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Maximum and minimum problems in secondary school education
81-98Views:29The 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. -
Square root in secondary school
59-72Views:107Although in Hungary, for decades, the calculation method of the square root of a real number is not in the mathematics curriculum, many of the taught concepts and procedures can be carried out using different square root finding methods. These provide an opportunity for students in secondary school to practice and deepen understand the compulsory curriculum. This article presents seven square-root- nding methods, currently teachable in secondary schools.
Subject Classification: A33, A34, F53, F54
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Diophantine equations concerning various means of binomial coefficients
71-79Views:19The main goal of this paper is to show by elementary methods, that there are infinitely many different pairs of binomial coefficients of the form (n C 2) such that also their arithmetic, geometric and harmonic means, resp. have the same form. We give all solutions for the arithmetic mean. We also give infinitely many non-trivial solutions for the arithmetic mean of three binomial coefficients satisfying some special conditions. The proofs require the solution of some other interesting Diophantine equations, too. Since the author is also a secondary school teacher, we use elementary methods that mostly can be discussed in secondary school, mainly within the framework of group study sessions. This explains why the means are generally analysed for two terms and for binomial coefficients with "lower" value 2, since further generalizations require substantially deeper mathematical methods which are beyond the frames of this paper. -
MRP tasks, critical thinking and intrinsic motivation to proving
149-168Views:23The lack of students' need for proof is often discussed. This is an important topic, on which quite a few others have written ([26], [27], [28], [17], [8]). Nevertheless, there is limited research knowledge about how teacher can participate in process of raising of students' intrinsic motivation to proving. In this article, we discuss relationships between intrinsic motivation to proving, critical thinking and special activity – engaging with so-called MRP tasks. We present here results of a research carried out by author in two elementary schools (21 classes, grade 5-9) in Ruzomberok, Slovakia. We identified the interesting relationship between students' dealing with MRP tasks and increasing of their intrinsic motivation to proving. -
Comparing various functions of the divisors of an integer in different residue classes
247-258Views:24The main goal of this paper is to investigate some problems related to the distribution of the divisors of a number in different residue classes. We study these questions modulo 3, and use mostly just elementary number theory. In some special cases, we demonstrate how this problem is related to other fields of maths, especially to combinatorics. Since the author is also a secondary school teacher, we use elementary methods that can be discussed in secondary school, mainly within the framework of group study sessions or in special maths classes. We do think that the investigation of these types of questions can motivate children to find their own way to create their own questions, and to get a deeper insight into problem solving by these experimentations. -
Solving Diophantine equations with binomial coefficients in study group sessions using both elementary and higher mathematical methods
1-12Views:19The paper can be considered as the continuation of [4] in the sense that we are studying Diophantine equations containing binomial coefficients. It was an important aspect that one should be able to discuss these problems — even if not in complete depth — also in high school study group sessions with the most talented students. We present various methods through several examples, which help the successful handling of other questions too, including problems in math competitions. Our discussion starts with the elementary treatment of easier problems, and then proceed gradually to more difficult questions which require higher mathematical methods. -
Comparative geometry on plane and sphere: didactical impressions
81-101Views:2Description of experiences in teaching comparative geometry for prospective teachers of primary schools. We focus on examples that refer to changes in our students' thinking, in their mathematical knowledge and their learning and teaching attitudes. At the beginning, we expected from our students familiarity with the basics of the geographic coordinate system, such as North and South Poles, Equator, latitudes and longitudes. Spherical trigonometry was not dealt with in the whole project. -
On some problems on composition of arithmetic functions
161-181Views:14The main goal of this paper is to investigate some problems related to the commutativity of the composition of arithmetic functions. The concept of commutativity arises many times in high school maths, so it is natural to study the composition of functions, namely the equation f(g(n)) = g(f(n)), where f and g are such well known arithmetic functions as d(n), φ(n), σ(n), ω(n), or Ω(n). We study various aspects of solvability: can we exhibit infinitely many solutions; can we determine every solution; can we find suitable values in the range of both functions f and g for which the equation is, or is not solvable, respectively. We need just the basic facts about the above functions,and we use only elementary methods in the proofs. We present some interesting questions, their solutions, and raise some unsolved problems. We found that this topic can be discussed well in secondary school, mainly within the framework of group study sessions as we had some classes with a group of kids in 9th grade. We summarize the experiences of this experiment in the last section. -
Teaching probability using graph representations
103-122Views:27The main objective of this paper is to present an elementary approach to classical probability theory, based on a Van Hiele type framework, using graph representation and counting techniques, highly suitable for teaching in lower and upper secondary schools. The main advantage of this approach is that it is not based on set theoretical, or combinatorial knowledge, hence it is more suitable for beginners and facilitates the transitions from level 0 to level 3. We also mention a few teaching experiences on different levels (lower secondary school, upper secondary school, teacher training, professional development, university students) based on this approach. -
Analysis of the affective factors of learning mathematics among teacher trainees
225-254Views:32The Hungarian National Core Curricula gives primacy to the development of abilities and the practical application of knowledge. The task of the training programme is primarily to prepare teacher trainees for the teaching and educating profession. As teachers, they are going to plan, organize, help, guide, control and evaluate the learning of mathematics of individuals and groups of students from the age of 6 to 10 (12), and cultivate their mathematical skills, thinking and positive attitude towards any mathematical activities. In order to train educators who are able to meet the above requirements on high standard, it is necessary to update the teacher training programme based on the trainees' preliminary knowledge and motivation level.
The key to learn about the child's mind and achieve conscious development is the systematization of factual knowledge and methodological awareness. The modern, flexible approach to subject pedagogy, based on pedagogy, psychology and epistemology, qualifies trainees to educate learners who understand and like mathematics. Therefore, it is essential to develop the trainees' positive approach to mathematics and arouse their demand for continuous professional improvement. (Programme of the four-year primary school teacher training, 1995.)
In our research we are looking for ways of ascertaining the starting parameters which have influence on the planning of the studies of mathematics and subject pedagogy. In this article we introduce a questionnaire by the means of which we collected information on the trainees' attitude and its changing towards mathematics. With the help of the analysis of the answers we paint a picture of the ELTE TÓFK (Eötvös Loránd University, Faculty of Elementary and Nursery School Teacher's Training) third year students' attitude to the subject, and we compare it to the tendencies noticed in the mass education. The energy invested in learning is influenced by the assumption of the relevance and importance of the subjects. Therefore we considered it also our task to reveal. Besides the students' attitude toward mathematics and their assumption about their own competence we have collected data also on their performance in the subject. Summarising the research results we show the advantages of the questionnaire, and summarise the observations which would indicate need for methodological changes in the mathematics teacher training. -
Prime building blocks in the mathematics classroom
217-228Views:134This theoretical paper is devoted to the presentation of the manifold opportunities in using a little-known but powerful mathematical manipulative, the so-called prime building blocks, originally invented by two close followers of Tamás Varga, to support discovery of various concepts in arithmetic in middle school, including the Fundamental Theorem of Arithmetic or as it is widely taught, prime factorization. The study focuses on a teaching proposal to show how students can learn about greatest common divisor (GCD) and least common multiple (LCM) with understanding, and meanwhile addresses internal connections and levels of abstractness within elementary number theory. The mathematical and methodological background to understanding different aspects of the concept prime property are discussed and the benefits of using prime building blocks to scaffold students’ discovery are highlighted. Although the proposal was designed to be suitable for Hungarian sixth graders, mathematical context and indications for the use of the manipulative in both primary and high school are given.
Subject Classification: F60, C30, E40, U60
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Nice tiling, nice geometry!?!
269-280Views:32The 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. -
Die Stichprobe als ein Beispiel dafür, wie im Unterricht die klassische und die bayesianische Auffassung gleichzeitig dargestellt werden kann
133-150Views:24Teaching statistics and probability in the school is a new challenge of the Hungarian didactics. It means new tasks also for the teacher- and in service-teacher training. This paper contains an example to show how can be introduced the basic notion of the inference statistics, the point- and interval-estimation by an elementary problem of the public pole. There are two concurrent theories of the inference statistics the so called classical and the Bayesian Statistics. I would like to argue the importance of the simultaneously introduction of both methods making a comparison of the methods. The mathematical tool of our elementary model is combinatorial we use some important equations to reach our goal. The most important equation is proved by two different methods in the appendix of this paper. -
Evaluating admission procedures for teacher education in Finland
231-243Views:28In 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. -
A proposed application of Monte Carlo method in teaching probability
37-42Views:32Pupils' misconception of probability often results from lack of experience. Combining the concept of probability and statistics, the proposed application is intended for the teachers of mathematics at an elementary school. By reformulating the task in the form of an adventure, pupils examine a mathematical problem, which is too difficult for them to solve by combinatorial method. By recommending the simulation of the problem, we have sought to provide pupils with valuable experience of experimenting, recording and evaluating data. -
The Project Method and investigation in school mathematics
241-255Views:27The Project Method (PM) is becoming more common in the teaching of mathematics. Most of the time, Project Method means solving open and relatively wide formulated problems for the application of particular mathematical topics and the solving of everyday life problems.
At present many experts in the theory of teaching mathematics advocate teaching activities as the characteristic for most mathematical work in the classroom. Thus, there is a question: whether it is possible or eventual desirable to use the PM for solving genuine mathematical problems. This paper deals with this question and discusses the connection between the PM and investigation of new mathematical knowledge for students. Our experience has shown that the PM in connection with investigations can be a useful and effective approach to teaching mathematics. -
Eine geometrische Interpretation der Ausgleichsrechnung
159-173Views:26Using real examples of applied mathematics in upper secondary school one has do deal with inaccurate measures. This will lead to over constrained systems of linear equations. This paper shows an instructive approach which uses methods of descriptive and computer aided geometry to get a deeper insight into the area of calculus of observations. Using a qualified interpretation one can solve problems of calculus of observations with elementary construction techniques of descriptive geometry, independent of the norm one uses. -
Pupils' meta-discursive reflection on their cooperation in mathematics: a case study
147-169Views:26This article addresses the issue of how 10–11 year old pupils in pairs can actively get involved in reforming their behavior as they reflect on their interaction in order to solve mathematical problems. We studied the opportunities offered for the development of meta-discursive reflection in a pair of pupils in two alternative environments: (1) pupils' observations and discussions on their video-recorded cooperation and (2) pupils' participation in playing and acting in a drama. The results of the research revealed three levels of the pupils' meta-discursive reflection on their interaction: (1) focusing on the achievement of personal goals, (2) focusing on partners' responsibility and (3) focusing on mutual responsibility. Both environments helped the pupils to improve their socio-mathematical interaction. -
On four-dimensional crystallographic groups
391-404Views:7In his paper [12] S. S. Ryshkov gave the group of integral automorphisms of some quadratic forms (according to Dade [6]). These groups can be considered as maximal point groups of some four-dimensional translation lattices in E^4. The maximal reflection group of each point group, its fundamental domain, then the reflection group in the whole symmetry group of the lattice and its fundamental domain will be discussed. This program will be carried out first on group T. G. Maxwell [9] raised the question whether group T was a reflection group. He conjectured that it was not. We proved that he had been right. We shall answer this question for other groups as well. Finally we shall give the location of the considered groups in the tables of monograph [4]. We hope that our elementary method will be useful in studying linear algebra and analytic geometry. Futhermore, 4-dimensional geometry with some visualisation helps in better understanding important concepts in higher-dimensional mathematics, in general. -
Zur Visualisierung des Satzes von Pythagoras
217-228Views:13In this article we make a study of a not-classical visualization of the theorem of Pythagoras using methods of elementary school geometry. We find collinear points, copoint straight lines and congruent pairs of parallelograms. The configuration of their midpoints induces a six-midpoint and a four-midpoint theorem. -
Decomposition of triangles into isosceles triangles II: complete solution of the problem by using a computer
275-300Views:26We solve an open decomposition problem in elementary geometry using pure mathematics and a computer programme, utilizing a computer algebra system. -
A mathematical and didactical analysis of the concept of orientation
111-130Views:31The development of spatial ability, in particular the development of spatial orientation is one of the aims of mathematics education.
In my work, I examine the concept of orientation, especially concepts of between, left, right, below, above, front, back, clockwise and anticlockwise. I analyze answers given for a simple orientation task prepared for elementary school pupils. I would like to call attention to the difficulties pupils have even in case of solving simple orientation problems.
We have different ways to know more about the crucial points of a concept, especially of the concept of orientation. In this study I bring out one of them. I analyze and make some didactical conclusions about the origin and the axiomatic structure of orientation. -
Mapping students’ motivation in a problem oriented mathematics classroom
111-121Views:60This 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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Compositions of dilations and isometries in calculator-based dynamic geometry
257-266Views:30In 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.
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