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Articles

Concreteness Fading of Algebraic Instruction: Effects on Learning

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References

  • Anderson, M. L. (2014). After phrenology: Neural reuse and the interactive brain. Cambridge, MA: MIT Press.
  • Anderson, M. L. (2015). Précis of after phrenology: Neural reuse and the interactive brain. Behavioral and Brain Sciences, 16, 1–22.
  • Banerjee, B., & Chandrasekaran, B. (2004, November). Perceptual and action routines in diagrammatic reasoning for entity-reidentification. Paper presented at the 24th Army Science Conference, Orlando, FL.
  • Barab, S., Zuiker, S., Warren, S., Hickey, D., Ingram‐Goble, A., Kwon, E. J., & Herring, S. C. (2007). Situationally embodied curriculum: Relating formalisms and contexts. Science Education, 91(5), 750–782.
  • Bernardo, A. B., & Okagaki, L. (1994). Roles of symbolic knowledge and problem-information context in solving word problems. Journal of Educational Psychology, 86(2), 212–220.
  • Braithwaite, D. W., Goldstone, R. L., van der Maas, H. L., & Landy, D. H. (2016). Non-formal mechanisms in mathematical cognitive development: The case of arithmetic. Cognition, 149, 40–55. doi:10.1016/j.cognition.2016.01.004
  • Bransford, J. D., & Schwartz, D. L. (1999). Rethinking transfer: A simple proposal with multiple implications. Review of Research in Education, 24, 61–100.
  • Bruner, J. S. (1966). Toward a theory of instruction. Cambridge, MA: Belknap.
  • Brunstein, A., Betts, S., & Anderson, J. R. (2009). Practice enables successful learning under minimal guidance. Journal of Educational Psychology, 101(4), 790–802.
  • Changizi, M. A., Zhang, Q., Ye, H., & Shimojo, S. (2006). The structures of letters and symbols throughout human history are selected to match those found in objects in natural scenes. The American Naturalist, 167(5), E117–E139. doi:10.1086/502806
  • Clark, A. (1998). Magic words: How language augments human computation. In P. Carruthers & J. Boucher (Eds.), Language and Thought: Interdisciplinary Themes, (pp. 162–183). Cambridge, UK: Cambridge University Press.
  • Common Core State Standards Initiative. (2010). Common Core State Standards for mathematics. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
  • Correll, M., Albers, D., Franconeri, S., & Gleicher, M. (2012). Comparing averages in time series data. In E. H. Chi & K. Höök (Eds.), Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (pp. 1095–1104). New York, NY: Association for Computing Machinery.doi:10.1145/2207676.2208556
  • Dehaene, S., & Cohen, L. (2007). Cultural recycling of cortical maps. Neuron, 56(2), 384–398. doi:10.1016/j.neuron.2007.10.004
  • Dehaene, S., Cohen, L., Morais, J., & Kolinsky, R. (2015). Illiterate to literate: Behavioural and cerebral changes induced by reading acquisition. Nature Reviews Neuroscience, 16(4), 234–244. doi:10.1038/nrn3924
  • Dörfler, W. (2006). Diagramme und Mathematikunterricht [Diagrammatic thinking: Affordances and constraints]. Journal Für Mathematik-Didaktik, 27(3/4), 200–219. doi:10.1007/BF03339039
  • Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26(1), 9–25. doi:10.1007/s10648-014-9249-3
  • Glenberg, A. M. (2011). How reading comprehension is embodied and why that matters. International Electronic Journal of Elementary Education, 4(1), 5–18.
  • Goldin-Meadow, S., Nusbaum, H., Kelly, S. D., & Wagner, S. (2001). Explaining math: Gesturing lightens the load. Psychological Science, 12(6), 516–522. doi:10.1111/1467-9280.00395
  • Goldstone, R. L., Landy, D., & Son, J. Y. (2008). A well-grounded education: The role of perception in science and mathematics. In M. DeVega, A. M. Glenberg, & A. C. Graesser (Eds.), Symbols and embodiment: Debates on meaning and cognition (pp. 327–355). Oxford, UK: Oxford University Press.
  • Goldstone, R. L., Landy, D., & Son, J. Y. (2010). The education of perception. Topics in Cognitive Science, 2, 265–284. doi:10.1111/tops.2010.2.issue-2
  • Goldstone, R. L., & Sakamoto, Y. (2003). The transfer of abstract principles governing complex adaptive systems. Cognitive Psychology, 46, 414–466. doi:10.1016/S0010-0285(02)00519-4
  • Goldstone, R. L., & Son, J. Y. (2005). The transfer of scientific principles using concrete and idealized simulations. Journal of the Learning Sciences, 14(1), 69–110. doi:10.1207/s15327809jls1401_4
  • Hayhoe, M. (2000). Vision using routines: A functional account of vision. Visual Cognition, 7(1–3), 43–64. doi:10.1080/135062800394676
  • Heeffer, A. (2013, March). Algebraic symbolism as a conceptual barrier in learning mathematics. Paper presented at the Sixth East Asia Regional Conference on Mathematics Education, Phuket, Thailand.
  • Horswill, I. (1995, August). Visual routines and visual search: A real-time implementation and an automata-theoretic analysis. Paper presented at the International Joint Conference on Artificial Intelligence, Montréal, QC.
  • Iverson, K. E. (2007). Notation as a tool of thought. ACM SIGAPL APL Quote Quad, 35(1–2), 2–31. doi:10.1145/1234321
  • Kaminski, J. A., & Sloutsky, V. M. (2012, November). Children’s acquisition of fraction knowledge from concrete versus generic instantiations. Paper presented at the XXXIV Annual Conference of the Cognitive Science Society, Sapporo, Japan.
  • Kaminski, J. A., & Sloutsky, V. M. (2013). Extraneous perceptual information interferes with children’s acquisition of mathematical knowledge. Journal of Educational Psychology, 105(2), 351–363. doi:10.1037/a0031040
  • Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2006, November). Do children need concrete instantiations to learn an abstract concept? Paper presented at the 28th Annual Conference of the Cognitive Science Society, Vancouver, BC.
  • Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2008, April 25). The advantage of abstract examples in learning math. Science, 320, 454–455. doi:10.1126/science.1154659
  • Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2009). Concrete instantiations of mathematics: A double-edged sword. Journal for Research in Mathematics Education, 40(2), 90–93.
  • Kaya, O. N. (2009). The nature of relationships among the components of pedagogical content knowledge of preservice science teachers: “Ozone layer depletion” as an example. International Journal of Science Education, 31, 961–988.
  • Kellman, P. J., Massey, C. M., Roth, Z., Burke, T., Zucker, J., Saw, A., … Wise, J. A. (2008). Perceptual learning and the technology of expertise: Studies in fraction learning and algebra. Learning Technologies and Cognition, 16(2), 356–405. doi:10.1075/p&c.16.2.07kel
  • Kellman, P. J., Massey, C. M., & Son, J. Y. (2010). Perceptual learning modules in mathematics: Enhancing students’ pattern recognition, structure extraction, and fluency. Topics in Cognitive Science, 2(2), 285–305. doi:10.1111/tops.2010.2.issue-2
  • Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
  • Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20, 274–287. doi:10.2307/749516
  • Kirshner, D., & Awtry, T. (2004). Visual salience of algebraic transformations. Journal for Research in Mathematics Education, 35(4), 224–257. doi:10.2307/30034809
  • Koedinger, K. R., Alibali, M. W., & Nathan, M. J. (2008). Trade-offs between grounded and abstract representations: Evidence from algebra problem solving. Cognitive Science, 32(2), 366–397. doi:10.1080/0364021070186393
  • Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. Journal of the Learning Sciences, 13(2), 129–164. doi:10.1207/s15327809jls1302_1
  • Kong, Q. P., Wong, N. Y., & Lam, C. C. (2003). Student engagement in mathematics: Development of instrument and validation of construct. Mathematics Education Research Journal, 15(1), 4–21. doi:10.1007/BF03217366
  • Landy, D., Allen, C., & Anderson, M. L. (2011). Conceptual discontinuity through recycling old processes in new domains. Commentary on Susan Carey: Précis of the Origin of Concepts. Behavioral and Brain Sciences, 33(6), 136–137.
  • Landy, D., Allen, C., & Zednik, C. (2014). A perceptual account of symbolic reasoning. Frontiers in Psychology, 5, 275. doi:10.3389/fpsyg.2014.00275
  • Landy, D., & Goldstone, R. L. (2007a). Formal notations are diagrams: Evidence from a production task. Memory & Cognition, 35, 2033–2040. doi:10.3758/BF03192935
  • Landy, D., & Goldstone, R. L. (2007b). How abstract is symbolic thought? Journal of Experimental Psychology: Learning, Memory, & Cognition, 33, 720–733.
  • Landy, D., & Goldstone, R. L. (2010). Proximity and precedence in arithmetic. The Quarterly Journal of Experimental Psychology, 63, 1953–1968.
  • Lochhead, J., & Mestre, J. P. (1988). From words to algebra: Mending misconceptions. In A. Cosford & A. Schulte (Eds.), The Idea of Algebra K-12: National Council of Teachers of Mathematics Yearbook (pp. 127–136). Reston, VA: National Council of Teachers of Mathematics.
  • Marghetis, T., & Núñez, R. (2013). The motion behind the symbols: A vital role for dynamism in the conceptualization of limits and continuity in expert mathematics. Topics in Cognitive Science, 5(2), 299–316. doi:10.1111/tops.12013
  • Marquis, J. (1988). Common mistakes in algebra. In A. F. Coxford & A. P. Shulte (Eds.), The ideas of algebra, K-12: National Council of Teachers of Mathematics yearbook (pp. 204–205). Reston, VA: National Council of Teachers of Mathematics.
  • Marshall, P. (2007, February). Do tangible interfaces enhance learning? Paper presented at the 1st International Conference on Tangible and Embedded Interaction, Baton Rouge, LA.
  • Martin, S. A., & Bassok, M. (2005). Effects of semantic cues on mathematical modeling: Evidence from word-problem solving and equation construction tasks. Memory & Cognition, 33(3), 471–478. doi:10.3758/BF03193064
  • Maruyama, M., Pallier, C., Jobert, A., Sigman, M., & Dehaene, S. (2012). The cortical representation of simple mathematical expressions. NeuroImage, 61, 1444–1460. doi:10.1016/j.neuroimage.2012.04.020
  • McNeil, N. M., & Fyfe, E. R. (2012). “Concreteness fading” promotes transfer of mathematical knowledge. Learning and Instruction, 22, 440–448. doi:10.1016/j.learninstruc.2012.05.001
  • Midgley, C., Maehr, M. L., Hruda, L. Z., Anderman, E., Anderman, L., Freeman, K. E., & Urdan, T. (2000). Manual for the Patterns of Adaptive Learning Scales. Ann Arbor: University of Michigan.
  • Nathan, M. (2012). Rethinking formalisms in formal education. Educational Psychologist, 47(2), 125–148. doi:10.1080/00461520.2012.667063
  • National Center for Education Statistics. (2011). The condition of education 2011 (Report No. NCES 2011-033). Retrieved from http://nces.ed.gov/pubs2011/2011033.pdf
  • National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.
  • Nogueira de Lima, R., & Tall, D. (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics, 67(1), 3–18. doi:10.1007/s10649-007-9086-0
  • Ottmar, E., Landy, D., & Goldstone, R. L. (2012). Teaching the perceptual structure of algebraic expressions: Preliminary findings from the Pushing Symbols intervention. In N. Miyake, D. Peebles, & R. P. Cooper (Eds.), Proceedings of the 34th Annual Conference of the Cognitive Science Society (pp. 2156–2161). Austin, TX: Cognitive Science Society.
  • Ottmar, E. R., Landy, D., Goldstone, R., & Weitnauer, E. (2015, July). Getting from here to there: Testing the effectiveness of an interactive mathematics intervention embedding perceptual learning. Paper presented at the 37th Annual Conference of the Cognitive Science Society, Pasadena, CA.
  • Pylyshyn, Z. W. (2000). Situating vision in the world. Trends in Cognitive Sciences, 4(5), 197–207. doi:10.1016/S1364-6613(00)01477-7
  • Radford, L., & Puig, L. (2007). Syntax and meaning as sensuous, visual, historical forms of algebraic thinking. Educational Studies in Mathematics, 66, 145–164. doi:10.1007/s10649-006-9024-6
  • Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2013). Math anxiety, working memory, and math achievement in early elementary school. Journal of Cognition and Development, 14(2), 187–202. doi:10.1080/15248372.2012.664593
  • Rao, R. P., & Ballard, D. H. (1995). An active vision architecture based on iconic representations. Artificial Intelligence, 78(1), 461–505. doi:10.1016/0004-3702(95)00026-7
  • 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(2), 346–362. doi:10.1037/0022-0663.93.2.346
  • Schwartz, D. L., Chase, C. C., Oppezzo, M. A., & Chin, D. B. (2011). Practicing versus inventing with contrasting cases: The effects of telling first on learning and transfer. Journal of Educational Psychology, 103(4), 759–775. doi:10.1037/a0025140
  • Sweller, J. (1994). Cognitive load theory, learning difficulty, and instructional design. Learning and Instruction, 4(4), 295–312.
  • Ullman, S. (1984). Visual routines. Cognition, 18(1–3), 97–159.
  • Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18(1), 37–54. doi:10.1016/S0193-3973(97)90013-7
  • Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. Retrieved from http://www.gutenberg.org/files/5740/5740-pdf.pdf

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