Reflections Of Zoltan P. Dienes On Mathematics Education

The name of Zoltan P. Dienes (1916- ) stands with those ofJean Piaget, Jerome Bruner, Edward Begle, and Robert Davis as legendary figures whose work left a lasting impression on the field of mathematics education. Dienes’ name is synonymous with the Multibase blocks which he invented for the teaching of place value. Among numerous other things, he also is the inventor of Algebraic materials and logic blocks, which sowed the seeds of contemporary uses of manipulative materials in instruction. Dienes’ place is unique in the field of mathematics education not only because of his theories on how mathematical structures can be effectively taught from the early grades onwards using manipulatives, games, stories and dance (e.g., Dienes, 1973, 1987), but also because of his tireless attempts for over 50 years to inform school practice through his fieldwork in the UK, Italy, Australia, Brazil, Canada, Papau New Guinea and the United States. Dienes’ theories on the learning of mathemat-ics have influenced many generations of mathematics education researchers, particularly those involved in the Rational Number Project and more recently those working in the models and modeling area of research. Dienes championed the use of collaborative group work and concrete materials, as well as goals such as democratic access to the process of mathematical thinking, long before the words “constructivism” and “equity” and “democratization” became fashionable. In this rare interview, Dienes reflects on his life, his work, the role of context, language and technology in mathematics teaching and learning today and on the nature of mathematics itself. Sriraman: It is an honor to be able to talk with you. I really appreciate the invitation to visit. Dienes: You have traveled so far . . . so I hope I am of some help. I can’t have very much longer on this planet. So it’s good you’re here. Sriraman: Your books have been very influential in my own work, many of your writings from the ’60s, and especially the one you wrote when you were in Adelaide. Dienes: With Jeeves, yes (see Dienes & Jeeves, 1965). Sriraman: Yes, particularly the innovative experiments you set up which investigate reasoning about isomorphic structures such as groups....Do you still believe this is the way to teach math- ematics, especially knowing that mathematics has become more and more applied in today’s world compared to the ’50s and ’60s. Dienes: Well! It depends on what you think in important and what one is after. Mathematics is characterized by structures, there is no denying this fact and in my opinion it is important to expose students to these structures as early as possible. This does not mean we tell them directly what these structures are but use mathematical games and other materials to help them discover and understand these structures. You have read about my theory of the six stages of learning (see Dienes, 2000b). And in this theory, the formalization stage comes at the very end. Sriraman: As you know, there have been theorists who think that such topics are too difficult at earlier developmental stages—al-though your work indicates otherwise. Piaget, for instance thought this type of thinking (structural thinking) was only possible at the stage of formal operations. Dienes: Children do not need to reach a certain developmental stage to experience the joy, or the thrill of thinking mathematically and experiencing the process of doing mathematics. We unfortunately do not give children the opportunities to engage in this type of thinking. One of the first things we should do in trying to teach a learner any mathematics is to think of different concrete situations with a common essence. (These situations) have just the properties of the mathematics cho-sen. Then . . . children will learn by acting on a situation. Introducing symbolic systems prematurely shocks the learner and impedes the learning of mathematics. Sriraman: What are your thoughts on Piaget’s theory? Dienes: [Dienes gets up and retrieves a manuscript] I was working with Piaget’s group of researchers at Institut Rousseau in Geneva I did not hear one consistent answer when I asked them what it means to be “operational”? . . . You can look at this manuscript and read what I asked Piaget. Sriraman: (reading from Dienes’ manuscript) “Is it so Monsieur Piaget, that a pre-operational child can operate on states to get to other states, but is unable to operate on an operator to get another operator, whereas an operational child can also operate on an operator, without having to think of the intervening states” Dienes: Yes, and Piaget agreed with my definition (Laughing). You can read about my conception of operationality in children yourself. It is a bit different from Piaget’s. Sriraman: You mentioned pre-mature use of symbolic systems in the teaching of mathematics earlier. I agree that notation is used too early without children completely understanding what it is they are being forced to represent and symbolize. I know you spent a year at Harvard with Jerome Bruner. Do you care to talk about this? Dienes: 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....I did have arguments with Bruner and his follow ers on this subject. I even invented a term “Symbol Shock” [Laughing] and there was disagreement with my approach from his camp. Sriraman: What got you interested in the teaching and learning of math-ematics? You come from the background of being a mathematician . . . Dienes: I explained it to some extent in that book (Dienes 2003). I thought it was strange how people didn’t understand math-ematics. What makes it so hard? Then I thought of things like . . . the distributive law for instance. It is very hard to explain this law to somebody who is not a mathematician, but you can invent some games which work in exactly the same way, which you can play. I thought why not try and see if you can do something like that with kids and see if they buy it. And they did. Sriraman: From the point of view of a university teacher educator who wishes to make a structural approach to learning more com-mon, how much mathematics do you think prospective teachers and teacher educators need to know before they can truly appreciate mathematical structures? Dienes: The answer to your question depends on what you mean by how much mathematics? There are several things that are important. One needs to be able think logically. How much mathematics one studies . . . depends . . . Sriraman: I think what I am trying to ask you is whether or not you think studying a lot of mathematics is important before starting to teach it. Dienes: It really depends on the person. Some are able to grasp the fundamental ideas very quickly. So, if one doesn’t study a whole lot of mathematics formally, but understands the material they have studied . . . it doesn’t matter. The real problem occurs when one doesn’t understand what mathematics is about in the first place and then tries to teach it. It is a question of depth. 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 percent-ages, 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: Recently, there have been initiatives by Richard Lesh which are guided by your principles of learning. His research group uses model eliciting activities and model development sequences in much the same way that you used concrete embodiments and multiple representations. But, his work focuses on simulations of “real life” situations more than on concrete manipulatives. Students often work in small groups, just as in your work; and, their work continues to focus on structure. What do you think of this approach? Dienes: Well, it is good to hear that others are making use of my learning principles. I emphasized small group work long before it became popular. Srir

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