SCAFFOLDING REVISITED: FROM TOOL FOR RESULT TO TOOL-AND-RESULT

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1 SCAFFOLDING REVISITED: FROM TOOL FOR RESULT TO TOOL-AND-RESULT Mike Askew King's College London & City College, New York The metaphor of scaffolding is popular in mathematics education, particularly in accounts purporting to examine the mediated nature of learning. Drawing on Holtzman and Newman s interpretation of Vygotsky I argue that scaffolding rests on a dualistic view that separates the knower from the known. In line with their work, I being to explore what an alternative metaphor development through performance (in the theatrical sense) might mean for mathematics education. INTRODUCTION Years ago, my colleagues, Joan Bliss and Sheila Macrae, and I conducted a study into the notion of scaffolding (Bruner 1985) in primary (elementary) school mathematics and science. At the time we had difficulty finding any evidence of scaffolding in practice! What we observed in lessons could just as easily be categorized as explaining or showing. Scaffolding, in the sense of providing a support for the learner, a support that could be removed to leave the structure of learning to stand on its own, was elusive (Bliss, Askew & Macrae, 1996). At the time this elusiveness seemed to be a result of two things. Firstly the nature of what was to be learnt. Many of the examples in relevant literature (for example, Rogoff (1990), Lave and Wenger (1991)) attend to learning that has clear, concrete outcomes. Becoming tailors or weavers means that the objects of the practice a jacket or a basket are apparent to the learner: the apprentice knows in advance of being able to do it, what it is that is being produced. In contrast, most (if not all) of mathematics is not known until the learning is done. Young children have no understanding of, say, multiplication, in advance of coming to learn about multiplication: the object of the practice only become apparent after the learning has taken place. (This is not to suggest that learners do not have informal knowledge that might form the basis of an understanding of multiplication, only that such informal knowledge is different and distinct from formal knowledge of multiplication.) Secondly, examples of scaffolding that did seem convincing focused on schooling situations that are close to apprenticeship models (Lave and Wenger, ibid.) in that the teacher-learner interaction is mainly one-to one. For example, Clay (1990) provides a strong Vygotskian account of Reading Recovery : a programme based on individual instruction. This led me to turn away from the Vygotskian perspectives all very nice in theory, but did they have much to offer the harried teacher of 30 or more students? Recently I ve returned to consider aspects of the work of Vygotsky, particularly as interpreted In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31 st Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp Seoul: PME. 2-33

6 included in this selection, and, like the others chosen, were given due warning of this so that they had time to prepare what they were going to say DISCUSSION Did the children learn about addition through this lesson? I cannot say. What was of concern was that they learnt that mathematics is learnable and that they were capable of performing it. Developmental learning involves learning act as a mathematician and the realisation that the choice to continue to act as mathematicians is available. Developmental learning is thus generative rather than aquisitional. As Holtman (1997) problematises it: Can we create ways for people to learn the kinds of things that are necessary for functional adaptation without stifling their capacity to continuously create for growth? This is a key question for mathematics education. In England, and elsewhere, policy makers are specifying the content and expected learning outcomes of mathematics education in finer and finer detail. For example, the introduction of the National Numeracy Strategy in England brought with it a document setting out teaching and learning objectives the Framework for Teaching Mathematics from Reception to Year 6 (DfEE, 1999) a year-by-year breakdown of teaching objectives. The objectives within the framework are at a level of detail far exceeding that of the mandatory National Curriculum (NC). The NC requirements for what 7- to 11-year-olds should know and understand in calculations is expressed in just over one page and, typically, include statements like: work out what they need to add to any two-digit number to make 100, then add or subtract any pair of two-digit whole numbers, handle particular cases of three-digit and four-digit additions and subtractions by using compensation or other methods (for example, , ) (Department for Education and Employment (DfEE), 1999a, p.25). In contrast, the Framework devotes over 50 pages to elaborating teaching objectives for calculation, at this the level of detail: Find a small difference between a pair of numbers lying either side of a multiple of 1000 For example, work out mentally that: = 15 by counting up 2 from 6988 to 6990, then 10 to 7000, then 3 to 7003 Work mentally to complete written questions like = 6004 = = 19 (Department for Education and Employment (DfEE), 1999b, Y456 examples, p 46) While teachers have welcomed this level of detail, there is a danger that covering the curriculum (in the sense of addressing each objective) becomes the over-arching goal of teaching, that acquisition of knowledge by learners becomes paramount and the curriculum content reified and fossilised. In particular the emphasis is on knowing rather than developing. Does coverage of pages of learning outcomes help students view themselves as being able to act as mathematicians? 2-38 PME

8 that tightly controlled (or if it is that the learning that emerges is limited and resistricted to being trained rather than playing a part). REFERENCES Bliss, J., Askew, M., & Macrae, S. (1996) Effective teaching and learning: scaffolding revisited. Oxford Review of Education. 22(1) pp Bruner, J. (1985) Vygotsky: a historical and conceptual perspective., in J. V. Wertsch (ed) Culture, Communication and Cognition: Vygotskian Perspectives. Cambridge: Cambridge University Press (pp ) Clay, M. M. & Cazden, C. B. (1990) A Vygotskian interpretation of Reading Recovery., in Moll, L. C. (ed) Vygotsky and Education: Instructional Implications and Applications of Sociohistorical Psychology. Cambridge: Cambridge University Press (pp ) Cole, M. (1996) Cultural Psychology: A Once and Future Discipline. Cambridge MA: Harvard University Press Department for Education and Employment (DfEE) (1999a) Mathematics in the National Curriculum. London: DfEE Department for Education and Employment (DfEE) (1999) The National Numeracy Strategy: Framework for teaching mathematics from Reception to Year 6. London: DfEE Fosnot, C. T. & Dolk, M. (2001) Young mathematicians at work: constructing number sense, addition and subtraction. Portsmouth, NH; Hiennemann Freudenthal, H. (1973) Mathematics as an educational task. Dordrecht: Reidel. Holtzman, L. (1997) Schools for growth: radical alternatives to current educational models. Mahwah, NJ & London: Lawrence Earlbaum Associates. Lave, J. & Wenger, E. (1991) Situated learning: Legitimate Peripheral Participation. Cambridge: Cambridge University Press (pp 21-34) Newman, F., & Holtzman, L. (1993) Lev Vygotsky; Revolutionary scientist. London: Routledge. Newman, F., & Holtzman, L. (1997) The end of knowing: A new development way of learning. London: Routledge. Rogoff, B. (1990) Apprenticeship in thinking: cognitive development in social context. New York: Oxford University Press Vygotsky (1978) Mind in society. Cambridge MA: Harvard University Press 2-40 PME

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