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This study seeks to answer the question: "What kinds of mental representation do children\ud project and how may these be associated with their level of achievement in elementary\ud arithmetic?". Drawing upon theories offering some explanation for the way in which\ud arithmetical activity is transformed into numerical concepts and those that hypothesise the\ud form and quality of mental representations the study suggests that qualitatively different\ud kinds of mental representation may be associated with qualitatively different kinds of\ud arithmetical behaviour.\ud The evidence is drawn from the classification and categorisation of data from two series of\ud semi-clinical interviews carried out with children aged eight to twelve who were at extremes\ud of numerical achievement. The first, a pilot study, largely concentrated on mental\ud representations associated with numerical concepts and skills. Its results suggest that\ud mental representations projected by children may have a disposition towards different kinds\ud of mental representation which transcends arithmetical and non-arithmetical boundaries.\ud Issues raised by this study, in conjunction with a re-appraisal of the psychological\ud evidence, informed the development of the main study. With a similar sample of children\ud this considered the relationship between children's projections, reports and descriptions of\ud mental representations in numerical and non-numerical contexts and in elementary\ud arithmetic. Words, pictures, icons and symbols stimulated the projection of these\ud representations. The evidence suggests that there is indeed a disposition towards the\ud formation of particular kinds of mental representation. low achievers' projected mental\ud representations which have descriptive emphasis. 'High achievers', whilst able to do the\ud same, also project those with relational characteristics, the frequency of which increases as\ud the stimulus becomes more 'language like'. This provides them with the flexibility to\ud oscillate between descriptive and abstract levels of thought.\ud The study indicates that qualitative different thinking in number processing is closely\ud associated to a disposition towards qualitatively different kinds of mental representation. Its\ud concluding comments suggest that these differences may have some considerable\ud implication for the received belief that active methods may supply all children with a basis\ud for numerical understanding.
4.3.1 The Disguised Nature of Mental Representations ......................... 70 4.3.2 Metacognition ................................................................. 71 4.3.3 Information Processing ......................................................
4.3.4 Importing from Psychology ................................................. 72 4.3.5 A Phenomenographical Orientation ...................... • .................. 74 4.3.6 Semi-Clinical Interviewing ............................ . . .................. -77-57 4.3.7 Reliability and Validity .........................................................
4.4.1 Refining the Approach .......................................................
4.4.2 Evoking Mental Representations ............................................ 81 4.4.3 Developing the Verbal Item Bank .......................................... 82 4.4.4 Presenting the Verbal Item Bank ......................... 84 4.4.5 The Visual Item Bank ....................................................... 86 4.4.6 Presenting the Visual Item Bank ........................................... 88 4.4.7 The Numerical Components_ ............................................ 89 Jaworski, B. (1988). 'Is" versus "seeing as": Constructivism and the mathematics classroom'. In D. Pimm (Ed.), Mathematics, Teachers and Children. London: Hodder and Staughton.
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