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International Electronic Journal of Mathematics Education

International Electronic Journal of Mathematics Education
Individual Differences in Conceptual and Procedural Fraction Knowledge: What Makes the Difference and What Does it Look Like?
Katja Lenz 1 * , Gerald Wittmann 1
1 University of Education Freiburg, Kunzenweg 21, 79117 Freiburg, GERMANY
* Corresponding Author
APA
In-text citation: (Lenz & Wittmann, 2021)
Reference: Lenz, K., & Wittmann, G. (2021). Individual Differences in Conceptual and Procedural Fraction Knowledge: What Makes the Difference and What Does it Look Like?. International Electronic Journal of Mathematics Education, 16(1), em0615. https://doi.org/10.29333/iejme/9282
AMA
In-text citation: (1), (2), (3), etc.
Reference: Lenz K, Wittmann G. Individual Differences in Conceptual and Procedural Fraction Knowledge: What Makes the Difference and What Does it Look Like?. INT ELECT J MATH ED. 2021;16(1), em0615. https://doi.org/10.29333/iejme/9282
Chicago
In-text citation: (Lenz and Wittmann, 2021)
Reference: Lenz, Katja, and Gerald Wittmann. "Individual Differences in Conceptual and Procedural Fraction Knowledge: What Makes the Difference and What Does it Look Like?". International Electronic Journal of Mathematics Education 2021 16 no. 1 (2021): em0615. https://doi.org/10.29333/iejme/9282
Harvard
In-text citation: (Lenz and Wittmann, 2021)
Reference: Lenz, K., and Wittmann, G. (2021). Individual Differences in Conceptual and Procedural Fraction Knowledge: What Makes the Difference and What Does it Look Like?. International Electronic Journal of Mathematics Education, 16(1), em0615. https://doi.org/10.29333/iejme/9282
MLA
In-text citation: (Lenz and Wittmann, 2021)
Reference: Lenz, Katja et al. "Individual Differences in Conceptual and Procedural Fraction Knowledge: What Makes the Difference and What Does it Look Like?". International Electronic Journal of Mathematics Education, vol. 16, no. 1, 2021, em0615. https://doi.org/10.29333/iejme/9282
Vancouver
In-text citation: (1), (2), (3), etc.
Reference: Lenz K, Wittmann G. Individual Differences in Conceptual and Procedural Fraction Knowledge: What Makes the Difference and What Does it Look Like?. INT ELECT J MATH ED. 2021;16(1):em0615. https://doi.org/10.29333/iejme/9282

Abstract

There is a general consensus that both conceptual and procedural knowledge are essential for students’ mathematical development. A common argument is that differences in mathematical performance are caused by differences in conceptual and procedural knowledge. Therefore, it is important to investigate to what extent such differences in conceptual and procedural knowledge are empirically evident at the level of individual students. Accordingly, the aim of the present study is to describe individual differences in conceptual and procedural knowledge using the example of fractions and to analyze their relationship to the covariates grade level, school type, school, class, gender, and general cognitive abilities. Data from 377 students in grades 8 and 9 from 18 classes at four schools in Germany was examined. A hierarchical cluster analysis showed five clusters which reflected individual differences in conceptual and procedural knowledge. The clusters were characterized by (a) equal strengths in conceptual and procedural knowledge, (b) relative strengths in procedural knowledge compared to conceptual knowledge, (c) relative weaknesses in procedural knowledge compared to conceptual knowledge. Cluster membership was not related to gender or grade level, whereas the school type, school, and grade level were relevant for cluster membership. A stronger correlation between conceptual knowledge and general cognitive abilities could only be confirmed to a limited extent.

Keywords

fractions individual differences conceptual knowledge procedural knowledge cluster analysis

References

  • Anderson, J. R., Funke, J., Neuser-von Oettingen, K., & Plata, G. (Eds.) (2013). Kognitive Psychologie [Cognitive Psychology]. Berlin: Springer. https://doi.org/10.1007/978-3-642-37392-3
  • Anderson, L. W., & Krathwohl, D. R. (Eds.) (2001). A taxonomy for learning, teaching, and assessing. A re-vision of Bloom’s taxonomy of educational objectives. New York: Longman
  • Baroody, A. J., & Gannon, K. E. (1984). The development of the commutativity principle and economical addition strategies. Cognition and Instruction, 1, 321-339. https://doi.org/10.1207/s1532690xci0103_3
  • Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of conceptual and procedural knowledge. Journal for Research in Mathematics Education, 38(2), 115-131. https://doi.org/10.2307/30034952
  • Bempeni, M., Poulopoulou, S., Tsiplaki, I., & Vamvakoussi, X. (2018). Individual differences in fractions’ conceptual and procedural knowledge: What about older students? Proceedings of PME 42, Umea, Schweden: PME.
  • Bortz, J., & Schuster, C. (2010). Statistik für Human- und Sozialwissenschaftler [Statistics for human and social scientists]. Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-642-12770-0
  • Briars, D., & Siegler, R. S. (1984). A featural analysis of preschoolers’ counting knowledge. Developmental Psychology, 20, 607-618. https://doi.org/10.1037/0012-1649.20.4.607
  • Bühl, A. (2016). SPSS 23. Einführung in die moderne Datenanalyse [Introduction to modern data analysis]. Hallbergmoos: Pearson.
  • Byrnes, J. P., & Wasik, B. A. (1991). Role of conceptual knowledge in mathematical procedural learning. Developmental psychology, 27, 777-786. https://doi.org/10.1037/0012-1649.27.5.777
  • Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2013). Applied multiple regression/correlation analysis for the behavioral sciences. New York, London: Routledge. https://doi.org/10.4324/9780203774441
  • Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34(4), 344-377. https://doi.org/10.1016/j.dr.2014.10.001
  • Döring, N., & Bortz, J. (2016). Forschungsmethoden und Evaluation in den Sozial- und Humanwissenschaften [Research methods and evaluation in the social and human sciences]. Berlin, Heidelberg, Germany: Springer. https://doi.org/10.1007/978-3-642-41089-5
  • Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Hamlett, C. L., & Cirino, P. T. (2013). Improving at-risk learners’ understanding of fractions. Journal of Educational Psychology, 105(3), 683-700. https://doi.org/10.1037/a0032446
  • Gabriel, F., Coche, F., Szucs, D., Carette, V., Rey, B., & Content, A. (2013). A componential view of children’s difficulties in learning fractions. Frontiers in Psychology, 4, 715. https://doi.org/10.3389/fpsyg.2013.00715
  • Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities 37(1), 4-15. https://doi.org/10.1177/00222194040370010201
  • Geary, D. C., Boykin, A. W., Embretson, S., Reyna, V., Siegler, R., Berch D. B., & Graban, J. (2008). Report of the task group on learning processes. In U.S. Department of Education (Ed.), The final report of the National Mathematics Advisory Panel. Washington D.C.: U.S. Department of Education.
  • Gilmore C. K., & Papadatou-Pastou M. (2009). Patterns of individual differences in conceptual understanding and arithmetical skill: A meta-analysis. Mathematical Thinking and Learning, 11(1-2), 25-40. https://doi.org/10.1080/10986060802583923
  • Hallett, D., Nunes, T., & Bryant, P. (2010). Individual differences in conceptual and procedural knowledge when learning fractions. Journal of Educational Psychology, 102, 395-406. https://doi.org/10.1037/a0017486
  • Hallett, D., Nunes, T., Bryant, P., & Thorpe, C. M. (2012). Individual differences in conceptual and procedural fraction understanding: The role of abilities and school experience. Journal of Experimental Child Psychology, 113(4), 469-486. https://doi.org/10.1016/j.jecp.2012.07.009
  • Hecht, S. A., & Vagi, K. J. (2010). Sources of group and individual differences in emerging fraction skills. Journal of Educational Psychology, 102, 843-859. https://doi.org/10.1037/a0019824
  • Hecht, S. A., & Vagi, K. J. (2012). Patterns of strengths and weaknesses in children’s knowledge about fractions. Journal of Experimental Child Psychology, 111(2), 212-229. https://doi.org/10.1016/j.jecp.2011.08.012
  • Hecht, S. A., Close, L., & Santisi, M. (2003). Sources of individual differences in fraction skills. Journal of Experimental Child Psychology, 86, 277-302. https://doi.org/10.1016/j.jecp.2003.08.003
  • Heller, K. A., & Perleth, C. (2000). KFT 4-12+ R Kognitiver Fähigkeitstest für 4. bis 12. Klassen. Manual [KFT 4-12+ R Cognitive Abilities Test4-12. Manual]. Göttingen: Hogrefe.
  • Herden, G., & Pallack, A. (2000). Zusammenhänge zwischen verschiedenen Fehlerstrategien in der Bruchrechnung. Empirische Erhebung über 244 SchülerInnen der Klassen sieben von Gymnasien [Student errors in working with fractions. Empirical investigation among 244 students of grade 7]. Journal für Mathematik-Didaktik, 21, 259-279. https://doi.org/10.1007/BF03338921
  • Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge. The case of mathematics (pp. 1-27). Hillsdale, NJ: Erlbaum.
  • Hiebert, J., & Wearne, D. (1996). Instruction, Understanding, and Skill in Multidigit Addition and Subtraction, Cognition and Instruction, 14(3), 251-283. https://doi.org/10.1207/s1532690xci1403_1
  • Holling, H., Preckel, F., & Vock, M. (2004). Intelligenzdiagnostik. Göttingen: Hogrefe.
  • Jordan, N. C., Hansen, N., Fuchs, L. S., Siegler, R. S., Gersten, R., & Micklos, D. (2013). Developmental predictors of fraction concepts and procedures. Journal of experimental child psychology, 116(1), 45-58. https://doi.org/10.1016/j.jecp.2013.02.001
  • Kerslake, D. (1986). Fractions: Children’s strategies and errors: A report of the strategies and errors in Secondary Mathematics Project. Windsor, UK: NFER-Nelson.
  • Kieren, T. E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. A. Lesh (Eds.), Number and measurement (pp. 101-150). Columbus, OH: Eric.
  • Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) (2001). Adding it up. Helping children learn mathematics. Washington, DC: National Academy Press.
  • Lenz, K., Dreher, A., Holzäpfel, L., & Wittmann, G. (2019a). Are conceptual and procedural knowledge empirically separable? - The case of fractions. British Journal of Educational Psychology. https://doi:10.1111/bjep.12333
  • Lenz, K., Holzäpfel, L., & Wittmann, G. (2019b). Aufgaben als Lerngelegenheiten für konzeptuelles und prozedurales Wissen zu Brüchen - Eine vergleichende Schulbuchanalyse [Tasks as learning opportunities for conceptual and procedural fraction knowledge -A comparative textbook analysis]. mathematica didactica 42(2), 105-121.
  • Lienert, G. A. (1969). Testaufbau und Testanalyse [Test construction and test analysis]. Weinheim: Beltz.
  • Lortie-Forgues, H., Tian, J., & Siegler, R. (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
  • Malle, G. (2004). Grundvorstellungen zu Bruchzahlen. Mathematik lehren, (123), 4-8.
  • Padberg, F., & Wartha, S. (2017). Didaktik der Bruchrechnung. Gemeine Brüche - Dezimalbrüche [Didactics of fractions. Common fractions - decimal fractions] Berlin, Germany: Springer Spektrum. https://doi.org/10.1007/978-3-662-52969-0
  • Peck, D., & Jencks, S. (1981). Conceptual issues in the teaching and learning of fractions. Journal for Research in Mathematics Education, 12, 339-348.
  • Prediger, S., Barzel, B., Leuders, T., & Hußmann, S. (2011). Systematisieren und Sichern. Nachhaltiges Lernen durch aktives Ordnen [Systemizing and consolidating knowledge. Lasting learning through active organization]. Mathematik lehren, 164, 2-9.
  • Reinhold, F. (2019). Wirksamkeit von Tablet-PCs bei der Entwicklung des Bruchzahlbegriffs aus mathematikdidaktischer und psychologischer Perspektive. Eine empirische Studie in Jahrgangsstufe 6. [Effectiveness of tablet PCs in the development of the fractional number concept from a mathematics educational and psychological perspective]. Wiesbaden: Springer Spektrum. https://doi.org/10.1007/978-3-658-23924-4
  • Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. In R. C. Kadosh & A. Dowker (Eds.), The Oxford handbook of numerical cognition (pp. 1102-1118). Oxford, UK, New York, NY: Oxford University Press. https://doi.org/10.1093/oxfordhb/9780199642342.013.014
  • Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics. An iterative process. Journal of Educational Psychology, 93, 346-362. https://doi.org/10.1037/0022-0663.93.2.346
  • Schneider, M., & Stern, E. (2010). The developmental relations between conceptual and procedural knowledge: A multimethod approach. Developmental Psychology, 46(1), 178-192. https://doi.org/10.1037/a0016701
  • Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations between conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47, 1525-1538. https://doi.org/10.1037/a0024997
  • Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107, 909-918. https://doi.org/10.1037/edu0000025
  • Son, J.-W., & Senk, S. L. (2010). How reform curricula in the USA and Korea present multiplication and division of fractions. Educational Studies in Mathematics, 74(2), 117-142. https://doi.org/10.1007/s10649-010-9229-6
  • Thorndike, R. L., & Hagen, E.P. (1971). Cognitive Abilities Test. Boston: Houghton-Mifflin.
  • Vamvakoussi, X., Bempeni, M., Poulopoulou, S., & Tsiplaki, I. (2019). Theoretical and methodological issues in the study of conceptual and procedural knowledge: Reflections on a series of studies on Greek secondary students’ knowledge of fractions. Educational Journal of the University of Patras UNESCO Chair, 6(2), 82-96.
  • Wiedenbeck, M., & Züll, C. (2010). Clusteranalyse. In C. Wolf & H. Best (Eds.), Handbuch der sozialwissenschaftlichen Datenanalyse [Manual of social science data analysis] (p. 525-552). Wiesbaden: VS Verlag für Sozialwissenschaften. https://doi.org/10.1007/978-3-531-92038-2_21
  • Yang, D., Reys, R., & Wu, L. (2010). Comparing the development of fractions in the fifth- and sixth-graders’ textbooks of Singapore, Taiwan, and the USA. School, Science and Mathematics 110(3), 118-127. https://doi.org/10.1111/j.1949-8594.2010.00015.x

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