Abstract
During the last decade several major shifts have occurred in the conceptualisation of mathematics as a domain, of mathematical competence as a goal for instruction, and of the way in which this competence should be acquired through schooling. This chapter begins with a summary of the general characteristics and principles underlying the ongoing world-wide reform of mathematics education. Afterwards it documents and illustrates how these general characteristics and principles permeate a major domain of the mathematics curricula for the elementary school, called ‘Number and Arithmetic’. Five related topics within this domain are discussed, namely: number concepts and number sense, the meaning of arithmetic operations, mastery of basic arithmetic facts, mental and written computation, and word problems as applications of the numerical and arithmetical knowledge and skills.
The final section lists some remaining issues and tasks for further curriculum research and development in the domain of number and arithmetic.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Australian Education Council: 1990, A National Statement on Mathematics for Australian Schools, Curriculum Co-operation, Carlton, Victoria.
Baroody, A., & Standifer, D.J.: 1993, ‘Addition and Subtraction in the Primary Grades’, in R.J. Jensen (ed.), Research Ideas for the Classroom. Early Childhood Mathematics, Macmillan, New York, 72–102.
Behr, M, Harel, G., Post, T., & Lesh, R.: 1992, ‘Rational Number, Ratio, and Proportion’, in D. A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 296–333.
Beishuizen, M.: 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.
Berger, A., Fischer, M., Hoffmann, M., Jüttemeier, M., Müller, G.N., & Wittmann, E.C.: 1994, Das Zahlenbuch. Mathematik im 1. Schuljahr, Ernst Klett Schulbuchverlag, Stuttgart.
Berger, A., Fischer, M., Hoffmann, M., Jüttemeier, M., Müller, G.N., & Wittmann, E.C.: 1995, Das Zahlenbuch. Mathematik im 2. Schuljahr, Ernst Klett Schulbuchverlag, Stuttgart.
Bossé, M.J.: 1995, ‘The NCTM Standards in the Light of the New Math Movement: A Warning’, Journal of Mathematical Behaviour, 14, 171–201.
Burkhardt, H., Fraser, R., & Ridgway, J.: 1990, The Dynamics of Curriculum Change’, in I. Wirszup & R. Streit (eds.), Developments in School Mathematics Education in the World (Proceedings of the Second UCSMP International Conference on Mathematics Education, 7–10 April 1988), National Council of Teachers of Mathematics, Reston, VA, 3–30.
California Board of Education: 1991, Mathematics Framework for California Public Schools (Preview edition), California State Department of Education, Sacramento.
Carpenter, T.P., Fennema, E. Peterson, P.L., Chiang, C, & Loef, M.: 1989, ‘Using Knowledge of Children’s Mathematical Thinking in Classroom Teaching: An Experimental Study’, American Educational Research Journal, 26, 499–531.
Carpenter, T.P., & Moser, J.M.: 1982, ‘The Development of Addition and Subtraction Problem Solving Skills’ in T.P. Carpenter, J.M. Moser, & T. Romberg (eds.), Addition and subtraction: A cognitive perspective, Erlbaum, Hillsdale, NJ, 9–24.
Carpenter, T., Hiebert, J., & Moser, J.M.: 1983, ‘The Effect of Instruction on Children’s Solutions of Addition and Subtraction Word Problems’, Educational Studies in Mathematics, 14, 55–72.
Clements, D.H.: 1983, ‘Training Effects on the Development and Generalisation of Piagetian Logical Operations and Knowledge of Number’, Journal of Educational Psychology, 76, 766–776.
Cobb, P.: 1995, ‘Cultural Tools and Mathematical Learning’, Journal for Research in Mathematics Education, 26, 362–385.
Cobb, P., Yackel, E., & Wood, T.; 1991, ‘Curriculum and Teacher Development: Psychological and Anthropological Perspectives’, in E. Fennema, T.P. Carpenter, & S.J. Lamon (eds.), Integrating Research on Teaching and Learning Mathematics, SUNY Press, Albany, NY, 92–131.
Cockcroft, W.H.: 1982, Mathematics Counts (Report of the Committee of Inquiry into the Teaching of Mathematics in Schools), Her Majesty’s Stationary Office, London.
Davis, R.B.: 1989, ‘The Culture of Mathematics and the Culture of Schools’, Journal of Mathematical Behaviour, 8, 143–160.
De Corte, E.: 1995, ‘Fostering Cognitive Growth: A Perspective from Research on Mathematics Learning and Instruction’, Educational Psychologist, 30, 37–46.
De Corte, E., Greer, B., & Verschaffet L.: 1996, ‘Mathematics Learning and Teaching’, in D. Berliner & R. Calfee (eds.), Handbook of Educational Psychology, Macmillan, New York.
De Corte, E., & Verschaffel, L.: 1985a, ‘Working with Simple Word Problems in Early Mathematics Instruction’, in L. Streefland (ed.), Proceedings of the Ninth International Conference for the Psychology of Mathematics Education. Vol 1. Individual Contributions, Research Group on Mathematics Education and Educational Computer Centre, Subfaculty of Mathematics, University of Utrecht Utrecht, The Netherlands, 304–309.
De Corte, E., & Verschaffel, L.: 1985b, ‘Writing Number Sentences to Represent Addition and Subtraction Word Problems’, in S. Damarin & M. Shelton (eds.), Proceedings of the Seventh Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, Ohio, 2–5 October 1985, Department of Psychology, Ohio State University, Columbus, OH, 50–56.
De Corte, E., & Verschaffel, L.: 1987, ‘The Effect of Semantic Structure on First Graders’ Solution Strategies of Elementary Addition and Subtraction Word Problems’, Joumal for Research in Mathematics Education, 18, 363–381.
De Corte, E., Verschaffel, L., & Greer, B.: 1994, ‘Learning and Instruction of Mathematics’, in T. Husen & T.N. Postlethwaite (eds.), The International Encyclopaedia of Education (second edition), Pergamon, Oxford, 3652–3655.
De Corte, E., Verschaffel, L., Janssens, V., & Joillet, L.: 1985, ‘Teaching Word Problems in the First Grade: A Confrontation of Educational Practice with Results of Recent Research’, in T. A. Romberg (ed.), Using Research in the Professional Life of Mathematics Teachers, Centre for Education Research, University of Wisconsin, Madison, WI, 3652–3655.
De Corte, E., Verschaffel, L., & Van Coillie, V.: 1988, ‘Influence of Number Size, Problem Structure and Response Mode on Children’s Solutions of Multiplication Word Problems’, Journal of Mathematical Behaviour, 7, 197–216.
Freudenthal, H.: 1978, Weeding and Sowing, Reidel, Dordrecht, The Netherlands.
Freudenthal, H.: 1983, Didactical Phenomenology of Mathematical Structures, Reidel, Dordrecht, The Netherlands.
Fuson, K.: 1992, ‘Research on Whole Number Addition and Subtraction’, in D.A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 243–275.
Gravemeijer, K.P.E.: 1994, Developing Realistic Mathematics Education, Freudenthal Institute, University of Utrecht, Utrecht, The Netherlands.
Greeno, J.G.: 1991, ‘Number Sense as Situated Knowing in a Conceptual Domain’, Journal for Research in Mathematics Education, 22, 170–218.
Greer, B.: 1992, ‘Multiplication and Division as Models of Situations’, in D.A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 276–295).
Greer, B.: 1993, The Modelling Perspective on Wor(l)d Problems’, Journal of Mathematical Behaviour, 12, 239–250.
Greer, B.: 1994, ‘Extending the Meaning of Multiplication and Division’, in G. Harel & J. Confrey (eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics, State University of New York Press, Albany, NY, 61–85.
Hart, K.M.: 1981, Children’s Understanding of Mathematics: 11–16, Murray, London.
Hembree, R„ & March, H.: 1993, ‘Problem Solving in Early Childhood. Building Foundations’, in R.J. Jensen (ed.), Research Ideas for the Classroom. Early Childhood Mathematics, Macmillan, New York, 151–170.
Hiebert, J., & Behr, M. (eds.): 1988, Number Concepts and Operations in the Middle Grades, Erlbaum, Hillsdale, NJ.
Hiebert, J., & Carpenter, T.P.: 1982, ‘Piagetian Tasks as Readiness Measures in Mathematics Instruction. A Critical Review’, Educational Studies in Mathematics, 13, 329–347.
Hughes, M.: 1986, Children and Number. Difficulties in Learning Mathematics, Basil Black-well, New York.
Kilpatrick, J.: 1987, ‘Problem Formulating: Where Do Good Problems Come from?’, in A.H. Schoenfeld (ed.), Cognitive Science and Mathematics Education, Erlbaum, Hillsdale, NJ, 123–147.
Kilpatrick, J.: 1994, ‘Mathematics Instruction: Contemporary Research’, in T. Husen & T.N. Postlethwaite (eds.), The International Encyclopaedia of Education (second edition), Per-gamon, Oxford, 3647–3652.
Kouba, V., & Franklin, K.: 1993, ‘Multiplication and Division: Sense Making and Meaning’, in R.J. Jensen (ed.), Research Ideas for the Classroom. Early Childhood Mathematics, Macmillan, New York, 103–126.
Lampert, M.: 1986, ‘Knowing, Doing, and Teaching Multiplication’, Cognition and Instruction, 3, 305–342.
Lampert, M.: 1990, ‘When the Problem is not the Question and the Solution is not the Answer: Mathematical Knowing and Teaching’, American Educational Research Journal, 27, 29–63.
McIntosh, A., Reys, B.J., & Reys, R.E.: 1992, ‘A Proposed Framework for Examining Basic Number Sense’, For the Learning of Mathematics, 12 (3), 2–8.
National Council of Teachers of Mathematics: 1989, Curriculum and Evaluation Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA.
National Council of Teachers of Mathematics: 1995, Assessment Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA.
Nesher, P.: 1980, ‘The Stereotyped Nature of School Word Problems’, For the Learning of Mathematics, 1, 41–48.
Nesher, P.: 1982, ‘Levels of Description in the Analysis of Addition and Subtraction’, in T.P. Carpenter, J.M. Moser, & T. Romberg (eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum, Hillsdale, NJ, 25–38.
Nesher, P.: 1988, ‘Multiplicative School Word Problems. Theoretical Approaches and Empirical findings’, in J. Hiebert & M. Behr (eds.), Number Concepts and Operations in the Middle Grades, Erlbaum, Hillsdale, NJ, 19–40.
Nesher, P., & Teubal, E.: 1975, ‘Verbal Cues as an Interfering Factor in Verbal Problem Solving’, Educational Studies in Mathematics, 6, 41–51.
Payne, J.N., & Huinker, D.M.: 1993, ‘Early Number and Numeration’, in R.J. Jensen (ed.), Research Ideas for the Classroom. Early Childhood Mathematics, Macmillan, New York, 43–71.
Piaget, J.: 1952, The Child’s Conception of Number, Routledge & Kegan Paul, London.
Rengerink, J.: 1983, De Staartdeling [Long division], Department of Education, University of Utrecht, Utrecht, The Netherlands.
Reusser, K.: 1988, ‘Problem Solving Beyond the Logic of Things: Contextual Effects on Understanding and Solving Word Problems’, Instructional Science, 17, 309–338.
Reusser, K.: 1996, ‘Every Word Problem Has a Solution. The Suspension of Reality and Sense-Making in School Mathematics’, Paper presented at the Annual Meeting of the American Educational Research Association, New York.
Riley, M.S., Greeno J.G., & Heller, J.I.: 1983, ‘Development of Children’s Problem-Solving Ability in Arithmetic’, in H.P. Ginsburg (ed.), The Development of Mathematical Thinking, Academic Press, New York, 153–196.
Romberg, T.A.: 1994, ‘Mathematics: Primary School Programs’, in T. Husen & T.N. Postlethwaite (eds.), The International Encyclopaedia of Education (second edition), Pergamon, Oxford, 3655–3661.
Romberg, T.A.: 1995, (ed.), Reform in School Mathematics and Authentic Assessment, State University of New York Press, Albany, NY.
Schoenfeld, A.H.: 1991, ‘On Mathematics as Sense-Making: An Informal Attack on the Unfortunate Divorce of Formal and Informal Mathematics’ in J.F. Voss, D.N. Perkins, & J.W. Segal (eds.), Informal Reasoning and Education, Erlbaum Hillsdale, NJ, 311–343.
Schoenfeld, A.H.: 1992, ‘Learning to Think Mathematically: Problem Solving, Metacogni-tion, and Sense-Making in Mathematics’, in D.A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 334–370.
Selter, C: 1994, Eigenproduktionen in Arithmetikunterricht der Primarstufe [Own Productions in Learning Elementary Arithmetic], Deutscher Universitätsverlag, Wiesbaden, Germany.
Seiter, C: 1995, From Teaching to Learning Mathematics, Keynote Lecture at the Panama Conference in Noordwijkerhout, The Netherlands.
Sowder, J.: 1992, ‘Estimation and Number Sense’, in D.A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 371–389.
Sowder, L.: 1988, ‘Children’s Solutions of Story Problems’, Journal of Mathematical Behaviour, 7, 227–238.
Stigler, J.W., Fuson, K.C., Ham, M., & Kim, M., 1986, ‘An Analysis of Addition and Subtraction Word Problems in U.S. and Soviet Elementary Mathematics Textbooks’, Cognition and Instruction, 3, 153–171.
Suydam, M., & Weaver, J.F.: 1975, ‘Research on Mathematics Learning’ in J.N. Payne (ed.), Mathematics Learning in Early Childhood, 37th Yearbook of the National Council of Teachers of Mathematics. National Council of Teachers of Mathematics, Reston, VA., 43–67
Ter Heege, H.: 1985, ‘The Acquisition of Basic Multiplication Skills’, Educational Studies in Mathematics, 16, 375–389.
Treffers, A.: 1987, Three Dimensions. A Model of Goal and Theory Description in Mathematics Education. The Wiskobas Project, Reidel, Dordrecht, The Netherlands.
Treffers, A.: 1991, ‘Meeting Numeracy at Primary School’, Educational Studies in Mathematics, 22, 333–352.
Treffers, A., de Moor, E., & Feys, E.: 1989, Proeve van een Nationaal Programma voor het Reken-Wiskundeonderwijs op de Basisschool. Deel I. Overzicht Einddoelen [Towards a National Mathematics Curriculum for the Elementary School. Part 1. Overview of the General Goals], Zwijssen, Tilburg, The Netherlands.
Treffers, A., & de Moor, E.: 1990, Proeve van een Nationaal Programma voor het Reken-Wiskundeonderwijs op de Basisschool Deel 2. Basisvaardigheden en Cijferen [Towards a National Mathematics Curriculum for the Elementary School. Part 2. Basic Skills and Written Computation], Zwijssen, Tilburg, The Netherlands.
Van den Brink, J.: 1989, Realistisch Rekenonderwijs aan Jonge Kinderen [Realistic Arithmetic Instruction for Young Children], Freudenthal Institute, University of Utrecht, Utrecht, The Netherlands.
Van den Heuvel-Panhuizen, M.: 1996, Assessment and Realistic Mathematics Education. (doctoral dissertation), CD-bèta Press, Utrecht, The Netherlands.
Van de Rijt, B.: 1996, Voorbereidende Rekenvaardigheid bij Kleuters [Early Mathematical Competence in Young Children.], (doctoral dissertation), University of Utrecht, Utrecht, The Netherlands.
Van de Walle, J., & Watkins, K.B.: 1993, ‘Early Development of Number Sense’, in R.J. Jensen (ed.), Research Ideas for the Classroom. Early Childhood Mathematics, Macmillan, New York, 127–151.
Vergnaud, G.: 1982, ‘A Classification of Cognitive Tasks and Operations of Thought Involved in Addition and Subtraction’, in T. Carpenter, J. Moser, & T. Romberg (eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum, Hillsdale, NJ, 39–59.
Vergnaud, G.: 1983, ‘Multiplicative Structures’, in R. Lesh & M. Landau (eds.), Acquisition of Mathematics Concepts and Processes, Academic Press, New York, 127–174.
Verschaffel, L.: 1995, ‘Visies op Reken/Wiskundeonderwijs op de Basisschool’ [Perspectives on Mathematics Education in the Primary School], in L. Verschaffel & E. De Corte (eds.), Naar een nieuwe reken/wiskundedidactiek voor de basisschool en de basiseducatie. Deel 1. Achtergronden [Toward a New Approach of Mathematics Education. Part 1. Theoretical Backgrounds and Perspectives], Studiecentrum voor Open Hoger Afstandsonderwijs (StO-HO), Brüssel, 95–128.
Verschaffel, L., & De Corte, E.: 1993, ‘A Decade of Research on Word-Problem Solving in Leuven: Theoretical, Methodological and Practical Outcomes’, Educational Psychology Review, 5, 239–256.
Verschaffel, L., & De Corte, E.: 1995, ‘Teaching Realistic Mathematical Modelling in the Elementary School. A Teaching Experiment with Fifth Graders’, Proceedings of the 19th International Conference for the Psychology of Mathematics Education. Vol. 3, Universidade Federal de Pernambuco, Graduate Program in Cognitive Psychology, Recife, Brazil, 105–112.
Verschaffel, L., & De Corte, E.: in press, ‘Word Problems. A Vehicle for Authentic Mathematical Understanding and Problem Solving in the Primary School?’ in P. Bryant & T. Nunes (eds.), How Do Children Learn Mathematics?, Erlbaum, Hillsdale, NJ.
Verschaffel, L., De Corte, E., & Lasure, S.: 1994, ‘Realistic Considerations in Mathematical Modelling of School Arithmetic Word Problems’, Learning and Instruction, 4, 273–294.
Vuurmans, A.-C, Klukhun, W., Gribling, S., & Nelissen, J.: 1986, Rekenwerk. Leerlingenboek 3a [Mathematics at Work. Student Book 3a], De Ruiter, Gorinchem, The Netherlands.
Willis, G.B., & Fuson, K.C.: 1988, Teaching Children to Use Schematic Drawings to Solve Addition and Subtraction Word Problems’, Journal of Educational Psychology, 80, 192–201.
Wittmann, E.C.: 1985, ‘Practising skills and reflection’, in Mathématiques pour Tous… à l’Âge de l’Ordinateur, Freudenthal Institute, Utrecht, The Netherlands, 60–72.
Yoshida, H., & Kuriyama, K.: 1995, ‘Linking Meaning of Symbols of Fractions to Problem Situations’, Japanese Psychological Research, 37, 229–239.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Kluwer Academic Publishers
About this chapter
Cite this chapter
Verschaffel, L., De Corte, E. (1996). Number and Arithmetic. In: Bishop, A.J., Clements, K., Keitel, C., Kilpatrick, J., Laborde, C. (eds) International Handbook of Mathematics Education. Kluwer International Handbooks of Education, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1465-0_4
Download citation
DOI: https://doi.org/10.1007/978-94-009-1465-0_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7155-0
Online ISBN: 978-94-009-1465-0
eBook Packages: Springer Book Archive