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Integrating elements of data science into high-school teaching: Naïve Bayes-classification algorithm and programming in Python
307-316Views:99Probability 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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Die aus der Studienzeit stammenden Aufzeichnungen des Johann Bolyai über die Würfelverdoppelung
307-316Views:27Hereinafter we are going to show that Bolyai Janos was preoccupied by the problem of the Duplication of the Cube, which was unknown until now by the rich Bolyai-literature.
This problem was solved using the Parabola, the Hyperbola and the Cissoide already in the ancient times. The Cissoide was created by Diocles especially for the constuction of the Duplication of the Cube without Compass and Straightedge. The hereinafter "deciphered" document of Bolyai was written during his university studies. In his study he presents the solutions discovered by then and tries to give a new one. We transcribed his notations to the present-day use and complemented it where it was necessary.
The mathematics historically background and the explication is very detailed described by Van derWaerden in Erwachende Wissenschaft [7], which is to find on English, German and Hungarian, too. That's because we dispense with this [8]. -
The far side of recursion
57-71Views:17Recursion is somewhat of an enigma, and examples used to illustrate the idea of recursion often emphasize three algorithms: Towers of Hanoi, Factorial, and Fibonacci, often sacrificing the exploration of recursive behavior for the notion that a "function calls itself". Very little effort is spent on more interesting recursive algorithms. This paper looks at how three lesser known algorithms of recursion can be used in teaching behavioral aspects of recursion: The Josephus Problem, the Hailstone Sequence and Ackermann's Function. -
Virtual manipulatives in inquiry-based approach of 3D problems by French 5th graders
229-240Views:71The 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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A role of geometry in the frame of competencies attainment
41-55Views:29We discuss aspects of the Education Reform from teaching to educational system. In this context we recognize some problems in recognition of some competencies that students need to achieve and we present how we have developed the measurement method of spatial abilities and problem solving competence. Especially, we investigate how students use spatial visualization abilities in solving various problems in other mathematical course. We have tested how students use their spatial abilities previously developed in geometry courses based on conceptual approach to solve a test based on procedural concept in Mathematical Analysis course. -
A proposed application of Monte Carlo method in teaching probability
37-42Views:38Pupils' 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. -
Würfel und Augensummen – ein unmögliches Paar
71-88Views:26It is well known that the values 2, 3, ..., 12 of the sum of eyes that appear when throwing two regular dice are not equally distributed. It can also be shown that no matter how the dice are falsified (or if only one of them is being manipulated) they can never reach the same probability concerning the sum of eyes ([8], 91 et seq.). This discovery can be generalized for n ≥ 2 dice. Various results of algebra and (real) calculus are used, so that a connection between two different mathematical fields can be realized. Such a connection is typical and often provides a large contribution for mathematics (because it frequently leads to a successful attempt of solving a special problem) and therefore examples of this sort should also be included in the mathematical education at schools as well as in the student teachers' university curriculum for the study of mathematics. -
Forming the concept of parameter with examples of problem solving
201-215Views:31Pupils 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. -
Interdisciplinary Secondary-School Workshop: Physics and Statistics
179-194Views:55The 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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Why some children fail? Analyzing a test and the possible signs of learning disorders in an answer sheet: dedicated to the memory of Julianna Szendrei
251-268Views:17Teachers and educators in mathematics try to uncover the background of the mistakes their students make for their own and their students' benefit. Doing this they can improve their teaching qualities, and help the cognitive development of their pupils. However, this improvement does not always support their students with learning disorders, since their problem is not caused by wrong attitude or lack of diligence. Therefore, it is the interest of a conscientious teacher to recognize whether the weaker performance of a student is caused by learning disorders, so the helping teacher can give useful advices. Although the teacher is not entirely responsible for the diagnosis, but (s)he should be be familiar with the possible symptoms in order to make suggestions whether or not to take the necessary test of the learning disorders.
In this article, through examining a test and the answer sheet of a single student, I show some signs that might be caused by learning disorders. -
Les mathématiques dans le grand public et dans l'enseignement: quelques éléments d'une analyse didactique
195-216Views:32The 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. -
Our duties in talent management in the light of the results of the International Hungarian Mathematics Competition of 2017
55-71Views:28The 4th International Hungarian Mathematics Competition held in Transcarpathia, Beregszász between April 28 and May 1, 2017, was organized by the Hungarian Carpathian Hungarian Teachers' Association (KMPSZ) and the Ferenc Rákóczi II. Transcarpathian Hungarian Institute (II. RFKMF).
The venue for the competition was the building of the Ferenc Rákóczi II. Transcarpathian Hungarian Institute. 175 students participated in the competition from Hungary, Romania, Serbia, Slovakia and Transcarpathia.
In this article, we are going to deal with the problems given in the two rounds to students in grades 5 and 6, and, in the light of expectations and performance, we make some suggestions for a more effective preparation of talented students on after-school lessons. -
Über die sogenannte Regel von de l’Hospital im Mathematikunterricht
193-208Views:7The aim of this paper is to provide an insight into the problems of the socalled indeterminate expressions, in order to make the students understand them better. The paper deals with the conditions and the proof of the theorem about the limit of a quotient of certain functions of one variable, usually named after l'Hospital. The question is of some interest, since the formulation of the result in several textbooks often appears redundant and the proof is more complex than necessary. First, the historical background is briefly sketched. Second, the theorem is formulated and justified, where three different, simple proof techniques are presented. Finally, possible applications are suggested for teaching, which are usually not treated in this problem area. -
Task variations for backtrack
107-120Views:77This article has been written for informatics teachers who want to issue back-track based tasks on their lessons or as homework or on competitions. We present a few methods to generate a more complicated problem from a simpler task, which will be more complex, and its solution needs a good idea or trick. Starting from an example, we lead the reader through increasingly di cult task variations.
Subject Classification: 97P50
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Shall we use one more representation? Suggestions about establishing the notion of recursion in teaching informatics in primary schools
209-229Views:27Among the most prominent developmental tasks of primary school education one finds increasing pupils' cognitive capacity with especial regard to observing, interpreting, coding and proving skills, which form an integral part of information and communication culture.
Info-technology (problem solving with the tools and methods of informatics), a subject matter within informatics, provides outstanding opportunities to reach the aims outlined above.
This study presents methodological ideas related to the subfield Algorithmization and data modelling of Info-technology. More specifically, it presents teaching methods to be applied while establishing the notion of recursion in grades 3–8 of primary education, and at the same time it also focuses on various realization possibilities of the prominent developmental tasks mentioned above. -
Kompetenzstreben und Kompetenzerwerb: Funktionale didaktische Fördermöglichkeiten durch Differenzierung und Individualisierung
1-52Views:32As a first glimpse of specific research endeavours the most important components of competence motivation are discussed in relation to didactical questions of gaining competence by inner differentiation and individualization: self-efficacy, optimal challenge, intrinsic motivation, exploration needs, internal attribution, self-determination motivation, defense of self-worth, self-concept, and achievement motivation. In this sense "competence" means ever changing standards of self-regulation of an individual interacting with the various cognitive and emotional demands of his/her environment.
In fulfilling these requirements a prototypical example of inner differentiation in mathematics instruction is given. This didactical elaboration is available as a selfinstructing unit in Hungarian and German language within the "Electronic periodical of the Department of Methodology of Mathematics" which can be reached under http://mathdid.inhun.com. -
Promoting a meaningful learning of double integrals through routes of digital tasks
107-134Views:179Within a wider project aimed at innovating the teaching of mathematics for freshmen, in this study we describe the design and the implementation of two routes of digital tasks aimed at fostering students' approach to double integrals. The tasks are built on a formative assessment frame and classical works on problem solving. They provide facilitative and response-specific feedback and the possibility to request different hints. In this way, students may be guided to the development of well-connected knowledge, operative and decision-making skills. We investigated the effects of the interaction with the digital tasks on the learning of engineering freshmen, by comparing the behaviours of students who worked with the digital tasks (experimental group, N=19) and students who did not (control group, N=19). We detected that students in the experimental group showed more exibility of thinking and obtained better results in the final exam than students in the control group. The results confirmed the effectiveness of the experimental educational path and offered us interesting indications for further studies.
Subject Classification: 97D40, 97U70, 44A45
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Conventions of mathematical problems and their solutions in Hungarian secondary school leaving exams
137-146Views:8Collecting and analyzing the conventions indispensable for interpreting mathematical problems and their solutions correctly assist successful education and objective evaluation. Many professional and didactic questions arose while collecting and analyzing these conventions, which needed clarification, therefore the materials involved concisely in the conventions enrich both the theory and practice of mathematics teaching. In our research we concentrated mainly on the problems and solutions of the Hungarian school leaving examinations at secondary level in mathematics. -
Writing a textbook – as we do it
185-201Views:17Recent surveys studying mathematics teaching show that there is a great variety in the level of mathematics teaching in Hungary. To increase efficiency (and decrease differences between schools) it is essential to create textbooks with new attitudes. The experiment we started after the PISA survey of 2000, produced a textbook that is new, in some sense even unusual in its attitude and methods. This paper presents the experiences we gained in the course of this work. -
Our digital education habits in the light of their environmental impact: the role of green computing in education
69-86Views:121With the increasing use of IT tools, the environmental impacts they generate have also increased. Education is increasingly relying on digital tools to become a major emitter of CO2 itself. Therefore, the task of education is to teach future generations how to use IT tools efficiently while being environmentally aware. In addition to some forms of green computing, we show the level and ratio of those teachers who have corresponding IT knowledge in the Hungarian education. In this study, we present the justification of the problem through a case study, which estimates the Internet traffic of a website streaming popular educational resources. In addition, we will examine the extent to which national and international educational organization and guidance documents address the development of digital environmentally aware thinking. Based on the content of this study, we suggest some considerations for content developers to decide if they really need to create the digital content.
Subject Classification: 97P99, 94-06, 94-02
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Heads or Tails gambling — what can be learned about probability?
15-41Views:23During the teaching of probability theory, a problem may appear whose solution requires the use of methods that are unfamiliar to secondary school students. In this paper, examples of methods that can resolve this difficulties are demonstrated, which could in future allow school students to tackle and solve a wide variety of problems involving probability. -
The application of modelling tasks in the classroom – why and how? with reflections on an EU teacher training course
231-244Views:34The aim of the article is to present the concept of mathematical modelling in the classroom. LEMA (Learning and Education in and through Modelling and Applications) was an EU Comenius funded project in which mathematics educators from six countries worked to produce materials to support teachers' professional development. A group of voluntary Hungarian mathematics teachers were taught modelling for a year and we were and still are given feedback continously. The article leads us from the general concept of mathematical modelling to its practice in the classroom. It presents difficulties that teachers have to face when doing modelling lessons and their students' reactions are also mentioned. We present sample tasks from the material of the teacher training course as well as tasks that were created by the participants. -
Mapping students’ motivation in a problem oriented mathematics classroom
111-121Views:64This 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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Trigonometric identities via combinatorics
73-91Views:71In this paper we consider the combinatorial approach of the multi-angle formulas sin nΘ and cos nΘ. We describe a simple "drawing rule" for deriving the formulas immediately. We recall some theoretical background, historical remarks, and show some topics that is connected to this problem, as Chebyshev polynomials, matching polynomials, Lucas polynomial sequences.
Subject Classification: 05A19
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Combinatorics – competition – Excel
427-435Views:31In 2001 the Informatics Points Competition of the Mathematics Journal for Secondary School Students (KÖMAL) was restarted [1]. The editors set themselves an aim to make the formerly mere programming competition a bit more varied. Therefore, every month there has been published a spreadsheet problem, a part of which was related to combinatorics. This article is intended to discuss the above mentioned problems and the solutions given to them at competitions. We will prove that traditional mathematical and programming tasks can be solved with a system developed for application purposes when applying a different way of thinking.