Procedural Knowledge-to-Conceptual Knowledge or Conceptual Knowledge-to-Procedural Knowledge: Which Order of Knowledge is Better for Conceptual Understanding and Procedural Fluency

Gliselle Faith L. Asilo-Ebisa^1, and Laila S. Lomibao²

¹Mathematics Department, Corpus Christi School, Cagayan de Oro City, Philippines

²Department of Mathematics Education, University of Science and Technology of Southern Philippines, Cagayan de Oro City, Philippines

Cite this paper:
Gliselle Faith L. Asilo-Ebisa and Laila S. Lomibao. Procedural Knowledge-to-Conceptual Knowledge or Conceptual Knowledge-to-Procedural Knowledge: Which Order of Knowledge is Better for Conceptual Understanding and Procedural Fluency. Journal of Innovations in Teaching and Learning. 2024; 4(1):1-6. doi: 10.12691/jitl-4-1-1

Abstract

This study explored the effectiveness of the two orders of instruction (conceptual-to-procedural knowledge and conceptual-to-procedural knowledge) to grade 10 students’ conceptual understanding and procedural fluency. A quasi-experimental control-group pretest-posttest design was used. The control group experienced a conceptual-to-procedural order of instruction which is the usual practice in the school while the experimental group experienced a procedural-to-conceptual order of instruction. Quade’s ANCOVA, a non-parametric statistical tool, was used to examine if there is a significant difference between the student’s conceptual understanding and procedural fluency when grouped according to order of instruction. A non-parametric statistical tool was utilized because the data was found to be not normally distributed. The pretest scores serve as a covariate in Quade’s ANCOVA to reduce the error variance and to eliminate systematic bias. Results showed that, there is no significant difference in the students’ conceptual understanding and procedural fluency in measures of central tendency, measures of dispersion, and measures of position when grouped according to the order of instruction. Moreover, students under the conceptual-to-procedural knowledge and procedural-to-conceptual knowledge performed similarly. Thus, the researchers recommend the interchangeable use of order of instruction. Since there is a bidirectional relationship between conceptual and procedural knowledge, the teachers are recommended not to stick with just one order of instruction but to use it iteratively.

Keywords:
conceptual knowledge procedural knowledge conceptual understanding procedural fluency Quade’s ANCOVA

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References:

[1]	OECD (2023), PISA 2022 Results (Volume I): The State of Learning and Equity in Education, PISA, OECD Publishing, Paris.
[2]	Mullis, I. V. S., Martin, M. O., Foy, P., Kelly, D. L., & Fishbein, B. (2020). TIMSS 2019 International Results in Mathematics and Science. Retrieved from Boston College, TIMSS & PIRLS International Study Center website: https://timssandpirls.bc.edu/timss2019/international-results/.
[3]	National Research Council. (2002). Helping Children Learn Mathematics. Mathematics Learning Study Committee, J Kilpatrick and J. Swafford, Editors. Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
[4]	Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346–362.
[5]	Ho, Thi. (2020). Measuring Conceptual Understanding, Procedural Fluency and Integrating Procedural and Conceptual Knowledge in Mathematical Problem Solving. International Journal of Scientific Research and Management. 8. 1334-1350. 10.18535/ijsrm/v8i05.el02.
[6]	J. Hechter, G. Stols & C. Combrinck (2022) The Reciprocal Relationship Between Conceptual and Procedural Knowledge—A Case Study of Two Calculus Problems, African Journal of Research in Mathematics, Science and Technology Education, 26: 2, 111-124.
[7]	Hurrell, D. P. (2021). Conceptual knowledge OR Procedural knowledge OR Conceptual knowledge AND Procedural knowledge: Why the conjunction is important for teachers. Australian Journal of Teacher Education, 46(2).
[8]	Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91(1), 175–189.
[9]	Cheung, J. J. H., Kulasegaram, K. M., Woods, N. N., & Brydges, R. (2019). Why Content and Cognition Matter: Integrating Conceptual Knowledge to Support Simulation-Based Procedural Skills Transfer. Journal of General Internal Medicine, 34(6), 969–977.
[10]	Sweller, J., van Merrienboer, J. J. G., & Paas, F. G. W. C. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10(3), 251–296.
[11]	Hiebert, J., & Wearne, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 199-224). Hillsdale, NJ: Erlbaum
[12]	SEI-DOST & MATHTED, (2011). Mathematics framework for Philippine basic education. Manila: SEI-DOST & MATHTED.
[13]	Grouws, D.A., & Cebulla, K.J. (2000). Improving Student Achievement in Mathematics. Educational Practices Series--4.
[14]	National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: National Council of Teachers of Mathematics
[15]	Thomas, L. (2020c, July 31). An Introduction to Quasi-Experimental Designs.
[16]	Rosenstein, L. D. (2019). Research Design and Analysis: A Primer for the Non-Statistician. Hoboken, NJ: Wiley.
[17]	Valmoria, E., & Tan, R. (2019). Sequences of Mathematical Tasks for Intensifying Procedural Fluency In Integer Operation. Work: Journal of Prevention, Assessment & Rehabilitation, 63(4), 457-460.
[18]	Haapasalo, L. (2003). The conflict between conceptual and procedural knowledge : Should we need to understand in order to be able to do, or vice versa ? Proceedings on the IXX Symposium of the Finnish Mathematics and Science Education Research Association, 1–20.
[19]	Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015)23. Not a one-way street: bidirectional relations between procedural and conceptual knowledge of mathematics4. Educational Psychology Review, 27(4), 587-597.
[20]	Fuson, K. C., Kalchman, M., & Bransford, J. D. (2005). Mathematical Understanding: An Introduction. In M. S. Donovan, & J. Bransford (Eds.), How Students Learn Mathematics in the Classroom (pp. 217-256). Washington DC: National Research Council.
[21]	Rittle-Johnson, B., & Koedinger, K. (2009). Iterating between lessons on concepts and procedures can improve mathematics knowledge. The British journal of educational psychology, 79 Pt 3, 483-500.
[22]	Hopkins, Sarah; O’Donovan, Richard (2019). Using complex learning tasks to build procedural fluency and financial literacy for young people with intellectual disability. Mathematics Education Research Journal.
[23]	Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47(6), 1525–1538.