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A Nim like game and a machine that plays it: a learning situation at the interface of mathematics and computer science
317-326Views:60The purpose of this work is to take a didactic look at a learning situation located at the interface between mathematics and computer science. This situation offers a first approach to the concept of artificial intelligence through the study of a reinforcement learning device. The learning situation, inspired by the Computer Science Unplugged approach, is based on a combinatorial game, along with a device that learns how to play this game. We studied the learning potential when the human players face the machine. After an a priori analysis using the Theory of Didactic Situations (TDS), we conducted a pre-experiment in order to strengthen our hypotheses. In this article, we will focus on the analysis of the didactic variables, the values we have chosen for these variables and their effects on students’ strategies.
Subject Classification: 97D99, 97K99, 97P80
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What does ICT help and does not help?
33-49Views:77Year 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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The role of representations constructed by students in learning how to solve the transportation problem
129-148Views:51The purpose of the research presented in this paper was to study the role of concrete and table representations created by students in learning how to solve an optimization problem called the transportation problem. This topic was learned in collaborative groups using table representations suggested by teachers in 2021. In 2022, the researchers decided to enrich the students’ learning environment with concrete objects and urged the students to use them to present the problem to be solved. The students did it successfully and, to be able to record it in their notebooks, they constructed a table representation by themselves without any help from their teacher. After that, they managed to solve the problem by manipulating the objects. At the same time, each step in the solution was presented with changes in the table. The students were assessed before (pre-test) and after collaborative learning (test) in both academic years. The pre-test results were similar, but the test results were better in 2022. Therefore, it can be concluded that using concrete and table representations constructed by students in learning how to solve transportation problems makes collaborative learning more constructivist and more effective than when they use only table representations suggested by their teachers.
Subject Classification: 97M10, 97M40
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Freudenthal fantasy on the bus, an American adaptation
133-142Views:29In the 1960’s two mathematicians, Hans Freudenthal in the Netherlands and Tamás Varga in Hungary, had argued that people learn mathematics by being actively involved and investigating realistic mathematical problems. Their method lives on in today’s teaching and learning through the various components of cooperative and active learning, by taking ownership in learning, and learning through student dialogue. The goal is to create a welcoming classroom atmosphere in which play takes the front seat. One such scenario is visiting various (animal) stations at the zoo by bus (illustrated by pictures). Passengers are getting on and off the bus at each station (illustrated by arrows), which is modeled on the open number line. This adapted and modified action research was carried out with 5-yearl-old children in public schools of Staten Island, NY in 2019.
Subject Classification: 97D40, 97F20, 97F30
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Capturing how students' abilities and teaching experiences affect teachers' beliefs about mathematics teaching and learning
195-212Views:79We developed an instrument to investigate the effect of students' abilities and teaching experiences on teachers' beliefs about teaching and learning of mathematics. In this pilot study, we used the instrument to measure the beliefs of 43 Indonesian math teachers and five additional teachers. Then, for further investigation, we interviewed those five additional teachers. Results from the 43 teachers' responses to the instrument show that in contrast to teachers with less than five years of teaching, teachers with more than five years elicit significantly different beliefs about mathematics teaching and learning in different contexts related to students' abilities. Teachers' reports in the further investigation indicate that teaching experiences with high and low ability students in teaching mathematics could be a possible explanation of this contrast.
Subject Classification: C20
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Inquiry based mathematics education and the development of learning trajectories
63-89Views:475This article is based on the panel on inquiry based mathematics education and the development of learning trajectories held at the VARGA 100 Conference. After an introduction presenting the theme and organization of the panel, this article focuses on the diversity of conceptualizations of inquiry based education existing today in mathematics education and their influence on the vision and development of learning trajectories. More precisely, it considers the conceptualizations respectively associated with Realistic Mathematics Education, Genetic Constructivism, Tamás Varga’s educational approach and the Anthropological Theory of the Didactic, presented by the panellists, and also shows the efforts undertaken in European projects to reach consensusal visions.
Subject Classification: 97C30Q, 97D10, 97D20, 97D30, 97D40, 97D50
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Connections between discovery learning through the Pósa Method and the secondary school leaving examination in three Hungarian mathematics classrooms
67-85Views:167The Pósa Method is a guided discovery learning method that has been used in Hungarian education in the form of extracurricular activities for "gifted" mathematics students. A four-year experiment implemented the method in three more "average" classrooms. This article reports on the relationship between the Pósa Method and the standardized secondary school leaving mathematics exam (Matura Exam in short) in Hungary. Data consists of students' survey responses, teacher interviews, and exam results from the three Hungarian classrooms who took part in the four-year experiment. We identify aspects of the Pósa Method that can benefit and hinder exam performance. In addition, we find that learning through the Pósa Method for the four years of high school has adequately prepared students for the exam.
Subject Classification: 97D44, 97D54, 97D64
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Rational errors in learning fractions among 5th grade students
347-358Views:37Our paper focuses on empirical research in which we map out the errors in learning fractions. Errors are often logically consistent and rule-based rather than being random. When people face solving an unfamiliar problem, they usually construct rules or strategies in order to solve it (Van Lehn, 1983). These strategies tend to be systematic, often make ‘sense’ to the people who created them but often lead to incorrect solutions (Ben-Zeev, 1996). These mistakes were named rational errors by Ben-Zeev (1996). The research aims to show that when learning fractions, students produce such errors, identified in the literature, and that students who make these kinds of mistakes achieve low results in mathematics tests. The research was done among 5th-grade students.
Subject Classification: 97C10, 97C30, 97C70, 97D60, 97D70, 97F50
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Mobile devices in Hungarian university statistical education
19-48Views:52The methodological renewal of university statistics education has been continuous for the last 30 years. During this time, the involvement of technology tools in learning statistics played an important role. In the Introduction, we emphasize the importance of using technological tools in learning statistics, also referring to international research. After that, we firstly examine the methodological development of university statistical education over the past three decades. To do this, we analyze the writings of statistics teachers teaching at various universities in the country. To assess the use of innovative tools, in the second half of the study, we briefly present an online questionnaire survey of students in tertiary economics and an interview survey conducted with statistics teachers.
Subject Classification: 97-01, 97U70, 87K80
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A constructive and metacognitive teaching path at university level on the Principle of Mathematical Induction: focus on the students' behaviours, productions and awareness
133-161Views:61We present the main results about a teaching/learning path for engineering university students devoted to the Principle of Mathematical Induction (PMI). The path, of constructive and metacognitive type, is aimed at fostering an aware and meaningful learning of PMI and it is based on providing students with a range of explorations and conjecturing activities, after which the formulation of the statement of the PMI is devolved to the students themselves, organized in working groups. A specific focus is put on the quantification in the statement of PMI to bring students to a deep understanding and a mature view of PMI as a convincing method of proof. The results show the effectiveness of the metacognitive reflections on each phase of the path for what concerns a) students' handling of structural complexity of the PMI, b) students' conceptualization of quantification as a key element for the reification of the proving process by PMI; c) students' perception of the PMI as a convincing method of proof.
Subject Classification: 97B40, 97C70
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Integrating Didactic Games in Higher Education: Benefits and Challenges
1-15Views:238In our paper, we study the reasons for the introduction of didactic games and the way of their application in higher education, especially in teaching mathematics. After describing the main characteristics and needs of Generation Z students, we outline the advantages and drawbacks of gamification and game-based learning, followed by some new aspects to their classification. The idea of device-based grouping arose because the most commonly used methods require IC tools. Gen Zs naturally accept gamified learning materials available on digital and mobile platforms, but we must not forget about traditional games either. In higher education, especially in the case of small-group teaching there should also be room for traditional, specialized didactic games, of which we focus on the benefits of card games.
Subject Classification: 97C70, 97D20, 97D40, 97U70
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A retrospective look at discovery learning using the Pósa Method in three Hungarian secondary mathematics classrooms
183-202Views:152While 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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Many paths lead to statistical inference: Should teaching it focus on elementary approaches or reflect this multiplicity?
259-293Views:50For 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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Promoting a meaningful learning of double integrals through routes of digital tasks
107-134Views:111Within 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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Mapping students’ motivation in a problem oriented mathematics classroom
111-121Views:40This 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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A computational thinking problem-thread for grade 7 students and above from the Pósa method
101-110Views:66Lajos Pósa has been developing his “learning through discovery” (Győri & Juhász, 2018) method since 1988. His weekend math camps are focused on fostering problem-solving skills and high-level mathematical-thinking skills in gifted students from grades 7 to 11. One of the core aspects of the method is the structure of the problems, all problems are part of a complex, intertwined, and rich network. In this article we analyze a computational thinking problem-thread and its role in the camps’s network of problems (Gosztonyi, 2019), and show some aspects of the method. The insights gained using this method can be useful in other contexts. The possible adaptation of the method to secondary and high schools is briefly discussed as well.
Subject Classification: 97D40
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Integrating elements of data science into high-school teaching: Naïve Bayes-classification algorithm and programming in Python
307-316Views:55Probability 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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Interdisciplinary Secondary-School Workshop: Physics and Statistics
179-194Views:36The 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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Looking back on Pólya’s teaching of problem solving
207-217Views:138This article is a personal reflection on Pólya's work on problem solving, supported by a re-reading of some of his books and viewing his film Let Us Teach Guessing. Pólya's work has had lasting impact on the goals of school mathematics, especially in establishing solving problems (including non-routine problems) as a major goal and in establishing the elements of how to teach for problem solving. His work demonstrated the importance of choosing rich problems for students to explore, equipping them with some heuristic strategies and metacognitive awareness of the problem solving process, and promoting 'looking back' as a way of learning from the problem solving experience. The ideas are all still influential. What has changed most is the nature of classrooms, with the subsequent appreciation of a supporting yet challenging classroom where students work collaboratively and play an active role in classroom discussion.
Subject Classification: 97D50, 97A30
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Categorising question question relationships in the Pósa method
91-100Views:42The doctoral research of the author – with a reverse didactic engineering (RDE) methodology – aims at reconstructing the theoretical background of the ‘intuitively developed’ Pósa method for inquiry-based learning mathematics (IBME) in Hungarian talent education. Preliminary results of the second step of this theorization is presented, which applies tools of the Anthropological Theory of the Didactic (ATD). A model is proposed for categorizing question-question relationship with 3 categories: helping question, follow-up question and question of a kernel. The first two of them are claimed to represent two types (relevant or not) of generating-derived questions relationship. The model is also a prospective tool for connected task- and curriculum design and analysis within IBME development.
Subject Classification: 97D20, 97D40, 97D50, 97E50, 97K30
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Mathematical Laboratory: Semiotic mediation and cultural artefacts in the mathematics classroom
183-195Views:55Aim of this presentation is to summarize the influence of Tamas Varga on the Italian research and practice concerning didactics of mathematics since the 70s of the 20th centuries. While being in Budapest for the Conference I noticed that this influence was not known by most Hungarian mathematics educators. I guess that also in Italy, only the teacher educators of my generation know Varga’s influence on the teaching and learning of mathematics in primary school. Hence I start from a brief summary of development of mathematics curriculum in Italy (mainly in primary school) in the last decades of the 20th century. I focus some elements that may be connected with Varga’s influence and, later, some recent development of them.
Subject Classification: 97G20, 97-U6, 97A40
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The time spent on board games pays off: links between board game playing and competency motivation
119-131Views:95The impact playing has on the development of thinking is an important topic of psychology of learning, brain research and mathematics didactics.
Our research is also connected to the aforementioned topic. We investigated the effects of playing board games on competence motivation and the development of mathematical competencies.
In this paper, we present the results of an experiment carried out in a secondary school class.
The experimental group spent one of three weekly mathematics lessons playing board games.
Apart from the several advantages of playing games in general, we can conclude that, based on the results of the national competence measurement, the mathematical competence of the students developed properly.
The readiness and the progress of the pupils were compared on the basis of input and output tests and an initial knowledge measurement and, at the same time, we compared their level of mathematical competence with the results of the national competence
measurement.Subject Classification: 97C70, 97D40
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Some Pythagorean type equations concerning arithmetic functions
157-179Views:41We investigate some equations involving the number of divisors d(n); the sum of divisors σ(n); Euler's totient function ϕ(n); the number of distinct prime factors ω(n); and the number of all prime factors (counted with multiplicity) Ω(n). The first part deals with equation f(xy) + f(xz) = f(yz). In the second part, as an analogy to x2 + y2 = z2, we study equation f(x2) + f(y2) = f(z2) and its generalization to higher degrees and more terms. We use just elementary methods and basic facts about the above functions and indicate why and how to discuss this topic in group study sessions or special maths classes of secondary schools in the framework of inquiry based learning.
Subject Classification: 97F60, 11A25
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Group Work at High School According to the Method of Tamás Varga
167-176Views:48The aim of our research is to develop students’ logical thinking. For this reason, Hungarian mathematics teachers need to be encouraged to try new methods which induce greater student involvement. Research all over the world prove that self-instruction or self-verbalizing has high effect on the learning process. This was one of the key elements of Tamás Varga’s experiment in high school. In our classroom experiments we are using a special cooperative method from Kagan among 14-18 years old students, called Sage and Scribe structure. We are looking for the answers to the following question: Does this method make mathematics lessons more enjoyable and more comfortable for students? Furthermore, we assume this structure could open the gate toward other collaborative and cooperative teaching technics.
Subject Classification: 97D40
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Introductory Computer Programming Courses in Mathematics Curriculum
19-30Views:66We 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