EXPLORING MATHEMATICAL REPRESENTATIONS IN SOLVING ILL-STRUCTURED PROBLEMS: THE CASE OF QUADRATIC FUNCTION

Ika Santia, Purwanto Purwanto, Akbar Sutawidjadja, Sudirman Sudirman, Subanji Subanji

Abstract


Mathematical representation has an essential role in solving mathematical problems. However, there are still many mathematics education students who have difficulty in representing ill-structured problems. Even though the ill-structured-problem-solving tasks designed to help mathematics education students understand the relevance and meaningfulness of what they learn, they also are connected with their prior knowledge. The focus of this research is exploring the used of mathematical representations in solving ill-structured problems involving quadratic functions. The topic of quadratic functions is considered necessary in mathematics teaching and learning in higher education. It's because many mathematics education students have difficulty in understanding these matters, and they also didn’t appreciate their advantage and application in daily life. The researchers' explored mathematical representation as used by two subjects from fifty-four mathematics education students at the University of Nusantara PGRI Kediri by using a qualitative approach. We were selected due to their completed all steps for solving the ill-structured problem, and there have different ways of solving these problems. Mathematical representation explored through an analytical framework of solving ill-structured issues such as representing problems, developing alternative solutions, creating solution justifications, monitoring, and evaluating. The data analysis used technique triangulation. The results show that verbal and symbolic representations used both subjects to calculate, detect, correct errors, and justify their answers. However, the visual representation used only by the first subject to detect and correct errors.


Keywords


mathematical representation; problem-solving; ill-structured problem; quadratic function

Full Text:

PDF

References


Akgün, L. (2011). Experiences of undergraduate students with literal symbols. Scientific Research and Essays, 6(7), 1489-1497. https://doi.org/10.5897/SRE10.075.

Anwar, R.B., Yuwono, I., Ashari, A.R., Sisworo, & Rahmawati, D. (2016). Mathematical representation by students in building relational understanding on concepts of area and perimeter of rectangle. Educational Research and Reviews, 11(21), 2002–2008. https://doi.org/10.5897/ERR2016.2813.

Bal, A.P. (2014). The examination of representations used by classroom teacher candidates in solving mathematical problems. Educational sciences: Theory and practice, 14(6), 2349-2365. https://doi.org/10.12738/estp.2014.6.2189.

Bossé, M.J., Adu-Gyamfi, K., & Chandler, K. (2014). Students’ differentiated translation processes. International Journal for Mathematics Teaching and Learning, 1–28.

Bannister, V.R.P. (2014). Flexible conceptions of perspectives and representations: an examination of pre-service mathematics teachers’ knowledge. International Journal of Education in Mathematics, Science and Technology, 2(3), 223–233. https://doi.org/10.18404/ijemst.23592.

Byun, J.N., Kwon, D.Y., & Lee, W.G. (2014). Development of ill-structured problems for elementary learners to learn by computer-based modeling tools. International Journal of Computer Theory and Engineering, 6(4), 292-296. https://doi.org/10.7763/IJCTE.2014.V6.877.

Caglayan, G., & Olive, J. (2010). Eighth grade students’ representations of linear equations based on a cups and tiles model. Educational Studies in Mathematics, 74(2), 143–162. https://doi.org/10.1007/s10649-010-9231-z.

Chapman, O. (2010). Teachers’ self-representations in teaching mathematics. Journal of Mathematics Teacher Education, 13(4), 289–294. https://doi.org/10.1007/s10857-010-9153-9.

Ge, X., & Land, S.M. (2003). Scaffolding students’ problem-solving processes in an Ill-structured task using question prompts and peer interactions. Educational Technology Research and Development, 51(1), 21–38. https://doi.org/10.1007/BF02504515.

Goldin-Meadow, S., & Beilock, S.L. (2010). Action’s influence on thought: The case of gesture. Perspectives on Psychological Science, 5(6), 664–674. https://doi.org/10.1177/1745691610388764.

Hong, J.Y., & Kim, M.K. (2016). Mathematical abstraction in the solving of ill-structured problems by elementary school students in Korea. Eurasia Journal of Mathematics, Science and Technology Education, 12(2), 267–281. https://doi.org/10.12973/eurasia.2016.1204a.

Janakievska, B., Stopanska, L.F., & Bogatinoska, D.J. (2012). Enhanced study of quadratic functions with GeoGebra and preparation for Calculus. Proceedings of the 8th International Symposium on Geometric Function Theory and Applications (pp. 1-6). Ohrid, R. Macedonia.

Jonassen, D.H. (1997). Instructional design model for well-structured and ill-structured problem-solving learning outcomes. Educational Technology Research and Development, 45(1), 65-95.

Jonassen, D.H. (2011). Learning to Solve Problems a Handbook for Designing Problem-Solving Learning Environments. London: Pfeiffer Publisher.

Kieran, C. (2004). Algebraic Thinking in the Early Grades: What Is It? Mathematics Educator, 8(1), 139–151. https://doi.org/10.1080/13670050.2017.1323445.

Kitchner, K.S. (1983). Cognition, metacognition, and epistemic cognition. Human Development, 26(4), 222–232. https://doi.org/10.1159/000272885.

López, J., Robles, I., & Martínez, P.R. (2016). Students’ understanding of quadratic equations. International Journal of Mathematical Education in Science and Technology, 47(4), 552–572. https://doi.org/10.1080/0020739X.2015.1119895.

Minarni, A., Napitupulu, E., & Husein, R. (2016). Mathematical understanding and representation ability of public junior high school in north sumatra. Journal on Mathematics Education, 7(1), 43-56. https://doi.org/10.22342/jme.7.1.2816.43-56.

Moreno, A.L., Hegedus, S.J., & Kaput, J.J. (2008). From static to dynamic mathematics: Historical and representational perspectives. Educational Studies in Mathematics, 68(2), 99–111. https://doi.org/10.1007/s10649-008-9116-6.

Muhtadi, D., Sukirwan, Warsito, & Prahmana, R.C.I. (2017). Sundanese ethnomathematics: mathematical activities in estimating, measuring, and making patterns. Journal on Mathematics Education, 8(2), 185-198. https://doi.org/10.22342/jme.8.2.4055.185-198.

NCTM. (2000). National Council of Teacher of Mathematics, Principles and Standards for School Mathematics. Retrieved at December 11th, 2014, from http://standards.nctm.org/document/appendix/numb.htm.

Nizar, H., Putri, R.I..I., & Zulkardi. (2018). Developing PISA-Like Mathematics Problem Using the 2018 Asian Games Football and Table Tennis Context. Journal on Mathematics Education, 9(2), 183-194. https://doi.org/10.22342/jme.9.2.5246.183-194.

Nizarrudin. (2014). Role of Multiple Representations in Mathematical Problem Solving. Conference on Mathematics and Natural Sciences Education Faculty, PGRI, Semarang University Indonesia, pp 163-168.

Palm, T. (2009). Gender Aspects of Sense Making in Word Problem Solving. Mathematical Modelling, 1(1), 59–76.

Pape, S.J., & Tchoshanov, M.A. (2001). The role of representation (s) in developing mathematical understanding. Theory into Practice, 40(2), 118–127. https://doi.org/10.1207/s15430421tip4002.

Putri, R.I.I., & Zulkardi. (2018). Higher-order thinking skill problem on data representation in primary school: A case study. Journal of Physics: Conference Series, 948(1), 012056. https://doi.org/10.1088/1742-6596/948/1/012056.

Rahmawati, D., Purwanto, Subanji, Hidayanto, E., & Anwar, R.B. (2017). Process of mathematical representation translation from verbal into graphic. International Electronic Journal of Mathematics Education, 12(4), 367–381.

Saleh, M., Prahmana, R.C.I., Isa, M., & Murni. (2018). Improving the reasoning ability of elementary school student through the Indonesian Realistic Mathematics Education. Journal on Mathematics Education, 9(1), 41-54. https://doi.org/10.22342/jme.9.1.5049.41-54.

Samo, M.A. (2009). Students’ perceptions about the symbols, letters and signs in Algebra and how do these affect their learning of Algebra: A case research in a government girls secondary school Karachi. International Journal for Mathematics Teaching and Learning, 35.

Schloeglmann, W. (2004). Routines in non-routine problem solving processes. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 4, (pp 161-168).

Schoenfeld, A.H. (1992). Learning To Think Mathematically: Sense-Making in Mathematics. In D. Grouws (Ed.), (pp. 334-370). New York: MacMillan.

Sinnott, J.D. (1989). A model for solution of ill-structured problems: Implications for everyday and abstract problem solving. In J. D. Sinnott (Ed.), Everyday problem solving: Theory and applications (pp. 72-99). New York: Praeger.

Swastika, G., As’ari, A., Irawan, E.B., Nusantara, T., Subanji, & Irawati, S. (2018). Representation translation analysis of junior high school student in mathematics problem solving. International Journal of Insights for Mathematics Teaching, 1(2), 115-129.

Villegas, J.L., Castro, E., & Gutiérrez, J. (2009). Representations in problem solving: a case research with optimization problems. Electronic Journal of Research in Educational Psychology, 7(1), 279–308.

Voss, J.F., & Post, T.A. (1988). On the solving of ill structured problems. In MTA.Chi, Glaser&M.J.Farr (Eds). The Nature of Expertise. Hilldale, NJ: Lawrence Erlbaum.

Wood, P.K. (1993). Inquiry systems and problem structured: Implication for cognitive. Development. Human Development, 26, 249-265.

Zhang, J. (1997). The nature problem of external in solving representations. Cognitive Science, 21(2), 179–217. https://doi.org/10.1207/s15516709cog2102_3.




DOI: https://doi.org/10.22342/jme.10.3.7600.365-378

Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.


Journal on Mathematics Education
Program S3(Doktor) Pendidikan Matematika FKIP Universitas Sriwijaya
Kampus FKIP Bukit Besar
Jl. Srijaya Negara, Bukit Besar
Palembang - 30139
email: jme@unsri.ac.id

p-ISSN: 2087-8885 | e-ISSN: 2407-0610
Journal on Mathematics Education is licensed under a Creative Commons Attribution 4.0 International License

View My Stats