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

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Abstract

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.

Résumé

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.

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References

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • Baroody, A.J., & Dowker, A. (Eds.). (2003). The development of arithmetic concepts and skills.Constructing adaptive expertise. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • 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 

  • 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.

  • 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.

    Article  Google Scholar 

  • 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 

  • Berar, I. (2004).Flexibility of cognitive processes — Indicator of giftedness. (Unpublished manuscript). Available at http://www.history-cluj.ro/SU/cercet/Berar/

  • Bereiter, C., & Scardamalia, M. (1993).Surpassing ourselves: An inquiry into the nature and implications of expertise. Chicago: Open Court.

    Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings.Journal for Research in Mathematics Education, 29, 41–62.

    Article  Google Scholar 

  • Boaler, J. (2000). Exploring situated insights into research and learning.Journal for Research in Mathematics Education, 39, 113–119.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Bransford, J. (2001).Thoughts on adaptive expertise (unpublished manuscript). Available at http://www.vanth.org/docs/ AdaptiveExpertise.pdf

  • 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 

  • Brownell, W.A. (1945). When is arithmetic meaningful?Journal of Educational Research, 38, 481–498.

    Google Scholar 

  • Bruner, J. (1986).Actual minds, possible worlds. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • 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 

  • Cajori, F. (1917).A history of elementary mathematics. New York: Macmillan.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Cattell, R. (1946). The riddle of perseveration.Journal of Personality, 14, 239–267.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • Crowley, K., & Siegler, R.S. (1993). Flexible strategy use in young children’s tic-tac-toe.Cognitive Science, 17, 531–561.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • Duncker, K. (1945). On problem solving.Psychological Monographs,58(5, Whole no 270).

  • Dunn, J.A. (1975). Tests of creativity in mathematics.International Journal of Mathematical Education in Science and Technology, 6, 327–332.

    Article  Google Scholar 

  • 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 Society

    Google Scholar 

  • Ellis, S. (1997). Strategy choice in sociocultural context.Developmental Review, 17, 490–524.

    Article  Google Scholar 

  • Engestrom, Y. (1987).Learning by expanding: An activity-theoretical approach to developmental research. Helsinki, Finland: Orienta-Konsultit Oy.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • Freudenthal, H. (1991).Revisiting mathematics education. Dordrecht: Reidel.

    Google Scholar 

  • Frobisher, L., & Threlfall, J. (1998).Teaching mental maths strategies. Heinemann: Oxford.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • Gravemeijer, K. (1994).Developing realistic mathematics education. Utrecht, The Netherlands: Freudenthal Institute, University of Utrecht.

    Google Scholar 

  • Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education.Mathematical Thinking and Learning, 6, 105–128.

    Article  Google Scholar 

  • 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 

  • Greer, B. (1997). Modelling reality in mathematics classrooms: The case of word problems.Learning and Instruction, 7, 293–307.

    Article  Google Scholar 

  • Groen, G.J., & Parkman, J.M. (1972). A chronometric analysis of simple addition.Psychological Review, 79, 329–343.

    Article  Google Scholar 

  • Guilford, J.P. (1967).The nature of human intelligence. New York: McGraw-Hill.

    Google Scholar 

  • 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 

  • 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 

  • Hatano, G., & Oura, Y. (2003). Reconceptualizing school learning using insight from expertise research.Educational Researcher, 32(8), 26–29.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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.

    Article  Google Scholar 

  • Hiebert, J. (Ed.). (1986).Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Ishida, J. (2002). Students’ evaluation of their strategies when they find several solutions.Journal of Mathematical Behavior, 21, 49–56.

    Article  Google Scholar 

  • Jausovec, N. (1994).Flexible thinking: An explanation for individual differences in ability, Cresskill, NJ: Hampton Press.

    Google Scholar 

  • Kahneman, D., Slovic, P., & Tversky, A. (1982).Judgment under uncertainty: Heuristics and biases. New York: Cambridge University Press.

    Google Scholar 

  • 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 

  • Kilpatrick, J., Swafford, J., & Findell, B. (2001).Adding it up. Helping children learn mathematics Washington, D.C.: National Academy Press.

    Google Scholar 

  • Klayman, J. (1985). Children’s decision strategies and their adaptation to task charecteristics.Organizational Behavior and Human Decision Processes, 35, 179–201.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Krutetskii, V.A. (1976).The psychology of mathematical abilities in school children Chicago: University of Chicago Press.

    Google Scholar 

  • Lave, J., & Wenger, E. (1991).Situated learning: Legitimate, peripheral participation. Cambridge: Cambridge University Press.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Lemaire, P., & Lecacheur, M. (2002). Children’s strategies in computational estimation.Journal of Experimental Child Psychology, 82, 281–304.

    Article  Google Scholar 

  • 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 

  • Leron, U., & Hazzan, O. (1997). The world according to Johny: A coping perspective in mathematics education.Educational Studies in Mathematics, 32, 265–292.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • Luwel, K., Verschaffel, L., & Lemaire, P. (2005). Children’s strategies in numerosity judgment.Cognitive Development, 20, 448–471.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • National Council of Teachers of Mathematics. (1989).Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • National Council of Teachers of Mathematics. (2000).Principles and standards for school mathematics. http://standards.nctm.org/document/index. htm

  • Nunes, T., Schliemann, A., & Carraher, D. (1993).Street mathematics and school mathematics. Cambridge, UK: Cambridge University Press.

    Google Scholar 

  • 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.

  • Perkins, D. (1995).Outsmarting IQ: The emerging science of learnable intelligence. New York: The Free Prees.

    Google Scholar 

  • 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 

  • Rogoff, B. (1990).Apprenticeship in thinking. New York: Oxford University Press.

    Google Scholar 

  • Schauble, L., & Glaser, R. (Eds.). (1996).Innovations in learning: New environments for education. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Selter, C. (1998). Building on children’s mathematics. A teaching experiment in grade three.Educational Studies in Mathematics, 36, 1–27.

    Article  Google Scholar 

  • Shrager, J., & Siegler, R.S. (1998). SCADS: A model of children’s strategy choices and strategy discoveries.Psychological Science, 9, 405–410.

    Article  Google Scholar 

  • Siegler, R.S. (1996).Emerging minds: The process of change in children’s thinking. Oxford: Oxford University Press.

    Google Scholar 

  • Siegler, R.S. (1998).Children’s thinking. New Jersey: Prentice Hall.

    Google Scholar 

  • Siegler, R.S. (2000). The rebirth of children’s learning.Child Development, 71, 26–35.

    Article  Google Scholar 

  • 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 

  • Siegler, R.S., & Jenkins, E. (1989).How children discover new strategies. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Sowder, J., Philipp, R., Armstrong, B., & Schappelle, B. (1998).Middle-grade teachers’ mathematical knowledge and its relationship to instruction. Albany, NY: SUNY.

    Google Scholar 

  • Spearman, C. (1927).The abilities of man, their nature and measurement. London: Macmillan.

    Google Scholar 

  • 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 

  • Stanonich, K.E., & West, R.F. (2000). Individual differences in reasoning: Implications for the rationality debate.Behavioral and Brain Sciences, 23, 645–726.

    Article  Google Scholar 

  • 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 

  • Star, J. (2005). Reconceptualizing procedural knowledge.Journal for Research in Mathematics Education, 36, 404–411.

    Google Scholar 

  • 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 

  • Threlfall, J. (2002). Flexible mental calculation.Educational Studies in Mathematics, 50, 29–47.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Treffers, A. (1987).Three dimensions. A model of goal and theory description in mathematics education. The Wiskobas project. Dordrecht, The Netherlands: Reidel.

    Google Scholar 

  • 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 

  • 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 

  • Van der Heijden, M.K. (1993).Consistentie van aanpakgedrag. [Consistency in solution behavior.] Lisse, The Netherlands: Swets & Zeitlinger.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • Wertheimer, M. (1945).Productive thinking. London: Tavistock.

    Google Scholar 

  • Wittmann, E.Ch. (1995). Mathematics education as a design science.Educational Studies in Mathematics, 29, 355–374.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • Woodward, J., Monroe, K., & Baxter, J. (2001). Enhancing student achievement on performance assessments in mathematics.Learning Disabilities Quarterly, 24 (Winter), 33–46.

    Article  Google Scholar 

  • Yackel, E., & Cobb, P. (1996). Classroom sociomathematical norms and intellectual autonomy.Journal for Research in Mathematics Education, 27, 458–477.

    Article  Google Scholar 

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Authors and Affiliations

  1. Centre for Instructional Psychology and Technology, Katholieke Universiteit Leuven, Vesaliusstraat 2, 3000, Leuven, Belgium

    Lieven Verschaffel, Koen Luwel, Joke Torbeyns & Wim Van Dooren

  2. Leuven Education College, Leuven, Belgium

    Joke Torbeyns

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  1. Lieven Verschaffel
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  2. Koen Luwel
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  3. Joke Torbeyns
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  4. Wim Van Dooren
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Correspondence to Lieven Verschaffel.

Additional information

This research was partially supported by Grant GOA 2006/01 “Developing adaptive expertise in mathematics education” from the Research Fund K.U. Leuven, Belgium.

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Verschaffel, L., Luwel, K., Torbeyns, J. et al. Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. Eur J Psychol Educ 24, 335–359 (2009). https://doi.org/10.1007/BF03174765

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