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Regula falsi in lower secondary school education II
121-142Views:236The 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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Transition from arithmetic to algebra in primary school education
225-248Views:217The main aim of this paper is to report a study that explores the thinking strategies and the most frequent errors of Hungarian grade 5-8 students in solving some problems involving arithmetical first-degree equations. The present study also aims at identifying the main arithmetical strategies attempted to solve a problem that can be solved algebraically. The analysis focuses on the shifts from arithmetic computations to algebraic thinking and procedures. Our second aim was to identify the main difficulties which students face when they have to deal with mathematical word problems. The errors made by students were categorized by stages in the problem solving process. The students' written works were analyzed seeking for patterns and regularities concerning both of the methods used by the students and the errors which occured in the problem solving process. In this paper, three prominent error types and their causes are discussed. -
Teaching sorting in ICT
101-117Views:179This article is aimed at considering how an algorithmic problem – more precisely a sorting problem – can be used in an informatics class in primary and secondary education to make students mobilize the largest possible amount of their intellectual skills in the problem solving process. We will be outlining a method which essentially forces students to utilize their mathematical knowledge besides algorithmization in order to provide an efficient solution. What is more, they are expected to use efficiently a tool that has so far not been associated with creative thinking. Sorting is meant to be just an example, through which our thoughts can easily be demonstrated, but – of course the method of education outlined can be linked to several other algorithmic problems, as well. -
Regula falsi in lower secondary school education
169-194Views:179The 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. -
Simple Variations on The Tower of Hanoi: A Study of Recurrences and Proofs by Induction
131-158Views:366The Tower of Hanoi problem was formulated in 1883 by mathematician Edouard Lucas. For over a century, this problem has become familiar to many of us in disciplines such as computer programming, algorithms, and discrete mathematics. Several variations to Lucas' original problem exist today, and interestingly some remain unsolved and continue to ignite research questions. Nevertheless, simple variations can still lead to interesting recurrences, which in turn are associated with exemplary proofs by induction. We explore this richness of the Tower of Hanoi beyond its classical setting to compliment the study of recurrences and proofs by induction, and clarify their pitfalls. Both topics are essential components of any typical introduction to algorithms or discrete mathematics.
Subject Classification: A20, C30, D40, D50, E50, M10, N70, P20, Q30, R20
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The use of different representations in teaching algebra, 9 th grade (14-15 years old)
29-42Views:162Learning Algebra causes many difficulties for students. For most of them Algebra means rote memorizing and applying several rules without understanding them which is a great danger in teaching Algebra. Using only symbolic representations and neglecting the enactive and iconic ones is a great danger in teaching Algebra, too. The latter two have a primary importance for average students.
In our study, we report about an action research carried out in a grade 9 class in a secondary school in Hungary.The results show that the use of enactive and iconic representations in algebra teaching develops the students' applicable knowledge, their problem solving knowledge and their problem solving ability. -
The effects of chess education on mathematical problem solving performance
153-168Views:218We 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. -
Some Remarks on History of Mathematical Problem Solving
51-64Views:149In 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:228In 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
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On an international training of mathematically talented students: assets of the 20 years of the “Nagy Károly Mathematical Student-meetings”
77-89Views:244The focus of this paper is to present the gems of the "Nagy Károly Mathematical Student-meetings" in Rév-Komárom (Slovakia) from 1991 to 2010. During these 20 years there was done a lot of work to train mathematically talented students with Hungarian mother tongue and to develop their mathematical thinking, and to teach them problem solving and heuristic strategies for successful acting on the competitions. We collected the most interesting problems and methods presented by the trainer teachers. -
Teaching of old historical mathematics problems with ICT tools
13-24Views:213The aim of this study is to examine how teachers can use ICT (information and communications technology) tools and the method of blended learning to teach mathematical problem solving. The new Hungarian mathematics curriculum (NAT) emphasizes the role of history of science, therefore we chose a topic from the history of mathematics, from the geometry of triangles: Viviani's Theorem and its problem field. We carried out our teaching experiments at a secondary school with 14-year-old students. Students investigated open geometrical problems with the help of a dynamic geometric software (GeoGebra). Their research work was similar to the historical way. -
Nice tiling, nice geometry!?!
269-280Views:140The squared papers in our booklets, or the squared (maybe black and white) pavements in the streets arise an amusing problem: How to deform the side segments of the square pattern, so that the side lines further remain equal (congruent) to each other? More precisely, we require that each congruent transformation of the new pattern, mapping any deformed side segment onto another one, leaves the whole (infinitely extended) pattern invariant (unchanged).
It turns out that there are exactly 14 types of such edge-transitive (or so-called isotoxal) quadrangle tilings, sometimes with two different forms (e.g. black and white) of quadrangles (see Figure 2). Such a collection of tiling can be very nice, perhaps also useful for decorative pavements in streets, in flats, etc.
I shall sketch the solution of the problem that leads to fine (and important) mathematical concepts (as barycentric triangulation of a polygonal tiling, adjacency operations, adjacency matrix, symmetry group of a tiling, D-symbol, etc). All these can be discussed in an enjoyable way, e.g. in a special mathematical circle of a secondary school, or in more elementary form as visually attractive figures in a primary school as well.
My colleague, István Prok [11] developed an attractive computer program on the Euclidean plane crystallographic groups with a nice interactive play (for free download), see our Figures 3-5.
A complete classification of such Euclidean plane tilings (not only with quadrangles) can be interesting for university students as well, hopefully also for the Reader (Audience). This is why I shall give some references, where you find also other ones.
Further problems indicate the efficiency of this theory now. All these demonstrate the usual procedure of mathematics and the (teaching) methodology as well: We start with a concrete problem, then extend it further, step-by-step by creating new manipulations, concepts and methods. So we get a theory at certain abstraction level. Then newer problems arise, etc.
This paper is an extended version of the presentation and the conference paper [7]. The author thanks the Organizers, especially their head Professor Margita Pavlekovic for the invitation, support and for the kind atmosphere of the conference. -
Virtual manipulatives in inquiry-based approach of 3D problems by French 5th graders
229-240Views:214The aim of this research is to study the appropriation of a 3D environment by learners in an a-didactical situation of problem solving. We try to evaluate the relevance of the virtual 3D environment in the development of students' cognitive and metacognitive abilities. We implanted a problem-solving activity related to a 3D cube situation with an empty part in the cube in different French primary school areas in May 2019. In the experimental group each learner works individually with a PC-computer where the virtual environment ANIPPO is implemented. In the control group the pupils work in a traditional class environment. We present the results of this pre-experimentation.
Subject Classification: 97D50, 97U60, 97U70
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Supporting the theory of math didactic using knowledge-measuring questions and analysis of the solutions
1-16Views:187New or rediscovered results presented in this paper are the results of the analysis of the problem sets used in the two-tier system secondary school final examination in mathematics, a system that was introduced in Hungary in 2005.
Many of the revealed problem arise in connection with misunderstanding the text of the problems. Causes of misinterpretation can be either that the text is lacking some important information, or that it should be interpreted not in word-to-word manner.
Theses and their argumentations presented here refer partly on the new types of problems (tests, non-standard mathematical contents), and partly on improvement of learning-teaching process in topics of equations and approximations. -
Pólya’s influence on (my) research
161-171Views:263In this article, I outline the influence of George Pólya's work on research in different areas and especially on mathematics education, namely heuristics and models of the problem-solving process. On a more personal note, I will go into some details regarding Pólya's influence on my own work in mathematical problem solving with a focus on the research project for my PhD thesis.
Subject Classification: 97xxx
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The Frobenius exchange problem on competitions and in classroom
203-218Views:57Let a_1, ..., a_n be relatively prime positive integers. The still unsolved Frobenius problem asks for the largest integer which cannot be represented as Σ x_i a_i with non-negative integers xi, and also for the number of non-representable positive integers. These and several related questions have been investigated by many prominent mathematicians, including Paul Erdős, and a wide range of partial results were obtained by various interesting methods differing both in character and difficulty. In this paper we give a self-contained introduction to this field through problems and comments suitable also for treatment in a class of talented students. -
Darstellungen und Vorstellungen und ihre Bedeutung für eine wirksame Metakognition beim Problemlösen und Begründen
195-220Views:155Metacognition has one of the highest effect sizes concerning successful learning. However metacognitive activities during task solving and problem solving are not directly obvious. But they can appear by writing someone's thoughts down. The following analysis, which focusses on the level of argumentation as well as on the way of derivation, shows that the quality of representation is an essential condition for the possibility of metacognition. -
A KöMaL problem in a new view
191-201Views:82The object of this paper is finding the general solution f : R^3 → R of the system of functional equations (1) valid for all x, y, z, t ϵ R. First f is expressed by a function of one variable which satisfies a system of two functional equations.This system is resolved by using an algebraic reformulation of the problem in terms of orbits and transversals. Finally the general solution of (1) is obtained. -
Interdisciplinary Secondary-School Workshop: Physics and Statistics
179-194Views:167The paper describes a teaching unit of four hours with talented students aged 15-18. The workshop was designed as a problem-based sequence of tasks and was intended to deal with judging dice whether they are regular or loaded. We first introduced the students to the physics of free rotations of rigid bodies to develop the physics background of rolling dice. The highlight of this part was to recognise that cubes made from homogeneous material are the optimal form for six-sided objects leading to equal probabilities of the single faces. Experiments with all five regular bodies would lead to similar results; nevertheless, in our experiments we focused on regular cubes. This reinsures that the participants have their own experience with the context. Then, we studied rolling dice from the probabilistic point of view and – step-by-step – by extending tasks and simulations, we introduced the idea of the chi-squared test interactively with the students. The physics and the statistics part of the paper are largely independent and can be also be read separately. The success of the statistics part is best described by the fact that the students recognised that in some cases of loaded dice, it is easier to detect that property and in other cases one would need many data to make a decision with small error probabilities. A physical examination of the dice under inspection can lead to a quick and correct decision. Yet, such a physical check may fail for some reason. However, a statistical test will always lead to reasonable decision, but may require a large database. Furthermore, especially for smaller datasets, balancing the risk of different types of errors remains a key issue, which is a characteristic feature of statistical testing.
Subject Classification: F90, K90, M50, R30
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Force of summation
185-199Views:155Programming theorems are important tools of programming methodology. By using analogous programming techniques, the solutions of different tasks can be created easily and fast based on programming theorems. Perhaps the summation is the simplest programming theorem that is widely-known among the programmers but once and for all the most various tasks can be solved by this theorem. The aim of the present paper is to investigate the summation programming theorem. Several different abstract levels of this theorem will be defined and the problem types that can be solved based on summation are going to be described. We will underline those points of a programming theorem that make a theorem general and that are not defined in advance, just later during its application, when the solution of a problem is derived from the theorem. -
Decomposition of triangles into isosceles triangles I: let the students ask bravely
163-184Views:146We report about working up an open geometric problem as a mathematical research with pupils of a mathematics camp. This paper shows the didactic aims and the methods we worked with, the didactic results. The second part of this paper gives a general solution of the problem, using pure mathematics and a computer programme. -
Teaching puzzle-based learning: development of basic concepts
183-204Views:386While computer science and engineering students are trained to recognise familiar problems with known solutions, they may not be sufficiently prepared to address novel real-world problems. A successful computer science graduate does far more than just program and we must train our students to reach the required levels of analytical and computational thinking, rather than hoping that it will just 'develop'. As a step in this direction, we have created and experimented with a new first-year level course, Puzzle-based Learning (PBL), that is aimed at getting students to think about how to frame and solve unstructured problems. The pedagogical goal is increase students' mathematical awareness and general problem solving skills by employing puzzles, which are educational, engaging, and thought provoking. We share our experiences in teaching such a course – apart from a brief discussion on our pedagogical objectives, we concentrate on discussing the presented material which covers (in two lectures) just one selected topic (pattern recognition). In this paper we present the ideas behind foundations for PBL and the material of the first of two lectures on pattern recognition, in which we address core concepts and provide students with sufficient exemplars to illustrate the main points. -
Why do we complicate the solution of the problem? reflection of Finnish students and teachers on a mathematical summer camp
405-415Views:200This paper deals with reactions and reflections of Finnish secondary school students and teachers on Hungarian mathematics teaching culture. The experiences were collected at a mathematics summer camp in Hungary. -
Integrating elements of data science into high-school teaching: Naïve Bayes-classification algorithm and programming in Python
307-316Views:256Probability theory and mathematical statistics are traditionally one of the most difficult chapters of mathematics to teach. One of the authors, Péter Princz has experience in teaching various topics via computer programming of the problem at hand as a class activity. The proposed method is to involve programming as a didactic tool in hard-to-teach topics. The intended goal in this case is to implement a naïve Bayes-classifier algorithm in Python and demonstrate the machine-learning capabilities of it by applying it to a real-world dataset of edible or poisonous mushrooms. The students would implement the algorithm in a playful and interactive way. The proposed incremental development process aligns well with the spirit of Tamás Varga who considered computers as modern tools of experimental problem solving as early as in the 1960s.
Subject Classification: 97D40, 97D50, 97K50, 97K99, 97M60, 97P40, 97P50, 97U50
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Challenges that a teacher-researcher faces during an action research – a case study
89-99Views:248This paper explores the dual role of the teacher-researcher in a four-year action research project focused on problem-based learning in mathematics. It highlights the challenges faced during the phases of planning, implementation, analysis, and reflection. Drawing on insights from the author’s experiences and observations based on both qualitative and quantitative data collection methods, the study identifies distinct challenges linked to the dual role, like differing design goals or subjective-objective voices. The author also proposes solutions to the identified challenges, such as collaboration with university experts and using reflective practices. Furthermore, the research underscores the beneficial impact of action research on enhancing teachers’ awareness and bridging the theory-practice gap, calling for further studies in this area.
Subject Classification: 97D99
