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3 - Developmental patterns in proportional reasoning

Published online by Cambridge University Press:  22 September 2009

By
Han Van Der Maas ,
Brenda Jansen  and
Maartje Raijmakers
Edited by
Andreas Demetriou  and
Athanassios Raftopoulos
Han Van Der Maas
Affiliation:
Department of Psychology, University of Amsterdam, The Netherlands
Brenda Jansen
Affiliation:
Department of Psychology, University of Amsterdam, The Netherlands
Maartje Raijmakers
Affiliation:
Department of Psychology, University of Amsterdam
Andreas Demetriou
Affiliation:
University of Cyprus
Athanassios Raftopoulos
Affiliation:
University of Cyprus
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Summary

The editors of this book asked the contributors to answer three basic questions about cognitive development: what, how and why. We will attempt to answer these questions by a careful study of one famous case of cognitive development: Piaget's balance scale task for assessing proportional reasoning.

In the past, we have investigated this task in various ways and we will take the present opportunity to summarize and integrate our main findings. We approach the study of cognitive development from a methodological point of view. In each of the forthcoming sections, we apply a new technique in order to get a new perspective on the developmental process.

We first shortly discuss the background of research on the balance scale task. In the second section we attempt to resolve the criticism of Siegler's (1981) rule theory, the main theory for balance scale reasoning, by introducing categorical latent structure models for rule assessment. In the third section we attempt to validate and extend this rule theory with response time (RT) measures. Using RTs allows the test of the rule theory in great detail and leads to a number of specifications and improvements of the theory. Fourth, we summarize our study of the transitions between the rules. Knowledge of these transitions is an important step in understanding the mechanisms involved in the development of proportional reasoning. We focus on a particular transition, from Rule I to Rule II, to investigate whether this transition is a genuine phase transition.

Type
Chapter
Information
Cognitive Developmental Change
Theories, Models and Measurement
 , pp. 118 - 156
Publisher: Cambridge University Press
Print publication year: 2005

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References

Anderson, J. R. and Lebiere, C. (1998). The atomic components of thought. Mahwah, NJ: Lawrence Erlbaum Associates, Inc
Bates, E. A. and Elman, J. L. (1993). Connectionism and the study of change. In Mark H. Johnson (ed.) Brain development and cognition. Oxford: Blackwell
Bijstra, J., Geert, P. and Jackson, S. (1989). Conservation and the appearance–reality distinction: what do children really know and what do they answer? British Journal of Developmental Psychology, 7, 43–53CrossRefGoogle Scholar
Boden, M. A. (1994). Piaget. London: Fontana Press
Boom, J., Hoijtink, H. and Kunnen, S. (2001). Rules in the balance: classes strategies, or rules for the balance scale task. Cognitive Development, 16, 717–35CrossRefGoogle Scholar
Braine, M. D. S. and Shanks, B. L. (1965). The development of conservation of size. Journal of Verbal Learning and Verbal Behavior, 4, 227–42CrossRefGoogle Scholar
Brainerd, C. J. (1973). Judgements and explanations as criteria for the presence of cognitive structures. Psychological Bulletin, 79, 172–9CrossRefGoogle Scholar
Brainerd, C. J. (1978). The stage question in cognitive-developmental theory. The Behavioral and Brain Sciences, 2, 173–213CrossRefGoogle Scholar
Brainerd, C. J. (1979). Markovian interpretations of conservation learning. Psychological Review, 86, 181–213CrossRefGoogle Scholar
Chletsos, P. N., Lisi, R., Turner, G. and McGillicuddy-De Lisi, A. V. (1989). Cognitive assessment of proportional reasoning strategies. Journal of Research and Development in Education, 23, 18–27Google Scholar
Clogg, C. C. (1995). Latent class models. In G. Arminger, C. C. Clogg and M. E. Sobel (eds.) Handbook of statistical modeling for the social and behavioral sciences. New York: Plenum PressCrossRef
Collins, L. M. and Wugalter, S. E. (1992). Latent class models for stage-sequential dynamic latent variables. Multivariate Behavioral Research, 27, 131–57CrossRefGoogle Scholar
Demetriou, A., Christou, C., Spanoudis, G. and Platsidou, M. (2002). The development of mental processing: efficiency, working memory, and thinking. Monographs of the Society for Research in Child Development, 67(1)
Donaldson, M. (1978). Children's minds. London: Fontana
Eckstein, S. G. (2000). Growth of cognitive abilities: dynamic models and scaling. Developmental Review, 20, 1–28CrossRefGoogle Scholar
Elbers, E. (1989). Het conservatie-experiment en de verwachting van het kind over de interaktie met de proefleider. Tijdschrift voor Ontwikkelings Psychologie, 16, 42–8Google Scholar
Ferretti, R. P. and Butterfield, E. C. (1986). Are children's rule-assessment classifications invariant across instances of problem types? Child Development, 57, 1419–28CrossRefGoogle ScholarPubMed
Ferretti, R. P. and Butterfield, E. C. (1989). Intelligence as a correlate of children's problem solving. American Journal on Mental Retardation, 93 (4), 424–33Google ScholarPubMed
Ferretti, R. P. and Butterfield, E. C. (1992). Intelligence-related differences in the learning, maintenance, and transfer of problem-solving strategies. Intelligence, 16(2), 207–23CrossRefGoogle Scholar
Ferretti, R. P., Butterfield, E. C., Cah, A. and Kerkman, D. (1985). The classification of children's knowledge: development on the balance-scale and inclined-plane tasks. Journal of Experimental Child Psychology, 39(1), 131–60CrossRefGoogle ScholarPubMed
Gilmore, R. (1981). Catastrophe theory for scientists and engineers. New York: Wiley
Gruen, G. E. (1966). Note on conservation: methodological and definitional considerations. Child Development, 37, 977–83CrossRefGoogle Scholar
Halford, G. S., Andrews, G., Dalton, C., Boag, C. and Zielinski, T. (2002). Young children's performance on the balance scale: the influence of relational complexity. Journal of Experimental Child Psychology, 81, 417–45CrossRefGoogle ScholarPubMed
Heinen, T. (1996). Latent class and discrete latent trait models: similarities and differences. Thousand Oaks, CA: Sage Publications
Inhelder, B. and Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. New York: Basic Books
Inhelder, B., Bovet, M., Sinclair, H. and Smock, C. D. (1966). On cognitive development. American Psychologist, 21, 160–4CrossRefGoogle Scholar
Jansen, B. R. J. and Maas, H. L. J. (1997). Statistical test of the rule assessment methodology by latent class analysis. Developmental Review, 17, 321–57CrossRefGoogle Scholar
Jansen, B. R. J. and Maas, H. L. J. (2001). Evidence for the phase transition from Rule I to Rule II on the balance scale task. Developmental Review, 21, 450–94CrossRefGoogle Scholar
Jansen, B. R. J. and Maas, H. L. J. (2002). Development of children's rule use on the balance scale task. Journal of Experimental Child Psychology, 81(4), 383–416CrossRefGoogle ScholarPubMed
Klahr, D. and Siegler, R. (1978). The representation of children's knowledge. In H. Reese and L. Lipsitt (eds.) Advances in child development and behavior (pp. 61–116). New York: Academic PressCrossRef
Kliman, M. (1987). Children's learning about the balance scale. Instructional Science, 15, 307–40CrossRefGoogle Scholar
Langeheine, R., Pannekoek, J. and van de Pol, F. (1995). Bootstrapping goodness-of-fit measures in categorical data analysis. CBS, Statistics Netherlands
Marini, Z. and Case, R. (1994). The development of abstract reasoning about the physical and social world. Child Development, 65, 147–59CrossRefGoogle Scholar
McClelland, J. L. (1989). Parallel distributed processing: implications for cognition and development. In M. G. M. Morris (ed.) Parallel distributed processing, implications for psychology and neurobiology (pp. 8–45). Oxford: Clarendon Press
McClelland, J. L. (1995). A connectionist perspective on knowledge and development. In T. J. Simon and G. S. Halford (eds.) Developing cognitive competence: new approaches to process modeling (pp. 157–204). Lawrence Erlbaum Associates
McClelland, J. L. and Jenkins, E. (1991). Nature, nurture, and connections: implications of connectionist models for cognitive development. In K. van Lehn (ed.) Architectures for intelligence: the twenty-second Carnegie Mellon Symposium on cognition (pp. 41–73). Pittsburgh, PA: Lawrence Erlbaum Associates
McFadden, G. T., Dufresne, A. and Kobasigawa, A. (1987). Young children's knowledge of balance scale problems. Journal of Genetic Psychology, 148, 79–94CrossRefGoogle Scholar
McGarrigle, J. and Donaldson, M. (1975). Conservation accidents. Cognition, International Journal of Cognitive Psychology, 3, 341–9CrossRefGoogle Scholar
Moore, C. and Frye, D. (1986). Context, conservation and the meanings of more. British Journal of Developmental Psychology, 4, 169–78CrossRefGoogle Scholar
Normandeau, S., Larivée, S., Roulin, J. L. and Longeot, F. (1989). The balance-scale dilemma: either the subject or the experimenter muddles through. Journal of Genetic Psychology, 150, 237–50CrossRefGoogle ScholarPubMed
Pauen, S. and Wilkening, F. (1997). Children's analogical reasoning about natural phenomena. Journal of Experimental Child Psychology, 67 (1), 90–113CrossRefGoogle ScholarPubMed
Phelps, E. and Damon, W. (1989). Problem solving with equals: peer collaboration as a context for learning mathematics and spatial contexts. Journal of Educational Psychology, 81 (4), 639–46CrossRefGoogle Scholar
Raijmakers, M. E. J., Koten, S. and Molenaar, P. C. M. (1996). On the validity of simulating stagewise development by means of PDP networks: application of catastrophe analysis and an experimental test of rule-like network performance. Cognitive Science, 20(1), 101–36CrossRefGoogle Scholar
Reese, H. W. (1989). Rules and rule-governance: cognitive and behavioristic views. In S. C. Hayes (ed.) Rule governed behavior: cognition, contingencies, and instructional control (pp. 3–84). New York: Plenum PressCrossRef
Richards, D. D. and Siegler, R. S. (1981). Very young children's acquisition of systematic problem-solving strategies. Child Development, 52, 1318–21CrossRefGoogle Scholar
Rindskopf, D. (1987). Using latent class analysis to test developmental models. Developmental Review, 7, 66–85CrossRefGoogle Scholar
Rose, S. A. and Blank, M. (1974). The potency of context in children's cognition: an illustration through conservation. Child Development, 45, 499–502CrossRefGoogle Scholar
Roth, W. M. (1991). The development of reasoning on the balance beam. Journal of Research in Science Teaching, 28, 631–45CrossRefGoogle Scholar
Sage, S. and Langley, P. (1983). Modeling cognitive development on the balance scale task. In A. Bundy (ed.) Proceedings of the eighth international joint conference on artificial intelligence (vol. 1, pp. 94–6). Karlsruhe, Germany
Schiff, W. (1983). Conservation of length redux: a perceptual–linguistic phenomenon. Child Development, 54, 1497–1506CrossRefGoogle Scholar
Schmidt, W. C. and Ling, C. X. (1996). A decision-tree model of balance scale development. Machine Learning, 24, 203–30CrossRefGoogle Scholar
Shultz, T. R., Schmidt, W. C., Buckingham, D. and Mareschal, D. (1995). Modeling cognitive development with a generative connectionist algorithm. In T. J. Simon and G. S. Halford (eds.) Developing cognitive competence: new approaches to process modeling (pp. 205–61). Hillsdale, NJ: Lawrence Erlbaum Associates
Siegal, M. (1991). Knowing children: experiments in conversation and cognition. essays in developmental psychology. Hove, UK: Lawrence Erlbaum Associates
Siegler, R. S. (1976). Three aspects of cognitive development. Cognitive Psychology, 8, 481–520CrossRefGoogle Scholar
Siegler, R. S. (1981). Developmental sequences within and between concepts. Monographs of the Society for Research in Child Development, 46(2), 1–74. (With commentary and reply from Sidney Strauss and Iris Levin.)CrossRefGoogle Scholar
Siegler, R. S. (1996). Emerging minds: the process of change in children's thinking. New York: Oxford University Press
Siegler, R. S. and Chen, Z. (1998). Developmental differences in rule learning: a microgenetic analysis. Cognitive Psychology, 36(3), 273–310CrossRefGoogle ScholarPubMed
Smedslund, J. (1963). Development of concrete transitivity of length in children. Child Development, 34, 389–405Google ScholarPubMed
Smedslund, J. (1969). Psychological diagnostics. Psychological Bulletin, 71, 234–48CrossRefGoogle ScholarPubMed
Smith, L. (1992). Judgements and justifications: criteria for the attribution of children's knowledge in Piagetian research. British Journal of Developmental Psychology, 10, 1–23CrossRefGoogle Scholar
Smith, L. (1993). Necessary knowledge: Piagetian perspectives on constructivism, Essays in Developmental Psychology Series. Hove, UK: Lawrence Erlbaum Associates
Strauss, S. and Levin, I. (1981). Commentary on Siegler's ‘Developmental sequences within and between concepts’. Monographs of the Society for Research in Child Development, 46 (2, serial no. 189), 75–81Google Scholar
Surber, C. F. and Gzesh, S. M. (1984). Reversible operations in the balance scale task. Journal of Experimental Child Psychology, 38, 254–74CrossRefGoogle Scholar
Taatgen, N. A. and Anderson, J. R. (2002). Why do children learn to say ‘broke’? A model of learning the past tense without feedback. Cognition 86 (2), 123–55CrossRefGoogle ScholarPubMed
Taatgen, N. A., van Rijn, H. and Zondervan, K. (2003). Explaining developmental transitions through constrained generalization. In preparation
Thelen, E. and Smith, L. B. (1994). A dynamic systems approach to the development of cognition and action. Cambridge, MA: MIT Press
Thom, R. (1975). Structural stability and morphogenesis: an outline of a general theory of models. Reading, MA: Benjamin
Thomas, H. and Lohaus, A. (1993). Modeling growth and individual differences in spatial tasks. Monographs of the Society for Research in Child Development, 58 (9, serial no. 237)CrossRef
Thorndike, E. L. (1913). Educational psychology: the psychology of learning (vol. 2). New York: Teachers College PressCrossRef
Turner, F. W. and Thomas, H. (2002). Bridging the gap between theory and model: a reflection on the balance scale task. Journal of Experimental Child Psychology, 81(4), 466–81CrossRefGoogle ScholarPubMed
van de Pol, F., Langeheine, R. and De Jong, W. (1996). PANMARK 3. Panel analysis using Markov chains. A latent class analysis program [User manual]. Voorburg, The Netherlands
Maas, H. L. J. (1995). Beyond the metaphor? Cognitive Development, 10, 621–42Google Scholar
Maas, H. L. J. and Jansen, B. R. J. (2003). What response times tell of children's behaviour on the balance scale task. Journal of Experimental Child Psychology, 85, 141–77CrossRefGoogle ScholarPubMed
Maas, H. L. J. and Molenaar, P. C. M. (1992). Stagewise cognitive development: an application of catastrophe theory. Psychological Review, 99, 395–417CrossRefGoogle ScholarPubMed
van der Maas, H. L. J. and Molenaar, P. C. M. (1996). Catastrophe analysis of discontinuous development. In A. von Eye and C. C. Clogg (eds.) Categorical variables in developmental research. Methods of analysis. San Diego, CA: Academic PressCrossRef
van der Maas, H. L. J., M. E. J. Raijmakers, Hartelman, P. A. I. and Molenaar, P. C. M. (1998). Reaction time as recovery time after pertubation. In Savelsbergh, G., van der Maas, H. L. J. and van Geert, P. (1999). Nonlinear analysis of developmental processes. Amsterdam: KNAW
Geert, P. (1998). A dynamic systems model of basic developmental mechanisms: Piaget, Vygotsky, and beyond. Psychological Review, 105(4), 634–77CrossRefGoogle Scholar
van Maanen, L., Been, P. H. and Sijtsma, K. (1989). The linear logistic test model and heterogeneity of cognitive strategies. In E. E. Roskam (ed.) Mathematical psychology in progress (pp. 267–88). Berlin, Germany: Springer-VerlagCrossRef
Rijn, H., Someren, M. and Maas, H. L. J. (2003). Modeling developmental transitions on the balance scale task. Cognitive Science, 27(2), 227–57CrossRefGoogle Scholar
Warton, P. and Bussey, K. (1988). Assisted learning: levels of support. British Journal of Developmental Psychology, 6 (2), 113–23CrossRefGoogle Scholar
Wilkening, F. and Anderson, N. H. (1982). Comparison of two rule-assessment methodologies for studying cognitive development and knowledge structure. Psychological Bulletin, 92, 215–37CrossRefGoogle Scholar
Wilson, M. (1989). Saltus: a psychometric model of discontinuity in cognitive development. Psychological Bulletin, 105, 276–89CrossRefGoogle Scholar

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