their final exams, this knowledge seems to disappear at their first year at university. We investigate the mathematical knowledge expected by the Hungarian universities and compare it to expectations of the National Core Curriculum. Based on the levelling tests of four universities we created a seven problem test for highschool students containing very basic problems required both by the universities and the National Core Curriculum. We analyse the results of the test.
Subject Classification: D34, D35
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Word problems in different textbooks at the early stage of teaching mathematics comparative analysis
31-49Views:151In a previous research, Csíkos and Szitányi (2019) studied teachers’ views and pedagogical content knowledge on the teaching of mathematical word problems. While doing so, they reviewed and compared Eastern European textbooks of Romania, Russia, Slovakia, Croatia, and Hungary to see how world problem-solving strategies are presented in commonly used textbooks. Their results suggested that teachers, in general, agreed with the approach of the textbooks regarding the explicit solution strategies and the types of word problems used for teaching problem-solving. They also revealed that the majority of the participants agreed that a word problem-solving algorithm should be introduced to the students as early as in the first school year. These results have been presented at the Varga 100 Conference in November 2019. As the findings suggested a remarkable similarity between the Eastern European textbook approaches, in the current study we decided to conduct further research involving more textbooks from China, Finland, and the United States.
Subject Classification: 97U20, 08A50
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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. -
Some Remarks on History of Mathematical Problem Solving
51-64Views:33In 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 undergraduate mathematics - a problem solving course for first year
183-206Views:102In this paper we describe a problem solving course for first year undergraduate mathematics students who would be future school teachers.
Subject Classification: 97B50, 97B70, 97D50, 97D60, 97F60, 97U30