ࡱ>  @mkbjbj >؝؝9`rrr8,4 F|,qEsEsEsEsEsEsE$GRIE-n(n(n(E4E111n(qE1n(qE11AC  ?ɐSr. BDtE0 FBJ 0J$CJC\ 1 &#HEE r0Xr MATHEMATICAL CREATIVITY AND ITS CONNECTION WITH MATHEMATICAL IMAGINATION DALIA ARALAS Abstract. This paper seeks to explore and formulate a notion of mathematical creativity, with a view to proposing a possible formulation of the concept which takes account of its connection with mathematical imagination. Such a discussion would be inadequate if undertaken without addressing related questions which bear on the formulation of mathematical creativity. This paper is in five parts and seeks to address four main questions: (i) What features of the notion of creativity may be of significance to the mathematical endeavour? (ii) What does a discussion of the nature of mathematics imply for the ways we might think about mathematical creativity? (iii) What does a discussion of mathematical imagination imply for the ways we might think about mathematical creativity? (iv)What might be a possible conceptualization of mathematical creativity? The fifth part concludes the paper, emphasizing the close affinity between mathematical creativity and mathematical imagination. Key words: Mathematical Creativity, Mathematical Imagination, Creative Agency, Creative Intentionality, Configurational Play INTRODUCTION The paper seeks to explore and formulate a notion of mathematical creativity, with a view to proposing a possible formulation of the concept which takes account of its connection with mathematical imagination. The paper addresses four main questions deemed important in formulating a conceptualization of mathematical creativity. CREATIVITY AND IMAGINATION This first part seeks to address a question: What features of the notion of creativity may be of significance to the mathematical endeavour? Some insights from multi-disciplinary research on creativity are discussed. It is noted that whilst there is a variety of perspectives and approaches to the study of creativity, some features, notably imagination, associated with the notion of creativity, may arguably be of prime significance for the purposes of extending valuable work in mathematics. The notion of creativity possesses a complexity that gives rise to a variety of views. An overview reveals some commonalities and differences amongst the voluminous amount of research on creativity, within psychology, philosophy and other disciplines. The differences are reflected in a multiplicity of views regarding the nature of creativity and in a variety of approaches to the measurement and testing of creativity (Boden, 2004; Cropley, 2001; Sternberg, 1988, 1999). Sternberg (1999), for example, discussed seven main approaches to the study of creativity, namely, mystical, psychoanalytic, pragmatic, psychometric, cognitive, social-personality and confluence approaches. The diversity of views prompted Torrance (1988: 43) to note that the notion of creativity defies precise definition as its nature is almost infinite. Be that as it may, there exists an important strand of agreement regarding central features of creativity, as discerned in an examination of some proposed definitions of creativity. Boden (2004: 1) states, Creativity is the ability to come up with ideas or artefacts that are new, surprising and valuable (emphasis in original). This resonates with a proposal by Cropley, for whom creativity has three main elements: novelty, effectiveness and ethicality (Cropley, 2001: 5-6). Similarly, in their review of research on creativity, Sternberg and Lubart (1999: 3) observed, Creativity is the ability to produce work that is both novel (i.e. original, unexpected) and appropriate (i.e. useful, adaptive concerning task constraints). On examination, the definitions offered by these researchers may be seen to associate creativity with the following main characteristics (i) newness, novelty, originality, (ii) surprise, unexpectedness, (iii) value, ethicality, (iv) effectiveness, appropriateness, usefulness. There are other questions regarding creativity which require further study, but at least an important question, regarding features which may be deemed central to the notion, has been addressed. A conceptualization of creativity, which takes into account the four main features mentioned above, is an important point of departure for research which seeks to formulate a notion of mathematical creativity. Upon analysis, it is clear that some of the features mentioned above value, ethicality, effectiveness, appropriateness, usefulness are emphasized in the formulation of educational objectives. Within educational contexts, research on creativity has largely been concerned with the creative enhancement of the quality of teaching and learning (Bruner 1962; Egan, 1992, 1967; Cropley, 2001; Presmeg, 2003; Sheffield, 2003). The creative promotion of greater empowerment amongst pupils and teachers with regard to learning and teaching mathematics implicates an engagement of imagination. To say this is to take into account three observations. First, research on aspects of creative teaching and learning, or of creative endeavour in general, demonstrate strong indications of a positive relationship between the enhancement of creativity and the fostering of imagination (Torrance, 1965, 1988; Egan, 1992, 1997; Pfenninger and Shubik, 2001). Second, an analysis of the assumptions underlying the notions of imagination and creativity reveals an affinity between both notions. As noted above, the notion of creativity is associated with the features newness, novelty, originality, surprise, unexpectedness. All these characteristics are closely linked with the notion of imagination. Third, the relationship between imagination and creativity is so intimate that many researchers seem to take for granted the interchangeable use of the terms imagination and creativity, but such usage does not make for clarity and has no basis in careful analysis. Such an analysis needs to take account of the distinction, as well as the close relationship, between creativity and imagination. Whilst an analysis of imagination is bound to a long philosophical tradition that has served to complicate, rather than simplify, the nature of imagination, it is also clear that progress may be made by selective renderings of the notion of imagination (Warnock 1976, 1994; Engell, 1981; White 1990; Kearney 1998, Chittick 1989). For example, for White, to imagine something is to think of it as possibly being so (White, 1990: 184). For Kearney (1998) imagination is discussed in multiple terms, including creation, re-creation, re-presentation, originality, discovery, freedom, divergence, possibility, innovation. Such renderings of the notion of imagination involve aspects which clearly relate to the features of creativity mentioned above. Clearly, whilst non-identical, imagination and creativity share a close affinity which is evident in educational research as well as philosophical analysis. VIEWS OF MATHEMATICS AND CREATIVE MATHEMATICAL WORK This second part seeks to address a question: What does a discussion of the nature of mathematics imply for the ways we might think about mathematical creativity? Insights from the literature, particularly the philosophy and history of mathematics and of mathematics education, regarding the diversity of views of mathematics, are discussed. The implications for mathematizing and the expansion of mathematical work are drawn. A sub-question is raised regarding the implications of particular conceptions of mathematics on the ways creativity is viewed in relation to the learning of mathematics. The endeavour to enquire into the nature of mathematics has demonstrated a complexity which makes it the subject of endless debate, giving rise to differences of views from classical times to the present age. This paper can only paint a sketch of these views. Plato granted mathematical objects the status of Forms which possess an independent existence beyond space and time. For Plato, mathematics was seen as an activity which dealt with representations of such Forms. For Aristotle, however, mathematics was deemed grounded in experienced reality and knowledge was gained from experimentation, observation and abstraction (Dossey, 1992: 40). Mathematics then, whether deemed a rational or an experimental endeavour, was seen as a subject which was a source of certainty. The late nineteenth and early twentieth centuries saw three perspectives of mathematics - namely logicism, intuitionism, formalism - struggle to deal with paradoxes in mathematical work. Logicists such as Frege and Russell sought to demonstrate that mathematics is based on, and likely identical to, logic (Bell, 1999; Dossey, 1992; Hodgkin, 2005). Intuitionists such as Brouwer rejected classical logic and affirmed only mathematical entitites which are constructed out of something given immediately in intuition and associated with constructive proofs (Bell, 1999: 205). Hilbert, most closely associated with formalism, developed the axiomatic method or metamathematical method. He sought to establish the consistency of mathematics, an objective for which Godel denied realization. These views of mathematics proved demonstrably inadequate in the search for foundations of mathematics (Bell, 1999; Dossey, 1992; Hodgkin, 2005; Benacerraf and Putnam, 1983). Latterly, other views of mathematics have been developed, of which constructivism and structuralism are the more familiar. The former comprises a range of positions, all of which stress the contributory role of subjectivities and mediational means, including language, in the active construction of mathematical knowledge (von Glasersfeld, 1995; Steffe and Gale, 1995; Ernest, 1998). For structuralists such as Resnik (1999) mathematics is deemed primarily a science of patterns in which mathematical objects do not occur in isolation but only in structures. This sketchy (within space constraints) overview of the most prevalent views of mathematics evidences a diversity of positions which underlie both mathematical knowledge and the practice of mathematizing. It may be worthwhile to consider that, rather than engaging in endless deliberations to choose one singular view deemed best representative of the nature of mathematics, educators would do well to use insights, drawn from all views, regarding important features of mathematics. It is worthwhile to incorporate these insights into the endeavour, within classroom contexts. Examples are given in the next section, of how insights drawn from considerations of multiple views regarding the nature of mathematics may promote the generation of all sorts of imaginative ways of grappling with puzzling situations. Clearly, there are a variety of ways in which insights from an understanding of the complex nature of mathematics may usefully shape the endeavour to promote mathematical creativity. MATHEMATICAL IMAGINATION AND CREATIVITY This third part seeks to address a question: What does a discussion of mathematical imagination imply for the ways we might think about mathematical creativity? Some particular insights from research which I have undertaken, based on empirical and theoretical work, on the nature of mathematical imagination, are discussed. I have developed a conceptualization of mathematical imagination which brings to the fore the importance of what I call configurational play. The latter involves notions of creative agency and creative intentionality. The implications for the extension and advancement of mathematical work are discussed. The research study on mathematical imagination (Aralas, 2004, 2007, forthcoming) raised and addressed questions about the imaginative ways pupils grapple with bewildering mathematical situations. The study, employing interrogative, recursive analytical interplays between the empirical and theoretical work undertaken, generated an analytical framework which proposes three main forms of mathematical imagination and four aspects which underlie the generation of these forms. For the purposes of this paper, it is important to note the significance of the aspects of creative agency and creative intentionality. The forms and aspects of mathematical imagination are related to the central theme of configurational play. When confronted with bewildering situations, pupils brought creative agency and creative intentionality to bear on the situations, thereby bringing about and enacting a variety of configurational plays. Pupils undertook actions and works, discernable as imaginative plays with, and on, and about, reconfigurations of situations, rendered as possible mathematical forms which were related to all sorts of linkages among resources - mathematical and extra-mathematical - in different spaces. The pupils configurational plays resulted in construals and renderings which were as often as not seen as unorthodox or erroneous from prevalent perspectives. The pupils configurational plays brought about shifts, discontinuities, transformations of perspectives and spaces. In brief, the research indicated the significance of mathematical imagination for the expansion of the mathematical endeavour in transformative ways. Part of the significance of this research for the educational promotion of mathematical creativity is that the research has generated an analytical framework of mathematical imagination which has importance in the way we might formulate a conceptualization of mathematical creativity. The framework generated in the study illuminates an analysis of the distinction, and the relationship, between mathematical imagination and mathematical creativity. It is noteworthy that mathematical imagination is made manifest in the generation of configurational plays which are aligned to interpretations of mathematical form, whether orthodox or unorthodox form. The works of pupils acting with mathematical imagination do not emphasize mainstream interpretations of effectiveness, appropriateness and usefulness which are of central importance for the evaluation of the creativity of mathematical works. Pupils configurational plays often spill over the boundaries set by the criteria of effectiveness and appropriateness in institutional ways. Arguably, the reach of mathematical imagination exceeds the boundaries which mark the areas wherein works of mathematical creativity are produced. It is noteworthy that the growth of imagination involves both convergence to, and divergence from, particular purposes or ends towards which the promotion of creativity strives. That is, mathematical imagination involve the generation of works deemed creative as well as those deemed non-creative, by prevalent criteria of effective, appropriate and useful works. Thus, it may be that what may be seen as, or definable as, mathematical creativity, is, on analysis, a part of the wider notion of mathematical imagination. Further, it may be that the endeavour to promote mathematical creativity needs take place within the wider endeavour of fostering mathematical imagination. The research (Aralas, forthcoming) indicates possible ways wherein the nature of the relationship between mathematical imagination and mathematical creativity may be discernable. For example, when pupils were confronted with puzzling situations which they had to resolve by the use of mathematical reasoning, pupils brought creative agency and creative intentionality to bear on the situations. The pupils construed and rendered situations in terms of diverse logic. The pupils configurational plays afforded them opportunities for unusual (interpretable as creative) work and for an appreciation of the diversity of mathematizing to varying degrees of logicality. Mathematical imagination in play enhanced pupils creative work and appreciation of the logical nature of mathematical reasoning, even if not always in accordance with prevalent views. Other examples involved pupils tackling questions on sets, angles and other situations. The pupils acted with mathematical imagination, they generated a variety of original (to them) configurational plays which were underlain by unusual or novel patterns. Whilst some of their works were conventionally incorrect, yet the pupils configurational plays afforded them opportunities for creative work and for an appreciation of the diversity of patterns or structures which are possible to generate. Mathematical imagination in play enhanced pupils creative work and appreciation of a diversity of structures underlying mathematical situations, drawing the pupils attention to the largely structural nature of mathematics. Thus, it may be discerned that employment of pupil imaginative engagement with mathematical situations provides important resources for the promotion of creative teaching and learning in mathematics classrooms. It is thus worthwhile considering the contribution of mathematical imagination, particularly with regard to the aspects of creative agency and creative intentionality, in formulating a conceptualization of mathematical creativity. A CONCEPTUALIZATION OF MATHEMATICAL CREATIVITY This fourth part seeks to address a question: What might be a possible conceptualization of mathematical creativity? With the recognition that the formulation of mathematical creativity to be proposed is one which fulfills human purposes within educational contexts, it is worthwhile considering what this might imply. Within educational contexts, the promotion of, and research into, creative teaching and learning, clearly aspires to undertake the practice of, and inquiry into, the ways creativity impacts on educational objectives related to greater empowerment on the part of pupils in raising the quality of their mathematical experience. These include the enhancement of thorough pupil understanding - of mathematical concepts, theories, algorithms, procedures, processes, forms, structures, situations as well as pupil ways of creative engagement with mathematical objects and processes, to produce works which are meaningful, significant and valuable. Such creative endeavour impinges on pupil achievement in academic spheres with consequences for achievement in other areas of their lives. Furthermore, the creative endeavour includes the enhancement of teacher effectiveness at promoting not only such objectives as pupil empowerment but also effectiveness at promoting conditions which facilitate such objectives. These considerations form an important part of the basis for formulating a definition of mathematical creativity. Clearly, in the endeavour to generate a conceptualization of mathematical creativity, it is worth taking into account insights drawn from a discussion of the above questions. These include the relationship between creativity and imagination, the nature of mathematics and the significance of mathematical imagination. The endeavour would also do well to take into account the strivings, aims, values of mathematics teachers and learners as well as the results of educational research which bear on such strivings, values and aims. Consequently, a possible formulation of mathematical creativity may be the following. Mathematical creativity is the imaginative, agential, intentional capacity (related to aspects of mathematical imagination, specifically creative agency and creative intentionality) to act upon, and to create with, an interplay of a variety of human and mathematical attributes, factors and conditions; where the fulfilment of such capacity may generate actions and mathematical processes, products, works which are, or potentially, deemed original, valuable and empowering. It is noted that such interplay may be related to shifts, reconfigurations and transformations of agential selves, perspectives, spaces as well as of mathematical works, whether process or product, in multiple senses in diverse ways. CONCLUSION In conclusion, the endeavour to formulate a conceptualization of mathematical creativity bears fruit when it draws on insights from a discussion of questions related to the relationship between creativity and imagination, the nature of mathematics, the significance of mathematical imagination and the strivings, values and aims which shape the mathematical engagement of pupils and teachers within educational contexts. The paper has alluded to the significance of an expanded conception of the mathematical endeavour which involves not only reason but also the poetics of mathematical imagination. The paper has thus undertaken an exploration and formulation of a conceptualization of mathematical creativity, drawn upon a discussion of its relationship with mathematical imagination. The conceptualization of mathematical creativity proposed opens up possibilities for further development and research into the mathematical endeavour which draws sustenance not only on reason but also on creativity and imagination. ACKNOWLEDGEMENT I am deeply grateful to Prof. Geoffrey Walford for his critical comments. REFERENCES Aralas, D. (2004) Investigating mathematical imagination. Poster 45 and Round Table Discussion at the Tenth International Congress on Mathematical Education (ICME-10), 4 - 11 July 2004, DTU Technical University of Denmark. Aralas, D. (2007) Extending video ethnographic approaches. In G. Walford (ed) Methodological Developments in Ethnography. Studies in Educational Ethnography. Vol. 12. Oxford: JAI, Elsevier. pp169-184. Aralas, D. (forthcoming) Investigating Mathematical Imagination: forms and aspects made manifest in analysis of pupil works. D. Phil Thesis. University of Oxford. Bell, J. (1999) The Art of the Intelligible. Dordrecht: Kluwer Academic. Benacerraf, P. and Putnam, H. (1983) Philosophy of Mathematics: selected readings. Cambridge: Cambridge University Press. Boden, M. A. (2004) The Creative Mind: myths and mechanisms. 2nd ed. London: Routledge. Bruner, J. S. (1962) The conditions of creativity. In H. E. Gruber, G. Terrell and M. Wertheimer (eds) Contemporary Approaches to Thinking. New York: Prentice-Hall. Chittick, W. C. (1989) The Sufi Path of Knowledge: Ibn al-Arabis metaphysics of imagination. Albany, NY: State University of New York Press. Cropley, A. (2001) Creativity in Education and Learning. London: Kogan Page. Dossey, J. (1992) The nature of mathematics: its role and influence. In D. A. Grouws, Handbook of Research on Mathematics Teaching and Learning (pp 39-48). New York: MacMillan. Egan, K. (1992) Imagination in Teaching and Learning: ages 8 to 15. London: Routledge. Egan, K. (1997) The Educated Mind. Chicago: Chicago University Press. Engell, J. (1981) The Creative Imagination. Cambridge, Mass.: Harvard University Press. Ernest, P. (1998) Social Constructivism as a Philosophy of Mathematics. New York: State University of New York Press. Hodgkin, L. (2005) A History of Mathematics: from Mesopotamia to Modernity. Oxford: Oxford University Press. Kearney, R. (1998) Poetics of Imagining: Modern to Post-modern. Edinburgh: Edinburgh University Press. Pfenninger, K. H. and Shubik, V. R. (2001) The Origins of Creativity. Oxford: Oxford University Press. Presmeg, N. (2003) Creativity, mathematizing and didactizing: Leen Streeflands work continues, Educational Studies in Math., 54, 127137. Resnik, M. (1999) Mathematics as a Science of Patterns. Oxford: Clarendon Press. Sheffield, L. (2003) Extending the Challenge in Mathematics: developing mathematical promise in K-8 students. Thousand Oaks: Corwin Press. Steffe, L. and J. Gale, J. (eds) (1995) Constructivism in Education. Hillsdale, New Jersey: Lawrence Erlbaum Sternberg, R. (ed) (1988) The Nature of Creativity. Cambridge: Cambridge University Press. Sternberg, R. (ed) (1999) Handbook of Creativity. Cambridge: Cambridge University Press. Torrance, E. (1965) Rewarding Creative Behaviour: experiments in classroom creativity. Englewood Cliffs, N. J.: Prentice-Hall. Torrance, E. (1988) The nature of creativity as manifest in testing. In R. Sternberg (ed) (1988) The Nature of Creativity. Cambridge: Cambridge University Press. Pp. 43 -75. Von Glasersfeld, E. (1995) Radical Constructivism: a way of knowing and learning. London: Falmer. Warnock, M. (1976) Imagination. London: Faber and Faber. Warnock, M. (1994) Imagination and Time. Oxford: Blackwell. White, A. (1990) The Language of Imagination. Oxford: Basil Blackwell. ABOUT THE AUTHOR Dalia Aralas Linacre College, University of Oxford St Cross Road, Oxford OX1 3JA, United Kingdom. -mails:  HYPERLINK "mailto:dalia.aralas@linacre.ox.ac.uk" dalia.aralas@linacre.ox.ac.uk  PAGE 554 PAGE 554 PAGE 554 PAGE 558 PAGE 554    Dalia Aralas Mathematical Creativity and its Connection With Mathematical Imagination PAGE 32 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 1 PAGE 31 ICME 11, Mexico, 2008 Proceedings of the Discussing Group 9 : Promoting Creativity for All Students in Mathematics Education The 11th International Congress on Mathematical Education Monterrey, Mexico, July 6-13, 2008 PAGE 23 ICME 11, Mexico, 2008 JKYbc4 > ?  %q ""e#22JKUUYY=Zʾ{lhaUUUUhEhE5CJaJ hEhEhEhEhECJaJmH sH hECJaJmH sH #hEhE6OJQJ^JmH sH hEhE6CJaJhEhECJaJhEhE56CJaJhEhE5CJ aJ hEhE5CJaJmH sH $hEhECJ$OJQJaJ$mH sH #hEhE6CJ OJQJ^JaJ !JKXY4 obZ$a$gdE $d\$a$gdE$ D%@lh<d\$]l^ha$gdE$ D%@P<d\$]P^a$gdE$ P<d\$]P^a$gdE $<d\$a$gdE$<d\$]^a$gdE dhxxgdE$<d\$]^a$gdE hlk  %\""e#%.222B5;BIJJK $xa$gdE $d\$a$gdE$a$gdEd\$gdE $ a$gdE $d\$a$gdEd\$gdEKPRUUU>XYYY=Z>ZIZ*[[\\$0^`0a$gdE$Ld\$^L`a$gdE$L^L`a$gdE$Lx^L`a$gdE $xa$gdE $d\$a$gdE$a$gdE $d\$a$gdE=Z>ZIZZZ[[\p\\\]3]o]]]]^@^r^^^ ____+`P`a````$agaaabSblbb c.cRccc di@iDiFiJiLifihiiii j[j$ee]e`ea$gd|*y,&#$+D,gd|*$a$gdE $a$gdE$a$gdEgdE$a$gdE $xa$gdEhhhy40JCJOJQJaJmHnHu hAoIhy0JCJOJQJaJ)jhAoIhy0JCJOJQJUaJhEhE5OJQJmH sH hy5OJQJmH sH hEhE5OJQJhyhEhyCJOJQJaJ[j\jhj~jjjjj$kGkHkTkjkkklkmk$a$gdE$a$gd:',&#$+D,gd$ 9r  ]a$gdIi$ 9r  ]a$gd52$a$gd|*$',&#$+D,a$gd:oj}jjjjFkGkHkIkOkPkRkSkTk[kikjkkklkmk~zlhEhjM5OJQJaJhw hIihyhyCJOJQJaJ%h>y40JCJOJQJaJmHnHu hAoIhy0JCJOJQJaJ)jhAoIhy0JCJOJQJUaJ h-Chyh-Chy56CJaJh-Chy56CJ aJ hyh;hyCJOJQJaJ< 00&P 1hP:p>y4. 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