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CALIBRATE and CAS/DGS resources
267-279Views:179The CALIBRATE project was initiated by the EU with the goal of expanding the use of ICT in education by increasing the amount of available learning resources via resource exchange. Although CAS/DGS can be used to easily create high quality learning resources which are also easily adaptable across national boundaries, such resources are difficult to find at CALIBRATE portals. We believe that this is due to CAS/DGS still being rather exotic to most of the people as well as with the common problem of finding existing appropriate resources. A possible solution is for CALIBRATE portals to properly equip existing and forthcoming CAS/DGS resources with suitable metadata and to provide some integration with CAS/DGS tools, enabling both beginners and power users to create and exchange CAS/DGS resources. -
Metacognition – necessities and possibilities in teaching and learning mathematics
69-87Views:299This article focuses on the design of mathematics lessons as well as on the research in mathematics didactics from the perspective that metacognition is necessary and possible.
Humans are able to self-reflect on their thoughts and actions. They are able to make themselves the subject of their thoughts and reflections. In particular, it is possible to become aware of one’s own cognition, which means the way in which one thinks about something, and thus regulate and control it. This is what the term metacognition, thinking about one’s own thinking, stands for.
Human thinking tends to biases and faults. Both are often caused by fast thinking. Certain biases occur in mathematical thinking. Overall, this makes it necessary to think slow and to reflect on one’s own thinking in a targeted manner.
The cognitive processes of thinking, learning and understanding in mathematics become more effective and successful when they are supplemented and extended by metacognitive processes. However, it depends on a specific design of the mathematics lessons and the corresponding tasks in mathematics.Subject Classification: 97C30, 97C70, 97D40, 97D50, 97D70
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Many paths lead to statistical inference: Should teaching it focus on elementary approaches or reflect this multiplicity?
259-293Views:303For 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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Experiences using CAS and multimedia int teaching vectorcalculus
363-382Views:108The 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. -
Teaching centroids in theory and in practice
67-88Views:280The main aim of this paper is to present an inquiry-based professional development activity about the teaching of centroids and to highlight some common misconceptions related to centroids. The second aim is to emphasize a major hindering factor in planning inquiry based teaching/learning activities connected with abstract mathematical notions. Our basic problem was to determine the centroid of simple systems such as: systems of collinear points, arbitrary system of points, polygons, polygonal shapes. The only inconvenience was that we needed practical activities where students could validate their findings and calculations with simple tools. At this point we faced the following situation: we have an abstract definition for the centroid of a finite system of points, while in practice we don't even have such systems. The same is valid for geometric objects like triangles, polygons. In practice we have triangular objects, polygonal shapes (domains) and not triangles, polygons. Thus in practice for validating the centroid of a system formed by 4,5,... points we also need the centroid of a polygonal shape, formed by an infinite number of points. We could use, of course, basic definitions, but our intention was to organize inquiry based learning activities, where students can understand fundamental concepts and properties before defining them. -
Promoting a meaningful learning of double integrals through routes of digital tasks
107-134Views:506Within 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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Teaching Java programming using case studies
245-256Views:259The paper deals with the technical background and the pedagogical issues of a specific implementation for the collection, assessment and archiving of the students' assignments written in Java. The implemented system automatically applies object-oriented metrics on the collected works in order to measure the characteristic features of the assignments. Tutors use these results for the detection of plagiarisms and for the selection of outstanding works. The paper interprets the measured values within a real Java course held in the 3rd term of the Informatics bachelor study programme at the technical university. Students have several case studies devoted to the simulation of the ATM (Automatic Teller Machine) at disposal. We conclude that the access to the analyzed pool of case studies, blended with the Sun Learning Connection license from the Sun Microsystems, Inc., is an effective way of teaching programming in Java. -
Solving word problems - a crucial step in lower secondary school education
47-68Views:413Algebra is considered one of the most important parts of Mathematics teaching and learning, because it lays the foundations of abstract thinking as well as reasoning abilities among the lower secondary school pupils who have just transited from the world of numbers and computations to the area of equalities, signs, symbols and letters. The present article focuses on the fact that how the transition from arithmetic to algebra can be made more smooth. We have concentrated our experiments towards the approach of algebraic reasoning and its utilities in filling the gap between arithmetic and beginning algebra in lower secondary school education.We also underline the importance of another approach in overcoming the challenges in the transition from arithmetic to algebra, to enhance and make algebraic learning more effective, with special considerations to word problem-solving processes. In our opinion, we have to go through three phases in the introducing of algebra in Grade 7 Mathematics education: Regula Falsi method (based only on numerical calculations); functional approach to algebra (which combines the numerical computation with letter-symbolic manipulation); and writing equations to word problems. The conclusions of the present article would be helpful to Mathematics teachers for applying themselves to develop the pupils’ interest in word problem-solving processes during algebra teaching classroom activities.
Subject Classification: 97B10, 97C30, 97C50, 97D10, 97D40
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Teaching puzzle-based learning: development of basic concepts
183-204Views:571While computer science and engineering students are trained to recognise familiar problems with known solutions, they may not be sufficiently prepared to address novel real-world problems. A successful computer science graduate does far more than just program and we must train our students to reach the required levels of analytical and computational thinking, rather than hoping that it will just 'develop'. As a step in this direction, we have created and experimented with a new first-year level course, Puzzle-based Learning (PBL), that is aimed at getting students to think about how to frame and solve unstructured problems. The pedagogical goal is increase students' mathematical awareness and general problem solving skills by employing puzzles, which are educational, engaging, and thought provoking. We share our experiences in teaching such a course – apart from a brief discussion on our pedagogical objectives, we concentrate on discussing the presented material which covers (in two lectures) just one selected topic (pattern recognition). In this paper we present the ideas behind foundations for PBL and the material of the first of two lectures on pattern recognition, in which we address core concepts and provide students with sufficient exemplars to illustrate the main points. -
Error analysis in teaching combinatorics: the development of prospective teachers’ confidence and problem-solving skills
103-125Views:91This study investigates the pedagogical potential of error analysis in the teaching of combinatorics within mathematics teacher education. Building on previous research that highlights the role of incorrectly worked sample solutions in cognitive, metacognitive, and affective learning processes, we conducted a mixed-methods study with prospective mathematics teachers at Eötvös Loránd University. Quantitative results from Likert-scale questionnaires (n = 26) indicate that regular analysis of incorrectly worked solutions substantially enhanced participants’ self-confidence, strengthened their problem-solving skills, and positively shaped their attitudes toward future teaching practice. Complementary qualitative data, analyzed through grounded theory, revealed five interrelated categories – self-reflection and confidence, discernment, deeper understanding, methodological surplus, and combinatorial surplus – that together explain the mechanisms through which error analysis supports professional growth. The findings suggest that systematic analysis of conceptual errors not only improves problem-solving competence but also fosters self-confidence, self-reflection, and teaching-related attitudes. By comparing our emergent model of error-analysis thinking with Schoenfeld’s problem-solving framework, we argue that “discernment” constitutes a distinctive and central dimension of error-based learning. The study contributes both theoretically, by refining models of mathematical problem solving, and practically, by offering concrete recommendations for integrating error analysis into mathematics teacher education curricula.
Subject Classification: 97C30, 97K20, 97D40, 97C70, 97C99
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Concept systematization with concept maps in data modelling
149-166Views:219An important goal of concept learning is that students can allocate concepts in the hierarchical system of concepts. In the data modelling course, first, we supported concept systematization with worksheets in which the students had to fill in the blank hierarchical figures of classification of the concepts or blank Venn diagrams describing the relationships between concepts. The hierarchical systems, however, are somewhat restricted to the description of connections. The filling in Venn diagrams did not deliver the expected result, so our attention turned to concept maps. In this paper we introduce the concept maps we drew. Then we evaluate the results of concept mapping survey conducted among students. The survey was done in three courses. We compare the results of our survey with the result of an earlier concept systematising survey. -
Computer cooking vs. problem solving
35-58Views:377Computer cooking is a task-related phenomenon where students (end-users) must blindly follow a long list of orders without any connection to the content of the problem, if there is any. Despite its low efficacy, this method is widely used and accepted in informatics both in the learning-teaching process and testing. The National Base Curriculum 2020 in Hungary is in complete accordance with the ‘Informatics Reference Framework for Schools’, but the course books hardly use the latest results of computer education research. The present paper provides examples of how the results of computer education research can be integrated into teaching-learning materials and classroom practices and discusses the effectiveness and consequences of the different solutions, where tool-centred approaches are compared to problem-focused solutions.
Subject Classification: 94-01
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Impact of teacher communication skills on students’ classroom engagement in mathematics learning
1-27Views:956The study investigated teachers’ communication skills in relation to students’ classroom engagement in mathematics learning. The study area is Makurdi Local Government Area in Benue State, Nigeria. This study adopted a cross-sectional research design. A sample of 34 teachers and 204 students were drawn from twenty schools. Two researcher-structured instruments were used for data collection: Mathematics Teacher’s Communication Skills Questionnaire (MTCSQ) and Students’ Engagement in Mathematics Questionnaire (SEMQ). Descriptive statistics, analysis of variance, and independent t-tests were used to address the research questions and test the hypotheses. It was found that there is significant difference among the mean ratings on behavioural, and emotional engagements of students in mathematics classes taught by teachers with poor, fair, and good communication skills. There is no significant difference among the mean ratings on combined and cognitive engagements of students in mathematics classes taught by teachers with poor, fair, and good communication skills. Equally found was that the differences between male and female students’ mean engagement in mathematics for poor, fair, and good communication skill classes were not statistically significant. It was then recommended that teacher communication skills should be fashioned in ways to accommodate and strengthen each component of students’ engagement.
Subject Classification: 97C70
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Research studies in didactics of mathematics supported by the Operant Motive Test
153-173Views:228The present paper reports a case-study which took place within an EUsupported international program organized for research and development of multi-grade schools (NEMED, [16] [26]). One of the main goals of the research was to develop the connection between disadvantageous social situations and the efficiency (success or failure) in learning mathematics especially from the point of view of average and above-average (talented) students: Why does the talent of children with socially disadvantageous background remain undiscovered? How can we make school mathematics more aware of hidden talents?
The author was looking for a didactical solution that compensated for social disadvantages without restricting the development of "average" students by using sociological, educational, psychological and mathematical (experimental and theoretical) studies in interaction with a series of experimental (hypothesis testing and exploratory) investigations.
We constructed tools and methods for exploration and experimental teaching, adapted to Hungarian conditions (Curriculum Development, teacher training, materials, interviews, Kuhl's motivation test, Malara's "researchers and practicing teachers in cooperation" method, etc., see [18], [20]).
The teaching materials and methodological guidelines are based on Bruner's representation theory (see [5]). The empirical research took place in 16 multi-grade schools located in different parts of the country. The author co-operated with nearly 250 students and 25 teachers for 3 years. In this paper we try to demonstrate how an Operant Motive Test can be involved in this research (see [18]). -
An improvement of the classification algorithm results
131-142Views:199One of the most important aspects of the precision of a classification is the suitable selection of a classification algorithm and a training set for a given task. Basic principles of machine learning can be used for this selection [3]. In this paper, we have focused on improving the precision of classification algorithms results. Two kinds of approaches are known: Boosting and Bagging. This paper describes the use of the first method – boosting [6] which aims at algorithms generating decision trees. A modification of the AdaBoost algorithm was implemented. Another similar method called Bagging [1] is mentioned. Results of performance tests focused on the use of the boosting method on binary decision trees are presented. The minimum number of decision trees, which enables improvement of the classification performed by a base machine learning algorithm, was found. The tests were carried out using the Reuters 21578 collection of documents and documents from an internet portal of TV Markíza. -
Models of impulsive phenomena: experiences with writing an interactive textbook
333-345Views:138"Take the textbook to computer" – is said quite often. Would it be so easy? If we start such a work, we meet a lot of trouble very soon. A book stored on a CD, read on the screen of computer and containing some hyperlinks does not become automatically electronic textbook. There are difficulties also in writing merely an electronic attachment to a classical book. In this paper, we deal with some important features (actually important from our point of view) of interactive mathematics textbooks, arising mathematical, didactical and technical problems. The "principles" are illustrated with examples taken from the book-CD "Models of Impulsive Phenomena". -
Removing the burden of syntax: developing computational thinking and algorithmic skills of STEM students
29-49Views:79In higher education, solving programming exercises using a high-level programming language is a standard approach for developing computational thinking and algorithmic skills. However, this method has its limitations: learning the syntax of a high-level programming language puts an extra cognitive load on students, preventing them from focusing on problem-solving. Furthermore, computational thinking is not limited to programming: STEM students can benefit more from solving problems within their own discipline, in different environments. This practical article proposes a collection of unplugged, semi-unplugged and plugged-in alternatives that can be used to develop the computational thinking and algorithmic skills of students.
Subject Classification: 97P99
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Teaching polygons in the secondary school: a four country comparative study
29-65Views:276This study presents the analysis of four sequences of videotaped lessons on polygons in lower secondary schools (grades 7 and 8) taught by four different teachers in four different countries (Belgium, Flanders, England, Hungary and Spain). Our study is a part of the METE project (Mathematics Educational Traditions in Europe). The aims and methodology of the project are described briefly in the introduction. In the next section of this paper we describe various perspectives on teaching and learning polygons which were derived from the literature, concerning the objectives, conceptual aspects and didactic tools of the topic. The next two sections introduce the main outcomes of our study, a quantitative analysis of the collected data and a qualitative description linked to the perspectives on teaching polygons. We conclude by discussing some principal ideas related to the theoretical and educational significance of this research work. -
Preliminary e ects of mathematics curriculum development for primary school student teachers in Sárospatak Comenius Campus
95-107Views:226Hungarian students' mathematics performance has been getting weaker in the past few years. A possible solution to stop this tendency is to develop curriculum. Therefore, Hungarian researchers have been refining a particular framework of curriculum development in primary school teacher training programmes. The national curriculum is designed on the assumption that learning can be broken into a sequence of levels and students can evenly succeed in gaining knowledge at successive levels. In this paper, we want to discuss how to reduce students' difficulties with different background to grow competence at successive levels. -
Statistical inference in school
265-273Views:110The paper explains a classroom example for convincing students about the utility and applicability of statistical methods in learning getting people's opinions. The emphasis is on convincing instead of proving. The necessary statistical data may be obtained from the Internet as a digital text. -
Introductory Computer Programming Courses in Mathematics Curriculum
19-30Views:315We present the results of surveys and curricular research on introductory computer programming courses that are required or recommended for mathematics degrees at U.S. colleges and universities. Our target schools were those with populations between 5,000 and 20,000 undergraduate students. A key result is a synopsis of programming languages in use in these introductory courses with Java, Python and C + + holding the top three spots. We found that 85% of the 340 schools in our pool require or recommend an introductory programming course as a component of a mathematics degree. Furthermore, most of these introductory programming courses are taught by faculty outside of the mathematics department. These results indicate that mathematics faculty value computer programming and should be actively involved in setting learning outcomes, incorporating skills and concepts learned in introductory programming courses into subsequent mathematics courses, and determining programming languages in use.
Subject Classification: 97D30, 97P20, 97P40
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Teaching probability theory by using a web based assessment system together with computer algebra
81-95Views:203In the course of Maths Basics 2, the Faculty of Economic Science students of Kaposvár University learn the classical chapters of Probability Theory, namely random variables and the well-known probability distributions. Our teaching experiences show that students' achievement is weaker in case of problems concerning continuous random variables. From school year 2012/13 we have had an opportunity to take Maple TA, the web-based test- and assessment system, into the course of education. It is sufficient for the users of Maple TA to have a browser. Maple computer algebra system, which runs on the server, assesses students' answers in an intelligent way, and compares them with the answers that are considered correct by the teacher. In our presentation we introduce some elements of Maple TA system, the didactic considerations the test sheets were made by, as well as our research results concerning the use of Maple TA. -
Blind versus wise use of CAS
407-417Views:250During 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. -
Teaching fractions at elementary level in the light of Hungarian mathematics textbooks in Romania
149-159Views:252According to the new curriculum in Romania, fractions are introduced in the second grade. The present study analyses Hungarian elementary mathematics textbooks on the topic of fractions focusing on the types of tasks in the textbooks, the significance of representations and the proportion of word problems. Additionally, the paper presents a questionnaire-based research on teachers’ opinion regarding the adequacy and sufficiency of the digital materials and exercises related to fractions in the textbooks.
Subject Classification: 97F40, 97F80, 97U20, 97U50
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Sage and scribe – asymmetrical pair work that can easily fit into any mathematics lesson, yet still have cooperative benefits
133-164Views:861This 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
