# Strategies and performance in elementary students’ three-digit mental addition

- 1.6k Downloads
- 4 Citations

## Abstract

The focus of this study is the relationship between students’ performance in mental calculation and the strategies they use when solving three-digit mental addition problems. The sample comprises 78 4th grade students (40 boys and 38 girls). Their mean age was 10 years and 4 months. The main novelties of the current research include (1) exploration of the relationship between strategy use and response time, (2) revealing the uniformity of the strategies used throughout the series of tasks, and (3) pointing out between-school differences in strategy use, but not in success rate or response time. Although connections between strategy use and success have been demonstrated, about half of the students insisted on one given strategy throughout the series of eight tasks. The results indicate that teachers developed their students’ mental claculation skills in a way such that some strategies became preferred and others ignored. In the discussion a comparison to previous research results and educational implications are provided.

## Keywords

Mental calculation Arithmetic Metacognition## Notes

### Acknowledgments

This research was supported by a grant from the Hungarian Scientific Research Fund (OTKA 81538), and by the MTA-SZTE Research Group on the Development of Competencies. Research assistants were Ágnes Csizmazia, Boglárka Dombi and Zsófia Henn.

Parts of this study were presented at the 36th Conference of the International Group for the Psychology of Mathematics Education (Csíkos, 2012), and in Hungarian as a book chapter (Csíkos, 2013).

## References

- Afflerbach, P., Pearson, P. D., & Paris, S. G. (2008). Clarifying differences between reading skills and reading strategies.
*The Reading Teacher, 61*, 364–373.CrossRefGoogle Scholar - Almasi, J. F. (2003).
*Teaching strategic processes in reading*. New York: Guilford Press.Google Scholar - Balassa, L., Csekné Szabó, K., & Szilas, Á. (2009).
*Harmadik matematikakönyvem*[My third mathematics book]. Celldömölk: Apáczai Kiadó.Google Scholar - Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.),
*The development of arithmetic concepts and skills*(pp. 1–34). Mahwah: Lawrence Erlbaum Associates.Google Scholar - Baroody, A. J., Cibulskis, M., Lai, M.-L., & Li, X. (2004). Comments on the use of learning trajectories in curriculum development and research.
*Mathematical Thinking and Learning, 6*, 227–260.CrossRefGoogle Scholar - Bartolini Bussi, M. G. (2011). Artefacts and utilization schemes in mathematics teacher education: Place value in early childhood education.
*Journal of Mathematics Teacher Education, 14*, 93–112.CrossRefGoogle Scholar - Ben-Zeev, T. (1996). When erroneous mathematical thinking is just as “correct”: The oxymoron of rational errors. In R. J. Sternberg & T. Ben-Zeev (Eds.),
*The nature of mathematical thinking*(pp. 55–79). Mahwah: Lawrence Erlbaum Associates.Google Scholar - Cankaya, O., LeFevre, J., & Sowinski, C. (2012).
*The influences of different number languages on numeracy learning*(Encyclopedia of Language and Literacy Development, pp. 1–8). London, ON: Western University.Google Scholar - Cannice, M. V. (2013). The right moves: Creating experiential management learning with chess.
*The International Journal of Management Education, 11*, 25–33.CrossRefGoogle Scholar - Cohen, J. (1969).
*Statistical power analysis for the behavioral sciences*. New York: Academic.Google Scholar - Csíkos, C. (2012). Success and strategies in 10 year old students’ mental three-digit addition. In T. Y. Tso (Ed.),
*Proceedings of the 36th Conference of the international Group for the Psychology of Mathematics Education*(Vol. 2, pp. 179–186). Taipei: PME.Google Scholar - Csíkos, C. (2013). A fejben számolás stratégiáinak vizsgálata háromjegyű számok összeadásával negyedik osztályos tanulók körében [Investigating the strategies of mental three-digit addition among 4th grade students]. In G. Molnár & E. Korom (Eds.),
*Az iskolai sikerességet befolyásoló kognitív és affektív tényezők értékelése*(pp. 31–45). Hungary: Nemzedékek Tudása Tankönyvkiadó.Google Scholar - Flavell, J. H. (1987). Speculations about the nature and development of metacognition. In F. E. Weinert & R. Kluwe (Eds.),
*Metacognition, motivation, and understanding*(pp. 21–29). Hillsdale: Erlbaum.Google Scholar - Foxman, D., & Beishuizen, D. (2002). Mental calculation methods used by 11-year-olds in different attainment bands: A reanalysis of data from the 197 APU survey in the UK.
*Educational Studies in Mathematics, 51*, 41–69.CrossRefGoogle Scholar - Frank, M. C., & Barner, D. (2012). Representing exact number visually using mental abacus.
*Journal of Experimental Psychology: General, 141*, 134–149.CrossRefGoogle Scholar - Fürst, A. J., & Hitch, G. J. (2000). Separate roles for executive and phonological components of working memory in mental arithmetic.
*Memory & Cognition, 28*, 774–782.CrossRefGoogle Scholar - Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., et al. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction.
*Journal for Research in Mathematics Education, 28*, 130–162.CrossRefGoogle Scholar - Gourgey, A. (1998). Metacognition in basic skill instruction.
*Instructional Science, 26*, 81–96.CrossRefGoogle Scholar - Hacker, D. J. (1998). Definitions and empirical foundations. In D. J. Hacker, J. Dunlosky, & A. C. Graesser (Eds.),
*Metacognition in educational theory and practice*(pp. 1–23). Mahwah: Erlbaum.Google Scholar - Harskamp, E., & Suhre, C. (2007). Schoenfeld’s problem solving theory in a student controlled learning environment.
*Computers & Education, 49*, 822–839.CrossRefGoogle Scholar - Heinze, A., Marschick, F., & Lipowsky, W. (2009). Addition and subtraction of three-digit numbers: Adaptive strategy use and the influence of instruction in German third grade.
*ZDM – The International Journal on Mathematics Education, 41*, 591–604.CrossRefGoogle Scholar - Hope, J. A., & Sherrill, J. M. (1987). Characteristics of unskilled and skilled mental calculators.
*Journal for Research in Mathematics Education, 18*, 99–111.Google Scholar - Kirk, E. P., & Ashcraft, M. H. (2001). Telling stories: The perils and promise of using verbal reports to study math strategies.
*Journal of Experimental Psychology: Learning, Memory, and Cognition, 27*, 157–175.Google Scholar - Laupa, M., & Becker, J. (2004). Coordinating mathematical concepts with the demands of authority: Children’s reasoning about conventional and second-order logical rules.
*Cognitive Development, 19*, 147–168.CrossRefGoogle Scholar - Lemaire, P., Lecacheur, M., & Farioli, F. (2000). Children’s strategy use in computational estimation.
*Canadian Journal of Experimental Psychology, 54*, 141–148.CrossRefGoogle Scholar - Miller, K. F., Kelly, M. K., & Zhou, X. (2005). Learning mathematics in China and the United States: Cross-cultural insight into the nature and course of mathematical development. In J. I. D. Campbell (Ed.),
*Handbook of mathematical cognition*(pp. 163–178). New York: Psychology Press.Google Scholar - Piazza, M., & Dehaene, S. (2004). From number neurons to mental arithmetic: The cognitive neuroscience of number sense. In M. S. Gazzaniga (Ed.),
*The cognitive neurosciences III*(pp. 965–975). Cambridge: The MIT Press.Google Scholar - Schillemans, V., Luwel, K., Bulté, I., Onghena, P., & Verschaffel, L. (2009). The influence of previous strategy use on individuals’ subsequent strategy choice: Findings from a numerosity judgment task.
*Psychologica Belgica, 49*, 191–205.CrossRefGoogle Scholar - Selter, C. (2001). Addition and subtraction of three-digit numbers: German elementary children’s success, methods and strategies.
*Educational Studies in Mathematics, 47*, 145–173.CrossRefGoogle Scholar - Siegler, R. S. (1987). The perils of averaging data over strategies: An example from children’s addition.
*Journal of Experimental Psychology: General, 116*, 250–264.CrossRefGoogle Scholar - Siegler, R. S., & Lin, X. (2010). Self-explanations promote children’s learning. In H. Salatas Waters & W. Schneider (Eds.),
*Metacognition, strategy use, and instruction*(pp. 85–112). New York: The Guilford Press.Google Scholar - Siegler, R. S., Adolph, K. E., & Lemaire, P. (1996). Strategy choices across the life span. In L. R. Reder (Ed.),
*Implicit memory and metacognition*(pp. 79–121). Mahwah: Erlbaum.Google Scholar - Szendrei, J. (2005).
*Gondolod, hogy egyre megy?*[Do you think it is all the same?]. Budapest: Typotex.Google Scholar - Threlfall, J. (1998). Are mental calculation strategies really strategies?
*Proceedings of the British Society for Research into Learning Mathematics, 18*(3), 71–76.Google Scholar - Threlfall, J. (2002). Flexible mental calculation.
*Educational Studies in Mathematics, 50*, 29–47.CrossRefGoogle Scholar - Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2006). The development of children’s adaptive expertise in the number domain 20 to 100.
*Cognition and Instruction, 24*, 439–465.CrossRefGoogle Scholar - Torbeyns, J., De Smedt, B., Stassens, N., Ghesquière, P., & Verschaffel, L. (2009a). Solving subtraction problems by means of indirect addition.
*Mathematical Thinking and Learning, 11*, 79–91.CrossRefGoogle Scholar - Torbeyns, J., Ghesquière, P., & Verschaffel, L. (2009b). Efficiency and flexibility of indirect addition in the domain of multi-digit subtraction.
*Learning and Instruction, 19*, 1–12.CrossRefGoogle Scholar - 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*, 335–359.CrossRefGoogle Scholar - Walter, J. G., & Gerson, H. (2007). Teachers’ personal agency: Making sense of slope through additive structures.
*Educational Studies in Mathematics, 65*, 203–233.CrossRefGoogle Scholar - Wiener, N. (1960/1999). Some moral and technical consequences of automation.
*Resonance, 4*, 80–88Google Scholar