Multi-digit Addition, Subtraction, Multiplication, and Division Strategies

• Marian Hickendorff
• Joke Torbeyns
• Lieven Verschaffel
Chapter

Abstract

This chapter provides an overview of the current findings about (the obstacles in) primary school children’s strategy use in the domain of multi-digit arithmetic. This involves addition, subtraction, multiplication, and division tasks in which at least one of the operands contains two or more digits. For both the additive and multiplicative domains, we provide a comprehensive framework for the classification of strategies, with two dimensions: (1) the operation that underlies the solution process and (2) the way the numbers are dealt with in computing the outcome (manipulating whole numbers or single digits). Empirical findings of children’s strategy use in the additive and multiplicative domain show that children use a variety of number-based strategies efficiently and adaptively before the introduction of the digit-based algorithms. The introduction of the digit-based algorithms seems a critical instructional event: children show a large tendency to use the digit-based algorithms once they are instructed, and they do so rather efficiently. The major obstacles children encounter in developing, selecting, or executing these strategies are their conceptual understanding, procedural fluency, and adaptive/flexible strategy selection.

Keywords

Multi-digit arithmetic Solution strategies Digit-based algorithm Number-based strategies Mental computation Strategy adaptivity Difficulties in strategy development

References

1. Ambrose, R., Baek, J.-M., & Carpenter, T. P. (2003). Children’s invention of multiplication and division algorithms. In The development of arithmetic concepts and skills: Constructive adaptive expertise (pp. 305–336). Google Scholar
2. Aragón, E., Canto, M. C., Marchena, E., Navarro, J. I., & Aguilar, M. (2017). Cognitive profile in learning mathematics with open calculation based on numbers (ABN) algorithm. Revista de Psicodidactica/Journal of Psychodidactics, 22(1), 1–14. Google Scholar
3. Baroody, A. J., Torbeyns, J., & Verschaffel, L. (2009). Young children’s understanding and application of subtraction-related principles. Mathematical Thinking and Learning, 11(1–2), 2–9. Google Scholar
4. Beishuizen, M. (1993). Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades. Journal for Research in Mathematics Education, 24(4), 294. Google Scholar
5. Blöte, A. W., van der Burg, E., & Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology, 93(3), 627–638. Google Scholar
6. Buijs, K. (2008). Leren vermenigvuldigen met meercijferige getallen [Learning to multiply with multidigit numbers]. Utrecht: Freudenthal Institute for Science and Mathematics Education.Google Scholar
7. Campbell, J. I. D., Xue, Q., & Campbell, I. D. (2001). Cognitive arithmetic across cultures. Journal of Experimental Psychology: General, 130(2), 299–315. Google Scholar
8. Cantlon, J. F., & Brannon, E. M. (2006). Adding up the effects of cultural experience on the brain. Trends in Cognitive Sciences, 11(1), 1–4. Google Scholar
9. Csíkos, C. (2016). Strategies and performance in elementary students??? Three-digit mental addition. Educational Studies in Mathematics, 91(1), 123–139. Google Scholar
10. Fagginger Auer, M. F., Hickendorff, M., & van Putten, C. M. (2016). Solution strategies and adaptivity in multidigit division in a choice/no-choice experiment: Student and instructional factors. Learning and Instruction, 41, 52–59. Google Scholar
11. Fagginger Auer, M. F., Hickendorff, M., Van Putten, C. M., Béguin, A. A., & Heiser, W. J. (2016). Multilevel latent class analysis for large-scale educational assessment data: Exploring the relation between the curriculum and students’ mathematical Strategies. Applied Measurement in Education, 29(2), 144–159. Google Scholar
12. Fuson, K. C. (2003). Developing mathematical power in whole number operations. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 68–94). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
13. Hatano, G. (2003). Foreword. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. xi–xiii). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
14. Heinze, A., Marschick, F., & Lipowsky, F. (2009). Addition and subtraction of three-digit numbers: Adaptive strategy use and the influence of instruction in German third grade. ZDM, 41(5), 591–604. Google Scholar
15. Hickendorff, M. (2013). The effects of presenting multidigit mathematics problems in a realistic context on sixth graders’ problem solving. Cognition and Instruction, 31(3), 314–344. Google Scholar
16. Hickendorff, M., Heiser, W. J., Van Putten, C. M., & Verhelst, N. D. (2009). Solution strategies and achievement in dutch complex arithmetic: Latent variable modeling of change. Psychometrika, 74(2), 331–350. Google Scholar
17. Hickendorff, M., Torbeyns, J., & Verschaffel, L. (2017). Grade-related differences in strategy use in multi-digit division in two instructional settings. Paper Submitted for Publication.Google Scholar
18. Hickendorff, M., van Putten, C. M., Verhelst, N. D., & Heiser, W. J. (2010). Individual differences in strategy use on division problems: Mental versus written computation. Journal of Educational Psychology, 102(2), 438–452. Google Scholar
19. Kamii, C., & Dominick, A. (1997). To teach or not to teach algorithms. The Journal of Mathematical Behavior, 16(1), 51–61. Google Scholar
20. Karantzis, I. (2010). Mental arithmetic calculation in the addition and subtraction of two-digit numbers: The case of third and fourth grade elementary school pupils. International Journal for Mathematics in Education, 3, 3–24 Retrieved from https://eclass.upatras.gr/modules/document/file.php/PDE1308/3οΆρθρο.pdf Google Scholar
21. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learning mathematics. Igarss 2014.
22. Larsson, K. (2016). Students â€™ understandings of multiplication (Doctoral dissertation). Stockholm University, Sweden.Google Scholar
23. Linsen, S., Torbeyns, J., Verschaffel, L., Reynvoet, B., & De Smedt, B. (2016). The association between symbolic and nonsymbolic numerical magnitude processing and mental versus algorithmic subtraction in adults. Acta Psychologica, 165, 34–42. Google Scholar
24. Luwel, K., Onghena, P., Torbeyns, J., Schillemans, V., & Verschaffel, L. (2009). Strengths and weaknesses of the choice/no-choice method in research on strategy use. European Psychologist, 14(4), 351–362. Google Scholar
25. Peltenburg, M., van den Heuvel-Panhuizen, M., & Robitzsch, A. (2012). Special education students’ use of indirect addition in solving subtraction problems up to 100—A proof of the didactical potential of an ignored procedure. Educational Studies in Mathematics, 79(3), 351–369. Google Scholar
26. Peters, G., De Smedt, B., Torbeyns, J., Ghesquière, P., & Verschaffel, L. (2013). Children’s use of addition to solve two-digit subtraction problems. British Journal of Psychology, 104(4), 495–511. Google Scholar
27. Peters, G., De Smedt, B., Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2014). Subtraction by addition in children with mathematical learning disabilities. Learning and Instruction, 30, 1–8. Google Scholar
28. Robinson, K. M. (2017). The understanding of additive and multiplicative arithmetic concepts. In D. C. Geary, D. B. Berch, R. J. Ochsendorf, & K. M. Koepke (Eds.), Acquisition of complex arithmetic skills and higher-order mathematics concepts (pp. 21–46). Elsevier Inc. https://www.elsevier.com/books/acquisition-of-complex-arithmetic-skills-and-higher-order-mathematics-concepts/geary/978-0-12-805086-6
29. Royal Dutch Society of Arts and Sciences. (2009). Rekenonderwijs op de basisschool. Analyse en sleutels tot verbetering [Mathematics education in primary school. Analysis and recommendations for improvement]. Amsterdam: KNAW.Google Scholar
30. Ruthven, K. (1998). The use of mental, written and calculator strategies of numerical computation upper primary pupils within a ‘calculator‐aware’ number curriculum. British Educational Research Journal, 24(1), 21–42.Google Scholar
31. Selter, C., Prediger, S., Nührenbörger, M., & Hußmann, S. (2012). Taking away and determining the difference–a longitudinal perspective on two models of subtraction and the inverse relation to addition. Educational Studies in Mathematics, 79(3), 389–408. Google Scholar
32. Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking. New York: Oxford University Press Retrieved from https://books.google.nl/books?hl=nl&lr=&id=lb-hjI0Et8kC&oi=fnd&pg=PR9&dq=siegler+emerging+minds&ots=0JyIMyFrGu&sig=7OGtuG8rkwfaZCoYaNqQodK_GtI#v=onepage&q=siegleremerging minds&f=false Google Scholar
33. Siegler, R. S. (2007). Cognitive variability. Developmental Science, 10(1), 104–109. Google Scholar
34. Siegler, R. S., & Lemaire, P. (1997). Older and younger adults’ strategy choices in multiplication: Testing predictions of ASCM using the choice/no-choice method. Journal of Experimental Psychology: General, 126(1), 71–92. Google Scholar
35. Star, J. R., Newton, K., Pollack, C., Kokka, K., Rittle-Johnson, B., & Durkin, K. (2015). Student, teacher, and instructional characteristics related to students’ gains in flexibility. Contemporary Educational Psychology, 41, 198–208. Google Scholar
36. Torbeyns, J., De Smedt, B., Stassens, N., Ghesquière, P., & Verschaffel, L. (2009). Solving subtraction problems by means of indirect addition. Mathematical Thinking and Learning, 11(1–2), 79–91. Google Scholar
37. Torbeyns, J., Hickendorff, M., & Verschaffel, L. (2017). The use of number-based versus digit-based strategies on multi-digit subtractions: 9–12-year-olds’ strategy use profiles and task performances. Learning and Individual Differences, 58(June 2016), 64–74. Google Scholar
38. Torbeyns, J., Peters, G., De Smedt, B., Ghesquière, P., & Verschaffel, L. (2016). Children’s understanding of the addition/subtraction complement principle. British Journal of Educational Psychology, 86(3), 382–396. Google Scholar
39. Torbeyns, J., Peters, G., De Smedt, B., Ghesquière, P., & Verschaffel, L. (2017). Subtraction by addition strategy use. Paper Submitted for Publication.Google Scholar
40. Torbeyns, J., Peters, G., De Smedt, B., Ghesquière, P., & Verschaffel, L. (2018). Subtraction by addition strategy use. Journal for Numerical Cognition, 4(1), 215–234. Google Scholar
41. Torbeyns, J., & Verschaffel, L. (2013). Efficient and flexible strategy use on multi-digit sums: A choice/no-choice study. Research in Mathematics Education. Google Scholar
42. Torbeyns, J., & Verschaffel, L. (2016). Mental computation or standard algorithm? Children’s strategy choices on multi-digit subtractions. European Journal of Psychology of Education, 31(2), 99–116. Google Scholar
43. Treffers, A. (1987). Integrated column arithmetic according to progressive schematisation. Educational Studies in Mathematics, 18(2), 125–145. Google Scholar
44. van den Heuvel-Panhuizen, M. (2008). Children learn mathematics. Rotterdam, the Netherlands: Sense Publishers.Google Scholar
45. van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 521–525). Dordrecht, Heidelberg/New York: Springer.Google Scholar
46. van den Heuvel-Panhuizen, M., Robitzsch, A., Treffers, A., & Köller, O. (2009). Large-scale assessment of change in student achievement: Dutch primary school students’ results on written division in 1997 and 2004 as an example. Psychometrika, 74(2), 351–365. Google Scholar
47. Van Putten, C. M., van den Brom-Snijders, P. A., & Beishuizen, M. (2005). Progressive Mathematization of long division strategies in Dutch primary schools. Journal for Research in Mathematics Education, 36(1), 44–73 Retrieved from http://www.jstor.org/stable/10.2307/30034920 Google Scholar
48. Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning pages (pp. 557–628). Greenwich: Information Age Publishing.Google Scholar
49. Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335–359. Google Scholar
50. Yang, D.-C., & Huang, K.-L. (2014). An intervention study on mental computation for second graders in Taiwan. The Journal of Educational Research, 107(1), 3–15. Google Scholar
51. Zhang, D., Ding, Y., Lee, S., & Chen, J. (2017). Strategic development of multiplication problem solving: Patterns of students’ strategy choices. The Journal of Educational Research, 110(2), 159–170. Google Scholar
52. Zhang, D., Xin, Y. P., Harris, K., & Ding, Y. (2014). Improving multiplication strategic development in children with math difficulties. Learning Disability Quarterly, 37, 15–30. Google Scholar

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

• Marian Hickendorff
• 1
Email author
• Joke Torbeyns
• 2
• Lieven Verschaffel
• 2
1. 1.Education and Child Studies, Leiden UniversityLeidenThe Netherlands
2. 2.Center for Instructional Psychology and Technology, KU LeuvenLeuvenBelgium