Nøgleord
Brøkregning
Grundskole
Matematisk forståelse
Citation/Eksport
Resumé
Hvordan kan vi støtte elevernes brøkforståelse og brøkregning? Artiklen er en indføring i det didaktiske problemfelt ækvivalens af brøker og brøkregning. Der præsenteres en teoretisk matematisk analyse af to forståelser af ækvivalens, som kaldes enhedsækvivalens og proportional ækvivalens. Vi vil analysere a) hvordan de to forskellige forståelser adskiller sig fra hinanden, og b) hvordan de optræder forskelligt inden for tre regnearter: addition, subtraktion og multiplikation. I denne bearbejdning og udvidelse af en større engelsk artikel af samme forfattere argumenterer vi for at et større kendskab til ækvivalens kan være essentielt for at udvikle fleksible regnestrategier inden for brøkregning.
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, s. 447‑455.
Behr, M.J., Wachsmuth, I., Post, T.R. & Lesh, R. (1984). Order and Equivalence of Rational Numbers: A Clinical Teaching Experiment. Journal for Research in Mathematics Education, 15, s. 323‑341.
Booth, J.L. & Newton, K.J. (2012). Fractions: Could They Really Be the Gatekeeper’s Doorman? Contemporary Educational Psychology, 37, s. 247‑253.
Cramer, K.A., Wyberg, T. & Levitt, S. (2009). Rational Number Project: Fraction Operations and Initial Decimal Ideas.
English, L. & Halford, G.S. (1995). Mathematics Education: Models and Processes. Routledge.
Fazio, L.K., DeWolf, M. & Siegler, R.S. (2016). Strategy Use and Strategy Choice in Fraction Magnitude Comparison. Journal of Experimental Psychology: Learning, Memory, and Cognition, 42, s. 1‑16. doi:10.1037/xlm0000153.
Hamdan, N. & Gunderson, E.A. (2017). The Number Line is a Critical Spatial Numerical Representation: Evidence from a Fraction Intervention. Developmental Psychology, 53, s. 587‑596. doi:10.1037/dev0000252.
Kamii, C. & Clark, F.B. (1995). Equivalent Fractions: Their Difficulty and Educational Implications. Journal of Mathematical Behavior, 14, s. 365‑378. doi:10.1016/0732‑3123(95)90035‑7.
Kieren, T.E. (1976). On the Mathematical, Cognitive, and Instructional Foundations of Rational Numbers. I: R.A. Lesh & D.A. Bradbard (red.), Number and Measurement: Papers from a Research Workshop (s. 101‑144). ERIC/SMEAC.
Moss, J. (2005). Pipes, Tubes, and Beakers: New Approaches to Teaching the Rational-Number System. I: M.S. Donovan & J.D. Bransford (red.), How Students Learn: Mathematics in the Classroom (s. 309‑349). The National Academies Press. doi:10.17226/11101.
Newton, K.J. (2008). An Extensive Analysis of Preservice Elementary Teachers’ Knowledge of Fractions. American Educational Research Journal, 45, s. 1080‑1110. doi:10.3102/0002831208320851.
Ni, Y. (2001). Semantic Domains of Rational Numbers and the Acquisition of Fraction Equivalence. Contemporary Educational Psychology, 26(3), s. 400‑417. doi:10.1006/ceps.2000.1072.
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), s. 27‑52. doi:10.1207/s15326985ep4001_3.
Pedersen, P.L. (2021). Learning and Understanding the Complexity of Fractions. https://vbn.aau.dk/ws/portalfiles/portal/441584586/PHD%20PLP%202%20E%20pdf.pdf.
Pedersen, P.L. & Bjerre, M. (2021). Two Concepts of Fraction Equivalence. Educational Studies in Mathematics, 107, s. 135‑157. doi:10.1007/s10649‑021‑10030‑7.
Pedersen, P. L., & Sunde, P. (2019). Students’ ability to compare fractions related to proficiency in the four operations. Eleventh Congress of the European Society for Research in Mathematics Education. Retrieved from https://hal.archives-ouvertes.fr/hal-02401060
Rau, M.A. & Matthews, P.G. (2017). How to Make ‘More’ Better? Principles for Effective Use of Multiple Representations to Enhance Students’ Learning about Fractions. ZDM – Mathematics Education, 49, s. 531‑544. doi:10.1007/s11858‑017‑0846‑8.
Schneider, M. & Siegler, R.S. (2010). Representations of the Magnitudes of Fractions. Journal of Experimental Psychology: Human Perception and Performance, 36, s. 1227‑1238. doi:10.1037/a0018170.
Sidney, P.G., Thompson, C.A. & Rivera, F.D. (2019). Number Lines, but Not Area Models, Support Children’s Accuracy and Conceptual Models of Fraction Division. Contemporary Educational Psychology, 58, s. 288‑298. doi:10.1016/j.cedpsych.2019.03.011. 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, s. 691‑697. doi:10.1177/0956797612440101.
Siegler, R.S., Fazio, L.K., Bailey, D.H. & Zhou, X. (2013). Fractions: The New Frontier for Theories of Numerical Development. Trends in Cognitive Sciences, 17, s. 13‑19. doi:10.1016/j.tics.2012.11.004.
Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49, 1994‑2004. doi:10.1037/a0031200
Stafylidou, S. & Vosniadou, S. (2004). The Development of Students’ Understanding of the Numerical Value of Fractions. Learning and Instruction, 14, s. 503‑518. doi:10.1016/j.learninstruc.2004.06.015.
Tian, J., & Siegler, R. S. (2017). Fractions learning in children with mathematics difficulties. Journal of Learning Disabilities, 50, 614‑620. doi:10.1177/0022219416662032
Torbeyns, J., Schneider, M., Xin, Z. & Siegler, R.S. (2015). Bridging the Gap: Fraction Understanding is Central to Mathematics Achievement in Students from Three Different Continents. Learning and Instruction, 37, s. 5‑13. doi:10.1016/j.learninstruc.2014.03.002.
Van Hoof, J., Lijnen, T., Verschaffel, L. & Van Dooren, W. (2013). Are Secondary School Students Still Hampered by the Natural Number Bias? A Reaction Time Study on Fraction Comparison Tasks. Research in Mathematics Education, 15, s. 154‑164. doi:10.1080/14794802.2013.797747.
Van Hoof, J., Verschaffel, L. & Van Dooren, W. (2017). Number Sense in the Transition from Natural to Rational Numbers. British Journal of Educational Psychology, 87, s. 43‑56. doi:10.1111/bjep.12134.
Winsløw, C. (2019). Professor: Vi skylder de danske børn og unge at lære dem matematik. Politiken. https://politiken.dk/debat/debatindlaeg/art7576181/Vi-skylder-de-danske-bOT1orn-og-unge-at-lOT1aere-dem-matematik.
Wong, M. (2010). Equivalent Fractions: Developing a Pathway of Students Acquisition of Knowledge and Understanding. I: L. Sparrow, B. Kissane & C. Hurst (red.), Shaping the Future of Mathematics Education: Proceedings of the 33rd Annual Conference of the Mathematics Education Research Group of Australasia (s. 673‑680). Fremantle: MERGA. https://files.eric.ed.gov/fulltext/ED521016.pdf.
Wong, M. & Evans, D. (2007). Students’ Conceptual Understanding of Equivalent Fractions. I: J. Watson & K. Beswick (red.), Mathematics: Essential Research, Essential Practice (s. 824‑833). Adelaide: MERGA.
Yoshida, H. & Sawano, K. (2002). Overcoming Cognitive Obstacles in Learning Fractions: Equal-Partitioning and Equal-Whole. Japanese Psychological Research, 44, s. 183‑195.
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