Resumé
Regnestrategier er vigtige for elevers matematiske udvikling, men forskning i strategier for rationale tal er begrænset. I denne artikel afdækker vi begrebet regnestrategier i relation til regning med brøker. Vi introducerer analysebegrebet fragmentbaseret metode og anvender dette i en analyse af fejlsvar knyttet til brøkopgaver i folkeskolens prøve i matematik uden hjælpemidler for 9. klasse. Analysen af fejlsvar viser omfanget af en mangelfuld talforståelse, og at der anvendes en fragment baseret metode. Undervisningen i brøker bør have som mål, at eleverne udvikler deres regnestrategier inden for brøker, så de mestrer et bredt udsnit af strategier og kan anvende disse adaptivt og fleksibelt
Referencer
Bailey, D.H., Hoard, M.K., Nugent, L. & Geary, D.C. (2012). Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology, 113(3), 447-455. https://doi.org/10.1016/j.jecp.2012.06.004
Bailey, D.H., Siegler, R.S. & Geary, D.C. (2014). Early predictors of middle school fraction knowledge. Developmental Science, 17(5), 775-785. https://doi.org/10.1111/desc.12155
Baroody, A.J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. I: A.J. Baroody & A. Dowker (red.), The development of arithmetic concepts and skills: Constructing adaptive expertise (s. 133). Lawrence Erlbaum Associates.
Booth, J.L., Newton, K.J. & Twiss-Garrity, L.K. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118(1), 110-118. https://doi.org/10.1016/j.jecp.2013.09.001
Bray, W.S. & Abreu-Sanchez, L. (2010). Using number sense to compare fractions. Teaching Children Mathematics, 17(2), 90-97. https://doi.org/10.5951/tcm.17.2.0090
Bütüner, S.Ö. (2017). Comparing the use of number sense strategies based on student achievement levels. International Journal of Mathematical Education in Science and Technology, 49(6), 824-855. https://doi.org/10.1080/0020739x.2017.1410738
Børne- og Undervisningsministeriet. (2019a). Matematik – læseplan. https://emu.dk/sites/default/files/2020-09/GSK_L%C3%A6seplan_Matematik.pdf
Børne- og Undervisningsministeriet. (2019b). Matematik – undervisningsvejledning. https://emu.dk/sites/default/files/2020-09/GSK_Vejledning_Matematik.pdf
Børne- og Undervisningsministeriet. (2021a). Data fra folkeskolens prøver i skriftlig matematik. Styrelsen for Undervisning og Kvalitet.
Børne- og Undervisningsministeriet. (2021b). Matematik FP9 – folkeskolens prøver – prøven uden hjælpemidler. Styrelsen for Undervisning og Kvalitet.
Børne- og Undervisningsministeriet. (2022a). Data fra folkeskolens prøver i skriftlig matematik. Styrelsen for Undervisning og Kvalitet.
Børne- og Undervisningsministeriet. (2022b). Matematik FP9 – folkeskolens prøver – prøven uden hjælpemidler. Styrelsen for Undervisning og Kvalitet
Børne- og Undervisningsministeriet. (2023a). Data fra folkeskolens prøver i skriftlig matematik. Styrelsen for Undervisning og Kvalitet.
Børne- og Undervisningsministeriet. (2023b). Matematik FP9 – folkeskolens prøver – prøven uden hjælpemidler. Styrelsen for Undervisning og Kvalitet.
Børne- og Undervisningsministeriet. (2024a). Data fra folkeskolens prøver i skriftlig matematik. Styrelsen for Undervisning og Kvalitet.
Børne- og Undervisningsministeriet. (2024b). Matematik FP9 – folkeskolens prøver – prøven uden hjælpemidler. Styrelsen for Undervisning og Kvalitet.
Carraher, T.N., Carraher, D.W. & Schliemann, A.D. (1985). Mathematics in the streets and in schools. British Journal of Developmental Psychology, 3(1), 21-29. https://doi.org/10.1111/j.2044835x.1985.tb00951.x
Christiansen, C. & Sorth-Olsen, J. (2017). Brøker! – Hvordan forstår man dem? I: M. Niss & U.T.
Jankvist (red.), Læringsvanskeligheder i matematik – hvordan kan de forstås og afhjælpes? (s. 39-57). Frydenlund.
Dela Cruz, A. & Dela Cruz, M. (2022). The influence of mathematical test anxiety and self-efficacy on students’ performance. Universal Journal of Educational Research, 1(1), 12-26. http://doi.org/10.17613/y0kfg-ajn34
Nationalt Center for Udvikling af Matematikundervisning. (2024). Fælles indsats for tal og algebra. https://matematikdidaktik.dk/tal-og-algebra
Geary, D.C., Hoard, M.K., Byrd-Craven, J. & DeSoto, M.C. (2004). Strategy choices in simple and complex addition: Contributions of working memory and counting knowledge for children with mathematical disability. Journal of Experimental Child Psychology, 88(2), 121-151. https://doi.org/10.1016/j.jecp.2004.03.002
Heinze, A., Star, J.R. & Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM Mathematics Education, 41, 535-540 https://doi.org/10.1007/s11858-009-0214-4
Hickendorff, M., Torbeyns, J. & Verschaffel, L. (2019). Multi-digit addition, subtraction, multiplication, and division strategies. I: A. Fritz, V.G. Haase & P. Räsänen (red.), International Handbook of mathematical learning difficulties (s. 543-560). Springer.
Hiebert, J. & Carpenter, T.P. (1992). Learning and teaching with understanding. I: D.A. Grouws (red.), Handbook of research on mathematics teaching and learning (s. 65-97). Macmillan.
Jóelsdóttir, L.B. & Sunde, P.B. (2024). Adaptivitet og fleksibilitet – addition og subtraktion med flercifrede tal. MONA, 2024(3), 7-23. https://doi.org/10.7146/mona.v24i3.149010
Jóelsdóttir, L.B., Sunde, P.B., Sunde, P. & Andrews, P. (2024). Routine and adaptive experts: Individual characteristics and their impact on multidigit arithmetic strategy flexibility and mathematics achievement. Journal of Numerical Cognition, 10, 1-20. https://doi.org/10.5964/jnc.14081
Jordan, N.C., Resnick, I., Rodrigues, J., Hansen, N. & Dyson, N. (2017). Delaware longitudinal study of fraction learning: Implications for helping children with mathematics difficulties. Journal of Learning Disabilities, 50(6), 621-630. https://doi.org/10.1177/0022219416662033
Kamii, C. & Dominick, A. (1997). To teach or not to teach algorithms. Journal of Mathematical Behavior, 16(1), 51-61. https://doi.org/10.1016/S0732-3123(97)90007-9
Kilpatrick, J., Swafford, J. & Findell, B. (red.). (2001). Adding it up: Helping children learn mathematics. National Academy Press.
Lortie-Forgues, H., Tian, J. & Siegler, R.S. (2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38, 201-221. https://doi.org/10.1016/j.dr.2015.07.008
McMullen, J. & Van Hoof, J. (2020). The role of rational number density knowledge in mathematical development. Learning and Instruction, 65, 1-9. https://doi.org/10.1016/j.learninstruc.2019.101228
Meert, G., Grégoire, J. & Noël, M.P. (2010). Comparing the magnitude of two fractions with common components: Which representations are used by 10- and 12-year-olds? Journal of Experimental Child Psychology, 107(3), 244-259. https://doi.org/10.1016/j.jecp.2010.04.00
Ni, Y. & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27-52. https://doi.org/10.1207/s15326985ep4001_3
Pedersen, P.L. (2021). Learning and understanding the complexity of fractions. Aalborg Universitetsforlag. https://doi.org/10.54337/aau441584586
Pedersen, P.L. & Bjerre, M. (2021). Two conceptions of fraction equivalence. Educational Studies in Mathematics, 107, 135-157.
https://doi.org/10.1007/s10649-021-10030-7
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
Siegler, R.S. & Jenkins, E. (1989). How children discover new strategies. Lawrence Erlbaum Associates.
Siegler, R.S. & Pyke, A.A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49(10), 1994-2004. https://doi.org/10.1037/a0031200
Sunde, P. (2022). Adaptivitet og fleksibilitet: Regnestrategier i de yngste klasser. MONA, 2022(2), 7-23. https://doi.org/10.7146/mona.v22i1.132755
Tian, J. & Siegler, R.S. (2017). Fractions learning in children with mathematics difficulties. Journal of Learning Disabilities, 50(6), 614-620. https://doi.org/10.1177/0022219416662032
Van Hoof, J., Degrande, T., Ceulemans, E., Verschaffel, L. & Van Dooren, W. (2018). Towards a mathematically more correct understanding of rational numbers: A longitudinal study with upper elementary school learners. Learning and Individual Differences, 61, 99-108. https://doi.org/10.1016/j.lindif.2017.11.010
Van Hoof, J., Vandewalle, J., Verschaffel, L. & Van Dooren, W. (2015). In search for the natural number bias in secondary school students’ interpretation of the effect of arithmetical operations. Learning and Instruction, 37, 30-38. https://doi.org/10.1016/j.learninstruc.2014.03.004
Vamvakoussi, X., Van Dooren, W. & Verschaffel, L. (2012). Naturally biased? In search for reaction time evidence for a natural number bias in adults. Journal of Mathematical Behavior, 31(3), 344-355. https://doi.org/10.1016/j.jmathb.2012.02.001
Verschaffel, L. (2023). Strategy flexibility in mathematics. ZDM Mathematics Education, 56(1), 115-126. https://doi.org/10.1007/s11858-023-01491-6

Dette værk er under følgende licens Creative Commons Navngivelse –Ikke-kommerciel (by-nc).
Copyright (c) 2026 Pernille Ladegaard Pedersen, Christian Gregersen Brix, Bent Sortkær
