CRITICAL THINKING SKILLS: ERROR IDENTIFICATIONS ON STUDENTS’ WITH APOS THEORY

Khoerul Umam, Ardi Dwi Susandi

Abstract


Purpose: Identifying students' errors in solving cases of critical thinking skills from two variables of linear equations (TVLE).

Research Methods: This was a qualitative study using a descriptive exploratory approach. The participants of the study were first-year students of mathematics education. The method used in this research is a test, interview, and triangulation.

Findings: The findings showed that the students have low critical thinking skills; therefore, they could not complete the task correctly. Based on the APOS theory, students' mistakes in completing math problems consist of four elements, namely: (1) the errors in interpreting; (2) the errors in understanding the concept; (3) the error in the procedures; and (4) the error in technical things. The student's response in this study was in the theoretical of APOS so that they could not reach a correct schema.

Implications for Research and Practice: The study results are expected to be beneficial in developing the activities in teaching TVLE so that the students will make less errors in completing critical thinking skills problems in mathematics. Therefore, further study in developing a teaching model for mathematics teaching to improve students' critical thinking skills is highly recommended.


References


A. D. Susandi, C. Sa’dijah, A. R. As’ari, and Susiswo, “What error happened to inferences of senior high school students using mathematical critical thinking ability?,” Int. J. Sci. Technol. Res., vol. 8, no. 9, pp. 507–511, 2019.

E. Aizikovitsh and M. Amit, “Evaluating an Infusion Approach to The Teaching of Critical Thinking Skills Through Mathematics,” Procedia - Soc. Behav. Sci., vol. 2, no. 2, pp. 3818–3822, 2010, doi: 10.1016/j.sbspro.2010.03.596.

A. R. As’ari, “Ideas For Developing Critical Thinking At Primary School Level,” in International Seminar on Addressing Higher Order Thinking: Critical Thinking Issues in Primary Education., 2014, no. 1, pp. 1–13, doi: 10.17809/14(2015)-12.

A. Dwi Susandi, C. Sa’Dijah, A. Rahman As’Ari, and Susiswo, “Students’ Critical Ability Of Mathematics Based On Cognitive Styles,” J. Phys. Conf. Ser., vol. 1315, no. 1, pp. 1–10, 2019, doi: 10.1088/1742-6596/1315/1/012018.

W. Schroyens, “Review Knowledge and Thought: An Introduction to Critical Thinking,” Exp. Psychol., vol. 52, no. 2, pp. 163–164, 1997, doi: 10.1027/1618-3169.52.2.163.

K. Y. L. Ku, “Assessing students’ critical thinking performance: Urging for measurements using multi-response format,” Think. Ski. Creat., vol. 4, no. 1, pp. 70–76, 2009, doi: 10.1016/j.tsc.2009.02.001.

H. A. Butler, “Halpern Critical thinking Assessment Predicts Real-World Outcomes of Critical Thinking,” Appl. Cogn. Psychol., vol. 26, no. 5, pp. 721–729, 2012, doi: 10.1002/acp.2851.

K. C. Yu, K. Y. Lin, and S. C. Fan, “An exploratory study on the application of conceptual knowledge and critical thinking to technological issues,” Int. J. Technol. Des. Educ., vol. 25, no. 3, pp. 339–361, 2015, doi: 10.1007/s10798-014-9289-5.

S. Maričić and K. Špijunović, “Developing Critical Thinking in Elementary Mathematics Education through a Suitable Selection of Content and Overall Student Performance,” Procedia - Soc. Behav. Sci., vol. 180, no. November 2014, pp. 653–659, 2015, doi: 10.1016/j.sbspro.2015.02.174.

H. A. Yacoubian, “A Framework for Guiding Future Citizens to Think Critically About Nature of Science and Socioscientific Issues,” Can. J. Sci. Math. Technol. Educ., vol. 15, no. 3, pp. 248–260, 2015, doi: 10.1080/14926156.2015.1051671.

H. Innabi and O. El Sheikh, “The Change in Mathematics Teachers’ Perceptions of Critical Thinking after 15 Years of Educational reform in Jordan,” Educ. Stud. Math., vol. 64, no. 1, pp. 45–68, 2007, doi: 10.1007/s10649-005-9017-x.

B. Bunyamin, K. Umam, and L. Lismawati, “Critical Review of M-Learning in Total Quality Management Classroom Practice in an Indonesian Private University,” Int. J. Interact. Mob. Technol., vol. 14, no. 20, pp. 76–90, 2020, doi: 10.3991/ijim.v14i20.15141.

Kusaeri and A. Aditomo, “Pedagogical beliefs about Critical Thinking among Indonesian mathematics pre-service teachers,” Int. J. Instr., vol. 12, no. 1, pp. 573–590, 2019, doi: 10.29333/iji.2019.12137a.

A. H. Dewantara, Zulkardi, and Darmawijoyo, “Assessing seventh graders’ mathematical literacy in solving pisa-like tasks,” J. Math. Educ., vol. 6, no. 2, pp. 39–49, 2015, doi: 10.22342/jme.6.2.2163.117-128.

F. Nursyahidah and I. U. Albab, “Investigating Student Difficulties on Integral Calculus Based on Critical Thinking Aspects,” J. Ris. Pendidik. Mat., vol. 4, no. 2, p. 211, 2017, doi: 10.21831/jrpm.v4i2.15507.

D. Uzel and S. M. Uyangor, “Attitudes of 7th class students toward mathematics in realistic mathematics education,” Int. Math. Forum, no. 39, pp. 1951–1959, 2006, doi: 10.12988/imf.2006.06172.

M. Schoenberger-Orgad and D. Spiller, “Critical thinkers and capable practitioners: Preparing public relations students for the 21st century,” J. Commun. Manag., vol. 18, no. 3, pp. 210–221, 2014, doi: 10.1108/JCOM-11-2012-0085.

R. H. Ennis, Critical Thinking. New Jersey: Prentice-Hall Inc., 1996.

Z. Ç. Özcan, “The relationship between mathematical problem-solving skills and self-regulated learning through homework behaviours, motivation, and metacognition,” Int. J. Math. Educ. Sci. Technol., vol. 47, no. 3, pp. 408–420, 2016, doi: 10.1080/0020739X.2015.1080313.

B. Torff, “Developmental changes in teachers’ beliefs about critical-thinking activities,” J. Educ. Psychol., vol. 97, no. 1, pp. 13–22, 2005, doi: 10.1037/0022-0663.97.1.13.

K. Yuliani and S. Suragih, “The Development Of Learning Devices Based Guided Discovery Model To Improve Understanding Concept And Critical Thinking Mathematically Ability Of Students At Islamic Junior High School Of Medan,” J. Educ. Pract., vol. 6, no. 24, pp. 116–128, 2015, [Online]. Available: http://search.proquest.com/docview/1773215035?accountid=8330.

E. Gaigher, J. M. Rogan, and M. W. H. Braun, “Exploring the development of conceptual understanding through structured problem-solving in physics,” Int. J. Sci. Educ., vol. 29, no. 9, pp. 1089–1110, 2007, doi: 10.1080/09500690600930972.

H. Kilic, “The nature of preservice mathematics teacher’s knowledge of students,” Procedia - Soc. Behav. Sci., vol. 9, no. 1, pp. 1096–1100, 2010, doi: 10.1016/j.sbspro.2010.12.292.

F. Hitt, “Student’s functional representations and conceptions in the construction of mathematical concepts. An exemple: the concept of limit,” Ann. didacqique Sci. Cogn., vol. 11, pp. 251–267, 2006.

S. Kiat, “Analysis of Students’ Difficulties in Solving Integration Problem,” Math. Educ., vol. 9, no. 1, pp. 39–59, 2005, doi: 10.20319/pijss.2016.s21.310324.

N. T. N. Lien et al., “Teachers’ feelings of safeness in school-family-community partnerships: Motivations for sustainable development in moral education,” Int. J. Eval. Res. Educ., vol. 10, no. 1, pp. 97–107, 2021, doi: 10.11591/ijere.v10i1.20798.

M. Marilyn, “Towards a Computational Theory of Algebraic Competence,” J. Math. Behav., vol. 3, no. 1, pp. 93–166, 1980.

I. Arnon, J. Cottrill, and E. Dubinsky, APOS Theory A Framework for Research and Curriculum Development in Mathematics Education. New York: Springer US, 2014.

D. Brijlall and Z. Ndlovu, “High school learners’ mental construction during solving optimization problems in calculus: A South African case study,” South African J. Educ., vol. 33, no. 2, pp. 1–18, 2013, doi: 10.15700/saje.v33n2a679.

N. Nasser, E. M. El Khouzai, and A. Zahidi, “Geometrical optic learning difficulties for Moroccan students during secondary/university transition,” Int. J. Eval. Res. Educ., vol. 10, no. 1, pp. 24–34, 2021, doi: 10.11591/ijere.v10i1.20639.

Nurrahmawati, C. Sa’dijah, Sudirman, and M. Muksar, “Assessing students’ errors in mathematical translation: From symbolic to verbal and graphic representations,” Int. J. Eval. Res. Educ., vol. 10, no. 1, pp. 115–125, 2021, doi: 10.11591/ijere.v10i1.20819.

E. D. Dubinsky and M. A. Mcdonald, “APOS: A Constructivist Theory Of Learning In Undergraduate Mathematics Education Research,” in The Teaching and Learning of Mathematics at University Level: An ICMI Study, The Netherlands: Kluwer Academic Publishers, 2001, pp. 275–282.

S. Chee, C. Tunku, A. Rahman, C. Phaik, K. Cheah, and T. A. Rahman College, “Teacher Perceptions of Critical Thinking Among Students and its Influence on Higher Education,” Int. J. Teach. Learn. High. Educ., vol. 20, no. 2, pp. 198–206, 2009, [Online]. Available: http://www.isetl.org/ijtlhe/.

G. S. Aktaş and M. Ünlü, “Critical Thinking Skills of Teacher Candidates of Elementary Mathematics,” Procedia - Soc. Behav. Sci., vol. 93, pp. 831–835, 2013, doi: 10.1016/j.sbspro.2013.09.288.

M. A. Alwadai, “Islamic Teachers’ Perceptions of Improving Critical Thinking Skills in Saudi Arabian Elementary Schools,” J. Educ. Learn., vol. 3, no. 4, pp. 37–48, 2014, doi: 10.5539/jel.v3n4p37.

M. Applebaum, “Activating Pre-Service Mathematics Teachers’ Critical Thinking.,” Eur. J. Sci. Math. Educ., vol. 3, no. 1, pp. 77–89, 2015.

R. Kumar.R and R. James, “Evaluation of Critical Thinking in Higher Education in Oman,” Int. J. High. Educ., vol. 4, no. 3, pp. 33–43, 2015, doi: 10.5430/ijhe.v4n3p33.

J. W. Creswell, Research Design: Qualitative, Quantitative and Mixed Method Approaches, vol. 3, no. 1. London: SAGE Publications, 2010.

V. a. Lambert and C. E. Lambert, “Qualitative Descriptive Research: An Acceptable Design,” Pacific Rim Int. J. Nurs. Res., vol. 16, no. 4, pp. 255–256, 2013, [Online]. Available: http://antispam.kmutt.ac.th/index.php/PRIJNR/article/download/5805/5064.

D. Mogari and M. Chirove, “Comparing grades 10-12 mathematics learners’ non-routine problem solving,” Eurasia J. Math. Sci. Technol. Educ., vol. 13, no. 8, pp. 4523–4551, 2017, doi: 10.12973/eurasia.2017.00946a.

E. S. Hastuti, L. Eclarin, and K. K. S. Dalam, “Kecemasan Siswa Sekolah Menengah Pertama Menyelesaikan Masalah SPLDV Pada Kelas Virtual Dalam,” Int. J. Progress. Math. Educ., vol. 1, no. 1, pp. 64–84, 2021, doi: 10.22236/ijopme.v1i1.6914.

Ebiendele Ebosele Peter, “Critical thinking: Essence for teaching mathematics and mathematics problem solving skills,” African J. Math. Comput. Sci. Res., vol. 5, no. 3, pp. 39–43, 2012, doi: 10.5897/ajmcsr11.161.

D. H. Hryciw and A. M. Dantas, “Scaffolded research-based learning for the development of scientific communication in undergraduate physiology students,” Int. J. Innov. Sci. Math. Educ., vol. 24, no. 1, pp. 1–11, 2016.

H. İ. Akyüz, S. Samsa Yetik, and H. Keser, “Effects of metacognitive guidance on critical thinking disposition,” Pegem Eğitim ve Öğretim Derg., vol. 5, no. 2, pp. 133–148, 2015, doi: 10.14527/pegegog.2015.007.

C. I. Lee, “An appropriate prompts system based on the Polya method for mathematical problem-solving,” Eurasia J. Math. Sci. Technol. Educ., vol. 13, no. 3, pp. 893–910, 2017, doi: 10.12973/eurasia.2017.00649a.

O. Serin, “The Critical Thinking Skills of Teacher Candidates Turkish Republic of Northern Cyprus Sampling,” Eurasian J. Educ. Res., no. 53, pp. 231–248, 2013, doi: 10.14689/ejer.2013.53.13.

J. Jumadi, R. Perdana, Riwayani, and D. Rosana, “The impact of problem-based learning with argument mapping and online laboratory on scientific argumentation skill,” Int. J. Eval. Res. Educ., vol. 10, no. 1, pp. 16–23, 2021, doi: 10.11591/ijere.v10i1.20593.

E. Türnüklü, F. Gündoǧdu Alayli, and E. N. Akkaş, “Investigation of prospective primary mathematics teachers’ perceptions and images for quadrilaterals,” Kuram ve Uygulamada Egit. Bilim., vol. 13, no. 2, pp. 1225–1232, 2013.

A. Gagatsis, C. Christou, and I. Elia, “The Nature of Multiple Representations in Developing Mathematical Relationships,” Quad. di Ric. Didatt., vol. 14, no. 1, pp. 150–159, 2004.

M. Trigueros and R. Martínez-Planell, “Geometrical representations in the learning of two-variable functions,” Educ. Stud. Math., vol. 73, no. 1, pp. 3–19, 2010, doi: 10.1007/s10649-009-9201-5.

S. Dündar, “Mathematics Teacher- Candidates’ Performance in Solving Problems with Different Representation Styles: The Trigonometry Example,” Eurasia J. Math. Sci. Technol. Educ., vol. 11, no. 6, pp. 1363–1378, 2015, doi: 10.12973/eurasia.2015.1396a.

P. Anne and T. M. Mangulabnan, “Assessing Translation Misconceptions Inside the Classroom: A Presentation of an Instrument and Its Results,” US-China Educ. Rev., vol. 3, no. 6, pp. 365–373, 2013.

S. W. Siyepu, “Analysis of Errors in Derivatives of Trigonometric functions,” Int. J. STEM Educ., vol. 2, no. 1, 2015, doi: 10.1186/s40594-015-0029-5.

D. W. Carraher, M. V. Martinez, and A. D. Schliemann, “Early algebra and mathematical generalization,” ZDM - Int. J. Math. Educ., vol. 40, no. 1, pp. 3–22, 2008, doi: 10.1007/s11858-007-0067-7.

Y. M. Alghazo and R. Alghazo, “Exploring Common Misconceptions and Errors about Fractions among College Students in Saudi Arabia,” Int. Educ. Stud., vol. 10, no. 4, p. 133, 2017, doi: 10.5539/ies.v10n4p133.

Z. Ç. Özcan, Y. İmamoğlu, and V. K. Bayraklı, “Analysis of sixth grade students’ think-aloud processes while solving a non-routine mathematical problem,” Kuram ve Uygulamada Egit. Bilim., vol. 17, no. 1, pp. 129–144, 2017, doi: 10.12738/estp.2017.1.2680.

K. Umam, T. Nusantara, I. N. Parta, E. Hidayanto, and H. Mulyono, “An Application of Flipped Classroom in Mathematics Teacher Education Programme,” Int. J. Interact. Mob. Technol., vol. 13, no. 03, p. 68, 2019, doi: 10.3991/ijim.v13i03.10207.

V. Septiany, S. E. Purwanto, and K. Umam, “Influence of learning on realistic mathematics ict-assisted mathematical problem solving skills students,” 2015.

E. S. Alim, K. Umam, and S. Wijirahayu, “The implementation of blended learning instruction by utilizing wechat application,” 2016.

K. Umam, Suswandari, N. Asiah, I. T. Wibowo, and S. Rohim, “The effect of think-pair-share cooperative learning model assisted with ICT on mathematical problem solving ability among junior high school students,” 2017.

P. Scroll and D. For, “An APOS analysis of natural science students’ understanding of integration,” J. Res. Math. Educ., vol. 3, no. 1, pp. 54–73, 2014, doi: 10.4471/redimat.2014.40.

A. H. Abdullah, N. L. Z. Abidin, and M. Ali, “Analysis of students’ errors in solving Higher Order Thinking Skills (HOTS) problems for the topic of fraction,” Asian Soc. Sci., vol. 11, no. 21, pp. 133–142, 2015, doi: 10.5539/ass.v11n21p133.

D. Wood, How Children Think and Learn. Oxford: Wiley-Blackwell, 1997.

R. Antonijević, “Cognitive activities in solving mathematical tasks: The role of a cognitive obstacle,” Eurasia J. Math. Sci. Technol. Educ., vol. 12, no. 9, pp. 2503–2515, 2016, doi: 10.12973/eurasia.2016.1306a.

K. C. Yu, K. Y. Lin, and S. F. Chang, “The development and validation of a mechanical critical thinking scale for high school students,” Eurasia J. Math. Sci. Technol. Educ., vol. 13, no. 5, pp. 1361–1376, 2017, doi: 10.12973/eurasia.2017.00675a.

C. Bergsten, J. Engelbrecht, and O. Kågesten, “Conceptual and procedural approaches to mathematics in the engineering curriculum - comparing views of junior and senior engineering students in two countries,” Eurasia J. Math. Sci. Technol. Educ., vol. 13, no. 3, pp. 533–553, 2017, doi: 10.12973/eurasia.2017.00631a.

P. Stapleton, “A survey of attitudes towards critical thinking among Hong Kong secondary school teachers: Implications for policy change,” Think. Ski. Creat., vol. 6, no. 1, pp. 14–23, 2011, doi: 10.1016/j.tsc.2010.11.002.

G. Aksu and N. Koruklu, “Determination the Effects of Vocational High School Students’ Logical and Critical Thinking Skills on Mathematics Success | Matematik Başarısı ile Tutum, Mantıksal Düşünme Yetenekleri ve Eleştirel Düşünme Eğilimleri Arasındaki Doğrudan ve Dolaylı İlişkile,” Egit. Arastirmalari - Eurasian J. Educ. Res., no. 59, pp. 181–206, 2015.




DOI: http://doi.org/10.11591/ijere.v11i1.21171

Refbacks

  • There are currently no refbacks.


Copyright (c) 2021 Institute of Advanced Engineering and Science

International Journal of Evaluation and Research in Education (IJERE)
p-ISSN: 2252-8822, e-ISSN: 2620-5440

View IJERE Stats

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