First-Year Undergraduate Students’ Ways of Thinking in Combinatorics
Abstract
The fundamental counting principle and counting problems, jointly called combinatorics in this study, commence in the last year of secondary school and proceeds in undergraduate studies and beyond. The South African secondary school situation reflect the same worldwide scenario where counting principles is regarded as difficult and poorly performed in the final national examinations. The purpose of this study was to explore undergraduate students’ mental constructions in solving counting problems against the backdrop of Grade 12 fundamental counting principles. The Action-Process-Object-Schema theory was used to describe undergraduate students’ thinking ways in combinatorics. A single case of a first-year class of 182 students was considered in this study, whereby they all wrote a task on combinatorics and a seven were further interviewed. The findings revealed that students were skilled at solving problems involving the counting principles, which was mainly step-by-step application of the formulae. This conception is at action level according to APOS theory, but the goal of teaching is to guide students to attain the object mental conception. Object conception allows for solving of diverse real-world counting problems and promotes mathematical thinking skills.
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References
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Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014). APOS theory: A framework for research and curriculum development in mathematics education. Springer Science & Business Media. https://doi.org/10.1007/978-1-4614-7966-6
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Awuah, F.K., & Folson, D. (2017). Grade 12 Learners’ Conceptual Knowledge in Probability and Counting Principles. International Journal of Computer Applications, 176(6).
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Batanero, C., Chernoff, E., Engel, J., Lee, H., & Sánchez, E. (2016). Research on Teaching and Learning Probability. ICME-13 Topical Surveys. https://doi.org/10.1007/978-3-319-31625-3_1
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:1002954428327
Batanero, C., & Sanchez, E. (2013). What is the Nature of High School Students’ Conceptions and Misconceptions about Probability? In Jones, G.A. (Ed.), Exploring Probability in School: Challenges for Teaching and Learning. New York, NY: Springer.
Caddle, M.C., & Brizuela, B.M. (2016). Multiplication Principle Problems and Permutation Problems. The Mathematics Teacher, 109(6), 463-467. https://doi.org/10.5951/mathteacher.109.6.0463
CadwalladerOlsker, T., Engelke, N., Annin, S., & Henning, A. (2012). Does a Statement of Whether Order Matters in Counting Problems Affect Students' Strategies? In (Eds.) S. Brown, S. Larsen, K. Marrongelle, and M. Oehrtman, Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education, February 23 - 25, Portland, Oregon.
Chikiwa, S., Westaway, L., & Graven, M. (2019). What mathematics knowledge for teaching is used by a Grade 2 teacher when teaching counting. South African Journal of Childhood Education, 9(1), a567. https://doi.org/10.4102/sajce.v9i1.567
Cook, S.A., & Tim Fukawa-Connelly, T. (2012). Toward A Description of Symbol Sense in Statistics. In (Eds.) S. Brown, S. Larsen, K. Marrongelle, and M. Oehrtman, Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education, February 23 - 25, Portland, Oregon.
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Dubinsky, E. (1984). The cognitive effect of computer experiences on learning abstract mathematical concepts. Korkeak Atk-Uutiset 2, 41-47.
Dubinsky, E., & McDonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In The teaching and learning of mathematics at university level (pp. 275-282). Springer. https://doi.org/10.1007/0-306-47231-7_25
Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 1. Educational Studies in Mathematics, 58, 335-359. https://doi.org/10.1007/s10649-005-2531-z
Ekol, G., & Mlotshwa, S. (2022). Investigating the cognitive demand levels in probability and counting principles learning tasks from an online mathematics textbook. Pythagoras, 43(1), a677. https://doi.org/10.4102/pythagoras.v43i1.677
Frenke, M.L., & Kazemi, E. (2001). Learning to teach mathematics: focus on student thinking. Theory into Practice, 40(2), 102-109.
Graven, M. (2016). When systemic interventions get in the way of localized mathematics reform. For the Learning of Mathematics, 36(1), 8–13.
Jalan, S., Nusantara, T., Subanji, S., & Chandra, T.D. (2016). Students’ thinking process in solving combination problems considered from assimilation and accommodation framework. Educational Research and Reviews, 11(16), 1494-1499. https://doi.org/10.5897/ERR2016.2811
Kazunga, C. & Bansial, S. (2017). Zimbabwean in-service mathematics teachers’ understanding of matrix operations. Journal of Mathematical Behavior, 47, 81-95.
Kimani, P.M., Gibbs, R.T., & Anderson, S.M. (2013). Restoring order to Permutations and Combinations. Mathematics Teaching in the Middle School, 18(7), 431-438.
Lamanna, L., Gea, M.M., & Batanero, C. (2022). Do Secondary School Students’ Strategies in Solving Permutation and Combination Problems Change with Instruction? Canadian Journal of Science, Mathematics and Technology Education, 22, 602-616. https://doi.org/10.1007/s42330-022-00228-z
Lockwood, E. (2012). A Model of Students’ Combinatorial Thinking: The Role of Sets of Outcomes. In (Eds.) S. Brown, S. Larsen, K. Marrongelle, and M. Oehrtman, Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education, February 23-25, Portland, Oregon.
Lockwood, E., & De Chenne, A. (2019). Enriching Students’ Combinatorial Reasoning through the Use of Loops and Conditional Statements in Python. International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-019-00108-2
Lockwood, E., & Erickson, S. (2016). Undergraduate students’ initial conceptions of factorials. International Journal of Mathematical Education in Science and Technology, 48(4), 499-519. https://doi.org/10.1080/0020739X.2016.1259517
Lockwood, E., & Reed, Z. (2020). Defining and demonstrating an equivalence way of thinking in enumerative combinatorics. The Journal of Mathematical Behavior, 58, 100780. https://doi.org/10.1016/j.jmathb.2020.100780
Lockwood, E., Swinyard, C.A., & Caughman, J.S. (2015). Patterns, Sets of Outcomes, and Combinatorial Justification: Two Students’ Reinvention of Counting Formulas. International Journal of Research in Undergraduate Mathematics Education, 1, 27-62. https://doi.org/10.1007/s40753-015-0001-2
Makinde, D.O. (2014). Teaching Permutation and Combination Using Play-way Method. Journal of Education and Practice, 5(28).
Makwakwa, E.G. (2012). Exploring problems experienced by Grade 11 mathematics in the teaching and learning of statistics. Masters dissertation, University of South Africa, Pretoria. http://hdl.handle.net/105009483
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). https://doi.org/10.29333/ejmste/12418.
Ndlovu, Z., Amin, N., & Samuel, M.A. (2017). Examining pre-service teachers’ subject matter knowledge of school mathematics concepts. Journal of Education, 70, 46-72.
Ndlovu, Z., & Brijlall, D. (2019). Pre-service mathematics teachers’ mental constructions when using Cramer’s rule. South African Journal of Education, 39(1), 1-13. https://doi.org/10.15700/saje.v39n1 a1550
Perrin, T. (2006). Probing students' thinking when introduced to formal combinatorics theory in Grade 12. Unpublished doctoral thesis. University of British Columbia, Vancouver.
Rycroft-Smith, L., Macey, D., & Rushton, N. (2020). What Does Research Suggest Around The Introduction Of Combinatorics? Cambridge Mathematics, 31.
Santagata, R., & & Lee, J. (2019). Mathematical knowledge for teaching and the mathematical quality of instruction: a study of novice elementary school teachers. Journal of Mathematics Teacher Education, 24, 33–60. https://doi.org/10.1007/s10857-019-09447-y
Simatwa, E.M.W. (2010). Piaget’s theory of intellectual development and its implication for instructional anagement at presecondary school level. Educ. Res. Rev. 5(7), 366-371.
Syamsuri, S., & Santosa, C.A.H.F. (2021). Thinking Structure of Students’ Understanding of Probability Concept in Term of APOS Theory. MaPan: Jurnal Matematika dan Pembelajaran, 9(1), 119-135. https://doi.org/10.24252/mapan.2021v9n1a8.
Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5-24. https://doi.org/10.1007/BF03217474.
Author (2021).
Taylor, N., & Taylor, S. (2013). Teacher knowledge and professional habitus. In N. Taylor, S. Van der Berg & T. Mabogoane (eds.), Creating effective schools, pp. 202–232, Pearson, Johannesburg.
Voskoglou, M.G. (2015). Fuzzy Logic in the APOS/ACE Instructional Treatment for Mathematics. American Journal of Educational Research, 3(3), 330-339. https://doi.org/10.12691/education-3-3-12.
Yee, L.E. (2023). Enumerating Daily Life with Counting Principles, Permutations, and Combinations. Yale-New Haven Teachers Institute, Yale University. https://teachers.yale.edu/curriculum/viewer/initiative_18.04.09_u
Yin, R.K. (2014). Case study research: design and methods (5th Ed.). Los Angeles: SAGE.
Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014). APOS theory: A framework for research and curriculum development in mathematics education. Springer Science & Business Media. https://doi.org/10.1007/978-1-4614-7966-6
Asiala, M., Cottrill, J., Dubinsky, E., & Schwingerdorf, K. E. (1997). The development of students graphical understanding of the derivative. Journal of Mathematical Behaviour, 16(4), 399-431. https://doi.org/10.1016/S0732-3123(97)90015-8
Awuah, F.K., & Folson, D. (2017). Grade 12 Learners’ Conceptual Knowledge in Probability and Counting Principles. International Journal of Computer Applications, 176(6).
Atagana, H.I., Mogari, L. D., Kriek, J., Ochonogor, E. C., Ogbonnaya, U. I., Dhlamini J. J., & Makwakwa, E. G. (2010). An intervention into teachers' and learners' difficulties in some topics in mathematics, science and technology: A report of the ISTE 2010 winter school. The Institute for Science and Technology Education, University of South Africa. Pretoria, South Africa.
Batanero, C., Chernoff, E., Engel, J., Lee, H., & Sánchez, E. (2016). Research on Teaching and Learning Probability. ICME-13 Topical Surveys. https://doi.org/10.1007/978-3-319-31625-3_1
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:1002954428327
Batanero, C., & Sanchez, E. (2013). What is the Nature of High School Students’ Conceptions and Misconceptions about Probability? In Jones, G.A. (Ed.), Exploring Probability in School: Challenges for Teaching and Learning. New York, NY: Springer.
Caddle, M.C., & Brizuela, B.M. (2016). Multiplication Principle Problems and Permutation Problems. The Mathematics Teacher, 109(6), 463-467. https://doi.org/10.5951/mathteacher.109.6.0463
CadwalladerOlsker, T., Engelke, N., Annin, S., & Henning, A. (2012). Does a Statement of Whether Order Matters in Counting Problems Affect Students' Strategies? In (Eds.) S. Brown, S. Larsen, K. Marrongelle, and M. Oehrtman, Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education, February 23 - 25, Portland, Oregon.
Chikiwa, S., Westaway, L., & Graven, M. (2019). What mathematics knowledge for teaching is used by a Grade 2 teacher when teaching counting. South African Journal of Childhood Education, 9(1), a567. https://doi.org/10.4102/sajce.v9i1.567
Cook, S.A., & Tim Fukawa-Connelly, T. (2012). Toward A Description of Symbol Sense in Statistics. In (Eds.) S. Brown, S. Larsen, K. Marrongelle, and M. Oehrtman, Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education, February 23 - 25, Portland, Oregon.
Department of Basic Education. (2011). Curriculum and Assessment Policy Statement Grades 10-12 Mathematics. Pretoria: Government Printers.
Department of Basic Education. (2021). National senior certificate Examination Diagnostic report 2020. Pretoria: Government Printer.
Department of Basic Education. (2022). National senior certificate Examination Diagnostic report 2021. Pretoria: Government Printer.
Department of Basic Education. (2023). National senior certificate Examination Diagnostic Report 2022. Pretoria: Government Printer.
Dubinsky, E. (1984). The cognitive effect of computer experiences on learning abstract mathematical concepts. Korkeak Atk-Uutiset 2, 41-47.
Dubinsky, E., & McDonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In The teaching and learning of mathematics at university level (pp. 275-282). Springer. https://doi.org/10.1007/0-306-47231-7_25
Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 1. Educational Studies in Mathematics, 58, 335-359. https://doi.org/10.1007/s10649-005-2531-z
Ekol, G., & Mlotshwa, S. (2022). Investigating the cognitive demand levels in probability and counting principles learning tasks from an online mathematics textbook. Pythagoras, 43(1), a677. https://doi.org/10.4102/pythagoras.v43i1.677
Frenke, M.L., & Kazemi, E. (2001). Learning to teach mathematics: focus on student thinking. Theory into Practice, 40(2), 102-109.
Graven, M. (2016). When systemic interventions get in the way of localized mathematics reform. For the Learning of Mathematics, 36(1), 8–13.
Jalan, S., Nusantara, T., Subanji, S., & Chandra, T.D. (2016). Students’ thinking process in solving combination problems considered from assimilation and accommodation framework. Educational Research and Reviews, 11(16), 1494-1499. https://doi.org/10.5897/ERR2016.2811
Kazunga, C. & Bansial, S. (2017). Zimbabwean in-service mathematics teachers’ understanding of matrix operations. Journal of Mathematical Behavior, 47, 81-95.
Kimani, P.M., Gibbs, R.T., & Anderson, S.M. (2013). Restoring order to Permutations and Combinations. Mathematics Teaching in the Middle School, 18(7), 431-438.
Lamanna, L., Gea, M.M., & Batanero, C. (2022). Do Secondary School Students’ Strategies in Solving Permutation and Combination Problems Change with Instruction? Canadian Journal of Science, Mathematics and Technology Education, 22, 602-616. https://doi.org/10.1007/s42330-022-00228-z
Lockwood, E. (2012). A Model of Students’ Combinatorial Thinking: The Role of Sets of Outcomes. In (Eds.) S. Brown, S. Larsen, K. Marrongelle, and M. Oehrtman, Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education, February 23-25, Portland, Oregon.
Lockwood, E., & De Chenne, A. (2019). Enriching Students’ Combinatorial Reasoning through the Use of Loops and Conditional Statements in Python. International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-019-00108-2
Lockwood, E., & Erickson, S. (2016). Undergraduate students’ initial conceptions of factorials. International Journal of Mathematical Education in Science and Technology, 48(4), 499-519. https://doi.org/10.1080/0020739X.2016.1259517
Lockwood, E., & Reed, Z. (2020). Defining and demonstrating an equivalence way of thinking in enumerative combinatorics. The Journal of Mathematical Behavior, 58, 100780. https://doi.org/10.1016/j.jmathb.2020.100780
Lockwood, E., Swinyard, C.A., & Caughman, J.S. (2015). Patterns, Sets of Outcomes, and Combinatorial Justification: Two Students’ Reinvention of Counting Formulas. International Journal of Research in Undergraduate Mathematics Education, 1, 27-62. https://doi.org/10.1007/s40753-015-0001-2
Makinde, D.O. (2014). Teaching Permutation and Combination Using Play-way Method. Journal of Education and Practice, 5(28).
Makwakwa, E.G. (2012). Exploring problems experienced by Grade 11 mathematics in the teaching and learning of statistics. Masters dissertation, University of South Africa, Pretoria. http://hdl.handle.net/105009483
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). https://doi.org/10.29333/ejmste/12418.
Ndlovu, Z., Amin, N., & Samuel, M.A. (2017). Examining pre-service teachers’ subject matter knowledge of school mathematics concepts. Journal of Education, 70, 46-72.
Ndlovu, Z., & Brijlall, D. (2019). Pre-service mathematics teachers’ mental constructions when using Cramer’s rule. South African Journal of Education, 39(1), 1-13. https://doi.org/10.15700/saje.v39n1 a1550
Perrin, T. (2006). Probing students' thinking when introduced to formal combinatorics theory in Grade 12. Unpublished doctoral thesis. University of British Columbia, Vancouver.
Rycroft-Smith, L., Macey, D., & Rushton, N. (2020). What Does Research Suggest Around The Introduction Of Combinatorics? Cambridge Mathematics, 31.
Santagata, R., & & Lee, J. (2019). Mathematical knowledge for teaching and the mathematical quality of instruction: a study of novice elementary school teachers. Journal of Mathematics Teacher Education, 24, 33–60. https://doi.org/10.1007/s10857-019-09447-y
Simatwa, E.M.W. (2010). Piaget’s theory of intellectual development and its implication for instructional anagement at presecondary school level. Educ. Res. Rev. 5(7), 366-371.
Syamsuri, S., & Santosa, C.A.H.F. (2021). Thinking Structure of Students’ Understanding of Probability Concept in Term of APOS Theory. MaPan: Jurnal Matematika dan Pembelajaran, 9(1), 119-135. https://doi.org/10.24252/mapan.2021v9n1a8.
Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5-24. https://doi.org/10.1007/BF03217474.
Author (2021).
Taylor, N., & Taylor, S. (2013). Teacher knowledge and professional habitus. In N. Taylor, S. Van der Berg & T. Mabogoane (eds.), Creating effective schools, pp. 202–232, Pearson, Johannesburg.
Voskoglou, M.G. (2015). Fuzzy Logic in the APOS/ACE Instructional Treatment for Mathematics. American Journal of Educational Research, 3(3), 330-339. https://doi.org/10.12691/education-3-3-12.
Yee, L.E. (2023). Enumerating Daily Life with Counting Principles, Permutations, and Combinations. Yale-New Haven Teachers Institute, Yale University. https://teachers.yale.edu/curriculum/viewer/initiative_18.04.09_u
Yin, R.K. (2014). Case study research: design and methods (5th Ed.). Los Angeles: SAGE.
Published
2024-01-02
How to Cite
TATIRA, Benjamin.
First-Year Undergraduate Students’ Ways of Thinking in Combinatorics.
Journal of Medives : Journal of Mathematics Education IKIP Veteran Semarang, [S.l.], v. 8, n. 1, p. 103 - 121, jan. 2024.
ISSN 2549-5070.
Available at: <https://e-journal.ivet.ac.id/index.php/matematika/article/view/2576>. Date accessed: 14 aug. 2025.
doi: https://doi.org/10.31331/medivesveteran.v8i1.2576.
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