Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education

  • Lieven VerschaffelEmail author
  • Koen Luwel
  • Joke Torbeyns
  • Wim Van Dooren


Some years ago, Hatano differentiated between routine and adaptive expertise and made a strong plea for the development and implementation of learning environments that aim at the latter type of expertise and not just the former. In this contribution we reflect on one aspect of adaptivity, namely the adaptive use of solution strategies in elementary school arithmetic. In the first part of this article we give some conceptual and methodological reflections on the adaptivity issue. More specifically, we critically review definitions and operationalisations of strategy adaptivity that only take into account task and subject characteristics and we argue for a concept and an approach that also involve the sociocultural context. The second part comprises some educational considerations with respect to the questions why, when, for whom, and how to strive for adaptive expertise in elementary mathematics education.

Key words

Adaptivity Arithmetic Flexibility Mathematics education Strategies 


Il y a quelques années, Hatano faisait le partage entre l’expertise routinière et adaptative, et plaidoyait avec force en faveur du développement et de la réalisation des programmes d’instruction qui visent spécialement ce dernier type d’expertise. Dans cette contribution nous réfléchissons sur un aspect de l’adaptativité, à savoir l’utilisation adaptative des stratégies de solution dans l’arithmétique de l’école primaire. Dans la première partie de cet article nous donnons quelques réflexions conceptuelles et méthodologiques sur la question d’adaptativité. Plus spécifiquement, nous analysons de façon critique les définitions et les opérationnalisations de l’adaptativité stratégique qui tiennent compte non seulement des caractéristiques de la tâche et de l’individu, mais nous plaidons aussi pour un concept et une approche méthodologique qui impliquent également le contexte socioculturel. La deuxième partie comporte quelques considérations éducatives concernant les questions pourquoi, quand, pour qui, et comment obtient-on l’expertise adaptive dans l’éducation élémentaire de mathématiques.


  1. Adelson, B. (1984). When novices surpass experts: The difficulty of a task may increase with expertise.Journal of experimental Psychology: Learning, Memory and Cognition, 10, 483–495.CrossRefGoogle Scholar
  2. Anghileri, J. (1999). issues in teaching multiplication and division. In I. Thompson (Ed.),Issues in teaching numeracy in primary schools (pp. 184–194). Buckingham, U.K.: Open University Press.Google Scholar
  3. Baroody, A.J. (1996). An investigative approach to the mathematics instruction of children classified as learning disabled. In D.K. Reid, W.P. Hresko et al. (Eds.),Cognitive approaches to learning disabilities (3rd ed. pp. 545–615). Austin, TX: PRO-ED.Google Scholar
  4. Baroody, A.J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A.J. Baroody & A. Dowker (Eds.),The development of arithmetic concepts and skills (pp. 1–34). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  5. Baroody, A.J., & Dowker, A. (Eds.). (2003). The development of arithmetic concepts and skills.Constructing adaptive expertise. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  6. Baroody, A.J., & Ginsburg, H.P. (1986). The relationship between initial meaningful learning and mechanical knowledge of arithmetic. In J. Hiebert (Ed.),Conceptual and procedural knowledge: The case of mathematics (pp. 75–112). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  7. Baroody, A.J., & Rosu, L. (2006).Adaptive expertise with basic addition and subtraction combinations — The number sense view. Paper presented at the Annual Meeting of the American Educational Research Association (April), San Fransisco, CA.Google Scholar
  8. Baxter, J.A., Woodward, J., & Olson, D. (2001). Effects of reform-based mathematics instruction on low achievers in five third-grade classrooms.Elementary School Journal, 101, 529–547.CrossRefGoogle Scholar
  9. Beishuizen, M. (2001). Different approaches to mastering mental calculation strategies. In I. Anghileri (Ed.),Principles and practices of arithmetic teaching (pp. 119–130). Buckingham: Open University Press.Google Scholar
  10. Berar, I. (2004).Flexibility of cognitive processes — Indicator of giftedness. (Unpublished manuscript). Available at Scholar
  11. Bereiter, C., & Scardamalia, M. (1993).Surpassing ourselves: An inquiry into the nature and implications of expertise. Chicago: Open Court.Google Scholar
  12. Bisanz, J. (2003). Arithmetical development. Commentary on chapters 1 through 8 and reflections on directions. In A.J. Baroody & A. Dowker (Eds.),The development of arithmetic concepts and skills (pp. 435–452). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  13. Blöte, A.W., Van der Burg, E., & Klein, A.S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: Instruction effects.Journal of Educational Psychology, 93, 627–638.CrossRefGoogle Scholar
  14. Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings.Journal for Research in Mathematics Education, 29, 41–62.CrossRefGoogle Scholar
  15. Boaler, J. (2000). Exploring situated insights into research and learning.Journal for Research in Mathematics Education, 39, 113–119.CrossRefGoogle Scholar
  16. Bottge, B.A. (1999). Effects of contextualized math instruction on problem solving of average and below-average achieving students.The Journal of Special Education, 33, 81–92.CrossRefGoogle Scholar
  17. Bottge, B.A., Heinrichs, M., Chan, S., & Serlin, R.C. (2001). Anchoring adolescents’ understanding of math concepts in rich problem solving environments.Remedial and Special Education, 22, 299–314.CrossRefGoogle Scholar
  18. Bottge, B.A., Heinrichs, M., Mehta, Z., & Hung, Y. (2002). Weighing the benefits of anchored math instruction for students with disabilities in general education classes.The Journal of Special Education, 35, 186–200.CrossRefGoogle Scholar
  19. Bransford, J. (2001).Thoughts on adaptive expertise (unpublished manuscript). Available at AdaptiveExpertise.pdfGoogle Scholar
  20. Brousseau, G. (1997).Theory of didactical situations in mathematics. N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield (Eds. and Trans.). Dordrecht, The Netherlands: Kluwer.Google Scholar
  21. Brownell, W.A. (1945). When is arithmetic meaningful?Journal of Educational Research, 38, 481–498.Google Scholar
  22. Bruner, J. (1986).Actual minds, possible worlds. Cambridge, MA: Harvard University Press.Google Scholar
  23. Buys, K. (2001). Progressive mathematsation: Sketch of a learning strand. In J. Anghileri (Ed.),Principles and practices of arithmetic teaching (pp. 107–118). Buckingham: Open University Press.Google Scholar
  24. Cajori, F. (1917).A history of elementary mathematics. New York: Macmillan.Google Scholar
  25. Cañas, J.J., Quesada, J.F., Antoli, A., & fajardo, I. (2003). Cognitive flexibility and adaptability to environmental changes in dynamic complex problem-solving tasks.Ergonomics, 46, 482–501.CrossRefGoogle Scholar
  26. Carr, M., & Jessup, D.L. (1997). Gender differences in first-grade mathematics strategy use: Social and metacognitive influences.Journal of Educational Psychology, 89, 318–328.CrossRefGoogle Scholar
  27. Cary, M., & Reder, L.M. (2002). Metacognition in strategy selection: Giving consciousness too much credit. In M. Izaute, P. Chambres, & P.J. Marescaux (Eds.),Metacognition: Process, function, and use (pp. 63–78). New York, NY: Kluwer.Google Scholar
  28. Cattell, R. (1946). The riddle of perseveration.Journal of Personality, 14, 239–267.CrossRefGoogle Scholar
  29. Cichon, D., & Ellis, J.G. (2003). The effects of MATH Connections on student achievement, confidence, and perception In S.L. Senk & D.R. Thompson (Eds.),Standards-based school mathematics curricula: What are they? What do students learn? (pp. 345–374). Mahwah, N.J.: Lawrence Erlbaum Associates.Google Scholar
  30. Clarke, D.J. (1996). Assessment. In A. Bishop, K. Clements, C. Keitel, & C. Laborde (Eds.),International handbook of mathematics education (part I, pp. 327–370). Dordrecht: Kluwer.Google Scholar
  31. Cobb, P., & Hodge, L.L. (2007). Culture, identity, and equity in the mathematics classroom. In N.S. Nasir & P. Cobb (Eds.),Diversity, equity, and access to mathematical ideas (pp. 159–171). New York: Teachers College Press.Google Scholar
  32. Coulson, R.L., Feltovich, P.J., & Spiro, R.J. (1989). Foundations of a understanding to the ultrastructural basis of myocardial failure. A reciprocation network of oversimplifications.Journal of Medicine and Philosophy, 14, 109–146.Google Scholar
  33. Crowley, K., & Siegler, R.S. (1993). Flexible strategy use in young children’s tic-tac-toe.Cognitive Science, 17, 531–561.CrossRefGoogle Scholar
  34. Dai, D.Y., & Sternberg, R.J. (2004). Beyond cognitivism: Toward an integrated understanding of intellectual functioning and development. In D.Y. Dai & R.J. Sternberg (Eds.),Motivation, emotion and cognition: Integrated perspectives on intellectual functioning and development (pp. 3–38). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  35. De Corte, E., & Verschaffel, L. (1987). The effect of semantic structure on first graders’ solution strategies of elementary addition and subtraction word problems.Journal for Research in Mathematics Education, 18, 363–381.CrossRefGoogle Scholar
  36. Duncker, K. (1945). On problem solving.Psychological Monographs,58(5, Whole no 270).Google Scholar
  37. Dunn, J.A. (1975). Tests of creativity in mathematics.International Journal of Mathematical Education in Science and Technology, 6, 327–332.CrossRefGoogle Scholar
  38. Elia, I., Gagatsis, A., & Deliyianni, E. (2005). A review of the effects of different modes of representation in mathematical problem solving. In A. Gagatsis, F. Spagnolo, Gr. Makrides, & V. Farmaki (Eds.),Proceedings of the 4th Mediterranean Conference on Mathematics Education: Vol. 1 (pp. 271–286). Palermo: University of Palermo, Cyprus Mathematical SocietyGoogle Scholar
  39. Ellis, S. (1997). Strategy choice in sociocultural context.Developmental Review, 17, 490–524.CrossRefGoogle Scholar
  40. Engestrom, Y. (1987).Learning by expanding: An activity-theoretical approach to developmental research. Helsinki, Finland: Orienta-Konsultit Oy.Google Scholar
  41. Engestrom, Y., & Miettinen, R. (1999). Introduction. In Y. Engestrom, R. Miettinen, & R.-J. Punamaki (Eds.),Perspectives on activity theory (pp. 1–18). Cambridge: Cambridge University Press.Google Scholar
  42. Feltovich, P.J., Spiro, R.J., & Coulson, R.L. (1997). Issues of expert flexibility in contexts characterized by complexity and change. In P.J. Feltovich, K.M. Ford, & R.R. Hoffman (Eds.),Expertise in context: Human and machine (pp. 125–146). Menlo Park, CA: AAAI Press.Google Scholar
  43. Frensch, P.A., & Sternberg, R.J. (1989). Expertise and intelligent thinking: When is it worse to know better? In R.J. Sternberg (Ed.),Advances in the psychology of human intelligence (vol. 5, pp. 157–188). Hillsdale, NJ: Erlbaum.Google Scholar
  44. Freudenthal, H. (1991).Revisiting mathematics education. Dordrecht: Reidel.Google Scholar
  45. Frobisher, L., & Threlfall, J. (1998).Teaching mental maths strategies. Heinemann: Oxford.Google Scholar
  46. Fuson, K., Carroll, W., & Drueck, J. (2000). Achievement results for second and third graders using the Standards-based curriculum. Everyday Mathematics.Journal for Research in Mathematics Education, 31, 277–295.CrossRefGoogle Scholar
  47. Geary, D.C. (2003). Arithmetical development: Commentary on Chapters 9 through 15 and future directions. In A.J. Baroody & A. Dowker (Eds.),The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 453–464). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  48. Goldin, G.A. (2003). Representation in school mathematics: A unifying research perspective. In J. Kilpatrick, M.G. Martin, & S. Schifter (Eds.),A research companion to principles and standards for school mathematics (pp. 275–286). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  49. Gravemeijer, K. (1994).Developing realistic mathematics education. Utrecht, The Netherlands: Freudenthal Institute, University of Utrecht.Google Scholar
  50. Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education.Mathematical Thinking and Learning, 6, 105–128.CrossRefGoogle Scholar
  51. Greeno, J.G., Collins, A.M., & Resnick, L.B. (1996). Cognition and learning. In D.C. Berliner & R.C. Calfee (Eds.),Handbook of educational psychology (pp. 15–46). New York, NY: Macmillan.Google Scholar
  52. Greer, B. (1997). Modelling reality in mathematics classrooms: The case of word problems.Learning and Instruction, 7, 293–307.CrossRefGoogle Scholar
  53. Groen, G.J., & Parkman, J.M. (1972). A chronometric analysis of simple addition.Psychological Review, 79, 329–343.CrossRefGoogle Scholar
  54. Guilford, J.P. (1967).The nature of human intelligence. New York: McGraw-Hill.Google Scholar
  55. Hatano, G. (1982). Cognitive consequences of practice in culture specific procedural skills.The Quartely Newsletter of the Laboratory of Comparative Human Cognition, 4, 15–18.Google Scholar
  56. Hatano, G. (2003). Foreword. In A.J. Baroody & A. Dowker (Eds.),The development of arithmetic concepts and skills (pp. xi-xiii). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  57. Hatano, G., & Oura, Y. (2003). Reconceptualizing school learning using insight from expertise research.Educational Researcher, 32(8), 26–29.CrossRefGoogle Scholar
  58. Heavey, L. (2003). Arithmetical savants. In A.J. Baroody & A. Dowker (Eds.),The development of arithmetic concepts and skills (pp. 409–434). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  59. Hecht, S.A. (2002). Counting on working memory in simple arithmetic when counting is used for problem solving.Memory and Cognition, 30, 447–455.Google Scholar
  60. Heirdsfield, A.M., & Cooper, T.J. (2002). Flexibility and inflexibility in accurate mental addition and subtraction: Two case studies.The Journal of Mathematical Behavior, 21, 57–74.CrossRefGoogle Scholar
  61. Hiebert, J. (Ed.). (1986).Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  62. Ishida, J. (2002). Students’ evaluation of their strategies when they find several solutions.Journal of Mathematical Behavior, 21, 49–56.CrossRefGoogle Scholar
  63. Jausovec, N. (1994).Flexible thinking: An explanation for individual differences in ability, Cresskill, NJ: Hampton Press.Google Scholar
  64. Kahneman, D., Slovic, P., & Tversky, A. (1982).Judgment under uncertainty: Heuristics and biases. New York: Cambridge University Press.Google Scholar
  65. Kaput, J. (1992). Technology and mathematics education. In D.A. Grouws (Ed.),Handbook for research on mathematics teaching and learning (pp. 515–556). New York: Macmillan.Google Scholar
  66. Kilpatrick, J., Swafford, J., & Findell, B. (2001).Adding it up. Helping children learn mathematics Washington, D.C.: National Academy Press.Google Scholar
  67. Klayman, J. (1985). Children’s decision strategies and their adaptation to task charecteristics.Organizational Behavior and Human Decision Processes, 35, 179–201.CrossRefGoogle Scholar
  68. Klein, A.S., Beishuizen, M., & Treffers, A. (1998). The empty number line in Dutch second grades: Realisticversus gradual program design.Journal for Research in Mathematics Education, 29, 443–464.CrossRefGoogle Scholar
  69. Koriat, A. (2000). Control processes in remembering. In E. Tulving & F.I.M. Craik (Eds.),The Oxford handbook of memory (pp. 333–346). New York, NY: Oxford University Press.Google Scholar
  70. Krutetskii, V.A. (1976).The psychology of mathematical abilities in school children Chicago: University of Chicago Press.Google Scholar
  71. Lave, J., & Wenger, E. (1991).Situated learning: Legitimate, peripheral participation. Cambridge: Cambridge University Press.Google Scholar
  72. Lemaire, P., & Lecacheur, M. (2001). Older and younger adults’ strategy use and execution in currency conversion tasks: Insights from French franc to Euro and Euro to French franc conversions.Journal of Experimental Psychology: Applied, 7, 195–206.CrossRefGoogle Scholar
  73. Lemaire, P., & Lecacheur, M. (2002). Children’s strategies in computational estimation.Journal of Experimental Child Psychology, 82, 281–304.CrossRefGoogle Scholar
  74. Lemaire, P., & Reder, L. (1999). What effects strategy selection in arithmetic? The example of parity and five effects on produet verification.Memory and Cognition, 27, 364–382.Google Scholar
  75. Leron, U., & Hazzan, O. (1997). The world according to Johny: A coping perspective in mathematics education.Educational Studies in Mathematics, 32, 265–292.CrossRefGoogle Scholar
  76. Lesh, R., & Doerr, H.M. (Eds.). (2003). Beyond constructivism.Models and modeling perspectives on mathematical problem solving, learning and teaching. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  77. Luchins, A.S., & Luchins, E.H. (1959).Rigidity of behavior — A variational approach to the effect of einstellung Eugene, OR: University of Oregon Books.Google Scholar
  78. Luwel, K., & Verschaffel, L. (2003). Adapting strategy choices to situational factors: The effect of time pressure on children’s numerosity judgement strategies.Psychologica Belgica, 43, 269–295.Google Scholar
  79. Luwel, K., Verschaffel, L., & Lemaire, P. (2005). Children’s strategies in numerosity judgment.Cognitive Development, 20, 448–471.CrossRefGoogle Scholar
  80. McClain, K., Cobb, P., & Bowers, J. (1998). A contextual investigation of three-digit addition and subtraction. In L.J. Morrow & M.J. Kenney (Eds.),The teaching and learning of algorithms in school mathematics (pp. 141–150). Reston: National Council of Teachers of Mathematics.Google Scholar
  81. Milo, B., & Ruijssenaars, A.J.J.M. (2002). Strategiegebruik van leerlingen in het speciaal basisonderwijs: Begeleiden of sturen? [Strategy instruction in special education: Guided of direct instruction?]Pedagogische Studiën, 79, 117–129.Google Scholar
  82. Moser Opitz, E. (2001). Mathematical knowledge and progress in the mathematical learning of children with special needs in their first year of school. InMATHE 2000. Selected papers (pp. 85–88). Dortmund, Germany: University of Dortmund, Department of Mathematics.Google Scholar
  83. National Council of Teachers of Mathematics. (1989).Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  84. National Council of Teachers of Mathematics. (2000).Principles and standards for school mathematics. htmGoogle Scholar
  85. Nunes, T., Schliemann, A., & Carraher, D. (1993).Street mathematics and school mathematics. Cambridge, UK: Cambridge University Press.Google Scholar
  86. Ontwikkelingsdoelen en eindtermen. Informatiemap voor de onderwijspraktijk. Gewoon basisonderwijs (1998). [Standards. Documentation for practitioners. Elementary education.] Brussel: Ministerie van de Vlaamse Gemeenschap, Departement Onderwijs, Afdeling Informatie en Documentatie.Google Scholar
  87. Perkins, D. (1995).Outsmarting IQ: The emerging science of learnable intelligence. New York: The Free Prees.Google Scholar
  88. 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 Prees.Google Scholar
  89. Rogoff, B. (1990).Apprenticeship in thinking. New York: Oxford University Press.Google Scholar
  90. Schauble, L., & Glaser, R. (Eds.). (1996).Innovations in learning: New environments for education. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  91. Selter, C. (1998). Building on children’s mathematics. A teaching experiment in grade three.Educational Studies in Mathematics, 36, 1–27.CrossRefGoogle Scholar
  92. Shrager, J., & Siegler, R.S. (1998). SCADS: A model of children’s strategy choices and strategy discoveries.Psychological Science, 9, 405–410.CrossRefGoogle Scholar
  93. Siegler, R.S. (1996).Emerging minds: The process of change in children’s thinking. Oxford: Oxford University Press.Google Scholar
  94. Siegler, R.S. (1998).Children’s thinking. New Jersey: Prentice Hall.Google Scholar
  95. Siegler, R.S. (2000). The rebirth of children’s learning.Child Development, 71, 26–35.CrossRefGoogle Scholar
  96. 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.Google Scholar
  97. Siegler, R.S., & Jenkins, E. (1989).How children discover new strategies. Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  98. Siegler, R.S., & Lemaire, P. (1997). Older and younger adults’ strategy choices in multiplication: Testing predictions of ASCM using the choice/no-choice method.Journal of Experimental Psychology: General, 126, 71–92.CrossRefGoogle Scholar
  99. Sowder, J., Philipp, R., Armstrong, B., & Schappelle, B. (1998).Middle-grade teachers’ mathematical knowledge and its relationship to instruction. Albany, NY: SUNY.Google Scholar
  100. Spearman, C. (1927).The abilities of man, their nature and measurement. London: Macmillan.Google Scholar
  101. Spiro, R.J. (1980). Constructive processes in prose comprehension and recall. In R.J. Spiro, B.C. Bruce, & W.F. Brewer (Eds.),Theoretical issues in reading comprehension (pp. 245–278). Hillsdale, NJ: Erlbaum.Google Scholar
  102. Stanonich, K.E., & West, R.F. (2000). Individual differences in reasoning: Implications for the rationality debate.Behavioral and Brain Sciences, 23, 645–726.CrossRefGoogle Scholar
  103. Straker, A. (1999). The National Numeracy project: 1996–99. In I. Thompson (Ed.),Issues in teaching numeracy in primary schools (pp. 39–48). Buckingham; UK: Open University Press.Google Scholar
  104. Star, J. (2005). Reconceptualizing procedural knowledge.Journal for Research in Mathematics Education, 36, 404–411.Google Scholar
  105. Thompson, I. (1999). Getting your head around mental calculation. In I. Thompson (Ed.),Issues in teaching numeracy in primary schools (pp. 145–156). Buckingham, UK: Open University Press.Google Scholar
  106. Threlfall, J. (2002). Flexible mental calculation.Educational Studies in Mathematics, 50, 29–47.CrossRefGoogle Scholar
  107. Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2004). Strategy development in children with mathematical disabilities: Insights from the choice/no-choice method and the chronological-age/ability-level-match design.Journal of Learning Disabilities, 37, 119–131.CrossRefGoogle Scholar
  108. Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2005). Simple addition strategies in a first-grade class with multiple strategy instruction.Cognition and Instruction, 23, 1–21.CrossRefGoogle Scholar
  109. Torbeyns, J., Arnaud, L., Lemaire, P., & Verschaffel, L. (2004). Cognitive change as strategic change. In A. Demetriou & A. Raftopoulos (Eds.),Cognitive developmental change: Theories, models and measurement (pp. 186–216). Cambridge, UK: Cambridge University Press.Google Scholar
  110. Treffers, A. (1987).Three dimensions. A model of goal and theory description in mathematics education. The Wiskobas project. Dordrecht, The Netherlands: Reidel.Google Scholar
  111. Treffers, A., De Moor, E., & Feijs, E. (1990).Proeve van een national programma voor het reken/wiksundeonderwijs op de basisschool. Deel 1. Overzicht einddoelen. [Towards a national curriculum for mathematics education in the elementary school. part 1. Overview of the goals.] Tilburg, The Netherlands: Zwijsen.Google Scholar
  112. Van den Heuvel-Panhuizen, M. (Ed.). (2001).Children learn mathematics. A learning-teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school. Groningen, The Netherlands: Wolters Noordhoff.Google Scholar
  113. Van der Heijden, M.K. (1993).Consistentie van aanpakgedrag. [Consistency in solution behavior.] Lisse, The Netherlands: Swets & Zeitlinger.Google Scholar
  114. Van Dooren, W., Verschaffel, L., Greer, B., & De Bock, D. (2006). Modelling for life: Developing adaptive expertise in mathematical modelling from an early age. In L. Verschaffel, F. Dochy, M. Boekaerts, & S. Vosniadou (Eds.),Instructional psychology: Past, present, and future trends. Sixteen essays in honour of Erik De Corte (pp. 91–112). Oxford, UK: Elsevier.Google Scholar
  115. Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F.K. Lester (Ed.),Second handbook of research on mathematics teaching and learning (pp. 557–628). Greenwich, CT: Information Age Publishing.Google Scholar
  116. Warner, L.B. (2005). Behaviors that indicate mathematical flexible thought (Doctoral dissertation. Rutgers, The State University of New Jersey, 2005).Dissertation Abstracts International, 66/01, 123.Google Scholar
  117. Warner, L.B., Davis, G.E., Alcock, L.J., & Coppolo, J. (2002). Flexible mathematical thinking and multiple representations in middle school mathematics.Mediterranean Journal for Research in Mathematics Education, 1(2), 37–61.Google Scholar
  118. Wertheimer, M. (1945).Productive thinking. London: Tavistock.Google Scholar
  119. Wittmann, E.Ch. (1995). Mathematics education as a design science.Educational Studies in Mathematics, 29, 355–374.CrossRefGoogle Scholar
  120. Wittmann, E.Ch., & Müller, G.N. (1990–1992).Handbuch Produktiver Rechenübungen. Vols 1 & 2 [Handbook of productive arithmetic exercises. Volume 1 & 2]. Düsseldorf und Stuttgart. Germany: Klett Verlag.Google Scholar
  121. Wittmann, E.Ch., & Müller, G.N. (2004).Das Zahlenbuch. Mathematik im 1. Schuljahr. [The book of numbers Mathematics in grade 1.] Düsseldorf und Stuttgart, Germany: Klett Verlag.Google Scholar
  122. Woodward, J., & Baxter, J. (1997). The effects of an innovative approach to mathematics on academically low achieving students in inclusive settings.Exceptional Children, 63, 373–388.Google Scholar
  123. Woodward, J., Monroe, K., & Baxter, J. (2001). Enhancing student achievement on performance assessments in mathematics.Learning Disabilities Quarterly, 24 (Winter), 33–46.CrossRefGoogle Scholar
  124. Yackel, E., & Cobb, P. (1996). Classroom sociomathematical norms and intellectual autonomy.Journal for Research in Mathematics Education, 27, 458–477.CrossRefGoogle Scholar

Copyright information

© Instituto Superior de Psicologia Aplicada, Lisbon, Portugal/ Springer Netherlands 2009

Authors and Affiliations

  • Lieven Verschaffel
    • 1
    Email author
  • Koen Luwel
    • 1
  • Joke Torbeyns
    • 1
    • 2
  • Wim Van Dooren
    • 1
  1. 1.Centre for Instructional Psychology and TechnologyKatholieke Universiteit LeuvenLeuvenBelgium
  2. 2.Leuven Education CollegeLeuvenBelgium

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