Assessing cognitive obstacles in learning number concepts: Insights from preservice mathematics teachers
DOI:
https://doi.org/10.23917/jramathedu.v10i3.8638Keywords:
Algebra learning, APOS theory, Cognitive obstacles, Conceptual understanding, Preservice mathematics teachersAbstract
Persistent difficulties in learning abstract algebraic concepts—particularly among preservice mathematics teachers—continue to hinder students’ mathematical development. While prior studies have documented general misconceptions, few have grounded their analysis in comprehensive learning theories. Addressing this gap, the present study adopts the APOS (Action, Process, Object, Schema) theoretical framework to examine the cognitive obstacles encountered in understanding logarithmic, matrix, and quadratic function concepts. This qualitative study employed a descriptive case study design involving six preservice mathematics teachers with varying levels of mathematical ability (high, moderate, and low). Data were collected through written responses, semi-structured interviews, classroom observations, and cognitive mapping. The findings revealed that most participants were at the action stage, relying on procedural steps without deep conceptual understanding. Key cognitive obstacles included errors in applying logarithmic properties, difficulties integrating logarithms with matrices, and an inability to perceive systems of equations as unified entities. Group discussions proved effective in helping participants transition through the learning stages. Collaborative interactions enabled participants to identify errors, correct misconceptions, and strengthen conceptual understanding through reflection and validation. Furthermore, the use of visual tools, graphical representations, and real-world contexts supported deeper conceptual integration. This study underscores the importance of implementing APOS-based instructional strategies, including group discussions, exploratory exercises, and problem-based learning, to facilitate transitions between stages. The implications of these findings highlight the need for developing APOS-based diagnostic tools and innovative instructional designs to address cognitive obstacles effectively.
References
Abakah, F., & Brijlall, D. (2024). Finding an Effective Assessment Approach to Enhance the Teaching and Learning of Circle Geometry. Africa Education Review, 20(3), 93–116. https://doi.org/10.1080/18146627.2024.2402772
Abu-Hilal, M. M., Abdelfattah, F. A., Alshumrani, S. A., Abduljabbar, A. S., & Marsh, H. W. (2013). Construct validity of self-concept in TIMSS’s student background questionnaire: A test of separation and conflation of cognitive and affective dimensions of self-concept among Saudi eighth graders. European Journal of Psychology of Education, 28(4), 1201–1220. https://doi.org/10.1007/s10212-012-0162-1
Adigun, O. T., Quadri, O. G., Jacob, U. S., & Ngubane, S. A. (2025). An Exploration of E-Teaching Preparedness of Pre-Service Teachers of Learners with Special Needs. Africa Education Review, 1–25. https://doi.org/10.1080/18146627.2024.2402767
Agustyaningrum, N., Husna, A., Hanggara, Y., Abadi, A. M., & Mahmudi, A. (2020). Analysis of Mathematical Proof Ability in Abstract Algebra Course. Universal Journal of Educational Research, 8(3), 823–834. https://doi.org/10.13189/ujer.2020.080313
Aksu, Z., KALAÇ, S., Özkaya, M., & KONYALIOĞLU, A. C. (2022). Pre-service Mathematics Teachers’ Conceptual Knowledge Related To Basic Concepts And Operations. Research on Education and Psychology, 6(2), 283–297. https://doi.org/10.54535/rep.1204295
Al-Adwan, A. S., Meet, R. K., Kala, D., Smedley, J., Urbaníková, M., & Al-Rahmi, W. M. (2024). Closing the divide: Exploring higher education teachers’ perspectives on educational technology. Information Development. https://doi.org/10.1177/02666669241279181
Allmon, W. D. (2011). Why Don’t People Think Evolution Is True? Implications for Teaching, In and Out of the Classroom. Evolution: Education and Outreach, 4(4), 648–665. https://doi.org/10.1007/s12052-011-0371-0
Alsina, Á., Pincheira, N., & Delgado-Rebolledo, R. (2024). The professional practice of designing tasks: how do pre-service early childhood teachers promote mathematical processes in early algebra? ZDM – Mathematics Education, 56(6), 1197–1210. https://doi.org/10.1007/s11858-024-01636-1
Andrews, D. A., Sekyere, E. O., & Bugarcic, A. (2020). Collaborative Active Learning Activities Promote Deep Learning in a Chemistry-Biochemistry Course. Medical Science Educator, 30(2), 801–810. https://doi.org/10.1007/s40670-020-00952-x
Angraini, L. M., Kania, N., & Gürbüz, F. (2024). Students’ Proficiency in Computational Thinking Through Constructivist Learning Theory. International Journal of Mathematics and Mathematics Education, 45–59. https://doi.org/10.56855/ijmme.v2i1.963
Anwar, L., Sa’dijah, C., Murtafiah, W., & Huljannah, M. (2023). Adversity quotient of Indonesian prospective mathematics teachers in solving geometry higher-order thinking skills problems. Journal on Mathematics Education, 15(1), 79–98. https://doi.org/10.22342/jme.v15i1.pp79-98
Arnawa, I. M., Yanita, Y., Yerizon, Y., Ginting, B., & Nita, S. (2021). Does The Use of APOS Theory Promote Students’ Achievement in Elementary Linear Algebra? International Journal of Instruction, 14(3), 175–186. https://doi.org/10.29333/iji.2021.14310a
Asghary, N., Afkhami, R., & Medghalchi, A. (2023). Developing a Framework for Evaluating Student’s Understanding at Figural Pattern Generalization. PNA. Revista de Investigación En Didáctica de La Matemática, 18(1), 57–76. https://doi.org/10.30827/pna.v18i1.16566
Ballard, J., Gamage, S., Winfield, L., & Mooring, S. (2023). Cognitive discourse during a group quiz activity in a blended learning organic chemistry course. Chemistry Teacher International, 5(3), 245–261. https://doi.org/10.1515/cti-2023-0007
Bardini, C., Pierce, R. U., & Stacey, K. (2004). Teaching linear functions in context with graphics calculators: Students’ responses and the impact of the approach on their use of algebraic symbols. International Journal of Science and Mathematics Education, 2(3), 353–376. https://doi.org/10.1007/s10763-004-8075-3
Borji, V., Erfani, H., & Font, V. (2020). A combined application of APOS and OSA to explore undergraduate students’ understanding of polar coordinates. International Journal of Mathematical Education in Science and Technology, 51(3), 405–423. https://doi.org/10.1080/0020739X.2019.1578904
Campo-Meneses, K. G., Font, V., García-García, J., & Sánchez, A. (2021). Mathematical Connections Activated in High School Students’ Practice Solving Tasks on the Exponential and Logarithmic Functions. Eurasia Journal of Mathematics, Science and Technology Education, 17(9), em1998. https://doi.org/10.29333/ejmste/11126
Chabursky, S., Gutt, J., & Guglhör-Rudan, A. (2024). Ukrainian refugee adolescents in Germany – A qualitative study on schooling experiences and the fulfillment of basic psychological needs. Children and Youth Services Review, 160, 107582. https://doi.org/10.1016/j.childyouth.2024.107582
Cinar, O. E., Rafferty, K., Cutting, D., & Wang, H. (2023). Using Augmented Reality to Enhance Learning and Understanding of Abstract Programming Concepts. 2023 9th International Conference on Virtual Reality (ICVR), 498–504. https://doi.org/10.1109/ICVR57957.2023.10169459
Copur-Gencturk, Y., & Doleck, T. (2021). Linking teachers’ solution strategies to their performance on fraction word problems. Teaching and Teacher Education, 101, 103314. https://doi.org/10.1016/j.tate.2021.103314
Cristea, T. S., Heikkinen, S., Snijders, C., Saqr, M., Matzat, U., Conijn, R., & Kleingeld, A. (2025). Dynamics of self-regulated learning: The effectiveness of students’ strategies across course periods. Computers & Education, 228, 105233. https://doi.org/10.1016/j.compedu.2025.105233
Dewi, I. L. K., Zaenuri, Z., Dwijanto, D., & Mulyono, M. (2021). Identification of Mathematics Prospective Teachers’ Conceptual Understanding in Determining Solutions of Linear Equation Systems. European Journal of Educational Research, volume-10-(volume-10-issue-3-july-2021), 1157–1170. https://doi.org/10.12973/eu-jer.10.3.1157
Dintarini, M. (2018). Understanding Logarithm: What are the Difficulties That Students Have? Proceedings of the 5th International Conference on Community Development (AMCA 2018). https://doi.org/10.2991/amca-18.2018.65
Dubinsky, E., & Wilson, R. T. (2013). High school students’ understanding of the function concept. The Journal of Mathematical Behavior, 32(1), 83–101. https://doi.org/10.1016/j.jmathb.2012.12.001
Farida, N., Sesanti, N. R., & Ferdiani, R. D. (2019). Tingkat Pemahaman Konsep dan Kemampuan Mengajar Mahasiswa pada Mata Kuliah Kajian dan Pengembangan Matematika Sekolah 2. MUST: Journal of Mathematics Education, Science and Technology, 4(2), 135. https://doi.org/10.30651/must.v4i2.2897
Faridayanti, I., Nasrullah, A., Hidayat, A., & Kania, N. (2025). The Impact of the Discovery Learning Model on Mathematical Communication Skills and Study Habits of Junior High School Students. Indonesian Journal of Teaching and Learning (INTEL), 4(1), 16–29. https://doi.org/10.56855/intel.v4i1.1330
Faustino, A. C., & Sales, E. R. de S. (2024). Pensamento algébrico nos anos iniciais: possibilidades com estudantes cegos e videntes. Prometeica - Revista de Filosofía y Ciencias, 31, 347–358. https://doi.org/10.34024/prometeica.2024.31.16476
Feriyanto, F., & Putri, R. O. E. (2020). Developing Mathematics Module Based on Literacy and Higher Order Thinking Skills (HOTS) Questions to Train Critical Thinking Ability of High School Students in Mojokerto. Journal of Physics: Conference Series, 1594(1). https://doi.org/10.1088/1742-6596/1594/1/012014
Fitriani, F., Siregar, Y. A., & Novitasari, W. (2021). Analisis kesulitan kemampuan komunikasi matematika mahasiswa menggunakan aplikasi google classroom pada matakuliah aljabar. Journal of Didactic Mathematics, 2(1), 18–25. https://doi.org/10.34007/jdm.v2i1.596
Fitriawan, D., Ijuddin, R., & Sayu, S. (2022). ANALISIS MATERI SULIT DIPAHAMI DAN MISKONSEPSI MAHASISWA DALAM MATA KULIAH KALKULUS INTEGRAL. Numeracy, 9(1), 27–38. https://doi.org/10.46244/numeracy.v9i1.1682
Hartono, S. (2025). Evolving trends in mathematical proof research in Indonesian mathematics education: A systematic review from design to data analysis. Multidisciplinary Reviews, 8(12), 2025372. https://doi.org/10.31893/multirev.2025372
Hasan, B. (2020). KESULITAN SISWA DAN SCAFFOLDING DALAM MENYELESAIKAN MASALAH GEOMETRI RUANG. Numeracy, 7(1), 49–64. https://doi.org/10.46244/numeracy.v7i1.998
Hlangwani, W., & Dhlamini, Z. B. (2024). Grade 12 learners’ understanding of differentiation: tracking first principle and power rule using the ACE teaching cycle. International Journal of Learning and Change, 16(6), 678–694. https://doi.org/10.1504/IJLC.2024.141954
Hyland, D., & O’Shea, A. (2022). How Well Do High-Achieving Undergraduate Students Understand School Algebra? Canadian Journal of Science, Mathematics and Technology Education, 22(4), 818–834. https://doi.org/10.1007/s42330-022-00256-9
Istanto, I., Fakhruddin, F., Yanto, H., & Pramono, S. E. (2024). Altruistic leadership: A quantitative analysis of its effects on school culture, teacher integrity, and performance in Muhammadiyah middle schools in Indonesia. International Journal of ADVANCED AND APPLIED SCIENCES, 11(10), 140–146. https://doi.org/10.21833/ijaas.2024.10.016
Jamil, N., Rosli, R., & Mahmud, M. S. (2025). Algebraic trends impact school mathematics education: A bibliometric review. Multidisciplinary Reviews, 8(9), 2025261. https://doi.org/10.31893/multirev.2025261
Jiang, L., Sun, Y., & Zhao, H. (2024). Grounded theory approach to non-market strategy and organizational resilience: the case of Chinese private enterprise. Kybernetes, 53(12), 5648–5670. https://doi.org/10.1108/K-03-2023-0386
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
Jung, H., Newton, J., & Maher, E. M. (2024). Conceptualizing algebraic connections: Honoring voices from future mathematics teachers and the instructors who teach them. International Electronic Journal of Mathematics Education, 19(4), em0797. https://doi.org/10.29333/iejme/15519
Kania, N., Fitriani, C., & Bonyah, E. (2023). Analysis of Students’ Critical Thinking Skills Based on Prior Knowledge Mathematics. International Journal of Contemporary Studies in Education (IJ-CSE), 2(1), 49–58. https://doi.org/10.56855/ijcse.v2i1.248
Kania, N., & Kusumah, Y. S. (2025). THE MEASUREMENT OF HIGHER-ORDER THINKING SKILLS: A SYSTEMATIC LITERATURE REVIEW. Malaysian Journal of Learning and Instruction, 22(1), 97–116. https://doi.org/10.32890/mjli2025.22.1.6
Kania, N., Kusumah, Y. S., Dahlan, J. A., Nurlaelah, E., Gürbüz, F., & Bonyah, E. (2024). Constructing and providing content validity evidence through the Aiken’s V index based on the experts’ judgments of the instrument to measure mathematical problem-solving skills. REID (Research and Evaluation in Education), 10(1), 64–79. https://doi.org/10.21831/reid.v10i1.71032
Keller, N. (2012). A tight quantitative version of Arrow’s impossibility theorem. Journal of the European Mathematical Society, 14(5), 1331–1355. https://doi.org/10.4171/jems/334
Kilp-Kabel, T., & Mädamürk, K. (2024). The developmental trajectories of math skills and its relation to math interest in Grades three and five. European Journal of Psychology of Education, 39(3), 2681–2699. https://doi.org/10.1007/s10212-024-00807-8
Kolar, V. M., & Hodnik, T. (2021). Mathematical literacy from the perspective of solving contextual problems. European Journal of Educational Research, 10(1), 467–483. https://doi.org/10.12973/EU-JER.10.1.467
Langi’, E. L., Juniati, D., & Abadi. (2023). Students as Prospective Teachers’ Understanding of Integral Based on the APOS Theory in Terms of Gender Difference. Journal of Higher Education Theory and Practice, 23(4). https://doi.org/10.33423/jhetp.v23i4.5897
Lee, Y., Mackenzie, M. J., & Lach, L. (2024). Toward a developmental transactional model of educational upward mobility. Review of Education, 12(3). https://doi.org/10.1002/rev3.70007
López-Martín, M. del M., Aguayo-Arriagada, C. G., & García López, M. del M. (2022). Preservice Elementary Teachers’ Mathematical Knowledge on Fractions as Operator in Word Problems. Mathematics, 10(3), 423. https://doi.org/10.3390/math10030423
Martín Molina, J. C., Maz Machado, A., Madrid, M. J., & Rodríguez Baiget, M. J. (2024). Comprehension of linear systems with two unknowns in secondary education. International Journal of Evaluation and Research in Education (IJERE), 13(6), 3914. https://doi.org/10.11591/ijere.v13i6.28206
Martínez-Planell, R., & Trigueros, M. (2020). Students’ understanding of Riemann sums for integrals of functions of two variables. The Journal of Mathematical Behavior, 59, 100791. https://doi.org/10.1016/j.jmathb.2020.100791
Mera, C., Delgado, C., Aragón, E., Menacho, I., Canto, M. D. C., & Navarro, J. I. (2022). Contributions of the psychology of mathematical cognition in early childhood education using apps. Frontiers in Psychology, 13. https://doi.org/10.3389/fpsyg.2022.913970
Moru, E. K. (2020). An APOS Analysis of University Students’ Understanding of Derivatives: A Lesotho Case Study. African Journal of Research in Mathematics, Science and Technology Education, 24(2), 279–292. https://doi.org/10.1080/18117295.2020.1821500
Moru, E. K., & Mathunya, M. (2022). A constructivist analysis of Grade 8 learners’ errors and misconceptions in simplifying mathematical algebraic expressions. JRAMathEdu (Journal of Research and Advances in Mathematics Education), 7(3), 130–144. https://doi.org/10.23917/jramathedu.v7i3.16784
Murre, J. M. J. (2023). How averaging individual curves transforms their shape: Mathematical analyses with application to learning and forgetting curves. Journal of Mathematical Psychology, 117, 102816. https://doi.org/10.1016/j.jmp.2023.102816
Mustafa, G., & Derya, C. (2016). A research on future mathematics teachers instructional explanations: The sample of Algebra. Educational Research and Reviews, 11(16), 1500–1508. https://doi.org/10.5897/ERR2016.2823
Mutambara, L. H. N., & Bansilal, S. (2022). A case study of in-service teachers’ errors and misconceptions in linear combinations. International Journal of Mathematical Education in Science and Technology, 53(11), 2900–2918. https://doi.org/10.1080/0020739X.2021.1913656
Mutambara, L. H. N., & Tsakeni, M. (2022). Cognitive obstacles in the learning of complex number concepts: A case study of in-service undergraduate physics student-teachers in Zimbabwe. Eurasia Journal of Mathematics, Science and Technology Education, 18(10), em2158. https://doi.org/10.29333/ejmste/12418
Mwakililo, P. S., Mauki, C. M., & Lyakurwa, S. E. (2025). Coping with dual roles: a case of adolescent student mothers in Tanzania’s secondary schools. Asian Education and Development Studies. https://doi.org/10.1108/AEDS-07-2024-0158
Ndiung, S., & Menggo, S. (2024). Project-Based Learning in Fostering Creative Thinking and Mathematical Problem-Solving Skills: Evidence from Primary Education in Indonesia. International Journal of Learning, Teaching and Educational Research, 23(8), 289–308. https://doi.org/10.26803/ijlter.23.8.15
Ndlovu, Z., & Brijlall, D. (2015). Pre-service Teachers’ Mental Constructions of Concepts in Matrix Algebra. African Journal of Research in Mathematics, Science and Technology Education, 19(2), 156–171. https://doi.org/10.1080/10288457.2015.1028717
Ndlovu, Z., & Brijlall, D. (2016). Pre-service Mathematics Teachers’ Mental Constructions of the Determinant Concept. International Journal of Educational Sciences, 14(1–2), 145–156. https://doi.org/10.1080/09751122.2016.11890488
Nga, N. T., Dung, T. M., Trung, L. T. B. T., Nguyen, T.-T., Tong, D. H., Van, T. Q., & Uyen, B. P. (2023). The Effectiveness of Teaching Derivatives in Vietnamese High Schools Using APOS Theory and ACE Learning Cycle. European Journal of Educational Research, volume-12-(volume-12-issue-1-january-2023), 507–523. https://doi.org/10.12973/eu-jer.12.1.507
Nhat, N. C., Oanh, T. T. K., & Hang, P. T. T. (2024). The Effect of Stem Education on Academic Performance: A Meta-Analysis Study. International Journal of Learning, Teaching and Educational Research, 23(11), 180–195. https://doi.org/10.26803/ijlter.23.11.9
Nikou, S. A., Perifanou, M., & Economides, A. A. (2024). Exploring Teachers’ Competences to Integrate Augmented Reality in Education: Results from an International Study. TechTrends, 68(6), 1208–1221. https://doi.org/10.1007/s11528-024-01014-4
Ningrum, H. U., Mulyono, M., & Isnarto, I. (2020). Mathematical Connection Ability Based on Self-Efficacy in IDEAL Problem Solving Model Assisted by ICT. Unnes Journal of …. https://journal.unnes.ac.id/sju/index.php/ujmer/article/view/32797
Okoye-Ogbalu, I., & Nnadozie, V. (2024). Subject competency and teacher knowledge: An exploration of second-year pre-service mathematics teachers’ difficulties in solving logarithmic problems using basic rules for logarithm. Journal of Mathematics and Science Teacher, 4(1), em054. https://doi.org/10.29333/mathsciteacher/13887
Oktaç, A., Vázquez Padilla, R., Ramírez Sandoval, O., & Villabona Millán, D. (2022). Transitional points in constructing the preimage concept in linear algebra. International Journal of Mathematical Education in Science and Technology, 53(5), 1170–1189. https://doi.org/10.1080/0020739X.2021.1968523
Ontiveros, Y. O., González, J. R., & Palencia, J. L. D. (2025). Integrating creative thinking and cultural insights through the anthropological theory of didactics in university algebra. Journal of Education and Learning (EduLearn), 19(2), 742–750. https://doi.org/10.11591/edulearn.v19i2.21461
Oppong, R. A., Adjei, E. A., Amoah, E. K., Okuleteh, E., & Larri, B. (2024). Analyses of Pre-Service Teachers’ Errors in Solving Algebraic Tasks in Ghana. International Journal of Mathematics and Mathematics Education, 155–174. https://doi.org/10.56855/ijmme.v2i3.1134
Osman, S., Abu, M., Mohammad, S., & Mokhtar, M. (2016). Identifying Pertinent Elements of Critical Thinking and Mathematical Thinking Used in Civil Engineering Practice in Relation to Engineering Education. The Qualitative Report. https://doi.org/10.46743/2160-3715/2016.2203
Padernal, R. E., & Tupas, S. V. (2024). First-year students’ mathematical skills: assessing and developing an innovative strategy for teaching mathematics. International Journal of Evaluation and Research in Education (IJERE), 13(5), 2890. https://doi.org/10.11591/ijere.v13i5.29182
Padmanabhan, G., De Saram, D. D., Schanandore, T. C., Schanandore, J., & Pieri, R. V. (2013). A surveying course as summer experience for. ASEE Annual Conference and Exposition, Conference Proceedings. https://www.scopus.com/inward/record.uri?eid=2-s2.0-85029034669&partnerID=40&md5=0362f492dc25b2a09f3ba38fec5a391a
Pape, S., Gunpinar, Y., JohnBull, R., & Bell, C. (2022). Advancing Algebra Achievement Using Affordances of Classroom Connectivity Technology: The Case for Noticing through Discussion. Mathematics, 10(15), 2641. https://doi.org/10.3390/math10152641
Pasigon, C. P. (2024). Mathematical Proficiency, Scientific Reasoning, Metacognitive Skills, and Performance of Learners in Physics: A Mathematical Model. International Journal of Learning, Teaching and Educational Research, 23(4), 252–278. https://doi.org/10.26803/ijlter.23.4.14
Pham, H. T., & Nguyen, N. T. V. (2023). Pedagogical student assessment tools for learning outcome assessment skills. Cypriot Journal of Educational Sciences, 18(1), 158–174. https://doi.org/10.18844/cjes.v18i1.8514
Pinter, L., & Siddiqui, M. F. H. (2024). Enhancing Calculus Learning through Interactive VR and AR Technologies: A Study on Immersive Educational Tools. Multimodal Technologies and Interaction, 8(3), 19. https://doi.org/10.3390/mti8030019
Rafi, I., & Retnawati, H. (2018). What are the common errors made by students in solving logarithm problems? Journal of Physics: Conference Series, 1097, 012157. https://doi.org/10.1088/1742-6596/1097/1/012157
Richland, L. E., & Simms, N. (2015). Analogy, higher order thinking, and education. In Wiley Interdisciplinary Reviews: Cognitive Science (Vol. 6, Issue 2, pp. 177–192). Wiley. https://doi.org/10.1002/wcs.1336
Rivard, S. (2024). Unpacking the process of conceptual leaping in the conduct of literature reviews. The Journal of Strategic Information Systems, 33(1), 101822. https://doi.org/10.1016/j.jsis.2024.101822
Romadhon, N. I., Amir, M. F., & Wardana, M. D. K. (2024). Assessing students’ mathematical reasoning in problem-based learning: a gender perspective. International Journal of Evaluation and Research in Education (IJERE), 13(6), 3763. https://doi.org/10.11591/ijere.v13i6.29580
Roulstone, A., Morsanyi, K., & Bahnmueller, J. (2024). Performance on curriculum-based mathematics assessments in developmental dyscalculia: the effect of content domain and question format. Psychological Research, 88(8), 2444–2454. https://doi.org/10.1007/s00426-024-02015-x
Saleemad, K., Noklang, S., & Dudsdeemaytha, J. (2022). Leadership Characteristics for Muslim School Leaders in a Multicultural Context in Thailand. International Journal of Instruction, 15(2), 293–306. https://doi.org/10.29333/iji.2022.15217a
Sari, I. P., Suryadi, D., Herman, T., Dahlan, J. A., & Supriyadi, E. (2024). Learning obstacles on fractions: A scoping review. Infinity Journal, 13(2), 377–392. https://doi.org/10.22460/infinity.v13i2.p377-392
Schillinger, T. (2021). Self-efficacy of Kindergarten Teachers’ Mathematical Instruction. Early Childhood Education Journal, 49(4), 623–632. https://doi.org/10.1007/s10643-020-01101-0
Schwarz, C. V., Ke, L., Salgado, M., & Manz, E. (2022). Beyond assessing knowledge about models and modeling: Moving toward expansive, meaningful, and equitable modeling practice. Journal of Research in Science Teaching, 59(6), 1086–1096. https://doi.org/10.1002/tea.21770
Siswono, T. Y. E., Hartono, S., Wachidul Kohar, A., Karim, K., & Lastiningsih, N. (2019). How do Prospective Teachers Manage Students’ Learning of Mathematics? TEM Journal, 677–685. https://doi.org/10.18421/TEM82-49
Soneira, C., Bansilal, S., & Govender, R. (2021). Insights into the reversal error from a study with South African and Spanish prospective primary teachers. Pythagoras, 42(1). https://doi.org/10.4102/pythagoras.v42i1.613
Susiswo, S., Retno Murniasih, T., Sa’dijah, C., Muksar, M., & Murtafiah, W. (2021). The Development of an Instrument on Negative Fractions to Measure the Cognitive Obstacle Based on Mental Mechanism Stages. TEM Journal, 1357–1362. https://doi.org/10.18421/TEM103-44
Suwardika, G., Sopandi, A. T., Indrawan, I. P. O., & Masakazu, K. (2024). A flipped classroom with whiteboard animation and modules to enhance students’ self-regulation, critical thinking and communication skills: a conceptual framework and its implementation. Asian Association of Open Universities Journal, 19(2), 135–152. https://doi.org/10.1108/AAOUJ-10-2023-0115
Taghizadeh Bilondi, M., & Radmehr, F. (2023). Students’ mathematical thinking of the tree concept: an integration of APOS with Tall’s three worlds of mathematics. Research in Mathematics Education, 1–22. https://doi.org/10.1080/14794802.2023.2292260
Tatira, B. (2021). Mathematics Education Students’ Understanding of Binomial Series Expansion Based on the APOS Theory. Eurasia Journal of Mathematics, Science and Technology Education, 17(12), em2035. https://doi.org/10.29333/ejmste/11287
Tatira, B., & Mukuka, A. (2024). Unpacking pre-service teachers’ conceptualization of logarithmic differentiation through the APOS theory. Eurasia Journal of Mathematics, Science and Technology Education, 20(12), em2541. https://doi.org/10.29333/ejmste/15655
Thomaidis, Y., & Tzanakis, C. (2022). Historical knowledge and mathematics education: a recent debate and a case study on the different readings of history and its didactical transposition. ZDM – Mathematics Education, 54(7), 1449–1461. https://doi.org/10.1007/s11858-022-01370-6
Travé González, G., Pozuelos Estrada, F. J., & Trave Gonzalez, G. (2017). How teachers design and implement intstructional materials to improve classroom practice. Intangible Capital, 13(5), 971. https://doi.org/10.3926/ic.1062
Trigueros, M., Badillo, E., Sánchez-Matamoros, G., & Hernández-Rebollar, L. A. (2024). Contributions to the characterization of the Schema using APOS theory: Graphing with derivative. ZDM – Mathematics Education, 56(6), 1093–1108. https://doi.org/10.1007/s11858-024-01615-6
Trigueros, M., Cabrera, A. C., & Aguilar, M. S. (2024). Mental constructions for the learning of the concept of vector space. ZDM – Mathematics Education, 56(7), 1417–1431. https://doi.org/10.1007/s11858-024-01640-5
Tuktamyshov, N. K., & Gorskaya, T. Y. (2024). APOS Theory in Learning Mathematics (Using Trigonometry as an Example). Integration of Education, 28(1), 111–124. https://doi.org/10.15507/1991-9468.114.028.202401.111-124
Umiralkhanov, A., Abylkassymova, A., Tuyakov, Y., Zhadrayeva, L., & Ardabayeva, A. (2024). Continuity in Teaching the Course Algebra and the Beginning of Analysis: Adaptation of Students to New Conditions. Qubahan Academic Journal, 4(3), 796–808. https://doi.org/10.48161/qaj.v4n3a1050
Usmayati, U., & Gürbüz, F. (2024). Empowering Students with Discovery Learning in Circle Geometry for Better Problem-Solving Uum. 01(01), 11–20.
Volikova, M. M., Armash, T. S., Yechkalo, Y. V, & Zaselskiy, V. I. (2019). Practical use of cloud services for organization of future specialists professional training. In K. A.E. & S. V.N. (Eds.), CEUR Workshop Proceedings (Vol. 2433, pp. 486–498). CEUR-WS. https://www.scopus.com/inward/record.uri?eid=2-s2.0-85072756023&partnerID=40&md5=43a0f48d8fd8a5e7ad478f8832d6ad22
Wester, J. S. (2021). Students’ Possibilities to Learn From Group Discussions Integrated in Whole-class Teaching in Mathematics. Scandinavian Journal of Educational Research, 65(6), 1020–1036. https://doi.org/10.1080/00313831.2020.1788148
Wilkins, J. L. M., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. International Journal of STEM Education, 5(1), 27. https://doi.org/10.1186/s40594-018-0119-2
Yerizon, D., Arnawa, I. M., Yanita, D., Ginting, B., & Nita, S. (2019). Students’ Errors in Learning Elementary Group Theory: A Case Study of Mathematics Students at Andalas University. Universal Journal of Educational Research, 7(12), 2693–2698. https://doi.org/10.13189/ujer.2019.071216
Yodiatmana, Y., & Kartini, K. (2022). Analysis of Student Errors in Solving Basic Logarithmic Problems Using Kastolan Error Analysis. Jurnal Gantang, 7(2), 129–136. https://doi.org/10.31629/jg.v7i2.4689
Yuanita, P., Maimunah, M., Kartini, K., & Arif, N. (2023). Analysis of SOLO taxonomy-based test instruments in approaching the Merdeka curriculum. JRAMathEdu (Journal of Research and Advances in Mathematics Education), 12–29. https://doi.org/10.23917/jramathedu.v8i1.1270
Zakaria, M. I., Hanri, C., Noer, S. H., Triana, M., & Widyastuti, W. (2025). 4C skills teaching activities for mathematics teachers: application of modified nominal group technique. Journal of Education and Learning (EduLearn), 19(2), 626–633. https://doi.org/10.11591/edulearn.v19i2.22337
Žakelj, A., Cotič, M., & Doz, D. (2024). Demetriou’s tests and levels of algebraic abilities and proportional reasoning in seventh, eighth, and ninth grades. European Journal of Science and Mathematics Education, 12(2), 326–334. https://doi.org/10.30935/scimath/14460
Zayyadi, M., Nusantara, T., Hidayanto, E., Sulandra, I. M., & As’ari, A. R. (2019). Exploring prospective student teacher’s question on mathematics teaching practice. Journal of Technology and Science Education, 9(2), 228. https://doi.org/10.3926/jotse.465
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