Sự sáng tạo trong năng lực mô hình hóa của sinh viên: khái niệm và đo lường

Educational Studies in Mathematics - Tập 109 - Trang 287-311 - 2021
Xiaoli Lu1, Gabriele Kaiser2,3
1School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China
2Universität Hamburg, Hamburg, Germany
3Australian Catholic University, Brisbane, Australia

Tóm tắt

Năng lực mô hình hóa hiện đang được đưa vào nhiều chương trình giảng dạy trên toàn thế giới và thường được công nhận là một cấu trúc phức tạp, định hướng theo quy trình. Do đó, việc đo lường hiệu quả nên bao gồm nhiều khía cạnh, chẳng hạn như những tiểu năng lực cần thiết trong suốt quá trình mô hình hóa. Xuất phát từ đặc điểm của các vấn đề mô hình hóa như những vấn đề thực tế mở và thường không xác định, chúng tôi đề xuất làm giàu khái niệm hiện tại về năng lực mô hình hóa toán học bằng cách đưa vào yếu tố sáng tạo, mà đóng vai trò quan trọng trong nhiều giai đoạn của quá trình mô hình hóa toán học nhưng chưa được chú ý nhiều trong thảo luận về mô hình hóa. Trong nghiên cứu được mô tả trong bài báo này, một công cụ mới để đánh giá cấu trúc được làm giàu này đã được phát triển và triển khai. Năng lực mô hình hóa bao gồm sự sáng tạo của sinh viên đã được đánh giá dựa trên tính đầy đủ của các mô hình và quy trình mô hình hóa được đề xuất, và tính thích hợp cũng như tính hoàn chỉnh của các phương pháp được đánh giá một cách chi tiết. Bằng cách điều chỉnh các phương pháp đo lường cho sự sáng tạo đã được phát triển trong thảo luận về giải quyết vấn đề, một số tiêu chí sáng tạo được chọn để đánh giá sự sáng tạo trong các phương pháp của sinh viên khi giải quyết các vấn đề mô hình hóa—cụ thể là tính hữu ích, độ phong phú và tính độc đáo. Nghiên cứu thực nghiệm đã được thực hiện trên 107 sinh viên Trung Quốc ở cấp trung học phổ thông, những người đã tham gia một trại mô hình hóa và độc lập giải quyết ba vấn đề mô hình hóa phức tạp. Kết quả cho thấy có mối tương quan đáng kể giữa độ phong phú và tính độc đáo trong hiệu suất của sinh viên qua tất cả các nhiệm vụ; tuy nhiên, mối quan hệ giữa tính hữu ích và hai khía cạnh sáng tạo còn lại không nhất quán. Tổng thể, kết quả của nghiên cứu hỗ trợ tầm quan trọng của việc đưa sự sáng tạo vào cấu trúc của năng lực mô hình hóa.

Từ khóa

#năng lực mô hình hóa #sự sáng tạo #đo lường #mô hình hóa toán học #giáo dục

Tài liệu tham khảo

Achmetli, K., Schukajlow, S., & Rakoczy, K. (2019). Multiple solutions to solve real-world problems and students' procedural and conceptual knowledge. International Journal of Science and Mathematics Education, 17, 1605–1625. Altman, D. G. (1991). Practical statistics for medical research. Chapman and Hall. Assmus, D., & Fritzlar, T. (2018). Mathematical giftedness and creativity in primary grades. In F. M. Singer (Ed.), Mathematical creativity and mathematical giftedness: Enhancing creative capacities in mathematically promising students (pp. 55–81). Springer. Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J. Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical Education. Intellectual and attitudinal challenges (pp. 73–96). Springer. Blum, W., & Leiß, D. (2005). Modellieren im Unterricht mit der “Tanken”- Aufgabe [Mathematical model building with the "refuelling"-problem]. Mathematik Lehren [Mathematics Teaching], 128, 18–21. Bonotto, C., & Santo, L. D. (2015). On the relationship between problem posing, problem solving, and creativity in the primary school. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing: from research to effective practice (pp. 103–123). Springer. Borromeo Ferri, R. (2018). Learning how to teach mathematical modelling in school and teacher education. Springer. Cai, J., & Hwang, S. (2002). Generalized and generative thinking in US and Chinese students’ mathematical problem solving and problem posing. The Journal of Mathematical Behavior, 21(4), 401–421. Chiu, M. S. (2009). Approaches to the teaching of creative and noncreative mathematical problems. International Journal of Science and Mathematics Education, 7, 55–79. Coxbill, E., Chamberlin, S. A., & Weatherford, J. (2013). Using model-eliciting activities as a tool to identify creatively gifted elementary mathematics students. Journal for the Education of the Gifted, 36(2), 176–197. Galbraith, P., & Stillman, G. (2006). A framework for identifying student blockages during transitions in the modelling process. ZDM—Mathematics Education, 38(2), 143–162. Guilford, J. P. (1977). Way beyond the IQ. Guide to improving intelligence and creativity. Creative Education Foundation. Haines, C., Izard, J., & Le Masurier, D. (1993). Modelling intentions realised: Assessing the full range of developed skills. In T. Breiteig, I. Huntley, & G. Kaiser-Meßmer (Eds.), Teaching and learning mathematics in context (pp. 200–211). Horwood. Hankeln, C. (2020). Mathematical modelling in Germany and France: A comparison of students’ modelling process. Educational Studies in Mathematics, 103, 209–229. Haylock, D. W. (1987). A framework for assessing mathematical creativity in school children. Educational Studies in Mathematics, 18(1), 59–74. Hébert, T. P., Cramond, B., Neumeister, K. L. S., Millar, G., & Silvian, A. F. (2002). E. Paul Torrance: His life, accomplishments, and legacy. Research Monograph Series. Retrieved April 21, 2021, from https://eric.ed.gov/?id=ED480289 Hersh, R., & John-Steiner, V. (2017). The origin of insight in mathematics. In R. Leikin & B. Sriraman (Eds.), Creativity and giftedness: Interdisciplinary perspectives from mathematics and beyond (pp. 135–146). Springer. Hong, E., & Milgram, R. M. (2010). Creative thinking ability: Domain generality and specificity. Creativity Research Journal, 22(3), 272–287. https://doi.org/10.1080/10400419.2010.503535 Ji, X. (2008). Kaocha “boluo zhong de shuxue”: Peiyang xuesheng shuxue jianmo nengli [Investigating “mathematics in pineapples”: Promoting students’ mathematical modelling skills]. Shuxue jiaoxue tongxun, 2008(5), 26–28. Kaiser, G. (2007). Modelling and modelling competencies in school. In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical Modelling (ICTMA12) Education, Engineering and Economics (pp. 110–119). Horwood. Kaiser, G. (2017). The teaching and learning of mathematical modelling. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 267–291). National Council of Teachers of Mathematics. Kaiser, G., & Brand, S. (2015). Modelling competencies: Past development and further perspectives. In G. A. Stillman, W. Blum, & M. S. Biembengut (Eds.), Mathematical modelling in education research and practice: Cultural, social and cognitive influences (pp. 129–149). Springer. Kaiser, G., & Stender, P. (2013). Complex modeling problems in co-operative, self-directed learning environments. In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice. International perspectives on the teaching and learning of mathematical modelling (pp. 277–293). Springer. Kattou, M., Christou, C., & Pitta-Pantazi, D. (2015). Mathematical creativity or general creativity? In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Conference of the European Society for Research in Mathematics Education (pp. 1016–1023). Charles University in Prague. Kattou, M., Kontoyianni, K., Pitta-Pantazi, D., & Christou, C. (2013). Connecting mathematical creativity to mathematical ability. ZDM—Mathematics Education, 45(2), 167–181. Klavir, R., & Gorodetsky, M. (2011). Features of creativity as expressed in the construction of new analogical problems by intellectually gifted students. Creative Education, 2(3), 164–173. Kupers, E., Lehmann-Wermser, A., McPherson, G., & van Geert, P. (2019). Children's creativity: A theoretical framework and systematic review. Review of Educational Research, 89(1), 93–124. Kwon, O. N., Park, J. S., & Park, J. H. (2006). Cultivating divergent thinking in mathematics through an open-ended approach. Asia Pacific Education Review, 7(1), 51–61. Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–145). Sense Publishers. Leikin, R. (2013). Evaluating mathematical creativity: The interplay between multiplicity and insight. Psychological Test and Assessment Modeling, 55(4), 285–400. Leikin, R., & Elgrably, H. (2020). Problem posing through investigations for the development and evaluation of proof-related skills and creativity skills of prospective high school mathematics teachers. International Journal of Educational Research, 102. https://doi.org/10.1016/j.ijer.2019.04.002 Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In A. Kelly & R. Lesh (Eds.), Handbook of research in mathematics and science education (pp. 113–149). Erlbaum. Lu, X., Cheng, J., Xu, B., & Wang, Y. (2019). Xuesheng shuxue jianmo suyang de pingjia gongju yanjiu [A research of the assessment tool of students’ mathematical modelling competency]. Kecheng Jiaocai Jiaofa [Curriculum, Teaching Materials, and Method], 39(2), 100–106. Ludwig, M., & Xu, B. (2010). A comparative study of modelling competencies among Chinese and German students. Journal für Mathematik-Didaktik, 31, 77–97. https://doi.org/10.1007/s13138-010-0005-z Maaß, K. (2006). What are modelling competencies? ZDM—Mathematics Education, 38, 113–142. https://doi.org/10.1007/BF02655885 Mayring, P. (2014). Qualitative content analysis: Theoretical foundation, basic procedures and software solution. Klagenfurt. https://nbnresolving.org/urn:nbn:de:0168-ssoar-395173 Ministry of Education of China. (2007). Putong gaozhong shuxue kecheng biaozhun (shiyan) [Mathematics curriculum standards for high schools (experimental)]. People’s Education Press. Ministry of Education of China. (2011). Yiwu jiaoyu shuxue kecheng biaozhun (2011 nian ban) [Mathematics curriculum standards for compulsory education (2011 version)]. Beijing Normal University Publishing Group. Ministry of Education of China. (2018). Putong gaozhong shuxue kecheng biaozhun (2017 nian ban) [Mathematics curriculum standards for high schools (2017 version)]. People’s Education Press. Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge. Niss, M., & Højgaard, T. (Eds.). (2011). Competencies and mathematical learning. Ideas and inspiration for the development of mathematics teaching and learning in Denmark. English translation of Danish original (2002). IMFUFA: Roskilde University, Denmark. Retrieved April 21, 2021, from https://www.researchgate.net/publication/270585013 Niss, M. A., & Højgaard, T. (2019). Mathematical competencies revisited. Educational Studies in Mathematics, 102(1), 9–28. https://doi.org/10.1007/s10649-019-09903-9 Pellegrino, J. W., & Hilton, M. L. (2012). Educating for life and work: Developing transferable knowledge and skills in the 21st century. National Research Council. The National Academies Press. Pitta-Pantazi, D., Kattou, M., & Christou, C. (2018). Mathematical creativity: Product, person, process and press. In F. M. Singer (Ed.), Mathematical creativity and mathematical giftedness: Enhancing creative capacities in mathematically promising students (pp. 27–53). Springer. Plucker, J. A., & Zabelina, D. (2009). Creativity and interdisciplinarity: One creativity or many creativities? ZDM – Mathematics Education, 41(1–2), 5–11. Pollak, H. O. (1977). The interaction between mathematics and other school subjects (including integrated courses). In H. Athen & H. Kunle (Eds.), Proceedings of the Third International Congress on Mathematical Education (pp. 255–264). Zentralblatt für Didaktik der Mathematik. Reiter-Palmon, R., Forthmann, B., & Barbot, B. (2019). Scoring divergent thinking tests: A review and systematic framework. Psychology of Aesthetics, Creativity, and the Arts, 13(2), 144–152. Runco, M. A. (2010). Divergent thinking, creativity, and ideation. In J. C. Kaufman & R. J. Sternberg (Eds.), The Cambridge handbook of creativity (pp. 413–446). Cambridge University Press. Schukajlow, S., Kaiser, G., & Stillman, G. (2018). Empirical research on teaching and learning of mathematical modelling: A survey on the current state-of-the-art. ZDM—Mathematics Education, 50, 5–18. Schukajlow, S., Krug, A., & Rakoczy, K. (2015). Effects of prompting multiple solutions for modelling problems on students’ performance. Educational Studies in Mathematics, 89(3), 393–417. Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM—Mathematics Education, 3, 75–80. Silver, E. A., & Cai, J. (2005). Assessing students’ mathematical problem posing. Teaching Children Mathematics, 12(3), 129–135. Sriraman, B. (2009). The characteristics of mathematical creativity. ZDM—Mathematics Education, 41(1–2), 13–27. Stender, P. (2017). The use of heuristic strategies in modelling activities. ZDM—Mathematics Education, 50, 315–326. Sternberg, R., & Lubart, T. (1999). The concept of creativity: Prospect and paradigm. In R. Sternberg (Ed.), A handbook of creativity (pp. 3–15). Cambridge University Press. Sternberg, R. J. (2017). School mathematics as a creative enterprise. ZDM—Mathematics Education, 49, 977–986. Stillman, G. (2011). Applying metacognitive knowledge and strategies in applications and modelling tasks at secondary school. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in Teaching and Learning of Mathematical Modelling (pp. 165–180). Springer. Stillman, G., Brown, J., & Galbraith, P. (2010). Identifying challenges within transition phases of mathematical modelling activities at year 9. In R. Lesh, P. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies ICTMA13 (pp. 385–398). Springer. Tabach, M., & Friedlander, A. (2018). Instances of promoting creativity with procedural tasks. In F. M. Singer (Ed.), Mathematical creativity and mathematical giftedness: Enhancing creative capacities in mathematically promising students (pp. 285–306). Springer. Torrance, E. P. (1966). Torrance tests of creative thinking: Norms-technical manual (Research ed.). Personnel Press. Van Harpen, X. Y., & Siraman, B. (2013). Creativity and mathematical problem posing: An analysis of high school students’ mathematical problem posing in China and the USA. Educational Studies in Mathematics, 82(2), 201–221. Vorhölter, K. (2018). Conceptualization and measuring of metacognitive modelling competencies: Empirical verification of theoretical assumptions. ZDM—Mathematics Education, 50(1-2), 343–354. Vorhölter, K. (2019). Enhancing metacognitive group strategies for modelling. ZDM—Mathematics Education, 51, 703–716. Wang, J., & Lu, X. (2018). Selection of content in high school mathematics textbooks: An international comparison. ZDM—Mathematics Education, 50, 813–826. Wang, Y. (2019). Shuxue benke shifansheng shuxue jianmo nengli shuiping de xianzhuang diaocha [Investigation on the mathematics modelling competency level of mathematics undergraduate teacher students: A study of four universities] (unpublished master’s thesis). East China Normal University. Wessels, H. (2014). Levels of mathematical creativity in model-eliciting activities. Journal of Mathematical Modelling and Application, 1(9), 22–40. Wong, N.-Y., Han, J., & Lee, P. Y. (2004). The mathematics curriculum: Toward globalization or westernization? In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How chinese learn mathematics: perspectives from insiders (pp. 27–70). World Scientific.