The Dienes method
Sándor Klein (prof. emeritus, University of Pécs)

   Zoltan P. Dienes is one of the most creative mathematics educators, who had an influence on elementary and secondary school education all over the world. He developed his theory of mathematics learning in the 50ies, and in his book “Building Up Mathematics”, published in 1960 he offered it to teachers and researchers. He conducted courses and carried out research in psychology and pedagogy of mathematics learning in the UK, Australia, Canada, the USA, Brazil, Chile, New Guinea and in almost every European country. He received Honorary Doctorate from five Universities ‒ the last one from the University of Pécs.

   His “principles of mathematics learning” (constructivity, dynamic, perceptual and mathematical variability) are well-known, just as well as the “six stages of learning” (free play, rule-based games, discovering the common structure, representation, symbolization, formalization). 

   The essence of the Dienes method is playful, empirical learning and most people know Dienes through his games (first of all the Multi-Base Arithmetic Blocks and the Logic Blocks ‒ originally an experimental tool developed by the Russian psychologist, Vygotsky).

   In spite of his many books, theories, games he was, first of all, a teacher. Entering into the classroom anywhere in the world and opening his suitcase full of “treasures” he was immediately at home ‒ the children accepted him as “one of them”. This is probably the most important thing we should learn from him.

On the origin and psycho-logic background of Zoltán P. Dienes’s Principles

András G. Benedek, Research Centre for the Humanities of the Hungarian Academy of Sciences

Zoltan P. Dienes extended Piaget’s theory of mathematical abstraction and gave a definition of the operational level of abstract thinking. In the Some Thoughts on the Dynamics of Learning Mathematics, one of the last works that he compiled by Christmas Day, 1995, he summed up his theory of abstraction through examples. Analyzing his widely known principles of learning and teaching mathematics in the light of these examples, I argue that behind his principles we can explore a psycho-semantic problem of communication that is a continuation of her mother’s, Valéria Dienes’s, “evologic” model. Her conception of semiotic meaning creation in the field embodied expression and dance implicitly influenced Zoltán P. Dienes’s principles with due modifications in the context of the psychology of mathematics. His principles and levels of abstraction demonstrated by his concrete heuristic practices exemplify a psycho-logic conception of understanding that extends the emerging model-theoretic semantics of the 1960ies to human activities and embodied experience. This extension helps us to see heuristics as the research of those ways, means, and tools that support mapping our symbolically expressible verbal thought processes to such operational and manipulative models and practices of problem-solving that we are able to put out from our heads. His extended notion of building abstractions includes not just logic or set-theoretic models but all kinds of extra-verbal structures and operations that promote the understanding of why we consider true what we are seeing. These human activities are based on embodied operational experience obtained by using objectified manipulatives just as on our psychological thought processes reproducing and inventing them.

The Moment of Discovery in Flow

Olga KISS, (Corvinus University, Budapest)

Flow, the perfect experience, is closely related to the concept of happiness. However, there is much more to flow than just subjective pleasure. When the flow occurs, our own intentions, abilities, and knowledge are aligned with what we do, the domain, the world, and indirectly, even those who deal with the same subject, the field. In scientific research, discovery is such a moment: it opens up a previously unknown side of truth. A new Gestalt is made up of data. This is what Kisiel calls the hermeneutical moment: when we suddenly understand something - which, of course, remains to be explained, unfolded and interpreted. A big discovery is more than a person's momentary “Aha!” experience. It is an event that can lead to a paradigm shift that can change our worldview, culture, and life. How is it possible? How is it visible in the connectedness and alignment of flow experience?

As a researcher, most of us are already content with knowing how to get to the "zone", how to re-create our own creativity. What matters to Csíkszentmihályi is why this causes us, humans, joy. Why does it feel so good to think, to discover, to create?

Forum of International Projects and Collaborations
TUSKA, Agnes (California State University, Fresno)

The goal of this forum is to highlight three highly successful projects through their connection with, and influence on, the work of the Department of Mathematics at California State University, Fresno (CSU Fresno).

The three projects - Development and Research in Early Math Education (DREME), Experience Workshop, and Inquiry Based Learning (IBL) rely on common principles of (1) deep engagement in rich mathematical activities and (2) opportunities to collaborate with peers in an equitable problem-solving environment. The focus of the research related to the projects is inquiry into student thinking.  

George Polya (1887-1985) and Zoltán Paul Dienes (1916-2014) were masters of mathematical heuristics and didactics based on the above principles. The workshops and summer institutes Polya conducted for teachers (particularly in California and in Switzerland) throughout the years have influenced the teaching and learning of mathematics for generations. Dienes played a key role in the development of experimental research-based mathematics education. He had a unique position in the research community as a well-trained mathematician and psychologist who also had extended experience in working with children and with school teachers worldwide.

Based on our long-term collaboration with, and observation of classroom practices of teachers in various Lesson Study grants, and by working together with teacher leaders in the San Joaquin Valley Mathematics Project (one of the state-funded California Mathematics Projects, http://www.fresnostate.edu/kremen/sjvmp/index.html), we realized that teachers need on-going, locally accessible support, that allow them to collaborate with experts as well as with peers, in implementing engaging and effective mathematics instruction. We plan to establish a “Polya Institute” with advisory board members from each of the three projects to promote mathematics educational research and practice at all levels in the Fresno region in the spirit of Polya and Dienes.

The major connection between DREME and CSU Fresno is the way we conduct our Math Circles for grade 3-4 students. Participants are expected to collaborate, explain their thinking, find alternative solutions, and critique the reasoning of their peers.  University students co-instruct sessions with university professors, gaining valuable insights to Cognitively Guided Instruction. The materials utilized by the Experience Workshop and the CSU Fresno’s STEM Summer Academies significantly overlap. Finally, the implementation of a problem-solving approach and active learning techniques of IBL became significant parts of a large variety of college-level mathematics instruction in Fresno.

Psychology of problem-solving and its role in education: from Sputnik shock to the web

Pléh Csaba CEU Cognitive Science, Budapest and Hungarian Academy of Sciences

vispleh@ceu.edu

Talk at the 2nd Heuristic Conference, Balatonfüred, 2019, Aug. 30

Overview / Áttekintés

Classic duality: analytic versus holistic                                                        Klasszikus kettősség: elemző és egészleges
Selz versus Wertheimer                                                                                 Selz versus Wertheimer

Direction and set in problem-solving                                                          Irány és beállítódás a feladatmegoldásban

Images and inner speech                                                                             Képek és motyogás a gondolkodásban

Selection theories                                                                                         Szelekciós elméletek

Logical inferential systems                                                                           Logikai következtetési rendszerek

Errors of deduction                                                                                       Deduktív hibák. Induktív hibák. 

Errors of induction, Wason                                                                           Induktív hibák, a Wason feladat

The Sputnik shock and the cognitive revolution                                         A szputnyik sokk és a kognitív forradalom

Changes in creatvity research                                                                       A kreativitás kutatás változásai

Implicit learning occurs pervasively in our life. A good example is how we learnt the grammar of our native language. We know this complex grammar perfectly and can use it in a fraction of a second of produce and comprehend sentences. Yet most people cannot describe any but the simplest rules of the grammar; even linguists do not have a complete grammar of any natural language! Another example is how we learn the rules of social engagement: We can explicitly describe many aspects of these rules, but in general we know more about getting on with people than we can freely articulate. Or consider the experience of listening to music. We can at once judge the style of the music (Beatles or Bach?) even before we know how we have done it; and we can like a style of music without having to be able to say what structural properties of the music lead us to like it. In Chapter Five, my grandfather gives an example from learning mathematics. After experience of a function determining the order of items in a sequence, children can successfully predict the next item while claiming they are just guessing. In fact, the development of implicit knowledge in people exposed to sequences has been extensively explored by psychologists interested in implicit learning (e.g. Reber, 1993).

Implicit learning is unconscious learning in the sense we are not conscious of the knowledge we have learnt. Explicit learning is learning that occurs when we are conscious of the knowledge we have learnt; we can then in general describe the knowledge that guides our actions. For example, knowledge of the recipe I follow in cooking a cake is often explicit. Following a worked maths example in order to solve a related problem would involve explicit knowledge. In Chapter Five, when people had learnt a particular group structure involving yellow and red cards, and could say "If I play a yellow, you show the same card that you showed before, but when I play a red, you change the colour", these people had demonstrated that they had acquired explicit knowledge.

Learning about a domain will often involve both implicit and explicit components. Given that one important aim of my grandfather's methods is to produce understanding, it is important to consider what role implicit and explicit learning may play in developing an understanding of a concept.

Beyond Symbols: Movement, Communication and Intuition in Valéria Dienes’s Philosophy, Psychology and Pedagogy

Brigitta BALOGH, Partium Christian University / Eötvös Loránd University - Doctoral School of Education

As one of the most many-sided and creative thinkers of 20^th century Hungary, Valéria Dienes has passed on us some very fruitful insights concerning the nature and upbuilding of human knowledge. As a trained mathematician, philosopher and movement artist, as well as creator of a unique „science of the movement” and movement system called „orchestics”, she combined in a very original way the perspectives and methodologies of the arts and sciences she practiced.

In my presentation, I will focus on the way how Dienes handles the link between conscious, symbol-based and/or conceptual knowledge on one hand, and (at least partly) unconscious, experience-based and non-conceptual knowledge on the other. Starting with two philosophical-psychological concepts of intuition, and relating them especially to mathematical cognition, I will outline Valéria Dienes’s approach on three levels: a) her philosophical anthropology and philosophy of communication, b) her psychological account of how human beings construct space and time by experiences of movement; c) pedagogical consequences of her insights concerning what she has called the „dead verbalism” of ordinary schooling.

http://euclidlab.org/

With the increasing popularity and performance of deep learning methods many scientific fields adapt deep learning to solve their challenges. Cognitive neuroscience is no exception: with specific modifications deep neural networks can be successfully used to analyze anatomic and functional brain images, as well as functional connectivity networks. At the Brain Imaging Centre of the Hungarian Academy of Sciences we work on neural network architectures that have the potential to advance cognitive and clinical brain research.

In Fresno, California, where we develop the Polya Center and Academy, we build on a tradition of Summer Camps admitting students from a diverse cultural background. They discover not only mathematics in group work but use it for their various interdisciplinary projects such as building 3D graphs, kites, etc. In this report, I spell out the advantages of combining group work, project-based learning and the Polya-style interdisciplinary approach to applied mathematics and give a summary of our proposals for the pursuit of the Polya Center and Academy including teacher training.

A kaliforniai Fresnoban, ahol a  Pólya Center és Akadémia terveinek megvalósítására törekszünk, nyári táborok több éves hagyományára építünk, melyek igen különböző kulturális hátterű tanulókat fogadnak. A tábor résztvevői nem csak metematikát fedezik fel csoport munka keretében, hanem azt, hogy hogyan alkalmazhatják azt olyan interdiszciplináris  mint a 3D gráfok készítése, vagy a sárkámyépítés. Beszámolok az alakuló Pólya Központ és Akadémia terveiről, nevelési, szakmai céljairól és módszereiről, melynek keretében vázolom a csoportmunka, a projekt alapú tanulás és a matematika Pólya-féle interdiszciplináris megközelítésének összeegyeztetési lehetőségeit tekintettel a tanár továbbképzés lehetőségeire is.