ࡱ>  @~Ibjbj >؝؝|?:::8r<Evv   BBBDDDDDDD$FR*ID,$>>,$,$D  4Dn(n(n(,$~  Dn(,$Dn(n(Y>@ j {z\:%ZE?BlD0E[?6I'I,@I@B:n(4 0!BBBDDD:(X: TEACHERS' CONCEPTIONS OF MATHEMATICAL CREATIVITY FOONG PUI YEE Abstract. How do teachers who are delivering the curriculum conceive of mathematical creativity? The purpose of this survey is to explore how mathematical creativity is seen by a group of teachers. A persons assumption and conception of what constitutes a thing or an idea could be socially significant and real in their consequence of actions or behaviours. The results painted a positive picture for mathematics education where majority of teachers conceived of mathematical creativity as embedded in the activities of mathematics lessons that have problem solving as a focus. Key words: teachers conception, mathematical creativity, problem solving INTRODUCTION In education it is an edict that all students should be given opportunities to develop their potentials to the fullest. If educators are of the view that creativity exists as a potential in every individual, then fostering creativity in all aspects of learning should be part of a tradition in the educational process of a student. So what is creativity and what should teachers foster? Among the many theories, Runco (2004)s theory of personal creativity is more appropriate for the educational development of individuals as it focuses on the individual and in particular on his or her interpretative capacities, choices, decisions, judgements and intentions. His view is that seeing creativity in strictly objective terms is problematic as very often the emphasis is on the creative product that has being judged as useful and original based on some social consensus views. The process which generated the product and the individuals perspectives are overlooked. This integrated view of creativity which consider the processes and personal properties would be more agreeable with educators where fostering the individual is a given. Cropley (1997) in his review on promoting creativity in the classroom identified ten cognitive aspects of creativity that teachers should strive to promote in students (p 92): Possession of a fund of general knowledge. Knowledge of one or more special fields. An active imagination. Ability to recover, discover or invent problems. Skill at seeing connections, overlaps, similarities and logical implications. Skill at making remote associations, accepting primary processes and forming new Gestalts. Ability to think up many ways to solve problems. A preference for accommodating rather than assimilating. Ability and willingness to evaluate their own work. Ability to communicate their results to other people. The idea of encouraging mathematical creativity implicitly through these aims is clearly attractive to curriculum developers and teachers. So how do the teachers who are delivering the curriculum conceive of mathematical creativity and its role in the learning objectives of the mathematics syllabus? How significant are their conceptions to their classroom practices? For the purpose of this discussion paper, the author presents the results of a small survey among pre-service and in-service teachers on their conceptions of mathematical creativity and its implication for study of classroom practices. SURVEY ON TEACHERS CONCEPTIONS It is important to recognize that the key figures responsible for changing the ways in which mathematics is taught and learned in the classroom are the teachers. How curricula innovation such as promoting creativity in the classroom is implemented depends in turn on teachers' conception of the innovation and the mathematics they are teaching. The purpose of this survey was to explore how mathematical creativity was conceived by a group of teachers. The focus was on what was seen as mathematical creativity by the pre-service teachers and experienced teachers. A persons assumption and conception of what constitutes a thing or an idea could be socially significant and real in their consequence of actions or behaviours. Like the term problem solving, there were differing definitions and how teachers interpreted and conceived of them would lead to different activities in the mathematics classroom. In this survey, the teachers were posed this question: What do you see in your minds eye when I say mathematical creativity?. They were not required to define it. They were supposed to write down what they saw as mathematical creativity and their answers would rely on their conception rather than perception. The process of conception would occur at a high cognitive level as what one could see should be drawn from ones previous experiences. Not everyone would necessarily see the same thing. By capturing different images of mathematical creativity from teachers-in-training to experienced teachers would contribute to a deeper reflection on different aspects as to how creativity could be promoted in the teaching and learning of mathematics. Twenty pre-service teachers doing their Postgraduate Diploma in Education and twenty in-service teachers doing their Master in Education degree participated in this survey. Their responses were analysed and categorised. Some expressions of their conceptions of mathematical creativity were: To use different methods to solve the same question and being able to explain the rationale behind it The ability to see Maths from a different angle. Solving maths problems unconventionally Coming up with ideas that most people would not have Extension of mathematics learning beyond the classroom, without use of textbook It is more on the teacher facilitating the learning and students discovering the wonders of mathematics themselves Some teachers contributed more than one statement of conceptions. The teachers conceptions were categorised into two perspectives: the learners perspective and the teachers perspective. In the learners perspective, mathematical creativity was seen from what students did in mathematics learning, while the teachers perspective was seen from what teachers did to promote it. Each statement of conceptions in the learners perspective was parsed into three components: the Action, the Context and the Creativity. The Action indicated what students did with illustrative verbs like solve, see, create, discover, learn etc. The Context indicated the situation of the action such as the problems, ideas, patterns, concepts etc., and the Creativity indicated the adjective or phrases that respondent had used to describe the event as creative. Table 1[Appendix A] highlighted some of these conceptions that would form a picture of how mathematical creativity played out in the perspective of the learners. Table 2 [Appendix A] showed the actions teachers took and the context that they used to promote mathematical creativity in the learners. Each table also presented the frequency of occurrences that the pre-service and in-service teachers expressed their conceptions. DISCUSSION OF RESULTS The picture that emerged from the teachers conceptions of mathematical creativity showed that majority of them saw it as an event where students solve problems that could be in the context of mathematics challenge problems, higher-order thinking tasks, or investigative tasks in different, non-conventional, unique or in ways that were not taught before. The concept of mathematical creativity for these teachers was closely related to mathematical problem solving where the creativity was found in the process of the solution. The creativity as described in the ways that problems were solved seemed to belong to the individuals as terms like unique to the person, personal touch and not taught before were used in some conceptions. More pre-service teachers conceived of mathematical creativity in this aspect than the experienced teachers. More of the experienced teachers than pre-service teachers conceived of mathematical creativity as creating, constructing or coming up with new, different or out-of-the-box ideas or patterns which could be interpreted as attributes of the learners. Two teachers saw mathematical creativity as in a person who thought like a master chess player or who applied multiple intelligence. As for students learning or discovering concepts and solution strategies, the creativity element was attributed to environmental factors such as beyond classroom, related to real-life, integrated with Arts or without use of textbooks. Three pre-service teachers conceived of mathematical creativity in students ability to explain, reflect and justify their ideas or solutions through sharing in group work. Table 2 showed more experienced teachers conceptions in the perspective of the teacher and teaching than the pre-service teachers. They conceived of teachers explaining concepts or facilitating solution strategies through use of games or ICT. Teachers taught without the use of textbook and related concepts to real-life. Teachers crafted open-ended problems. Teachers assessed with alternative methods and teachers catered to different learning styles. These were images of what teachers did to promote mathematical creativity in the students. To summarise this small survey, the conceptions of majority of the teachers reflect common beliefs that mathematics is about problem solving and problem solving is a creative process. However, it would seem that giving students problems to solve as the means for mathematical creativity maybe seen to be too narrow a conception. Often in many mathematics classrooms, when students are given problems to solve, it is usually within very familiar context for applying techniques that have been taught. It will be worrisome if teachers were to think that creativity is only found in those students who have it in them to think or solve problems unique to themselves or in ways that were not taught before. However, there are teachers who believe that creating or posing problems is also a creative problem solving process. Mathematical creativity is not just about what pupils do but also what teachers do as reflected in some of the experienced teachers conceptions of what teachers can do to offer opportunities for student creativity. These conceived actions of the teachers such as teachers explain, facilitate, assess, craft and cater to different learning styles can be found as activities embedded in regular classrooms. If the premise that ones conceptions of learning and teaching can evoke behaviours which make the conceptions a reality is true, then mathematical creativity like problem solving will be an integral part of everyday mathematics activity. In this way teachers will be mindful in choosing, adapting or creating problems that would offer scope for creativity whereby students are encouraged to be creative with the tools available in them. CONCLUSION AND FUTURE WORK This small scale survey paints a very positive picture for mathematics education where majority of teachers conceive of mathematical creativity as embedded in the activities of mathematics lessons that have problem solving as the focus. Although the notion of teachers deliberately or unknowingly inhibiting student mathematical creativity maybe unacceptable, there are evidences to show that such incidents have occurred and have consequences. In a study on general creativity education, Tan (2004), finds that teachers self-reports of classroom experiences pertaining to enhancing creativity shows distinct discrepancies between their understandings of knowing what is good and their act of doing what may work. She learns that teachers seldom conduct learning activities they believe could enhance creativity. Instead they spend most of their instructional time carrying out routine and teacher-centred activities. Teachers attributes their inconsistent behaviours to reasons such as insufficient curricular time, inadequate support and low self-confidence in managing such activities. In particular to the mathematics classroom, there is reason to believe that teachers conceptions of mathematical problem solving could also have constraints on their effort to promote creativity in this aspect. For many teachers in Singapore they believe that they do teach problem solving in mathematics all the time especially through mathematics word problems and the use of targeted heuristics for certain challenging problems. This conception of problem solving can be viewed from Schroeder & Lesters (1989) interpretation of the different roles of problem solving: teaching for problem solving, teaching about problem solving, and teaching via problem solving. According to Foong (2002) teachers in Singapore from primary to secondary level used problems for its role in teaching for problem solving where the emphasis is on learning mathematics for the main purpose of applying it to solve problems after learning a particular topic. There is very little scope for creativity if students were just learning to use taught techniques. Explicit heuristics instructions as teaching about problem solving are being practiced in the mathematics classroom where emphasis is on teaching the heuristics recommended in the syllabus and targeting them at certain non-routine problems. Future work will need to equip teachers with a clearer picture of these different roles of problem solving especially the role of teaching mathematics via problem solving that used problem-based investigative and open-ended tasks. Open-ended problem-based and problem posing activities do allow for student creative problem solving. More classroom studies should be conducted to observe teacher practices for their alignment of their conceptions of problem solving and creativity in mathematics. REFERENCES Cropley, A.J., (1997). Fostering creativity in the classroom. In M.A.Runco (ed.). The creativity research handbook. Vol.1. pp83-115. NJ: Hampton Press. Foong, P.Y. (2002). Roles of Problem to Enhance Pedagogical Practices in the Singapore Classrooms. The Mathematics Educator. Vol. 6, No.2, 15-31. Runco, M. (2004). Personal creativity and culture. In L. Sing, H.H. Hui, & Y.C. Ng. (eds.). Creativity when east meets west. pp. 9-23. Singapore: World Scientific. Schroeder, T.L. & Lester, F.K. (1989). Developing understanding in mathematics via problem solving. In P.R. Trafton & A.P. Shulte (Eds.), New directions for elementary school mathematics: 1989 yearbook. Reston, Va: National Council of Teachers of Mathematics. Tan, A. G. (2004). Singapores creativity education. In L. Sing, H.H. Hui, & Y.C. Ng. (eds.). Creativity when east meets west. pp. 277-305. Singapore: World Scientific. ABOUT THE AUTHOR Foong Pui Yee, Ph.D. Mathematics and Mathematics Education Academic Group National Institute of Education Nanyang Technological University 1, Nanyang Walk Singapore 637616 Tel: (65)67903913 Email: puiyee.foong@nie.edu.sg APPENDIX A Table 1: Teachers conceptions of mathematical creativity in the learners perspective Action Context CreativityPre-service teachers: frequency count of conceptionsIn-service teachers frequency count of conceptionsSolve/ FindProblems (Challenge, investigation, higher-order, open-ended) New, Differently, Non-conventionally, alternative, Uniquely, Not taught before, Variety, More than 1 way, Simply & efficiently149Create/ Construct/ Design/coming up withIdeas, knowledge, games, puzzles New thingsNew way, different, alternative, fun, out-of-the-box, variety, individual touch35See/ Visual/ Think/ AnalyseIdeas, patternUnconventionally, uniquely, like a master chess player; multiple-intelligence,23Learn/Discover/ConnectMathematics concepts, solution strategies, Relate to real-life, integrate with Art, beyond classroom, without textbook44Explain/Reflect/ JustifyConcepts/ solutionsInteractively; groupwork30 Table 2: Teachers conceptions of mathematical creativity in the teachers perspective Action ContextPre-service teachers: frequency count of conceptionsIn-service teachers frequency count of conceptionsTeach/Explain/ FacilitateConcepts/ Solutions36UseGames/ ICT22Craft/ ConstructProblems, open-ended problems01AssessAlternative methods02Cater to different pupilsLearning styles02  PAGE 126 PAGE 126 PAGE 126 PAGE 126 PAGE 126    Pui Yee Foong Teachers' Conceptions of Mathematical Creativity PAGE 138 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 2 PAGE 139 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 133 ICME 11, Mexico, 2008  5D = H 34\]stMN~ѽyk\y\P\P\P\P\P\P\P\PhpgCJaJmH sH h *#hpgCJaJmH sH h *#hpg56CJaJh *#hpg6CJaJmH sH h *#hpg6CJaJh *#hpgCJaJh *#hpg5CJaJ!hpgCJ$OJQJ\aJ$mH sH 'h *#hpgCJ$OJQJ\aJ$mH sH h *#hpgCJ$OJQJ\aJ$ h *#hpgOJQJaJmH sH hpgOJQJaJmH sH 56DE 5^uO$ & F0xd\$`0a$gdpg $d\$a$gdpgd\$gdpg$d\$]^a$gdpg $d\$a$gdpgd\$gdpg|G}I!"#lL    X { # ###92:2T28889$9'999:ƻٍٍٍ~h *#hpg56CJ]aJh *#hpg56CJaJh *#hpg5CJaJmH sH h *#hpg6CJaJmH sH h *#hpgmH sH hpgmH sH h *#hpgCJaJh *#hpg5CJaJhpgCJaJmH sH h *#hpgCJaJmH sH 1#'LF # # d\$^gdpg$d\$^a$gdpg$ & p@ P !d\$a$gdpgd\$gdpg $d\$a$gdpg $d\$a$gdpg$ & F0xd\$`0a$gdpg ###+5262:2U26===5>>l?r@A*$ & 0` P@01$7$8$H$^`0a$gdpg$0*$^`0a$gdpg$0^`0a$gdpg $d\$a$gdpg $d\$a$gdpg $d\$a$gdpg::7<;<<<=====>5>>>>$?D?l???6@p@r@@@A BBBǾ泧}l^VhpgCJaJh *#hpg5CJ\aJ!h *#hpgB*CJ_HaJphh *#hpgCJaJmHsHh *#hpg6@CJaJh *#hpg@CJaJh *#hpg6CJaJh *#hpgCJaJhH+5CJaJh *#hpg5CJaJmH sH h *#hpg56CJ]aJh *#hpg5CJaJh *#hpg56CJaJAA/ADAyAAAAAA B BBBqBsBzB{BBBB $$Ifa$gd0B x`gdpg $x`a$gdpg $d\$a$gdpg$a$gdpg $d\$a$gdpg$d\$`a$gdpgBBBQBpBqBEE3F4FzG{G|G}GGGGGGGGGGGGGGGGGGGGGGGGGGGGGλyuyuyuyuhjhUh0JmHnHu h0Jjh0JU hpghjM h *#hpghiMhpg56 hpg56hpgmH sH hpg56mH sH hShpg56mH sH "hk7hpg56CJaJmH sH "hk7hk756CJaJmH sH )BBBBBBB $$Ifa$gd0BBBCDCECyCCG>5555 $Ifgdk7 $Ifgd0Bkd$$IfTlr ~Tr a t0!44 laTCCCCC4kd$$IfTlr ~Tr a t0!44 laT $$Ifa$gd0B $Ifgdk7CCD DpDrDtD $$Ifa$gd0B $Ifgdk7 $Ifgd0BtDuDDDDI@77 $Ifgdk7 $Ifgd0Bkd^$$IfTlr ~Tr a t0!44 laTDDDD E=4 $Ifgd0Bkd $$IfTlr ~Tr a t0!44 laT $$Ifa$gd0B E7EEEEE4kd$$IfTlr ~Tr a t0!44 laT $$Ifa$gd0B $Ifgdk7EEEEEE $$Ifa$gd0B $Ifgdk7 $Ifgd0BEEE4FI>0 $ xa$gdpg B xgdpgkde$$IfTlr ~Tr a t0!44 laT4F5F*B*ph4@4 Header  9r 4 @4 Footer  9r .)@!. 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