How we use what we learn in Math: An integrative account of the development of commutativity.
Main Article Content
Abstract
One of the crucial issues in mathematics development is how children acquire mathematical concepts and procedures. Most researchers now agree that this knowledge develops iteratively (e.g., Resnick, 1992). However, little is known about how well this knowledge is integrated into a more abstract concept and how children come to spontaneously apply such concepts. Expertise research suggests that spontaneously spot and use a principle whenever it applies requires well-integrated conceptual and procedural knowledge. Here, we report a method allowing to asses procedural and conceptual knowledge about the commutative principle in an unobtrusive manner. In two different tasks, procedural and conceptual knowledge of second and third graders as well as adult students were assessed independently and without any hint concerning commutativity. Results show that, even though second graders according to our measures already possessed procedural and conceptual knowledge about commutativity, the knowledge assessed in these two tasks was unrelated. An integrated relation between the two measures first emerged with some of the third graders and was further strengthened for adult students.
Article Details
How to Cite
Haider, H., Eichler, A., Hansen, S., Vaterrodt, B., Gaschler, R., & Frensch, P. (2014). How we use what we learn in Math: An integrative account of the development of commutativity. Frontline Learning Research, 2(1), 1–21. https://doi.org/10.14786/flr.v2i1.37
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References
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Baroody, A. J. & Gannon, K. E. (1984). The development of the commutativity principle and economical addition strategies. Cognition & Instruction, 1, 321-339.
Baroody, A. J., Ginsburg, H. P. & Waxman, B. (1983). Children's use of mathematical structure. Journal for Research in Mathematics Education, 14, 156-168.
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Bisanz, J., Watchorn, R. P. D., Piatt, C., & Sherman, J. (2009). On “Understanding” Children's Developing Use of Inversion. Mathematical Thinking and Learning, 11, 10–24.
Briars, D. & Siegler, R. S. (1984). A featural analysis of preschoolers' counting knowledge. Developmental Psychology, 20, 607-618.
Canobi, K. H. (2004). Individual differences in children's addition and subtraction knowledge. Cognitive Development, 19, 81-93.
Canobi, K. H. (2005). Children's profiles of addition and subtraction understanding. Journal of Experimental Child Psychology, 92, 220-246.
Canobi, K. H., Reeve, R. A. & Pattison, P. E. (1998). The role of conceptual understanding in children's addition problem solving. Developmental Psychology, 34, 882-891.
Canobi, K. H., Reeve, R. A. & Pattison, P. E. (2002). Young children’s understanding of addition concepts. Educational Psychology, 22, 513-532.
Canobi, K. H., Reeve, R. A. & Pattison, P. E. (2003). Patterns of knowledge in children’s addition. Developmental Psychology, 39, 521–534.
Cowan, R. & Renton, M. (1996). Do they know what they are doing? Children's use of economical addition strategies and knowledge of commutativity. Educational Psychology, 16, 407-420.
Farrington-Flint, L., Canobi, K.H., Wood, C. & Faulkner, D. (2010). Children’s patterns of reasoning about reading and addition concepts. British Journal of Developmental Psychology, 28, 427–448
Frensch, P. A. & Haider, H. (2008). Transfer and Expertise: The Search for Identical Elements. In H. L. Roediger, III (Ed.), Cognitive Psychology of Memory. Vol. [2] of Learning and Memory: A Comprehensive Reference (pp. 579-596) Oxford: Elsevier.
Fuson, K. C. (1988). Children's counting and concepts of number. New York, NY: Springer-Verlag Publishing.
Gaschler, R., Vaterrodt, B., Frensch, P. A., Eichler, A. & Haider, H. (under revision). Spontaneous usage of different shortcuts based on the commutativity principle – evidence for knowledge integration in primary school.
Geary, D. C. (2006). Development of Mathematical Understanding. In W. Damon, R. M. Lerner, & N. Eisenberg (Eds.), Handbook of child psychology: Social, emotional, and personality development (Vol. 3, pp. 777–810). John Wiley and Sons.
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Geary, D. C., Brown, S. C. & Samaranayake, V. A. (1991). Cognitive addition: A short longitudinal study of strategy choice and speed-of-processing differences in normal and mathematically disabled children. Developmental Psychology, 27, 787-797.
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, 121–151.
Gelman, R. (1990). First principles organize attention to and learning about relevant data: Number and the animate-inanimate distinction as examples. Cognitive Science, 14, 79-106.
Gelman, R. & Gallistel, C. R. (1978). The child's understanding of number. In. Cambridge, MA: Harvard University Press.
Gelman, R. & Meck, E. (1983). Preschoolers' counting: Principles before skill. Cognition, 13, 343-359.
Gentner, D. & Toupin, C. (1986). Systematicity and surface similarity in the development of analogy. Cognitive Science, 10, 277-300.
Goldman, S. R., Mertz, D. L. & Pellegrino, J. W. (1989). Individual differences in extended practice functions and solution strategies for basic addition facts. Journal of Educational Psychology, 81, 481-496.
Goldstone, R. L., & Sakamoto, Y. (2003). The transfer of abstract principles governing complex adaptive systems. Cognitive Psychology, 46, 414-466.
Hannula, M. M., & Lehtinen, E. (2005). Spontaneous focusing on numerosity and mathematical skills of young children. Learning and Instruction, 15, 237-256.
Hannula, M. M., Lepola, J., & Lehtinen, E. (2010). Spontaneous focusing on numerosity as a domain-specific predictor of arithmetical skills. Journal of Experimental Child Psychology, 107, 394-406.
Haider, H. & Frensch, P. A. (1996). The role of information reduction in skill acquisition. Cognitive Psychology, 30, 304-337.
Haider, H. & Rose, M. (2007). How to investigate insight: A proposal. Methods, 42, 49–57.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introduc- tory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Erlbaum.
Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2008). Learning Theory: The Advantage of Abstract Examples in Learning Math. Science, 320, 454–455.
Koedinger, K. R. & Anderson, J. R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. Cognitive Science, 14, 511-550.
LeFevre, J.-A., Smith-Chant, B. L., Fast, L., Skwarchuk, S.-L., Sargla, E., Arnup, J. S., et al. (2006). What counts as knowing? The development of conceptual and procedural knowledge of counting from kindergarten through grade 2. Journal of Experimental Child Psychology, 93, 285-303.
McMullen, J.A., Hannula-Sormunen, M.M., & Lehtinen, E. (2011). Young children’s spontaneous focusing on quantitative aspects and their verbalizations of their quantitative reasoning. In Ubuz, B. (Ed.). Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education, 3, pp. 217-224. Ankara, Turkey: PME.
McNeil, N. M. (2007). U-shaped development in math: 7-year-olds outperform 9-year-olds on equivalence problems. Developmental Psychology, 43, 687–695.
Prather, R. W., & Alibali, M. W. (2009). The development of arithmetic principle knowledge: How do we know what learners know? Developmental Review, 29, 221–248.
Resnick, L. B. (1992). From protoquantities to operators: Building mathematical competence on a foundation of everyday knowledge. In G. Leinhardt, R. T. Putnam & R. A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 373-429). Hillsdale, NJ, UK: Lawrence Erlbaum Associates, Inc.
Rittle-Johnson, B. & Siegler, R. S. (1998). The relation 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., 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.
Robinson, K. M. & Dubé, A. K. (2009). Children’s understanding of addition and subtraction concepts. Journal of Experimental Child Psychology, 103, 532–545.
Robinson, K. M., & Ninowski, J. E. (2003). Adults’s understanding of inversion concepts: How does performance on addition and subtraction inversion problems compare to performance on multiplication and division inversion problems? Canadian Journal of Experimental Psychology, 57 321-330.
Schneider, M. &. S. E. (2010). The Developmental Relations Between Conceptual and Procedural Knowledge: A Multimethod Approach. Developmental Psychology, 46, 178–192.
Siegler, R. S. (1987). The perils of averaging data over strategies: An example from children's addition. Journal of Experimental Psychology: General, 116, 250-264.
Siegler, R. S. & Araya, R. (2005). A computational model of conscious and unconscious strategy discovery. In R. V. Kail (Ed.), Advances in child development and behavior (vol. 33) (pp. 1-42). Oxford, UK: Elsevier.
Siegler, R. S. & Jenkins, E. (1989). How children discover new strategies. Hillsdale, NJ, UK: Lawrence Erlbaum Associates, Inc.
Siegler, R. S. & Stern, E. (1998). Conscious and unconscious strategy discoveries: A microgenetic analysis. Journal of Experimental Psychology: General, 127, 377-397.
Sloutsky, V. M. & Fisher, A. V. (2008). Attentional learning and flexible induction: How mundane mechanisms give rise to smart behaviors. Child Development, 79, 639-651.
Sophian, C. & Adams, N. (1987). Infants' understanding of numerical transformations. British Journal of Developmental Psychology, 5, 257-264.
Sophian, C., Harley, H. & Manos Martin, C. S. (1995). Relational and representational aspects of early number development. Cognition & Instruction, 13, 253-268.
Star, J.R. (2004, April). The development of flexible procedural knowledge in equation solving. Paper presented at the annual meeting of the American Educational Research Association, San Diego.
Star, J. R. & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31, 280-300.
Starkey, P. & Gelman, R. (1982). The development of addition and subtraction abilities prior to formal schooling in arithmetic. In T. P. Carpenter, J. M. Moser & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 99–116). Hillsdale, NJ: Erlbaum.
Baroody, A. J. (1984). More precisely defining and measuring the order-irrelevance principle. Journal of Experimental Child Psychology, 38, 33-41.
Baroody, A. J., Feil, Y. & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38, 115-131.
Baroody, A. J. & Gannon, K. E. (1984). The development of the commutativity principle and economical addition strategies. Cognition & Instruction, 1, 321-339.
Baroody, A. J., Ginsburg, H. P. & Waxman, B. (1983). Children's use of mathematical structure. Journal for Research in Mathematics Education, 14, 156-168.
Bisanz, J., & LeFevre, J. (1992). Understanding elementary mathematics. In J. D. Campbell (Ed.) , The nature and origins of mathematical skills (pp. 113-136). Oxford England: North-Holland.
Bisanz, J., Watchorn, R. P. D., Piatt, C., & Sherman, J. (2009). On “Understanding” Children's Developing Use of Inversion. Mathematical Thinking and Learning, 11, 10–24.
Briars, D. & Siegler, R. S. (1984). A featural analysis of preschoolers' counting knowledge. Developmental Psychology, 20, 607-618.
Canobi, K. H. (2004). Individual differences in children's addition and subtraction knowledge. Cognitive Development, 19, 81-93.
Canobi, K. H. (2005). Children's profiles of addition and subtraction understanding. Journal of Experimental Child Psychology, 92, 220-246.
Canobi, K. H., Reeve, R. A. & Pattison, P. E. (1998). The role of conceptual understanding in children's addition problem solving. Developmental Psychology, 34, 882-891.
Canobi, K. H., Reeve, R. A. & Pattison, P. E. (2002). Young children’s understanding of addition concepts. Educational Psychology, 22, 513-532.
Canobi, K. H., Reeve, R. A. & Pattison, P. E. (2003). Patterns of knowledge in children’s addition. Developmental Psychology, 39, 521–534.
Cowan, R. & Renton, M. (1996). Do they know what they are doing? Children's use of economical addition strategies and knowledge of commutativity. Educational Psychology, 16, 407-420.
Farrington-Flint, L., Canobi, K.H., Wood, C. & Faulkner, D. (2010). Children’s patterns of reasoning about reading and addition concepts. British Journal of Developmental Psychology, 28, 427–448
Frensch, P. A. & Haider, H. (2008). Transfer and Expertise: The Search for Identical Elements. In H. L. Roediger, III (Ed.), Cognitive Psychology of Memory. Vol. [2] of Learning and Memory: A Comprehensive Reference (pp. 579-596) Oxford: Elsevier.
Fuson, K. C. (1988). Children's counting and concepts of number. New York, NY: Springer-Verlag Publishing.
Gaschler, R., Vaterrodt, B., Frensch, P. A., Eichler, A. & Haider, H. (under revision). Spontaneous usage of different shortcuts based on the commutativity principle – evidence for knowledge integration in primary school.
Geary, D. C. (2006). Development of Mathematical Understanding. In W. Damon, R. M. Lerner, & N. Eisenberg (Eds.), Handbook of child psychology: Social, emotional, and personality development (Vol. 3, pp. 777–810). John Wiley and Sons.
Geary, D. C., Boykin, A. W., Embretson, S., Reyna, V., Siegler, R., Berch, D. B., et al. (2008). Report of the task group on learning processes. In National mathematics advisory panel, reports of the task groups and subcommittees (pp. 4-1–4-211).
Geary, D. C., Brown, S. C. & Samaranayake, V. A. (1991). Cognitive addition: A short longitudinal study of strategy choice and speed-of-processing differences in normal and mathematically disabled children. Developmental Psychology, 27, 787-797.
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, 121–151.
Gelman, R. (1990). First principles organize attention to and learning about relevant data: Number and the animate-inanimate distinction as examples. Cognitive Science, 14, 79-106.
Gelman, R. & Gallistel, C. R. (1978). The child's understanding of number. In. Cambridge, MA: Harvard University Press.
Gelman, R. & Meck, E. (1983). Preschoolers' counting: Principles before skill. Cognition, 13, 343-359.
Gentner, D. & Toupin, C. (1986). Systematicity and surface similarity in the development of analogy. Cognitive Science, 10, 277-300.
Goldman, S. R., Mertz, D. L. & Pellegrino, J. W. (1989). Individual differences in extended practice functions and solution strategies for basic addition facts. Journal of Educational Psychology, 81, 481-496.
Goldstone, R. L., & Sakamoto, Y. (2003). The transfer of abstract principles governing complex adaptive systems. Cognitive Psychology, 46, 414-466.
Hannula, M. M., & Lehtinen, E. (2005). Spontaneous focusing on numerosity and mathematical skills of young children. Learning and Instruction, 15, 237-256.
Hannula, M. M., Lepola, J., & Lehtinen, E. (2010). Spontaneous focusing on numerosity as a domain-specific predictor of arithmetical skills. Journal of Experimental Child Psychology, 107, 394-406.
Haider, H. & Frensch, P. A. (1996). The role of information reduction in skill acquisition. Cognitive Psychology, 30, 304-337.
Haider, H. & Rose, M. (2007). How to investigate insight: A proposal. Methods, 42, 49–57.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introduc- tory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Erlbaum.
Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2008). Learning Theory: The Advantage of Abstract Examples in Learning Math. Science, 320, 454–455.
Koedinger, K. R. & Anderson, J. R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. Cognitive Science, 14, 511-550.
LeFevre, J.-A., Smith-Chant, B. L., Fast, L., Skwarchuk, S.-L., Sargla, E., Arnup, J. S., et al. (2006). What counts as knowing? The development of conceptual and procedural knowledge of counting from kindergarten through grade 2. Journal of Experimental Child Psychology, 93, 285-303.
McMullen, J.A., Hannula-Sormunen, M.M., & Lehtinen, E. (2011). Young children’s spontaneous focusing on quantitative aspects and their verbalizations of their quantitative reasoning. In Ubuz, B. (Ed.). Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education, 3, pp. 217-224. Ankara, Turkey: PME.
McNeil, N. M. (2007). U-shaped development in math: 7-year-olds outperform 9-year-olds on equivalence problems. Developmental Psychology, 43, 687–695.
Prather, R. W., & Alibali, M. W. (2009). The development of arithmetic principle knowledge: How do we know what learners know? Developmental Review, 29, 221–248.
Resnick, L. B. (1992). From protoquantities to operators: Building mathematical competence on a foundation of everyday knowledge. In G. Leinhardt, R. T. Putnam & R. A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 373-429). Hillsdale, NJ, UK: Lawrence Erlbaum Associates, Inc.
Rittle-Johnson, B. & Siegler, R. S. (1998). The relation 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., 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.
Robinson, K. M. & Dubé, A. K. (2009). Children’s understanding of addition and subtraction concepts. Journal of Experimental Child Psychology, 103, 532–545.
Robinson, K. M., & Ninowski, J. E. (2003). Adults’s understanding of inversion concepts: How does performance on addition and subtraction inversion problems compare to performance on multiplication and division inversion problems? Canadian Journal of Experimental Psychology, 57 321-330.
Schneider, M. &. S. E. (2010). The Developmental Relations Between Conceptual and Procedural Knowledge: A Multimethod Approach. Developmental Psychology, 46, 178–192.
Siegler, R. S. (1987). The perils of averaging data over strategies: An example from children's addition. Journal of Experimental Psychology: General, 116, 250-264.
Siegler, R. S. & Araya, R. (2005). A computational model of conscious and unconscious strategy discovery. In R. V. Kail (Ed.), Advances in child development and behavior (vol. 33) (pp. 1-42). Oxford, UK: Elsevier.
Siegler, R. S. & Jenkins, E. (1989). How children discover new strategies. Hillsdale, NJ, UK: Lawrence Erlbaum Associates, Inc.
Siegler, R. S. & Stern, E. (1998). Conscious and unconscious strategy discoveries: A microgenetic analysis. Journal of Experimental Psychology: General, 127, 377-397.
Sloutsky, V. M. & Fisher, A. V. (2008). Attentional learning and flexible induction: How mundane mechanisms give rise to smart behaviors. Child Development, 79, 639-651.
Sophian, C. & Adams, N. (1987). Infants' understanding of numerical transformations. British Journal of Developmental Psychology, 5, 257-264.
Sophian, C., Harley, H. & Manos Martin, C. S. (1995). Relational and representational aspects of early number development. Cognition & Instruction, 13, 253-268.
Star, J.R. (2004, April). The development of flexible procedural knowledge in equation solving. Paper presented at the annual meeting of the American Educational Research Association, San Diego.
Star, J. R. & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31, 280-300.
Starkey, P. & Gelman, R. (1982). The development of addition and subtraction abilities prior to formal schooling in arithmetic. In T. P. Carpenter, J. M. Moser & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 99–116). Hillsdale, NJ: Erlbaum.