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FOSTERING CREATIVITY BY ESTABLISHING
THE CONDITIONS FOR COMPLEX EMERGENCE
KIM BESWICK
Abstract: This paper builds on the premise that classrooms are complex systems and explores how teachers might use understandings of complexity science and, in particular, conditions recognised as fostering complex emergence, to establish the kinds of mathematics classroom environments in which creative thinking is likely to occur. It is argued that creativity is a property of both the classroom environment and of the thinking of individuals and that the fostering the former promotes the latter.
Key words: Creativity, Creative thinking, Complexity science, Creative environment.
CREATIVITY
In this paper creativity is used as defined by Plucker and Beghetto ADDIN EN.CITE Plucker20041609, p.15616095Plucker, Jonathon A.Beghetto, Ronald A.Sternberg, Robert J.Grigorenko, Elena L.Singer, Jerome L.Why creativity is domain general, why it looks domain specific, and why the distinction does not matterCreativity: From potential to realization153-167creativity2004Washington DCAmerican Psychological Association(2004, p.156). That is,
Creativity is the interplay between ability and process by which an individual or group produces an outcome or product that is both novel and useful as defined within some social context.
The social aspect of this understanding of creativity encompasses additions, such as communication and ethicality, to the criteria of usefulness and novelty, that have been proposed by others such as Cropley ADDIN EN.CITE Cropley1997161016105Cropley, A. J.Runco, Mark A.Fostering creativity in the classroom: General principlesThe creativity research handbook: Volume one83-114creaticityfostering creativityteaching creativity1997Cresskill, NJHampton Press(1997). Creativity is both domain specific and generic partly because of the role of social context in assessing the novelty and usefulness of a product ADDIN EN.CITE Plucker2004160916095Plucker, Jonathon A.Beghetto, Ronald A.Sternberg, Robert J.Grigorenko, Elena L.Singer, Jerome L.Why creativity is domain general, why it looks domain specific, and why the distinction does not matterCreativity: From potential to realization153-167creativity2004Washington DCAmerican Psychological Association(Plucker & Beghetto, 2004). For example, the creativity of a school students solution to a mathematics problem is likely to be judged in a social milieu that includes peers and at least a person with a degree of expertise in the discipline, likely in educational contexts, to be regarded as more than the individual who devised the solution. In such contexts the solutions novelty will be judged against the experiences of the group and in view of the teachers expectations of his/her students abilities to solve such problems in certain ways. In a different social context, such as a different grade level, in an English class, or among professional mathematicians the assessment of the creativity of the solutions would likely be different. In addition, an individuals propensity to display creativity is related to such things as his/her expertise and/or skills, motivation and self-confidence ADDIN EN.CITE Cropley1997161016105Cropley, A. J.Runco, Mark A.Fostering creativity in the classroom: General principlesThe creativity research handbook: Volume one83-114creaticityfostering creativityteaching creativity1997Cresskill, NJHampton Press(Cropley, 1997), all of which are likely to vary according to the particular domain concerned. Importantly these represent cognitive, social, and personal characteristics which must all be attended to if students creative potentials are to be realised. Cropley (1997, p. 98) claimed that teachers who create classroom environments that foster creativity,
Encourage students to learn independently.
Have a cooperative, integrative style of teaching.
Motivate their students to master factual knowledge so that their students have a solid base for divergent thinking.
Delay judging students ideas until they have been thoroughly worked out and clearly formulated.
Encourage flexible thinking in students.
Promote self-evaluation in students.
Take students suggestions and questions seriously.
Offer students opportunities to work with a wide variety of materials and under many different conditions.
Help students to cope with frustration and failure, so that they have the courage to try the new and unusual.
COMPLEXITY SCIENCE
The notion of complex systems arose from the recognition that traditional ways of studying the world were not adequate for all phenomena ADDIN EN.CITE Davis20031467146717Davis, BrentSimmt, ElaineUnderstanding learning systems: Mathematics education and complexity scienceJournal for Research in Mathematics EducationJournal for Research in Mathematics Education137-342systemslearningmathematics learningmathematics educationcomplexity2003137-167(Davis & Simmt, 2003). For example, although Newtonian mechanics provides powerful descriptions of simple systems involving interactions between very small numbers of objects, increasing the numbers of objects results in systems that are complicated and to which probabilistic models need to be applied. Complex systems are not just very complicated but are inherently different from complicated systems and demand quite different tools for their analysis ADDIN EN.CITE Davis20031467146717Davis, BrentSimmt, ElaineUnderstanding learning systems: Mathematics education and complexity scienceJournal for Research in Mathematics EducationJournal for Research in Mathematics Education137-342systemslearningmathematics learningmathematics educationcomplexity2003137-167(Davis & Simmt, 2003). The interacting entities in complex systems are autonomous (at least to some degree), often living, and it is from their interactions that characteristics of the collective that are not attributable to the actions of any particular individuals emerge. That is, in complex systems transcendent properties emerges from the interactions of autonomous agents and without any imposed leadership or authority. Davis and Simmt (2003) argue that this makes it reasonable to talk about complex systems in anthropomorphic terms (for example, an uncooperative class).
Nestedness is often a feature of complex systems. Davis and Simmt (2003) illustrated this with the example of cells, organs, individuals, social groups, and society more broadly which can all be considered to be complex systems which emerge from the interactions of agents which comprise the entities of the previous level. Complex systems have also been described as self-organising, self-maintaining, self-renewing, and structurally determined ADDIN EN.CITE Davis2004147514756Davis, BrentInventions of teaching: A genealogy350philosophies of teachingphilosophyteachingtheoriesmetaphysicsphysicalrationalismempiricismmysticismreligionintersubjectivityinterobjectivitystructuralismpoststructuralismcomplexity scienceecology2004Mahwah, NJLawrence Erlbaum(Davis, 2004). The last of these refers to the fact that a complex systems reaction to a particular influence is not wholly determined by the nature of that influence, but is also related to the prior learning, of the system. This means that a complex systems reaction to the same stimulus on different occasions is unpredictable the system learns. Miller, McDaniel, Crabtree and Stange ADDIN EN.CITE Miller20011477147717Miller, William L.McDaniel, Reuben R.Crabtree, Benjamin F.Stange, Kurt C.Practice jazz: Understanding variation in family practices using complexity scienceThe journal of family practiceThe journal of family practice5010complexity sciencemedical practice2001(2001) described complex systems as engaged in improvisation and sense-making in response to changes in the environment by means of multiple feedback loops and nonlinear interactions among agents. These allow systems to self-organise and to adapt to changing circumstances ADDIN EN.CITE Miller20011477147717Miller, William L.McDaniel, Reuben R.Crabtree, Benjamin F.Stange, Kurt C.Practice jazz: Understanding variation in family practices using complexity scienceThe journal of family practiceThe journal of family practice5010complexity sciencemedical practice2001(Miller, McDaniel, Crabtree, & Stange, 2001) as well as to maintain themselves.
Davis and colleagues ADDIN EN.CITE Davis20041475i.e., 14756Davis, BrentInventions of teaching: A genealogy350philosophies of teachingphilosophyteachingtheoriesmetaphysicsphysicalrationalismempiricismmysticismreligionintersubjectivityinterobjectivitystructuralismpoststructuralismcomplexity scienceecology2004Mahwah, NJLawrence ErlbaumDavis20031467146717Davis, BrentSimmt, ElaineUnderstanding learning systems: Mathematics education and complexity scienceJournal for Research in Mathematics EducationJournal for Research in Mathematics Education137-342systemslearningmathematics learningmathematics educationcomplexity2003137-167Davis20051478147817Davis, BrentSumara, DennisComplexity science and educational action research: towards a pragmatics of transformationEducational action researchEducational action research453-464133complexity scienceaction researcheducationchange2005(i.e., Davis, 2004; Davis & Simmt, 2003; Davis & Sumara, 2005) have suggested five necessary but not sufficient, interdependent conditions for the emergence of complexity from the interactions of a collective. These are:
Diversity which refers to variations among the agents in a collective that provide a range of possibilities for novel responses. In classroom contexts diversity is always present.
Redundancy which refers to commonalities among the agents of a system that enable them to interact meaningfully. It can be seen as balancing diversity whilst also contributing to the robustness of the system by ensuring that various agents within it are able to compensate for one anothers weaknesses.
Enabling constraints that organise and focus the activity of the collective but still allow its diversity to be expressed. The actions of individual agents cannot be completely random if complexity is to emerge.
Decentralised control that emphasises that complexity cannot be planned nor its outcomes entirely predicted. Rather, it emerges from the shared endeavours and interactions of autonomous agents.
Neighbour interactions which refer to interactions between agents in the system. In educational settings it refers principally to interactions between ideas.
USING THE CONDITIONS OF COMPLEX EMEREGENCE TO FOSTER CREATIVITY
Plucker and Beghettos (2004) defined creativity in terms of the outcome of either individual or group ability and activity. It ahs also been noted that both individuals and groups of individuals, such as those that comprise school classes, can be considered as complex systems the former nested in the latter. In this section it is argued that creative products can emerge from both of these systems and that in applying the conditions for complex emergence to the classroom they are simultaneously fostered within individual students. Each is considered in turn.
Diversity: This is inevitably present in classrooms and is recognised and capitalised on by the use of cooperative teaching strategies by which students are encouraged to communicate their thinking, to work through their ideas and those of other students, and where consideration of multiple solution methods are encouraged (flexible thinking) (Cropley, 1997). These interactions simultaneously provide individual students with access to a greater diversity of ideas than they would have on their own and hence diversity within their own knowledge schemas is enhanced. Encouraging students to develop their knowledge bases also enhances the available diversity to the extent that the knowledge of individuals is distinct.
Redundancy: Shared knowledge contributes to redundancy in the class. The sorts of interactions that capitalise on the diversity among students include opportunities to work with diverse materials in and in diverse conditions as recommended by Cropley (1997). These also contribute to redundancy by adding to the accumulation of shared ideas and experiences that make future interactions increasingly meaningful. At the same time individuals understandings are made richer, and opportunities for the formation of connections between ideas are enhanced.
Enabling constraints: These provide a degree of safety within which students can exercise their imaginations. In mathematics classrooms they could be in the form of providing problems that are sufficiently constrained to encourage thinking in productive ways and about important mathematical ideas, but not so constrained that their solution is reduced to trivial mimicking of a procedure. Enabling constraints can also be understood in terms of the behavioural norms that govern classroom interactions in such a way that it is safe for students to communicate their thinking. This would include Cropleys (1997) exhortation for teachers to delay judgement of students ideas and extend it to the students as well. In terms of the individual enabling constraints may take the form of appropriately disciplined and reflective thinking including the capacity to evaluate their own thinking and progress (Cropley, 1997).
Decentralised control: This is consistent with the encouragement of independent learning (Cropley, 1997) as well as with the expression of diversity. In terms of the individual it balances the constraints of disciplined thought with associations between disparate ideas and novel approaches to problems. Plucker and Beghetto (2004) claimed that creativity is enhanced by moving back and forth between thinking that is, and is not, confined to the particular task at hand. The effect is to enhance the diversity of ideas brought to bear on the problem.
Neighbour interactions: Interactions between ideas are encouraged through the expression of the diversity of ideas and possibilities available to both the individual and to the group. An environment in which judgment is delayed (Cropley, 1997) allows ideas to remain in play and hence open to use.
CONCLUSION
Complexity science provides a potentially useful framework for teachers to use to guide their thinking about structuring their classroom environments and teaching to maximise the chances of creativity emerging. The conditions for complexity are consistent with recommendations from the creativity research. In addition, they encourage both the creativity of the class and of individuals within it, recognising that creativity can be characteristic of both. Most importantly complexity science provides a way to approach the fostering of creativity that recognises its inherent unpredictability and the consequent fact that there can be no recipe for its development. Seels (2003) notion of watchful anticipation captures the role of the teacher having put in place the conditions for the emergence of creativity.
REFERENCES
ADDIN EN.REFLIST Cropley, A. J. (1997). Fostering creativity in the classroom: General principles. In M. A. Runco (Ed.), The creativity research handbook: Volume one (pp. 83-114). Cresskill, NJ: Hampton Press.
Davis, B. (2004). Inventions of teaching: A genealogy. Mahwah, NJ: Lawrence Erlbaum.
Davis, B., & Simmt, E. (2003). Understanding learning systems: Mathematics education and complexity science. Journal for Research in Mathematics Education, 34(2), 137-.
Davis, B., & Sumara, D. (2005). Complexity science and educational action research: towards a pragmatics of transformation. Educational action research, 13(3), 453-464.
Miller, W. L., McDaniel, R. R., Crabtree, B. F., & Stange, K. C. (2001). Practice jazz: Understanding variation in family practices using complexity science. The journal of family practice, 50(10).
Plucker, J. A., & Beghetto, R. A. (2004). Why creativity is domain general, why it looks domain specific, and why the distinction does not matter. In R. J. Sternberg, E. L. Grigorenko & J. L. Singer (Eds.), Creativity: From potential to realization (pp. 153-167). Washington DC: American Psychological Association.
ABOUT THE AUTHOR
Dr Kim Beswick
University of Tasmania
Faculty of Education
Launceston 7250, Tasmania
Australia
e-mail: HYPERLINK "mailto:[email protected]" [email protected]
PAGE 126 PAGE 126 PAGE 126 PAGE 126 PAGE 126
Kim Beswick
Fostering Creativity by Establishing the Conditions for Complex Emergence
PAGE 130
DG 9: Promoting Creativity for All Students in Mathematics Education, Section 2
PAGE 131
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 127
ICME 11, Mexico, 2008
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