International Electronic Journal of Mathematics Education

International Electronic Journal of Mathematics Education
Multiplicative Thinking: ‘Pseudo-procedures’ are Enemies of Conceptual Understanding
AMA 10th edition
In-text citation: (1), (2), (3), etc.
Reference: Hurst C, Hurrell D. Multiplicative Thinking: ‘Pseudo-procedures’ are Enemies of Conceptual Understanding. INT ELECT J MATH ED. 2020;15(3), em0611. https://doi.org/10.29333/iejme/8567
APA 6th edition
In-text citation: (Hurst & Hurrell, 2020)
Reference: Hurst, C., & Hurrell, D. (2020). Multiplicative Thinking: ‘Pseudo-procedures’ are Enemies of Conceptual Understanding. International Electronic Journal of Mathematics Education, 15(3), em0611. https://doi.org/10.29333/iejme/8567
Chicago
In-text citation: (Hurst and Hurrell, 2020)
Reference: Hurst, Chris, and Derek Hurrell. "Multiplicative Thinking: ‘Pseudo-procedures’ are Enemies of Conceptual Understanding". International Electronic Journal of Mathematics Education 2020 15 no. 3 (2020): em0611. https://doi.org/10.29333/iejme/8567
Harvard
In-text citation: (Hurst and Hurrell, 2020)
Reference: Hurst, C., and Hurrell, D. (2020). Multiplicative Thinking: ‘Pseudo-procedures’ are Enemies of Conceptual Understanding. International Electronic Journal of Mathematics Education, 15(3), em0611. https://doi.org/10.29333/iejme/8567
MLA
In-text citation: (Hurst and Hurrell, 2020)
Reference: Hurst, Chris et al. "Multiplicative Thinking: ‘Pseudo-procedures’ are Enemies of Conceptual Understanding". International Electronic Journal of Mathematics Education, vol. 15, no. 3, 2020, em0611. https://doi.org/10.29333/iejme/8567
Vancouver
In-text citation: (1), (2), (3), etc.
Reference: Hurst C, Hurrell D. Multiplicative Thinking: ‘Pseudo-procedures’ are Enemies of Conceptual Understanding. INT ELECT J MATH ED. 2020;15(3):em0611. https://doi.org/10.29333/iejme/8567

Abstract

Multiplicative thinking is widely accepted as a critically important ‘big idea’ of mathematics that underpins much mathematical understanding beyond the primary years. It is therefore important to ensure not only that children can think multiplicatively, but that they can do so at a conceptual rather than procedural level. This paper reports on a large study of 530 primary school children in Australia, New Zealand and the United Kingdom. The research question was “To what extent do children of 10 and 11 years of age understand what happens to digit values when numbers are multiplied and divided by powers of ten?” A written multiplicative thinking quiz was administered and one section of four questions asked students to multiply and divide two digit whole and decimal numbers by a power of ten and then explain what happened to the numbers. Correct response rates for the four calculations ranged from 38.3% to 61.7%. Response rates for appropriate explanations about what happened to the numbers ranged from 2.6% to 5.5%. Most students who attempted to explain what happened did so at a ‘pseudo-procedural’ level with responses such as ‘a zero is added’ or ‘we take off the zero’. The students who did explain it conceptually did so in terms of the digits moving a place for each power of ten. The implication is that teaching of multiplication and division needs to be done at a conceptual level, with attention paid to the underlying mathematical structure, rather than relying on ‘pseudo-procedures’ such as ‘adding a zero’ which are unsustainable and will likely lead to errors.

References

  • Alcock, L., Ansari, D., Batechelor, S., Bison, M., De Smedt, B., Gilmore, C., … Weber, K. (2016). Challenges in mathematical cognition: A collaboratively-derived research agenda. Journal of Numerical Cognition, 2(1), 20-41. https://doi.org/10.5964/jnc.v2i1.10
  • Anthony, G., & Walshaw, M. (2002). Swaps and switches: Students’ understandings of commutativity. In B. Barton, K. C. Irwin, M. Pfannkuch, & M. O. J. Thomas (Eds.). Mathematics Education in the South Pacific (Proceedings of the 25th annual conference of the Mathematics Education Research Group of Australasia, Auckland) pp. 91-99. Sydney: MERGA. Retrieved from https://merga.net.au/Public/Publications/Annual_Conference_Proceedings/2002_MERGA_CP.aspx
  • Anthony, G., & Walshaw, M. (2009). Effective pedagogy in mathematics. Belley, France: United Nations Educational, Scientific and Cultural Organisation. Retrieved from http://www.ibe.unesco.org/publications.htm
  • Australian Curriculum, Assessment and Reporting Authority (ACARA) (2020). Australian curriculum mathematics - Version 8.3. Retrieved from http://www.australiancurriculum.edu.au/Mathematics/Curriculum/F-10
  • Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38, 115-131. Retrieved from https://www-jstor-org.dbgw.lis.curtin.edu.au/stable/pdf/30034952.pdf
  • 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
  • Clements, D., & Sarama, J. (2019) From Children’s Thinking to Curriculum to Professional Development to Scale: Research Impacting Early Maths Practice. In G. Hine, S. Blackley, & A. Cooke (Eds.). Mathematics Education Research: Impacting Practice (Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia) pp. 36-48. Perth: MERGA. Retrieved from https://merga.net.au/Public/Publications/Annual_Conference_Proceedings/2019-MERGA-conference-proceedings.aspx
  • diSessa, A. A., Gillespie, N. M., & Esterly, J. B. (2004). Coherence versus fragmentation in the development of the concept of force. Cognitive Science, 28, 843-900. https://doi.org/10.1207/s15516709cog2806_1
  • Downton, A., Russo, J., & Hopkins, S. (2019). The case of disappearing and reappearing zeros: A disconnection between procedural knowledge and conceptual understanding. In G. Hine, S. Blackley, & A. Cooke (Eds.). Mathematics Education Research: Impacting Practice (Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia) pp. 236-243. Perth: MERGA. Retrieved from https://merga.net.au/Public/Publications/Annual_Conference_Proceedings/2019-MERGA-conference-proceedings.aspx
  • Givvin, K. B., Stigler, J. W., & Thompson, B. J. (2011). What community college developmental mathematics students understand about mathematics, Part II: The interviews. The MathAMATYC Educator, 2(3), 4-18. Retrieved from https://doi.org/10.1080/00461520.2012.667065
  • Hiebert, J. (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale, N.J.: Erlbaum.
  • Hiebert, J. (1999). Relationships between research and the NCTM standards. Journal for Research in Mathematics Education, 30(1), 3-19. https://doi.org/10.2307/749627
  • Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371-404). Charlotte, NC: Information Age.
  • 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). Hillsdale, NJ: Erlbaum.
  • Hurst, C. (2017). Children have the capacity to think multiplicatively, as long as . . . European Journal of STEM Education, 2(3), 1-14. https://doi.org/10.20897/ejsteme/78169
  • Hurst, C. (2018). A tale of two kiddies: A Dickensian slant on multiplicative thinking. Australian Primary Mathematics Classroom, 23(1), 31-36. Retrieved from https://search-informit-com-au.dbgw.lis.curtin.edu.au/documentSummary;dn=485064898800183;res=IELHSS
  • Miller, S. P., & Hudson, P. J. (2007). Using evidence-based practices to build mathematics competence related to conceptual, procedural, and declarative knowledge. Learning Disabilities Research and Practice, 22(1), 47-57. Retrieved from https:///doi.org/10.1111/j.1540-5826.2007.00230.x
  • National Governors Association Center for Best Practices, Council of Chief State School Officers (NGA Center). (2010). Common core state standards for mathematics. Retrieved from http://www.corestandards.org/the-standards
  • Pesek, D. D., & Kirshner, D., (2000). Interference of instrumental instruction in subsequent relational learning. Journal for Research in Mathematics Education, 31, 524-540. https://doi.org/10.2307/749885
  • Puchner, L., Taylor, A., O’Donnell, B., & Fick, K. (2010). Teacher learning and mathematics manipulatives: A collective case study about teacher use of manipulatives in elementary and middle school mathematics lessons. School Science and Mathematics, 108(7), 313-325. Retrieved from https://doi.org/10.1111/j.1949-8594.2008.tb17844.x
  • Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics for instruction. Hillsdale: Erlbaum.
  • Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the Conceptual Structure of Mathematics, Educational Psychologist 47(3), 189-203. https://doi.org/10.1080/00461520.2012.667065
  • Rittle-Johnson, B. (2017). Developing mathematics knowledge. Child Development Perspectives, 11(3), 184-190. https://doi.org/10.1111/cdep.12229
  • Rittle-Johnson, B. Schneider, M., & Star, J. (2015). Not a one-way street: Bi-directional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27. https://doi.org/10.1007/s10648-015-9302-x
  • Rittle-Johnson, B., & Koedinger, K. R. (2009). Iterating between lessons concepts and procedures can improve mathematics knowledge. British Journal of Educational Psychology, 79, 483-500. Retrieved from https://doi.org/10.1348/000709908X398106
  • Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge in mathematics. In R. Cohen Kadosh & A. Dowker (Eds.), Oxford handbook of numerical cognition (pp. 1102-1118). Oxford, UK: Oxford University Press. https://doi.org/10.1093/oxfordhb/9780199642342.013.014
  • Rittle-Johnson, B., Fyfe, A.M., & Loehr, E.R. (2016). Improving Conceptual and Procedural Knowledge: The Impact of Instructional Content Within A Mathematics Lesson. British Journal of Educational Psychology, 86(4), 576-591. https://doi.org/10.1111/bjep.12124
  • 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, 346-362. https://doi.org/10.1037/0022-0663.93.2.346
  • Ross, S. (2002). Place value: Problem solving and written assessment. Teaching Children Mathematics, March 2002, 419-423. Retrieved from https://search-proquest-com.dbgw.lis.curtin.edu.au/docview/214137991/abstract/14585DF896874B80PQ/9?accountid=10382
  • Schneider, M., & Stern, E. (2009). The Inverse Relation of Addition and Subtraction: A Knowledge Integration Perspective. Mathematical Thinking and Learning, 11, 92-101. Retrieved from: https://doi.org/10.1080/10986060802584012
  • Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations between conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47(6), 1525-1538. https://doi.org/10.1037/a0024997
  • Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early Predictors of High School Mathematics Achievement. Psychological Science, 23(7), 691-697. https://doi.org/10.1177/0956797612440101
  • Siemon, D., Bleckly, J., & Neal, D. (2012). Working with the Big Ideas in Number and the Australian Curriculum: Mathematics. In B. Atweh, M. Goos, R. Jorgensen, & D. Siemon (Eds.), Engaging the Australian National Curriculum: Mathematics - Perspectives from the Field (pp. 18-45). Online Publication: Mathematics Education Research Group of Australasia. Retrieved from https://merga.net.au/Public/Public/Publications/Engaging_the_Australian_curriculum_mathematics_book.aspx
  • Siemon, D., Breed, M., Dole, S., Izard, J., & Virgona, J. (2006). Scaffolding Numeracy in the Middle Years - Project Findings, Materials, and Resources (Final Report), Victorian Department of Education and Training and the Tasmanian Department of Education. Retrieved from http://www.eduweb.vic.gov.au/edulibrary/public/teachlearn/student/snmy.ppt
  • Skemp, R. (1976). Relational and instrumental understanding. Mathematics Teaching in the Middle School, 12(2), 88-95. Retrieved from http://www.jstor.org/stable/41182357
  • Sowder, J. T. (1998). What are the “math wars” in California all about? Reasons and perspectives. Retrieved from http://staff.tarleton.edu/brawner/coursefiles/579/Math%20Wars%20in%20California.pdf
  • Stigler, J. W., Givvin, K. B., & Thompson, B. J. (2010). What community college developmental mathematics students understand about mathemDDatics. Mathematics Teacher, 1(3), 4-16. Retrieved from https://www.researchgate.net/publication/260908914
  • Warren, E., & English, L. (2000). Primary school children’s knowledge of arithmetic structure. In J. Bana & A. Chapman (Eds.), Mathematics Education beyond 2000 (Proceedings of 23rd annual conference of the Mathematics Education Research Group of Australasia, Fremantle). pp. 624-631. Sydney: MERGA. Retrieved from https://merga.net.au/Public/Publications/Annual_Conference_Proceedings/2000_MERGA_CP.aspx

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