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Interdisciplinary Secondary-School Workshop: Physics and Statistics
179-194Views:170The 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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Many paths lead to statistical inference: Should teaching it focus on elementary approaches or reflect this multiplicity?
259-293Views:243For statistics education, a key question is how to design learning paths to statistical inference that are elementary enough that the learners can understand the concepts and that are rich enough to develop the full complexity of statistical inference later on. There are two ways to approach this problem: One is to restrict the complexity. Informal Inference considers a reduced situation and refers to resampling methods, which may be completely outsourced to computing power. The other is to find informal ways to explore situations of statistical inference, also supported with the graphing and simulating facilities of computers. The latter orientates towards the full complexity of statistical inference though it tries to reduce it for the early learning encoun-ters. We argue for the informal-ways approach as it connects to Bayesian methods of inference and allows for a full concept of probability in comparison to the Informal Inference, which reduces probability to a mere frequentist concept and – based on this – restricts inference to a few special cases. We also develop a didactic framework for our analysis, which includes the approach of Tamás Varga.
Subject Classification: 97K10, 97K70, 97K50, 97D20
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Approximated Poncelet configurations
163-176Views:154In this short note we present the approximate construction of closed Poncelet configurations using the simulation of a mathematical pendulum. Although the idea goes back to the work of Jacobi ([17]), only the use of modern computer technologies assures the success of the construction. We present also some remarks on using such problems in project based university courses and we present a Matlab program able to produce animated Poncelet configurations with given period. In the same spirit we construct Steiner configurations and we give a few teaching oriented remarks on the Poncelet grid theorem. -
Informatics as a particular field of education
283-294Views:223Informatics education can be discussed at various levels. There is informatics education at the university, there is professional informatics training and there is public informatics education. In the following article we are going to deal with the latter, that is we are going to discuss what areas of informatics should be introduced to students within the frame of the informatics subject in primary and secondary education.
Knowledge in connection with informatics can be grouped from different points of view. We consider the following points to be acceptable: according to scopes of knowledge. [1, 2] -
Sage and scribe – asymmetrical pair work that can easily fit into any mathematics lesson, yet still have cooperative benefits
133-164Views:740This article uses a case study experiment to learn the characteristics of a pair work, called the sage and scribe method (Kagan, 2008). We also wished to explore the positive and negative effects of the systematic application of this single cooperative element without any other structural changes during the lessons. In the case study experiment, we asked two teachers, accustomed to traditional frontal teaching methods, to substitute individual work tasks in their standard lesson plans with the sage and scribe method. Our experiments indicate that this method wastes insignificant time, requires little extra effort on the part of the teacher, yet has many of the positive effects of cooperative methods: in our experiments, students received immediate feedback, corrected each other’s mistakes, learned from each other in meaningful discussions and engaged in collaborative reasoning to address emerging problems.
Subject Classification: 97D40
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Blind versus wise use of CAS
407-417Views:194During my courses for mathematics major students I often use technology linked to the arising problems. In such cases I noted that some students were used to learn just some procedures, which made them able to solve (partially) some problems and when they got the result, they accepted it passively and did not relate it to the initial problem.
In this paper I outline a strategy and investigate some simple exercises about how to develop a critical attitude towards the results obtained by technology in an introductory course to CAS.
I believe that wise use of technology offers an effective method in teaching mathematics, without reducing the students' mental contribution. -
Delusions in informatics education
151-161Views:109In the following article our intention is to try to introduce the negative ideas that exist today in Hungary regarding informatics education within the secondary education system. [Zs] As far as we know, these delusions are characteristic of not only Hungary, but we believe that we should look for our own mistakes, that is why we refer to Hungarian examples.
We have examined the informatic knowledge taught in the first 10 years of secondary education, the possible curriculum of the general informatics subject.
To reach our aim, first we have to deviate a bit from our original topic, because without this, it would be more difficult to understand the core subject of the article. In the deviation we will explain what is called informatics, what is called informatics subject. Then we will deal with the main topic and in the summary we will explain what we believe is the aim of general informatics education. -
Implementation opportunities of the Moodle learning management system in virtual environment the Sloodle project
275-293Views:110Using e-learning was firstly appeared in companies' sphere. It should be very useful if learning management systems were applied. Nowadays e-learning is used in different fields and gives useful informations in case of basics and its knowledge. It is essential to know the arranging technics and applicated handling methods of some supporting learning management systems of e-learning. The Moodle is the best-known learning management system.
The Second Life is one of the virtual environments which is useful in learning-teaching methods that is used in most educational institute all over the world. Sloodle is an open source project which connects the Second Life with Moodle learning management system. Sloodle is a kind of "bridge" in which different kind of activities and registering and provided in both Moodle and Second Life.
In our department, University of Debrecen Health Faculty of Nyíregyháza ILIAS learning management system has operated since February, 2008. In the interest of higher level education we decided to use and made available some courses through Moodle learning management system.
Some tools of Sloodle will be presented in our article. It will be the first study for our research in which we would use the Moodle learning management system, the virtual environment of Second Life and the project of Sloodle itself. Our article will contain the starting details and its statistical confirmation of our Sloodle project. We like to demonstrate that the results of the Sloodle-aided group are significantly better than the results of the control group in the most cases. -
Teaching sorting in ICT
101-117Views:190This 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. -
Un point d'heuristique important et mal connu: la particularisation
235-245Views:157Cet 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. -
"How to be well-connected?" An example for instructional process planning with Problem Graphs
145-155Views:224Teachers’ 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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Equivalences of some forms of the change of variable formula and the fundamental theorem of calculus
269-279Views:122We discuss an interplay between some versions of the Change of Variable Theorem and the Fundamental Theorem of Calculus for the Riemann integral. We show that the two theorems are equivalent, and that for both theorems to be true it suffices to assume two particular formulas derived from them. In the realm of teaching, this material might be among our interests.
