Supporting the emergence of mathematical knowledge through problem posing

Simone Passarella(1*),

(1) University of Padova
(*) Corresponding Author

Abstract


Problem posing represents a valuable strategy to create a bridge between mathematics classroom activities and everyday-life experiences. Despite the value of problem posing activities as opportunities for measuring students’ mathematical learning outcomes, more research is needed in investigating if and how problem posing could support the introduction of new mathematical knowledge promoting the development of mathematical concepts.  The aim of this paper is to start investigating how problem posing can extend students’ mathematical knowledge. After having introduced the notion of emergent problem posing, some results from a teaching experiment conducted in a primary school class are reported. The design of the teaching experiment was explicated through the development of the three components of a Hypothetical Learning Trajectory: learning goal; hypothetical learning process; learning activities. Results from the study indicate that semi-structured problem posing activities that start from a suitable artifact could support the emergence of new mathematical knowledge, supporting students’ in re-inventing mathematical strategies to solve problems posed by themselves.  However, further research is necessary, especially in: supporting the notion of emergent problem posing with more teaching experiments; investigating the role of different artifacts in supporting the process of emergent problem-posing; evaluating which characteristics an artifact should have in order to support the process of emergent problem posing; examining possible relations between students’ abilities and emergent problem posing performances.


Keywords


mathematical problem posing; emergent problem posing; hypothetical learning trajectory; realistic mathematics education

Full Text:

PDF

References


Bonotto, C. (2005). How Informal Out-of-School Mathematics can Help Students Make Sense of Formal In-School Mathematics: The Case of Multiplying by Decimal Numbers, Mathematical Thinking and Learning, 7(4), 313-344.

Bonotto, C. (2013). Artifacts as sources for problem-posing activities. Educational Studies in Mathematics, 83(1), 37-55.

Bonotto C., & Dal Santo L. (2015). On the Relationship Between Problem Posing, Problem Solving, and Creativity in the Primary School. In: Singer

Brown, S.I., & Walter, M.I. (1990). The art of problem posing. Hillsdale: Lawrence Erlbaum.

Cai, J., & Hwang, S. (2002). Generalized and generative thinking in U.S. and Chinese students’ mathematical problem solving and problem posing. Journal of Mathematical Behaviour, 21(4), 401-421.

Cai J., Hwang, S., Jiang, C. & Silber, S. (2015). Problem-posing Research in Mathematics Education: Some Answered and Unanswered Questions. In: Singer F., F. Ellerton N., Cai J. (Eds) Mathematical Problem Posing. Research in Mathematics Education. Springer, New York, NY.

Chairuddin, C., & Farman, F. (2019). Comparison of the effectiveness of scientific approach and problem-solving approach in problem-based learning in class IX of SMP Negeri 3 Pangsid. Journal of Mathematics Education, 4(2), 69–75. DOI: 10.31327/jomedu.v4i2.1010

Christou, C., Mousoulides, N., Pittalis, M., Pitta-Pantazi, D., & Sriraman, B. (2005). An empirical taxonomy of problem posing processes. Zentralblatt fur Didaktik der Mathematik, 37(3), 149-158.

Djidu, H., & Jailani, J. (2017). Model pembelajaran kalkulus SMA berbasis masalah untuk meningkatkan kemampuan berpikir tingkat tinggi siswa (V. Indiani (ed.)). Parama Publishing.

Ellerton, N.F. (1986). Children’s made up mathematics problems: A new perspective on talented mathematicians. Educational Studies in Mathematics, 17, 261-271.

Ellerton, N. F., Singer, F. M., & Cai, J. (2015). Problem posing in mathematics: Reflecting on the past, energizing the present, and foreshadowing the future. In Mathematical problem posing (pp. 547-556). Springer, New York, NY. DOI: 10.1007/978-1-4614-6258-3_26

Freudenthal, H. (1973). Mathematics as an educational task, Dordrecht: D. Reidel.

Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Dordrecht: Kluwer.

Gellert, U., & Jablonka, E. (2007). Mathematization-demathematization. In U. Gellert & E. Jablonka (Eds.), Mathematization and demathematization: Social, philosophical and educational ramifications (pp. 1-18). Rotterdam: Sense Publishers.

Gravemeijer, K. (1999). How emergent models may foster the construction of formal mathematics. Mathematical Thinking and Learning, 1(2), 155-177.

Gravemeijer, K. & Doorman, M. (1999). Context Problems in Realistic Mathematics Education: A Calculus Course as an Example. Educational Studies in Mathematics, 39(1-3), 111-129.

Greer, B., Verschaffel, L. & Mukhopadhyay (2007). Modelling for life: mathematics and children’s experience. In W. Blum et al. (Eds), Modelling and applications in mathematics education (pp. 89-98). New ICMI Studies no. 10. New York: Springer.

Hansen, R., & Hana, G. M. (2015). Problem posing from a modelling perspective. In Mathematical Problem Posing (pp. 35-46). Springer, New York, NY. DOI: 10.1007/978-1-4614-6258-3_2

Kopparla, M., Bicer, A., Vela, K., Lee, Y., Bevan, D., Kwon, H., Caldwell, C., Capraro, M.M., & Capraro, R. M. (2018). The effect of problem-posing intervention types on elementary students’ problem-solving. Educational Studies, 45(6), 7708-725.

Klaasen, K., & Doorman, M. (2015). Problem Posing as Providing Students with Content-Specific Motives. In: Singer F., F. Ellerton N., Cai J. (Eds) Mathematical Problem Posing. Research in Mathematics Education, (pp. 215-240). Springer, New York, NY.

Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Hillsdale, NJ: Lawrence Erlbaum Associates.

Leung, S. (1997). On the role of creative thinking in problem posing. Zentralblatt fur Didaktik der Mathematik, 97(2), 48-52.

Leung, S. S., & Silver, E. A. (1997). The role of task format, mathematics knowledge, and creative thinking on the arithmetic problem posing of prospective elementary school teachers. Mathematics Education Research Journal, 9(1), 5-25.

Mashuri, S., Djidu, H., & Ningrum, R. K. (2019). Problem-based learning dalam pembelajaran matematika: Upaya guru untuk meningkatkan minat dan prestasi belajar siswa. Pythagoras: Jurnal Pendidikan Matematika, 14(2), 112–125. DOI: 10.21831/pg.v14i2.25034

Nasruddin, N., Mashuri, S., & Jahring, J. (2019). Implementation of Team Assited Individualization Instructional Strategies Supported by Geogebra Software to Improve Mathematical Problem Solving Ability. Jurnal Pendidikan Progresif, 9(1), 16-21.

Polya, G. (1954). Mathematics and plausible reasoning (2 vols). Princeton, NJ: Princeton University Press.

Silver, E. A. (1994). On mathematical problem solving. For the learning of mathematics, 14(1), 19-28.

Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 29(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(5), 521-539.

Simon, M.A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145.

Sriraman, B. (2009). The characteristics of mathematical creativity. ZDM, 41, 13-27.

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

Suendarti, M., & Liberna, H. (2019). Effectiveness of the realistic mathematics education approach to the ability to solve mathematical problems. Journal of Mathematics Education, 4(1), 27–32. Retrieved from http://usnsj.com/index.php/JME/article/download/4.1.27-32/pdf_1

Van Harpen, X.Y., & Presmeg, N.C. (2013). An Investigation of Relationships between Students’ Mathematical Problem-Posing Abilities and Their Mathematical Content Knowledge. Educational Studies in Mathematics, 83(1), 117-132.

Verschaffel, L., Greer, B. & De Corte, E. (2000). Making sense of word problems, Lisse, The Netherlands: Sweets & Zeitlinger.

Xie, J, & Masingila, J.O. (2017). Examining Interactions between Problem Posing and Problem Solving with Prospective Primary Teachers: A Case of Using Fractions. Educational Studies in Mathematics, 96, 101-118.

Yuan, X., & Sriraman, B. (2010). An exploratory study of relationships between students’ creativity and mathematical problem-posing abilities. In B. Sriraman and K.H. Lee (Eds.). The Elements of Creativity and Giftedness in Mathematics, 5-28.




DOI: https://doi.org/10.31327/jme.v5i2.1259

Article Metrics

Abstract view : 449 times
PDF - 246 times

Refbacks

  • There are currently no refbacks.


Copyright (c) 2020 Simone Passarella


INDEXING DATABASE