Location of Repository

Qualitatively different approaches to simple arithmetic

By Edward Martin Gray


This study explores the qualitative difference in performance between those who are more successful and those who are less successful in simple arithmetic.\ud In the event that children are unable to retrieve a basic number combination the study identifies that there is a spectrum of performance between children who mainly use procedures, such as count-all in addition and take-away in subtraction, to those who handle simple arithmetic in a much more flexible way.\ud Two independent studies are described The first contrasts the performances of children in simple arithmetic. It considers teacher selected pupils of different ability from within each year group from 7+ to 12+. It takes a series of snapshots of different groups of children and considers their responses to a series of simple number combinations. This first experiment shows qualitatively different thinking in which the less successful children are seen to focus more on the use of procedures and in the development of competence in utilising them. The more successful appear to have developed a flexible mode of thinking which is not only capable of stimulating their selection of more efficient procedures but, the procedures they select are then used in an efficient and competent way.\ud However, the use of procedures amongst the more successful is seen to be only one of two alternative approaches that they use. The other approach involves the flexible use of mathematical objects, numbers, that are derived from encapsulated processes. The below-average children demonstrate little evidence of the flexible use of encapsulated processes. \ud It is the ability of the more able children to demonstrate flexibility through the use of efficient procedures and/or the use of encapsulated processes that stimulates the development of the theory of procepts. This theory utilises the duality which is ambiguously inherent in arithmetical symbolism to establish a framework from which we may identify the notion of proceptual thinking.\ud The second study considers the development of a group of children over a period of nearly a year. This study relates to aspects of the numerical component of the standardised tests in mathematics which form part of the National Curriculum. It provides the data which gives support to the theory and provides evidence to confirm the snap shots taken of children at the age of 7+ and 8+. It indicates that children who possess procedural competence may achieve the same level of attainment as those who display proceptual flexibility at one level of difficulty but they may not possess the appropriate mental tools to cope with the next.\ud The evidence of the study supports the hypothesis that there is a qualitative difference in children's arithmetical thinking

Topics: LB1501, QA
OAI identifier: oai:wrap.warwick.ac.uk:2309

Suggested articles



  1. (1991). (a), 1be primary mathematics textbook. intermediary in the cycle of change'.
  2. (1977). (Ed): 1983.7he Development of Mathematical 7hinking. Ncw York: Aca&-mic Pw-%& Ginsburg. If.:
  3. (1985). (General Editor):
  4. (1972). A chronometric analysis of simple addition', doi
  5. (1983). A developmental theory of number understanding', In
  6. (1980). A Hierarchy of understanding in mathematics',
  7. (1987). A model of the cognitive meaning of mathematical expressions, doi
  8. (1976). A structural Foundation for Tomorrow's Education', To Understand is to Invent, doi
  9. A.: 1976,7he Psychology of Mathematical Abilities doi
  10. (1987). Ability Stereotyping in Mathematics'. doi
  11. (1987). About Constructivism', In
  12. (1960). Ae process of education.
  13. (1984). Algebra: Children's Strategies and Errors.
  14. (1991). An analysis of diverging approaches to simple arithmetic: preference and its consequences'. doi
  15. (1987). An Analysis of Three Models of Early Number Development'. doi
  16. (1975). An experimental test of rive process models for subtraction', doi
  17. (1987). An investigation of young children's arithmetic contexts. doi
  18. (1977). Can preschool children invent addition algorithms? '. doi
  19. (1988). Children solving exchange problems in addition and subtraction: 7he consequence of limited security in number combinations.
  20. (1982). Children's Counting Types: Philosophy, 77zeory and Applications,
  21. (1984). Children's difficulties in subtraction: Some cause and questions'. doi
  22. (1984). Children's difficulties with school mathematics'. In
  23. (1986). Children's mastery of written numerals and the construction of basic number concepts',
  24. (1974). Children's solution processes in arithmetic word problems', doi
  25. (1981). Children's understanding of mathematics 11-16. doi
  26. (1983). Children's use of mathematical structure'. doi
  27. (1981). Classes, collections, and distinctive features: Alternative strategies for solving inclusion problems'. doi
  28. (1973). Classrooms Observed. London: Routledge and Kegan Paul.
  29. (1988). Cognitive and metacognitive shifts',
  30. (1982). Cognitive development and childrcn's solutions to vcrbal arithrnctic problems'. Journalfor Research in Afalhemadd Edwcarion. 13,83-98. 110111nds- R-. (Chicf Editor):
  31. (1976). Cognitive development, an information processing view.
  32. (1973). Comments on mathematical Education'. In doi
  33. (1983). Complex mathematical Cognition.
  34. (1986). Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis', In doi
  35. (1983). Conceptual Entities'.
  36. (1993). Count-on: 71c Parting of the Ways for Simple Arithmetic'.
  37. (1979). Critical Variables in Mathematics Education.. doi
  38. (1971). Cultural Roots of the Curriculum', In
  39. (1990). Developing understanding in trigonometry in boys and girls using a computer to link numerical and visual representations. Unpublished Ph.
  40. (1951). Developmental trends in arithmetic',
  41. (1982). Differential effects of the symbol systems ofarithmetic and geometry on the interpretation ofalgebraic symbols'. Paper presented at the meeting of the American Educational Research Association,
  42. (1991). Duality, ambiguity & flexibility in successful mathematical thinking.
  43. E.: 1983(b), Ile Notion of Principle: Tle case of Counting', In
  44. (1968). Educational Psychology: A Cognitive View. doi
  45. (1991). Encouraging versatile thinking in algebra using the computer', doi
  46. Examination and Assessment Council: 1992a, Standard Assessment Task, Teacher's Handbook, Key Stage 1.
  47. Examination and Assessment Council: 1992b, Standard Assessment Task,
  48. Examination and Assessment Council: 1992c, Standard Assessment Task,
  49. (1976). From communication to Curriculum.
  50. (1989). From counting to arithmetic: The development of early number skills', doi
  51. (1982). Generating number combinations: Rote process or problem solving? '
  52. (1970). Genetic Epistemology, doi
  53. (1979). Goals of learning and qualities of understanding,
  54. (1989). How children discover new strategies. doi
  55. (1985). Instruction on derived fact strategies in addition and subtraction', doi
  56. (1988). Is- versus -seeing as-: constructivism and the mathematics classroom'. In
  57. (1993). Known Fact 2 Count-on 0 Errpr InSAT: 3 Derived Fact I Count-all Time ise'conds) given In italics (1)7 Strategy and time at
  58. (1978). L.: 1922,7he psychology of arithmetic.
  59. (1982). Low Attainers in Mathematics: Policies and Practices in Schools. London:
  60. (1985). Mastery of basic number combinations: Internalisation of relationship or facts'. doi
  61. (1967). Mathematics and conditions of learning.
  62. (1982). Mathematics Counts. Report of the Committee of Inquiry into the Teaching of Mathematics in Schools.
  63. (1966). Mathematics in the Primary SchooL London: Macmillan. Dubinsky E.: 199 1, 'Reflective Abstraction'.
  64. (1982). Mathematicsfor Schools, An Integrated Series. (Second Edition),
  65. (1982). Me development of mental arithmetic: A chronometric approach'. Developmental Review, doi
  66. (1985). Mental multiplication skill: Structure, process and acquisition'. doi
  67. (1982). Mind sets in elementary school mathematics'.
  68. (1983). Models of Understanding', Zentralblattpr Didal-rik der Alathematik,
  69. (1988). Multiple perspectives', Paper prepared for Sixth International Congress on Mathematical Education,
  70. (1974). Number development in young children', doi
  71. (1989). Nurneracy without schooling'.
  72. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on processes and objects as different sides of the same coin', doi
  73. (1967). Organising and memorisizing: Sw&es in the psychology of teaming and reaching. Ncw York Hafncr.
  74. (1980). Problem solving and education'. doi
  75. (1981). Problem structure and first-grade children's initial solution processes for simple addition and subtraction problems'. doi
  76. (1935). Psychological considerations in the learning and teaching of arithmetic'.
  77. (1992). Race, ethnicity, social class, language and achievement in mathematics.
  78. (1992). Rational Number, Ratio and Proportion'.
  79. (1976). Relational understanding and instrumental understanding', doi
  80. (1991). role of conceptual entities and their symbols in building advanced mathematical thinking'. doi
  81. (1991). Rules without reasons as processes without objects: the case of equations and inequalities',
  82. (1990). Solution strategies: Subtraction number facts', doi
  83. (1987). Speaking Mathematically: Communications in mathematics classrooms.
  84. Strategy and time at inteMiew 202
  85. (1984). Strategy choices in addition: How do children know what to doT,
  86. (1905). Suggestions for the consideration of teachers and others concerned in the work- of Public Elementary Schools.
  87. (1962). Teacher Understanding and pupil efficiency in mathematics-a study of relationship'.
  88. (1986). Teaching children to add by counting on with finger patterns', doi
  89. (1986). Teaching children to subtract by counting up', doi
  90. (1982). The acquisition and elaboration of the number word sequence'. doi
  91. (1984). The acquisition of addition and subtraction concepts in grades one through three'. doi
  92. (1983). The acquisition of early number word meanings: A conceptual analysis and review'.
  93. (1962). The acquisition of knowledge',
  94. (1965). The child's conception of number (C. doi
  95. (1929). The Child's Conception of the World. London: Roudedge & Kegan Paul. doi
  96. (1988). The Child's maturity to Learn Mathematics in the School Situation'.
  97. (1986). The Child's Understanding of Number. (2nd ý Edition). Cambridge MA,
  98. (1992). The culture of the mathematics classroom: An unknown quantity', doi
  99. (1982). The development of addition and subtraction problem solving skills'. In doi
  100. (1928). The development of children's number ideas in the primary grades. Chicago: doi
  101. (1983). The development of children's problem solving ability in arithmetic'.
  102. (1987). The development of counting strategies for single digit addition'. doi
  103. (1990). The effect of superfluous information on children's solution of story arithmetic problems. doi
  104. (1986). The emergence of information retrieval strategies in numerical cognition: A development study', Cognition and Instruction. doi
  105. (1985). The Equilibrium of Cognitive Structures. Cambridge MA:
  106. (1991). The falsifiability criterion and refutation by mathematical induction',
  107. (1956). The magical number seven, plus or minus two: some limits on our capacity for processing information', doi
  108. (1986). The Psychology of leaming mathematics, (2nd Edition).
  109. (1976). The Psychology of Memory. doi
  110. (1965). The psychology of productive (creative) thinking,
  111. (1986). The relationship between initial meaningful and mechanical knowledge of arithmetic'.
  112. (1973). The role of quantification operators in development of the conservation of quantity', doi
  113. (1983). The transition from counting-all to counting-on in addition, doi
  114. (1987). The World of mathematics: Dream, myth or realityT In
  115. (1988). Theoretical Frameworks and Empirical Facts in the Psychology of Mathematics Education',
  116. (1989). To Inculcate versus to Elicit Knowledge'.
  117. (1968). Toward a theory of instruction. doi
  118. (1989). Transition from Operational to Structural Conception: The notion of function revisited,
  119. (1987). Understanding the mathematics teacher. - A study ofpractice in first schools.
  120. (1986). Understanding the number concepts in low attaining 7-9 year olds. Part 1. Development of descriptive framework and diagnostic instrument'. doi
  121. (1986). Understanding the number concepts in low attaining 7-9 year olds. Part IL The Teaching Studies'. doi
  122. (1987). What constructivism might be in mathematics education', In
  123. (1985). Young children "invent arithmetic.

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.