Article Sidebar

Download
PDF
Statistic
Read Counter : 460 Download : 267

Downloads

Download data is not yet available.

Main Article Content

Abstract

The algebraic structure is one of the axiomatic mathematical materials that consists of definitions and theorems. Learning algebraic structure will facilitate the development of logical reasoning, hence facilitating the study of other aspects of axiomatic mathematics. Even with this, several researchers say a lack of algebraic structure sense is a source of difficulty in acquiring algebraic structures. This study aims to examine a pattern of sequences of problem-solving paths in algebra, which is an illustration of learners' algebraic structure sense so that it can be utilized to enhance the ability to solve problems involving algebraic structure. This study employed a qualitative descriptive approach. Students who have received abstract algebra courses were chosen to serve as subjects. The instruments include tests based on algebraic structure sense, questionnaires, and interviews. This study reveals the sequence of paths used by students in the structure sense process for group materials, i.e., path of construction–analogy (constructing known mathematical properties or objects, then analogizing unknown mathematical properties or objects), path of analogy–abstraction (analogizing an unknown mathematical property or object with consideration of the initial knowledge, then abstracting a new definition), path of abstraction-construction (abstracting the definition of the extraction of a known mathematical structure or object, then constructing a new mathematical structure or object), and path of formal-construction (constructing the structure of known and unknown mathematical properties or objects through the logical deduction of a familiar definition). In general, the student's structure sense path for solving problems of group material begins with construction, followed by analogy, abstraction, and formal construction. Based on these findings, it is suggested that there is a way for lecturers to observe how students develop algebraic concepts, particularly group material, so that they can employ the appropriate strategy while teaching group concepts in the future.

Keywords

Algebraic Structure Sense Group Mathematics Path Sequence

Article Details

Creative Commons License

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

How to Cite
Junarti, Zainudin, M., & Utami, A. D. (2022). The sequence of algebraic problem-solving paths: Evidence from structure sense of Indonesian student. Journal on Mathematics Education, 13(3), 437–464. https://doi.org/10.22342/jme.v13i3.pp437-464
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX

References

  1. Bintoro, H. S., Sukestiyarno, Y. L., Mulyono, & Walid. (2021). The spatial thinking process of the field-independent students based on action-process-object-schema theory. European Journal of Educational Research, 10(4), 1807-1823. https://doi.org/10.12973/eu-jer.10.4.1807
  2. Bone, E., Bouck, E., & Witmer, S. (2021). Evidence-based systematic review of literature on algebra instruction and interventions for students with learning disabilities. Learning Disabilities, 19(1), 1–22.
  3. Bruner, J. S. (1966). Toward a theory of instruction (Vol. 59). Harvard University Press.
  4. Capaldi, M. (2014). Non-traditional methods of teaching abstract algebra. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 24(1), 12-24. https://doi.org/10.1080/10511970.2013.821427
  5. ÇETIN, Ö. F. (2021). The importance of algebra teaching; daily life variables and number systems corresponding to these variables. International Journal of New Trends in Arts, Sports & Science Education, 10(5), 297–315.
  6. Cresswell, J.W. (2017). Research Design. Sage
  7. Dewi, I. L. K., Zaenuri, Dwijanto, & Mulyono. (2021). Identification of mathematics prospective teachers’ conceptual understanding in determining solutions of linear equation systems. European Journal of Educational Research, 10(3), 1157-1170. https://doi.org/10.12973/eu-jer.10.3.1157
  8. Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27(3), 267-305. https://doi.org/10.1007/BF01273732
  9. Harel, G., & Tall, D. (1991). The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics, 11(1), 38-42. https://www.jstor.org/stable/40248005
  10. Hoch, M. (2003). Structure sense. The 3rd Conference of the European Researchers in Mathematics Education.
  11. Hoch, M., & Dreyfus, T. (2005). Students’ difficulties with applying a familiar formula in an unfamiliar context. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. Melbourne, Australia: PME. Vol. 3, 145-152.
  12. Hoch, M., & Dreyfus, T. (2010). Developing katy’s algebraic structure sense. Proceedings of CERME 6, 529–538.
  13. Hwang, J., Riccomini, P. J., & Morano, S. (2019). Examination of cognitive processes in effective algebra problem-solving interventions for secondary students with learning disabilities. Learning Disabilities, 17(2), 205–222.
  14. Junarti, Sukestiyarno, Y. L., Mulyono, & Dwidayati, N. K. (2019). The profile of structure sense in abstract algebra instruction in an Indonesian mathematics education. European Journal of Educational Research, 8(4), 1081–1091. https://doi.org/10.12973/eu-jer.8.4.1081
  15. Junarti, Sukestiyarno, Y. L., Mulyono, & Dwidayati, N. K. (2020). The process of structure sense of group prerequisite material: A case in indonesian context. European Journal of Educational Research, 9(3), 1047–1061. https://doi.org/10.12973/EU-JER.9.3.1047
  16. Kamacı, H. (2021). Linear Diophantine fuzzy algebraic structures. Journal of Ambient Intelligence and Humanized Computing, 12(11), 10353-10373. https://doi.org/10.1007/s12652-020-02826-x
  17. Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40(2), 173-196. https://doi.org/10.1023/A:1003606308064
  18. Mason, T., Stephens M., & Watson A. (2009). Appreciating mathematical structure for all. Mathematics Education Research Journal, 21(2), 10-32. https://files.eric.ed.gov/fulltext/EJ883866.pdf
  19. Novotná, J., & Hoch, M. (2008). How structure sense for algebraic expressions or equations is related to structure sense for abstract algebra. Mathematics Education Research Journal, 20(2), 93–104. https://doi.org/10.1007/BF03217479
  20. Novotná, J., Stehlíková, N., & Hoch, M. (2006). Structure sense for university algebra. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Prague, Czech Republic: PME. Vol. 4, 249-256.
  21. Nu, A. T. (2019). Algebra thinking process on vocational school students in completing line problems. KOLOKIUM Jurnal Pendidikan Luar Sekolah, 7(2), 75–87. https://doi.org/10.24036/kolokium-pls.v7i2.350
  22. Oktac, A. (2016). Abstract algebra learning: Mental structures, definitions, examples, proofs and structure sense. Annales De Didactique Et De Sciences Cognitives, 21, 297 -316.
  23. Ralston, N. C., Li, M., & Taylor, C. (2018). The development and initial validation of an assessment of algebraic thinking for students in the elementary grades. Educational Assessment, 23(3), 211–227. https://doi.org/10.1080/10627197.2018.1483191
  24. Schneider, S., Beege, M., Nebel, S., Schnaubert, L., & Rey, G. D. (2021). The cognitive-affective-social theory of learning in digital environments (CASTLE). Educational Psychology Review, 34, 1-38. https://doi.org/10.1007/s10648-021-09626-5
  25. Schüler-Meyer, A. (2017). Students’ development of structure sense for the distributive law. Educational Studies in Mathematics, 96(1), 17-32. https://doi.org/10.1007/s10649-017-9765-4
  26. Simpson, A., & Stehlikova, N. (2006). Apprehending mathematical structure: A case study of coming to understand a commutative ring. Educational Studies in Mathematics, 61(3), 347-371. https://doi.org/10.1007/s10649-006-1300-y
  27. Souminen, A. L. (2018). Abstract algebra and secondary school mathematics connections as discussed by mathematicians and mathematics educators. In Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers. Springer.
  28. Stehlíková, N. (2004). Structural Understanding in Advanced Mathematical Thinking. Prague: Charles University in Prague, Faculty of Education.
  29. Taban, J. G., & Cadorna, E. A. (2018). Structure sense in algebraic expressions and equations of groups of students. Journal of Educational and Human Resource Development, 6, 140–154.
  30. Usiskin, Z. (2001). Teachers’ mathematics: A collection of content deserving to be a field. Mathematics Teacher, 6(1), 86–98.
  31. Wasserman, N. H. (2014). Introducing algebraic structures through solving equations: Vertical content knowledge for k-12 mathematics teachers. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate, 24(3), 191–214. http://doi.org/10.1080/10511970. 2013.857374
  32. Wasserman, N. H. (2017). Exploring how understandings from abstract algebra can influence the teaching of structure in early algebra. Mathematics Teacher Education and Development, 19(2), 81–103.
  33. Widodo, S. A., Irfan, M., Trisniawati, T., Hidayat, W., Perbowo, K. S., Noto, M. S., & Prahmana, R. C. I. (2020). Process of algebra problem-solving in formal student. Journal of Physics: Conference Series, 1657(1), 012092. https://doi.org/10.1088/1742-6596/1657/1/012092
  34. Wilburne, J. M., & Long, M. (2010). Secondary pre-service teachers’ content knowledge for state assessments: Implications for mathematics education programs. Issues in the Undergraduate Mathematics Preparation of School Teachers, 1(January), 1–13.