An Investigation of Mathematical Problem Posing Skills of Gifted Students

This study aimed to investigate the mathematical problem posing skills of gifted students. The participants of the study, designed as a case study, were 55 middle school students (20 sixth grade, 17 seventh grade, 18 eighth grade) who were studying at Science and Art Center in a city in the Eastern Anatolia region. Data were collected through a problem posing form which includes a semi-structured problem posing task in which the students were asked to make up three problems (easy, moderately difficult, and difficult) about three different figures given. The students’ responses to the problem posing task were analyzed with descriptive analysis method. Results showed that almost all of the problems posed by students were mathematical problems. Seventh and eighth-grade students posed more non-mathematical problems than sixth-grade students. Results also revealed that the students mostly posed extensive problems (related to further steps beyond the three given figures) in easy, moderately difficult and difficult tasks. Problem posing rates of the students with the level of difficulty that progresses hierarchically as desired were found to be quite low in the progression analysis of problems’ difficulty level.

Özel Yetenekli Öğrencilerin Matematiksel Problem Kurma Becerilerinin İncelenmesi

Bu çalışmada özel yetenekli öğrencilerin matematiksel problem kurma becerilerinin incelenmesi amaçlanmıştır. Çalışmada, durum çalışması kullanılmıştır. Çalışmanın katılımcılarını Türkiye’nin Doğu Anadolu Bölgesi’ndeki bir ilde bulunan Bilim ve Sanat Merkezi’nde öğrenim görmekte olan 55 ortaokul (20 altıncı sınıf, 17 yedinci sınıf, 18 sekizinci sınıf) öğrencisi oluşturmaktadır. Veri toplama aracı olarak, yarı-yapılandırılmış bir problem kurma görevinden oluşan problem kurma formu kullanılmıştır. Problem kurma görevinde öğrencilerden, verilen üç farklı şekil ile ilgili basit, orta ve zor düzeyde üç farklı problem kurmaları istenmiştir. Özel yetenekli öğrencilerin problem kurma görevine verdikleri yanıtlar betimsel analiz yöntemiyle incelenmiştir. Bulgulara göre, özel yetenekli öğrencilerin kurduğu problemlerin tamamına yakınının matematiksel problemler olduğu tespit edilmiştir. Yedinci ve sekizinci sınıf seviyelerinde matematiksel olmayan problem kuran öğrenci sayısının altıncı sınıfa göre fazla olduğu belirlenmiştir. Özel yetenekli öğrencilerin kolay, orta ve zor problem kurma görevlerinde ağırlıklı olarak geniş kapsamlı problemler (verilen üç şeklin ötesinde daha ileri basamaklarla ilgili problemler) kurduğu görülmüştür. Problemlerin zorluk düzeyi ilerleme analizinde, özel yetenekli öğrencilerin istenen şekilde hiyerarşik ilerleyen zorluk düzeyine sahip problem kurma oranlarının oldukça düşük olduğu sonucuna varılmıştır.

PDF

___

Altun, M. (2015). Teaching mathematics for education faculties and primary teachers (19th ed.). Bursa: Alfa Aktuel.

Amit, M. & Neria, D. (2008). “Rising to the challenge”: Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students. ZDM Mathematics Education, 40, 111–129.

Arikan, E. E. & Unal, H. (2015). Investigation of problem-solving and problem posing abilities of seventhgrade students. Educational Sciences, Theory & Practice, 15(5), 1403-1416.

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. 373–404). Cham, Switzerland: Springer International Publishing.

Benedicto, C., Jaime, A., & Gutiérrez, A. (2015). Análisis de la demanda cognitiva de problemas de patrones geométricos. In C. Fernández, M. Molina, & N. Planas (Eds.), Investigación en educación matemática XIX (pp. 153–162). Alicante, Spain: SEIEM.

Bozkurt, A. & Karsligil-Ergin, G. (2018). Students’ achievement and mathematical thinking in process of problem solving and problem posing. E-International Journal of Educational Research, 9(3), 1-33.

Cai, J. (2003). Singaporean students' mathematical thinking in problem solving and problem posing: An exploratory study. International Journal of Mathematical Education in Science and Technology, 34(5), 719-737.

Cai, J., Chen, T., Li, X., Xu, R., Zhang, S., Hu, Y., et al. (2019). Exploring the impact of a problem-posing workshop on elementary school mathematics teachers’ problem posing and lesson design. International Journal of Educational Research. https://doi.org/10.1016/j.ijer.2019.02.004 Online First.

Cai, J., Moyer, J. C., Wang, N., Hwang, S., Nie, B., & Garber, T. (2013). Mathematical problem posing as a measure of curricular effect on students’ learning. Educational Studies in Mathematics, 83, 57–69.

Cai, J. & Hwang, S. (2019). Learning to teach mathematics through problem posing: Theoretical considerations, methodology, and directions for future research. International Journal of Educational Research. https://doi.org/10.1016/j.ijer.2019.01.001 Online First.

Canturk-Gunhan, B., Gecici, M. E., & Gunkaya, B. (2019). The effect of problem posing based mathematics teaching on students' success: A meta-analysis study. Necatibey Faculty of Education Electronic Journal of Science and Mathematics Education, 13(2), 1042-1062.

Chen, T. & Cai, J. (2019). An elementary mathematics teacher learning to teach using problem posing: A case of the distributive property of multiplication over addition. International Journal of Educational Research. https://doi.org/10.1016/j.ijer.2019.03.004 Online First.

Dai, D. Y., Moon S. M., & Feldhusen, J. F. (1998). Achievement motivation and gifted students: A social cognitive perspective. Educational Psychologist, 33(2-3), 45-63.

Davis, G. A. & Rimm, S. B. (2004). Education of the gifted and talented. Boston, MA: Pearson Education Press.

English, L. D. (2019). Teaching and learning through mathematical problem posing: Commentary. International Journal of Educational Research. https://doi.org/10.1016/j.ijer.2019.06.014 Online First.

Erdogan, A. & Yemenli, E. (2019). Gifted students’ attitudes towards mathematics: a qualitative multidimensional analysis. Asia Pacific Education Review, 20, 37–52.

Erdogan, F. & Erben, T. (2018). Investigation of gifted students’ problem posing abilities requiring arithmetical operations with natural numbers. İnonu University Journal of the Faculty of Education, 19(3), 534-546.

Espinoza, J., Lupiáñez J. L., & Segovia, I. (2013). Características del talento matemático asociadas a la invención de problemas. Revista Científica, número especial octubre 2013, 190-195.

Espinoza, J., Lupiáñez, J. L., & Segovia, I. (2016). The posing of arithmetic problems by mathematically talented students. Electronic Journal of Research in Educational Psychology, 14(2), 368-392.

Freiman, V. (2018). Complex and open-ended tasks to enrich mathematical experiences of kindergarten students. In F. M. Singer (Ed.), Mathematical creativity and mathematical giftedness: Enhancing creative capacities in mathematically promising students (pp. 373–404). Cham, Switzerland: Springer International Publishing.

Fritzlar, T. & Karpinski-Siebold, N. (2012). Continuing patterns as a component of algebraic thinking—An interview study with primary school students. In Pre proceedings of the 12th International Congress on Mathematical Education (pp. 2022–2031). Seoul, South Korea: ICMI. Retrieved January 12, 2018, from http://www.icme12.org/data/ICME12_Pre-proceedings.zip.

Gagné, F. (2003). Transforming gifts into talents: The DMGT as a developmental theory. In N. Colangelo & G. A. Davis (Eds), Handbook of gifted education (pp. 60-74). Boston MA: Allyn and Bacon, Inc.

Goldberg, S. R. (2008). An exploration of intellectually gifted students’ conceptual views of mathematics. Unpublished doctorate dissertation, Columbia University, USA.

Gutierrez, A., Benedicto, C., Jaime, A., & Arbona, E. (2018). The cognitive demand of a gifted student’s answers to geometric pattern problems. In F. M. Singer (Ed), Mathematical creativity and mathematical giftedness (pp. 196-198). Cham, Switzerland: Springer International Publishing.

Guzel, R. & Biber, A.Ç. (2019). The effect of the problem posing approach for academic success in the teaching of ınequalities. Kastamonu Education Journal, 27(1), 199-208.

Hu, H. (2019) Implementing resilience recommendations for policies and practices in gifted curriculum. Roeper Review, 41(1), 42-50.

Johnson, D. T. (2000). Teaching mathematics to gifted students in a mixed ability classroom. Reston, VA: ERIC Clearinghouse on Disabilities and Gifted Education.

Kesan, C., Kaya, D., & Guvercin, S. (2010). The effect of problem posing approach to the gifted student’s mathematical abilities. International Online Journal of Educational Sciences, 2(3), 677-687.

Kılıc, Ç. (2019). Investigation of the performance of the middle school students in the posing of problems that can be solved by the looking for a pattern strategy. Kastamonu Education Journal, 27(2), 647-656.

Korkmaz, E. & Gur, H. (2006). Determining of prospective teachers’ problem posing skills. Journal of Balıkesir University Institute of Science and Technology, 8(1), 64-74.

Krutetskii, V. A. (1976). The psychology of mathematical abilities in school children. Chicago, IL: University of Chicago Press.

Leikin, R. (2009). Bridging research and theory in mathematics education with research and theory in creativity and giftedness. In. R. Leikin, A. Berman & B. Koichu (Eds.), Creativity in mathematics and education of gifted students (pp. 385-411). Rotterdam: Sense Publishers.

Leikin, R. (2011). The education of mathematically gifted students: Some complexities and questions. The Mathematics Enthusiast, 8(1-2), 167–188.

Leikin, R. (2015). Problem posing for and through investigations in a dynamic geometry environment. In F. M. Singer, N. Ellerton & J. Cai (Eds), Problem posing: From research to effective practice (pp. 373–391). Dordrecht: Springer.

Leikin, R., Koichu, B., & Berman, A. (2009). Mathematical giftedness as a quality of problem-solving acts. R. Leikin, A. Berman & B. Koichu (Eds), Creativity in mathematics and the education of gifted students (pp. 115–128). Rotterdam: Sense Publishers.

Leikin, R., Koichu, B., Berman, A., & Dinur, S. (2017a). How are questions that students ask in high level mathematics classes linked to general giftedness? ZDM Mathematics Education, 49(1), 65-80.

Leikin, R., Leikin, M., Paz-Baruch, N., Waisman, I., & Lev, M. (2017b). On the four types of characteristics of super mathematically gifted students. High Ability Studies, 28(1), 107-125.

Levenberg, I. & Shaham, C. (2014). Formulation of word problems in geometry by gifted pupils. Journal for the Education of the Young Scientist and Giftedness, 2(2), 28-40.

Leung, S. S. (2013). Teachers implementing mathematical problem posing in the classroom: Challenges and strategies. Educational Studies in Mathematics, 83(1), 103-116.

Liljedahl, P. & Sriraman, B. (2006). Musings on mathematical creativity. For the Learning of Mathematics, 26(1), 20–23.

Merriam, S. B. (1998). Qualitative research and case study applications in education. San Francisco, CA: Jossey-Bass.

Miles, M. B., Huberman, A. M., & Saldana, J. (2014). Qualitative data analysis. CA:SAGE.

Miller, R. C. (1990). Discovering mathematical talent. Reston, VA: Eric Clearinghouse on Handicapped and Gifted Children.

Ministry of National Education. (2018). Mathematics curriculum (Primary and secondary 1, 2, 3, 4, 5, 6, 7 and 8 grades). Ankara: MoNE Publ.

Muijs, D. (2004). Doing quantiative research in education with SPSS. London: Sage

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics Reston, Va: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics (2016). Providing opportunities for students with exceptional mathematical promise: A position of the national council of teachers of mathematics. Reston: NCTM.

Nolte, M. (2018). Twice-exceptional students: Students with special needs and a high mathematical potential. In F. M. Singer (Ed.), Mathematical creativity and mathematical giftedness (pp. 199-225). Cham, Switzerland: Springer International Publishing.

Ozcelik, T. (2017). Efficiency of differentiated mathematics curriculum designed for gifted and talented students. Unpublished doctorate dissertation, Hacettepe University, Ankara.

Poulos, A. & Mamona-Downs, J. (2018). Gifted students approaches when solving challenging mathematical problems. In F. M. Singer (Ed), Mathematical creativity and mathematical giftedness (pp. 309-341). Cham, Switzerland: Springer International Publishing.

Renzulli, J. S. (2012). Reexamining the role of gifted education and talent development for the 21st century: A four-part theoretical approach. Gifted Child Quarterly, 56(3), 150–159.

Sheffield, L. J. (2003). Development of mathematical promise. In S. Pfeiffer & L. Limburg-Weber (Eds), Early gifts: Recognizing and nurturing children’s talents (pp. 59-81). Waco, TX: Prufrock Press.

Sheffield, L. J. (2018). Commentary paper: A reflection on mathematical creativity and giftedness. In F. M. Singer (Ed), Mathematical creativity and mathematical giftedness (pp. 405-428). Cham, Switzerland: Springer International Publishing.

Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 3, 75–80.

Silver, E. A. & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27, 521–539.

Singer, F. M., Ellerton, N., & Cai, J. (2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83(1), 1–7.

Singer, F. M., Ellerton, N., & Cai, J. (2015). Mathematical problem posing: From research to effective practice. New York: Springer.

Singer, F. M., Sheffield, L., Freiman, V., & Brandl, M. (2016). Research on and activities for mathematically gifted students. New York: Springer Nature.

Singer, F. M., Sheffield, L. J., & Leikin, R. (2017a). Advancements in research on creativity and giftedness in mathematics education: Introduction to the special issue. ZDM Mathematics Education, 49(1), 4-12.

Singer, F. M., & Voica, C. (2015). Is problem posing a tool for identifying and developing mathematical creativity? In F. M. Singer, N. Ellerton & J. Cai (Eds), Mathematical problem posing: From research to effective practice (pp. 141–174). New York: Springer.

Singer, F. M., Voica, C., & Pelczer, I. (2017b). Cognitive styles in posing geometry problems: implications for assessment of mathematical creativity. ZDM Mathematics Education, 49(1), 37-52.

Smedsrud, J. (2018) Mathematically gifted accelerated students participating in an ability group: A qualitative interview study. Front. Psychol., 9, 1-12.

Sowell, E. J., Zeigler, A. J., Bergwall, L., & Cartwright, R. M. (1990). Identification and description of mathematically gifted students: A review of empirical research. Gifted Child Quarterly, 34, 147–154.

Sriraman, B. (2005). Are giftedness and creativity synonyms in mathematics. The Journal of Secondary Education, 17(1), 20–36.

Stoyanova, E. & Ellerton, N. F. (1996). A framework for research into students’ problem posing in school mathematics. In P. Clarkson (Ed), Technology in mathematics education (pp. 518–525). Melbourne: Mathematics Education Research Group of Australasia.

Subotnik, R. F., Robinson, A., Callahan, C. M., & Gubbins, E. J. (2012). Malleable minds: Translating insights from psychology and neuroscience to gifted education. Storrs: University of Connecticut, NRCGT.

Turhan, B. & Guven, M. (2014). The effect of mathematics ınstruction with problem posing approach on problem solving success, problem posing ability and views towards mathematics. Cukurova University Faculty of Education Journal, 43(2), 217-234.

Van Tassel-Baska, J. & Stambaugh, T. (2006). Comprehensive curriculum for gifted learners (3rd ed.). Boston: Pearson Education Inc.

Voica, C. & Singer, F. M. (2013). Problem modification as a tool for detecting cognitive flexibility in school children. ZDM, 45(2), 267–279.

Voica, C. & Singer, F. M. (2014). Problem posing: A pathway to identifying gifted students. In MCG8 Proceedings (pp. 119–124). Univ. of Denver, Colorado, USA.

Wagner, H. & Zimmermann, B. (1986). Identification and fostering of mathematically gifted students. In A. Cropley, K. Urban, H. Wagner & W. Wieczerkowski (Eds), Giftedness: A continuing world-wide challenge (pp.273-287). New York: Trillium Pres.

Xu, B., Cai, J., Liu, Q., & Hwang, S. (2019). Teachers’ predictions of students’ mathematical thinking related to problem posing. International Journal of Educational Research. https://doi.org/10.1016/j.ijer.2019.04.005. Online First.

Yazgan-Sag, G. (2019). A theoretical view to mathematical giftedness. National Education, 48(221), 159-174.

Yin, R. K. (2017). Case study research and applications: Design and methods. Sage Publications.

Young, A. E. & Worrell, F. C. (2018). Comparing metacognition assessments of mathematics in academically talented students. The Gifted Child Quarterly, 63(2), 259-275.

Yuan, X. & Sriraman, B. (2011). An exploratory study of relationships between students’ creativity and mathematical problem posing abilities. In B. Sriraman & K. Lee (Eds). The elements of creativity and giftedness in mathematics (pp. 5–28). Rotterdam, the Netherlands: Sense.