| | | |

## Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks?

#### Erhan BİNGÖLBALİ [1] , Ferhan BİNGÖLBALİ [2]

This study explores whether mathematics tasks in primary school mathematics textbooks provide opportunities for divergent and convergent thinking. Four mathematics textbooks (one from each of first to fourth grades) are examined for this purpose. A task is divided into three segments for the analysis and the segments are named as the beginning, the intermediary, and the end. These segments are analysed in terms of the numbers of entry points, solution methods, and correct outcomes respectively. The modes of the segments enable us to identify six different tasks. Tasks that definitively have an open-ending (multiple correct outcomes) are considered to have divergent thinking features and those which have only one correct outcome are deemed to have convergent thinking characteristics. The study reveals that the textbooks provide opportunities for both divergent and convergent thinking, yet more chances are particularly given for convergent thinking. The findings are discussed in relation to divergent and convergent thinking alongside creativity and some implications are provided for textbooks studies.
Divergent thinking, Convergent thinking, Mathematics textbook analysis, Creativity, Open-ended tasks
• Altay, M. K., Gümüş, F. Ö., Yaman, H., Özer, A. & Akar, Ş. Ş. (2018). First grade primary school mathematics textbook. Ankara: MHG Publication.
• Atlı, A., Doğangüzel, E. E., Güneş, A. & Şahin, N. (2018). Second grade primary school mathematics textbook. Ankara: MEB State Books.
• Balka, D. S. (1974). Creative ability in mathematics. Arithmetic Teacher, 21(7), 633-636.
• Bennevall, M. (2016). Cultivating creativity in the mathematics classroom using open-ended tasks: A systematic review. Retrieved from http://www.diva-portal.org/smash/get/diva2:909145/FULLTEXT01.pdf
• Bingolbali, E. (2019). An analysis of questions with multiple solution methods and multiple outcomes in mathematics textbooks. International Journal of Mathematical Education in Science and Technology, DOI: 10.1080/0020739X.2019.1606949.
• Bowen, G. A. (2009). Document analysis as a qualitative research method. Qualitative Research Journal, 9(2), 27-40.
• Brophy, D. R. (2001). Comparing the attributes, activities, and performance of divergent, convergent, and combination thinkers. Creativity Research Journal, 13(3-4), 439-455.
• Cai, J. (2000). Mathematical thinking involved in US and Chinese students' solving of process-constrained and process-open problems. Mathematical Thinking and Learning, 2(4), 309-340.
• Cropley, A. (2006). In praise of convergent thinking. Creativity Research Journal, 18(3), 391-404.
• Eisenmann, T., & Even, R. (2011). Enacted types of algebraic activity in different classes taught by the same teacher. International Journal of Science and Mathematics Education, 9, 867–891
• Foster, C. (2015). The convergent–divergent model: An opportunity for teacher–learner development through principled task design. Educational Designer, 2(8), 1-25.
• Genç, N., Güleç, H., Şahin, N. & Taşcı, S. (2018). Third grade primary school mathematics textbook. Ankara: MEB State Books.
• Glasnovic Gracin, D. (2018). Requirements in mathematics textbooks: a five-dimensional analysis of textbook exercises and examples. International Journal of Mathematical Education in Science and Technology, 49(7), 1003-1024.
• Guilford, J.P. (1967). The nature of human intelligence. McGraw-Hill, New York.
• Haggarty, L., & Pepin, B. (2002). An investigation of mathematics textbooks and their use in English, French and German classrooms: Who gets an opportunity to learn what? British Educational Research Journal, 28(4), 567–590.
• Han, S. Y., Rosli, R., Capraro, R. M., & Capraro, M. M. (2011). The textbook analysis on probability: The case of Korea, Malaysia and US textbooks. Research in Mathematical Education, 15(2), 127-140.
• Haylock, D. (1987). A framework for assessing mathematical creativity in schoolchildren. Educational Studies in Mathematics, 18, 59-74.
• Haylock, D. (1997). Recognising mathematical creativity in schoolchildren. ZDM, 29(3), 68-74.
• Haylock, D. W. (1985). Conflicts in the assessment and encouragement of mathematical creativity in schoolchildren. International Journal of Mathematical Education in Science and Technology, 16(4), 547-553.
• Imai, T. (2000). The influence of overcoming fixation in mathematics towards divergent thinking in open-ended mathematics problems on Japanese junior high school students. International Journal of Mathematical Education in Science and technology, 31(2), 187-193.
• Japardi, K., Bookheimer, S., Knudsen, K., Ghahremani, D. G., & Bilder, R. M. (2018). Functional magnetic resonance imaging of divergent and convergent thinking in Big-C creativity. Neuropsychologia, 118, 59-67.
• Kasar, N (2013). To what extent alternative solution methods and different question types are given place in mathematics teaching?: Examples from real classroom practices (Unpublished master’s thesis). University of Gaziantep: Gaziantep.
• Kim, K. H. (2008). Meta‐analyses of the relationship of creative achievement to both IQ and divergent thinking test scores. The Journal of Creative Behavior, 42(2), 106-130.
• Kwon, O. N., Park, J. H., & Park, J. S. (2006). Cultivating divergent thinking in mathematics through an open-ended approach. Asia Pacific Education Review, 7(1), 51-61.
• Lee, K. H. (2017). Convergent and divergent thinking in task modification: a case of Korean prospective mathematics teachers’ exploration. ZDM, 49(7), 995-1008.
• Leikin, R., & Lev, M. (2007, July). Multiple solution tasks as a magnifying glass for observation of mathematical creativity. In Proceedings of the 31st International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 161-168). Seoul, Korea: The Korea Society of Educational Studies in Mathematics.
• Leikin, R., & Levav-Waynberg, A. (2008). Solution spaces of multiple-solution connecting tasks as a mirror of the development of mathematics teachers’ knowledge. Canadian Journal of Science, Mathematics, and Technology Education, 8(3), 233-251.
• Leung, S. S. (1997). On the open-ended nature in mathematical problem posing. In E. Pehkonen (Ed.) Use of Open-Ended Problems in Mathematics Classroom (pp. 26-33). Finland: University of Helsinki.
• Levav-Waynberg, A., & Leikin, R. (2012). The role of multiple solution tasks in developing knowledge and creativity in geometry. The Journal of Mathematical Behavior, 31(1), 73-90.
• Nohda, N. (2000). Teaching by Open-Approach Method in Japanese Mathematics Classroom. In T. Nakahara & M. Koyama (Eds.), Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (PME), 24(1), 39-55. Retrieved from http://files.eric.ed.gov/fulltext/ED466736.pdf
• Olsher, S., & Even, R. (2014). Teachers editing textbooks: Changes suggested by teachers to the math textbook they use in class. In K. Jones, C. Bokhove, G. Howson, & L. Fan (Eds.), Proceedings of the International Conference on Mathematics Textbook Research and Development (ICMT-2014) (pp. 43–48). Southampton: University of Southampton.
• Özçelik, U. (2018). Fourth grade primary school mathematics textbook. Ankara: Ata Publication.
• Patton, M. Q. (2015). Qualitative research & evaluation methods: Integrating theory and practice (4th ed.). Thousand Oaks, CA: Sage.
• Pehkonen, E. (1997). Introduction to the concept “open-ended problem. In E. Pehkonen (Ed.) Use of Open-Ended Problems in Mathematics Classroom (pp. 7-11). Finland: University of Helsinki.
• Reitman, W. (1965). Cognition and thought. New York: Wiley.
• Robitaille, D. F., Schmidt, W. H., Raizen, S. A., McKnight, C. C., Britton, E. D., & Nicol, C. (1993). Curriculum frameworks for mathematics and science (Vol. 1). Vancouver, Canada: Pacific Educational Press.
• Shen, W., Hommel, B., Yuan, Y., Chang, L., & Zhang, W. (2018). Risk-taking and creativity: Convergent, but not divergent thinking is better in low-risk takers. Creativity Research Journal, 30(2), 224-231.
• Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 29(3), 75-80.
• Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: From research to practice. Mathematics Teaching in the Middle School, 3(4), 268-275.
• Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbook. Dordrecht, The Netherlands: Kluwer.
• Wronska, M. K., Bujacz, A., Gocłowska, M. A., Rietzschel, E. F., & Nijstad, B. A. (2019). Person-task fit: Emotional consequences of performing divergent versus convergent thinking tasks depend on need for cognitive closure. Personality and Individual Differences, 142, 172-178.
• Wu, X., Yang, W., Tong, D., Sun, J., Chen, Q., Wei, D., Zhang, Q., Zhang, M.,& Qiu, J. (2015). A meta‐analysis of neuroimaging studies on divergent thinking using activation likelihood estimation. Human Brain Mapping, 36(7), 2703-2718.
• Yang, D. C., Tseng, Y. K., & Wang, T. L. (2017). A comparison of geometry problems in middle-grade mathematics textbooks from Taiwan, Singapore, Finland, and the United States. Eurasia Journal of Mathematics Science and Technology Education, 13(7), 2841-2857.
• Zaslavsky, O. (1995). Open-ended tasks as a trigger for mathematics teachers' professional development. For the Learning of Mathematics, 15(3), 15-20.
• Zhu, Y., & Fan, L. (2006). Focus on the representation of problem types in intended curriculum: A comparison of selected mathematics textbooks from Mainland China and the United States. International Journal of Science and Mathematics Education, 4(4), 609-626.
Primary Language en Social Articles Orcid: 0000-0001-5373-9341Author: Erhan BİNGÖLBALİ (Primary Author)Institution: AFYON KOCATEPE UNIVERSITYCountry: Turkey Orcid: 0000-0003-0847-1328Author: Ferhan BİNGÖLBALİ Institution: Independent ScholarCountry: Turkey Publication Date : June 15, 2020
 Bibtex @research article { ijcer689555, journal = {International Journal of Contemporary Educational Research}, issn = {}, eissn = {2148-3868}, address = {}, publisher = {Mustafa AYDIN}, year = {2020}, volume = {7}, pages = {240 - 252}, doi = {10.33200/ijcer.689555}, title = {Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks?}, key = {cite}, author = {Bi̇ngölbali̇, Erhan and Bi̇ngölbali̇, Ferhan} } APA Bi̇ngölbali̇, E , Bi̇ngölbali̇, F . (2020). Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks? . International Journal of Contemporary Educational Research , 7 (1) , 240-252 . DOI: 10.33200/ijcer.689555 MLA Bi̇ngölbali̇, E , Bi̇ngölbali̇, F . "Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks?" . International Journal of Contemporary Educational Research 7 (2020 ): 240-252 Chicago Bi̇ngölbali̇, E , Bi̇ngölbali̇, F . "Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks?". International Journal of Contemporary Educational Research 7 (2020 ): 240-252 RIS TY - JOUR T1 - Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks? AU - Erhan Bi̇ngölbali̇ , Ferhan Bi̇ngölbali̇ Y1 - 2020 PY - 2020 N1 - doi: 10.33200/ijcer.689555 DO - 10.33200/ijcer.689555 T2 - International Journal of Contemporary Educational Research JF - Journal JO - JOR SP - 240 EP - 252 VL - 7 IS - 1 SN - -2148-3868 M3 - doi: 10.33200/ijcer.689555 UR - https://doi.org/10.33200/ijcer.689555 Y2 - 2020 ER - EndNote %0 International Journal of Contemporary Educational Research Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks? %A Erhan Bi̇ngölbali̇ , Ferhan Bi̇ngölbali̇ %T Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks? %D 2020 %J International Journal of Contemporary Educational Research %P -2148-3868 %V 7 %N 1 %R doi: 10.33200/ijcer.689555 %U 10.33200/ijcer.689555 ISNAD Bi̇ngölbali̇, Erhan , Bi̇ngölbali̇, Ferhan . "Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks?". International Journal of Contemporary Educational Research 7 / 1 (June 2020): 240-252 . https://doi.org/10.33200/ijcer.689555 AMA Bi̇ngölbali̇ E , Bi̇ngölbali̇ F . Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks?. International Journal of Contemporary Educational Research. 2020; 7(1): 240-252. Vancouver Bi̇ngölbali̇ E , Bi̇ngölbali̇ F . Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks?. International Journal of Contemporary Educational Research. 2020; 7(1): 240-252.

Authors of the Article