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Application of computer algebra systems in automatic assessment of math skills
395-408Views:149Mathematics is one of those areas of education, where the student's progress is measured almost solely by testing his or her ability of problem solving. It has been two years now that the authors develop and use Web-based math courses where the assessment of student's progress is fully automatic. More than 150 types of problems in linear algebra and calculus have been implemented in the form of Java-driven tests. Those tests that involve symbolic computations are linked with Mathematica computational kernel through the Jlink mechanism. An individual test features random generation of an unlimited number of problems of a given type with difficulty level being controlled flat design time. Each test incorporates the evaluation of the student's solution. Various methods of grading can be set at design time, depending on the particular purpose that a test is used for (self-assessment or administrative exam). Each test is equipped with the correct solution presentation on demand. In those problems that involve a considerable amount of computational effort (e.g. Gauss elimination), additional special tools are offered in a test window so that the student can concentrate on the method of solution rather than on arithmetic computations. (Another obvious benefit is that the student is thus protected from the risk of frustrating computational errors). Individual tests can be combined into comprehensive exams whose parameters can be set up at design time (e.g., number of problems, difficulty level, grading system, time allowed for solution). The results of an exam can be automatically stored in a database with all authentication and security requirements satisfied. -
How the derivative becomes visible: the case of Daniel
81-97Views:129This paper reports how an advanced 11th-grade student (Daniel) perceived the derivative from a graph of a function at a task-based interview after a short introduction to the derivative. Daniel made very impressive observations using, for example, the steepness and the increase of a graph as well as the slope of a tangent as representations of the derivative. He followed the graphs sequentially and, for example, perceived where the derivative is increasing/decreasing. Gestures were an essential part of his thinking. Daniel's perceptions were reflected against those of a less successful student reported previously [Hähkiöniemi, NOMAD 11, no. 1 (2006)]. Unlike the student of the previous study, Daniel seemed to use the representations transparently and could see the graph as a representation of the derivative. -
Examining relation between talent and competence through an experiment among 11th grade students
17-34Views:141The areas of competencies that are formable, that are to be formed and developed by teaching mathematics are well-usable in recognizing talent. We can examine the competencies of a student, we can examine the competencies required to solve a certain exercise, or what competencies an exercise improves.
I studied two exercises of a test taken by students of the IT specialty segment of class 11.d of Jedlik Ányos High School, a class that I teach. These exercises were parts of the thematic unit of Combinatorics and Graph Theory. I analysed what competencies a gifted student has, and what competencies I need to improve while teaching mathematics. I summarized my experience about the solutions of the students, the ways I can take care of the gifted students, and what to do to the less gifted ones. -
Teaching student teachers: various components of a complex task
55-72Views:78In this paper we summarize various aspects of teacher training and teaching student teachers (mainly concerning teachers of upper secondary school and High school). We stress several hints and recommendations to better achieve the obviously important aim: they should learn doing, understanding and teaching mathematics!
Of course, our view is particularly influenced by European traditions, but we think most of them equally apply to teacher training and teaching student teachers elsewhere. Neither is the paper meant to give an all sided overview about the problem field of teacher education as a whole, nor does it contain provocative, completely new ideas. We just want to describe our view of some aspects, based primarily on our personal experience in the mentioned field. -
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:167Teachers 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. -
Proof step analysis for proof tutoring - a learning approach to granularity
325-343Views:148We present a proof step diagnosis module based on the mathematical assistant system Ωmega. The task of this module is to evaluate proof steps as typically uttered by students in tutoring sessions on mathematical proofs. In particular, we categorise the step size of proof steps performed by the student, in order to recognise if they are appropriate with respect to the student model. We propose an approach which builds on reconstructions of the proof in question via automated proof search using a cognitively motivated proof calculus. Our approach employs learning techniques and incorporates a student model, and our diagnosis module can be adjusted to different domains and users. We present a first evaluation based on empirical data. -
How to teach computer programming if our goal is the International Olympiad in Informatics
13-25Views:144Nowadays if a student in Hungary (age between 17-20 years old) wants to be the member of the Hungarian selected team (has four members) to participate in the International Olympiad in Informatics (IOI), first, he has to qualify himself in the first fifteen of the National Secondary School Competition (OKTV) in the programming category after the III. round. Then he should be in the first four place after the sixth round of the Selection Competition. Being successful is necessary that the student wants to start studying computer programming at least in the 9th school year and he needs a teacher who prepares him. In the last nine years three students of the author have participated in five Olympics and two of them won gold and bronze medals. This article wants to demonstrate the methods that a teacher needs to use to teach students in 9th school year for computer programming, to be the member of the Hungarian National Team after three or four years. -
Some thoughts on a student survey
41-59Views:121The paper analyzes a survey of college students and describes its major findings. The object of the survey, involving 154 students, was to discover and highlight the problems that arise in taking the course Economic Mathematics I. The paper, as the summary of the first phase of a research project, wishes to present these problems, ways that may lead out of them, and possible means of help that can be offered to those taking the course. -
Assimilation of mathematical knowledge using Maple
321-331Views:113For more than four years we have been teaching a Maple course at University of Debrecen for prospective mathematics teachers. The aim of the course is that students get some experience on mathematical visualization with Maple. At the last part of the course the student is provided with a problem of geometrical flavor. Within three or four weeks he/she must obtain a solution. In this paper we present and analyze two of student projects: rotation of the hypercube and drawing of complex functions. The concluding remark is that most of the students will profit from using Maple for such type of problems: it helps to assimilate mathematical knowledge. -
On an international training of mathematically talented students: assets of the 20 years of the “Nagy Károly Mathematical Student-meetings”
77-89Views:217The 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. -
Mechanisms for teaching introductory programming using active learning
407-421Views:88One of the requirements of teaching introductory programming to students whose branch of learning is engineering or science is bridging the gap between in-class lectures and real-world applications. Traditional passive approaches to lecturing often focus on the syntax of a language with little or no discussion of the process involved in using the language to design algorithms to solve real-world problems. One way of overcoming the limitations of traditional lecturing is by tailoring lectures towards becoming more student-oriented, a pedagogical methodology known as active learning. This paper explores mechanisms for implementing active learning in introductory programming courses in computer science. -
What does ICT help and does not help?
33-49Views:247Year by year, ICT tools and related teaching methods are evolving a lot. Since 2016, the author of the present lines has been looking for a connection between them that supports the development of mathematical competencies and could be integrated into Transcarpathian minority Hungarian language education too. As a doctoral student at the University of Debrecen, I experienced, for example, how the interactive whiteboard revolutionized illustration in Hungarian mathematics teaching, and how it facilitated students' involvement. During my research of teaching in this regard, in some cases, the digital solution had advantageous effects versus concrete-manipulative representation of
Bruner's too.
At the same time, ICT "canned" learning materials (videos, presentations, ...) allow for a shift towards repetitive learning instead of simultaneous active participation, which can be compensated for by the "retrieval-enhanced" learning method.
I have conducted and intend to conduct several research projects in a Transcarpathian Hungarian primary school. In the research so far, I examined whether, in addition to the financial and infrastructural features of the Transcarpathian Hungarian school, the increased "ICT-supported" and the "retrieval-enhanced" learning method could be integrated into institutional mathematics education. I examined the use of two types of ICT devices: one was the interactive whiteboard, and the other was providing one computer per student.
In this article, I describe my experiences, gained during one semester, in the class taught with the interactive whiteboard on the one hand, and in the class taught according to the "retrieval-enhanced" learning method on the other hand.
I compare the effectiveness of the classes to their previous achievements, to each other, and to a class in Hungary.Subject Classification: 97U70
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Evaluating admission procedures for teacher education in Finland
231-243Views:168In 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. -
Engineering and Economic Mathematics for Engineering Management Students
35-50Views:137In this article we describe the first part of a case study, which was made with 48 Engineering Management students. The participants of the case study were MSc level students at the Szent István University, Gödöllő. We looked for methods by which we can support the most important components of competence motivation and the development of mathematical and other key competences during the mathematics lessons and individual learning. Another goal of our research was to get reliable information about students learning methods and their awareness of self-efficacy, furthermore their achievement in the subject of Engineering and Economic Mathematics. Detailed assistance was provided for the students in the e-learning portal. Knowledge tests, questionnaire and personal interviews with the students were also used. As an example we introduce one of the knowledge tests connected with the first half of the course about linear programming and graph theory. We detail its didactical background and show the results of the students. -
Visualisation in geometry education as a tool for teaching with better understanding
337-346Views:318In primary and secondary geometry education, some problems exist with pupils’ space thinking and understanding of geometric notions. Visualisation plays an important role in geometry education, and the development of pupils’ visualisation skills can support their spatial imagination. The authors present their own thoughts on the potential of including visualisation in geometry education, based on the analysis of the Hungarian National Core Curriculum and Slovak National Curriculum. Tasks for visualisation are also found in international studies, for example the Programme for International Student Assessment (PISA). Augmented reality (AR) and other information and communication technology (ICT) tools bring new possibilities to develop geometric thinking and space imagination, and they also support mathematics education with better understanding.
Subject Classification: 97U10, 97G10
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Software engineering education in cooperation with industrial partners
133-148Views:131This paper presents our experiences on teaching software engineering in teams which are organized around different R+D projects. These long-running, innovative projects are carried out in cooperation with industrial partners, and are supported by student exchange. While MSc and PhD students work together with faculty staff members on the projects in an industrial-like environment, the students develop skills that would be otherwise very hard for them to obtain. The methodological contributions of the paper are illustrated by, and substantiated with, the description of a concrete software engineering project. -
Interactive web portals in mathematics
347-361Views:200Many 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. -
Mathematics in Good Will Hunting II: problems from the students perspective
3-19Views:206This is the second part of a three paper long series exploring the role of mathematicians and of the mathematical content occurring in popular media. In particular we analyze the drama film Good Will Hunting. Here we investigate the mathematical content of the movie by considering the problems appearing in it. We examine how a mathematician or a mathematics student would solve these problems. Moreover, we review how these problems could be integrated into the higher education of Hungary. -
A retrospective look at discovery learning using the Pósa Method in three Hungarian secondary mathematics classrooms
183-202Views:328While the Pósa Method was originally created for mathematical talent management through extracurricular activities, three "average" public secondary school classrooms in Hungary have taken part in a four-year experiment to implement the Pósa Method, which is based on guided discovery learning of mathematics. In this paper, we examine the students' and teachers' reflections on the Pósa Method, and how student perspectives have changed between their first and last year of high school. Overall, teachers and students had a positive experience with the Pósa Method. Furthermore, our research indicated that this implementation has met several objectives of the Pósa Method, including enjoyment of mathematics and autonomous thinking.
Subject Classification: 97D40
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Better understanding mathematics by algorithmic thinking and computer programming
295-305Views:267Tamás Varga’s mathematics education experiment covered not just mathematics, but also other related topics. In many of his works he clearly stated that computer science can support the understanding of mathematics as much as mathematics supports informatics. On the other hand, not much later than the introduction of the new curriculum in 1978, personal computers started to spread, making it possible to teach informatics in classes and in extracurricular activities. Varga’s guided discovery approach has a didactic value for other age groups as well, not only in primary school. Its long-lasting effect can be observed even in present times. Having reviewed several educational results in the spirit of Tamás Varga, we have decided to present an extracurricular course. It is an open study group for age 12-18. Students solve problems by developing Python programs and, according to our experiences, this results in a deeper understanding of mathematical concepts.
Subject Classification: 97B10, 97B20, 97D50, 97N80, 97P20, 97P30, 97P40, 97P50, 97U70
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Mathematician Judita Cofman (1936–2001)
91-115Views:174Judita Cofman was the first generation student of mathematics and physics at Faculty of Philosophy in Novi Sad, Serbia, and the first holder of doctoral degree in mathematical sciences at University of Novi Sad. Her Ph.D. thesis as well as her scientific works till the end of 70's belong to the field of finite projective and affine planes and the papers within this topic were published in prestigious international mathematical journals. She dedicated the second part of her life and scientific work to didactic and teaching of mathematics and to work with young mathematicians. -
Würfel und Augensummen – ein unmögliches Paar
71-88Views:143It 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. -
Fehleranalyse beim Lösen von offenen Aufgaben Ergebnisse einer empirischen Studie in der Grundschule
83-113Views:150Open problems play a key role in mathematics education, also in primary school. However, children in primary school work in many relations in a different way from learner in secondary school. Therefore, the (possibly) first confrontation with an open task could be problematical. Within the framework of an international paper and pencil test it was examined how far children of primary school notice the openness of a task and which mistakes they do during working on that task. In particularly are meant by openness different interpretations of the task, which all lead to a set of numbers with more than one element as a result. For evaluation, a common classification system was adapted by slightly modification of the original system. -
Prime building blocks in the mathematics classroom
217-228Views:298This 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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An e-learning environment for elementary analysis: combining computer algebra, graphics and automated reasoning
13-34Views:382CreaComp is a project at the University of Linz, which aims at producing computer-supported interactive learning units for several mathematical topics at introductory university level. The units are available as Mathematica notebooks. For student experimentation we provide computational, graphical and reasoning tools as well. This paper focuses on the elementary analysis units.
The computational and graphical tools of the CreaComp learning environment facilitate the exploration of new mathematical objects and their properties (e.g., boundedness, continuity, limits of real valued functions). Using the provided tools students should be able to collect empirical data systematically and come up with conjectures. A CreaComp component allows the formulation of precise conjectures and the investigatation of their validity. The Theorema system, which has been integrated into the CreaComp learning environment, provides full predicate logic with a user-friendly twodimensional syntax and a couple of automated reasoners that produce proofs in an easy-to-read and natural presentation. We demonstrate the learning situations and the provided tools through several examples.
