Children's understanding of the inverse relation between multiplication and division

Baroody, 1994, An evaluation of evidence supporting fact-retrieval models, Learning and Individual Differences, 6, 1, 10.1016/1041-6080(94)90013-2 Baroody, 2003, The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge, 1 Baroody, 1986, The relationship between initial meaningful and mechanical knowledge of arithmetic, 75 Baroody, 1983, Children's use of mathematical structure, Journal for Research in Mathematics Education, 14, 156, 10.2307/748379 Baroody, 2007, Preschoolers’ understanding of the addition–subtraction inversion principle: A Taiwanese sample, Mathematical Thinking and Learning, 9, 131, 10.1080/10986060709336813 Berch, 2005, Making sense of the number sense: Implications for children with mathematical disabilities, Journal of Learning Disabilities, 38, 333, 10.1177/00222194050380040901 Bisanz, 1990, Strategic and nonstrategic processing in the development of mathematical cognition, 213 Bisanz, 1992, Understanding elementary mathematics, 113 Bryant, 1999, Children's understanding of the relation between addition and subtraction: Inversion, identity, and decomposition, Journal of Experimental Child Psychology, 74, 194, 10.1006/jecp.1999.2517 Canobi, 2004, Individual differences in children's addition and subtraction knowledge, Cognitive Development, 19, 81, 10.1016/j.cogdev.2003.10.001 Cowan, 1996, Even more precisely assessing children's understanding of the order-irrelevance principle, Journal of Experimental Child Psychology, 62, 84, 10.1006/jecp.1996.0023 Cowan, 1996, Do they know what they are doing? Children's use of economical addition strategies and knowledge of commutativity, Educational Psychology, 16, 407, 10.1080/0144341960160405 Dubé, A. K., & Robinson, K. M. (2009). Skills underlying inversion shortcut use: The role of analogical reasoning and working memory. Manuscript submitted for publication. Geary, 1994 Gersten, 2005, Early identification and interventions for students with mathematical disabilities, Journal of Learning Disabilities, 38, 293, 10.1177/00222194050380040301 Gilmore, 2006, Investigating children's understanding of inversion using the missing number paradigm, Cognitive Development, 21, 301, 10.1016/j.cogdev.2006.03.007 Gilmore, 2008, Children's understanding of the relationship between addition and subtraction, Cognition, 107, 932, 10.1016/j.cognition.2007.12.007 Heirdsfield, 2004, Factors affecting the process of proficient mental addition and subtraction: Case studies of flexible and inflexible computers, The Journal of Mathematical Behavior, 23, 443, 10.1016/j.jmathb.2004.09.005 Klein, 2000, Preschoolers doing arithmetic: The concepts are willing but the working memory is weak, Canadian Journal of Experimental Psychology, 54, 105, 10.1037/h0087333 LeFevre, 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, 10.1016/j.jecp.2005.11.002 MacDonald, 2000, Type I error rate comparisons of post hoc procedures for I x J chi-square tables, Educational and Psychological Measurement, 60, 735, 10.1177/00131640021970871 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Piaget, 2001, The inversion of arithmetic operations, 69 Rasmussen, 2003, Use of the mathematical principle of inversion in young children, Journal of Experimental Child Psychology, 85, 89, 10.1016/S0022-0965(03)00031-6 Rittle-Johnson, 2001, Developing conceptual understanding and procedural skill in mathematics: An iterative process, Journal of Educational Psychology, 93, 346, 10.1037/0022-0663.93.2.346 Robinson, 2001, The validity of verbal reports in children's subtraction, Journal of Educational Psychology, 93, 211, 10.1037/0022-0663.93.1.211 Robinson, K. M., & Dubé, A. K. (in press). A microgenetic study of the multiplication and division inversion concept. Canadian Journal of Experimental Psychology. Robinson, 2003, Adults’ 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, 10.1037/h0087435 Robinson, 2006, Children's understanding of the arithmetic concepts of inversion and associativity, Journal of Experimental Child Psychology, 94, 349, 10.1016/j.jecp.2006.03.004 Sherman, 2007, Evidence for use of mathematical inversion by three-year-old children, Journal of Cognition and Development, 8, 333, 10.1080/15248370701446798 Siegler, 1987, The perils of averaging data over strategies: An example from children's addition, Journal of Experimental Psychology: General, 116, 250, 10.1037/0096-3445.116.3.250 Siegler, 1989, Hazards of mental chronometry: An example from children's subtraction, Journal of Educational Psychology, 81, 497, 10.1037/0022-0663.81.4.497 Siegler, 2005, A computational model of conscious and unconscious strategy discovery, Vol. 33, 1 Siegler, 1995, Variation, selection, and cognitive change, 31 Siegler, 1984, Strategy choices in addition and subtraction: How do children know what to do?, 229 Siegler, 1998, Conscious and unconscious strategy discoveries: A microgenetic analysis, Journal of Experimental Psychology: General, 127, 377, 10.1037/0096-3445.127.4.377 Squire, 2004, Does the cue help? Children's understanding of multiplicative concepts in different problem contexts, British Journal of Educational Psychology, 74, 515, 10.1348/0007099042376364 Starkey, 1982, The development of addition and subtraction abilities prior to formal schooling in arithmetic, 99 Vilette, 2002, Do young children grasp the inverse relationship between addition and subtraction?, Cognitive Development, 17, 1365, 10.1016/S0885-2014(02)00125-9