Zoltan Dienes

Introduction

My grandfather and I share some things in common. We are both called Zoltan Dienes, for example! In addition, we both have an interest in how people learn. I have been particularly interested in how people can learn about sometimes complex structures when they do not necessarily know that they are learning and they cannot say what it is they have learnt. This sort of learning is called implicit learning (see Berry and Dienes, 1993). What I would like to do is to discuss the relation between implicit learning and my grandfather's principles of education.

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.

What is it to understand a concept?

 What is it to understand something? This is a surprisingly difficult question. But we can say some things. We can say that understanding, our grasp of something, admits of degrees. We can understand a concept more or less well. The extent of our understanding of a concept is directly related to how systematically we can use the concept. A person's use of a concept is systematic if they can use the concept in all the contexts to which it applies. If you tell me the definition of a mathematical group, I can understand the conditions which you specify, and that alone gives me a degree of the grasp of the concept. If I can recognise the group structure of various concrete situations then I demonstrate a deeper understanding. Similarly, I can demonstrate my understanding of a cardinal number, e.g. the number six, by counting a set of objects and telling you, in answer to the question "How many are there?" that there are six. If however,  after you have spread the same objects out on the table I believe there are more than six objects now (as Piaget showed young children do), then my failure to grasp there are still six objects there reveals a lack of full understanding of what it is for there to be six objects. I am not able to apply the concept in all the contexts to which it is applicable.

 Intellectual discovery often consists in seeing old patterns in surprising incarnations; surprising because we did not know they were there before. That is, the creative act of intellectual discovery often consists in the identification of a well-known pattern, or mathematical concept, in a new guise. The person who makes the discovery demonstrates their deep understanding of the concept; creativity and understanding go together. A person has a deep understanding of a concept if they can apply it in novel contexts. If a person does not recognise the pattern in situations where the pattern does in fact manifest, the person's grasp of the concept is not so great. If I think that a quadratic equation is just something I know how to deal with when the teacher sets me certain types of problems to solve, and I just use the standard formula for the solution then I do have some hold on what a quadratic is. But because I will fail to identify this pattern in many situations where it may be  - hidden in plain sight - my grasp of the concept is limited.

These comments apply not just to mathematics. To quote from the philosopher Millikan (1984), who was not explicitly talking about mathematics: "My hold on what I am thinking of depends upon how versatile and how reliable I am in performing acts of correct identification of the referents of my thought .. . . having the capacity to identify what I am thinking of via percepts as well as via thoughts and language is another way to increase my hold on what I am thinking of. . . if I can recognise you on the street, or recognise your voice on the phone as well as knowing your name and something about you, I have better hold on who I am thinking of when I am thinking of you than if I cannot (p. 252)."  According to Millikan (1998), having a concept means being able to identify it, more or less reliably, in its different instantiations. Millikan proposed her account as a theory of having concepts of substances; I have found Millikan’s approach very useful in trying to understand mathematical understanding as well.

Understanding and implicit and explicit knowledge

Understanding mathematics cannot just be the ability to manipulate symbols according to mechanical rules. Searle (1980) employs his "Chinese room argument" to caricature the notion that understanding could be produced by following explicit rules. Searle considered the case of understanding stories. Take an English person who understands no Chinese. Lock him in a room where there is a huge rule book written in English. He is given a piece of paper with a story in Chinese on and then a series of questions about the story, all in Chinese. He has no idea what any of the Chinese characters mean, but the rule book tells him how to manipulate the symbols so as to produce an answer to each question; in fact, an answer that a Chinese person would recognise as the correct answer to the question. The person can follow all the rules perfectly, and thereby produce correct answers, but he has no idea what any character means, or what any of his answers mean. So understanding is not just following rules.

Understanding is lacking for the person in the Chinese Room because the person could not identify the referent of any of the characters by any perceptual (visual, tactile, etc) means, even if the person were actively manipulating the referent in question. For example, the person could be following rules appropriately for the character for "horse"  in answer to questions written in Chinese; but the person would not know that the character could be used to name a horse the person was looking at, riding on, etc.  Further, the person could not identify the real-world patterns referred to by combinations of characters (phrases, sentences, e.g. "this is a horse") by any perceptual means; when looking at a horse they could not identify this situation as a referent of the sentence "this is a horse".

Grasping what a word means does not just entail being able to give an explicit definition. I show my understanding of the definite article ("the") and indefinite articles (e.g. "a") by using them appropriately in the contexts to which they apply; I identify each of these individual contexts as appropriate for using the words whenever I use the word. It is not at all necessary that I be able to give an explicit (dictionary) definition to demonstrate my understanding. In fact, if a person whose native language does not have articles (e.g. Chinese) comes later in life to learn English, they learn the dictionary definition and often can recite the definition better than most English people. Nonetheless, their USE of articles is often inappropriate; the dictionary rules just don't cover the subtleties of actual language use. That is, often our grasp of how to identify appropriate situations for using words is entirely implicit.

If all I could do was give a dictionary definition for every word I used, then I would be like the person in the Chinese Room: I would know how to relate a lot of symbols together, but I would have no understanding. Something must take the free-floating symbols and connect them to the world; that is, to "ground" them in the real world. There must be a mechanism that goes from our perceptual interaction with the world to the symbols (Harnad, 1990).

In general our understanding of the symbol is not exhausted by our explicit knowledge of the conditions of using the symbol (since such explicit knowledge just consists in uttering more symbols); that is, our understanding must be more than explicit definitions and descriptions. Therefore, the connection between perception and symbol consists of implicit knowledge. Here are some examples of how our perceptual mechanisms yield categories and labels in ways we cannot describe. By what features do you recognise particular people's faces? Consider what happens when someone has a haircut, or changes their glasses, or makes some other minor change; often we cannot say what has changed, at least for some moments. Nonetheless, what it is that has been changed has been registered somewhere inside us in order for us to know that there has been change. What makes a face beautiful? Even plastic surgeons have had a hard time discerning the relevant facial properties; it seems in fact that perception of beauty has a lot to do with the implicit perception of the so-called Golden Ratio, a particular mathematical relationship (Bates & Cleese, 2001). Much of the way we categorise the world is based on implicit knowledge.

Some ways by which we identify the referents of words, or identify the appropriate conditions for using a word, are implicit. But some ways are also explicit.  Sometimes we do appropriately use words and concepts by use of knowledge we can explicitly state. You might ask why did I call Tom a bachelor? I might answer that is because Tom is an unmarried adult male, thereby giving an explicit account of my use of the word bachelor.

Understanding a concept is for many concepts shown by being able to identify the referent of the concept in any of the contexts to which it is applicable. There must be methods of identification that involve implicit knowledge in order to connect with the real world; but other methods of identification can be explicit (Millikan, 1984). Sometimes we require both implicit and explicit identification methods. I might show my understanding of how to ride a bike by riding a bike. But I could still claim "I do not understand how I ride this bike" because I cannot say what I do to control it and stay upright, other than I just do it. So sometimes understanding involves making explicit our implicit knowledge. Likewise, when asked by a second-language student of English, why do I use a definite article there, I might after reflection consider that I do not really understand how definite articles are used - because I could not make explicit my implicit knowledge.

A person has a good grasp of some structure in mathematics if they can identify it across its different possible perceptual embodiments (and hence use that structure in at least many of the contexts in which that structure is relevant). When a person has a good understanding, such identification would in general consist of both implicit and explicit processes.

Understanding could initially consist of simply learning implicit and explicit procedures for dealing with the structure in many different contexts.  Developing such skills can be valuable per se.  But for my grandfather's methods, specific ways of dealing with particular embodiments are merely the stepping stones to developing an understanding of the concept per se. In fact, initially, these contextually-embedded processes do not identify the same structure as being the same in the different embodiments. The person simply acquires practical knowledge - implicit or explicit -  of dealing with the structure-in-context. For example, the person may in fact be dealing with a structure isomorphic to the symmetries of the cube, but the person only sees it as dealing with a situation involving, say, a father, daughter and son, and various interactions between them. The person at this point does not see the relation between this structure and the cube. Once the person has built up contextually-embedded knowledge about the structure in different embodiments, the person can be asked to map between the embodiments. When the person has successfully noticed the isomorphism between the embodiments, the perception of the commonality of the embodiments is the person's first grasp of the abstract mathematical structure itself. (This is what Millikan, 1998, also requires for us to have a concept: We must be able to identify it as the same thing, as we track it across different contexts.) When the person has noted the mapping between the embodiments, their knowledge of how to deal with each of the different contexts becomes identification procedures for the structure in question:  Now the particular embodiments are seen AS embodiments of the same structure. At this step, the person has acquired an understanding of the structure, an ability to see it in different contexts, viz those the person has just been exposed to.  This last step, of establishing the isomorphism, is very important, but sometimes missed out in student-centred contextual learning in practice.

Implicit and explicit knowledge in learning particular embodiments

Now I want to turn to some details of the role implicit and explicit knowledge plays as a person learns a mathematical structure in a particular concrete embodiment. Consider the person learning the symmetries of the cube but as embodied in a game about a father, a daughter, and a son, and their interactions. In one sense of implicit, the structure of the symmetries of the cube is implicit in the knowledge the person has about the father, daughter, and son; but this is a different sense of implicit than the one used in the implicit learning literature. The person does not have knowledge which is unconscious about the relation between the cube and the domain of the father, daughter and son; the person does not have any knowledge, conscious or unconscious, so no implicit knowledge of the cube is involved. The person just has knowledge about the structure of the family. The knowledge might be explicit (the person can say why certain operations yield certain results) or implicit (for example, the person might believe they are just guessing); but whatever its form, it is about the father, daughter, and son.  Is there an optimal mix of implicit and explicit learning in learning about a  particular embodiment?

Perhaps surprisingly, the development of implicit and explicit knowledge in children has only just started to be investigated by experimental psychologists. But it appears that in some cases, knowledge is originally learnt implicitly and only later is made explicit in the normal course of development (e.g. Karmiloff-smith, 1992; Clements & Perner, 1994; Ruffman et al, 2001). For example, Clements and Perner and Ruffman et al looked at how children understand that other people can have false beliefs. According to a story the children heard, a mouse falsely believes where some cheese is; when the mouse next looks, will he look where the child knows the cheese actually is, or where the mouse last saw the cheese placed? Children before about three years old state that the mouse will look where the cheese really is; older children appreciate the mouse can have a false belief, and state the mouse will look where the mouse last saw it placed. There is a period of a few months before they fully understand false belief, when children will state the mouse will look where the cheese actually is (that is, their verbal response fails to indicate the children understand false belief), but the children themselves spontaneously look to where the mouse should falsely believe the cheese is (i.e. the location children should indicate if they understand that others can have false beliefs). The looking was taken as an indication of developing implicit knowledge of false belief, implicit knowledge that presaged the later explicit knowledge. Further, Ruffman and his colleagues have shown that it is this initial implicit stage in understanding false belief that is missing in autistic children. These children can come to explicitly understand about other people’s beliefs and false beliefs, but it is an effortful hard process; most of us can just see intuitively how people can come to have true and false beliefs. Karmiloff-smith developed a general theory of learning in children; she argued that a typical sequence in many domains was for the child to first acquire implicit knowledge, and then, after the child has mastered the task behaviourally, the knowledge gradually becomes explicit.

Sometimes the development of knowledge goes in the opposite direction, from explicit and conscious to unconscious, or at least from consciously controlled to automatic. For example, when we learn to tie our shoelaces, we may laboriously work out each movement; but after many repetitions practicing the task we can perform it without thinking. Sometimes explicit knowledge may serve as a scaffold to help relevant implicit knowledge to develop, as when learning the grammar of a second language explicitly. It is an open question of what the optimal strategy is educationally in terms of the mix of implicit and explicit knowledge encouraged at different points in time (and with what individual differences?).  This is a question that experimental psychologists interested in implicit learning are only just beginning to investigate.

Mathews and colleagues (1988) tried to teach a type of rule structure to adults either in an implicit way, an explicit way, or a mixture of both. People were presented with strings of letters, like the following example:

XCSS.TPVV

There were always an ordered set of four letters, a period, then another ordered set of four letters. Any of the letters X, T, P, C, S and V could occur in any order in the first set. The letters in the second set were fixed by this choice. The letter in the first position of the first set determined the letter in the first position in the second set, the second position determined the second position, and so on. The rules were X goes with T, C with P, and S with V.

Mathews encouraged an implicit form of learning by not telling people there were rules and just asking them to try to remember particular strings for short periods of time. He encouraged an explicit form of learning by asking subjects to try to work out the rules and indicate the parts of a string where rules may be violated. Some subjects just learnt under conditions encouraging implicit learning; others under conditions encouraging explicit learning; and others learnt for half the time under implicit conditions followed by half the time under explicit conditions. It was the latter subjects, encouraged to learn first implicitly and then explicitly, that showed the greatest subsequent ability to classify new strings. Mathews points out that it is not known how general this synergistic effect of implicit and explicit learning is, but it is consistent with the claim that “forming a hypothesis too early may prevent best use of the information available to a person” (Mathews et al, 1988, p. 1098). Sometimes it is indeed useful to simply be exposed to stimuli and implicitly absorb the structure before trying to be analytical, even in adults.

The above results at least urge caution in the practice of making children approach tasks analytically as a matter of routine. Getting an intuitive feel for materials, or the way a game works, may be a vital preliminary stage for at least some children in at least some domains.

Knowledge is fully explicit when the person represents the knowledge as knowledge that they have  (see Dienes & Perner, 1999). In this case, the person can say with confidence e.g. that the next element in the sequence is a red triangle, or (at the next level) that the same structure applies in the different embodiments. If the person does not possess fully explicit knowledge, they may just believe they are guessing (even though they consistently answer questions correctly).

Work with adults learning artificial grammars indicates that often subjects come to be able to answer consistently and correctly while believing they have no knowledge whatsoever, they are simply guessing. With more experience on the task, their confidence comes to match their performance more and more closely, even when they are not given feedback by an external person as to whether they are being correct or not. When learning artificial grammars, people do not need verbal exhortations to be more or less confident in order to bring confidence in line with performance; simple experience is enough. Implicit knowledge will gradually be made explicit with experience (though often there remains a small or large residue of purely implicit knowledge).

Once a person has acquired various identification procedures for the concept-in-context, by my grandfather’s methods, they are then encouraged to attempt an appropriate mapping between the domains (to establish they can track the same structure across the contexts). To what extent can this happen implicitly or explicitly?

Implicit and explicit knowledge in forming an abstract concept

Often, when one really understands a concept one can generate novel explanations, novel applications, and identifications in novel contexts. This can happen more or less explicitly. In one way of achieving this, the person could simply determine analogies with existing embodiments that are known about. The mental procedures that determine these analogies constitute the person's understanding of the abstract structure itself. The knowledge is implicit in one sense: The abstract structure is represented implicitly in the procedures used in identifying the structure in different domains by the process of analogy. When using analogy, the person may be able to say that there is a common structure, without being able to say what the structure is, except by example.

Alternatively, the person could form an internal mental model, or representation, of the structure, where elements and relations are represented by variables that can be bound by different perceptual embodiments, and the model's function is to represent the particular structure abstractly. This constitutes an explicit representation of the structure itself.

So the abstract structure can be represented implicitly or explicitly as an abstract structure. In either case, the knowledge can be implicit or explicit as knowledge. For example, knowledge by analogy would be fully implicit as knowledge if the person did not know that analogy was the process being used and they believed they were just guessing. Alternatively, the person may be quite aware of using analogy and know that they have this knowledge.

Real mathematicians, of course, have explicit knowledge of explicit abstract structures. But real mathematical thinking rarely proceeds by strict formal manipulation of rules and symbols. Koedinger and Anderson (1990) found that expert mathematicians, even in planning something so outwardly formal as a mathematical proof, engaged in intuitive leaps of inference enabled by concrete, perceptually-based knowledge representations. Fully analytic knowledge of abstract structures is supported by perceptual knowledge of particular embodiments, even in the expert – perhaps, especially in the expert. Consistently, Heffernan & Koedinger (1997) propose what they call the inductive support hypothesis:  “Formal knowledge grows out of prior inductive experience which provides a semantic foundation in the form of back-up strategies for retrieving, checking, or rederiving formal knowledge.”

Thus, knowledge of different embodiments of a mathematical structure actually enables a person to find out more about the mathematical structure. Each embodiment provides it’s own type of inspiration and suggestions about properties of the structure.  Explicit knowledge of the embodiments helps in consciously thinking about the structure; implicit knowledge of the embodiments may play a role in intuitive leaps that seem to come from nowhere (Davies, 1992).

Conclusion

I have tried to provide a framework for seeing how the distinction between implicit and explicit knowledge relates to learning mathematics, especially in light of my grandfather’s methods. I have found it helpful to view having the concept of a certain mathematical structure to be the ability to track that structure across different embodiments of it (cf Millikan, 1998). This naturally leads to a method of education: Present the structure in different embodiments and also make sure the person identifies it as the same structure across the embodiments (and, my grandfather specifies: in that order, at least at a first pass at grasping the concept). A person can have means of identification that are implicit or explicit. Ultimately, there must be both forms of knowledge: The explicit to provide the analytical understanding, but also the implicit to ground the knowledge in the real world and give it meaning.

I am myself no expert in children’s understanding of mathematics in particular (I am an expert in implicit learning generally), but I am glad to see experts’ views that fit this approach. Nunes & Bryant (1996) provide an excellent overview of how children’s understanding of mathematics develops in definite ways that allow them eventually to track the same mathematical structures across different situations. Koedinger and others from Anderson’s research group at Carnegie Mellon have explicitly addressed the role of implicit knowledge and processes and perceptually-embedded knowledge in mathematical thinking and education. From a practical point of view, we have made great strides, but finding the right mix of encouraging implicit and explicit knowledge for different domains for different individuals is still in good measure up to the implicit knowledge of teachers.

References

Bates, B. & Cleese, J. (2001). The human face. BBC Consumer Publishing (Books).

Berry, D. C. and Dienes, Z.  (1993).  Implicit learning: Theoretical and empirical issues.  Hove: Lawrence Erlbaum.

Clements, W. & Perner, J. (1994). Implicit understanding of belief. Cognitive Development, 9, 377-397.

Davies, P. (1992). The mind of God: Science and the search for ultimate meaning. Penguin.

Dienes, Z., & Perner, J. (1999). A theory of implicit and explicit knowledge.  Behavioural and Brain Sciences, 22, 735-755.

Harnad, S. (1990) The Symbol Grounding Problem. Physica D 42: 335-346.

Heffernan, N. & Koedinger, K.R. (1997).  The composition effect in symbolizing: The role of symbol production vs. text comprehension.  In Proceedings of the Nineteenth Annual Conference of the Cognitive Science Society, (pp. 307-312). Hillsdale, NJ:  Erlbaum.

Karmiloff-Smith, A. (1992). Beyond Modularity. MIT Press, Cambridge, MA.

Koedinger, K.R., & Anderson, J.R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. Cognitive Science, 14, 511-550.

Mathews, R.C., Buss, R.R., Stanley, W.B., Blanchard-Fields, F., Cho, J.R., & Druhan, B. (1989).Role of implicit and explicit processes in learning from examples: A synergistic effect. Journal of Experimental Psychology: Learning, Memory & Cognition, 15, 1083-1100.

Millikan, R. G. (1984). Language, Thought, and Other Biological Categories. Bradford Books/MIT Press.

Millikan, R G.. (1998). A Common Structure for Concepts of Individuals, Stuffs, and Basic Kinds: More Mama, More Milk and More Mouse. Behavioral and Brain Sciences 22, 55-65.

Nunes, T.  & Bryant, P. (1996). Children doing mathematics. Blackwell.

Reber, A.S. (1993). Implicit learning and tacit knowledge: An essay on the cognitive unconscious. New York: Oxford University Press.

Ruffman, T., Garnham, W., Import, A., & Connolly, D. (in press). Does eye direction indicate implicit sensitivity to false belief?: Charting transitions in knowledge. Journal of Experimental Child Psychology.

Searle, J R. (1980). Minds, brains and programs. The Behavioral and Brain Sciences, 3, 417-457.

Csaba Pléh (member of the Hungarian Academy of Sciences and CEU), Psychology of problem-solving and its role in education: from sputnik shock to the web 

Ekaterina Babintseva (University of Pennsylvania, USA), Creative Thinking Control: Lev Landa and his Algo-Heuristic Theory

Zoltan Dienes (University of Sussex), The role of implicit and explicit knowledge in understanding mathematics

Katalin Munkácsy (Eötvös Loránd University) – Eleonóra Stettner (Kaposvár University), Computer Assisted Heuristic Learning of Analysis

Agnes Tuska (Department of Mathematics California State University, Fresno) Forum of International Projects and Collaborations

George Pólya and the Heuristic Tradition
Fascination with Genius in Central Europe

Previous part

„How To Solve It” 

Hungarian-born mathematician George Pólya (1887-1985) was one of those who channeled the Hungarian and, more broadly speaking, European school tradition into American education in a series of books and articles, starting with his 1945 book How to Solve It[19]. In 1944 Pólya remembered the time when, at the turn of the century in Hungary,

he was a student himself, a somewhat ambitious student, eager to understand a little mathematics and physics. He listened to lectures, read books, tried to take in the solutions and facts presented, but there was a question that disturbed him again and again: ‘Yes, the solution seems to work, it appears to be correct; but how is it possible to invent such a solution? Yes, this experiment seems to work, this appears to be a fact; but how can people discover such facts? And how could I invent or discover such things by myself?‘[20]

Pólya came from a distinguished family of academics and professionals. His father, Jakab, an eminent lawyer and economist provided the best education for his children. They included George‘s brother, Jenõ Pólya, the internationally recognized professor of surgery and honorary member of the American College of Surgeons[21]. George Pólya first studied law, later changing to languages and literature, then philosophy and physics, to settle finally on mathematics, in which he received his Ph.D. in 1912. He was a student of Lipót Fejér, whom Pólya considered one of the key people who influenced Hungarian mathematics in a definitive way.

Pólya felt that Fejér, the competitive examination in mathematics and the mathematical journal Középiskolai Matematikai Lapok were responsible for the development of a large number of major mathematicians in Hungary[22].

For emancipated Jews in Hungary, who received full rights as citizens in 1867, it was the Hungarian Law 1867:XII that made it possible, among other things, to become teachers in high schools and even professors at universities. This is one of the reasons that lead to the explosion of mathematical talent in Hungary, just as happened in Prussia after the emancipation of Jews in 1812[23]. John Horváth of the University of Maryland was one who pointed out the overwhelming majority of Jewish mathematicians in Hungary in the early part of the 20th century.

Culture in the second half of the nineteenth century became a matter of very high prestige in Hungary, where the tradition to respect scientific work started to loom large after the Austro-Hungarian Ausgleich (Compromise) in 1867 between Austria and Hungary. For sons of aspiring Jewish families, a professorship at a Budapest university or membership in the Hungarian Academy of Letters and Science promised entry into the Hungarian élite and eventual social acceptance in Hungarian high society, an acknowledged way to respectability. Pursuing scientific professions, particularly mathematics, secured a much desired social position for sons of Jewish-Hungarian families, who longed not only for emancipation, but for full equality in terms of social status and psychological comfort. Thus, in many middle class Jewish families, at least one of the sons was directed into pursuing a career in academe.

Distinguished scientists such as Manó Beke, Lipót Fejér, Mihály Fekete, Alfréd Haar, Gyula and Dénes König, Gusztáv Rados, Mór Réthy, Frigyes Riesz and Lajos Schlesinger belonged to a remarkable group of Jewish-Hungarian mathematical talents, who, after studying at major German universities, typically Göttingen or Heidelberg, became professors in Hungary‘s growing number of universities before World War I. Several of them, like Gyula König and Gusztáv Rados, even became university presidents at the Technical University of Budapest. There were several renowned scientists active in related fields, such as physicist Ferenc Wittmann, engineer Donát Bánki and others. Mathematicians were also needed outside the academoc world: just before the outbreak of World War I George Pólya was about to join one of Hungary‘s big banks, at the age of 26, with a Ph.D. in mathematics and a working knowledge of four foreign languages in which he already publisched important articles[24].

Despire what we know about the social conditions which nurtured and even forced out the talent of these many extraordinary scientists, how this occured still remains somewhat mysterious. Stanislaw Ulam recorded an interesting quote from John von Neumann when describing their 1938 journey to Hungary in his Advdentures of a Mathematician.

I returned to Poland by train from Lillafüred, traveling through the Carpathian foothills. ... This whole region on both sides of the Carpathian Mountains, which was part of Hungary, Czechoslovakia, and Poland, was the home of many Jews. Johnny [von Neumann] used to say that all the famous Jewish scientists, artists, and writers who emigrated from Hungary around the time of the first World War came, either directly or indirectly, from these little Carpathian communities, moving up to Budapest as their material conditions improved. The [Nobel Laureate] physicist I[sidor] I[saac] Rabi[25] was born in that region and brought to America as an infant. Johnny used to say that it was a coincidence of some cultural factors which he could not make precise: an external pressure on the whole society of this part of Central Europe, a feeling of extreme insecurity in the individuals, and the necessity to produce the unusual or else face extinction[26].

An interesting fact about Jewish-Hungarian geniuses at the turn of the century was that several of them could multiply huge numbers in their head. This was true of von Kármán, von Neumann and Edward Teller. Von Neumann, in particular commanded extraordinary mathematical abilities. Nevertheless, there is no means available to prove that this prodigious biological potential was more present in Hungary at the turn of the century than elsewhere in Europe[27].

Similarly, heuristic thinking was also a common tradition that many other Hungarian mathematicians and scientists shared. John Von Neumann‘s brother remembered the mathematician‘s „heuristic insights” as a specific feature that evolved during his Hungarian childhood and appeared explicity in the work of the mature scientist[28].

Von Neumann‘s famous high school director, physics professor Sándor Mikola, made a special effort to introduce heuristic thinking in the elementary school curriculum in Hungary already in the 1900s[29].

Fejér drew a numer of gifted students to his circle, such as Mihály Fekete, Ottó Szász, Gábor Szegõ and, later, Paul Erdõs. His students remembered Fejér‘s lectures and seminars as „the center of their formative circle, its ideal and focal point, its very soul”. „There was harcly an intelligent, let alone a gifted, student who could exempt himself from the magic of his lectures. They could not resist imitatting his stress patterns and gestures, such was his personal impact upon them.”[30] George Pólya remembered Fejér‘s personal charm and personal drive to have been responsible for his great impact: „F[ejér] influenced more than any other single person the development of math[ematic]‘s Hungary. ...”[31]

In Budapest, Pólya was one of the founders, along with Károly Polányi, of the student society called Galilei Kör [Galileo Circle], where he lectured on Ernst Mach. The Galileo Circle (1908-1918) was the meeting place of radical intellectuals, mostly Jewish college students from the up and coming Budapest families of a new bourgeoisie. Members of the circle became increasingly radical and politicized. Oddly enough, the Communists of 1919 found it far too liberal, while the extremist right-wing régime of Admiral Horthy considered it simply Jewish. In a Hungary of varied totalitarian systems, the radical-liberal tradition remained unwanted[32]. Soon however, Pólya went to Vienna where he served the academic year of 1911, after receiving his doctorate in mathematics in Budapest. In 1912-13 he wen to Göttingen, and later to Paris and Zurich, where he took an appointment at the Eidgenössiche Technische Hochschule (Swiss Federal Institute of Technology). He became full professor at the ETH in 1928.

A distinguished mathematician, Pólya drew on several decades of teaching mathematics based on new approaches to problem solving, first as a professor in Zurich, Switzerland, and later in his life at Stanford, California. It was in Zurich that Pólya and fellow Hungarian Gábor Szegõ started their long collaboration by signing a contract in 1923 to publish their much acclaimed joint collection of Aufgaben und Lehrsätze aus der Analysis[33]. Considered a mathematical masterpiece even today, Aufgaben und Lehrsätze took several years to complete, and it continues to impress mathematicians not only with the range and depth of the problems contained in it, but also with its organization: to group the problems not by subject but by solution method was a novelty[34]. His primary concern had always been to provide and maintain „an independence of reasoning during problem solving”[35], an educational goal he declared to be of paramount impoprtance when addressing the Swiss Association of Professors of Mathematics in 1931. Several of his articles on the subject preceded this lecture, probably the earliest being published in 1919[36]. Pólya had provided a model for problem solving by the time he was in Berne, Switzerland, suggesting „a systematic collection of rules and methodological advices”, which he considered „heuristics modernized”[37].

Pólya was active in a number of important fields of mathematics, such as theory probability, complex analysis, combinatorics, analytic number theory, geometry, and mathematical physics. In the United States after 1940, and at Stanford as of 1942, Pólya became the highest authority on the teaching of problem solving in mathematics.

With his arrival at the United States, Pólya started a new career based on his new found interst in teaching and in heuristics[38]. He developed several new courses such as his „Mathematical Methods in Science”, which he first offered in the Autumn 1945 Quarter at Stanford, introducing general and mathematical methods, deduction and induction, the relationship between mathematics and science, as well as the „use of physical intuition in the solution of mathematical problems”[39]. In his popular and often repeated Mathematics 129 course on „How to Solve the Problem?” Pólya taught mathematical invention and mathematical teaching, quoting Samuel Butler:

All the inventions that the world contains
Were not by reason first found out, nor brains
But pass for theirs, who had the luck to light
Upon them by mistake or oversight[40].

He surveyed all aspects of a problem, general and specific, restating it in every possible way and pursued various courses that might lead to solving it. He studied several ways to prove a hypothesis or modify the plan, always focusing on finding the solution. He compiled a characteristic list of „typical questions for this course”, which indeed contained his most important learnings from a long European shcooling[41].

In a course on heuristics he focused on problems and solutions, using methods from classical logic to heuristic logic. Offering the course alternately as Mathematics 110 and Physical Sciences 115, he sought to attract a variety of students, including those in education, psychology and philosophy[42]. The courses were based on Pólya‘s widely used textbook How to Solve It.

In due curse, Pólya published several other books on problem solving in mathematics such as the two-volume Mathematics and Plausible Reasoning (1954), and Mathematical Discovery, in 1965. Both became translated into many languages[43]. Towards the end of his career his „profound influence of mathematical education” was internationally recognized[44].

Pólya‘s significance in general methodology seems to have been his proposition to interpret heuristics as problem solving, more specifically, the search for those elements in a given problem that may help us find the right solution[45]. For Pólya, heuristics equaled „Enfindungskunst”, a way of inventive or imaginative power, the ability to invent new strategems of learning, and bordered not only on mathematics and philosophy but also psychology and logic. In this way a centuries-old European tradition was renewed and transplanted into the United States where Pólya had teremendous influence on subsequent generations of teachers of mathematics well into the 1970s. In 1971 the aged mathematician received an honorary degree at the University of Waterloo where he addressed the Convocation, appropriately calling for the use of „heuristic proofs”: „In a class for future mathematicians you can do something more sophisticated: You may peresent first a heuristic proof, and after that a strict proof, the main idea of which was foreshadowed by the heuristic proof. You may so do something important for your students: You may teach them to do research[46].” „Heuristics should be given a new goal”, Pólya argued, „that should in no way belong to the realm of the fantastic and the utopian”[47].

Problem wolving for Pólya was seen as „one third mathematics and two thirds common sense”[48]. This was a tactic which he emphatically suggested for teachers of mathematics in American high schools. If the teaching of mathematics neglects this tactic, he commented, it misses two important goals: „It fails to give the right attitude to future users of mathematics, and it fails to offer an essential ingredient of general education to future non-users of mathematics[49].”

Throughout his career as a teacher he strongly opposed believing in what authorities profess. Teachers and principals, he argued, „should use their own experience and their own judgment.[50]” His matter-of-fact, experience-based reasoning has been repeatedly described in books and articles. He even made two films on the teaching ot mathematics („Let Us Teach Guessing”, a prize winner at the American Film Festival in 1968; „Guessing and Proving”, based on an Open University Lecture, Reading, 1962)[51]. The most simple and straightforward summary of his ideas on teaching was presented in the preface of a course that he gave at Stanford and subsequently published in 1967. Pólya‘s description is the best introduction to heuristic thinking.

Start from something that is familiar or useful or challenging: From some connection with the world around us, from the prospect of some application, from an intuitive ide.

Don‘t be afraid of using colloquial alnguage when it is more suggestive than the conventional, precise terminology. In fact, do not introduce technical terms before the student can see the need for them.

Do not enter too early or too far into the heavy details of a proof. Give first a general idea or just the intuitive germ of the proof.

More generally, realize that the natural way to learn is to learn by stages: First, we want to see an outline of the subject, to perceive some concrete source or some possible use. Then, graudally, as soon as we can see more use and connections and interest, we take more willingly the trouble to fill in the details[52].

Pólya had lasting influence on a variety of thinkers in and beyond mathematics. The first curriculum recommendation of the National Council of the Teachers of Mathematics suggested that „problem solving be the focus of school mathematics in the 1980s [in the U.S.]. The 1980 NCTM Yearbook, published as Problem Solving in School Mathematics, the Mathematical Association of America‘s Compendia of Applied Problems and the new editor of the American Mathematical Monthly, P.R. Halmos, all called for more use of problems in teaching in 1980[53]. Philosophers Imre Lakatos, who described mathematical heuristics as his main field of interest in 1957, acknowledging his debt to Pólya‘s influence, and particularly to How to Solve It, which he translated into Hungarian.

Critics, however, like mathematician Alan H. Schoenfeld, pointed out that wile Pólya‘s influence extended „far beyond the mathematics education community”, „the scientific status of Pólya‘s work on problem solving strategies has been more problematic.” Students and instructors often felt that the heuristics-based approach rarely improved the actual problem-solving performance itself. Researchers in artificial intelligence claimed that they were unable to write problem solving programs using Pólya‘s heuristics. „We suspect the strategies he describes epiphenomenal rather than real”. Recent work in cognitive science, however, has provided methods for making Pólya‘s strategies more accessible for problem solving instruction. New studies have provided clear evidence that students can significantly improve their problem-solving performance through heuristics. „It may be possible to program computer knowledge structures capable of supporting heuristic problem-solving strategies of the type Pólya described.”

The Stanford Mathematics Competition

Initiated jointly by Professors George Pólya and Gabor Szegõ, one of the most significant Hungarian contributions to the teaching of mathematics was the introduction of the Stanford Mathematics Competition for high school students. Modelled after the Eötvös Competition organized in Hungary from 1894 on, the main purpose of the competition was to discover talent, and to revive the competitive spirit of the Eötvös Competition, which Szegõ himself won in 1912. This contest was held annually for over 30 years until it was terminated in 1928. Stress was laid on inherent cognitive ability and insight rather than upon memorization and speed. Those who were able to go beyond the question posed were given additional credit. Those who were cognizant of the preponderance of Hungarian mathematicians were tempted to speculate upon the relationship between the Eötvös Prize and „the mathematical fertility of Hungary”. Winners of the Eötvös Prize have included Lipót Fejér, Theodore von Kármán, Alfréd Haar, George Pólya, Frigyes Riesz, Gabor Szegõ, and Tibor Radó.

The Stanford competition was started in 1946 and discontinued in 1965 wen the Stanford Department of Mathematics turned more towards graduate training. When first started, the Stanford Ecamination was administered to 322 participants in 60 California high schools. The las examination, in 1965, was administered to about 1200 participants in over 150 larger schools in seven states from Nevada to Montana. The Stanford Univesity Competitive Examination in Mathematics emphasized „originality and insight rather than routine competence”. Even a typical question required a high degree of ingenuity and the winning student was asked „to demonstrate research ability”.

Organizers of the competition thought of mathematics „not necessarily as an end in itself, but as an adjunct necessary to the study of any scientific subject”. It was suggested that ability in mathematical reasoning correlated with succdess in higher education in any field. Also, the discovery of singularly gifted students helped identify the originality of mind displayed by grappling with difficult problems: mathematical ability was regarded as an index of general capacity. Those responsible for the competition were firmly convinced that Dan early manifestation of mathematical ability is a definite indication of excepcional intelligence and suitability for intellectual leadeship.” Several of the winners of the Stanford competition did not go into mathematics but went on to specialize in electrical engineering (1946), physics (1947), biology (1948) and geology (1956).

It is intersting to note that by introducing Pólya‘s article about the 1953 Stanford Competitive Examination, the California Mathematics Council Bulletin found it important to make a connection between „the best interests of democracy” and the need „that our superior students be challenged by courses of appropriate content, encouraged to prograss in accordance with their capacities”. It seems as if the Competitive Examination was viewed by some as reflecting the dangerously mounting international tensions, somewhat forecasting the era of the Sputnik fears yet to come. Speaking at the National Council of Teachers of Mathematics in 1956, Gábor Szegõ articulated this opinion when declaring that „much is said in these days about the pressing need for science and engineering graduates. Our view is that the nation needs just as well good humanists, lawyers, economists, and politicla scientists in its present struggle. This is a view which can be defended, I think, in very strong terms.” (This ominous reference was dropped from a similar introduction by 1957.)

Through its long and distinguished tenure the Stanford examination proved to be a pioneer in the discovery of mathematical talent not only in California and the West Coast, but nationally. To this day, George Pólya is best remembered in the United States as one who introduced European models of competitive educational methods of problem solving in mathematics. He served as one of the several bridges that linked Central European as well as distinctly Hungarian patterns of thinking and reasoning with American achievements in problem solving. The study of Pólya may reveal some of the Central European origins of heuristic thinking in the United States.

Notes

19. G. Pólya, How to Solve It. A New Aspect of Mathematical Method (Princeton, N.J.: Princeton University Press, 1945). How to Solve it has never been out of print and has sold well over 1 million copies. It has been translated into 17 languages, probably a record for a modern mathematics book. Gerald L. Alexanderson, „Obituary. George Pólya”, Bulletin of the London Mathematical Sociaty, Vol. 19, 1987, p. 563, 603.

20. G. Pólya, „How to Solve It”, op. cit., p. vi.

21. Vilmos Milkó, Pólya Jenõ emlékezete [In memoriam Jenõ Pólya], Archivum Chirurgicum, Vol. 1, No. 1, 1948, p. 1.

22. G. Pólya, „Leopold Fejér”, Journal of the London Mathematical Society, Vol. 36., 1961. p. 501.

23. R. Hersch and V. John-Steiner, „A Visit to Hungarian Mathematics”, Ms., pp. 35-37.

I received a copy of this arcitle from Professor Gerald L. Alexanderson of the Department of Mathematics, Santa Clara University, Santa Clara, CA. John Horváth compared this explosion of Jewish talent after the Jewish emancipation to the surprising number of sons of Protestant ministers entering the mathematical profession in Hungary after World War II, „Those kids would have become Protestant ministers, just as the old ones would have become rabbis... mathematics is the kind of occupation where you sit at your desk and read. Instead of reading the Talmud, you read proofs and conjectures. It‘s really a very similar occupation”. R. Hersch and V. John-Steiner, op. cit., p. 37.

24. György Pólya to Baron Gyula Madarassy-Beck, Paris, February 23, 1914. I am grateful to Professor Gerald Alexanderson of the University of Santa Clara for showing me this document as well as his collection of Pólya documents that were to be transferred to the George Pólya Papaers, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

25. Nobel Prize in Physics, 1944.

26. S. M. Ulam, Adventures of a Mathematician (New York: Scribner‘s, 1976), p. 11.

Cf. Tibor Fabian, „Carpathians Were a Cradle of Scientists”, Princeton, NJ. November 16, 1989, The New York Times, December 2, 1989. -- George Pólya‘s nephew John Béla Pólya had an even more surprising, though cautious proposition to make. He suggested that through George Pólya‘s mother, Anna Deutsch (1853-1939), Pólya was realted to Eugene Wigner and Edward Teller, „who are thought to have” acestry originating from the same region between the towns of Arad and Lugos in Transsylvania (the Hungary, today Roumania). Though this relationship is not yet documented and should be taken at this point merely as a piece of Pólya family legend, it is nonethelles an interesting reflexion of the strong belief in the productivity of the Jewish community in North-Eastern Hungary and Transsylvania in terms of maethematical talent. John Béla Pólya, „Notes on George Pólya‘s family”, attached to John Béla Pólya to Gerald L. Alexanderson, Greensborough, Australia, July 28, 1986. -- I am deeply grateful to Gerald L. Alexanderson of Santa Clara University, Santa Clara, CA. for his generous and highly informative support of my research on George Pólya.

27. Norman Macrae, John von Neumann (New York: Pantheon, 1992), p. 9. J. M. Rosenberg, Computer Prophets (New York: Macmillan, 1969) p. 155. ff. Edward Teller and Alan Brown, The Legacy of Hiroshima (Garden City: Doubleday, 1962) p. 160. Cf. William O. McCagg, op. cit.. 211.

28. Nicholas A. Vonneuman, John Von Neumann as Seen by His Brother(Meadowbrook, PA, 1987), p. 44.

29. Sándor Mikola, „Die heuristischen Methode in Unterricht der Mathematik der unteren Stufe”, in: E. Beke und. S. Mikola, Hrg., Abhandlungen über die Reform des mathematischen Unterrichts in Ungarn (Leipzig und Berlin: Teubner, 1911), pp. 57-73.

30. Gábor Szegõ, „[Lipót Fejér],”, MS. Gábor Szegõ Papers, SC 323, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

31. [Lecture outline, n.d. unpublished MS] George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

32. The full history of the Galileo Circle is yet to be written. Important documents were published by Zoltán Horváth in 1971: „A Galilei Körre vonatkozó ismeretlen dokumentumok”, Századok, Vol. 105, No. 1, 1971, pp. 95-104. Cf. György Litván, Magyar gondolat--szabad gondolat [Hungarian Thought-- Free Thought] (Budapest: Mwagvetõ, 1978); cf. György Litván, „Jászi Oszkár, A magyar progresszió és a nemzet”, [Oscar Jaszi, Hungarian Progressives and the Nation] in Endre Kiss, Kristóf János Nyíri, eds., A magyar gondolkodás a századelõn [Hungarian Philosophy at the Turn of the Century], (Budapest: Kossuth, 1977). Litván pointed out that while similar social science organizations, such as Társadalomtudományi Társaság or Húszadik Század had a fair number of gentile contributors, the Galileo Circle almost exclusively drew upon progressive Jewish students. Cf. Attila Pók, A magyarországi radikális demokrata ideológia kialakulása. A „Huszadik Század” társadalomszemléléete (1900-1907) [The Rise of Democratic Radicalism in Hungary: the Social Concept of Huszadik Század (1900-1907)] (Budapest: Akadémiai Kiadó, 1990) p. 152-165.

33. Georg Pólya-Gábor Szegõ, Aufgaben und Lehrsätze aus der Analysis (Berlin: Springer, 1925, new editions: 1945, 1954, 1964, 1970-71), Vols. I-II; Translations: English, 1972-76; Bulgarian, 1973; Russian, 1978; Hungarian, 1980-81.

34. Gerald L. Alexanderson, „Obituary. George Pólya”, op. cit., pp. 562.

35. G. Pólya, „Comment chercher la solution d‘un problème de mathématiques?” L0enseignement mathématique, 30e année, 1931, Nos. 4-5-6.

36. G. Pólya, „Geometrische Darstellung einer Gedankenkette”, Schweizerische Pädagogische Zeitschrift, 1919.

37. G. Pólya, „Comment chercher”, op. cit.

38. Gerald L. Alexanderson, „Obituary. George Pólya”, Bulletin of the London Mathematical Society, Vol. 19, 1987, p. 563; on „Pólya the mathematician and teacher”, see pp. 566-570.

39. Paul Kirkpatrick, Acting Dean, School of Physical Sciences, Stanford University, Course outline, September 4, 1945, George Pólya Papaers, SC 337, 87-137, Box 2, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

40. G. Pólya, „Elementary Mathematics from Higher Point of View”, Mathematics 129, George Pólya Papers, SC 337, 86-036, Box 1, Folder 9, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA. Cf. Samuel Butler, „Miscellaneous Thoughts”, in: The Poems of Samuel Butler, Vol. II. (Chiswick: C. Willingham, 1822), p. 281.

41. G. Pólya, „Elementary Mathematics from a Higher Point of View”, Survey of Typical Questions, George Pólya Papers, SC 337, 87-147, Box 3, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA. -- Pólya was indeed very well read and liked to show his erudition by quoting Socrates, Descartes, Leibniz, Kant, Herbert Spencer, Thomas Arnold, J. W. Goethe, Leonhard Euler and his famous colleagues, such as Albert Einstein, adn many others. George Pólya Papers, SC 337, 87-034. Box 1&3, 87-147, Box 2, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

42. Untitled course description, nd. George Pólya Papers, SC 337, 86-036, Box 1, Folder 3, Department of Special Collekctions and University Archives, Stanford University Libraries, Stanford, CA.

43. G. Pólya Mathematics and Plausible Reasoning (Princeton, N.J.: Princeton University Press, 1954, 2nd ed. 1968), Vols. 1-2. Translations: Bulgarian, 1970; French, 1957-58; German, 1962-63; Japanese, 1959; Roumanian, 1962; Russian, 1957, 1975; Spanish, 1966; Turkish, 1966. G. Pólya, Mathematical Discovery. On Understanding, Learning and Teaching Problem Solving(New York - London - Sidney: John Wiley and Sons, 1965, Vols. 1-2; combined paperback ed. 1981), Translations: Bulgarian, 1968; French, 1967; German, 1966, 1967, 1979, 1983; Hungarian, 1969, 1979, Italian, 1970-71, 1979, 1982; Japanese, 1964; Polish, 1975; Roumanian, 1971; Russian 1970, 1976, Cf. Gerald L. Alexanderson, „Obituary, George Pólya”, op. cit. pp. 604-605.

44. A good example was the Second International Congress on Mathematical Education at the University of Exeter, England. Cf. The invitation sent to Pólya by the Chairman of the Congress, Professor Sir James Lighthill, FRS, June 24, 1971. [Cambridge] George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

45. G. Pólya, „Die Heuristik. Versuch einer vernünftigen Zielsetzung”, Der Mathematikunterricht, Heft 1/64 (Stuttgart: Ernst Klett, 1964); cf. ‘L‘Heuristique est-elle un sujet d‘étude raisonnable?‘, La méthode dans les sciences modernes, ‘Travail et Méthode‘, numéro hors série, pp. 279-285.

46. G. Pólya, „Guessing and Proving”. Adddress delivered at the Convocation of the University of Waterloo, October 29, 1971. George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

47. G. Pólya, „Die Heuristik”, op. cit., p.5

48. George Pólya, Untitled note, n.d., George Pólya Papers, SC 337, 87-034, Box 1, department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

49. G. Pólya, „Formation, Not Only Information”, Address at the Mathematical Association of America, George Pólya Papers, SC 337, 87--034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

50. George Pólya tob Robert J. Griggin, Stanford, June 12, 1962. George Pólya Papers, SC 337, 87-034, Box 2, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

51. George Pólya to Anthony E. Mellor, Harper and Row, Stanford, March 11, 1974; Stanford University News Service, Februar 17, 1969. George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.

52. George Pólya, „Preface”, MS George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and Unviersity Archives, Stanford University Libraries, Stanford, CA.

53. Imre Lakatos to Dr Maier (Rockefeller Foundation), Cambridge, England, May 5, 1957. George Pólya Papers, SC 337, 87-137, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.--In turn, Pólya expressed his admiration for Lakatos‘s „Proofs and Refutations”, and recommended him as Professor of Logic at the London School of Economics and Political Science, „with special reference to the Philosophy of Mathematics”. George Pólya to Walter Adams (LSE), Stanford, CA, January 13, 1969, George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.