Mathematics Teachers' Understanding of the Algorithms for Solving Quadratic Function Problems

Emmanuel Ibeawuchi; Ugorji Ogbonnaya

doi:10.18326/hipotenusa.v7i2.5536

Authors

Emmanuel Ibeawuchi
Ugorji Ogbonnaya University of Pretoria

DOI:

https://doi.org/10.18326/hipotenusa.v7i2.5536

Keywords:

knowledge of meaning behind algorithms, subject matter knowledge, quadratic function

Abstract

This study investigated secondary school mathematics teachers' understanding of the mathematical concepts underlying the algorithms used in teaching quadratic functions. It examined three constructs: i) why the graph of a parabola is a curve and not a straight line; ii) why the solution of a quadratic equation remains unchanged when the equation is multiplied by -1; and iii) the justification for the counter-intuitive shift of the parabola in horizontal transformations. A qualitative case study design was employed, involving five secondary school mathematics teachers selected through a combination of purposive and convenience sampling. Data were collected using a Subject Matter Knowledge Questionnaire (SMKQ) that focused on teachers' understanding of quadratic functions and their underlying algorithms. Responses were analysed through conventional qualitative content analysis. Findings revealed that while most teachers demonstrated adequate procedural skills, they exhibited limited conceptual understanding of the meaning underlying the algorithms in the constructs investigated. Their reasoning often reflected a focus on knowing how rather than understanding why, indicating a gap between procedural fluency and conceptual depth. The study concludes that strengthening teachers' conceptual understanding of mathematical algorithms is essential for improving instructional quality and learner outcomes. It recommends targeted professional development programmes that integrate both procedural and conceptual aspects of mathematical knowledge.

References

Agbozo, K. K. & Fletcher, J. A. (2020). Pre-service teachers’ understanding of selected concepts of fractions. European Journal of Education Studies, 7(2), 90-111. http://dx.doi.org/10.46827/ejes.v0i0.2941

Akhter, N., Akhtar, M., & Abaidullah, M. (2015). The perceptions of high school mathematics problem-solving teaching methods in mathematics education. Bulletin of Education and Research, 37(1), 1–23.

Aziz, T. A., Pramudiani, P., & Purnomo, Y. W. (2018). Differences between Quadratic Equations and Functions: Indonesian Pre-Service Secondary Mathematics Teachers' Views. Journal of Physics: Conference Series, 948(1) 012043).

Ball, D. L. (1991). Research on teaching mathematics: making subject matter knowledge part of the equation. In J. Brophy (Ed.), Advances in research on teaching: Vol. 2 (pp 1-47). JAI Press.

Cahyono, D., & Rusiadi, R. (2025). The role of the teacher as a facilitator in the learning process: A review of educational psychology. International Journal of Teaching and Learning (INJOTEL), 3(1), 205-212.

Canobi, K. H. (2009). Concept-procedure interactions in children's addition and subtraction. Journal of Experimental Child Psychology, 102, 131-149. https://doi.org/10.1016/j.jecp.2008.07.008

Copur-Gencturk, Y., & Tolar, T. (2022). Mathematics teaching expertise: A study of the dimensionality of content knowledge, pedagogical content knowledge, and content-specific noticing skills. Teaching and Teacher Education, 114, 103696. https://doi.org/10.1016/j.tate.2022.103696

Department of Basic Education. (2022). National senior certificate: Diagnostic report, 2022 - Part 1. Government Printers.

Department of Basic Education. (2023). National senior certificate: Diagnostic report, 2023 – Part 1. Government Printers.

Department of Basic Education. (2024). National senior certificate: Diagnostic report, 2024 – Part 1. Government Printers.

Elo, S., Kääriäinen, M., Kanste, O., Pölkki, T., Utriainen, K., & Kyngäs, H. (2014). Qualitative content analysis: A focus on trustworthiness. SAGE Open, 4(1), 1–10. https://doi.org/10.1177/2158244014522633

Even, R., & Tirosh, D. (1995). Subject-matter knowledge and knowledge about students as sources of teacher presentations of the subject matter. Educational Studies in Mathematics, 29, 1-20. https://doi.org/10.1007/BF01273897

Lincoln, Y. S., & Guba, E. E. (1986). Research, evaluation, and policy analysis: Heuristics for disciplined inquiry. Review of Policy Research, 5(3), 546-565.

Febriani, A. S., Auliya’Bernadine, N., Friyana, S. B. E., & Rofiki, I. (2024). Teaching Profile of mathematics teachers on classroom management: A study on quadratic equations. Journal of Authentic Research on Mathematics Education (JARME), 6(1), 43-54. https://doi.org/10.37058/jarme.v6i1.8904

Gautam, K. K., & Agarwal, R. (2023). The new generation teacher: teacher as a facilitator. International Journal of Creative Research Thoughts, 11(7), 2320-2882.

Groth, R. & Bergner, J. (2006). Preservice elementary teachers conceptual & procedural knowledge of Mean, Median & Mode. Mathematical Thinking & Learning, 8(1): 37-63. https://doi.org/10.1207/s15327833mtl0801_3

Lindström, M., Johansson, S., & Borger, L. (2025). Does formal teacher competence matter for students’ mathematics achievement? Results from Swedish TIMSS 2019. Educational research and evaluation, 30(1-2), 137-166. https://doi.org/10.1080/13803611.2024.2367486

Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Erlbaum.

Hsieh, H. F., & Shannon, S. E. (2005). Three approaches to qualitative content analysis. Qualitative health research, 15(9), 1277-1288. https://doi.org/10.1177/1049732305276687

Ibeawuchi, E., & Ngoepe, M. (2012) Investigating grade 11 learners’ misconceptions in understanding of quadratic functions in some South African’s schools. In Proceedings: Towards Effective Teaching and Meaningful Learning in Mathematics, Science and Technology. ISTE International Conference on Mathematics, Science and Technology Education. 21-25 October 2012. Mopani Camp in Kruger National Park, Limpopo, South Africa.

Malambo, P., Van Putten, S., Botha, J. J., & Stols, G. H. (2019). Dysfunctional functions: the case of Zambian mathematics education students. Eurasia Journal of Mathematics, Science and Technological Education, 15(1), em 1651. http://doi.org/10.29333/ejmste/99510

Memnun, D. S., Aydin, B., Dinç, E., Çoban, M., & Sevindik, F. (2015). Failures and Inabilities of High School Students about Quadratic Equations and Functions. Journal of Education and Training Studies, 3(6), 50-60. https://doi.org/10.11114/jets.v3i6.918

Newton, J., Alvey, C., & Hudson, R. (2022). Investigating Mathematics Pre-Service Teachers' Knowledge for Teaching: Focus on Quadratic Equations. Mathematics Teacher Education and Development, 24(2), 86-110.

New York State Education Department. (2005) Learning Standards for Mathematics. Retrieved on 15 July 2009 from http://www.emsc.nysed.gov/ciai/mst/mathstandards/intro.html.

Ning, Y., Zhang, C., Xu, B., Zhou, Y., & Wijaya, T. T. (2024). Teachers’ AI-TPACK: Exploring the relationship between knowledge elements. Sustainability, 16(3), 978. https://doi.org/10.3390/su16030978

Nixon, R. S., & Bennion, A. (2025). Reflection time and valuing science: Elementary teachers' science subject matter knowledge development during teaching experience. Science Education, 109(1), 59-81. https://doi.org/10.1002/sce.21902

Nyaaba, M., & Zhaı, X. (2024). Generative AI professional development needs for teacher educators. Journal of AI, 8(1), 1-13. https://doi.org/10.61969/jai.1385915

Reddy, V., Visser, M., Winnaar, L., Arends, F., Juan, A and Prinsloo, C.H., & Isdale, K. (2016). TIMSS 2015: Highlights of Mathematics and Science Achievement of Grade 9 South African Students. Human Sciences Research Council.

Rittle-Johnson, B. & Schneider, M. (2014). Developing conceptual & procedural knowledge in mathematics. In R. Cohen Kadosh & A. Dowker (Ed.), Oxford handbook of numerical cognition. Oxford Academic. https://doi.org/10.1093/oxfordhb/9780199642342.013.014

Schliemann, A. D., Carraher, D. W., & Teixidor-i-Bigas, M. (2022). Teacher development structured around reasoning about functions. International Journal of Science and Mathematics Education, 20(4), 793-816. https://doi.org/10.1007/s10763-021-10169-y

Shenton, A. K. (2004). Strategies for ensuring trustworthiness in qualitative research projects. Education for information, 22(2), 63-75.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational researcher, 15(2), 4-14. https://doi.org/10.3102/0013189x015002004

Siregar, T., Fauzan, A., Yerizon, Y., & Syafriandi, S. (2025). Designing Mathematics Teaching through Deep Learning Pedagogy: Toward Meaningful, Mindful, and Joyful Learning. Journal of Deep Learning, 188-202.

Stake, R. E. (1994). Case studies. In Norman K. Denzin and Yvonna S. Lincoln (Eds.), Handbook of Qualitative Research (4th Ed.) (pp. 236-247). Sage.

Suominen, 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 (pp. 149-173). Springer.

Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. Compendium for research in mathematics education, 421.

Tobin, R. T. (2010). Descriptive case study. In A.J. Mills, G. Durepos, & E. Wiebe (Eds.), Encyclopedia of Case Study Research. Sage.

Ubah, I. J. A., & Bansilal, S. (2018). Pre-service mathematics teachers’ knowledge of mathematics for teaching: quadratic functions. Problems of Education in the 21st Century, 76(6), 847.

Venkat, H., & Spaull, N. (2015). What do we know about primary teachers’ mathematical content knowledge in South Africa? An analysis of SACMEQ 2007. International Journal of Educational Development, 41, 121–130. https://doi.org/10.1016/j.ijedudev.2015.02.002

Yaşa, G. K. (2022). Secondary Mathematics Teachers' Subject Matter Knowledge of Quadratic Functions and Its Contribution to Student Learning Outcomes (Doctoral dissertation, Middle East Technical University (Turkey)).

Zazkis, R. (2010). What have I learned: mathematical insights and pedagogical implications. In Learning Through Teaching Mathematics: Development of Teachers' Knowledge and Expertise in Practice (pp. 91-110). Springer.

Zazkis, R., Liljedahl, P., & Gadowsky, K. (2003). Conceptions of function translation: Obstacles, intuitions, and rerouting. The Journal of Mathematical Behavior, 22(4), 435-448. https://doi.org/10.1016/j.jmathb.2003.09.003