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Building on Children's Mathematics - a Teaching Experiment in Grade Three

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Abstract

There rarely seems to be any disagreement that teaching should build on children's mathematics, but there are many possible ways to put this demand in concrete terms. In this sense, the present paper describes a teaching experiment in grade three based on Treffers' idea of vertical mathematisation. Central components of the developed course are discussed, with focus on the use of the empty number line and of maths diaries as well as on the idea of conducting maths conferences. The teaching experiment is illustrated by means of children's own productions and finally put into a broader context as contributing to the design science approach of mathematics education.

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REFERENCES

  • Baroody, Arthur J.: 1986, ‘The value of informal approaches to mathematics instruction and remediation’. Arithmetic Teacher 33 (No. 5): 14-18.

    Google Scholar 

  • Baroody, Arthur J.: 1987, Children's Mathematical Thinking. New York.

  • Beck, Erwin et al.: 1992, Projekt Eigenst ändige Lerner (Project autonomous learners). St. Gallen.

  • Becker, Jerry P. and Selter, Christoph: 1996, ‘Elementary school practices’, in Alan Bishop et al. (eds.), International Handbook on Mathematics Education. Dordrecht, pp. 511-564.

  • Beishuizen, M., van Putten, C. M. and van Mulken, F.: ‘Mental arithmetic and strategy use with indirect number problems up to one hundred, Learning and Instruction (in press).

  • Beishuizen, Meindert: 1993, ‘Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades’, Journal for Research in Mathematics Education 24: 294-323.

    Google Scholar 

  • Bird, Marion: 1983, Generating Mathematical Activity in the Classroom.Leicester.

  • Borasi, Raffaella and Rose Barbara, J: 1989, ‘Journal writing and mathematics instruction’, Educational Studies in Mathematics 20: 347-365.

    Google Scholar 

  • Burns, Marilyn: 1987, A Collection of Math Lessons from Grade 3 through 6. San Francisco.

  • Burns, Marilyn: 1991, ‘Introducing division through problem-solving experiences’, Arithmetic Teacher 38 (No. 8): 14-18.

    Google Scholar 

  • Burton, Grace M.: 1985, ‘Writing as a way of knowing in a mathematics eduction class’, Arithmetic Teacher 33 (No. 4): 40-45.

    Google Scholar 

  • Carpenter, Thomas P. and Moser, James M.: 1982, ‘The development of addition and subtraction problem-solving skills, in Thomas P. Carpenter, James M. Moser and Tom A. Romberg (eds), Addition and Subtraction: a Cognitive Perspective. Hillsdale, NJ, 9-24.

  • Carpenter, Thomas P. and Moser, James M.: 1984, ‘The acquisition of addition and subtraction concepts in grades one through three’, Journal for Research in Mathematics Education (No. 3): 179-202.

    Google Scholar 

  • Cobb, Paul et al.: 1995, ‘Mathematizing and symbolizing: the emergence of chains of signification in one first-grade classroom’, paper presented at the 6th EARLI-Conference. Nijmegen.

  • Cooper, T. J., Heirdsfield, A. M. and Irons, C. J., (1996), ‘Years 2 and 3 children's mental addition and subtraction strategies for 2-and 3-digit word problems and algorithmic exercises’, paper presented at Topic Group 1 at ICME8, Sevilla.

  • Del Campo, Gina and Clements, Ken (M. A.): 1990, ‘Expanding the modes of communication in mathematics classrooms’, Journal f ür MathematikDidaktik 11 (No. 1): 45-79.

    Google Scholar 

  • Eisner, Elliot: 1991, The Enlightened Eye. Qualitative Inquiry and the Enhancement of Educational Practice. New York.

  • Freudenthal, Hans: 1968, ‘Why to teach mathematics so as to be useful?’, Educational Studies in Mathematics 1: 3-8.

    Google Scholar 

  • Freudenthal, Hans: 1991, Revisiting Mathematics Education. China Lectures, Dordrecht.

  • Fuson, Karen: 1992, ‘Research on whole number addition and subtraction’, in Douglas A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning. New York, 243-275.

  • Fuson, Karen and Smith, Steven T.: Supporting multiple 2digit conceptual structures and strategies in the classroom: Issues of conceptual supports, language, and instructional design, in: Ernest van Lieshout et al. (ed.), The Role of Contexts and Models in the Development of Mathematical Strategies and Procedures. Utrecht (in press).

  • Gallin, Peter and Ruf, Urs: 1993, ‘Sprache und Mathematik in der Schule. Ein Bericht aus der Praxis. (Language and mathematics in school. A report from teaching practice)’, Journal für Mathematik-Didaktik 14 (No. 1): 3-33.

    Google Scholar 

  • Ginsburg, Herbert D.: 1977, Children's Arithmetic. The Learning Process. New York.

  • Gravemeijer, Koeno: 1990, ‘Context problems and realistic mathematics instruction’, in Koeno Gravemeijer; Marja van den Heuvel and Leen Streefland, Contexts, Free Productions, Tests, and Geometry in Realistic Mathematics Education. Utrecht, pp. 10-32.

  • Gravemeijer, Koeno: 1991, ‘An instruction-theoretical reflection on the use of manipulatives’, in Leen Streefland (ed.), Realistic Mathematics Education in Primary School. On the occasion of the opening of the Freudenthal Institute. Utrecht, pp. 57-76.

  • Gravemeijer, Koeno: 1993, ‘The empty number line as an alternative means of representation for addition and subtraction, in Jan de Lange et al. (eds.), Innovation in Mathematics Education by Modelling and Applications. Chichester, pp. 141-149.

  • Gravemeijer, Koeno: 1994, Developing Realistic Mathematics Education. Utrecht.

  • Graves, Donald H.: 1983, Writing: Teachers and Children at Work.Exeter.

  • N. H. Jones, G. A., Thornton, C. A. and Putt, I. J., (1994), ‘A model for nurturing and assessing multi digit number sense among first grade children’, Educational Studies in Mathematics 27: 117-143.

    Google Scholar 

  • Klein, Ton, Beishuizen, Meindert and Treffers, Adri, ‘The empty numberline in Dutch second grades under two conditions: A ‘realistic’ versus ‘gradual’ program design’, Journal for Research in Mathematics Education (in press).

  • Krauthausen, Günter: 1993, ‘Kopfrechnen, halbschriftliches Rechnen, schriftliche Normalverfahren, Taschenrechner: Für eine Neubestimmung des Stellenwertes der vier Rechenmethoden. (On the relative importance of different types of computation)’, Journal für Mathematik-Didaktik (No. 3/4): 189-219.

    Google Scholar 

  • Lampert, Magdalene: 1986, ‘Teaching multiplication’, Journal of Mathematical Behavior 5: 241-280.

    Google Scholar 

  • Lampert, Magdalene: 1990, ‘Connecting inventions with conventions’, in Leslie P. Steffe and Terry Wood (eds.), Transforming Children's Mathematics Education. New Jersey, 253-265.

  • Madell, Rob: 1985, ‘Children's natural processes’, Arithmetic Teacher, 32 (No. 7): 20-22.

    Google Scholar 

  • NCTM (National Council of Teachers of Mathematics): 1989, Curriculum and Evaluation Standards for School Mathematics.Reston, Va.

  • Nunes, Terezinha, Schlieman, Analucia and Carraher, David: 1993, Street Mathematics and School Mathematics. Cambridge.

  • Olivier, Alwyn, Murray, Hanlie and Human, Piet: 1990, ‘Building on young children's informal mathematical knowledge’, in: George Booker, Paul Cobb and Teresa N. Mendicuti (eds.), Proceedings of PME -14. Mexico. Vol. III, pp. 3-10.

  • Peshkin, Alan: 1985, ‘Virtous subjectivity: in the participant observer's I s’, in David Berg and Kenwyn K. Smith (eds), Exploring Clinical Methods for Sound Research. Beverly Hills, pp. 267-286.

  • Radatz, Hendrik: 1980, Fehleranalysen im Mathematikunterricht (Analysis of Children's Errors in Mathematics). Braunschweig.

  • Reys, Robert et al.: 1995, ‘Mental computation performance and strategy use of Japanese students in grades 2, 4, 6 and 8’, Journal for Research in Mathematics Education 26: 304-326.

    Google Scholar 

  • Riley, M. S., Greeno, J. G. and Heller, J. I.: 1983, ‘Development of children's problem solving ability in arithmetic, in Herbert P. Ginsburg (ed.), The Development of Mathematical Thinking. New York, pp. 153-196.

  • Selter, Christoph and Spiegel, Hartmut: 1997, Wie Kinder rechnen (How children do arithmetic). Leipzig.

  • Selter, Christoph: 1996, ‘Going one's own way’, in Corine van der Boer and Maarten Dolk (eds), Modellen, Meten en Meetkunde. Paradigma's van Adaptief Onderwijs. Utrecht, 87-99.

  • Sierpinska, Anna and Kilpatrick, Jeremy: 1997, Mathematics education as a research domain: a search for identity. Dordrecht.

  • Spitta, Gudrun: 1992, Schreibkonferenzen—ein Weg vom spontanen Schreiben zum bewuβten Verfassen von Texten in Klasse 3 und 4 (Writing conferences — from spontaneous writing to conscious producing texts in grades 3 and 4). Frankfurt.

  • Strauss, Anselm L.: 1991, Grundlagen qualitativer Sozialforschung. Datenanalyse und Theoriebildung in der empririschen soziologischen Forschung (Foundations of qualitative sociological research). München.

  • Streefland, Leen: 1991, Fractions in Realistic Mathematics Education. A paradigm of developmental research. Dordrecht.

  • Sundermann, Beate and Selter, Christoph: 1995, ‘Halbschriftliches Rechnen auf eigenen Wegen’ (Flexible arithmetic on own ways), in Gerhard N. Mü ller and ErichCh. Wittmann (eds.): Mit Kindern rechnen. Frankfurt, 165-178.

  • Thompson, Ian: 1994, ‘Young children's idiosyncratic written algorithms for addition’, Educational Studies in Mathematics 26: 323-345.

    Google Scholar 

  • Treffers, Adrian: 1987a, Three dimensions. A model of goal and theory description in mathematics instruction—the Wiskobas project. Dordrecht.

  • Treffers, Adrian: 1987b, ‘Integrated column arithmetic according to progressive schematisation’, in: Educational Studies in Mathematics 18: 125-145.

    Google Scholar 

  • Treffers, Adrian: 1991, ‘Didactical background of a mathematics program for primary education’, in Leen Streefland (ed.), Realistic Mathematics Education in primary school. On the occasion of the opening of the Freudenthal Institute. Utrecht, pp. 21-56.

  • Treffers, Adrian: 1993, ‘Wiskobas and Freudenthal: Realistic Mathematics Education’, Educational Studies in Mathematics 25 (1/2): 89-108.

    Google Scholar 

  • Treffers, Adrian and de Moor, Ed: 1990, Proeve van een nationaal programma voor het rekenwiskundeonderwijs op de basisschool. Deel 2: Basisvaardigheden en cijferen (Outline for a national curriculum for mathematics in primary school. Part 2: Basic skills and algorithms). Tilburg.

  • Vuurmans, Anne Cos (ed.): 1991, Rekenen tot honderd—nascholing zorgverbreding speerpunt rekenen (Arithmetic up to 100 — inservice course spearhead arithmetic). "s Hertogenbosch.

  • Walther, Gerhard: 1982, ‘Acquiring mathematical knowledge’, in Mathematics Teaching (101): 10-12.

    Google Scholar 

  • Walther, Gerhard: 1984, ‘Mathematical activity in an educational context: a guideline for primary mathematics teacher training’, in Robert Morris (ed.), Studies in mathematics education, Vol. 3—The Mathematical Education of Primary School Teachers. Paris, pp. 69-88.

  • Wittmann, Erich Christian: 1995, ‘Mathematics education as a ‘design science’, Educational Studies in Mathematics 29: 355-374.

    Google Scholar 

  • Wittmann, Erich Christian and Müller, Gerhard Norbert: 1992, Handbuch produktiver Rechenübungen. Band 2: Vom halbschriftlichen zum schriftlichen Rechnen (Handbook on productive practicing of arithmetic. Vol. 2: Flexiblemethods and standard algorithms).

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  1. Pädagogische Hochschule Heidelberg, Fakultät III Im Neuenheimer Feld 561, 69120, Heidelberg, Germany

    Christoph Selter

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Selter, C. Building on Children's Mathematics - a Teaching Experiment in Grade Three. Educational Studies in Mathematics 36, 1–27 (1998). https://doi.org/10.1023/A:1003111425076

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