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Educational Studies in Mathematics

, Volume 16, Issue 2, pp 181–204 | Cite as

Proportional reasoning: A review of the literature

  • Francoise Tourniaire
  • Steven Pulos
Article
  • 622 Downloads

Abstract

This paper presents a review of the research on proportional reasoning. Methodologies used in proportional reasoning studies are presented first. The discussion is then organized around the following topics: strategies use to solve proportion problems, including erroneous strategies; factors that influence performance on proportion problems, both task-related and subject-related; training studies. The discussion is accompanied by suggestions for educational and research applications.

Keywords

Research Application Training Study Proportional Reasoning Reasoning Study Proportion Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Adi H. and Pulos S.: 1980, ‘Individual differences and formal operational performance of college students’,Journal for Research in Mathematics Education 11(2), 150–156.Google Scholar
  2. Anderson, L. H.: 1979,Development Effects in Learning Hierarchy Structure for Problems Involving Proportional Reasoning, unpublished doctoral dissertation, University of California, Berkeley.Google Scholar
  3. Bart W. M.: 1972, ‘Construction and validation of formal reasoning instruments’,Psychological Reports 30, 663–670.Google Scholar
  4. Biemiller, A.: 1981,A Neo-piagetian Approach to Development During the Formal-operational Period, paper presented at the Eleventh Annual International Conference on Piagetian Theory and its Implications for the Helping Professions, University of Southern California, Los Angeles.Google Scholar
  5. Brendzel, S: 1980,Proportional Reasoning and Visual Spatial Ability, paper presented at the annual meeting of the National Association for Research in Science Teaching, Boston, Mass.Google Scholar
  6. Capon N and Kuhn K.: 1979, ‘Logical reasoning in the supermarket: Adult females' use of a proportional strategy in an everyday context’,Developmental Psychology,15(4), 450–452;6, 544–573.Google Scholar
  7. Case R.: 1978, ‘Piaget and beyond: Toward a developmentally based theory and technology of instruction’, in R. Glaser (ed.),Advances in Instructional Psychology, LEA, Hillsdale, New Jersey.Google Scholar
  8. Case R.: 1979, ‘Intellectual development and instruction: a neo-Piagetian view’, in A. Lawson (ed.),The Psychology of Teaching for Thinking and Creativity, ERIC, Columbus, Ohio.Google Scholar
  9. Case R.: 1980, ‘Implications of neo-piagetian theory for improving the design of instruction’, in J. R. Kirby and J. B. Biggs (eds),Cognitive Development and Instruction, Academic Press, New York.Google Scholar
  10. Cloutier R and Goldschmid M. L.: 1976, ‘Individual differences in the development of formal reasoning’,Child Development 47, 1097–1102.Google Scholar
  11. Cloutier R. and Goldschmid M. L.: 1978, ‘Training proportionality through peer instruction’,Instructional Science 7, 127–142.Google Scholar
  12. Falk, R.: 1978,Analysis of the Concept of Probability in Young Children, paper presented at the Second Conference of IGPME.Google Scholar
  13. Fischbein E., Pampu I., and Manzat I.: 1970, ‘Comparison of ratios and the chance concept in children’,Child Dvelopment 41, 377–389.Google Scholar
  14. Furman, I.: 1981,The Development of Problem-solving Strategies: A Neo-piagetian Analysis of Children's Performance in a Balance Task, unpublished doctoral dissertation, University of California, Berkeley.Google Scholar
  15. Gold, A. P.: 1978,Cumulative Learning Versus Cognitive Development: A Comparison of Two Different Theoretical Bases for Planning Remedial Instruction in Arithmetic, unpublished doctoral dissertation, University of California, Berkeley.Google Scholar
  16. Hart K. M.: 1981,Children's Understanding of Mathematics: 11–16, John Murray Ltd., London.Google Scholar
  17. Horn J. L.: 1982, ‘The ageing of human abilities’, in B. Wolman (ed.),Handbook of Developmental Psychology, Prentice-Hall, Englewood Cliffs, New Jersey.Google Scholar
  18. Horowitz, L.: 1982,Visualization and Arithmetic Problem Solving, paper presented at the annual meeting of the American Educational Research Association, Los Angeles.Google Scholar
  19. Inhelder B. and Piaget J.: 1958,The Growth of Logical Thinking from Childhood to Adolescence, Basic Books, New York.Google Scholar
  20. Jackson S.: 1965, ‘The growth of logical thinking in normal and subnormal children’,British Journal of Educational Psychology 35, 225–258.Google Scholar
  21. Joyce L. K.: 1977, ‘A study of formal reasoning in elementary education majors’,Science Education 61(2), 153–158.Google Scholar
  22. Juraschek W. A. and Grady M. T.: 1981, ‘Format variations on equilibrium in the balance’,Journal of Research in Science Teaching 18(1), 47–49.Google Scholar
  23. Karplus R.: 1981, ‘Education and formal thought: A modest proposal’, in E. Siegel (ed.),New Directions in Piagetian Theory and Practice, Lawrence Erlbaum Associates, Hillsdale N.J.Google Scholar
  24. Karplus R., Karplus E., Formisano M., and Paulsen A.: 1979, ‘Proportional reasoning and control of variables in seven countries’, in J. Lochheed and J. Clement (eds.),Cognitive Process Instruction, The Franklin Institute Press, Philadelphia.Google Scholar
  25. Karplus R., Karplus E., and Wollman W.: 1974, ‘Intellectual development beyond elementary school. IV: Ratio, the influence of cognitive style’,School Science and Mathematics 74(6), 476–482.Google Scholar
  26. Karplus, R., Pulos, S., and Stage, E. K.: 1981,Proportional Reasoning in Early Adolescents: Comparison and Missing Value Problems in Three Schools, paper presented at the Third Annual Conference for the Psychology of Mathematics Education, Minneapolis, Minnesota.Google Scholar
  27. Karplus R., Pulos S., and Stage E. K.: 1983a, ‘Proportional reasoning of early adolescents’, in R. Lesh and M. Landau (eds.),Acquisition of Mathematics Concepts and Processes, Academic Press, New York.Google Scholar
  28. Karplus R., Pulos S., and Stage E. K.: 1983b, ‘Early adolescents' proportional reasoning on “rate” problems’,Educational Studies in Mathematics 14, 219–233.Google Scholar
  29. Karplus, R., Pulos, S., and Stage, E. K.: in press, ‘A study of early adolescents' mathematical reasoning’,Journal of Mathematical Behavior.Google Scholar
  30. Kieren T. E. and Nelson D.: 1978, ‘The operator construct of rational numbers in childhood and adolescence—an exploratory study’,The Alberta Journal of Educational Research 24(1), 22–30.Google Scholar
  31. Kieren T. E. and Southwell B.: 1979, ‘The development in children and adolescents of the construct of rational numbers as operators’,The Alberta Journal of Educational Research 25(4), 234–247.Google Scholar
  32. Kurtz B. and Karplus R.: 1979, ‘Intellectual development beyond elementary school. VII: Teaching for proportional reasoning’,School Science and Mathematics 79(4), 387–289.Google Scholar
  33. Lawson A. E.: 1977, ‘Relationships among performances on three formal operations tasks’,Journal of Psychology 96, 235–241.Google Scholar
  34. Lawson A. E.: 1979, ‘Relationships among performances on group-administered items of formal reasoning’,Perceptual and Motor Skills 48, 71–78.Google Scholar
  35. Lawson A. E.: 1982, ‘Formal reasoning, achievement, and intelligence: an issue of importance’,Science Education 66(1), 77–83.Google Scholar
  36. Lawson A. E. and Nordland F. H.: 1976, ‘Factor structure of some Piagetian tasks’,Journal of Research in Science Teaching 13(5), 461–466.Google Scholar
  37. Lawson, W. E. and Wollman, W. T.: 1975,Teaching Proportions and Intellectual Development—an experiment, AESOP Report No. ID-30, Lawrence Hall of Science.Google Scholar
  38. Lawson A. E. and Wollman W. T.: 1977, ‘Cognitive level, cognitive style, and value judgement’,Science Education 61(3), 397–407.Google Scholar
  39. Linn M. C. and Pulos S.: 1983, ‘Aptitude and experience influences on proportional reasoning during adolescence: focus on male-female differences’,Journal for Research in Mathematics Education 14(1), 30–46.Google Scholar
  40. Linn M. C. and SwineyJr J. F.: 1981, ‘Individual differences in formal thought’,Journal of Educational Psychology 73(2), 274–286.Google Scholar
  41. Lovell K.: 1961, ‘A follow-up study on Inhelder and Piaget's the Growth of Logical Thinking’,British Journal of Psychology 52(2), 143–153.Google Scholar
  42. Martorano S. C.: 1977, ‘A developmental analysis of performance on Piaget's formal operations tasks’,Developmental Psychology 13(6), 666–672.Google Scholar
  43. Newton, R., Copie, W., and Tobin, K. G.: 1981,Patterns of reasoning: Proportional Reasoning, paper presented at the annual meeting of the National Association for Research in Science Teaching, Grossinger, New York.Google Scholar
  44. Noelting, G.: 1975,Stages and Mechanisms in the Development of Proportions in the Child and Adolescent, paper presented at the Fifth Interdisciplinary Seminar on Piagetian Theory and its Implications for the Helping Professions, University of Southern California, Los Angeles.Google Scholar
  45. Noelting G.: 1980a, ‘The development of proportional reasoning and the ratio concept: Part I—Differentiation of stages’,Educational Studies in Mathematics 11, 217–253.Google Scholar
  46. Noelting G.: 1980b, ‘The development of proportional reasoning and the ratio concept: Part 11—Problem structure at successive stages; problem solving strategies and the mechanism of adaptive restructuring’,Educational Studies in Mathematics 11, 331–363.Google Scholar
  47. Pallrand G. J.: 1977,The Development of Formal Thought, National Association for Research in Science Teaching, Cincinnati, Ohio.Google Scholar
  48. Pallrand G. J.: 1979, ‘The transition to formal thought’,Journal of Research in Science Teaching 16(5), 445–451.Google Scholar
  49. Pascual-Leone J., Goodman D., Ammon P., and Subleman I.: 1978, ‘Piagetian theory and neo-Piagetian analysis as psychological guides in education’, in J. M. Gallagher and J. A. Easley (eds.),Knowledge and Development, Vol. 2, Plenum, New York.Google Scholar
  50. Piaget J., Grize J. B., Szeminska A., and Vinh Bang: 1968,Epistémologie et Psychologie de la Fonction, Presses Universitaires de France, Paris.Google Scholar
  51. Piaget J. and Inhelder B.: 1951,La Genèse de l'idée de hasard chez l'enfant, Presses Universitaires de France, Paris.Google Scholar
  52. Pitt, R. B. and Brouwer-janse, M.: 1981,Proportional Reasoning in Adolescents and Adults. Paper presented at the Society for Research in Child Development Biennal Meeting, Boston, Mass.Google Scholar
  53. Pulos S., Karplus R., and Stage E. K.: 1981, ‘Generality of proportional reasoning in early adolescents: Content effects and individual differences’,Journal of Early Adolescence 1, 257–264.Google Scholar
  54. Pulos, S., Stage, E. K., and Karplus, R.: 1980, ‘Cognitive correlates of proportional reasoning in early adolescence’,Proceedings of the 4th International Conference for the Psychology of Mathematics Education, 143–149.Google Scholar
  55. Pulos S., Stage E. K., and Karplus R.: 1982, ‘Setting effects in mathematical reasoning of early adolescents’,Journal of Early Adolescence 2(1), 39–60.Google Scholar
  56. Quintero, A. H. and Schwartz, J. L.: 1982,The Development of the Concept of Ratio in Children, DBRE Working Paper Series (WP-15).Google Scholar
  57. de Ribaupierre A., and Pascual-Leone J.: 1979, ‘Formal operations andM-power: A neo-Piagetian investigation’, in D. Kuhn (ed.),Intellectual Development Beyond Childhood, Jossey-Bass, San Francisco.Google Scholar
  58. Ricco G.: 1982, ‘Les première acquisitions de la notion de fonction linèaire chez l'enfant de 7 à 11 ans,’Educational Studies in Mathematics 13(3), 289–327.Google Scholar
  59. Riebman, B. and Overton, W.: 1977,Reflection-impulsivity and the Utilization of Formal Operational Thought, paper presented at the biennal meeting of the Society for Research in Child Development, New Orleans.Google Scholar
  60. Roberge J. J. and Flexer B. K.: 1979, ‘Further examination of formal operational reasoning abilities’,Child Development 50, 478–484.Google Scholar
  61. Ross R. J.: 1973, ‘Some empirical parameters of formal thinking’,Journal of Youth and Adolescence 2(2), 167–177.Google Scholar
  62. Rupley, W. H.: 1981,The Effects of Numerical Characteristics on the Difficulty of Proportional Problems. Unpublished doctoral dissertation, University of California, Berkeley.Google Scholar
  63. Siegler R. S.: 1976, ‘Three aspects of cognitive development’,Cognitive Psychology 8, 481–520.Google Scholar
  64. Siegler R. S.: 1978, ‘The origins of scientific reasoning’, in R. S. Siegler (ed.),Children's thinking: What develops? Lawrence Erlbaum Associates, Hillsdale, N.J.Google Scholar
  65. Stage, E. K. Karplus, R., and Pulos, S.: 1980, ‘Social content of early adolescents' proportional reasoning’,Proceedings of the 4th International Conference for the Psychology of Mathematics Education, 150–157.Google Scholar
  66. Tourniaire, F.: 1984,Proportional Reasoning in Grades Three, Four and Five, Unpublished doctoral dissertation, University of California, Berkeley.Google Scholar
  67. Van den Brink, J.: 1978,Young Children (6–8): Ratio and Proportion, paper presented at the Second Conference of IGPME.Google Scholar
  68. Vergnaud G.: 1979, ‘The acquisition of arithmetical concepts’,Educational Studies in Mathematics 10, 263–274.Google Scholar
  69. Vergnaud G.: 1980, ‘Didactics and acquisition of multiplicative structures in secondary schools’, in W. F. Archenbold, R. H. Driver, A. Orton, and G. Wood-Robinson (eds.),Cognitive Development Research in Science and Mathematics, The University of Leeds, Leeds, England.Google Scholar
  70. Vergnaud G.: 1983, ‘Multiplicative structures’, in R. Lesh and M. Landau (eds.),Acquisition of Mathematics Concepts and Processes, Academic Press, New York.Google Scholar
  71. Wersan, N.: 1981,Utilizing a Self-generated Visual Art Strategy to Facilitate Proportional Problem Solving in Mathematics, Unpublished doctoral dissertation, Rutgers University.Google Scholar
  72. Wilkening F and Anderson N. H.: 1982, ‘Comparison of two rule assessment methodologies for studying cognitive development and knowledge structure’,Psychological Bulletin 92(1), 215–237.Google Scholar
  73. Witken H. A. and Goodenough D. R.: 1981,Cognitive Style: Essence and Origins, International Universities Press, New York.Google Scholar
  74. Wollman W. and Lawson A. E.: 1978, ‘The influence of instruction on proportional reasoning in seventh graders’,Journal of Research in Science Teaching 15(3), 227–232.Google Scholar
  75. Wozny, C. D. and Cox, D. L.: 1975,The Effects of Task Differences on the Assessment of Formal Operational Thinking, Paper presented at the annual meeting of the American Educational Research Association, Washington, D.C.Google Scholar

Copyright information

© D. Reidel Publishing Company 1985

Authors and Affiliations

  • Francoise Tourniaire
    • 1
  • Steven Pulos
    • 1
  1. 1.Group in Science and Mathematics Education, Lawrence Hall of ScienceUniversity of CaliforniaBerkeleyUSA

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