Zoltan Paul Dienes: The life and legacy of a maverick mathematician
Agnes Tuska
Department of Mathematics, California State University, Fresno, USA
Abstract
The study of the life and work of Zoltan Paul Dienes allows us to understand some major reforms and setbacks of mathematics education in Hungary and in an international arena during the last 150 years. In addition, based on the study of the writings of Dienes, I highlight necessary conditions for the effective teaching and learning of mathematics and for developing in learners the joy of “doing mathematics”.
Introduction
Zoltan Paul Dienes (1916-2014) had a long and productive life. He was trained as a mathematician and as a psychologist, and worked on mathematics education-related projects all around the world. He summarized his major accomplishments in mathematics education during an interview with Bharath Sriraman in Wolfville, Nova Scotia on April 25, 2006 in the following way:
My emphasis was on the use of mathematical games with appropriate learning aids (manipulatives), work, and communication in small groups with the teacher overseeing these groups. …What I have been doing for over 50 years is not so much outside social issues but critical thinking about what mathematics is and what it can be used for and to have it presented as fun, as play, and in this sense it can be self-motivating because it is in itself a fun activity. I have critiqued mathematics being presented as a boring repetitious activity as opposed to a way to think. So it is not so much critical thinking of social issues but as a way to train the mind [e.g., Dienes & Golding, 1966], understand patterns and relationships, in ways that are playful and fun. (Sriraman & Lesh, 2007 p. 62-64)
He is considered to be the inventor of the Dienes Multi -base Arithmetic Blocks and many other games and materials that embodied mathematical concepts. He was an early pioneer in what was later to be called sociocultural perspectives and democratization of learning (Sriraman & Lesh, 2008, p. 2). He founded the Journal of Structural Learning, worked with Jerome Bruner on the Harvard Mathematics Learning Project for a whole academic year, and was the director of the Centre de Recherches en Psychomathématiques at the Université de Sherbrooke in Quebec for over a decade. Dienes founded, and compiled the report of the International Study Group for Learning Mathematics on mathematics in primary education in 1966. This initiative of UNESCO played a crucial role in changing math education (“New Math”) around the world.
While some of the ideas and reforms of Dienes evolved and became common practice in mathematics instruction, there are many that are yet to be implemented on a large scale. Although Dienes remained active even in his nineties working and promoting his findings, many of his collaborators and admirers had died decades earlier. My intention is to help to keep his legacy alive with this paper. I only highlight a few aspects of his rich and productive life, but I hope that I will be able to convey his brilliance, charm, enthusiasm, and resourcefulness as researcher, instructor, and life-long learner.
Family background
Zoltan Paul Dienes was born in Budapest, Hungary, on September 11, 1916. He was the second son of the mathematician Paul Dienes (1882-1952) and the philosopher, choreographer, dance teacher, and creator of the dance theory orkesztika (orchestics), Valéria Dienes (née V. Geiger, 1879-1978). He and his wife, Tessa, had five children together as fruits of their 68 years of marriage.
Zoltan Paul Dienes enjoyed writing not only as part of his scholarly work, but also about all aspects of his life and thoughts. He published some of his poetry, too (Dienes, 2003a). Many details of his life, travels, and family events are well-documented in his autobiography (Dienes, 2003b).
One obviously key element of his life is that he was a child of two brilliant parents, who had immense curiosity not only towards mathematics, but all kinds of aspects of life, and were connected to the intellectual elite of Europe. As a consequence, Zoltan grew up living in many different countries, learning to communicate quickly in various languages. As his parents divorced, he lived with his mother in Hungary, Italy, and France, but, at age fifteen, he moved to England to live with his father. Many schools have refused taking a child with no knowledge of the English language. Dienes recalls:
My father had a long telephone call with Bill Curry, the headmaster, who said he would come and see me in two weeks’ time. My father said to me, ‘You have two weeks to learn English! Go to it!’……
‘Have you read anything in English?’ asked Mr. Curry. ‘Yes, I am reading David Copperfield,’ I answered in a very thick Hungarian accent. ‘I have read about half of the book. ‘Tell me about it,’ said Mr. Curry. In somewhat broken English, I tried to give him an idea of the plot to the point where I was in the story. ‘How long have you been studying English?’ was the next query. ‘I started two weeks ago,’ I replied. At this point he indicated that the interview was at an end and went to talk with my father in another room. After he had departed, my father came towards me smiling, finding me already at work on David Copperfield, and informed me that I could indeed go to Dartington Hall School. So the continuation of my education was assured and I could look forward to some interesting times in one of the most avant-garde schools in the country. (Dienes, 2003b, p 97-98)
Therefore, in The Power of Mathematics, Dienes may reflect on his own childhood experiences when he writes:
We must consider for a moment the difference between natural and artificial learning, for this provides important clues to the understanding of children’s difficulties when they are confronted with artificial scholastic learning situations. If a child is taken to a foreign country where his mother-tongue is not spoken, within a few months he can speak the new language as well as his newly acquired friends, whereas his parents may still be struggling with grammar years later, trying to learn the language
‘properly’. It is, of course, the child who has learnt it ‘properly’, and the reason is that he learnt it naturally. Fortunately, it is impossible to learn skating or riding a bicycle from a book, or many people would have a try. The ‘fiddling around’ with the data is the only way of making sure that you do not eventually take a tumble on the ice or fall off your bicycle. Natural learning is not invariably preferable to artificial learning; it is however a priori probable that it will be more effective. (Dienes, 1964, p. 24)
Dienes carried the memories of the independent “modern education” (Dienes, 2003b, p. 53) of his youth that flourished in his parents’ circles throughout his life. They formed the “hard core” of his later
“maverick mathematics” that he spread as “playing mathematics for fun” throughout Europe, Australia, South America, Papua New Guinea, or with First Nations and Métis classes in Manitoba, training Peace Corps workers to teach in the Philippines. He developed his methods as a teaching practice to be used at universities of Australia (Adelaide University) and Canada (Université de Sherbrooke in Quebec), and many other places, researching the psychology of mathematics and problem solving.
World Citizen
Dienes did not only travel around visiting many parts of the world, but truly lived in those places, immersing in the local culture. He had an extraordinary ability to communicate with diverse learners, partially because he cared to learn about the background of the learners. In an interview, Dienes gave an example of his cultural sensitivity teaching logic:
Context is very important. In the work I’ve done in different parts of the world, I’ve always tried to put things in practical terms. It somewhat depends on the local culture in which you are operating. It wouldn’t be the same in the United States as in India or China or New Guinea. … In New Guinea, I
came across a tribe in whose language there was nothing for the concept of “either/or”. [… ] How do you teach logic if you don’t know about “either/or”. So I had to work out a way of making sure the kids understood. I did this by [tapping my arm] which meant it was correct; and, this [nodding head] meant no. So, with attribute blocks, a child would produce an answer to a question, and I would say [tapping my arm] that it was okay, and another child would produce another block, a different answer to the same question which was okay and I would again say [tapping my arm] that it was okay. That flabbergasted them. How could the answer be either this block or that block. … This was how I managed to teach the notion of “either or” in a culture where they had no words for such a concept. … In New Guinea, there is no “either/or” because the tribal system was so strict. You do THIS, and under these circumstances, you do THIS [under other circumstances]. And that’s it. And God help you if you don’t [Laughing]. (Sriraman & Lesh, 2007 p. 63)
Dienes was a master of providing opportunities to reflect on physical and mental actions on concrete, manipulative materials that – according to his construction principle - led to the formation of mathematical relations. For example, he arranged a field trip to the local shore in order to prepare the ground for sophisticated mathematical concepts in the mind of young children:
At one of the schools, I was trying to introduce some topological ideas into the curriculum. In the case of children in elementary school age, this meant being aware of insides and outsides, connectedness and separateness and suchlike. Paper and pencil work on this did not seem to mean much to the children, so I took them to the shore of a sandy lagoon and asked them to make a harbour. They made several harbours and also made some ‘connections’, so that ‘boats’ could go from one stretch of water to another. The whole complex of islands and canals became extremely complicated before we finished the project! I would ask two children to be two crocodiles and stand in the water and then asked them if they could ‘swim’ to each other without going over dry land.
Afterwards they had to be wallabies and they had to decide whether they could get to each other without jumping over some water. After these experiences, the paper and pencil work became much more meaningful and the children really enjoyed making their own mazes for imaginary crocodiles and wallabies. (Dienes, 2003b, p. 328)
Special connection with Hungary
Dienes stayed connected with his country of origin. He loved singing Hungarian folk songs, and visited Hungary many times. He also paid particular attention and contributed to the development of mathematics education in Hungary. For example, Rózsa Péter’s Playing with Infinity, that was first published in Hungary in 1957, was translated into English by Dienes and got published in England in 1961.
Dienes worked closely with Tamás Varga on “New Math” in Hungary during the 1960s. They collaboratively examined, evaluated, and perfected the uses of many manipulatives in instruction, such as, for example, Vigotsky’s logic blocks (Servais & Varga, 1971, pp. 38-46), Cuisenaire rods, and Dienes’
Multi -base Arithmetic Blocks (Servais & Varga, 1971, p. 107).
Multi-base Arithmetic Blocks and other manipulatives
Using various manipulatives as tools for conveying mathematical concepts and for solving problems is a natural part of communication and thinking for many people. However, most of these manipulatives were ad-hoc inventions, not common part of the “toolboxes” of teachers around the world. One of the greatest legacy of the mathematics educational reforms of the 1960’s is that classroom sets of manipulatives were designed, and workshop series were organized to help teachers learn ways of effectively using them. The most widely analysed and promoted manipulatives of the 1960’s were the Cuisenaire Rods, the Geoboards, and the Multi -base Arithmetic Blocks.
Georges Cuisenaire (1891–1975), also known as Emile-Georges Cuisenaire, was a Belgian elementary school teacher who invented the set of Cuisenaire rods as a mathematics teaching aid. In the 1950’s, Caleb Gattegno (1911–1988) became the main advocate of Cuisenaire rods. Gattegno is considered to be the designer of Geoboards. He invented and promoted innovative approaches to teaching and learning mathematics (Visible & Tangible Math), foreign languages (The Silent Way) and reading (Words in Color). He was a prolific writer, authoring more than 120 books and hundreds of articles mostly on the topics of education and human development. These manipulatives were mostly used in the teaching of young children, but, for example, George Polya used Cuisenaire rods and geoboards in instructing graduate level mathematics classes, too (Taylor & Taylor, 1993, p. 75).
One of the early promoters of familiarizing young students with wooden geometrical objects - some of which were similar to Dienes’ Multi -base Arithmetic Blocks – was Friedrich Froebel (1782-1852), a German educator, the inventor of the word and the institute of “kindergarten” as the “Play and Activity Institute for small children”. Children were to play, sing, dance, and use any other means of active learning as a preparation for their lives. Caroline Pratt further popularized the unit blocks in the early 1900’s. The Montessori system (Montessori, 1965), illustrating division with rulers, squares and cubes has been in use since 1914 (Servais & Varga, 1971, p. 105). The use of these kinds of blocks were known in Hungary during the childhood of Dienes, too. The anecdote below includes some misinformation, marked in bold (“Diene’s” should be “Dienes’”, “Zolton” should be “Zoltan”, and Zoltan P. Dienes was originally from Hungary, only worked in Australia), but these distortions may be due to the fact that the information was based on a phone conversation of the authors with Mary Laylock in 1992:
Mary Laylock, a California mathematics teacher and consultant, was leading an in-service workshop at Stanford when Polya stopped by to observe. The students in her class were on the floor diligently pursuing an activity in which they were performing arithmetic operations in bases other than ten using Diene’s Blocks. Diene’s Blocks are similar to Cuisenaire Rods and were developed by Zolton P.
Dienes, who was originally from Australia. Polya had to leave before he could speak to Laylock about her work, but he told a colleague, “Tell Mary that is just the way I learned my numbers, using blocks like those, seventy years ago in Hungary.” (Taylor & Taylor, 1993, p.75)
The parents of Zoltan P. Dienes were contemporaries of George Polya. Moreover, the father, Paul Dienes not only knew Polya through Lipót Fejér and from the Galilei Circle in Hungary, but collaborated with Polya later, too. For example, they both were authors in the first issue of a mathematics journal that was started by G.H. Hardy (Albers et. al., 2015, p. 96). It is very likely that the parents learned their numbers using blocks just as Polya did. Since Zoltan was eager to learn math from his parents (he was ahead of the math he encountered in school), it is very likely that the mathematics education he received was based on the progressive math experiences his parents lived through around the turn of the century, marked by the names of Mór Kármán, Manó Beke, László Rátz, and Gyula Kőnig, (Szénássy, 1992, pp. 217-218), and many other eminent teachers and scholars, supported by wise educational policies introduced by József Eötvös, Loránd Eötvös, and others.
Major Reforms and Setbacks
The unique contributions of Zoltan Paul Dienes related to the use of arithmetic blocks were to (1) systematically use blocks with various bases (hence the name “Multi-base Arithmetic Blocks”), therefore highlighting the structure of place-value systems for recording numbers and doing operations with them, depending on the base number chosen, (2) conduct experimental studies on the ways students learn, struggle with, and master the use of those structures with observable and recordable actions by the learner via the use of manipulatives, and (3) demonstrate the replicability of results by conducting the same kinds of activities in classrooms all around the world.
The variation of the base number in the blocks provides a good example of Dienes’
principle of multiple embodiment (Dienes, 1964, p. 40). A good example of the experimental studies related to the Multi -base Arithmetic Blocks is the book reporting on the work of the “Mathematics Learning Project” led by Jerome Bruner and Dienes (Dienes, 1963). The planning of the experiments and the discussion of the data collected in the five experimental groups took place in frequent seminars held at the Harvard University Center for Cognitive Studies during the 1960-1961 academic year. Dienes explains the function of play in mathematical thinking in this book as follows:
To sum up, we shall speak of manipulative play, representational play and rule-bound play. Manipulative play is an activity which could be described as exploration, since the subject is hardly aware of the exploratory process in the beginning, but awareness increases with the accumulation of experience.
Representational play takes place when objects or people are assigned properties different from those they in fact have. Imagination is an essential component of this kind of play. Rule-bound play is essentially
‘playing the game’, meaning that choices are limited in some circumscribed way by the rules of the game.
(Dienes, 1963, pp. 23-24)
As a major educational implication, Dienes called for further international collaboration in his experimental study report:
It is clear from this account of the present state of research (which is by no means exhaustive), that future research should be both interdisciplinary and international. The problems involved in studying the learning of mathematics open up psychological, mathematical and philosophical questions, as well as practical educational questions such as classroom organization, punishment-reward systems and the like.
No one person can be equally conversant in all these disciplines. It is also clear that there is so little fundamental work in progress relevant to the evolution of satisfactory models that co-ordination of effort is an urgent necessity. […] The point to bear in mind is that research alone will not meet the demands of the situation, however essential such research might be. The psychologists will have to listen not only to the mathematicians, as they have begun to do already both in Geneva and at Harvard Center, but also to practising teachers, training-college lecturers, administrators of school systems and so on who may need to start their own centres where work relevant to their special roles may be satisfactorily studied. (Dienes, 1963, pp. 176-177)
Dienes did more than his fair share of the work outlined above. Besides working directly with teachers and children in many countries, he also completed a doctoral degree in psychology besides his existing doctorate in mathematics, founded the International Study Group for Learning Mathematics, and compiled the report of the group’s work on mathematics in primary education in 1966. This initiative, funded by UNESCO played a crucial role in changing math education (“New Math”) all over the world. The results of the projects of Dienes and Jeeves (Dienes & Jeeves, 1965), along the works of Piaget, Bruner, Bartlett, Skemp, Suppes, Pescarini, Varga, and other major researchers, were tested through experimental studies and in classroom instruction. The report provides the most comprehensive list of evidence-based best practices of the 1960’s around the world, with specific recommendations for teacher training (International Study Group for Mathematics Learning, 1966, pp. 107-124).
Dienes clearly recognized that elementary school teachers need and deserve mathematical content enhancement connected with pedagogical content knowledge, developed in the context of teaching as action research. During the 1960’s, many researchers and mathematicians worked side by side with classroom teachers, and conducted intensive workshop series for large groups of teachers. The above cited workshop story of Mary Laylock shows the teacher leader’s connectedness with on-going educational research in the use of manipulatives, and close collaboration with a caring local mathematician (Polya) in the process of empowering teacher colleagues. Or, for example, Hungarian elementary teachers went through intensive training that
prepared them for instruction through workshop activities in the early 1970’s if they were to use the educational materials and framework developed under the leadership of Tamás Varga.
Unfortunately, many of the teacher training institutions were not prepared for such kind of work. The “top down-approach” of changing the curriculum on teachers with expectations of using methods and conveying knowledge that they were shaky on, produced many set-backs. Many mathematicians looked down on teachers for not knowing “enough” math, and many educational leaders wanted to take the math out of the classrooms because it was considered to be too hard. In order to replace math, they requested teachers to comply with ever-changing classroom management techniques, as a way of keeping teachers busy, and making them feel deficient.
Classroom teachers were considered empty vessels that needed to be filled by a heavy dose of math or by a heavy dose of methods by “experts”. Many teachers acted similarly in their classrooms. Mathematics was also frequently used as a “gate keeper” to deny access of learners to advanced studies.
Conclusion
Dienes was a maverick mathematician and mathematics educator. He considered mathematics a human activity that can be experienced and enjoyed by everybody. His legacy is to build a friendly, democratic approach to the teaching and learning of mathematics. In contrast to elitist or inferiority complex-based education, his approach was uplifting and respectful towards all learners.
Learning mathematics can improve our communication, culture and decision making, and can help solving our problems. The teacher’s role is to engage students in communications, plays, and activities for teaching informed decision making and problem solving. Math is our symbolized communication tool for investigating patterns and form constructs (i.e. structures that are built together for further communication and problem solving).1 Math is needed to make just (as in justice) and wise decisions in problems facing our society, because for democratic decision making we need to be able to convince others about the feasibility of our propositions and fit them into the existing construct (proofs), to build “common sense”, or to show that the existing construct does not make sense in our expanding world, contradicts with our practices and experiences, and does not allow space for our needs, so we need to build a better construct. Dienes understood well the changing demands towards mathematics education:
You can learn mathematics simply as a utility and learn how to use it. That happened during the 18th and 19th centuries, during the Industrial revolution. It became necessary for people to read instructions, to do simple number work, because it was economically necessary. (But), all you had to do was learn certain tricks. To add, to multiply, get percentages, a little bit of fractions and so on. But, the situation today is different economically than it was say 150 years ago. It was good enough then to know just how to do the tricks. But it is not good enough anymore for doing the work we do now in most jobs. So, we need to know a little more mathematics. Now as to what type of mathematics we need to know I suppose it doesn’t matter very much because most mathematics you learn, if you understand it, will teach you a way of thinking … structural thinking. Thinking in structures, how structures fit into one another. How do they relate to each other and so on. Now, whether you learn that in linear algebra or in infinite series or any other area…As long as you get the idea of what mathematical thinking is like, you can apply it to all sorts of other situations. (Sriraman & Lesh, 2007 p. 62-63)
The original Multi -base Arithmetic Blocks have mostly been replaced by interlocking cubes, that allow to build models for understanding arithmetic in any base system, and by algebra tiles, for generalizing to “base x system” the concepts of arithmetic, in the cabinets of many teachers. The evolving technology also opens up many new learning opportunities. For example, the dynamic view offered by educational films of Nicolet
1 For the communication theoretic background of Dienes’s work in the context of his didactic principles, see Benedek (2018).
in the 1940’s (Servais & Varga, 1971, pp. 101-102) can be enhanced or replaced by hands-on experiences with dynamic software, such as, for example, SketchUp and GeoGebra.
The brilliant insight of Dienes, requesting interdisciplinary and international collaboration among researchers and practitioners for the study of the learning of mathematical thinking remains as appropriate today as it was in the 1960’s. Organizing Networks of Excellences for the study of complex issues seems to be a “hot” new approach. Dienes not only envisioned and designed a very effective network, but played a leading role in all aspects of the work. The implementation of the suggestions made in his report would greatly enhance mathematics education in the whole world. Pierre-Simon Laplace is attributed this exhortation: “Read Euler, read Euler, he is the master of us all!” Paraphrasing him, I am happy to suggest to mathematics educators: “Read Dienes, read Dienes, he is the master of us all!”
References
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