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Illustrated analysis of Rule of Four using Maple
383-404Views:163Rule of Four, as a basic didactic principle, was formulated among the NCTM 2000 standards (see [14]) and since then it is quoted by numerous books and publications (see [4], [9], [12]). Practically we can say it is accepted by the community of didactic experts. The usage of the Rule of Four, however, has been realized mainly in the field of calculus, in fact certain authors restrict the wording of the principle to the calculus itself (e.g. [3]).
Calculus is a pleasant field, indeed. A sequence of values of a function provides us with example for numeric representation, while the formula and the graph of the function illustrate symbolic and graphical representations, respectively. In the end by wording the basic features of the function on natural language we gain textual representation.
This idyllic scene, however, becomes more complex when we leave the frame of calculus. In this paper we investigate the consequences of the usage of Rule of Four outside calculus. We discuss the different types of representations and show several examples which make the multiple features of representation evident. The examples are from different fields of mathematics and are created by the computer algebra system Maple, which turns out to be an excellent tool for illustration and visualization of the maim features of mathematical objects.
Next we introduce the concept of basic representation and rational representation, which is considered as the mathematical notion of "didactic usable" or "didactic rational" representation. In the end we generalize the notion of numeric representation, which leads us a more widely usable didactic principle which can be considered as a generalization of Rule of Four. -
Word problems in different textbooks at the early stage of teaching mathematics comparative analysis
31-49Views:363In 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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Zur Visualisierung des Satzes von Pythagoras
217-228Views:161In 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. -
Equivalence and range of quadratic forms
123-129Views:95If two quadratic forms are equivalent, that is, if there is a linear transformation with integer coefficients and determinant 1 or −1 which takes one form to the other, then their ranges are the same and also their determinants are the same. The result of the paper is that for positive definite binary quadratic forms the converse is also true. Namely, if two positive definite binary quadratic forms of the same determinant have the same range, then they are equivalent. The arguments are guided by geometric considerations. -
The appearance of the characteristic features of the mathematical thinking in the thinking of a chess player
201-211Views:242It is more and more important in 21st century's education that not only facts and subject knowledge should be taught but also the ways and methods of thinking should be learnt by students. Thinking is a human specificity which is significant both in mathematics and chess. The exercises aimed at beginner chess players are appropriate to demonstrate to students the mathematical thinking of 12-14 year-old students.
Playing chess is an abstract activity. During the game we use abstract concepts (e.g. sacrifice, stalemate). When solving a chess problem we use logical quantifiers frequently (e.g. in the case of any move of white, black has a move that...). Among the endgames we find many examples (e.g. exceptional draw options) that state impossibility. Affirmation of existence is frequent in a mate position with many moves. We know there is a mate but the question in these cases is how it can be delivered.
We present the chess problem on beginners' level although these exercises appear in the game of advanced players and chess masters too, in a more complex form. We chose the mathematical tasks from arithmetic, number theory, geometry and the topic of equations. Students encounter these in classes, admission exams and student circles. Revealing the common features of mathematical and chess thinking shows how we can help the development of students' mathematical skills with the education of chess. -
Teaching integral transforms in secondary schools
241-260Views:197Today, Hungarian students in the secondary schools do not know the idea of complex numbers, and they can not integrate except those ones who learn mathematics in advance level. Without this knowledge we can teach Fourier transform for students. Why should we teach Fourier transform (FT) or Wavelet transform (WT) for them? To teach image file formats like JPEG, (JPEG2000) we need to talk about integral transforms. For students who are good in computer programming, writing the program of 1D FT or 2D FT is a nice task. In this article we demonstrate how we can teach Fourier and Wavelet transform for students in secondary school. -
Professional Competence in science education
129-137Views:142The article begins with a brief introduction aimed at sensitizing the reader to the perception of a trend in Mathematics and Computer Science Education publications towards empirical studies. Contrary to the stated trend, the characterization of Professional Competence is intended to serve as the guiding concept for the paper. The role of Professional Competence is discussed in various areas incorporating context-relevant publications in consecutive chapters. The discussion starts with the area of material development, covering Educational Standards and ends with Didactic Principles.
Subject Classification: 97xxx, 94xxx
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Typical mistakes in Mental Cutting Test and their consequences in gender differences
385-392Views:267Spatial ability of first year university students is measured and evaluated in this paper. We used standard Mental Cutting Test (MCT), where a body is given by perspective view and correct cross section has to be chosen. While gender differences in MCT are reported by several papers including our earlier results, much less known are the reasons of these differences. Here we show that typical mistakes (answers to problems which are close to be correct) can be one of the possible reasons, since female students made typical mistakes in some cases more frequently than males. -
Un point d'heuristique important et mal connu: la particularisation
235-245Views:191Cet article est consacré á la présentation d'un point d'heuristique d'une grande importance et sur lequel on insiste trés peu dans notre enseignement. C'est donc une cause fréquente d'échec pour de nombreux éléves. Il s'agit du procédé consistant á particulariser lorsqu'on dispose d'une hypothése dont l'énoncé commence par "quel que soit...". Plusieurs exemples dans divers domaines des mathématiques sont proposés.
This article is devoted to the presentation of a point of heuristics of a great importance, and on which we do not lay much emphasis in our teaching. Then, it is a frequent cause of failure for many pupils. It concerns the followings process: to particularize when we dispose of an hypothesis that begins "For any...". Several examples in various domains of mathematics are proposed. -
Maximum and minimum problems in secondary school education
81-98Views:267The 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. -
Proof without words: Knopp series for (pi)
451-452Views:219There are many expressions for number π as infinite series or infinite product (see for example [1, 2, 3]). In [1] the following series for number π is attributed to K. Knopp:... -
Experiences using CAS and multimedia int teaching vectorcalculus
363-382Views:104The 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. -
The investigation of students' skills in the process of function concept creation
249-266Views:245Function is a basic concept of mathematics, in particular, mathematical analysis. After an analysis of the function concept development process, I propose a model of rule following and rule recognition skills development that combines features of the van Hiele levels and the levels of language about function [11]. Using this model I investigate students' rule following and rule recognition skills from the viewpoint of the preparation for the function concept of sixth grade students (12-13 years old) in the Ukrainian and Hungarian education system. -
Forming the concept of parameter with examples of problem solving
201-215Views:198Pupils are encountering difficulties with learning algebra. In order for them to understand algebraic concepts, particularly the concept of parameter it was decided by the teacher of mathematics and Information Technology to integrate the teaching of these two subjects. The aim of this study is to investigate whether, and to what degree, software can be useful in process of forming the concept of parameter. This longitudinal study was conducted in a junior high school (13-16 year old children) using different computer programs. -
Les mathématiques dans le grand public et dans l'enseignement: quelques éléments d'une analyse didactique
195-216Views:176The paper looks for reaction of the public at large that is people out of educational system, concerning the mathematical exercises. We can see some results about:
• impact of terms on the motivation
• the effects of the traditional didactic on the method to resolve a problem.
Résumé. Cet article cherche la réactions du grand public c.a.d. de personnes hors systéme scolaire, de nombreuses années aprés avoir terminé leurs études vis á vis des exercises mathématiques.
Nous présentons quelques résultats concernant les points suivants:
– l'impact de l'« habillage » d'un énoncé sur la motivation
– les effets de l'absence d'un contrat didactique traditionnel sur la maniére de résoudre un probléme. -
Online tests in Comprehensive Exams – during and after the pandemic
77-93Views:395The Covid-19 pandemic accelerated the development of electronic (e-learning) assessment methods and forced their use worldwide. Many instructors and students had to familiarize themselves with the form of distance education. During and since Covid-19 in Hungary, at the Faculty of Engineering of the University of Debrecen, the written part of the Comprehensive Exam in Mathematics is organized in a computer lab of the university using an online test. Our goal is that the results of the tests may be as reliable as possible in terms of measuring the students’ knowledge, and thus the grades given based on the test results would be realistic. In this paper, we show the analysis of a sample written exam and compare the real exam results of students who were prepared for the comprehensive exam during Covid-19 and who have participated in face-to-face education since then. The tools provided by the Moodle system necessary for comparison are also presented.
Subject Classification: 97D40, 97D70, 97U50
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Strategies used in solving proportion problems among seventh-grade students
101-127Views:162In the 2023/2024 school year, 146 seventh-grade Hungarian students (aged 12-13) participated in our classroom experiment on solving proportion problems. At the beginning and the end of the teaching phase, both the experimental and the control groups solved a test. Regarding the answers of the students, in the pre- and post-test mostly consisting of word problems, we examined the success of solving the problems, as well as the solution strategies. For this, we used the strategies of proportional thinking that already exist in the literature of mathematical didactics. We intended to answer the following questions: To what extent and in which ways do the different types of problems and texts influence the solution strategies chosen by the students? How successfully do seventh-grade students solve proportion problems?
Subject Classification: 97D50, 97F80
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Problems of computer-aided assessment of mathematical knowledge
41-52Views:210Although conventional written and oral exams are dominant in assessment nowadays, computer-aided assessment is developing dynamically. There are several assessment systems, but most of them evaluate only multiple choice questions and even the most sophisticated ones cannot follow the process of thinking of students in detail. Why is it? In this article I will analyse the difficulties of the implementation of assessment system focused primarily on mathematics questions and present some of my experience related to the eMax system, developed at Óbuda University. -
Proof without words: four circles
307-309Views:195Theorem. Let O, P and Q be three points on a line, with P lying between O and Q. Semicircles are drawn on the same side of the line with diameters OP, PQ and OQ. An arbelos is the figure bounded by these three semicircles. Draw the perpendicular… -
A first course in computer science: languages and goals
137-152Views:119The College Board Advanced Placement exam in computer science will use the language Java starting in fall 2003. The language chosen for this exam is based on the language commonly taught in introductory computer science courses at the university level. This article reviews the purpose of an introductory course and the various suggestions for the curriculum of introductory courses published by the Association for Computing Machinery. It then proposes that such a course stress foundational concepts over specific language syntax, and then provides a list of such foundational concepts and related topics. Based on this fundamental curriculum, the article recommends C++ as the most appropriate language. An appendix provides a sample syllabus. -
On the past of a famous theorem: the predecessors of a theorem of Pythagoras
255-267Views:219The well-known Theorem of Pythagoras asserts a relation among the sides of any right-angled triangle. It can be found any secondary school textbook. An interesting question whether this result due to the Pythagoreans from the VIth century BC, or it was known in earlier civilizations. The first answer is a vague yes. According to the legends the Egyptian rope-stretchers used a triangle with sides 3,4,5 units to create right angle. But are there real evidences that this result was known earlier? We will argue that in almost all river-valley civilizations it was known and used. -
The requirements in statistics education – comparison of PISA mathematical tasks and tasks from the mathematical textbooks in the field of statistics
263-275Views:216This work presents the results of the analysis of both PISA items and Croatian mathematical textbooks in the field of statistics.
The analysis shows that PISA's released statistics problems have in many ways different mathematical requirements from the requirements of textbook problems in the statistics chapters, with respect to the mathematical activities, complexity and in the forms of questions. The textbook analysis shows that mathematical examples and problems often require operation and interpretation skills on a reproductive or connections level. Statistics textbook problems are given in the closed-answer form. The results also show that while PISA puts strong emphasis on the statistics field, in the current Croatian curriculum this field is barely present. These discrepancies in requirements and portion of statistics activities surely affect the results of Croatian pupils on PISA assessment in the field of mathematical literacy. -
"Frontier algorithms"
139-152Views:194In this paper we present a new method to compare algorithm design strategies. As in case of frontier towns the cultures blend, the so called "frontier algorithms" are a mixture of different programming techniques like greedy, backtracking, divide and conquer, dynamic programming. In case of some of them the frontier character is hidden, so it has to be discovered. There are algorithms that combine different techniques purposively. Furthermore, determining the programming technique the algorithm is using can be a matter of point of view. The frontier algorithms represent special opportunities to highlight particular characteristics of the algorithm design strategies. According to our experience the frontier algorithms fit best to the revision classes. -
A mathematical and didactical analysis of the concept of orientation
111-130Views:387The 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. -
Interactive web portals in mathematics
347-361Views:306Many of the recent problems in higher education (less contact seminars, the heterogeneity and the increasing number of our students) call for new instructional methods. At University of Szeged we have developed a mathematical web portal which can offer a solution for such problems among the changing circumstances. This freely available, easy-to-use web-surface supports interactive mathematical problem-solving and student self assessment. Our computer program cooperates with a lot of free software (computer algebra systems, formula parsers, converters, word processors). WebMathematics Interactive has been available for the public since June 2002 on its web page http://wmi.math.u-szeged.hu.
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