An Overview of Mental Calculation Strategies and the Frequency of Their Application
Keywords:
mental calculation, categorization of mental calculation strategies, school Mathematics, teaching mental calculation, frequency of application of mental calculation strategies
Abstract
The paper aims to explore mental calculation strategies and the frequency of their application in tasks. In the paper, we categorized mental calculation strategies related to four basic mathematical operations, looked at several different strategies used by students in a test containing mental calculation tasks, and presented an overview of the interview results. We also analyzed age-related differences in the number of strategies applied by participants. It was found that Mathematics in school does not always contribute to the development of and flexibility in the use of mental calculation strategies. One apparent reason is student preference for previously acquired written algorithms.
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References
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Baranyai, T., Egri, E., Molnár, A. É., and Zsoldos-Marchis, I. (2019). Mental calculation strategies used by pre-service primary school teachers. In L. Gómez Chova, A. López Martínez and I. Candel Torres (Eds.), Proceedings of the 11th International Conference on Education and New Learning Technologies (8717–8724). Valencia: IATED Academy.
Beishuizen, M., and Anghileri, J. (1998). Which mental strategies in the early number curriculum? A comparison of British ideas and Dutch views. British Educational Research Journal, 24(5), 519–538.
Beishuizen, M., Van Putten, C. M., and Van Mulken, F. (1997). Mental arithmetic and strategy use with indirect number problems up to one hundred. Learning and Instruction, 7(1), 87–106.
Blöte, A. W., van der Burg, E., and Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology, 93(3), 627–638.
Bruinsma, B. (Ed.). (1969). Nieuw rekenen. Algemene Inleiding [New arithmetic. General introduction]. Baarn, The Netherlands: Bosch en Keuning.
Cambridge University press (2018). Cambridge Dictionary. Cambridge. https://dictionary.cambridge.org/ Retrieved June, 14, 2018
Caney, A., Watson, J. M. (2003). Mental computation strategies for part-whole numbers. In J. Hattie (Ed.), NZARE AARE Conference 2003: educational research, risks and dilemmas (1–13). University of Auckland Press.
Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., Empson, S. B. (1997). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for research in mathematics education, 29(1), 3–20.
Cindric, M., Zorica, I. M., Juric J. (2019). From “calculation in mind” till “mental calculation”. In Kolar-Begović Z., Kolar-Šuper R. Jukić Matić L. (Eds.), Towards New Perspectives on Mathematics Education, (74–82). Element.
Forsyth, M., and Mangan, L. (2014). Collins English Dictionary: Complete and Unabridged. UK: Harper Collins.
Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., ... Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130–162.
Hartnett, J. (2007). Categorisation of mental computation strategies to support teaching and to encourage classroom dialogue. Mathematics: Essential Research, Essential Practice, 1, 345–352.
Heirdsfield, A. M. (2005). One Teacher’s Role in Promoting Understanding in Mental Computation. In Chick, H. L. and Vincent, J. L. (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, 3, 113–120. Melbourne: PME.
Heirdsfield, A. M. (2011). Teaching mental computation strategies in early mathematics. Young Children, 66(2), 96–102.
Heirdsfield, A. M., Cooper, T., Mulligan, J., and Irons, C. (1999). Children’s mental multiplication and division strategies. In Zaslavsky O. (Ed.), Proceedings of 23rd Conference of the International Group for the Psychology of Maths Education (89–96). Haifa: Technion.
Hope, J. A. (1987). A case study of a highly skilled mental calculator. Journal for Research in Mathematics Education, 18(5), 331–342.
Korten, L. (2017). “The fostering of flexible mental calculation in an inclusive mathematics classroom during Mutual Learning.” CERME 10, Feb 2017, Dublin, Ireland. ffhal-01873470f.
Leksikografski zavod Miroslav Krleža. (2013). Hrvatska enciklopedija, mrežno izdanje. Retrieved from http://enciklopedija.hr/
Lemonidis, C. (2016). Mental Computation and Estimation: Implications for mathematics, education research, teaching and learning. Abingdon, OX: Routledge.
Maghfirah, M., and Mahmudi, A. (2018). Number sense: The result of mathematical experience. Journal of Physics: Conference Series, 1097(1), 012141.
Mead, K. C. (2014). Exploring Everyday Math without Using Technology (Doctoral dissertation).
McIntosh, A., Bana, J., and Farrell, B. (1995). Mental Computation in School Mathematics: Preference, Attitude and Performance of Students in Years 3, 5, 7, and 9. Mathematics, Science and Technology Education Centre (MASTEC), Edith Cowan University, Perth, Western Australia, Australia.
McIntosh, A., Reys, B. J., and Reys, R. E. (1997a). Number Sense: Simple effective number sense experiences, Grades 1-2. USA: Dale Seymour Publication.
McIntosh, A., Reys, B., Reys, R., Bana, J., and Farrell, B. (1997b). Number Sense in School Mathematics: Student performance in four countries. Perth, Australia: Mathematics, Science Technology Education Centre, Edith Cowan University.
Mulligan, J. T., and Mitchelmore, M. C. (1997). Young children’s intuitive models of multiplication and division. Journal for Research in Mathematics Education, 23(3), 309–330.
Murphy, C. (2004). How do children come to use a taught mental calculation strategy? Educational Studies in Mathematics, 56(1), 3–18.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and Standards for School Mathematics. NCTM: Reston, VA.
Nunes, T., Dorneles, B. V., Lin, P. J., and Rathgeb-Schnierer, E. (2016). Teaching and Learning about Whole Numbers in Primary School. Springer Nature.
OECD (Organisation for Economic Co-operation and Development). (2003). Mathematics Teaching and Learning Strategies in PISA. Paris: OECD Publishing.
Qualifications and Curriculum Authority (QCA). (1999). Teaching Mental Calculation Strategies: Guidance for teachers at Key Stages 1 and 2. Retrieved from: https://silo.tips/queue/the-national-numeracy-strategy?queue_id=-1v=1596432539u=NzcuMjE2LjE3My4xNzU=
Rathgeb-Schnierer, E., and Green, M. (2013). Flexibility in mental calculation in elementary students from different math classes. In B. Ubuz, Ç. Haser, and M. A. Mariotti (Eds.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education (353–362). Ankara, Turkey: Middle East Technical University.
Rathgeb-Schnierer, E., and Green, M. G. (2019). Developing flexibility in mental calculation. Educação Realidade, 44(2).
Rechtsteiner-Merz, C., and Rathgeb-Schnierer, E. (2015). Flexible mental calculation and “Zahlenblickschulung”. In K. Krainer and N. Vondrovà (Eds.), CERME 9-Ninth Congress of the European Society for Research in Mathematics Education (354–360). Prague, Czech Republic, Charles University.
Rechtsteiner, C., and Rathgeb-Schnierer, E. (2017). Zahlenblickschulung” as approach to develop flexibility in mental calculation in all students. Journal of Mathematics Education, 10(1), 1–16.
Schröder, U. (2007). Förderung der Metakognition. In J. Walter, and F. B. Wember (Eds.), Entdeckendes Lernen im Mathematikunterricht der Schule für Lernbehinderte (271–311). Göttingen, Germany: Hogrefe.
Rezat, S. (2011). Mental calculation strategies for addition and subtraction in the set of rational numbers. In M. Pytlack, T. Rowland, and E. Swodoba (Eds.), Proceedings of CERME-7 (pp. 396-405). Rzeszow, Poland: CERME.
Schütte, S. (2004). Die matheprofis 1. Lehrermaterialieen. Ausg. D., München: Oldenbourg.
Selter, C. (2000). Vorgehensweisen von Grundschüler (inne) n bei Aufgaben zur Addition und Subtraktion im Zahlenraum bis 1000. Journal für Mathematik-Didaktik, 21(3), 227–258.
Selter, C. (2009). Creativity, flexibility, adaptivity, and strategy use in mathematics. ZDM, 41(5), 619–625.
Sharma, M. C. (2001). Matematika bez suza. Lekenik: Ostvarenje.
Sowder, J. T. (1988). Mental computation and number comparison: Their roles in the development of number sense and computational estimation. Number concepts and operations in the middle grades, 182-197.THIS ENTRY IS INCOMPLETE. IT NEEDS THE NAMES OF EDITORS, THE PUBLISHER AND PLACE OF PUBLICATION
Star, J. R., and Newton, K. J. (2009). The nature and development of experts’ strategy flexibility for solving equations. ZDM Mathematics Education, 41(5), 557–567.
Suren, N., and Kandemir, M. A. (2020). The effects of mathematics anxiety and motivation on students’ mathematics achievement. International Journal of Education in Mathematics, Science and Technology, 8(3), 190–218.
Thompson, I. (1999). Mental calculation strategies for addition and subtraction: Part 1. Mathematics in school, 28(5), 2–4.
Thompson, I. (2000). Mental calculation strategies for addition and subtraction: Part 2. Mathematics in school, 29(1), 24–26.
Threlfall, J. (2002). Flexible mental calculation. Educational Studies in Mathematics, 50, 29–47.
Threlfall, J. (2009). Strategies and flexibility in mental calculation. ZDM, 41(5), 541–555.
Torbeyns, J., Verschaffel, L., and Ghesquière, P. (2005). Simple addition strategies in a first-grade class with multiple strategy instruction. Cognition and Instruction,23(1), 1–21.
Van den Heuvel-Panhuizen, M. (2000). Mathematics education in the Netherlands: A guided tour. Freudenthal Institute Cd-rom for ICME9. Utrecht: Utrecht University.
Verschaffel, L., Luwel, K., Torbeyns, J., and Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335–359.
Vidić, T., Klasnić, I., and Đuranović, M. (2022). The Role of Teachers’ Support and Enthusiasm in Predicting Mathematics Anxiety and Confidence among Students. Journal of Elementary Education, 15(1), 51–69.
Zhou, H., Aheto, D. L., Gao, Q., and Chen, W. (2021). Mathematical calculation ability of primary school children: A comparative study between Ghana and China. Journal of Psychology in Africa, 31(3), 286–291.
Baranyai, T., Egri, E., Molnár, A. É., and Zsoldos-Marchis, I. (2019). Mental calculation strategies used by pre-service primary school teachers. In L. Gómez Chova, A. López Martínez and I. Candel Torres (Eds.), Proceedings of the 11th International Conference on Education and New Learning Technologies (8717–8724). Valencia: IATED Academy.
Beishuizen, M., and Anghileri, J. (1998). Which mental strategies in the early number curriculum? A comparison of British ideas and Dutch views. British Educational Research Journal, 24(5), 519–538.
Beishuizen, M., Van Putten, C. M., and Van Mulken, F. (1997). Mental arithmetic and strategy use with indirect number problems up to one hundred. Learning and Instruction, 7(1), 87–106.
Blöte, A. W., van der Burg, E., and Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology, 93(3), 627–638.
Bruinsma, B. (Ed.). (1969). Nieuw rekenen. Algemene Inleiding [New arithmetic. General introduction]. Baarn, The Netherlands: Bosch en Keuning.
Cambridge University press (2018). Cambridge Dictionary. Cambridge. https://dictionary.cambridge.org/ Retrieved June, 14, 2018
Caney, A., Watson, J. M. (2003). Mental computation strategies for part-whole numbers. In J. Hattie (Ed.), NZARE AARE Conference 2003: educational research, risks and dilemmas (1–13). University of Auckland Press.
Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., Empson, S. B. (1997). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for research in mathematics education, 29(1), 3–20.
Cindric, M., Zorica, I. M., Juric J. (2019). From “calculation in mind” till “mental calculation”. In Kolar-Begović Z., Kolar-Šuper R. Jukić Matić L. (Eds.), Towards New Perspectives on Mathematics Education, (74–82). Element.
Forsyth, M., and Mangan, L. (2014). Collins English Dictionary: Complete and Unabridged. UK: Harper Collins.
Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., ... Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130–162.
Hartnett, J. (2007). Categorisation of mental computation strategies to support teaching and to encourage classroom dialogue. Mathematics: Essential Research, Essential Practice, 1, 345–352.
Heirdsfield, A. M. (2005). One Teacher’s Role in Promoting Understanding in Mental Computation. In Chick, H. L. and Vincent, J. L. (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, 3, 113–120. Melbourne: PME.
Heirdsfield, A. M. (2011). Teaching mental computation strategies in early mathematics. Young Children, 66(2), 96–102.
Heirdsfield, A. M., Cooper, T., Mulligan, J., and Irons, C. (1999). Children’s mental multiplication and division strategies. In Zaslavsky O. (Ed.), Proceedings of 23rd Conference of the International Group for the Psychology of Maths Education (89–96). Haifa: Technion.
Hope, J. A. (1987). A case study of a highly skilled mental calculator. Journal for Research in Mathematics Education, 18(5), 331–342.
Korten, L. (2017). “The fostering of flexible mental calculation in an inclusive mathematics classroom during Mutual Learning.” CERME 10, Feb 2017, Dublin, Ireland. ffhal-01873470f.
Leksikografski zavod Miroslav Krleža. (2013). Hrvatska enciklopedija, mrežno izdanje. Retrieved from http://enciklopedija.hr/
Lemonidis, C. (2016). Mental Computation and Estimation: Implications for mathematics, education research, teaching and learning. Abingdon, OX: Routledge.
Maghfirah, M., and Mahmudi, A. (2018). Number sense: The result of mathematical experience. Journal of Physics: Conference Series, 1097(1), 012141.
Mead, K. C. (2014). Exploring Everyday Math without Using Technology (Doctoral dissertation).
McIntosh, A., Bana, J., and Farrell, B. (1995). Mental Computation in School Mathematics: Preference, Attitude and Performance of Students in Years 3, 5, 7, and 9. Mathematics, Science and Technology Education Centre (MASTEC), Edith Cowan University, Perth, Western Australia, Australia.
McIntosh, A., Reys, B. J., and Reys, R. E. (1997a). Number Sense: Simple effective number sense experiences, Grades 1-2. USA: Dale Seymour Publication.
McIntosh, A., Reys, B., Reys, R., Bana, J., and Farrell, B. (1997b). Number Sense in School Mathematics: Student performance in four countries. Perth, Australia: Mathematics, Science Technology Education Centre, Edith Cowan University.
Mulligan, J. T., and Mitchelmore, M. C. (1997). Young children’s intuitive models of multiplication and division. Journal for Research in Mathematics Education, 23(3), 309–330.
Murphy, C. (2004). How do children come to use a taught mental calculation strategy? Educational Studies in Mathematics, 56(1), 3–18.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and Standards for School Mathematics. NCTM: Reston, VA.
Nunes, T., Dorneles, B. V., Lin, P. J., and Rathgeb-Schnierer, E. (2016). Teaching and Learning about Whole Numbers in Primary School. Springer Nature.
OECD (Organisation for Economic Co-operation and Development). (2003). Mathematics Teaching and Learning Strategies in PISA. Paris: OECD Publishing.
Qualifications and Curriculum Authority (QCA). (1999). Teaching Mental Calculation Strategies: Guidance for teachers at Key Stages 1 and 2. Retrieved from: https://silo.tips/queue/the-national-numeracy-strategy?queue_id=-1v=1596432539u=NzcuMjE2LjE3My4xNzU=
Rathgeb-Schnierer, E., and Green, M. (2013). Flexibility in mental calculation in elementary students from different math classes. In B. Ubuz, Ç. Haser, and M. A. Mariotti (Eds.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education (353–362). Ankara, Turkey: Middle East Technical University.
Rathgeb-Schnierer, E., and Green, M. G. (2019). Developing flexibility in mental calculation. Educação Realidade, 44(2).
Rechtsteiner-Merz, C., and Rathgeb-Schnierer, E. (2015). Flexible mental calculation and “Zahlenblickschulung”. In K. Krainer and N. Vondrovà (Eds.), CERME 9-Ninth Congress of the European Society for Research in Mathematics Education (354–360). Prague, Czech Republic, Charles University.
Rechtsteiner, C., and Rathgeb-Schnierer, E. (2017). Zahlenblickschulung” as approach to develop flexibility in mental calculation in all students. Journal of Mathematics Education, 10(1), 1–16.
Schröder, U. (2007). Förderung der Metakognition. In J. Walter, and F. B. Wember (Eds.), Entdeckendes Lernen im Mathematikunterricht der Schule für Lernbehinderte (271–311). Göttingen, Germany: Hogrefe.
Rezat, S. (2011). Mental calculation strategies for addition and subtraction in the set of rational numbers. In M. Pytlack, T. Rowland, and E. Swodoba (Eds.), Proceedings of CERME-7 (pp. 396-405). Rzeszow, Poland: CERME.
Schütte, S. (2004). Die matheprofis 1. Lehrermaterialieen. Ausg. D., München: Oldenbourg.
Selter, C. (2000). Vorgehensweisen von Grundschüler (inne) n bei Aufgaben zur Addition und Subtraktion im Zahlenraum bis 1000. Journal für Mathematik-Didaktik, 21(3), 227–258.
Selter, C. (2009). Creativity, flexibility, adaptivity, and strategy use in mathematics. ZDM, 41(5), 619–625.
Sharma, M. C. (2001). Matematika bez suza. Lekenik: Ostvarenje.
Sowder, J. T. (1988). Mental computation and number comparison: Their roles in the development of number sense and computational estimation. Number concepts and operations in the middle grades, 182-197.THIS ENTRY IS INCOMPLETE. IT NEEDS THE NAMES OF EDITORS, THE PUBLISHER AND PLACE OF PUBLICATION
Star, J. R., and Newton, K. J. (2009). The nature and development of experts’ strategy flexibility for solving equations. ZDM Mathematics Education, 41(5), 557–567.
Suren, N., and Kandemir, M. A. (2020). The effects of mathematics anxiety and motivation on students’ mathematics achievement. International Journal of Education in Mathematics, Science and Technology, 8(3), 190–218.
Thompson, I. (1999). Mental calculation strategies for addition and subtraction: Part 1. Mathematics in school, 28(5), 2–4.
Thompson, I. (2000). Mental calculation strategies for addition and subtraction: Part 2. Mathematics in school, 29(1), 24–26.
Threlfall, J. (2002). Flexible mental calculation. Educational Studies in Mathematics, 50, 29–47.
Threlfall, J. (2009). Strategies and flexibility in mental calculation. ZDM, 41(5), 541–555.
Torbeyns, J., Verschaffel, L., and Ghesquière, P. (2005). Simple addition strategies in a first-grade class with multiple strategy instruction. Cognition and Instruction,23(1), 1–21.
Van den Heuvel-Panhuizen, M. (2000). Mathematics education in the Netherlands: A guided tour. Freudenthal Institute Cd-rom for ICME9. Utrecht: Utrecht University.
Verschaffel, L., Luwel, K., Torbeyns, J., and Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335–359.
Vidić, T., Klasnić, I., and Đuranović, M. (2022). The Role of Teachers’ Support and Enthusiasm in Predicting Mathematics Anxiety and Confidence among Students. Journal of Elementary Education, 15(1), 51–69.
Zhou, H., Aheto, D. L., Gao, Q., and Chen, W. (2021). Mathematical calculation ability of primary school children: A comparative study between Ghana and China. Journal of Psychology in Africa, 31(3), 286–291.
Published
2023-09-14
How to Cite
Jurić J., & Pjanić K. (2023). An Overview of Mental Calculation Strategies and the Frequency of Their Application. Journal of Elementary Education, 16(3), 277-294. https://doi.org/10.18690/rei.16.3.3087
Section
Scientific Articles
Copyright (c) 2023 Josipa Jurić, Karmelita Pjanić
This work is licensed under a Creative Commons Attribution 4.0 International License.