Abstract
Knowledge representations that result from practicing problem solving can be expected to differ from knowledge representations that emerge from explicit verbalizing of principles and rules. We examined the degree to which the two types of learning improve problem-solving knowledge and verbal explanation knowledge in classroom instruction. We presented algebraic addition and multiplication problems to 153 sixth graders randomly assigned to two conditions. Students in the explicit learning condition had to verbally compare contrasted algebra problems. Students in the implicit learning condition had to generate and solve new problems. On three follow-up tests over 10 weeks, students in the explicit learning condition exhibited better problem-solving knowledge than students in the implicit learning condition, as well as some advantages in verbal concept knowledge. Implicit learning showed some advantages on not directly taught but incidentally learned aspects. Overall, this outcome favors the explicit learning of concepts. Explicit comparison fostered student performance on non-verbal and verbal measures, indicating that verbalization facilitates effective comparison.
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References
Aleven, V. A., & Koedinger, K. R. (2002). An effective metacognitive strategy: learning by doing and explaining with a computer-based cognitive tutor. Cogni Sci, 26(2), 147–179. https://doi.org/10.1207/s15516709cog2602_1.
Alfieri, L., Nokes, T. J., & Schunn, C. D. (2013). Learning through case comparisons: a metaanalytic review. Educational Psychologist, 48(2), 87–113. https://doi.org/10.1080/00461520.2013.775712.
Anderson, J. R. (1995). Learning and memory: an integrated approach. New York: Wiley.
Anderson, J. R. (1996). The architecture of cognition. Cambridge: Harvard University Press.
Anderson, J. R., & Lebiere, C. (1998). The atomic components of thought. Mahwah: Lawrence Erlbaum Associates.
Atkinson, R. K., Renkl, A., & Merrill, M. M. (2003). Transitioning from studying examples to solving problems: effects of self-explanation prompts and fading worked-out steps. Journal of Educational Psychology, 95(4), 774–783. https://doi.org/10.1037/0022-0663.95.4.774.
Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38(2), 115–131.
Birnbaum, M. S., Kornell, N., Bjork, E. L., & Bjork, R. A. (2013). Why interleaving enhances inductive learning: the roles of discrimination and retrieval. Memory and Cognition, 41(3), 392–402. https://doi.org/10.3758/s13421-012-0272-7.
Bou-Llusar, J. C., & Segarra-Ciprés, M. (2006). Strategic knowledge transfer and its implications for competitive advantage: an integrative conceptual framework. Journal of Knowledge Management, 10(4), 100–112. https://doi.org/10.1108/13673270610679390.
Carvalho, P. F., & Goldstone, R. L. (2014). Effects of interleaved and blocked study on delayed test of category learning generalization. Frontiers in Psychology, 5(936), 1–11. https://doi.org/10.3389/fpsyg.2014.00936.
Chi, M. T. H. (2000). Self-explaining expository texts: the dual processes of generating inferences and repairing mental models. In R. Glaser (Ed.), Advances in instructional psychology: educational design and cognitive science (Vol. 5, pp. 161–238). Mahwah: Erlbaum.
DeKeyser, R. (2003). Implicit and explicit learning. In C. J. Doughty & M. H. Long (Eds.), Handbook of second language acquisition (pp. 313–348). Oxford: Blackwell Publishing.
Eid, M., Lischetzke, T., Nussbeck, F. W., & Trierweiler, L. I. (2003). Separating trait effects from trait-specific method effects in multitrait-multimethod models: a multiple-indicator CT-C(M-1) model. Psychology Methods, 8(1), 38–60. https://doi.org/10.1037/1082-989X.8.1.38.
Fiedler, K. (2011). Voodoo correlations are everywhere—not only neuroscience. Perspectives on Psychological Science, 6(2), 163–171. https://doi.org/10.1177/1745691611400237.
Field, A. (2009). Discovering statistics using SPSS (3rd ed.). London: Sage.
Gentner, D. (2010). Bootstrapping the mind: analogical processes and symbol systems. Cognitive Science, 34(5), 752–775. https://doi.org/10.1111/j.1551-6709.2010.01114.x.
Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning and transfer: a general role for analogical encoding. Journal of Educational Psychology, 95(2), 393–408. https://doi.org/10.1037/0022-0663.95.2.393.
Gentner, D., Loewenstein, J., Thompson, L., & Forbus, K. D. (2009). Reviving inert knowledge: analogical abstraction supports relational retrieval of past events. Cognitive Science, 33(8), 1343–1382. https://doi.org/10.1111/j.1551-6709.2009.01070.x.
Holland, P. W., & Wainer, H. (2012). Differential item functioning. New York: Routledge.
Horn, W. (1983). Leistungsprüfsystem LPS (2. Auflage ed.). Göttingen: Hogrefe.
Kamii, C., & Dominick, A. (1997). To teach or not to teach algorithms. Journal of Mathematical Behavior, 16(1), 51–61. https://doi.org/10.1016/S0732-3123(97)90007-9.
Kang, S. H. K., & Pashler, H. (2012). Learning painting styles: spacing is advantageous when it promotes discriminative contrast. Applied Cognitive Psychology, 26(1), 97–103. https://doi.org/10.1002/acp.1801.
Karmiloff-Smith, A. (1994). Précis of beyond modularity: a developmental perspective on cognitive science. Behavioral and Brain Sciences, 17, 693–745. https://doi.org/10.1017/S0140525X00036621.
Koedinger, K. R., & Aleven, V. A. (2007). Exploring the assistance dilemma in experiments with cognitive tutors. Educational Psychology Review, 19(3), 239–264. https://doi.org/10.1007/s10648-007-9049-0.
Koedinger, K. R., Corbett, A. T., & Perfetti, C. (2012). The knowledge-learning-instruction framework: bridging the science-practice chasm to enhance robust student learning. Cognitive Science, 36(5), 757–798. https://doi.org/10.1111/j.1551-6709.2012.01245.x.
Körner, C. (2005). Concepts and misconceptions in comprehension of hierarchical graphs. Learning and Instruction, 15, 281–296. https://doi.org/10.1016/j.learninstruc.2005.07.003.
Kurtz, K. J., Miao, C. H., & Gentner, D. (2001). Learning by analogical bootstrapping. Journal of the Learning Sciences, 10(4), 417–446. https://doi.org/10.1207/S15327809JLS1004new_2.
Matthews, P. G., & Rittle-Johnson, B. (2009). In pursuit of knowledge: comparing self-explanations, concepts, and procedures as pedagogical tools. Journal of Experimental Child Psychology, 104(1), 1–21. https://doi.org/10.1016/j.jecp. 2008.08.004.
McCloskey, M., & Cohen, N. J. (1989). Catastrophic interference in connectionist networks: the sequential learning problem. The Psychology of Learning and Motivation, 24, 109–165. https://doi.org/10.1016/S0079-7421(08)60536-8.
Mitchell, C., Nash, S., & Hall, G. (2008). The intermixed-blocked effect in human perceptual learning is not the consequence of trial spacing. Journal of Experimental Psychology: Learning, Memory, and Cognition, 34(1), 237–242. https://doi.org/10.1037/0278-7393.34.1.237.
O’Neill, K. A., McPeek, W. M., & Wild, C. L. (1993). Differential item functioning on the graduate management admission test. ETS Research Report Series, 1993(2), 1–46. https://doi.org/10.1002/j.2333-8504.1993.tb01546.x.
Podsakoff, P. M., MacKenzie, S. B., Lee, J.-Y., & Podsakoff, N. P. (2003). Common method biases in behavioral research: a critical review of the literature and recommended remedies. Journal of Applied Psychology, 88(5), 879–903. https://doi.org/10.1037/0021-9010.88.5.879.
Rau, A. R., Aleven, V. A., & Rummel, N. (2013). Interleaved practice in multi-dimensional learning tasks: which dimension should we interleave? Learning and Instruction, 23(1). https://doi.org/10.1016/j.learninstruc.2012.07.003.
Renkl, A. (1997). Learning from worked-out examples: a study on individual differences. Cognitive Science, 21(1), 1–29. https://doi.org/10.1016/S0364-0213(99)80017-2.
Renkl, A. (2005). The worked-out example principle in multimedia learning. In R. E. Mayer (Ed.), Cambridge handbook of multimedia learning (pp. 229–247). Cambridge, England: Cambridge University Press.
Renkl, A. (2015). Different roads lead to Rome: the case of principle-based cognitive skills. Learning: Research and Practice, 1(1), 79–90. https://doi.org/10.1080/23735082.2015.994255.
Rittle-Johnson, B. (2006). Promoting transfer: effects of self-explanation and direct instruction. Child Development, 77(1), 1–15. https://doi.org/10.1111/j.1467-8624.2006.00852.x.
Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural, knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574. https://doi.org/10.1037/0022-0663.99.3.561.
Rittle-Johnson, B., & Star, J. R. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101(3), 529–544. https://doi.org/10.1037/a0014224.
Rittle-Johnson, B., & Star, J. R. (2011). The power of comparison in learning and instruction: learning outcomes supported by different types of comparisons. Psychology of Learning and Motivation: Cognition in Education, 55, 199–225. https://doi.org/10.1016/B978-0-12-387691-1.00007-7.
Rittle-Johnson, B., Star, J., & Durkin, K. (2012). Developing procedural flexibility: are novices prepared to learn from comparing procedures? British Journal of Educational Psychology, 82(3), 436–455. https://doi.org/10.1111/j.2044-8279.2011.02037.x.
Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27(4), 587–597. https://doi.org/10.1007/s10648-015-9302-x.
Rohrer, D. (2012). Interleaving helps students distinguish among similar concepts. Educational Psychology Review, 24(3), 355–367. https://doi.org/10.1007/s10648-012-9201-3.
Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics practice problems boosts learning. Instructional Science, 35(6), 481–498. https://doi.org/10.1007/s11251-007-9015-8.
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.
Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475–522. https://doi.org/10.1207/s1532690xci1604_4.
Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy discoveries: a microgenetic analysis. Journal of Experimental Psychology-General, 127(4), 377–397. https://doi.org/10.1037/0096-3445.127.4.377.
Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: an experimental study on computational estimation. Journal of Experimental Child Psychology, 102(4), 408–426. https://doi.org/10.1016/j.jecp.2008.11.004.
Sun, R., Slusarz, P., & Terry, C. (2005). The interaction of the explicit and the implicit in skill learning: a dual-process approach. Psychological Review, 112(1), 159–192. https://doi.org/10.1037/0033-295X.112.1.159.
Sun, R., Mathews, R. C., & Lane, S. M. (2007). Implicit and explicit processes in the development of cognitive skills: a theoretical interpretation with some practical implications for science instruction. In E. M. Vargios (Ed.), Educational psychology research (pp. 1–26). New York: Nova Science Publishers.
Sweller, J. (1994). Cognitive load theory, learning difficulty, and instructional design. Learning and Instruction, 4(4), 295–312. https://doi.org/10.1016/0959-4752(94)90003-5.
Sweller, J. (2006). The worked example effect and human cognition. Learning and Instruction, 16(2), 165–169. https://doi.org/10.1016/j.learninstruc.2006.02.005.
Taylor, K., & Rohrer, D. (2010). The effect of interleaving practice. Applied Cognitive Psychology, 24(6), 837–848. https://doi.org/10.1002/acp.1598.
VanLehn, K., Jones, R. M., & Chi, M. T. H. (1992). A model of the self-explanation effect. Journal of the Learning Sciences, 2, 1–59. https://doi.org/10.1207/s15327809jls0201_1.
von Aufschnaiter, C., & Rogge, C. (2010). Misconceptions or missing conceptions. Eurasia Journal of Mathematics, Science and Technology Education, 6(1), 3–18.
Williams, J. J., & Lombrozo, T. (2010). The role of explanation in discovery and generalization: evidence from category learning. Cognitive Science, 34(5), 776–806. https://doi.org/10.1111/j.1551-6709.2010.01113.x.
Wong, R. M. F., Lawson, M. J., & Keeves, J. (2002). The effects of self-explanation training on students’ problem solving in high-school mathematics. Learning and Instruction, 12(2), 233–262. https://doi.org/10.1016/S0959-4752(01)00027-5.
Wulf, G., & Shea, C. H. (2002). Principles derived from the study of simple skills do not generalize to complex skill learning. Psychonomic Bulletin & Review, 9(2), 185–211. https://doi.org/10.3758/BF03196276.
Ziegler, E., & Stern, E. (2014). Delayed benefits of learning elementary algebraic transformations through contrasted comparisons. Learning and Instruction, 33, 1–16. https://doi.org/10.1016/j.learninstruc.2014.04.006.
Ziegler, E., & Stern, E. (2016). Consistent advantages of contrasted comparisons: algebra learning under direct instruction. Learning and Instruction, 41(1), 41–51. https://doi.org/10.1016/j.learninstruc.2015.09.006.
Acknowledgements
We are grateful to Sara Ziegler for her assistance in the implementation of the study in the school classes and Sara Ziegler and Adrienne Suter for their assistance in the evaluation of the test materials.
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Ziegler, E., Edelsbrunner, P.A. & Stern, E. The Relative Merits of Explicit and Implicit Learning of Contrasted Algebra Principles. Educ Psychol Rev 30, 531–558 (2018). https://doi.org/10.1007/s10648-017-9424-4
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DOI: https://doi.org/10.1007/s10648-017-9424-4