Exploring students’ procedural fluency and written adaptive reasoning skills in solving open-ended problems

Stephanie Gayle B. Andal & Rose R. Andrade

Volume 2, Issue 1, March 2022

Abstract

Developing students’ mathematical skills requires both procedure and reasoning. However, the declination of possessing these skills is still evident today. Hence, this study aimed to describe the students’ procedural fluency in terms of accuracy, flexibility, and efficiency and written adaptive reasoning in terms of explanation and justification in solving open-ended problems. The study employed descriptive-correlational design through purposive sampling of thirty students from a National High School in Laguna, Philippines. The quantitative data revealed that in procedural fluency, students can quickly submit a complete solution leading to correct answer. However, they fail to provide two or more solutions in solving open-ended problems. The results also showed that students can clearly explain the problem but struggle to justify their solution. Moreover, procedural fluency is positively correlated to their adaptive reasoning. Consequently, students with an average level of mathematical achievement scored significantly higher than those at a low mathematical level in terms of flexibility. Pedagogical implications suggest that problem-solving activities for students should not solely focus on getting the correct procedures and answers. Further, it is recommended that teachers should expose students in open-ended problems and allow them to try and justify their own unique solutions irrespective of their mathematical achievement.

Keywords

Author information & Contribution

Stephanie Gayle B. Andal. Laguna State Polytechnic University. San Pablo City Campus.
Rose R. Andrade. Corresponding author. Laguna State Polytechnic University. San Pablo City Campus.

Disclosure statement

Funding

Institutional Review Board Statement

Declaration

Notes

Acknowledgement

References

Achmetli, K., & Schukajlow, S. (2019). Multiple Solutions, The Experience of Competence, and Interest. Affect and Mathematics Education. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-030-13761-8_3

Albab, A., & Wangguway, Y. (2020). Profile of Students’ Creative and Innovative Thinking in Solving Open-ended Mathematics Problems about the Coffee Plantation. Journal of Physics: Conference Series 1538. https://doi.org/10.1088/1742-6596/1538/1/012071

Al-Balasi, R., & Barham, A. (2010). The effect of using multiple mathematical representations on the attainment of mathematical concepts among basic eighth grade students and their ability to solve verbal problems. Dirasat: Educational Sciences(37)1, 1-13.

Al-Shuaibi, Abdulghani. (2014). The Importance of Education.

Andrews, N., Ingram, J., & Pitt, A. (2019). When students offer explanation without the teacher explicitly asking them to. Retrieved from Educ Stud Math: http://doi.org/10.1007/s10649-018-9873-9.

Aprianti, C. (2014). Mathematical études: embedding opportunities for developing procedural fluency within rich mathematical contexts. International Journal of Mathematical Education in Science and Technology, 44(5), 765–774. https://doi.org/10.1080/0020739X.2013.770089

Arigbabu, A. (2013). Effects of framing and team individualised instructional strategies on senior secondary school students’ attitudes toward mathematics. Acta Didactica Napocensia, 6(1), 1-22.

Asmida, A., Sugiatno, S., & Hartoyo, A. (2018). Developing The Mathematics Conceptual Understanding And Procedural Fluency Through Didactical Anticipatory Approach Equipped With Teaching Aids. Journal of Education, Teaching and Learning, 367. http://dx.doi.org/10.26737/jetl.v3i2.796

Awofala, A. (2017). Assessing Senior Secondary School Students’ Mathematical Proficiency as Related to Gender and Performance in Mathematics in Nigeria. International Jounal of Research in Education and Science (IJRES), 3(2), 488-502. https://doi.org/10.21890/ijres.327908

Bautista, R. G. (2012). The Students Procedural Fluency and Well-Written Explanation on Constructed Response Tasks in Physics. Journal of Technology and Science Education, vol. 3, 49-56. https://doi.org/10.3926/jotse.68

Bas, V. C. (1977). The Pragmatics of Explanation. American Philosophical Quarterly Vol. 14, No. 2, 143-150.

Belecina, Rene R. & Jose M. Ocampo, Jr. (2018). “Effecting Change on Students’ Critical Thinking in Problem Solving” in EDUCARE: International Journal for Educational Studies, Volume 10(2), February, pp.109-118. Bandung, Indonesia and BS Begawan, Brunei Darussalam: Minda Masagi Press owned by ASPENSI and BRIMAN Institute, ISSN 1979-7877. https://doi.org/10.2121/edu-ijes.v10i2.949

Bernard, M., & Chotimah, C. (2018). Improve student mathematical reasoning with open-ended approach using VBA for powerpoint. AIP Conference Proceeding, 1-2. https://doi.org/10.1063/1.5054417

Brezavšˇcek, A., Jerebic, J., Rus, G., & Žnidaršiˇc, A. (2020). Factors Influencing Mathematics Achievement of University Students of Social Sciences. emathematics, 50-62. https://doi.org/10.3390/math8122134

Carey, E.; Hill, F.; Devine, A.; Sz˝ ucs, D. (2017). The Modified Abbreviated Math Anxiety Scale: A Valid and Reliable Instrument for Use with Children. Front. Psychol. 8.

Dewey, J. (1938) Experience and Education. New York: Collier Books.

Dewi, K., Waluya, S. B., Rachmad, & Firmasari, S. (2020). Adaptive reasoning and procedural fluency in three-dimensional. Journal of Physics: Conference Series 1511.

Eko, Y., Prabawanto, S., & Jupri, A. (2018). The role of writing justification in mathematics concept: the case of trigonometry. Journal of Physics: Conference Series 1097. https://doi.org/10.1088/1742-6596/1097/1/012146

Fortes, E.C., & Andrade, R.R. (2019). Mathematical Creativity in Solving Non-Routine Problems. The Normal Lights, 13(1), 108 – 135

Foster, C. (2017). Developing Mathematical Fluency: Comparing Exercises. Educ Stud Math (2018) 97:121–141. https://doi.org/10.1007/s10649-017-9788-x

Gerber, B., Cavallo, A., & Marek, E. (2001). Relationships among informal learning environments, teaching procedures and scientific reasoning ability. International Journal of Science Education Volume 23, 2001 – Issue 5, 535-549. https://doi.org/10.1080/095006901750162892

Glass, A., & Kang, M. (2020). Fewer students are benefiting from doing their homework: an eleven-year study. Educattional Psychology, DOI: 10.1080/01443410.2020.1802645.

Hadiastuti, D., Soedjoko, E., & Mulyono. (2019). Analysis of Mathematical Representation Ability Based on Students’ Thinking Style in Solving Open-Ended Problems. Unnes Journaal of Mathematics Education 8 (3) 2019: 195-201. https://doi.org/10.15294/ujme.v8i3.34189

Inayah, S., Septian, A., & Fazrianto, R. (2020). Student Procedural Fluency in Numerical Method Subjects. Desimal: Jurnal Matematika Vol 3 No 1 (2020), 53-64. https://doi.org/10.24042/djm.v3i1.5316

K-12 Mathematics Education Curriculum Guide Mathematics (2016). Department of Education.

Kattuo, M., Kontoyianni, K., Pitta-Pantazi, D., & Christou, C. (2012). Connecting mathematical creativity to mathematical ability. ZDM Mathematics Education. doi:10.1007/s11858-012-0467-1

Kilpatrick, J., Swafford, J., & Findell, a. B. (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press.

Klavir, R., & Hershkovitz, S. (2008). Teaching and Evaluating ‘Open-Ended’ Problems.

Kwon, O., Park, J., & Park, J. (2016). Cultivating divergent thinking in mathematics through an open-ended approach. Asia Pacific Education Review, 7(1), 51–61.
https://doi.org/10.1007/BF03036784

Laswadi, Kusumah, Y., Darwis, S., & Afgani, J. (2016). Developing Conceptual Understanding and Procedural Fluency for Junior High School Students Through Model-Facilitated Learning. European Journal of Science and Mathematics Education Vol. 4, No. 1, 2016, 67‐74.

Mellony, G., & Stott, G. (2012). Conceptualising Procedural Fluency as a Spectum of Proficiency. South Africa: Rhodes University.

Milos, C. (2014). Open-Ended Tasks and Questions in Mathematics.

Muin, A., Hanifahand, S., & Diwidian, F. (2018). The effect of creative problem solving on students’ mathematical. Journal of Physics: Conference Series 948, 1.

Mushtaq, F. (2013). Mathematics achievements among high school students: Gender difference in math achievements. Karlstads Universitet, 112-123.

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston VA: Author.

Nold, H. (2017). Using Critical Thinking Teaching Methods to Increase Student Success: An Action Research Project. International Journal of Teaching and Learning in Higher Education, 17-32. doi:https://files.eric.ed.gov/fulltext/EJ1136016.pdf

Noureen, G., Awan, R.-U.-N., & Irfan, I. (2015). Assessment of Students’ Problem-Solving Proficiency in Mathematics at Grade VI. Pakistan Journal of Education, 2.

Pentang, J. T., Ibañez, E. D., Subia, G. S., Domingo, J. G., Gamit, A. M., & Pascual, L. E. (2021). Problem-solving performance and skills of prospective elementary teachers in Northern Philippines. Journal of Hunan University Natural Sciences, 48(1), 122-132.

Programme for International Student Assessment (PISA). (2018) Retrieved on 4 December 2019 from http://www.pisa.oecd.org/

Rizki, H., Frentika, D., & Wijaya, A. (2017). Exploring students’ adaptive reasoning skills and van Hiele levels of geometric thinking: A case study in geometry. Journal of Physics: Conf. Series 983. https://doi.org/10.1088/1742-6596/983/1/012148

Rodriguez, C., & Bonner, E. (2018). The Impact of Teacher Questioning and Open-ended Problems on Mathematical Communication. The Journal of Teacher Action Research.

Samuelsson, J. (2010). The Impact of Teaching Approaches on Students’ Mathematical Proficiency in Sweden. International Electronic Journal of Mathematics Education Vol.5, No.2. https://doi.org/10.29333/iejme/250

Schukajlow, S., & Krug, A. (2014). Do Multiple Solutions Matter? Prompting Multiple Solutions, Interest, Competence, and Autonomy. Journal for Research in Mathematics Education Vol.45 No.4, 497-533. https://doi.org/10.5951/jresematheduc.45.4.0497

Stupel, M., & Ben-Chaim, D. (2013). One problem, multiple solution: How multiple proofs can connect several areas of mathematics. Far East Journal of Mathematical Education, 11(2), 129.

Syukriani, A., Juniati, D., & Siswono, T. (2017). Investigating adaptive reasoning and strategic competence: Difference male and female. AIP Conference Proceedings 1867, 020033 (2017). https://doi.org/10.1063/1.4994436

Thomas, S. (2018). Practical reasoning in natural language (3rd ed). Englewood Cliffs: NJ:Prentice-Hall.

Yuniatri, Y., Kusumah, Y., Suryadi, D., & Kartasasma, B. (2017). The Effectiveness of Open-Ended Problems Based Analytic-Synthetic Learning on the Mathematical Creative Thinking Ability of Pre-Service Elementary School Teachers. International Electronic Journal of Mathematics Education, 655-666. https://doi.org/10.29333/iejme/640

Wijaya, A. (2018). How do Open-Ended Problems Promote Mathematical Creativity? A Reflection of Bare Mathematics Problem and Contextual Problem. IOP Conf. Series: Journal of Physics: Conf. Series 983 (2018). https://doi.org/10.1088/1742-6596/983/1/012114

Cite this article:

Andal, S.B. & Andrale, R.R. (2022). Exploring students’ procedural fluency and written adaptive reasoning skills in solving open-ended problems. International Journal of Science, Technology, Engineering and Mathematics, 2(1), 1-25. https://doi.org/10.53378/352872