Level of combinatorial thinking in solving mathematical problems

Abstract

Combinatorial thinking is an important reasoning process in building one's knowledge and experience. The purpose of this study is to describe the characteristics of the level of combinatorial thinking in solving mathematical problems. The subjects of the study were 40 students of Elementary Teacher Education Department (PGSD): 20 students of the second semester and the others of the sixth semester. The reason for choosing subjects from these two levels is to meet all levels of combinatorial thinking. All research subjects were given test questions about combinatorial problems. From 40 subjects, five students were selected to be interviewed as they had fulfilled all five levels of combinatorial thinking. The data validity was conducted by triangulation through recording interview results and comparing it with data from students' written test results to ensure the validity and reliability of this research. The results show that there are five levels of combinatorial thinking in solving mathematical problems: investigating “some cases’, systematically checking cases, using the calculation order, systematically generating all cases, and changing the problem into another combinatorial problem. Level one is the identification of the possibility of students’ understanding the questions incorrectly, or vice versa, already can answer the questions with systematic procedures but the results are less precise. Level two is conducting systematic checking about students' understanding of the combination material. Besides, it also concerns about the ability to answer problems systematically using diagram trees. Level three is students are able to apply the calculation orders, which are addition and multiplication. Level four is systematically generating all cases about the ability to calculate possibilities without schematic, drawings, or diagrams. Level five is changing the problem into another combinatorial problem, it is the ability to calculate possibilities with complex problems Based on the research findings, it turns out there is another level of combinatorial thinking, which is using the calculation order and this is found between level two and level three. The researchers recommend further research to explore more on the application of calculation order.

Keywords

• Combinatorial thinking level
• Mathematics education
• Mathematical problems
• Students of elementary teacher

References

1. Al-Mutawah, M. A., Thomas, R., Eid, A., Mahmoud, E. Y., & Fateel, M. J. (2019). Conceptual Understanding, Procedural Knowledge and Problem-Solving Skills in Mathematics : High School Graduates Work Analysis and Standpoints. International Journal of Education and Practice 7(3), 258–273. https://doi.org/10.18488/journal.61.2019.73.258.273.al
2. Balera, J. M., & Júnior, V. A. S. (2017). An Algorithm for Combinatorial Interaction Testing: Definitions and Rigorous Evaluations. Journal of Software Engineering Research and Development 5(1), 2-41. https://doi.org/10.1186/s40411-017-0043-z.
3. Batanero, C., Navarro-Pelayo, V., & Godino, J. D. (1997). Effect of the Implicit Combinatorial Model on Combinatorial Reasoning in Secondary School Pupils. Educational Studies in Mathematics 32(2), 181–199. https://doi.org/10.1023/A
4. Cresswell, J. (2012). Educational Research: Planning, Conducting and Evaluating Qualitative and Quantitative Research (4th ed.). Boston: Pearson Education Inc.
5. Cuevas, O., Larios, V., Peralta, J. X., & Jiménez, A. R. (2018). Mathematical Knowledge of Students who Aspire to Enroll in Engineering Programs. International Electronic Journal of Mathematics Education 13(3), 161–169. https://doi.org/10.12973/iejme/3832.
6. Eizenberg, M., M., & Zaslavsky, O. (2004). Students’ Verification Strategies for Combinatorial Problems. Mathematical Thinking and Learning, 6(1): 15–36. https://doi.org/10.1207/s15327833mtl0601_2.
7. English, L.D. (1991). Young children’s combinatoric strategies. Educational Studies in Mathematics, 22(5), 451–474. https://doi.org/10.1007/BF00367908
8. English, L.D. (2005). Combinatorics and the Development of Children’s Combinatorial Reasoning. Exploring Probability in School: Challenges for Teaching and Learning, 121–141. https://doi.org/10.1007/0-387-24530-8_6.

9. Golafshani, N. (2003). Understanding Reliability and Validity in Qualitative Research. The Qualitative Report 8(4): 597-606.
10. Hayashi, T., & Ohsawa, Y. (2013). Processing Combinatorial Thinking: International Journal of Knowledge and Systems Science, 4(3), 14–38. https://doi.org/10.4018/ijkss.2013070102
11. Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Describing Levels and Components of a Math-Talk Learning Community. Journal for Research in Mathematics Education, 35(2), 81. https://doi.org/10.2307/30034933
12. Jazim, Anwar, B. A., & Rahmawati, D. (2017). The Use of Mathematical Module Based on Constructivism Approach as Media to Implant the Concept of Algebra Operation. International Electronic Journal of Mathematics Education 12(3): 579–583.
13. Kamau, L. M., Kimani, P., & Muthoni, P. (2016). Factors that Influence Teachers' Perceptions of Information Communication And Technology (ICT) in Mathematics Teaching in Kenyan Secondary Schools. International Journal of Education and Practice 4(4), 154–166. https://doi.org/10.18488/journal.61/2016.4.4/61.4.154.166.
14. Lay, Y. F. (2009). Logical Thinking Abilities among Form 4 Students in the Interior Division of Sabah, Malaysia. Journal of Science and Mathematics Education in Southeast Asia, 32, 161–187. Retrieved from http://www.recsam.edu.my/R&D_Journals/YEAR2009/dec2009vol2/logicalthinking(161-187).pdf%5Cnhttp://131.211.208.19/login?auth=eng&url=http://ovidsp.ovid.com/ovidweb.cgi?T=JS&CSC=Y&NEWS=N&PAGE=fulltext&D=eric3&AN=EJ910939
15. Lockwood, E. (2012). Counting Using Sets of Outcomes. Mathematics Teaching in the Middle School 18(3): 125-132. http://10.5951/mathteacmiddscho.18.3.0132.
16. Lockwood, E. (2013). A model of students’ combinatorial thinking. Journal of Mathematical Behavior, 32(2), 251–265. https://doi.org/10.1016/j.jmathb.2013.02.008.
17. Malloy, C. E., & Jones, M.G. (1998). An Investigation of African American Students’ Mathematical Problem Solving. Journal for Research in Mathematics Education, 29(2): 191–196.
18. Melusova, J., & Vidermanova, K. (2015). Upper-secondary Students’ Strategies for Solving Combinatorial Problems. Procedia - Social and Behavioral Sciences, 197(February), 1703–1709. https://doi.org/10.1016/j.sbspro.2015.07.223
19. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM. NRC. (1989). Everybody Counts: A Report to the Nation on the Future of Mathematics Education. Washington, D.C.: National Academy Press.
20. Pamungkas, A. S., & Khaerunnisa, E. (2020). The analysis of students statistical literacy based on prior knowledge and mathematical self esteem. Journal for the Mathematics Education and Teaching Practices 1(1)): 43–51. Retrieved from https://dergipark.org.tr/en/pub/jmetp/issue/55820/707759.
21. Pizlo, Z., & Li, Z. (2005). Solving combinatorial problems: The 15-puzzle. Memory & Cognition 33(6), 1069–1084. https://doi.org/10.3758/BF03193214
22. Pramusinta, Y., Setyosari, P., Widiati, U., & Kuswandi, D. (2019). Exploring Metacognitive and Critical Thinking Skills of Pre-Service Elementary School Teachers through Discovery Learning Method by Integrating Various Cognitive Styles. Journal for the Education of Gifted Young Scientists 7(4): 999–1017. https://doi.org/10.17478/jegys.614028.
23. Rezaie, M., & Gooya, Z. (2011). What do I mean by combinatorial thinking? Procedia - Social and Behavioral Sciences, 11,122–126. https://doi.org/10.1016/j.sbspro.2011.01.046.
24. Rosidin, U., Suyatna, A., & Abdurrahman, A. (2019). A Combined HOTS-Based Assessment/STEM Learning Model to Improve Secondary Students’ Thinking Skills: A Development and Evaluation Study. Journal for the Education of Gifted Young Scientists 7(2), 435–448. https://doi.org/10.17478/jegys.518464.
25. Setianingsih, R., Sa’dijah, C., As’ari, A. R., & Muksar, M. (2017). Investigating Fifth- Grade Students ’ Construction of Mathematical Knowledge through Classroom Discussion. International Electronic Journal of Mathematics Education 12(4), 383–396.
26. Silwana, A., Subanji, Manyunu, M., & Rashahan, A. A. (2021). Students' Responses Leveling in Solving Mathematical Problem Based on SOLO Taxonomy Viewed from Multiple Intelligences. Indonesian Journal on Learning and Advanced Education (IJOLAE) 3(1), 1–16. https://doi.org/10.23917/ijolae.v3i1.10528
27. Suyono, S. M., Roekhan, & Harsiati, T. (2019). Critical Thinking Patterns of First-Year Students in Argumentative Essay. Journal for the Education of Gifted Young Scientists 7(3), 683–697. https://doi.org/http://dx.doi.org/10.17478/jegys.605324.
28. Tsai, Y. L., Chang, C. K. (2008). Using Combinatorial Approach to Improve Students’ Learning of the Distributive Law and Multiplicative Identities. International Journal of Science and Mathematics Education 7(3), 501-531. https://doi.org/10.1007/s10763-008-9135-x.
29. Yuberti, Y., Rantika, J., Irwandani,I., & Prasetyo, A. E. (2019). The Effect of Instructional Design Based on Learning cycle 7E Model with Mind Map Technique to the Students' Critical Thiking Skills. Journal of Gifted Education and Creativity 6(3), 175-191.
30. Yuen, G. (2008). Problem Solving Strategies Students Use when Solving Combinatorial Problems. A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of. Master of Arts in the Faculty of Graduate Studies the University of British Columbia.
31. Yuli, T., Siswono, E., Hartono, S., Kohar, A. W., Karim, K., & Lastiningsih, N. (2019). How do Prospective Teachers Manage Students’ Learning of Mathematics ? 8(2), 677–685. https://doi.org/10.18421/TEM82-49