Optativo 2 Final Proceedings Mathematical Creativity

8/9/2019 Optativo 2 Final Proceedings Mathematical Creativity

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RESEARCH REPORTS

Proceedings of The 5thInternational Conference on Creativity in Mathematicsand the Education of Gifted Students.

Haifa, Israel, February 24-28, 2008

MATHEMATICAL CREATIVITY: IN THE WORDS OF

THE CREATORS

PETER LILJEDAHL

Simon Fraser University, Canada

What is the genesis of mathematics? What mechanisms govern the act ofmathematical creativity? This"is a problem which should intensely interest

the psychologist. It is the activity in which the human mind seems to takethe least from the outside world, in which it acts or seems to act only ofitself and on itself" (Poincar, 1952, p. 46). It should also intensely interestthe mathematics educator, for it is through mathematical creativity that wesee the essence of what it means to 'do' and learn mathematics. In this articleI explore the topic of mathematical creativity along two fronts, the first ofwhich is a brief synopsis of the history of work in this area. This is thenfollowed by a glimpse at a study designed to elicit from prominentmathematicians ideas and thoughts on their own encounters withmathematical creativity.

HISTORICAL BACKGROUND

In 1908 Henri Poincar (18541912) gave a presentation to the FrenchPsychological Society in Paris entitled 'Mathematical Creation'. This presentation,as well as the essay it spawned, stands to this day as one of the most insightful, andthorough treatments of the topic of mathematical creativity and invention. Inspiredby this presentation, Jacques Hadamard (1865-1963) began his own empiricalinvestigation into mathematical invention. The results of this seminal workculminated in a series of lectures on mathematical invention at the cole Libre desHautes Etudes in New York City in 1943. These talks were subsequently publishedas The Psychology of Mathematical Invention in the Mathematical Field(Hadamard,1945).

Hadamard's treatment of the subject of invention at the crossroads of mathematicsand psychology is an extensive exploration and extended argument for the

existence of unconscious mental processes. To summarize, Hadamard took theideas that Poincar had posed and, borrowing a conceptual framework for thecharacterization of the creative process in general, turned them into a stage theory.This theory still stands as the most viable and reasonable description of the process


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of mathematical invention. In what follows I present this theory, referenced notonly to Hadamard and Poincar, but also to some of the many researchers who'swork has informed and verified different aspects of the theory.

MATHEMATICAL INVENTION

The phenomenon of mathematical invention, although marked by suddenillumination, consists of four separate stages stretched out over time, of whichillumination is but one part. These stages are initiation, incubation, illumination,

and verification (Hadamard, 1945). The first of these stages, the initiation phase,consists of deliberate and conscious work. This would constitute a person'svoluntary, and seemingly fruitless, engagement with a problem and becharacterized by an attempt to solve the problem by trolling through a repertoire ofpast experiences (Bruner, 1964; Schn, 1987). This is an important part of theinventive process because it creates the tension of unresolved effort that sets up theconditions necessary for the ensuing emotional release at the moment ofillumination (Davis & Hersh, 1980; Feynman, 1999; Hadamard, 1945; Poincar,1952; Rota, 1997). Following the initiation stage the solver, unable to come to asolution stops working on the problem at a conscious level (Dewey, 1933) andbegins to work on it at an unconscious level (Hadamard, 1945; Poincar, 1952).This is referred to as the incubation stage of the inventive process and it isinextricably linked to the conscious and intentional effort that precedes it. After theperiod of incubation a rapid coming to mind of a solution, referred to asillumination, may occurs. This is accompanied by a feeling of certainty (Poincar,1952) and positive emotions (Barnes, 2000; Burton 1999; Rota, 1997). Withregards to the phenomenon of illumination, it is clear that this phase is themanifestation of a bridging that occurs between the unconscious mind and theconscious mind (Poincar, 1952), a coming to (conscious) mind of an idea orsolution. However, what brings the idea forward to consciousness is unclear. Thereare theories on aesthetic qualities of the idea (Sinclair, 2002; Poincar, 1952),effective surprise/shock of recognition (Bruner, 1964), fluency of processing(Whittlesea and Williams, 2001), or breaking functional fixedness (Ashcraft,1989). Regardless of the impetus, the correctness of the emergent idea is evaluatedduring the fourth and final stage verification.


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HADAMARD RESURECTED

For the study presented here a portion of Hadamard's original questionnaire wasused to elicit from contemporary mathematicians ideas and thoughts on their ownencounters with the phenomenon of mathematical discovery1. Hadamard's originalquestionnaire contained 33 questions pertaining to everything from personal habits,to family history, to meteorological conditions during times of work (Hadamard,1945). From this extensive and exhaustive list of questions the five that mostdirectly related to the phenomena of mathematical discovery and creation were

selected. They are: Would you say that your principle discoveries have been the result of deliberate

endeavour in a definite direction, or have they arisen, so to speak,spontaneously? Have you a specific anecdote of a moment ofinsight/inspiration/illumination that would demonstrate this? [Hadamard # 9]

How much of mathematical creation do you attribute to chance, insight,inspiration, or illumination? Have you come to rely on this in any way?[Hadamard# 7]

Could you comment on the differences in the manner in which you work whenyou are trying to assimilate the results of others (learning mathematics) ascompared to when you are indulging in personal research (creatingmathematics)? [Hadamard # 4]

Have your methods of learning and creating mathematics changed since you

were a student? How so? [Hadamard # 16]

Among your greatest works have you ever attempted to discern the origin of theideas that lead you to your discoveries? Could you comment on the creativeprocesses that lead you to your discoveries? [Hadamard # 6]

These questions, along with a covering letter, were then sent to 150 prominentmathematicians (see below) in the form of an email.

Hadamard set excellence in the field of mathematics as a criterion for participationin his study. In keeping with Hadamard's standards, excellence in the field ofmathematics was also chosen as the primary criterion for participation in this study.

1I would like to acknowledge the contribution made by Peter Borwein with respect to thecollection of data for this study.


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As such, recipients of the survey were selected based on their achievements in theirfield as recognized by their being honored with prestigious prizes or membershipin noteworthy societies. In particular, the 150 recipients were chosen from thepublished lists of the Fields Medalists, the Nevanlinna Prize winners, as well as themembership list of the American Society of Arts & Sciences. The 25 recipients,who responded to the survey, in whole or in part, have come to be referred to as the'participants' in this study.

The responses were initially sorted according to the survey question they weremost closely addressing. In addition, a second sorting of the data was done

according to trends that emerged in the participants' responses, regardless of whichquestion they were in response to. This was a much more intensive and involvedsorting of the data in an iterative process of identifying themes, coding for themes,identifying more themes, recoding for the new themes, and so on. In the end, therewere 12 themes that emerged from the data, each of which can be attributed to oneof the four stages presented in the previous section. Some of these themes serve toconfirm the work of Hadamard while others serve to extend it. Given the limitedspace available here, in what follows I provide a verybrief summaryof three of thethemes that extend our understanding of various stages of the inventive process.

The De-Emphasis of Details

A particularly strong theme that emerged from this study was the role that detail

does NOT play in the incubation phase. Many of the participants mentioned howdifficult it is to learn mathematics by attending to the details, and how much easierit is if the details are de-emphasized.

Stephen: Understanding others is often a painful process until one suddenly goesbeyond the details and sees whole what's going on.

Mark: There is not much difference. More precisely, I seldom study or learnmathematics in detail.

In some cases, this also manifested itself as a strategy for problem solving andresearch.

Carl: Get the basics of the problem firmly and thoroughly into the head. Afterthat, an hour or two each day of thinking on it is all that's needed for

progress. [..] For that reason, I started some 20 years ago to ask students(and colleagues) wanting to tell me some piece of mathematics to tell medirectly, perhaps with some gestures, but certainly without the aid of ablackboard. While that can be challenging, it will, if successful, put theproblem more firmly and cleanly into the head, hence increases thechances for understanding. I am also now more aware of the fact that


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others under the name of "the chance hypothesis" and is nicely demonstrated inDan's comment.

Dan: And relevant ideas do pop up in your mind when you are taking a shower,and can pop up as well even when you are sleeping, (many of these ideasturn out not to work very well) or even when you are driving. Thus whileyou can turn the problem over in your mind in all ways you can think of,try to use all the methods you can recall or discover to attack it, there isreally no standard approach that will solve it for you. At some stage, ifyou are lucky, the right combination occurs to you, and you are able to

check it and use it to put an argument together.Extrinsic chance, on the other hand, deals with the luck associated with a chancereading of an article, a chance encounter, or a some other chance encounter with apiece of mathematical knowledge, any of which contribute to the eventualresolution of the problem that one is working on. This is best demonstrated by thewords of Mark and Carl.

Mark: Do I experience feelings of illumination? Rarely, except in connectionwith chance, whose offerings I treasure. In my wandering life betweenconcrete fields and problems, chance is continually important in twoways. A chance reading or encounter has often brought an awareness ofexisting mathematical tools that were new to me and allowed me to returnto old problems I was previously obliged to leave aside. In other cases, a

chance encounter suggested that old tools could have new uses that helpedthem expand.

Carl: But chance is a major aspect: what papers one happens to have read, whatdiscussions one happens to have struck up, what ideas one's students arestruck by (never mind the very basic chance process of insemination thatproduced this particular mathematician).

Once again, the idea that mathematical discovery often relies on the fleeting andunpredictable occurrences of chance encounters is starkly contradictory to theimage projected by mathematics as a field reliant on logic and deductive reasoning.Extrinsic chance, in particular, is an element that has been largely ignored in theliterature.

CONCLUSIONSMathematical creativity and invention are aspects of 'doing' mathematics that havelong been accepted as standing outside of the theories of "logical forms" (Dewey,1938, p.103). That is, they rely on the extra-logical processes of insight andillumination as opposed to the logical process of deductive and inductive


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reasoning. This study confirms this understanding as well as adds to this cohort ofextra-logical processes the role of chance. In addition, this study also comments onthe initiation phase of the inventive process in showing that it is best facilitatedthrough the de-emphasizes of details and transmission of knowledge throughtalking and (re)creation. As such, this study also contributes to our understandingof the larger contexts of 'doing' and learning mathematics.

REFERENCES

Ashcraft, M. (1989).Human memory and cognition. Glenview, Illinois: Scott, Foresman and

Company.Bruner, J. (1964).Bruner on knowing. Cambridge, MA: Harvard University Press.

Burton, L. (1999). The practices of mathematicians: What do they tell us about coming toknow mathematics?Educational Studies in Mathematics, 37(2), 121-143.

Davis, P. & Hersch, R. (1980). The mathematical experience. Boston, MA: Birkhauser.

Dewey, J. (1933).How we think. Boston, MA: D.C. Heath and Company.

Dewey, J. (1938).Logic: The theory of inquiry. New York, NY: Henry Holt and Company.

Feynman, R. (1999). The pleasure of finding things out. Cambridge, MA: PerseusPublishing.

Hadamard, J. (1954). The psychology of invention in the mathematical field. New York, NY:Dover Publications.Poincar, H. (1952). Science and method. New York, NY: DoverPublications, Inc.

Poincar, H. (1952). Science and method.New York, NY: Dover Publications Inc.Rota, G. (1997).Indiscrete thoughts.Boston, MA: Birkhauser.

Schn, D. (1987). Educating the reflective practitioner. San Fransisco, CA: Jossey-BassPublishers.

Sinclair, N. (2002). The kissing triangles: The aesthetics of mathematical discovery.International Journal of Computers for Mathematics Learning, 7(1), 45-63.

Whittlesea, B. & Williams, L. (2001). The discrepancy-attribution hypothesis: The heuristicbasis of feelings of familiarity.Journal for Experimental Psychology: Learning, Memory,and Cognition, 27(1), 3-13

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RESEARCH REPORTS

Proceedings of The 5thInternational Conference on Creativity in Mathematics

and the Education of Gifted Students.Haifa, Israel, February 24-28, 2008

CONCEPTIONS OF GIFTEDNESS IN SCHOOL

MATHEMATICS A SWEDISH CONTEXT

LINDA MATTSSON

Mathematical Sciences, Gteborg University, Sweden

Abstract

This report presents an ongoing study focusing on conceptions ofmathematical giftedness as these are expressed in three different, but closelyrelated Swedish domains: the public domain, the research domain, and thedomain of practicing teachers. These domains will be examined with respectto the prevailing debates on mathematical giftedness, as well as to conceptsused and issues of interest to the parties concerned.

At the time of the conference, preliminary results from the ongoing project

will be presented.

INTRODUCTION

One factor which makes the national discussion about high ability students inSwedish schools difficult is the fact that we do not have an accepted usage of termsto denote and characterise high ability students. Thus conceptions of giftedness areinterpreted and expressed differently in various settings. Further, conceptions ofgiftedness are seldom made clear, leaving debates held and assertions made hard tofollow and even harder to refute. These vague conceptions of giftedness may leadto a situation when biases against high ability students, as well as myths about thisgroup, will make the discussion even more blurred. Thus in order to promote amore structured educational development for high ability students in mathematics,we need to gain a more thorough understanding about the prevalent conceptions

held by the parties engaged in discussions about education for mathematicallygifted students.


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The modern study of giftedness has a long past (Mnks, Heller & Passow, 2000),and the definition of giftedness has been expressed in a variety of ways (Mnks &Mason, 2000). Most of the recent models of conceptions of giftedness are liberal,and researchers agree that there is such a thing as specific giftedness (Mayer,2005). However, the conceptions of mathematical giftedness may be described in avariety of ways (Wieczerkowski, Cropley & Prado, 2000), but are often defined asindividual abilities in mathematics (for an overview see Sriraman, 2005). Addingto the variety of terms and characterizations used, are the many conceptions used inpractical settings (see e.g. European Commission, 2006). Hence we end up with anumber of concepts and conceptions used for describing and providing forgiftedness. To summarize, there exist many interpretations of giftedness, manycontroversies of how to provide for the group of high ability students, as well as alaxity of terminology for denoting this group (see e.g. Gang, 1995). However, inspite of the many unsolved and remaining questions, there is a major differencebetween the uncertainty as expressed in the international debate about giftedstudents and the conceptions of mathematical giftedness, and the wavering andvague discussion of mathematical giftedness in Sweden. The ongoing Swedishstudy on gifted education in mathematics will bring some clarity to this field.

FOCUS OF RESEARCH

In the ongoing study, focus is put on conceptions of mathematical giftedness inschool as these are expressed within three domains the public domain, theresearcher domain, and the domain of practicing teachers. The public domain is anarea dominated by political actors, government agencies, and other authorities.Within the research domain we find researchers involved in the investigation anddiscussion about gifted education in Sweden. In the domain of practicing teacherswe find the agents who have the responsibility to arrange educational activities forall students.

The object of this study is to identify the prevalent debates and issues ofinterests regarding mathematical giftedness within each of these domains. A furtherpurpose is to elucidate underlying conceptions of giftedness held within eachdomain, and specifically to investigate the conceptions of mathematical giftedness.

METHOD

Data collection

Data will be gathered from selected sources representing the three domains. Thepublic domain expresses itself through e.g. curricula, recommendations andgovernmental documents. Data will thus be collected from such sources, syllabusesof mathematics, budget proposals, as well as, from documents from such as


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authorities as The Swedish National Agency for School Improvement and TheSwedish National Agency for Education.

The research field refers to contributions to the field of gifted education and to thenational debate on gifted education in Sweden. Since such research contribution israre, this study will comprise all research efforts. Data will be collected fromjournals, magazines, public newspapers, and applications for funding.

The domain of practicing teachers refers to debates and discussions held amongteachers working at the upper secondary school level. Information about thediscussions in this domain will be gathered in three different ways: throughquestionnaires sent to randomly selected mathematics teachers in the Swedishupper secondary school, through recordings of informal talks with teachers atselected schools, and through studies of applications for funding for educationaldevelopment projects.

Analysis

Data will be analysed using Content analysis (Weber, 1990; Krippendorf 1980).This means that each of the given texts, i.e. documents and interviews, will bebroken down and coded into different themes of concepts of mathematicalgiftedness or high ability in mathematics. Using relational analysis, the conceptsare then explored in order to find meaningful relationships between them. Theresult of the analysis of the content within each theme is then compared to thetheme content obtained from the other sources of data, thus making it possible to

emphasise general themes and to draw conclusions about variations of conceptionsof mathematical giftedness within and among domains.

At the time of the conference, preliminary results from the ongoing project will be

presented.

REFERENCES

European Comission.(2006). Specific educational measures to promote all forms ofgiftedness at school in Europe. (Working document).

Gagn, F. (1995). From giftedness to talent: A developmental model and its impact on thelanguage of the field.Roeper Review, 18 (2), 103-111.

Krippendorf, K. (1980). Content analysis: An introduction to its methodology.Beverly Hills:Sage Publications.

Mayer, R. E. (2005). The scientific study of giftedness. In Sternberg, R. J. & Davidson, J.E.(Eds.), Conceptions of giftedness. 2nded. (pp. 437-447). New York: Cambridge.Mnks, F. J., Heller, K. A. & Passow, A. H. (2000). The study of giftedness: Reflections on

where we are and where we are going. In Heller, K. A., et al. (Ed.), Internationalhandbook of giftedness and talent. 2nded. (pp. 839-863). Oxford: Elsevier.


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Mnks, F. J. & Mason, E. J. (2000). Developmental psychology and giftedness: Theoriesand research. In Heller, K. A., et al. (Ed.), International handbook of giftedness andtalent.2nded. (pp. 141-155). Oxford: Elsevier.

Sriraman, B. (2005). Are giftedness and creativity synonyms in mathematics?. The Journalof Secondary Gifted Education, 1, 20-36.

Weber, R. P. (1990). Basic content analysis, second edition. Newbury Park: SagePublications.

Wieczerkowski, W., Cropley, A. J. & Prado, T. M. (2000). Nurturing talents/gifts inmathematics. In Heller, K. A., et al. (Ed.), International handbook of giftedness andtalent.2nded. (pp. 141-155). Oxford: Elsevier


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CREATIVITY OR GIFTEDNESS?

HARTWIG MEISSNER

Westf. Wilhelms-Univ. Muenster, Germany

Abstract

We confront young children and adults with mathematical challenges andobserve their behavior. Some problem solvers are creative, some are gifted,some are both. But what does it mean to be creative or to be gifted? Howcan we distinguish? Is it necessary at all to distinguish? Analyzing themental processes we discover two types of mental activities or behavior. Onthe one hand we distinguish a logical and conscious mode of action and onthe other an intuitive and mainly unconscious kind of behavior. Theysometimes complement one another but sometimes also conflict with eachother. Analyzing an example where different individuals were working onthe same mathematical challenge we can identify explicitly those differentmental processes. In the oral presentation at the conference we will presentsome more examples.

CREATIVITY

What does creativity mean? Many experts from different disciplines give variousdescriptions, but there is no standardized answer (Meissner 2003, 2005) 1.Creativity is a highly complex phenomenon, creativity cannot be described via along list of isolated items. To develop and further creativity in mathematicseducation teachers and students need more than a correct and solid mathematicalknowledge. Specific environments are necessary.

In our research group in Muenster we try to concentrate on three aspects. First, wemust further individual and social components, like motivation, curiosity, self-confidence, flexibility, engagement, humor, imagination, happiness, acceptance of

1see also http://wwwmath1.uni-muenster.de/didaktik/u/meissne/WWW/creativity.htm


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oneself and others, satisfaction, success, ... We need a competitive atmospherewhich still allows spontaneous actions and reactions, we need responsibilitycombined with voluntariness, ... We need tolerance and freedom, for theindividuals to express their views, and within the group to further a fruitfulcommunication. We need profound discussions as well as intuitive or spontaneousinputs and reactions.

Second, we need challenging problems. They must be fascinating, interesting,exciting, thrilling, important, or provoking. Open ended problems are welcome orchallenging problems with surprising contexts and results. We must connect theproblems with the individual daily life experiences of the students, we must meet

their fields of experiences and their interest areas. The students must be able toidentify themselves with the problem and its possible solution(s).

And third, the children must develop important abilities. They must learn toexplore and to structure a problem, to invent own or to modify given techniques, tolisten and argue, to define goals, and to cooperate in teams. We need children whoare active, who discover and experience, who enjoy and have fun, who guess andtest, and who can laugh on own mistakes. That means, another step to furthercreative thinking is to further the development of these abilities. But they aredemanding abilities and not simple skills. They rely and depend on a complexsystem of cognitive processes.

GIFTEDNESS

aboveaverage ability AC

(top 15-20%) creativityA C

ACT

GIFTEDNESSAT CT

T

task commitment& motivation

Fig. 1. Three-Ring Model of Giftedness (Renzulli)


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Similar to creativity there are many different descriptions or definitions of whatgiftedness might mean, see for example Sternberg e. a. (2005), Heller e. a.(1993), or Kiesswetter e. a. (1988). Probably the most common model of giftednessis the Three-Ring Model from Renzulli, see Fig. 1. To describe giftedness Renzulliconcentrates on the three components

G1 aboveaverage ability (upper 15 20%)

G2 creativity (original thinking, ingenuity, divergent thinking, ...)

G3 task commitment and intrinsic motivation (energy and perspiration for asuccessful problem solving and for mastering specific performance areas,

enjoying challenges, ...)Putting these three interlocking clusters of traits together, giftedness then appearsas the interaction among these three clusters. And with regard to the topic of thisconference it is interesting to see that creativity is one of the necessary conditionsfor giftedness. But looking at Fig. 1 we also can see that creativity in RenzullisThree-Ring Model is not necessarily a subset, neither of giftedness, nor of aboveaverage abilities nor of task commitment & motivation. Summarizing, creativityand giftedness are strongly related to each other. But also non gifted children canbe creative2.

We also see the strong relationship between creativity and giftedness when wecompare the properties for creativity described above with the indicators formathematical giftedness given by Kaepnick (1998). According to Kaepnick

mathematical giftedness is characterized by certain mathematics related abilities(K1) and general human aspects (K2):

K1 Necessary mathematics related abilities are . . .being mathematically sensitive (for numbers, figures, operations, structures,esthetical aspects),being original and having fantasy in mathematical activities,remembering mathematical facts,ability to structure mathematical facts,ability to switch levels of representation,reversible thinking and transfer,visual / spatial thinking.

2Several examples are given in the Proceedings from the recent conferences on creativityin mathematics education.


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K2 Necessary general human aspects are . . .being highly mentally active,being intellectually inquisitive,task commitment and motivation,enjoying problem solving,ability to work with concentration,perseverance,independence,ability to cooperate.

Comparing these K1 and K2 properties for giftedness with the properties for

creativity we see a close connection. We therefore will analyze the relatedcognitive processes.

MENTAL PROCESSES

A successful problem solving depends on the cognitive structure the problemsolver has. An appropriate internal representation or concept image isnecessary, a powerful Vorstellung(MEISSNER2002). These Vorstellungen3are likemental scripts or frames or micro worlds. They are personal and individual. Wedistinguish two kinds of Vorstellungen, which we call spontaneous Vorstellungen(S1) and reflective Vorstellungen. (S2). Thus we refer to a polarity in thinkingwhich already was discussed before by many other authors4and which is reflectedin more detail in Dual-Process-Theories in cognitive psychology5.

Reflective Vorstellungen (S2) may be regarded as an internal mental copy of a netof knowledge, abilities, and skills, a net of facts, relations, properties, etc. wherewe have a conscious access to. Reflective Vorstellungen mainly are the result of ateaching. The development of reflective Vorstellungen certainly is in the center of

3We will use the German word Vorstellung instead of the vague expression internalrepresentation.

4 e.g. VYGOTZKI (1978) talks about spontaneous and scientific concepts, G INSBURG(1977) compares informal work and written work, or STRAUSS (1982) discusses acommon sense knowledge vs. a cultural knowledge. STRAUSSespecially has pointed

out that these two types of knowledge are quite different by nature, that they developquite differently, and that sometimes they interfere and conflict (U-shapedbehavior).

5for example see Kahnemann & Frederick (2005) or Leron & Hazzan (2006).


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mathematics education. Here a formal, logical, deterministic, and analyticalthinking is the goal. To reflect and to make conscious are the important activities.

Most teachers or students or even researchers in mathematics often are unaware oftheir spontaneous and intuitive Vorstellungen. In the mathematics education classroom often we more or less do not realize or even ignore or suppress intuitive orspontaneous ideas. The traditional mathematics education does not emphasizeunconsciously produced feelings or reactions. In mathematics education there is nospace for informal pre-reflections, for an only general or global or overall view, orfor uncontrolled spontaneous activities. Guess and test or trial and error are notconsidered to be a valuable mathematical behavior in the class room. But all these

components are necessary to develop spontaneousor common-sense Vorstellungen(S1). These Vorstellungen mainly develop unconsciously or intuitively. They arefast and automatic and need not much working memory. But they also are hard tochange or to overcome.

When we discuss creativity and giftedness especially these S1 components arenecessary. Creative or gifted children use their common-sense knowledge veryintensively. They very often react unconsciously, they spontaneously develop newideas, and often we can detect a cognitive jump. These observations allow anadditional interpretation of Fig. 1: In the Three-Ring Model from Renzulli thesubsets A and AT mainly represent the analytical, logical S2 components whilethe subsets C and CT can be interpreted as the area of spontaneous, intuitive andoften unconscious S1 components. Thus we can summarize:

Creativity basically determines the intuitive part of giftedness, while Giftedness needs an effective interplay of intuition and conscious

knowledge.

We will analyze that interplay of Vorstellungen along an example, where childrenand adults were working on a multiplication problem.


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AN EMPIRICAL EXAMPLE

Fig. 2. Decimal Grid Task

Children as well as adults had to work on the following task (individually, thendiscussion): In the grid of Fig. 2, select a path from A to B. Change the directionafter each numbered step. Multiply together (with a calculator) the decimalnumbers6on each step you go along. Find the path with the smallest product. Markeach path in one of the small grids. You have 4 trials.

Very often we observe the same behavior, we will summarize: Getting theworksheet and a calculator most of the subjects start immediately pressing buttons(intuitive, common sense activity, no logical or analytical waste of time). Weargue that common sense Vorstellungen related to the keywords multiply andcalculator and small product become dominant. Analyzing the more detailedanswers we can identify different aspects of an unconscious Vorstellungmultiplication makes bigger, like the smaller the factor, the smaller the productor as less factors as possible. Often we get the following smallest products:

6In Germany a comma is used instead of a decimal point.


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Product Explanation Strategy

(1) 62.121024 0.7x2.1x3.1x6.4x7.1x0.3 I always took thesmallest number

(2) 1.89 2.5x8.4x0.3x0.3 as less steps aspossible

(3) 0.3564 13.2x0.5x0.6x0.3x0.3 I looked for manyzeros (decimals0,__)

We also see many unconscious and unsystematic trials. When we ask we getanswers like I do not really remember why I did that (intuitively searching for astrategy). We write down at the black board several of the given smallestproducts, then a challenge starts: Try to explain the paths from others, how didyou get that result, why did you choose that path, ...? New strategies andexperiences come up, some intuitive strategies become conscious and get reflected:

(4) I was looking for small detours (instead 13.2 go 0.7x5.9, instead 5.4 go0.5x0.6, etc.).

(5) After 0.7x5.9 we can continue with small factors x0.5x0.6x0.3x0.3=0.11151.

(6) I discovered a circle (... x0.6x0.8x0.3x ...).

(7) I tried to use a small path more than once (small path means decimal < 1).

(8) I run through the circle many times (intuitive concept of limit).We will summarize, creativity or giftedness? At the beginning there were(unconscious) fixed mental frames. Then some creative ideas opened thediscussion (see 4 8), some cognitive jumps happened (see 4, 7). The intuitiveVorstellung multiplication makes bigger became conscious and got reflected andgot changed into the analytical Vorstellung multiplication makes bigger withfactors bigger than 1 and smaller with factors smaller than 1. Creativity wasnecessary, but giftedness?

REFERENCES

Ginsburg, H. (1977). Childrens Arithmetic. New York: Van Nostrand.Heller, K. A., Moenks, F. J., Passow, H. A. (1993). International Handbook of Research and

Development of Giftedness and Talent. Oxford.Kahneman, D., Frederick, S. (2005). A Model of Heuristic Judgement. In Holyoak, K.J.,Morrison, R.J. (Eds.), The Cambridge Handbook of Thinking and Reasoning, pp. 267 -293. UK: Cambridge University Press.

Kaepnick, F. (1998). Mathematisch begabte Kinder: Modelle, empirische Studien undFoerderprojekte fuer das Grundschulalter. Peter Lang, Frankfurt am Main.


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Kiesswetter, K. (1988, Ed.). Das Hamburger Modell zur Identifizierung und Foerderung vonmathematisch besonders befaehigten Schuelern. Berichte aus der Forschung Bd. 2.Fachbereich Erziehungswissenschaft, Universitt Hamburg.

Leron, U., Hazzan, O. (2006). The Rationality Debate: Application of Cognitive Psychologyto Mathematics Education. Educational Studies in Mathematics, Vol. 62/2, pp. 105 126.

Meissner, H. (2002). Einstellung, Vorstellung, and Darstellung. Proceedings of the 26thConference of the International Group for the Psychology of Mathematics Education,Vol. 1, pp. 156 161. University of East Anglia, Norwich UK.

Meissner, H. (2003). Stimulating Creativity. Proceedings of the Third InternationalConference "Creativity in Mathematics Education and the Education of Gifted Students",pp. 30 - 36. Rousse, Bulgaria.

Meissner, H. (2005). Challenges to Provoke Creativity. Proceedings of the 3rd East AsiaRegional Conference on Mathematics Education (EARCOME 3), Symposium onCreativity and Mathematics Education. Shanghai, China.

Renzulli, J. S. (1978). What makes giftedness? Reexamining a definition. Phi Delta Kappa,60,pp. 180-184, 261.

Renzulli, J. S. (1998). The Three-Ring Conception of Giftedness. In: Baum, S. M., Reis, S.M., Maxfield, L. R. (Eds.). Nurturing the gifts and talents of primary grade students.Mansfield Center, CT: Creative Learning Press.

Sternberg, R. J., Davidson, J. E. (2005, Eds.). Conceptions of Giftedness - Second Edition;Cambridge University Press, Cambridge, pp. 246 279.

Strauss, S. (1982, Ed.). U-shaped Behavioral Growth. Academic Press, New York.Vygotsky, L. S. (1978). Mind in Society: The Development of Higher Psychological

Processes. Cambridge, MA: Harvard University Press.


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RESEARCH REPORTS



INTELLECTUAL COURAGE AND

MATHEMATICAL CREATIVITY1

NITSA MOVSHOVITZ-HADARTechnion, Israel Institute of Technology

This paper attempts at looking into mathematical works with reference tothe actions they involve, searching for traces of intellectually courageousmoves that presumably lead to findings, and looking into the history ofmathematics as it is related to this issue. Consequently, some conclusionsabout education of the mathematically gifted young generation are drawn.

George Polya (1954)

INTRODUCTION

Research of the characteristics of mathematically talented and gifted students in thepast decade yielded new instruments for identifying such students (e.g. E.L. Mann,2005). In addition, there is a visible tendency in the teacher education literature(e.g. Silver 1997) to focus on the need to encourage students to take risks ofmaking mistakes, particularly in calculations and also in problem solving, asdiscouraging risk taking may inhibit the development of mathematical creativity.Research jargon includes a lot about curiosity, and challenge but very little aboutthe courage it takes to actually do mathematics

Unlike Polya (ibid), the majority of practicing mathematicians are not concernedwith the human creative processes that are involved in learning and knowingmathematics and in its actual creation. Very few leave a record about the various

1 I wish to thank Dr. Michael Fried, Dr. Richard Getz, Prof. Larry Lesser, Prof.

Diane Resek, Dr. Alla Shmukler, and Prof. John Webb for their enlighteningcomments on an earlier version of this paper.


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Nitsa Movshovitz-Hadar

174 CMEG-5

actions they had taken in the process of unfolding their contributions to thedevelopment of mathematics. It is the results and their establishment that largelyoccupy the minds and the writings of the mathematics community. Mainstreamphilosophy of mathematics focuses, in general, on mathematical foundations,dealing mainly with the nature of mathematics as an intellectual discipline and itslogic-based development. Thus, the study of mathematical knowledge acquisitionand the psychology underlying mathematics-in-the-making, remains, almost solely,the interest of mathematics educators and cognitive scientists. These communitiesembrace a postmodern, humanist form of constructivism, tending to focus onepistemological issues, which do not always apply to the needs of practicing

educators.This paper attempts at looking into mathematical works with reference to theactions they involve, searching for traces of intellectually courageous moves thatpresumably lead to findings, and looking into the history of mathematics as it isrelated to this issue. Consequently, some conclusions about education of themathematically gifted young generation are drawn.

A FEW WORDS ABOUT COURAGE

Let us distinguish between three kinds of courage: Physical courage, Socialcourage andIntellectualcourage.

Physical courage usually involves taking bodily risk of pain, injury or evendeath. (E.g. The courage needed in order to save endangered human life, or inorder to jump into cold water).

Socialcourage is usually a part of some kind of conscience-driven action thatinvolves taking risk of social acceptance or of some kind of personal penaltysuch as being fired or sent to jail. (E.g. The courage needed to express atheisticattitudes in a religious society, or the courage to expose corruption in one'sworkplace).

Intellectualcourage is something else. It involves one's inner persistency andmotivation to expand one's own understanding of a matter or illuminate it froma different direction. It usually involves no risk, nether physical pain norpenalty. Intellectual courage has nothing to do with one's body orconsciousness. It requires cognitive self confidence and insight. The only riskin making such a move is that it may not lead anywhere. Then some intellectual

courage is needed in order to abandon it and change course. The play on word


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Intellectual Courage and

Mathematical Creativity

CMEG-5 175

in Leunig's cartoon, breaking the word understanding to two parts andconnecting it to the 'No standing' road sign, is a great example of an intellectualcourageous move by the artist2. Escher's drawings belong to this category in adifferent manner. The rest of this paper examines a few examples ofintellectually courageous moves carried out by great mathematicians.

Before we delve into a few examples of intellectually courageous moves inmathematics, let us note that the history of mathematics knew many intellectual"battles" and hence is strewn with actions of social courage by mathematicians,who were often emotionally involved. To name a few:

The fight over admitting the negative numbers; (For an original way ofexposing it, see: Gavin Hitchcock (2004).)

The attack by Bishop Berkeley (1685-1753) on Newton's fluxions and Leibniz'sinfinitesimals; (For more details see for example, Davis and Hersh (1981).)

The Vibrating String Controversy settled by Jean le Rond d'Alembert (1717 1783); (For more details see Grattan-Guinness (1970).)

2 "No standing" was a part of the three tier parking strategy (No Stopping, No Standing,

No Parking) in NSW Australian. It was repealed in 2006.Michael Leunig is a great Australian artist/poet, who kindly gave permission to usethis cartoon in mathematical expositions. I wish to thank my colleague DavidEasdown from the University of Sydney mathematics department for sharing it withme.


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176 CMEG-5

The necessity felt by Sophie Germain (1776-1831) to write using a malepseudonym (Antoine Leblanc) in order to pave her way into the maledominated mathematics community in which the common social attitude wasthat Mathematics was no place for women (Katz(1993));

The long and arduous struggle Georg Cantor (1845-1918)had to withstand withhis contemporaries in order to defend his controversial view of the infinite andtransfinite (this, eventually, had an influence on his mental health. For moredetails see for example Dunham (1990), Maor (1991).);

The dispute between the Intuitionists and the Formalists over the legitimacy of

proof by contradiction ("reduction ad absurdum"); Logicists vs. Empiricists' differences over the foundations of mathematical

knowledge being a-priory analytically derivable from logic or empiricallydiscovered;

Present days turbulence over computer assisted proof, among philosophers,logicians and mathematicians concerned about the concept of truth.

INSTANCES OF INTELLECTUALLY COURAGEOUS MOVES INMATHEMATICS

Unknown Hindu mathematician (8th

century): Designating Nothing as

Something

The invention of a symbol (0) for nothing is traditionally attributed to the Hindu inthe 8thcentury. The Persian mathematician al-Khowarizmi (780-850) described acompleted Hindu system in his book dated A.D. 825. (Eves (1983), Lecture 15).

The invention of a symbol for no quantity was an extraordinarily intellectuallycourageous move. Of course, the notion of nothing was not new to the Hindu. Thetrue need for its designation by a symbol was raised probably in the process oftransition from abacus arithmetic to written calculation. An empty rod on theabacus caused trouble in this process. Using the symbol zero solved it. Introducingzero as a placeholder in the decimal number system was the breakthrough fordeveloping written calculations.

Even nowadays zero presents a challenge not only to school children. Many adults

find it difficult to relate to zero as a regular whole number that represents aquantity, and to consider its parity and other additive and multiplicative properties,as the quantity it represents is null. "I do not have anything to say" and "I havenothing to say" are not immediately perceived as equivalent. Furthermore, "havingnoting" sounds quite different than "having zero", as "having zero" is in a wayanalogous to having some positive quantity. Seife (2000) tells the story of Zero


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CMEG-5 177

from its birth in Babylonian times, till present days. "It is the story of the peoplewho battled over the meaning of the mysterious number." (ibid)

Rafael Bombelli (1526-1573): Wild guessing, possibly nonsensical

Bombelli, the Italian mathematician, set out to resolve Cardan's confusion aboutthe solution of 3 15 4x x= + . Cardan found the three (real) roots for this equation:4, 2 3 . However, applying his general formula for solving cubic equations of

the form 3x ax b= + , he got a seemingly fourth solution:3 32 121 2 121x= + + which tantalized his mind as he knew this was

impossible. Moreover, this result was "meaningless" as roots of negative integershad no right to exist in those days.

Bombelli had a wild guess: He noticed that two of the solutions: 2 3 , and also

the two addends 2 121, 2 121+ of this strange solution, differ only in thesign of the square root. Hence, by analogy, he dared thinking that the same might

hold for their cubic roots: 3 2 121+ and 3 2 121 . Thus, he assumed

3 2 121 a b+ = + and 3 2 121 a b =

for some real numbers a, b. Making another, possibly nonsensical move by

analogy, he applied real number arithmetic to some negative number roots, and hewas able to show that a=2, b=1 satisfy this system3. Back into Cardan's formula,Bombelli resolved Cardan's problem showing that the third solution, 4, coincideswith Cardan's weird root as

3 32 121 2 121 2 1 2 1 4x= + + = + + = . In so doing, Bombellialso opened the door for imaginary numbers arithmetic. (For more details seeKleiner (1988)). "I respect conscious guessing because it comes from the besthuman qualities: courage and modesty", says Imre Lakatos (1976, p.30), and hefurther suggests that this is how new mathematical knowledge develops.

3 As we presently know:3(2 1) 8 12 1 6 1 2 11 1 2 121+ = + = + = + ,

and similarly 3(2 1) 2 121 = .


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178 CMEG-5

Claude de Meziriac (1581-1638): Solving a problem through its generalization

According to Drrie (1965), in a book published in 1624 The Frenchmathematician Claude Gaspard Bachet de Meziriac solved the 4-weight problemby ignoring some given data thus approaching it through a more general case. Theoriginal problem states:

A merchant had a 40-kg measuring weight that broke into four pieces as theresult of a fall. When the pieces were subsequently weighed, it was foundthat the weight of each piece was a whole number of kilograms and that thefour pieces could be used to weigh every integral weight between 1 and 40

kg. What were the weights of the pieces?

As I pointed out in an earlier paper (Hadar and Hadass (1982b)) the naturaltendency is to focus on the numerical data 4, and 40. This may turn out to be quitefrustrating as it may lead to a dead end, while departing from the given numbersand reformulating the original problem as a particular case of a more generalproblem leads quite easily to the solution employing an inductive line of reasoning.The more general problem being:

For any natural number n, find at least one set of integral weights with whichany integral load from 1 to n kg can be weighed; evaluate the differentpossible sets for each value ofn(e.g. in terms of economy).

Leonhard Euler (1707-1783): Connecting apparently disconnected areas (andmore)

It is impossible to choose one example that demonstrates Euler's mathematicalcourage. Many books are devoted to the analysis of his multitude of mathematicaltalents and unbelievable prolific writing (For example see: Dunham (1999), andOpera Omnia available on line at http://www.math.dartmouth.edu/~euler.) Thefollowing example is sufficiently short to earn its inclusion here. Euler attacked acentury old unsolved problem of determining the sum of reciprocals of squares of

all positive integers:2

1

1

n n

= , known to his predecessors, Leibniz and the Bernoulli

brothers to be convergent to a number less than 2. According to Dunham (1990),

"with a boldness to match his brilliance" Euler discovered and proved the sum toconverge to 2/ 6 . To do so Euler had to go through a sequence of courageousmoves which represent his great mathematical vision and intuition. First of all hehad to abandon his curiosity driven straightforward computational approach, thatfailed to lead him anywhere beyond the fact that the sum was tending to a peculiarnumber as 1.6449, which made no sense to him. Challenged Euler then tied the


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CMEG-5 179

ends of seemingly unconnected facts: He recalled from Trigonometry and Calculusthe Taylor series expansion of the trigonometric function:

3 5 7 9

sin ...3! 5! 7! 9!

x x x xx x= + + , which he changed to

2 4 6

1 ...3! 5! 7!

x x xx

+ +

.

Keeping in his mind that for a finite polynomial ( )P x of degree n with n roots

1 2 3, , ,..., na a a a , if (0) 1P = , then ( )P x can be factored into:

1 2 3

( ) 1 1 1 1n

x x x xP x

a a a a

=

, he then attempted to solve

2 4 6

( ) 1 ... 03! 5! 7!

x x xf x = + + = for which clearly f(0) = 1, in order to factor it

accordingly, although this is NOT a finite polynomial. Euler observed that fornonzero values ofx, solving

2 4 6

2 4 6 1 ... sin3! 5! 7!( ) 1 ... 03! 5! 7!

x x x

x x x xf x x

x x

+ +

= + + = = =

amounts to solving sinx= 0, yieldingx k= , k=1,2,3, for the sought roots

of f(x)= 0. He had the foolhardiness or healthy intuition to (luckily successfully)extend to infinite polynomials the theorem that he knew holds for finitepolynomials, obtaining

( ) 1 1 1 1 1 1 ...2 2 3 3

x x x x x xf x

=

and therefore

2 4 6 2 2 2

2 2 21 ... 1 1 1 ...

3! 5! 7! 4 9

x x x x x x

+ + =

.

At this point Euler virtuosic manipulation of the infinite product in the right hand

side turned this equation from equality between a sum and a product into anequality of two infinite sums of like powers ofx:

2 4 62 4 6

2 2 2

1 1 11 ... 1 ... (...) (...) ...

3! 5! 7! 4 9

x x xx x x

+ + = + + + + +

.


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180 CMEG-5

Equating like powers, particularly the coefficients of x2, and further algebraicmanipulations lead him to the surprising result:

.6

...25

1

16

1

9

1

4

11

2

=+++++

The above is one of many examples of Euler's guts to attack problems that wereknown to be difficult or nearly impossible to solve. His fruitful search for pairs ofamicable numbers is one of the many others. G. B. Dantzig (1914-2005), the fatherof linear programming, who as a student at Berkeley solved two then unsolved

problem only because he came late to Jerzy Neyman's class and assumed that theproblems on the blackboard were mandatory homework, would probably attributeEuler's courage to "positive thinking". The belief that it is doable and the innerconviction that I can do it, are the two characteristics of positive thinking Dantzigemphasized in a 1986 interview (Albers and Reid (1986).)

Euler exhibited intellectual courage in many more ways of which we'll mentionjust his problem posing nerve (e.g. - is it possible to walk through all the 7 bridgesof Konigsberg without crossing any of them twice, coming back to the point ofdeparture?), and his obsession with searching for patterns (e.g. looking for aproperty common to allconvex polyhedra) which meant phrasing bold conjectureson a regular basis.

It is difficult to conclude a section about Euler without mentioning one more aspectof his creativity that is his bravery to give a conventional entity, such as the circle,a fresh look by expanding its definition. Euler (1778) considered circular arcrotors as a part of his geometry investigations of triangular curves. An extension ofthe definition of a circle, the ultimate rotor crosscut shape and curve of constantwidth, was needed in order to do it.

Karl Weierstrass (1815-1897): Constructing a counter intuitive counter

example, a monstrous function

Unlike Euler, Wierestrass did not publish much; nevertheless he earned the title"the father of modern analysis" possibly due to his rigorous logical reasoning andhis talent as a lecturer and tutor to his students. His creative mathematics wascarried in parallel to 15 years of high school teaching prior to his appointment atage 40 to the University of Berlin. His search for rigor in general and particularly

for polishing the notion of function, which was a center of attention in his life-timeperiod, gave rise to an extremely courageous move, forming a "monstrous"function according to those days perception, continuous alas nowhere-

differentiable: ( )1

( ) cosn n

n

f x b a x

=

= , where a, b are chosen such that ais an


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CMEG-5 181

odd integer, b is a real number in (0, 1), and their product ab > 1+3/2. Such afunction gave a blow to the intuitive perception of a continuous function that restedon the "smooth" graphic representation of any such analytic expression, andbrought a change in the commonly accepted concept of function. (For more detailssee, for example, Kleiner (1989).)Such a process ofevolving concept definition iswell known in the history of mathematics. Not always, however, the appearance ofa counter example (which was usually an overwhelming experience to thecommunity) yielded an immediate change.

The Pythagoreans, for example, were NOT courageous enough to adapt their

notion of numbers when they discovered that a segment of length 2 presented a"pathological" case. This shocking discovery that there are incommensurablemagnitudes was secretly kept and cleverly bypassed by the Greek mathematicians.Legend has it (Eves (1983) Lecture 6) that the Pythagoreans banished Hippasusand erected a tomb for him, for his impiety in disclosing the secret to outsiders.The irrational numbers were admitted centuries later, not without a bitter debate.Somewhat similar fate was that of the question of independence of the ParallelAxiom, in Euclid's time, and the question of provability of Cantor's 1870continuum hypothesis in set theory of modern time. The former had to wait till the1800s when Bernhard Rieman (1826-1866) and Nikolai Lobachevsky (1792-1856)courageously introduced Elliptic and hyperbolic non-Euclidean geometriesrespectively, implying the independence of Euclid's 5th axiom. The latter wasresolved in part by Kurt Gdel (1906-1978) who in 1940 showed it was impossible

to disprove the continuum hypothesis, and finally completed in 1963 by PaulCohen (1934-2007) who showed that the continuum hypothesis is independent ofthe usual Zermelo-Fraenkel axioms of set theory. This solved in an unexpectedway the first of the 23 problems that David Hilbert (1862-1943) posed in his veryinfluential address to the International Mathematical Union in 1900. Hilbertexamined the broad gamut of unsolved mathematical problems and identified 23 ofthem as the most important challenges for mathematicians in the century ahead - ayet another great example of intellectually courageous move.

Felix Hausdorff (1868 1942): Making a counterintuitive interconnection:

Dimension needs not be an integer

Euclidean space dimension d assumes the integer values 1, 2, or 3. If we take aEuclidean object (a line segment, a square, a box) of dimension d,and reduce itslinear unit size by rin each spatial direction (namely making it 1/r of its original

size), its measure (length, area, or volume) becomes dM r= as shown in Table 1.In 1918 the German mathematician Felix Hausdorff treated dM r= algebraically,as follows: log logdM r M d r= = . Consequently he made an


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182 CMEG-5

intellectually courageous move suggesting that d need not be an integer. Since

r

Md

log

log= it could be a fraction, he claimed! Benot Mandelbrot, who was born in

1924 into a world that had recognized Hausdorff dimension, employed thisgeneralized treatment of dimension for his 1977 publication: Fractals: Form,chance and dimension (Mandelbrot (1977).)

Table 1: The measure (M) of a Euclidean shape of dimension (d) if reduced by r in each

direction

CONCLUDING REMARKS: EDUCATING MATHEMATICALLYGIFTED STUDENTS

The examples given above are but a few from a huge assortment of examples that

the history of mathematics and contemporary mathematics can provide. They werepicked to demonstrate some intellectually courageous moves in mathematics asexhibited in great mathematicians' work. A partial list of such moves includes:Making a guess, employing a non-conventional mechanism, dismissing of somedata in order to clarify a complicated situation, adding an auxiliary component thatshed light on an otherwise obscured picture, raising a conjecture, making an

Euclidean DimensiondReductionfactorr 1 2 3

1

(Original

size)

M=1M=1

M=1

2M= 21=2

M= 22=4

M= 23=8

3

M= 31=3

M= 32=9 M= 33=27


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CMEG-5 183

assumption, connecting apparently unconnected mathematical facts, defining a newconcept or extending an existing definition, constructing a counter example,identifying interesting problems, and more. This is by no mean an exhaustive set ofmoves or examples.

There are many school level mathematical tasks that call for similar intellectuallycourageous moves. E.g. Is there something in common to all trianglescircumscribed by one circle? Is there anything in common to all triangles? To allquadrilaterals? To all polygons? In general, there is a good chance that searchingfor a common property will yield some hypothesis testing and thereby require

courageous moves. Analogous thinking is another promising direction (For a fewexamples see also: Movshovitz-Hadar (1988, 1993), Hadar and Hadass (1981a,1981b, 1982a).)

Looking into the future of mathematics, the question is how to educate a bravegeneration of mathematically talented youngsters? More precisely, whatmathematical experiences are appropriate for various age groups to empower themto take intellectually courageous moves? And not less important, what kind ofteachers' attitudes and classroom atmosphere is sufficiently nourishing yetadequately open to give them the strength to overcome a concern about the scrutinythat their innovative ideas may be subjected to? To avoid impediment of creativethoughts? To act courageously in mathematics?

The significance of these questions is their practical implications. An accumulative

set of valid mathematical tasks for various levels of school knowledge, calling forintellectually courageous moves, needs be compiled, tested, analyzed, and finallydisseminated by this respectable research community to those who assume thedaily challenge of and responsibility for dealing with the mathematically gifted andtalented students. These teachers and mentors need to develop the ability toidentify intellectually courageous moves in their students' work, and give extracredit for such moves.

Digging into the history of mathematics (i.e. from antiquity tillyesterday) toidentify great mathematicians' intellectually courageous moves, following themodest attempt in this paper can shed light on more types of moves of this sort,possibly classifying them in a more systematic way, and may also serve as eye-opener for teachers and mentors in identifying such moves carried out by theirnovice talented students.

REFERENCES

Albers, D. and Reid, C., (1986): "An Interview of George B. Dantzig: The Father of LinearProgramming",College Mathematics Journal, 17, 4 pp. 293-314.


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Cohen, P. J. (1963): The independence of the continuum hypothesis. Proceedings of theNational Academy of Science. USA: Part I, Vol.50, 1143-1148; Part II, Vol.51 (1964),105-110

Davis, P. J. & Hersh, R. (1981): The Mathematical Experience, Birkhuser, Boston (pp.241-245)

Drrie, H. (1965): 100 great problems of Elementary Mathematics, Dover Publications,New York, pp.7-9.

Dunham, W. (1990): Journey Through Genius, the great theorems of mathematics, JohnWiley & sons inc., New York. (pp. 147-154).

Dunham, W. (1999): Euler: The Master of us all, The Mathematical Association of America,P. O. Box 91112, Washington, D.C.

Eves, H. (1983): Great Moments in Mathematics, Vol. 1: Before 1650, Lecture 6: Resolutionof the first Crisis,, and Lecture 15: Blockhead. MAA Dolciani Mathematical ExpositionNo. 5, The Mathematical Association of America.

Euler, L. (1778): "De Curvis Triangularibus", Originally published in Acta AcademiaeScientarum Imperialis Petropolitinae, 1778, 1781, pp. 3-30. Reprinted in Opera Omnia,Series 1, Volume 28, pp. 298 321, retrieved from:http://www.math.dartmouth.edu/~euler, number E513.

Gdel, K. (1940): The consistency of the axiom of choice and of the generalized continuumhypothesis,Princeton University Press.

Grattan-Guinness, I. (1970): The Development of the Foundations of Mathematical Analysisfrom Euler to Riemann, MIT Press.

Hadar, N. & Hadass R. (1981a): An Analysis of Teaching Moves in the Teaching of the SineTheorem via Guided Discovery. International Journal of Mathematics Education inScience and Technology, 12, 1, pp. 101-108.

Hadar, N. & Hadass R. (1981b): The Road to Solving a Combinatorial Problem is Strewnwith Pitfalls.Educational Studies in Mathematics,12, 4, pp. 435-443.

Hadar, N. & Hadass R. (1982a): Another Approach to Similar Triangles. MathematicsTeaching, 101, pp. 33-35.

Hadar, N. & Hadass R. (1982b): "The 4-Weight Problem - A Pedagogical Account".Mathematics in School, 11, 5, pp. 32-34.

Hilbert, D. (1900) : "Cantor's problem of the cardinal number of the continuum". Originallypublished in German, "Cantors Problem von der Mchtigkeit des Continuums".Gttinger

Nachrichten 1900, 263-264. English Translation retrieved from:http://aleph0.clarku.edu/~djoyce/hilbert/problems.html

Hitchcock, G. (2004): "Alien Encounters", Convergence, MAA on-line magazine wheremathematics, history and teaching interact. Retrieved from:

http://mathdl.maa.org/convergence/1/convergence/1/?pa=content&sa=viewDocument&nodeId=435

Katz, V. J. (1993): A History of Mathematics, an introduction, Harper Collins CollegePublishers, NY, p. 591.

Kleiner, I. (1988): "Thinking the unthinkable",Mathematics Teacher, 81, 7, pp. 583-592.


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Kleiner, I. (1989): "Evolution of the Function Concept: A Brief survey", CollegeMathematics Journal, 20, pp. 282-300.

Lakatos, I. (1976): Proofs and Refutations, the Logic of Mathematical Discovery, NewYork: Cambridge University press.

Mandelbrot, B. B. (1977): Fractals: Form, Chance and Dimension. San Francisco, CA: W.H. Freeman and Company, 1977, xviii+265 pp. (A shorter French version published1975, Paris: Flammarion).

Mann, E. L. (2005):Mathematical Creativity and School Mathematics: Indicators of MathematicalCreativity in Middle School Students. Ph.D. dissertation, University of Connecticut. Retrievedfrom: http://www.gifted.uconn.edu/siegle/Dissertations/Eric%20Mann.pdf

Maor, E. (1991): To Infinity and Beyond, A Cultural History of the Infinity (Chapter 10),Princeton University Press. New Jersey.

Movshovitz-Hadar, N. (1988): School Mathematics Theorems - an Endless Source ofSurprise. For the Learning of Mathematics, Vol. 8, No. 3. Pp. 34-40.

Movshovitz-Hadar, N. (1993): A Constructive Transition from Linear to QuadraticFunctions, School Science and Mathematics.Vol. 93, no. 6, pp. 288 - 298.

Polya, G. (1954):Mathematics and Plausible Reasoning, Princeton U. Press, p. 8.

Seife, C. (2000):Zero: The Biography of a Dangerous Idea, Penguin Books

Silver, E.A. (1997): "Fostering Creativity Through Instruction in Mathematical ProblemSolving and Problem Posing."International Review on Mathematical Education, 29, pp.75-80.

):,) .1974 , , " ".


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