Articles
Published: 2024-05-30

# Characteristics of pre-service mathematics teacher when solving convergent sequence problems

Pendidikan Matematika UIN Datokarama Palu
Problem solving approach Mathematical representation Convergent sequence Case study

## Abstract

[English]: Convergent sequences pose a challenge for students to comprehend in real analysis courses. Problem-solving based learning can serve as an alternative approach for imparting the understanding of convergent sequences. The objective of this research is to provide a description of the activities undertaken by students when solving problems related to convergent sequences, focusing on specific characteristics of their problem-solving approaches. This study used a qualitative methodology, namely a case study design, with a sample size of 14 participants who were recruited using purposive sampling. The collection of data was conducted via tests and interviews. The findings of this study indicate that the participants can be classified into dominant representation groups and non-dominant representation groups. The subjects who are verbally dominant tend to express ideas using precise language, possess the ability to elaborate on concepts, and demonstrate logical reasoning and argumentation skills. The subjects who are visually dominant tend to analyze or convert visual representations throughout the process of problem-solving. Meanwhile, those who are symbolically dominant tend to approach problem-solving by breaking down the difficulties into multiple solution phases that are conceptually organized. The subjects who are not dominant in a certain type of representation show flexibility in understanding the problem by using a variety of representations that are appropriate for their situation and level of knowledge. The results of this study can serve as a guide for constructing educational approaches, taking into account the characteristics of students when solving problems related to convergent sequences.

[Bahasa]: Barisan konvergen merupakan salah satu konsep yang sulit dipahami peserta didik pada matakuliah analisis real. Pembelajaran berbasis pemecahan masalah dapat menjadi alternatif untuk menanamkan konsep barisan konvergen. Penelitian ini bertujuan untuk mendeskripsikan aktivitas peserta didik dalam memecahkan masalah barisan konvergen berdasarkan representasi matematis yang mereka kembangkan. Kajian ini menggunakan pendekatan kualitatif dengan jenis studi kasus yang melibatkan 14 responden yang dipilih secara purposive sampling. Data dikumpulkan menggunakan instrumen tes dan pedoman wawancara. Hasil penelitian ini menunjukkan bahwa responden dapat dikategorikan menjadi kelompok representasi dominan dan kelompok representasi tidak dominan. Responden yang dominan verbal cenderung menyampaikan ide melalui kata-kata yang jelas, mampu merincikan konsep serta menyusunnya dengan argumen dan pemikiran yang logis. Responden yang dominan visual cenderung menginterpretasikan atau menerjemahkan tampilan visual dalam membangun tahapan pemecahan masalah. Adapun responden dominan simbolik cenderung memecahkan masalah dengan menguraikannya menjadi beberapa langkah penyelesaian yang terstruktur secara konseptual. Karakteristik responden yang tidak dominan pada satu tipe representasi menunjukkan fleksibilitas mahasiswa dalam memahami masalah dengan menggunakan beragam representasi yang disesuaikan dengan kondisi dan pengetahuan yang dimiliki. Temuan penelitian ini dapat menjadi acuan untuk merancang skenario pembelajaran berdasarkan karakteristik individu dalam pemecahan masalah barisan konvergen.

## References

1. Adeoye, M. A., & Jimoh, H. A. (2023). Problem-solving skills among 21st-century learners toward creativity and innovation ideas. Thinking Skills and Creativity Journal, 6(1), 52–58. https://doi.org/10.23887/tscj.v6i1.62708
2. Afriyani, D., Sa’dijah, C., Subanji, S., & Muksar, M. (2018). Characteristics of students’ mathematical understanding in solving multiple representation task based on solo taxonomy. International Electronic Journal of Mathematics Education (IEJME), 13(3), 281–287. https://doi.org/10.12973/iejme/3920
3. Angraini, L. M., & Wahyuni, A. (2021). The effect of concept attainment model on mathematical critical thinking ability. International Journal of Instruction, 14(1), 727–742. https://doi.org/10.29333/iji.2021.14144a
4. Arnal-Palacián, M., Claros-Mellado, J., & Sánchez-Compaña, M. T. (2020). Infinite limit of sequences and its phenomenology. International Electronic Journal of Mathematics Education (IEJME), 15(3), 1–13. https://doi.org/10.29333/iejme/8279
5. Bartle, R. G., & Sherbert, D. R. (2011). Introduction to real analysis. In University of Illinois, Urbana-Champaign (4th ed.). John Wiley & Sons, Inc.
6. Çelik, H. C., & Özdemir, F. (2020). Mathematical thinking as a predictor of critical thinking dispositions of pre-service mathematics teachers. International Journal of Progressive Education, 16(4), 81–98. https://doi.org/10.29329/ijpe.2020.268.6
7. Cooper, J. L., Sidney, P. G., & Alibali, M. W. (2018). Who benefits from diagrams and illustrations in math problems ? Ability and attitudes matter. Applied Cognitive Psychology, 38, 24–38. https://doi.org/10.1002/acp.3371
8. Creswell, J. W., & Creswell, J. D. (2018). Research design qualitative, quantitative, and mixed methods approaches (5th ed.). SAGE Publications, Inc.
9. Darling-Hammond, L., Flook, L., Cook-Harvey, C., Barron, B., & Osher, D. (2020). Implications for educational practice of the science of learning and development. Applied Developmental Science, 24(2), 97–140. https://doi.org/10.1080/10888691.2018.1537791
10. Demircioglu, T., Karakus, M., & Ucar, S. (2023). Developing students’ critical thinking skills and argumentation abilities through augmented reality–based argumentation activities in science classes. In Science and Education (Vol. 32, Issue 4). Springer Netherlands. https://doi.org/10.1007/s11191-022-00369-5
11. Diarni, I. M., Ikhsan, M., Zaura, B., Johar, R., & Mailizar, M. (2023). Profile of students’ mathematical understanding through diagnostic tests viewed from multiple intelligences. Jurnal Didaktik Matematika, 10(1), 77–92. https://doi.org/10.24815/jdm.v10i1.31965
12. Ebissa, L. (2020). Improving geometric concepts perceived difficult to first year linear mathematics students ’ of Kemissie College of Teachers Education in 2019 G . C . International Journal of Creative Research Thoughts (IJCRT), 8(11), 276–288.
13. Fatmawati, A., Zubaidah, S., Mahanal, S., & Sutopo, S. (2022). Representation skills of students with different ability levels when learning using the LCMR model. Pegem Journal of Education and Instruction, 13(1), 177–192. https://doi.org/10.47750/pegegog.13.01.20
14. Feriyanto. (2017). The ability of students’ mathematical proof in determining the validity of argument reviewed from gender differences. Journal of Physics: Conference Series, 947(1), 1–6. https://doi.org/10.1088/1742-6596/947/1/012042
15. Fraenkel, J. R., Wallen, N. E., & Hyun, H. H. (2012). How to Design and Evaluate Research in Education (8th ed.). McGraw-Hill.
16. Fries, L., Son, J. Y., Givvin, K. B., & Stigler, J. W. (2021). Practicing connections : a framework to guide instructional design for developing understanding in complex domains. Educational Psychology Review, 33, 739–762. https://doi.org/10.1007/s10648-020-09561-x
17. Hadiastuti, D. I., Soedjoko, E., & Universitas, M. (2019). Analysis of mathematical representation ability based on students’ thinking style in solving open-ended problems. Unnes Journal of Mathematics Education, 8(3), 195–201. https://doi.org/10.15294/ ujme. v8i3.34189
18. Hariyani, M., Suherman, S., Andriani, M., & Herawati, H. (2023). The Importance of mathematical representation ability for elementary school students: a literature review and its implications. Syekh Nurjati International Conference on Elementary Education, 1(0), 38–46. https://doi.org/10.24235/sicee.v1i0.14579
19. Herizal, H., Suhendra, S., & Nurlaelah, E. (2019). The ability of senior high school students in comprehending mathematical proofs. Journal of Physics: Conference Series, 1157(2), 2–7. https://doi.org/10.1088/1742-6596/1157/2/022123
20. Himmah, M., & Rahaju, E. B. (2021). Analysis of student’s mathematics representation in solving mathematics problems based on spatial cognitive style. MATHEdunesa, 10(2), 189–199. https://doi.org/10.26740/mathedunesa.v10n2.ppdf_189-199
21. Isnani, Waluya, S. B., Rochmad, Dwiyanto, & Asih, T. S. N. (2021). Analysis of problem-solving difficulties at limits of sequences. Journal of Physics: Conference Series, 1722(1). https://doi.org/10.1088/1742-6596/1722/1/012033
22. Iwuanyanwu, P. N. (2022). What students gain by learning through argumentation. International Journal of Teaching and Learning in Higher Education, 34(1), 97–107. http://www.isetl.org/ijtlhe/
23. Kola, M., & Molise, H. (2023). Assessing the implementation of critical thinking skills in the university: a focus on technology education. E-Journal of Humanities, Arts and Social Sciences, 4(5), 500–515. https://doi.org/10.38159/ehass.2023451
24. Kristianto, E., Mardiyana, & Saputro, D. R. S. (2019). Analysis of students’ error in proving convergent sequence using newman error analysis procedure. JIOP Conf. Series: Journal of Physics: Conf. Series, 1180(012001), 1–7. https://doi.org/10.1088/1742-6596/1180/1/012001
25. Kumar Shah, R. (2019). Effective constructivist teaching learning in the classroom. Shanlax International Journal of Education, 7(4), 1–13. https://doi.org/10.34293/education.v7i4.600
26. L.Man, Y., Asikin, M., & Sugiman. (2022). Systematic literature review: student’s mathematical representation ability in mathematics learning. Daya Matematis : Jurnal Inovasi Pendidikan Matematika, 10(1), 36–44. https://doi.org/10.26858/jdm.v10i1.26821
27. Langoban, M. A. (2020). What makes mathematics difficult as a subject for most students in higher education? International Journal of English and Education, 9(3), 214–220.
28. Lima, P. D. S. N., Silva, L. D. A., Felix, I. M., & Brandao, L. D. O. (2019). Difficulties in basic concepts of mathematics in higher education: a systematic review. Proceedings - Frontiers in Education Conference, FIE, 2019-Octob(November). https://doi.org/10.1109/FIE43999.2019.9028658
29. Mainali, B. (2021). Representation in teaching and learning mathematics. International Journal of Education in Mathematics, Science and Technology, 9(1), 1–21. https://doi.org/10.46328/ijemst.1111
30. Maslihah, S., Waluya, S. B., Rochmad, R., & Suyitno, A. (2020). The role of mathematical literacy to improve high order thinking skills. Journal of Physics: Conference Series, 1539(1), 1–6. https://doi.org/10.1088/1742-6596/1539/1/012085
31. Munna, A. S., & Kalam, M. A. (2021). Teaching and learning process to enhance teaching effectiveness: literature review. International Journal of Humanities and Innovation (IJHI), 4(1), 1–4. https://doi.org/10.33750/ijhi.v4i1.102
32. Mutammam, M. B., & Wulandari, E. N. (2023). Profile of junior high school students’ symbol sense thinking. JTAM (Jurnal Teori Dan Aplikasi Matematika), 7(2), 509–521. https://doi.org/10.31764/jtam.v7i2.12622
33. Mutodi, P., & Mosimege, M. (2021). Learning mathematical symbolization: conceptual challenges and instructional strategies in secondary schools. Bolema - Mathematics Education Bulletin, 35(70), 1180–1199. https://doi.org/10.1590/1980-4415v35n70a29
34. Novianti, M., & Retnawati, H. (2019). Student-teacher’s perception of mathematical representation in mathematics learning. Journal of Physics: Conference Series, 1320(1), 1–6. https://doi.org/10.1088/1742-6596/1320/1/012106
35. Nurdin, N., Assagaf, S. F., & Arwadi, F. (2021). Students’ understanding on formal definition of limit. Journal of Physics: Conference Series, 1752(012082), 1–4. https://doi.org/10.1088/1742-6596/1752/1/012082
36. Parame-Decin, M. B. (2023). Visual representations in teaching mathematics. Sprin Journal of Arts, Humanities and Social Sciences, 2(05), 21–30. https://doi.org/10.55559/sjahss.v2i05.107
37. Peňalber, M. D. (2023). The practice of gardner’s multiple intelligences theory in the classroom. Journal for Educators, Teachers and Trainers, 14(4), 62–74. https://doi.org/10.47750/jett.2023.14.04.006
38. Ponnusamy, S. (2012). Sequences: Convergence and Divergence. In Foundations of mathematical analysis (pp. 23–71). Springer Science+Business Media. https://doi.org/10.1007/978-0-8176-8292-7
39. Prayitno, S., Lu’luilmaknunn, U., Sridana, N., & Subarinah, S. (2021). Analyzing the ability of mathematics students as prospective mathematics teachers on multiple mathematical representation. Proceedings of the 2nd Annual Conference on Education and Social Science (ACCESS), 556, 309–313. https://doi.org/10.2991/assehr.k.210525.096
40. Rabih, M. N. A. (2017). On convergence criteria for sequences. IJRR International Journal of Research & Review, 4(5), 87–91. https://doi.org/10.4444/ijrr.1002/375
41. Ramírez-Uclés, R., & Ruiz-Hidalgo, J. F. (2022). Reasoning, representing, and generalizing in geometric proof problems among 8th grade talented students. Mathematics, 10(789), 1–21. https://doi.org/10.3390/math10050789
42. Rashid, Y., Rashid, A., Warraich, M. A., Sabir, S. S., & Waseem, A. (2019). Case study method: a step-by-step guide for business researchers. International Journal of Qualitative Methods, 18, 1–13. https://doi.org/10.1177/1609406919862424
43. Ratumanan, T. G., Ayal, C. S., & Tupamahu, P. Z. (2022). Mathematical representation ability of mathematics education study program students. Jurnal Pendidikan Matematika (JUPITEK), 5(1), 50–59. https://doi.org/10.30598/jupitekvol5iss1pp50-59
44. Rosa, M., D’Ambrosio, Ubiratan; Orey, D. C., Shirley, L., Alangui, W. V., & Palhares, Pedro; Gavarrete, M. E. (2016). Current and future perspectives of ethnomathematics as a program (ICME-13 To). Springer Open. https://doi.org/10.1007/978-3-319-30120-4_1
45. Ross, K. A. (2013). Elementary analysis : the theory of calculus (2nd ed.). Springer Science+Business Media. https://doi.org/10.1016/S0049-237X(09)70531-2
46. Schüler-Meyer, A. (2020). Mathematical routines in transition: facilitating students’ defining and proving of sequence convergence. Teaching Mathematics and Its Applications: An International Journal of the IMA, 39(Januari), 237–247. https://doi.org/10.1093/teamat/hrz019
47. Sebsibe, A. S., & Feza, N. N. (2020). Assessment of students ’ conceptual knowledge in limit of functions. International Electronic Journal of Mathematics Education, 15(2), 1–15. https://doi.org/10.29333/iejme/6294
48. Subedi, A. (2020). Graduate level students’ techniques and difficulties in proving theorems of abstract algebra. Education and Development, 30(1), 99–112. https://doi.org/10.3126/ed.v30i1.49515
49. Tuna, A., Biber, A. C., & Korkmaz, S. (2019). What do teacher candidates know about the limits of the sequences? Journal of Curriculum and Teaching, 8(3), 132–142. https://doi.org/10.5430/jct.v8n3p132
50. Utami, A. P., Mardiyana, & Pramudya, I. (2021). Visual students: how their representation in problem solving? Proceedings of the International Conference of Mathematics and Mathematics Education (I-CMME 2021), 597, 31–41. https://doi.org/10.2991/assehr.k.211122.005
51. Utomo, D. P., & Syarifah, D. L. (2021). Examining mathematical representation to solve problems in trends in mathematics and science study: voices from indonesian secondary school students. International Journal of Education in Mathematics, Science and Technology, 9(3), 540–556. https://doi.org/10.46328/IJEMST.1685
52. Vintere, A. (2018). A constructivist approach to the teaching of mathematics to boost competences needed for sustainable development. Rural Sustainability Research, 39(334), 2–7. https://doi.org/10.2478/plua-2018-0001
53. Widiati, I., & Sthephani, A. (2018). Difficulties analysis of mathematics education students on the real analysis subject. The 6th South East Asia Design Research International Conference (6th SEA-DR IC), 1088, 1–5. https://doi.org/10.1088/1742-6596/1088/1/012037
54. Žakelj, A., & Klančar, A. (2022). The role of visual representations in geometry learning. European Journal of Educational Research, 11(3), 1393–1411. https://doi.org/10.12973/eu-jer.11.3.1393
55. Zorzos, M., & Avgerinos, E. (2023). Research on visualization in probability problem solving. Eurasia Journal of Mathematics, Science and Technology Education, 19(4), 1–10. https://doi.org/10.29333/EJMSTE/13066

### How to Cite

Nursupiamin, & Rochaminah, S. (2024). Characteristics of pre-service mathematics teacher when solving convergent sequence problems. Beta: Jurnal Tadris Matematika, 17(1), 1–18. https://doi.org/10.20414/betajtm.v17i1.617