American Journal of Educational Research
ISSN (Print): 2327-6126 ISSN (Online): 2327-6150 Website: Editor-in-chief: Ratko Pavlović
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American Journal of Educational Research. 2019, 7(3), 212-218
DOI: 10.12691/education-7-3-4
Open AccessArticle

On the Procedural-Conceptual Based Taxonomy and Its Adaptation to the Multi-Dimensional Approach SPUR to Assess Students’ Understanding Mathematics

Ho Thi Minh Phuong1,

1Department of Mathematics, Quy Nhon University, Quy Nhon, Vietnam

Pub. Date: March 07, 2019

Cite this paper:
Ho Thi Minh Phuong. On the Procedural-Conceptual Based Taxonomy and Its Adaptation to the Multi-Dimensional Approach SPUR to Assess Students’ Understanding Mathematics. American Journal of Educational Research. 2019; 7(3):212-218. doi: 10.12691/education-7-3-4


In this paper we propose a new cognitive taxonomy which is so-called the PCK taxonomy (based on Procedural and Conceptual Knowledge) and adapt this taxonomy to the multi-dimensional approach SPUR (Skills, Properties, Uses, Representations) in analyzing the written mathematics assessments given at some high schools located in Binh Dinh province, Vietnam. Base on the findings we discuss and propose methods to establish the written mathematics assessments in order to assess students’ understanding mathematics more accurately.

mathematics assessment understanding mathematics SPUR cognitive taxonomy procedural knowledge conceptual knowledge

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[1]  Thompson, D. R. and Senk, S. L., 2008, A multi-dimensional approach to understanding in mathematics textbooks developed by UCSMP. Paper presented in Discussion Group 17 of the International Congress on Mathematics Education. Monterrey, Mexico.
[2]  Thompson, D. R. and Kaur, B., 2011, Using a multi-dimensional approach to understanding to assess students’ mathematical knowledge. In B. Kaur & K. Y. Wong (Eds.), Assessment in the mathematics classroom, (pp. 17-32). Singapore: World Scientific Publishing.
[3]  Bleiler, S.K. and Thompson, D.R., 2013, “Multidimensional assessment of CCSSM,” Teaching Children Mathematics, 19(5), 292-300.
[4]  Wong, L. F., & Kaur, B., 2015, “A study of mathematics written assessment in Singapore secondary schools”, The Mathematics Educator, 16(1), 1-26.
[5]  Khaw, A. H. R. and Kaur, B., 2017, “A study of mathematics homework in Singapore Secondary Two classrooms,” The Mathematics Educator, 17(1), 29-56.
[6]  Hiebert, J. and 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). Hillsdale, NJ: Lawrence Erlbaum Associates.
[7]  Baroody, A. J. and Ginsburg, H. P., 1986, The relationship between initial meaningful and mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 75-112). Hillsdale, NJ, US: Lawrence Erlbaum Associates, Inc.
[8]  Rittle-Johnson, B. and Siegler, R. S., 1998, The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75-328). Hove, UK: Psychology Press.
[9]  Rittle-Johnson, B. and Alibali, R.M., 1999, Conceptualand procedural knowledge of mathematics: Does one lead to another? Journal of Educational Psychology, 91, 175-189.
[10]  Barr, C., Doyle, J. M. C., Teresa D. L. and Carol D., 2003, “There is More to Math: A Framework for Learning and Math Instruction”, Waterloo Catholic District School Board.
[11]  Arslan, S., 2010, “Traditional instruction of differential equations and conceptual learning,” Teaching Mathematics and its Applications, 29 (2), 94-107.
[12]  Rittle-Johnson, B. and Schneider, M., 2015, Developing conceptual and procedural knowledge in mathematics. In R. Cohen Kadosh and A. Dowker (Eds), Oxford handbook of numerical cognition, 1102-1118, UK: Oxford University Press.
[13]  Piper, B., Ralaingita, W., Akach, L. and King, S., 2016, “Improving procedural and conceptual mathematics outcomes: evidence from a randomised controlled trial in Kenya,” Journal of Development Effectiveness, 8, (3), 404-422.
[14]  Bloom, B. S., Engelhart, M. D., Furst, E. J., Hill, W. H. and Krathwohl, D. R., 1956, Taxonomy of educational objectives: Handbook I: Cognitive domain. New York: David McKay.
[15]  Skemp, R., 1976, “Relational Understanding and Instrumental Understanding,” Mathematics Teaching, 77. 20-26.
[16]  Hiebert, J. and Carpenter, T. P., 1992, Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York: Mcmillan.
[17]  Barmby, P., Harries, T., Higgins, S. and Suggate, J., 2007, How can we assess mathematical understanding? In Woo, J. H., Lew, H. C., Park, K. S. and Seo, D. Y. (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, 41-48.
[18]  Baroody, A. J., 2003, The Development of Adaptive Expertise and Flexibility: The Integration of Conceptual and Procedural Knowledge. In A. J. Baroody and A. Dowker (Eds.). The Development of Arithmetic Concepts and Skills, 1-34.