MATHEMATICAL CREATIVITY THROUGH TEACHERS PERCEPTIONS

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1 MATHEMATICAL CREATIVITY THROUGH TEACHERS PERCEPTIONS Maria Kattou, Katerina Kontoyianni, & Constantinos Christou University of Cyprus This study examines elementary school teachers conceptions of creativity, focused in mathematics. The study was conducted among 47 elementary school teachers in Cyprus, using a questionnaire of four open-ended questions. The results revealed that while teachers acknowledge the importance of their role as individuals and as professionals in fostering mathematical creativity, they report several factors of the educational system that inhibit the manifestation of mathematical creativity, excluding themselves from accepting any responsibility. Implications for researchers and policy makers are outlined. INTRODUCTION The year 2009 was declared as the Year of Creativity and Innovation by the European Community. This fact reveals that policy makers globally have started to recognize the importance of creative thinking as an investment in their country s future (Craft, 2007). Nevertheless, any effort to foster creativity in classrooms will ultimately depend on the teacher. Thus, teachers conceptions and knowledge of creativity are of great importance to researchers and policy makers and should be taken into account in developing students creativity (Runco & Johnson, 2002). THEORETICAL FRAMEWORK Creativity as a construct Creativity is a complex construct and as such it has been defined in several ways. Torrance (1995) defines creativity as a product of fluency, flexibility, originality and elaboration. Fluency is the ability of producing many ideas (Gil, Ben-Zvi & Apel, 2007). Flexibility refers to the number, the degree and the focus of approaches that are observed in a solution (Gil, Ben-Zvi & Apel, 2007). At the same time, the term originality refers to the possibility of holding extraordinary, new and unique ideas (Gil, Ben-Zvi & Apel, 2007), and elaboration refers to the ability of extension, improvement or format of an idea (Mann, 2006). Teachers perceptions on the enhancement of creativity According to Best and Thomas (2007), one of the components to ensure creative teaching in mathematics is teachers professional and personal domain. Thus, the theoretical framework of the present study is organized based on the abovementioned distinction. Professional domain refers to teachers role and actions during teaching In Tzekaki, M., Kaldrimidou, M. & Sakonidis, C. (Eds.). Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 1, pp. XXX-YYY. Thessaloniki, Greece: PME. 1-1

2 in order to enhance creativity. Horng and colleagues (2005) argued that teachers should serve more as facilitators, learning partners, inspirers or navigators than as lecturers. In a study conducted by Shriki (2008), teachers believe that a creative environment should include open-ended activities and non-routine problems that give students freedom to apply imaginative ideas and find novel methods or solutions. Similarly, teachers believe that the avoidance of assignments focusing on rote thinking facilitates the development of students' creativity (Fleith, 2000). Furthermore, the use of a variety of teaching methods and aids such as technology, consist of key factors that improve students creativity (Horng et al., 2005). Moreover, teachers believe that students cooperation with classmates of similar interests fosters creativity (Fleith, 2000). The personal domain includes a variety of personality traits such as self-confidence, openness to experience, fantasy orientation, imagination and flexibility of thoughts (Horng et al., 2005). Gardner s (1994) and Horng and his colleagues (2005) research studies emphasized the communicative nature of the teacher. In particular, professional relationships and interactions with colleagues were mentioned by teachers as aspects that contribute to the development of teachers creativity. Perceptions on the barriers inhibiting creativity Despite the fact that most teachers acknowledge the importance of creativity, still many of them do not include it in their teaching. Specifically, teachers identified the following factors which hinder creativity: the use of one right answer, no mistakes, ignored ideas, competition, evaluation, and insufficient knowledge (Fleith, 2000; Shriki, 2008). Other inhibiting characteristics mentioned by teachers include strict discipline, drill work, emphasis on curriculum and lack of time due to various external pressures such as covering the syllabus and helping students succeed in exams (Fleith, 2000; Shriki, 2008). Consequently, teachers tend to emphasize memorization and rote thinking in teaching rather than creativity. PURPOSE OF THE STUDY Despite the importance of fostering creativity in elementary school settings, the literature review indicates that little attention has been paid to teachers conceptions regarding creativity (Diakidoy & Phtiaka, 2001). The existing body of research focused on teachers views concerning characteristics of creative individuals, practices or environments. This study adds to the research literature some evidence about teachers perceptions on mathematical areas which encourage the development of students creativity by providing specific examples of tasks. Therefore, the present study purports to: (1) reveal teachers perceptions with respect to the characteristics and practices of a creative teacher, (2) recommend mathematical areas considered by the teachers to be more suitable for facilitating creativity, by presenting specific examples of creative mathematical tasks and (3) reveal teachers 1-2 PME

3 perceptions regarding the opportunities offered by the Cypriot elementary educational system to foster mathematical creativity. METHOD Sample - Procedure Data were collected through an anonymous questionnaire that was administered to 47 randomly selected teachers in elementary schools in Cyprus. These teachers varied in their teaching experience from 1 to 19 years. Most of them (55.32%) had long teaching experience (10-19 years). The questionnaire was disseminated to teachers by hand or through and it was returned in the same way. Instrument The research instrument consisted of four open-ended questions. Teachers were asked not only to answer the questions but to justify their responses (see Figure 1). The justifications and the examples provided by teachers were analyzed through content analysis (Weber, 1990) i.e., the collected data were broken down and coded into different concepts/labels. After clustering these labels, categories were generated, thus making it possible to emphasize general themes and draw conclusions about teachers conceptions regarding mathematical creativity. 1. What sort of characteristics and practices would you identify in a mathematically creative teacher? 2. To what extent do you use creative activities/strategies during mathematics teaching? Are there any mathematical areas that are more suitable for fostering creativity and others that are not appropriate? 3. Please provide an example of mathematical activity that you consider to be creative. 4. What do you think about the opportunities offered by the Cypriot elementary educational system to foster mathematical creativity? Justify your answer. Figure 1: Items of the questionnaire. RESULTS The results of the study are presented according to the four items of the questionnaire. Question 1: Characteristics and practices of a mathematically creative teacher In an attempt to describe a mathematically creative teacher, participants provided a variety of characteristics that could be categorized in two groups: personality traits and professional abilities (Table 1). These categories are in accord with Best and Thomas s (2007) notion of teachers professional and personal domain. Regarding professional abilities, the participants focused on the variety of activities, teaching methods and manipulatives used during teaching (53.19%). Teachers believed that PME

4 certain pedagogical approaches such as inquiry learning (51.07%), the use of openended tasks (38.30%) and differentiation of teaching according to students needs (21.28%) are effective ways in working towards creativity. Concepts such as cooperative learning (17.02%), use of technological tools (17.02%) and realistic mathematics (12.77%) where mentioned by teachers as strategies necessary to differentiate the curriculum for creative students. Teachers put less emphasis on their mathematical knowledge (12.77%) and their role as facilitators (8.51%). In addition, teachers considered originality (55.32%), flexibility (42.55%) and imagination (36.17%) as the most important characteristics that referred to personality traits. Furthermore, teachers perceived the openness to new ideas (28.79%), perseverance (21.28%), and divergent thinking (19.15%) as the factors that foster creativity. A percentage of 10.64% of teachers suggested communication, either with students or with other teachers, as a characteristic of a creative teacher. Finally, very few of them (8.51%) proposed critical thinking as an important factor that contributes to creativity. Personality traits N (%) Professional abilities N (%) Originality 26 (55.32) Use of a variety of activities, 25 (53.19) teaching methods and manipulatives Flexibility 20 (42.55) Investigation-discovery of knowledge 24 (51.07) Imagination 17 (36.17) Use of open-ended tasks 18 (38.30) Open mindedness 14 (28.79) Opportunities for all students- 10 (21.28) Differentiation Perseverance 10 (21.28) Combination of cooperative and 8 (17.02) individual work Divergent thinking 9 (19.15) Use of technology 8 (17.02) Communicative nature 5 (10.64) Use of realistic problems 6 (12.77) Critical thinking 4 (8.51) Mathematics knowledge-efficacy 6 (12.77) Teacher as facilitator- coordinator 4 (8.51) Table 1: Characteristics and practices of a mathematically creative teacher. Question 2: The use of creative activities/strategies during mathematics teaching The second question requested teachers to note whether they use creative activities/strategies during mathematics teaching and to mention mathematical areas they consider to be more suitable for facilitating creativity. The majority of teachers (65.96%) stated that very often they use creative activities during their daily lessons and believed that all mathematical topics are suitable for fostering creativity (78.72%). In their own words: I believe that creativity can be applied in all mathematical areas, because creativity characterizes both students and tasks. Thus, all mathematical activities could be 1-4 PME

5 considered to be creative, if the teacher wishes to. For example, by modifying a mathematical task (think of another method, find as many solutions as you can) you could change the aim of the activity. On the other hand, students can work with an activity in their own way, indicating original solution paths. The following extracts show the emphasis placed by teachers on their role in developing creative activities for students; specifically, they focus on the availability of suitable activities and manipulatives: We can use creative activities in all mathematical areas, as long as there is enough time and the appropriate manipulatives are provided. Since the teacher has a collection of appropriate tasks that promote originality and flexibility, all mathematical areas can foster creativity. If teachers know well the mathematical content, then creativity can be found in all mathematical areas. Systematic preparation is needed. In total, teachers acknowledged problem solving (12.77%) and geometry (12.77%) as the most popular mathematical areas to apply creative tasks. A teacher referring to problem solving reported that Through problem solving, students imagination and flexibility is developed, until the proper solution is found. In contrast, 17 teachers (36.17%) thought that algorithms and routine problems are not recommended for creative tasks: In my opinion, during teaching the algorithms for the four operations, you can not apply creativity, since what you teach is mechanistic. Less or even no creativity can be applied in closed activities where the way of solution is given. Question 3: Examples of creative mathematical activities The mathematical examples provided by teachers were divided into two categories; creative and non-creative mathematical tasks. Torrance s features of fluency, flexibility, originality and elaboration had to characterize the tasks in order to be considered creative. For the present study, only indicative creative mathematical tasks will be presented. A significant proportion of teachers (40.43%) provided examples of creative mathematical activities derived form Numbers and operations such as the following task, which allows for multiple solutions and novel responses: Look at the numbers: 23, 20, 15, 2. Which number does not belong here? Why? Participants also provided creative examples from Geometry (27.66%) such as covering an area in different original ways. In addition, eight teachers suggested examples related to problem solving and problem posing. Some indicative openended problems are the following: A bear weights 500Kg. How many children weigh the same as a bear? PME

6 We can give students information about airport tickets, accommodation etc. Then we can ask them to decide which will be the best choice for their vacations and to justify their answers. Question 4: Opportunities for mathematical creativity offered by the Cypriot elementary educational system The majority of teachers (83%) consider that mathematics, as presented and taught in Cyprus, are not connected with creativity. Thus, they provided several barriers that inhibit mathematical creativity, as shown in Table 2. Barriers N (%) Focus on the answer than in the process 14 (29.79) Content to be covered 13 (27.66) Restricted time 12 (25.53) Textbooks and activities provided 8 (17.02) Teachers training 7 (14.89) Suitable teaching material 2 (4.26) Table 2: Barriers inhibiting mathematical creativity As shown in Table 2, the pressure of covering content (27.66%) in limited time (25.53%) causes teachers to focus on the right answer (29.79%), ignoring the process followed and to limit in exercises provided in textbooks (17.02%). A teacher s view that summarizes the abovementioned barriers follows: Teachers usually adopt the textbook activities, without differentiating them and without challenging the students. They are more interested in the presentation of typical solutions. They are not particularly concerned in helping students broaden their thinking; instead they insist on preparing them for examinations or cover the suggested content form the material. Another barrier for not fostering creativity in mathematics is the lack of in-service training (14.89%) as well as the lack of suitable material (4.26%): In my opinion, teachers have not been informed about the importance of creative activities and their use in mathematics. I believe that training courses that involve definitions of creativity, examples of mathematical creative activities, as well as evaluation methods are required. DISCUSSION The aim of the present study was to investigate teachers perceptions regarding the characteristics and the practices of a mathematically creative teacher, in order to foster creativity in students. Moreover, factors of the Cypriot elementary educational system which teachers conceive as barriers toward mathematical creativity have been outlined. 1-6 PME

7 With regard to the characteristics and the practices of a mathematically creative teacher, Cypriot elementary school teachers proposed both professional and personality characteristics. The proposed professional characteristics include general pedagogical approaches, such as cooperative learning, learning through inquiry and the use of technology, verifying similar findings (Fleith, 2000; Horng, et al., 2005). Moreover, teachers perceive that the use of open-ended tasks, differentiation and the variety in activities, teaching methods and manipulatives can contribute to the manifestation of creativity. It can be assumed that teachers consider that through teaching, creative potential might be maximized in all students. With respect to personality traits, teachers mainly referred to Torrance s characteristics (1995) of creative behaviour, namely originality and flexibility. In addition, imagination and the ability to being open to new ideas were also mentioned, enhancing the findings of Horng and colleagues (1996). Nevertheless, professional relationships and interactions with colleagues were not strongly perceived to be a possible facilitator to creativity from Cypriot teachers, despite related findings (Gardner, 1994; Horng et al, 2005). It can be deduced that teachers focus only on themselves as personalities and individuals and do not extend to work relationships which include third persons. Regarding teaching creatively, the majority of participants believed that they apply creative activities while teaching mathematics. Despite our intention to suggest mathematical areas considered to be more suitable for fostering creativity, teachers noted that all mathematical areas can serve to this end. When teachers asked to provide specific examples, problem solving was considered to be an ideal way of fostering creativity, verifying Shriki s findings (2008). At the same time, teachers perceive that the teaching of algorithms does not allow creativity. Based on the abovementioned results, it is argued that teachers believe that what really matters in relation to mathematical creativity is not the mathematical content itself, but rather the teacher and his/her practices. Even though teachers were asked to express their opinion regarding opportunities offered by the Cypriot educational system to foster creativity, their responses focused on specific barriers inhibiting the implementation of creative mathematical activities. Mathematics curriculum in combination with textbooks represents the main obstacle, enhancing Shriki s findings (2008). In an attempt to cover mathematical content in restricted time, teachers ignore the use and influence of mathematical activities in students understanding. Teachers responses might be explained by the lack of preservice and in-service training related to creativity and by the inadequacy of mathematics curriculum and textbooks to provide opportunities to foster creativity in educational settings. For this reason, new mathematical textbooks should be developed that will give the opportunity for a creative implementation of mathematical content. In addition, further research is required for the development of related teacher training. PME

8 This study offers researchers an insight into teachers perceptions of mathematical creativity and the way it is incorporated in everyday teaching. Summing up, it can be deduced that the results of the present study give strong evidence to support that although elementary school teachers consider themselves as a key factor in developing mathematical creativity, at the same time they do not accept any responsibility as one of the reasons of hindering creativity. Instead, they focus only on aspects of the educational system. References Best, B. & Thomas, W. (2007). The Creative Teaching and Learning Toolkit. Continuum International Publishing Group: New York. Craft, A. (2007). Possibility thinking in the early years and primary classroom. In A. G. Tan (Ed.), Creativity: A handbook for teachers. Singapore: World Scientific. Diakidoy, I.A., & Phtiaka, H. (2001). Teachers beliefs about creativity. In S. S. Nagel (Ed.), Handbook of policy creativity: Creativity from diverse perspectives (pp ). Huntington, NY: Nova Science. Fleith, S. D. (2000). Teacher and student perceptions of creativity in the classroom environment. Roeper Review, 22(3), Gardner, H. (1994). Creating Minds: An Anatomy of Creativity Seen Through the Lives of Freud, Einstein, Picasso, Stravinsky, Eliot, Graham, and Gandhi, Basic Books, New York. Gil, E., Ben-Zvi, D., & Apel, N. (2007).What is hidden beyond the data? Helping young students to reason and argue about some wider universe. In D. Pratt & J. Ainley (Eds.), Reasoning about Informal Inferential Statistical Reasoning: A collection of current research studies. Proceedings of the Fifth International Research Forum on Statistical Reasoning, Thinking, and Literacy (SRTL-5), University of Warwick, UK, August, Horng, J., Hong, J., ChanLin, L., Chang, S., & Chu, H. (2005). Creative teachers and creative teaching strategies. International Journal of Consumer Studies. 29(4), Mann, E. (2006). Creativity: The Essence of Mathematics. Journal for the Education of the Gifted, 30(2), Runco, M. A. & Johnson, D. J. (2002). Parents and teachers implicit theories of children s creativity: A cross-cultural perspective. Creativity Research Journal, 14(3/4), Shriki, A. (2008). Towards promoting creativity in mathematics of pre-service teachers: The case of creating a definition. In R. Leikin (Ed.), Proceedings of the 5th International Conference on Creativity in Mathematics and the Education of Gifted Students (p.p ). Haifa, February, Torrance, E. P. (1995). The beyonders in why fly? A philosophy of creativity. Norwood, NJ: Ablex. Weber, R.P. (1990). Basic content analysis, 2 nd Edition. Newbury Park: Sage Publications. 1-8 PME

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