Do First Graders Make Efficient Use of External Number Representations? The Case of the Twenty-Frame

REFERENCES

• Aunio, P., & Niemivirta, M. (2010). Predicting children's mathematical performance in grade one by early numeracy. Learning and Individual Differences, 20, 427–435.
   Web of Science ®Google Scholar
• Baddeley, A.D. (2000). The episodic buffer: A new component of working memory? Trends in Cognitive Science, 4, 417–423.
   PubMed Web of Science ®Google Scholar
• Baddeley, A.D., & Hitch, G. (1974). Working memory. In G.H. Bower (Ed.), The psychology of learning and motivation: Advances in research and theory (Vol. 32, pp. 89–195). New York, NY: Academic Press.
   Google Scholar
• Beishuizen, M. (1985). Evaluation of the use of structured materials in the teaching of primary mathematics. In B.S. Alloway & G.M. Mills (Eds.), New directions in education and training technology: Aspects of educational technology (Vol. 32, pp. 246–258). London, UK: Kogan Page.
   Google Scholar
• Beishuizen, M. (1993). Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades. Journal for Research in Mathematics Education, 24, 294–323.
   Web of Science ®Google Scholar
• Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46(1), 3–18.
   PubMed Web of Science ®Google Scholar
• Chao, S., Stigler, J.W., & Woodward, J.A. (2000). The effects of physical materials on kindergartners’ learning of number concepts. Cognition and Instruction, 18, 285–316.
   Web of Science ®Google Scholar
• Clements, D.H. (1999). Subitizing: What is it? Why teach it? Teaching Children Mathematics, 5, 400–405.
   Google Scholar
• Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2–33.
   Web of Science ®Google Scholar
• De Corte, E., & Verschaffel, L. (1985). Beginning first graders’ initial representation of arithmetic word problems. Journal of Mathematical Behavior, 4, 3–21.
   Google Scholar
• De Smedt, B., Janssen, R., Bouwens, K., Verschaffel, L., Boets, B., & Ghesquière, P. (2009). Working memory and individual differences in mathematics achievement: A longitudinal study from first grade to second grade. Journal of Experimental Child Psychology, 103, 186–201.
   PubMed Web of Science ®Google Scholar
• DeStefano, D., & Lefevre, J.A. (2004). The role of working memory in mental arithmetic. European Journal of Cognitive Psychology, 16, 353–386.
   Web of Science ®Google Scholar
• Dornheim, D. (2008). Prädiktion von Rechenleistung und Rechenschwäche: Der Beitrag von Zahlen-Vorwissen und allgemein-kognitiven Fähigkeiten [Predicting calculation ability and disability: The influence of pre-knowledge of numbers and general cognitive abilities]. Berlin, Germany: Logos.
   Google Scholar
• Ericsson, K.A., & Simon, H.A. (1980). Verbal reports as data. Psychological Review, 87, 215–251.
   Web of Science ®Google Scholar
• Frank, M.C., & Barner, D. (2012). Representing exact number visually using mental abacus. Journal of Experimental Psychology: General, 141, 134–149.
   PubMed Web of Science ®Google Scholar
• Gravemeijer, K., Lehrer, R., van Oers, B., & Verschaffel, L. (Eds.). (2002). Symbolizing, modeling and tool use in mathematics education. Dordrecht, The Netherlands: Kluwer Academic.
   Google Scholar
• Gravemeijer, K.P. E. (1994). Developing realistic mathematics education. Utrecht, The Netherlands: Freudenthal Institute.
   Google Scholar
• Hatano, G., Amaiwa, S., & Shimizu, K. (1987). Formation of a mental abacus for computation and its use as a memory device for digits: A developmental study. Developmental Psychology, 23, 832–838.
   Web of Science ®Google Scholar
• Hatano, G., Myake, Y., & Binks, M.G. (1977). Performance of expert abacus operators. Cognition, 5(1), 47–55.
   Web of Science ®Google Scholar
• Janvier, C. (Ed.) (1987). Problems of representation in the teaching and learning of mathematics. Hillsdale, NJ: Erlbaum.
   Google Scholar
• Krajewski, K., Grüßing, M., & Peter-Koop, A. (2009). Die Entwicklung mathematischer Kompetenzen bis zum Beginn der Grundschulzeit [The development of mathematical competence before the start of primary school]. In A. Heinze & M. Grüßing (Eds.), Mathematiklernen vom Kindergarten bis zum Studium [Learning mathematics from kindergarten to university] (pp. 17–34). Münster, Germany: Waxmann.
   Google Scholar
• Krajewski, K., & Schneider, W. (2009). Early development of quantity to number-word linkage as a precursor of mathematical school achievement and mathematical difficulties: Findings from a four-year longitudinal study. Learning and Instruction, 19, 513–526.
   Web of Science ®Google Scholar
• Krauthausen, G., & Scherer, P. (2007). Einführung in die Mathematikdidaktik [Introduction to mathematics education] (3rd ed.). München, Germany: Spektrum Akademischer Verlag.
   Google Scholar
• Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: A study of 8–9-year-old students. Cognition, 93, 99–125.
   PubMed Web of Science ®Google Scholar
• Linn, M.C., & Petersen, A.C. (1985). Emergence and characterization of sex differences in spatial ability: A meta-analysis. Child Development, 56, 1479–1498.
   PubMed Web of Science ®Google Scholar
• Lohman, D.F. (1988). Spatial abilities as traits, processes, and knowledge. In R.J. Sternberg (Ed.), Advances in the psychology of human intelligence (pp. 181–248). Hillsdale, NJ: Erlbaum.
   Google Scholar
• Lorenz, J.H. (1998). Anschauung und Veranschaulichungsmittel im Mathematikunterricht. Mentales visuelles Operieren und Rechenleistung [Visualization and visual tools in the mathematics classroom. Mental visual operations and arithmetic] (2nd ed.). Gottingen, Germany: Hogrefe.
   Google Scholar
• Lorenz, J.H. (2007). Hamburger Rechentest. Manual [Manual of the Hamburger Rechentest]. Hamburg, Germany: Freie und Hansestadt Hamburg, Behörde für Bildung und Sport.
   Google Scholar
• Luwel, K., Lemaire, P., & Verschaffel, L. (2005). Children's strategies in numerosity judgment. Cognitive Development, 20, 448–471.
   Web of Science ®Google Scholar
• Luwel, K., Verschaffel, L., Onghena, P., & De Corte, E. (2000). Children's strategies for numerosity judgment in square grids of different sizes. Psychologica Belgica, 40, 183–209.
   Web of Science ®Google Scholar
• Luwel, K., Verschaffel, L., Onghena, P., & De Corte, E. (2001). Strategic aspects of children's numerosity judgment. European Journal of Psychology of Education, 16, 233–255.
   Web of Science ®Google Scholar
• Mandler, G., & Shebo, B.J. (1982). Subitizing: An analysis of its component processes. Journal of Experimental Psychology: General, 111, 1–22.
   PubMed Web of Science ®Google Scholar
• McGuire, P., Kinzie, M.B., & Berch, D.B. (2012). Developing number sense in pre-k with five-frames. Early Childhood Education Journal, 40, 213–222.
   Google Scholar
• Obersteiner, A., Reiss, K., & Ufer, S. (2013). How training on exact or approximate mental representations of number can enhance first-grade students’ basic number processing and arithmetic skills. Learning and Instruction, 23, 125–135.
   Web of Science ®Google Scholar
• Padberg, F. (2007). Didaktik der Arithmetik für Lehrerausbildung und Lehrerfortbildung [Didactics of arithmetic for teacher education and teacher professional development] (3rd ed.). Heidelberg, Germany: Spektrum Akademischer Verlag.
   Google Scholar
• Petermann, F., & Petermann, U. (Eds.) (2008). HAWIK-IV. Hamburg-Wechsler-Intelligenztest für Kinder–IV. Übersetzung und Adaptation der WISC-IV von David Wechsler [HAWIK-IV. Hamburg-Wechsler-Test of intelligence for children-IV. Translation and adaptation of the WISC-IV by David Wechsler] (2nd ed.). Bern, Switzerland: Verlag Hans Huber, Hogrefe.
   Google Scholar
• Rasmussen, C., & Bisanz, J. (2005). Representation and working memory in early arithmetic. Journal of Experimental Child Psychology, 91, 137–157.
   PubMed Web of Science ®Google Scholar
• Schneider, M., Heine, A., Thaler, V., Torbeyns, J., De Smedt, B., Verschaffel, L.,  …  Stern, E. (2008). A validation of eye movements as a measure of elementary school children's developing number sense. Cognitive Development, 23, 409–422.
   Web of Science ®Google Scholar
• Schneider, W., Eschman, A., & Zuccolotto, A. (2002). E-Prime user's guide. Pittsburgh, PA: Psychology Software Tools.
   Google Scholar
• Schnotz, W., & Kürschner, C. (2008). External and internal representations in the acquisition and use of knowledge: Visualization effects on mental model construction. Instructional Science, 36, 175–190.
   Web of Science ®Google Scholar
• Sowell, E.J. (1989). Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics Education, 20, 498–505.
   Web of Science ®Google Scholar
• Stigler, J.W. (1984). “Mental abacus”: The effect of abacus training on Chinese children's mental calculation. Cognitive Psychology, 16, 145–176.
   Web of Science ®Google Scholar
• Thurstone, L.L. (1950). Some primary abilities in visual thinking. Proceedings of the American Psychological Society, 94, 517–521.
   Google Scholar
• Treffers, A. (2001). Grade 1 (and 2): Calculation up to twenty. In M. van den Heuvel-Panhuizen (Ed.), Children learn mathematics (pp. 43–60). Utrecht, The Netherlands: Freudenthal Institute, University of Utrecht.
   Google Scholar
• Verschaffel, L., & De Corte, E. (1996). Number and arithmetic. In A. Bishop, K. Clements, C. Keitel, & C. Laborde (Eds.), International handbook of mathematics education. Part I (pp. 99–138). Dordrecht, The Netherlands: Kluwer.
   Google Scholar
• Verschaffel, L., De Corte, E., de Jong, T., & Elen, J. (Eds.) (2010). Use of representations in reasoning and problem solving. Analysis and improvement. London, UK: Routledge.
   Google Scholar
• Verschaffel, L., De Corte, E., Lamote, C., & Dhert, N. (1998). The acquisition and use of an adaptive strategy for estimating numerosity. European Journal of Psychology of Education, 13, 347–370.
   Web of Science ®Google Scholar