Individual differences in students’ knowing and learning about fractions: Evidence from an in-depth qualitative study
Main Article Content
Abstract
We present the results of an in-depth qualitative study that examined ninth graders’ conceptual and procedural knowledge of fractions as well as their approach to mathematics learning, in particular fraction learning. We traced individual differences, even extreme, in the way that students combine the two kinds of knowledge. We also provide preliminary evidence indicating that students with strong conceptual fraction knowledge adopt a deep approach to mathematics learning (associated with the intention to understand), whereas students with poor conceptual fraction knowledge adopt a superficial approach (associated with the intention to reproduce). These findings suggest that students differ in the way they reason and learn about fractions in systematic ways and could be used to inform future quantitative studies.
Article Details
How to Cite
Bempeni, M., & Vamvakoussi, X. (2015). Individual differences in students’ knowing and learning about fractions: Evidence from an in-depth qualitative study. Frontline Learning Research, 3(1), 18–35. https://doi.org/10.14786/flr.v3i1.132
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References
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Clarke, D. M., & Roche Α. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72(1), 127-138. doi: 10.1007/s10649-009-9198-9
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Entwisle, N. & McCune V. (2004). The conceptual bases of study strategy inventories. Educational Psychology Review, 16, 325-345. doi: 10.1007/s10648-004-0003-0
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Gilmore, C. K., & Bryant, P. (2008). Can children construct inverse relations in arithmetic? Evidence for individual differences in the development of conceptual understanding and computational skill. British Journal of Developmental Psychology, 26, 301–316. doi: 10.1348/026151007X236007
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Hallett, D., Nunes, T., & Bryant, P. (2010). Individual differences in conceptual and procedural knowledge when learning fractions. Journal of Educational Psychology, 102, 395–406. doi: 10.1037/a0017486
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Peck, D. M., & Jencks, S. M. (1981). Conceptual issues in the teaching and learning of fractions. Journal for Research in Mathematics Education, 12(5), 339-348.
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Rittle-Johnson, B., & Siegler, R. S. (1998). The relations between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75-110). East Sussex, UK: Psychology Press.
Rittle-Johnson, & B., Schneider, M. (in press). Developing conceptual and procedural knowledge of mathematics. In R. Kadosh & A. Dowker (Eds), Oxford Handbook of Numerical Cognition. Oxford Press.
Schneider, M., & Stern, E. (2010). The developmental relations between conceptual and procedural knowledge: A multimethod approach, Developmental Psychology, 46, 178-192. doi: 10.1037/a0016701
Schneider. M., Rittle-Johnson B, & Star J. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Journal of Developmental Psychology, 47, 1525-1538. doi: 10.1037/a0024997
Siegler, R, Pyke, A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49, 1994–2004. doi: 10.1037/a0031200
Silver, E. A. (1986). Using conceptual and procedural knowledge: A focus on relationships. In J. Hiebert and P. Lefevre (Eds.), Conceptual and procedural knowledge: The case of mathematics. New Jersey: Erlbaum Associates.
Smith, J. (1995). Competent reasoning with rational numbers. Cognition and Instruction, 13(1), 3-50. doi: 10.1207/s1532690xci1301_1
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36. doi: 10.1007/BF00302715
Stathopoulou, C., & Vosniadou, S. (2007). Conceptual change in physics and physics-related epistemological beliefs: A relationship under scrutiny. In S. Vosniadou, A. Baltas, & X. Vamvakoussi (Eds.), Reframing the conceptual change approach in learning and instruction (pp. 145-164). Oxford, UK: Elsevier. doi: 10.1016/j.cedpsych.2005.12.002
Vamvakoussi X. & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ reasoning about rational numbers and their notation. Cognition & Instruction, 28(2), 181-209. doi: 10.1080/07370001003676603
Yang. D., C., Reys, R., & Reys, B. (2007). Number sense strategies used by pre-service teachers in Taiwan. International Journal of Science and Mathematics Education, 7(2), 383-403. doi: 10.1007/s10763-007-9124-5
Canobi, K. H. (2004). Individual differences in children's addition and subtraction knowledge. Cognitive Development, 19, 81-93. doi: 10.1016/j.cogdev.2003.10.001
Canobi, K. H., Reeve, R. A., & Pattison, P. E. (2003). Patterns of knowledge in children’ s addition. Developmental Psychology, 39, 521-534. doi: 10.1037/0012-1649.39.3.521
Clarke, D. M., & Roche Α. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72(1), 127-138. doi: 10.1007/s10649-009-9198-9
Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (ed.) Advanced Mathematical Thinking (pp. 95-123). Kluwer: Dordrecht, doi: 10.1007/0-306-47203-1_7
Entwisle, N. & McCune V. (2004). The conceptual bases of study strategy inventories. Educational Psychology Review, 16, 325-345. doi: 10.1007/s10648-004-0003-0
Faulkenberry, T. J. (2013). The conceptual/procedural distinction belongs to strategies, not tasks: A comment on Gabriel et al.(2013). Frontiers in Psychology, 4, 1-2. doi: 10.3389/fpsyg.2013.00820
Gilmore, C. K., & Bryant, P. (2006). Individual differences in children’s understanding of inversion and arithmetical skill. British Journal of Educational Psychology, 76, 309–331. doi: 10.1348/000709905X39125
Gilmore, C. K., & Bryant, P. (2008). Can children construct inverse relations in arithmetic? Evidence for individual differences in the development of conceptual understanding and computational skill. British Journal of Developmental Psychology, 26, 301–316. doi: 10.1348/026151007X236007
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, 25-40. doi: 10.1080/10986060802583923
Gray, E., Tall. D. (1994). Duality, Ambiguity, and Flexibility: A Proceptual view of simple Arithmetic. Journal for Research in Mathematics Education 25(2), 407-428.
Hallett, D., Nunes, T., & Bryant, P. (2010). Individual differences in conceptual and procedural knowledge when learning fractions. Journal of Educational Psychology, 102, 395–406. doi: 10.1037/a0017486
Haapasalo, L., & Kadijevich, D. (2000). Two types of mathematical knowledge and their relation. JMD -- Journal for Mathematic-Didaktik, 21, 139-157. doi: 10.1007/BF03338914
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, 469-486. doi: 10.1016/j.jecp.2012.07.009
Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition & Instruction, 14(3), 251-283. doi: 10.1207/s1532690xci1403_1
Kerslake, D. (1986). Fractions: Children’s strategies and errors: A report of the strategies and errors in Secondary Mathematics Project. Windsor, UK: NFER–Nelson.
McIntosh, A., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12, 2-8.
McMullen, J., Laakkonen, E., Hannula-Sormumen M., & Lehtinen E. (in press). Modeling developmental trajectories of rational number. Learning and Instruction. doi:10.1016/j.learninstruc.2013.12.004.
Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30, 122-147.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
Peck, D. M., & Jencks, S. M. (1981). Conceptual issues in the teaching and learning of fractions. Journal for Research in Mathematics Education, 12(5), 339-348.
Resnick, L. B. (1982). Syntax and semantics in learning to subtract. In T. P. Carpenter, J. M. Moser & T. A. Romburg (Eds.), Addition and subtraction: A cognitive perspective (pp. 136–155). Hillsdale, NJ: Erlbaum.
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. doi: 10.1037/0022-0663.93.2.346
Rittle-Johnson, B., & Siegler, R. S. (1998). The relations between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75-110). East Sussex, UK: Psychology Press.
Rittle-Johnson, & B., Schneider, M. (in press). Developing conceptual and procedural knowledge of mathematics. In R. Kadosh & A. Dowker (Eds), Oxford Handbook of Numerical Cognition. Oxford Press.
Schneider, M., & Stern, E. (2010). The developmental relations between conceptual and procedural knowledge: A multimethod approach, Developmental Psychology, 46, 178-192. doi: 10.1037/a0016701
Schneider. M., Rittle-Johnson B, & Star J. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Journal of Developmental Psychology, 47, 1525-1538. doi: 10.1037/a0024997
Siegler, R, Pyke, A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49, 1994–2004. doi: 10.1037/a0031200
Silver, E. A. (1986). Using conceptual and procedural knowledge: A focus on relationships. In J. Hiebert and P. Lefevre (Eds.), Conceptual and procedural knowledge: The case of mathematics. New Jersey: Erlbaum Associates.
Smith, J. (1995). Competent reasoning with rational numbers. Cognition and Instruction, 13(1), 3-50. doi: 10.1207/s1532690xci1301_1
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36. doi: 10.1007/BF00302715
Stathopoulou, C., & Vosniadou, S. (2007). Conceptual change in physics and physics-related epistemological beliefs: A relationship under scrutiny. In S. Vosniadou, A. Baltas, & X. Vamvakoussi (Eds.), Reframing the conceptual change approach in learning and instruction (pp. 145-164). Oxford, UK: Elsevier. doi: 10.1016/j.cedpsych.2005.12.002
Vamvakoussi X. & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ reasoning about rational numbers and their notation. Cognition & Instruction, 28(2), 181-209. doi: 10.1080/07370001003676603
Yang. D., C., Reys, R., & Reys, B. (2007). Number sense strategies used by pre-service teachers in Taiwan. International Journal of Science and Mathematics Education, 7(2), 383-403. doi: 10.1007/s10763-007-9124-5