References

Acredolo, C. (1982). Conservation-nonconservation: Alternative explanations. In c.J. Brainerd (Ed.), Progress in cognitive development research: Vol. 1. Children's logical and mathematical cognition (pp. 1-31). New York: Springer-Verlag.

Ashcraft, M. H., & Fierman, B. A. (1982). Mental addition in third, fourth, and sixth graders. Journal of Experimental Child Psychology, 33, 216-234.

Ashcraft, M. H., & Stazyk, E. H. (1981). Mental addition: A test of three verification models. Memory & Cognition, 9, 185-196.

Avesar, C., & Dickerson, D. J. (in press). Children's judgments of relative number by one to one correspondence: A planning perspective. Journal of Experimental Child Psychology.

Baron, J., Lawson, G., & Siegel, L. S. (1975). Effects of training and set size on children's judgments of number and length. Developmental Psychology, 11, 583-588.

Baroody, A. J. (1982). Evaluating the validity of Brainerd's cardinality task. Child Study Journal, 12, 79-87.

Baroody, A. J. (1983). The development of procedural knowledge: An alternative explanation for chronometric trends of mental arithmetic. Developmental Review, 3,225-230.

Baroody, A. J. (1984). Children's difficulties in subtraction: Some causes and ques­tions. Journal for Research in Mathematics Education, 15, 203-213.

Baroody, A. J. (1986a). Basic counting principles used by mentally retarded children. Journal for Research in Mathematics Education, 17, 382-389.

Baroody, A. J. (1986b). Counting ability of moderately and mildly handicapped children. Education and Training of the Mentally Retarded, 21, 289-300.

Baroody, A. J. (1987 a) Children's mathematical thinking: A developmental framework for preschool, primary, and special education teachers. New York: Teachers

420 References

College Press. Baroody, A. J. (1987b). The development of counting strategies for single-digit addi­

tion. Journal for Research in Mathematics Education, 18, 141-157. Baroody, A. J., & Ginsburg, H. P. (1983). The effects of instruction on children's

understanding of the "equals" sign. The Elementary School Journal, 84, 199-212. Baroody, A. J., & Ginsburg, H. P. (1984, April). TMR and EMR children's ability to

learn counting skills and principles. Paper presented at the annual meeting of the American Educational Research Association, New Orleans.

Baroody, A. J., & Ginsburg, H. P. (1986). The relationship between initial mean­ingful and mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 75-112). Hillsdale, NJ: Lawrence Erlbaum.

Baroody, A. J., Ginsburg, H. P., & Waxman, B. (1983). Children's use of mathematical structure. Journal of Research in Mathematics Education, 14, 156-168.

Baroody, A. J., & Price, J. (1983). The development of the number-word sequence in the counting of three-year-olds. Journal for Research in Mathematics Education, 14,361-368.

Bebout, H. C. (1986). First-graders' levels of verbal problem performance. In Pro­ceedings of the Tenth International Conference on the Psychology of Mathematics Education (pp. 1-6). London: University of London Institute of Education.

Beckwith, M., & Restle, F. (1966). Processes of enumeration. Psychological Review, 73, 437-444.

Behr, M., Erlwanger, S., & Nichols, E. (1980). How children view the equals sign. Mathematics Teaching, 92, 13-15.

Beilin, H. (1969). Stimulus and cognitive transformation in conservation. In D. Elkind & J. H. Flavell (Eds.), Studies in cognitive development: Essays in honor of Jean Piaget (pp. 409-438). New York: Oxford University Press.

Beilin, H. (1975). The child psychology series: Experimental and theoretical analyses of child behavior. Studies in the cognitive basis of language development. New York: Academic Press.

Bell, M. S., & Burns, J. (1981a). Counting, numeration, and arithmetic capabilities of primary school children. Proposal submitted to National Science Foundation.

Bell, M. S., & Burns, J. (1981b). [Primary school children's performance on counting, numeration, and arithmetic tasks.] Unpublished raw data.

Bell, M. S., Fuson, K. c., & Lesh, R. A. (1976). Algebraic and arithmetic structures: A concrete approach for elementary school teachers. New York: The Free Press.

Blevins, B., Mace, P. G., Cooper, R. G., Starkey, P., & Leitner, E. (1981, April). What do children know about addition and subtraction? Paper presented at the biennial meeting of the Society for Research in Child Development, Boston.

Blevins-Knabe, B., Cooper, R. G., Starkey, P., Mace, P. G., & Leitner, E. (1987). Preschoolers sometimes know less than we think: The use of quantifiers to solve addition and subtraction tasks. Bulletin of the Psychonomic Society, 25, 31-34.

Botvin, G. J., & Murray, F. B. (1975). The efficacy of peer modelling and social conflict in the acquisition of conservation. Child Development, 46, 796-799.

Brainerd, C. J. (1973). Mathematical and behavioral foundations of number. Journal of General Psychology, 88, 221-281.

Brainerd, C. J. (1977). Feedback, rule knowledge, and conservation learning. Child Development, 48, 404-411.

References 421

Brainerd, C. J. (1979a). Markovian interpretations of conservation learning. Psycho­logical Review, 86, 181-213.

Brainerd, C. J. (1979b). The origins of the number concept. New York: Praeger. Brainerd, C. J., & Hooper, F. H. (1975). A methodological analysis of developmental

studies of identity conservation and equivalence conservation. Psycyhological Bulletin, 82, 725-737.

Brainerd, C. J., & Hooper, F. H. (1978). More on the identity-equivalence sequences: An update and some replies to Miller. Psychological Bulletin, 85, 70-75.

Brainerd, C. J. & Kingma, J. (1984). Do children have to remember to reason? A fuzzy-trace theory of transitivity development. Developmental Review, 4, 311-377.

Briars, D. J., & Fuson, K. C. (1979). [Developmental changes in spatial strategies for remembering already counted objects.] Unpublished raw data.

Briars, D. J., & Fuson, K. C. (1980). [Developmental changes in the interpretation of getting two different count words for the same set of objects.] Unpublished raw data.

Briars, D. J., & Larkin, J. H. (1984). An integrated model of skills in solving elementary word problems. Cognition and Instruction, I, 245-296.

Briars, D. J., & Siegler, R. S. (1984). A featural analysis of preschooler's counting knowledge. Developmental Psychology, 20, 607-618.

Bruner, J. S., Olver, R.R., & Greenfield, P. M. (1966). Studies in cognitive growth. New York: John Wiley & Sons.

Brush, L. R. (1978). Preschool children's knowledge of addition and subtraction. Journal for Research in Mathematics Education, 9, 44-54.

Bryant, P. E. (1974). Perception and understanding of young children: An experi­mental approach. New York: Basic Books.

Carpenter, T. P. (1975). Measurement concepts of first- and second-grade students. Journal for Research in Mathematics Education, 6, 3-14.

Carpenter, T. P., Bebout, H. c., & Moser, J. M. (1985, March). The representation of basic addition and subtraction word problems. Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL.

Carpenter, T. P., & Moser, J. M. (1983a). The acquisition of addition and subtraction concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics: Concepts and Processes (pp. 7-44). New York: Academic Press.

Carpenter, T. P., & Moser, J. M. (1983b). Addition and subtraction questions: How they develop (project paper). Madison: Wisconsin Center for Education Research.

Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Educa­tion, 15, 179-202.

Case, R. (1985a). A developmentally based approach to the problem of instructional design. In S. F. Chipman, J. W. Segal, & R. Glaser (Eds.), Thinking and learning skills: Vol. 2. Research and open questions (pp. 545-562). Hillsdale, NJ: Lawrence Erlbaum Associates.

Case, R. (1985b). Intellectual development: Birth to adulthood. Orlando, FL: Academic Press, Inc.

Case, R., Kurland, M., & Daneman, M. (1979, March). Operational efficiency and the growth of M-space. Paper presented at the biennial meeting of the Society for Re­search in Child Development, San Francisco.

Case, R., & Sandieson, R. (1986). Horizontal and vertical enrichment: A develop­mental approach to the teaching of central conceptual skills (First Year Report).

422 References

Ontario Institute for Studies in Education. Case, R., & Sandieson, R. (1987, May). A developmental approach to the isolation

and teaching of central conceptual skills in middle school mathematics and science. Paper presented at the National Science Foundation conference on middle grade mathematics, Dekalb, IL.

Chi, M.T.H., & Klahr, D. (1975). Span and rate of apprehension in children and adults. Journal of Experimental Child Psychology, 19, 434-439.

Clements, D. H. (1984). Training effects on the development and generalization of Piagetian logical operations and knowledge of number. Journal of Educational Psy­chology, 76, 766-776.

Cobb, P. (1985, April). Children's concepts of addition and subtraction: From number to part-whole. Paper presented at the annual meeting of the American Educational Research Association, Chicago.

Cobb, P. (1986, April). Counting types and word problems. Paper presented at the annual meeting of the American Educational Research Association, San Francisco.

Cobb, P. (1987). An analysis of three models of early number development. Journal of Research in Mathematics Education, 18, 163-179.

Comiti, c., Bessot, A., & Pariselle, C. (1980). Analyse de comportements d'eleves du cours preparatoire confrontes a une tache de construction d'un ensemble equipotent a un ensemble donne. Recherches en didactique des mathbnatiques. 1, 171-217.

Cooper, R. G. (1984). Early number development: Discovering number space with addition and subtraction. In C. Sophian (Ed.), Origins of cognitive skills (pp. 157-192). Hillsdale, NJ: Lawrence Erlbaum.

Cowan, R. (1979a). Performance in number conservation tasks as a function of the number of items. British Journal of Psychology, 70, 77-81.

Cowan, R. (1979b). A reappraisal of the relation between performances of quantitative identity and quantitative equivalence conservation tasks. Journal of Experimental Child Psychology, 28, 68-80.

Cowan, R. (1984). Children's relative number judgments: One-to-one correspon­dence, recognition of noncorrespondence, and the influence of cue conflict. Journal of Experimental Child Psychology, 38, 515-532.

Cowan, R. (1987). When do children trust counting as a basis for relative number judgments? Journal of Experimental Child Psychology, 43, 328-345.

Cuneo, D. O. (1982). Children's judgments of numerical quantity: A new view of early quantification. Cognitive Psychology, 14, 13-44.

Davis, H., & Bradford, S. A. (1986). Counting behavior by rats in a simulated natural environment. Ethology, 73, 265-280.

Davis, H. & Memmott, J. (1982). Counting behavior in animals: A critical evaluation. Psychological Bulletin, 92, 547-571.

Davis, H., & Perusse, R. (1987). Numerical competence in animals: Definitional issues, current evidence and a new research agenda. Unpublished manuscript, Uni­versity of Guelph, Ontario.

Davydov, V. V., & Andronov, V. P. (1981). Psychological conditions of the origina­tion of ideal actions (Project Paper 81-2), (English translation), Madison: Wisconsin Research and Development Center for Individualized Schooling, The University of Wisconsin.

De Corte, E., & Verschaffel, L. (1985a). Beginning first graders' initial representation of arithmetic word problems. Journal of Mathematical Behavior, J, 3-21.

References 423

De Corte, E., & Verschaffel, L. (1985b, April). An empirical validation of computer models of children's word problem solving. Paper presented at the annual meeting of the American Educational Research Association, Chicago.

De Corte, E., & Verschaffel, L. (1985c) 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 (pp. 304-309). Utrecht, The Netherlands: State University of Utrecht.

De Corte, E., & Verschaffel, L, & De Win, L. (1985) Influence of rewording verbal problems on children's problem representations and solutions. Journal of Educa­tional Psychology, 77, 460-470.

Dienes, Z. P. (1960). Building up mathematics (4th ed.) London: Hutchinson Educa­tional, Ltd.

Durkin, K., Shire, B., Riem, R., Crowther, R. D., & Rutter, D. R. (1986). The social and linguistic context of early number word use. British Journal of Developmental Psychology, 4, 269-288.

Elkind, D. (1967). Piaget's conservation problems. Child Development, 38, 15-27. Elkind, D., & Schoenfeld, E. (1972). Identity and equivalence conservation at two age

levels. Developmental Psychology, 6, 529-533. Frye, D., Braisby, N., Lowe, J., Maroudas, c., & Nicholls, J. (1986). Young

children's understanding of counting and cardinality. Manuscript submitted for publication.

Fuson, K. C. (1979a, November). Counting solution procedures in addition and sub­traction. Paper presented at the Wisconsin Wingspread Conference on the Initial Learning of Addition and Subtraction Skills, Racine, WI.

Fuson, K. C. (1979b). The development of self-regulating aspects of speech: A re­view. In G. Zivin (Ed.), The development of self-regulation through private speech (pp. 135-217). New York: John Wiley & Sons.

Fuson, K. C. (1982a). An analysis of the counting-on solution procedure in addition. In T. P. Carpenter, 1. M. Moser, & T. A. Romberg (Eds.), Addition and subtrac­tion: A cognitive perspective (pp. 67-81). Hillsdale, NJ: Lawrence Erlbaum.

Fuson, K. C. (1982b). The counting-on solution procedure: Analysis and empirical results. Unpublished manuscript. Northwestern University.

Fuson, K. C. (1984). More complexities in subtraction. Journalfor Research in Mathe­matics Education, IS, 214-225.

Fuson, K. C. (1986a). Roles of representation and verbalization in the teaching of multi-digit addition and subtraction. European Journal of Psychology of Educa­tion, 1, 35-56.

Fuson, K. C. (1986b). Teaching children to subtract by counting up. Journal for Re­search in Mathematics Education, 17, 172-189.

Fuson, K. C. (1986c, November). Effects of collection terms on class inclusion and on number tasks. Paper presented at the annual meeting of the Psychonomics Society, New Orleans.

Fuson, K. C. (1987, April). Teaching the general multi-digit addition and subtraction algorithms to first and second graders. Paper presented at the biennial meeting of the Society for Research in Child Development, Baltimore, Maryland.

Fuson, K. c., & Darling, C. L. (1984). A study of the effectiveness of the graphics tablet and an Apple lie microcomputer in recording errors in children's counting. Unpub­lished manuscript.

Fuson, K. c., & Hall, 1. W. (1983). The acquisition of early number word meanings.

424 References

In H. Ginsburg (Ed.), The development of children's mathematical thinking (pp. 49-107). New York: Academic Press.

Fuson, K. c., Lyons, B. G., Pergament, G. G., Hall, J. W., & Kwon, Y. (in press). Effects of collection terms on class inclusion and on number tasks. Cognitive Psychology.

Fuson, K. c., & Mierkiewicz, D. (1980, April). A detailed analysis of the act of counting. Paper presented at the annual meeting of the American Educational Research Association, Boston.

Fuson, K. c., Pergament, G. G., & Lyons, B. G. (1985). Collection terms and pre­schoolers' use of the cardinality rule. Cognitive Psychology, 17, 315-323.

Fuson, K. c., Pergament, G. G., Lyons, B. G., & Hall, J. W. (1985). Children's con­formity to the cardinality rule as a function of set size and counting accuracy. Child Development, 56, 1429-1439.

Fuson, K. c., & Richards, J. (1980, April). Children's construction of the counting numbers: From a .Ipew to a directional chain. Paper presented at the annual meeting of the American Educational Research Association, Boston.

Fuson, K. c., Richards, J., & Briars, D. J. (1982). The acquisition and elaboration of the number word sequence. In C. Brainerd (Ed.), Progress in cognitive develop­ment research: Vol I. Children's logical and mathematical cognition (pp. 33-92). New York: Springer-Verlag.

Fuson, K. c., & Secada, W. G. (1983). The development of number word sequence skills. In J. Bergeron & N. Herscovics (Eds.), Proceedings of the Fifth Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 266-274). Montreal: University of Montreal.

Fuson, K. c., & Secada, W. G. (1986). Teaching children to add by counting on with finger patterns. Cognition and Instruction, 3, 229-260.

Fuson, K. c., Secada, W. G., & Hall, J. W. (1983). Matching, counting, and con­servation of numerical equivalence. Child Development, 54,91-97.

Fuson, K. c., Stigler, J. W., & Bartsch, K. (in press). Grade placement of addition and subtraction topics in China, Japan, the Soviet Union, Taiwan, and the United States. Journal of Research in Mathematics Education.

Fuson, K. c., & Willis, G. B. (1986). First and second graders' performance on compare and equalize word problems. In Proceedings of the Tenth International Conference on the Psychology of Mathematics Education (pp. 19-24). London: University of London Institute of Education.

Fuson, K. c., & Willis, G. B. (1987). [Teaching addition and subtraction word problems via schematic drawings]. Unpublished raw data.

Fllson, K. c., & Willis, G. B. (in press). Subtracting by counting up: More evidence. Journal for Research in Mathematics Education.

Gelman, R. (1969). Conservation acquisition: A problem of learning to attend to relevant attributes. Journal of Experimental Child Psychology, 7, 167-187.

Gelman, R. (1972). The nature and development of early number concepts. In H. W. Reese (Ed.), Advances in child development and behavior (Vol. 7, pp. 115-167). New York: Academic Press.

Gelman, R. (1982a). Accessing one-to-one correspondence: Still another paper about conservation. Journal of Psychology, 73, 209-220.

Gelman, R. (1982b). Basic numerical abilities. In R. J. Sternberg (Ed.), Advances in /il<' psychology of intelligence (Vol. 1 pp. 181-205). Hillsdale, N. J.: Lawrence hlbaum.

References 425

Gelman, R., & Baillargeon, R. (1983). A review of some Piagetian concepts. In P. H. Mussen (Ed.), Handbook of child psychology: Vol. 3. Cognitive Development (pp. 167-230). New York: John Wiley & Sons, Inc.

Gelman, R., & Gallistel, C. R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.

Gelman, R., & Meek, E. (1983). Preschoolers' counting: Principles before skill. Cognition, 13, 343-359.

Gelman, R., & Meek, E. (1986). The notion of principle: The case of counting. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 29-57). Hillsdale, NJ: Lawrence Erlbaum.

Gelman, R., Meek, E., & Merkin, S. (1986). Young children's numerical competence. Cognitive Development, 1, 1-29.

Ginsburg, H. P. (1977). Children's arithmetic: The learning process. New York: D. Van Nostrand.

Ginsburg, H. P., & Russell, R. L. (1981). Social class and racial influences on early mathematical thinking. Monographs of the Society for Research in Child Develop­ment, 46, (6, Serial No. 193).

Greco, P. (1962). Quantite et quotite. In P. Greco & A. Morf (Eds.), Structures numeriques elementaires: Etudes d'epistemologic genetique (Vol. 13, pp. 1-70). Paris: Presses Universitaires de France.

Greeno, J. G., Riley, M. S., & Gelman, R. (1984). Conceptual competence and chil­dren's counting. Cognitive Psychology, 16, 94-143.

Groen, G. J., & Parkman, J. M. (1972). A chronometric analysis of simple addition. Psychological Review, 79, 329-343.

Gullen, G. E. (1978). Set comparison tactics and strategies of children in kinder­garten, first grade, and second grade. Journal for Research in Mathematics Educa­tion, 9, 349-360.

Hamann, M. S., & Ashcraft, M. H. (1985). Simple and complex mental addition across development. Journal of Experimental Child Psychology, 40, 49-72.

Hatano, G. (1982). Learning to add and subtract: A Japanese perspective. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 211-223). Hillsdale, NJ: Lawrence Erlbaum.

Herscovics, N., Bergeron, J. c., & Bergeron, A. (1986). The kindergartner's per­ception of the invariance of number under various transformations. In G. Lappan & R. Evan (Eds.), Proceedings of the eighth annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (pp. 28-34). East Lansing, MI.

Hiebert, J. (1981, April). Young children's solution processes for verbal addition and subtraction problems: The effect of the position of the unknown set. Paper presented at the 59th annual meeting of the National Council of the Teachers of Mathematics, St. Louis.

Hodges, R., & French, L. (1987, April). Classes and collections: Their influence on class-inclusion, cardinality, and number conservation. Paper presented at the annual meeting of the American Educational Research Association, Washington, DC.

Houlihan, D. M., & Ginsburg, H. P. (1981). The addition methods of first- and second-grade children. Journal for Research in Mathematics Education, 12, 95-106.

Hudson, T. (1983). Correspondences and numerical differences between joint sets.

426 References

Child Development, 54, 84-90. Hudson, M., Guthrie, K. H., & Santilli, N. R. (1982) . The use of linguistic and non­

linguistic strategies in kindergarteners' interpretations of 'more' and 'less.' Journal of Child Language, 9, 125-138.

Jensen, E . M., Reese, E. P., & Reese, T. W. (1950). The subitizing and counting of visually presented fields of dots. The Journal of Psychology, 30, 363- 392.

Kamii, C. K. (1985) . Young children reinvent arithmetic: Implications of Piaget's theory. New York : Teachers College Press.

Katz, H., & Beilin, H. (1976). A test of Bryant's claims concerning the young child's understanding of quantitative invariance. Child Development, 47, 877-880.

Kaufman, E. L., Lord, M. W., Reese, T. W., & Volkmann , J. (1949). The discrimina­tion of visual number. American Journal of Psychology, 62, 498-·525.

Kaye, D. B. , Post , T. A., Hall, Y. c., & Dineen , J. T. (1986). Emergence of infor­mation-retrieval strategies in numerical cognition: A developmental study. Cogni­tion and Instruction, 3, 127-150.

Kingma, J., & Roelinga, U. (1984). Task sensitivity and the sequence of development in seriation, ordinal correspondence, and cardination. Genetic Psychology Mono­graphs, 110, 181-205.

Kintsch, W., & Greeno, J . G . (1985). Understanding and solving word arithmetic problems. Psychological Review, 92, 109-129.

Klahr, D. (1984). Transition processes in quantitative development . In R. J. Sternberg (Ed.), Mechanisms of cognitive development (pp. 101-139). New York: W. H. Freeman.

Klahr, D. , & Wallace, J. G. (1973) . The role of quantification operators in the development of conservation of quantity. Cognitive Psychology, 4, 301-327.

Klahr, D., & Wallace, J. G. (1976). Cognitive development: An information processing view. Hillsdale, NJ: Lawrence Erlbaum.

Kwon, Y. (1986). Correspondence errors in children's counting. Unpublished manu­script, Northwestern University.

LaPointe , K., & O'Donnell , J . P. (1974). Number conservation in children below age six: Its relationship to age, perceptual dimensions, and language comprehension. Developmental Psychology, 10, 422-428.

Lawson, G. , Baron, J . , & Siegel , L. (1974) . The role of number and length cues in children's quantitative judgments. Child Development, 45, 731-736.

Lieberthal, E. M. (1979). Complete book of fingermath. New York: McGraw-HilI. Luria , A. R. (1959). The directive function of speech in development and dissolution.

Part 1: Development of the directive function of speech in early childhood. Word, 15,341-351.

Luria, A. R. (1961) . The role of speech in the regulation of normal and abnormal be­haviour (1. Tizard, Ed.), Oxford: Pergamon.

Mandler, G. , & Shebo, B. J. (1982) . Subitizing: An analysis of its component pro­cesses. Journal of Experimental Psychology: General, 111,1-22.

Markman, E. M. (1979). Classes and collections: Conceptual organization and numerical abilities. Cognitive Psychology, 1, 395-411.

Matsuzawa, T. (1985). Use of numbers by a chimpanzee. Nature, 315,57-59. McCloskey, M. , Sokol, S. M., & Goodman, R. A. (1986). Cognitive processes in

verbal-number production: Inferences from the performance of brain-damaged subjects. Journal of Experimental Psychology: General, 115, 307-330.

McGarrigle, J . , & Donaldson, M. (1975). Conservation accidents. Cognition, 3,314-350.

References 427

McLaughlin, J. A. (1981). Development of children's ability to judge relative numerosity. Journal of Experimental Child Psychology, 31, 103-114.

Melnick, G. I. (1973). A mechanism for transition of concrete to abstract cognitive processes. Child Development, 44, 599-605.

Michie, S. (1984a). Number understanding in preschool children. British Journal of Educational Psychology, 54, 245-253.

Michie, S. (1984b). Why preschoolers are reluctant to count spontaneously. British Journal of Developmental Psychology, 2, 347-358.

Michie, S. (1985). Development of absolute and relative concepts of number in pre­school children. Developmental Psychology, 21,247-252.

Millar, c., & Mackay, C. K. (1978). Conflict between cues in number conservation tasks. British Journal of Educational Psychology, 50, 188-191.

Miller, K. (1984). Child as the measurer of all things: Measurement procedures and the development of quantitative concepts. In C. Sophian (Ed.), Origins of cognitive skills (pp. 193-288). Hillsdale, NJ: Lawrence Erlbaum.

Miller, K., & Gelman, R. (1983). The child's representation of number: A multi­dimensional scaling analysis. Child Development, 54, 1470-1479.

Miller, K., & Stigler, J. W. (1987). Counting in Chinese: Cultural variation in a basic cognitive skill. Cognitive Development, 2, 279-305.

Miller, P. H. (1978). Stimulus variables in conservation: An alternative approach to assessment. Merrill-Palmer Quarterly, 24, 141-160.

Miller, P. H., Heldmeyer, K. H., & Miller, S. A. (1975). Facilitation of conservation of number in young children. Developmental Psychology, 11, 253.

Miller, P. H., & Heller, K. A. (1976). Facilitation of attention to number and con­servation of number. Journal of Experimental Child Psychology, 22, 454-467.

Miller, S. A. (1977). A disconfirmation of the quantitive identity-quantitative equiv­alence sequence. Journal of Experimental Child Psychology, 24, 180-189.

Miura, I. T. (1987). Mathematics achievement as a function of language. Journal of Educational Psychology, 79, 79-82.

Modgil, S., & Modgil, C. (1976). Piagetian research: Compilation and commentary (Vols. 2, 4, 7, 8, 12). London: NEFR Publishing.

Moore, c., & Frye, D. (1986a). Context, conservation and the meanings of 'more.' British Journal of Developmental Psychology, 4, 169-178.

Moore, c., & Frye, D. (1986b). The effect of experimenter's intention on the child's understanding of conservation. Cognition, 22, 283-298.

Murray, F. B. (1981). The conservation paradigm: The conservation of conservation research. In I. E. Sigel, D. M. Brodzinsky, & R. M. Golinkoff (Eds.), New directions in Piagetian theory and practice. Hillsdale, NJ: Lawrence Erlbaum.

Nesher, P. (1980). The stereotyped nature of school word problems. For the Learning of Mathematics, 1, 41-48.

Pepperberg, J. (1987). Evidence for conceptual quantitative abilities in the African Grey Parrot: Labelling of cardinal sets. Ethology, 75, 37-61.

Pergament, G. G., & Fuson, K. C. (1982). Effects of size of set, homogeneity of objects, and collection nouns on children's accurate counting and use of the car­dinality rule. In S. Wagner (Ed.), Proceedings of the Sixth Annual Conference for the Psychology of Mathematics Education-North America (pp. 78-84). Athens, GA: University of Georgia.

Piaget, J. (1965). The child's conception of number. New York: W. W. Norton. (Original work published 1941).

Piaget, J. (1967). Cognitions and conservations: Two views. Contemporary Psychol-

428 References

ogy, 12, 532-533. Piaget, J. (1968a). On the development of memory and identity, Barre, MA: Clark

University Press. Piaget, l. (1968b). Quantification, conservation, and nativism. Science, 162, 976-979. Piaget, J. (1977). Some recent research and its link with a new theory of groupings and

conservations based on commutability. In R. W. Rieber & K. Salzinger (Eds.), The roots of American psychology: Historical influence and implications for the future. Annals of the New York Academy of Sciences, 291,350-357.

Piaget, l. (1978). Recherches sur la generalisation (Etudes d'epistemologie genetique, XXXVI). Paris, Presses Universitaires de France.

Piaget, l. (1979). Correspondences and transformations. In Frank B. Murray (Ed.), The impact of Piagetian theory (pp. 17-28). Baltimore: University Park Press.

Pollio, H. R., & Whitacre, J. D. (1970). Some observations on the use of natural num­bers by preschool children. Perceptual and Motor Skills, 30, 167--174.

Potter, M. c., & Levy, E. 1. (1968). Spatial enumeration without counting. Child Development, 39, 265-273.

Pringle, R., & Hennessey, A. (1987). The role of size and density in number equiv­alence judgment: Chronometric and developmental analyses. Manuscript sub­mitted for publication.

Pufall, P. B., & Shaw, R. E. (1972). Precocious thoughts on number: The long and the short of it. Developmental Psychology, 7, 62-69.

Rathmell, E. C. (1978). Using thinking strategies to teach the basic facts. In M. Suydam & R. Reys (Eds.), Developing computational skills: 1978 Yearbook of the National Council of Teachers of Mathematics (pp. 13-38). Reston, VA: NCTM.

Rathmell, E. C. (1986). Helping children learn to solve story problems. In A. Zollman, W. Speer, & J. Meyer (Eds.), The Fifth Mathematics Methods Confer­ence Papers (pp. 101-109). Bowling Green, OH: Bowling Green State University.

Resnick, L. B. (1983). A developmental theory of number understanding. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 109-151). New York: Academic Press.

Resnick, L. B., & Neches, R. (1984). Factors affecting individual differences in learning ability. In R. Sternberg (Ed.), Advances in the psychology of human intelligence (Vol. 2, pp. 275-323). Hillsdale, NJ: Lawrence Erlbaum.

Riley, M. S., Greeno, J. G., & Heiler, J. 1. (1983). Development of children's problem-solving ability in arithmetic. In H. Ginsburg (Ed.), The development of mathematical thinking (pp. 153-196). New York: Academic Press.

Rogoff, B., & Wertsch, l. V. (1984). Children's learning in the zone of proximal development: New directions for child development. San Francisco: lossey-Bass.

Russac, R. J. (1978). The relation between two strategies of cardinal number: Cor­respondence and counting. Child Development, 49, 728-735.

Russell, B. (1903). The principles of mathematics. Cambridge, England: Cambridge University Press.

Russell, B. (1919). Introduction to mathematical philosophy. London: Allen and Unwin.

Saxe, G. B. (1977). A developmental analysis of notational counting. Child Develop­ment, 48, 1512-1520.

Saxe, G. B. (1979). Developmental relations between notational counting and number conservation. Child Development, 50, 180-187.

Saxe, G. B. (1982). Culture and the development of numerical cognition: Studies

References 429

among the Oksapmin of Papua New Guinea. In C. J. Brainerd (Ed.), Progress in cognitive development research: Vol. 1. Children's logical and mathematical cogni­tion (pp. 157-176). New York: Springer-Verlag.

Saxe, G. B. , Guberman, S. R., & Gearhart, M. (in press) . Social and developmental processes in children's understanding of number. Monographs of the Society for Research in Child Development.

Saxe, G. 8., & Kaplan, R. (1981). Gesture in early counting: A developmental analysis. Perceptual and Motor Skills, 53,851-854.

Saxe, G. B., Sadeghpour, M., & Sicilian, S. (in press). Developmental differences in children 's understanding of number word conventions. Journal for Research in Mathematics Education.

Schaeffer, B., Eggleston, V. H., & Scott, J. L. (1974). Number development in young children. Cognitive Psychology, 6, 357-379.

Schonfeld, I. S. (1986). The Genevan and Cattell-Horn conceptions of intelligence compared: Early implementation of numerical solution aids: Developmental Psychology, 22, 204-212.

Secada, W. G. (1982, March). The use of counting for subtraction. Paper presented at the annual meeting of the American Educational Research Association, New York.

Secada, W. G. (1984). Counting in sign: The number string, accuracy and use. Un­published doctoral dissertation, Northwestern University.

Secada, W. G., Fuson, K. c., & Hall, J. W. (1983). The transition from counting-all to counting-on in addition. Journal for Research in Mathematics Education, 14, 47-57.

Siegel, A. W. , Goldsmith, L. T ., & Madson, C. R. (1982). Skill in estimation problems of extent and numerosity. Journal for Research in Mathematics Educa­tion, 13,211-232.

Siegel, L. S. (1977). The cognitive basis of the comprehension and production of rela­tional terminology. Journal of Experimental Child Psychology, 24, 40-52.

Siegel, L. S. (1978). The relationship of langauge and thought in the preoperational child: A reconsideration of nonverbal alternatives to Piagetian Tasks. In L. S. Siegel & c. J . Brainerd (Eds.), Alternatives to Piaget: Critical essays on the theory (pp. 43-63). New York: Academic Press.

Siegel, L. S. (1982). The development of quantIty concepts: Perceptual and linguistic factors . In C. J. Brainerd (Ed.), Progress in cognitive development research: Vol. 1. Children 's logical and mathematical cognition (pp. 123-155). New York : Springer­Verlag.

Siegel, L. S., & Brainerd, C. J. (1978). Alternatives to Piaget: Critical essays on the theory. New York: Academic Press.

Siegel, L. S., & Goldstein, A. G. (1969). Conservation of number in young children: Recency versus relational response strategies. Developmental Psychology, 2, 128-130.

Siegler, R. S. (1981). Developmental sequences within and between concepts. Mono­graphs of the Society for Research in Child Development, 46 (2, Serial No. 189).

Siegler, R. S. (in press). The perils of averaging data over strategies: An example from children's addition. Journal of Experimental Psychology.

Siegler , R. S., & Campbell, J. (in press) . Diagnosing individual differences in strategy choice procedures . In N. Frederiksen (Ed.) , Diagnostic monitoring of knowledge and skill acquisition. Hillsdale, N.1.: Lawrence Erlbaum.

430 References

Siegler, R. S., & Robinson, M. (1982). The development of numerical understand­ings. In H. W. Reese & L. P. Lipsitt (Eds.), Advances in child development and behavior (Vol. 16, pp. 242-312). New York: Academic Press.

Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction. How do children know what to do. In C. Sophian (Ed.), Origins of cognitive skills (pp. 229-293). HilJsdale, NJ: Lawrence Erlbaum.

Silverman, I. W., & Briga, J. (1981). By what processes do young children solve smalJ number conservation problems? Journal of Experimental Child Psychology, 32, 115-126.

Silverman, I. W., & Rose, A. P. (1982). Compensation and conservation. Psycho­logical Bulletin, 91, 80-101.

Smither, S. J., Smiley, S. S., & Rees, R. (1974). The use of perceptual cues for number judgment by young children. Child Development, 45, 693-699.

Starkey, P. (1981). Young children's performance in number conservation tasks: Evidence for a hierarchy of strategies. Journal of Genetic Psychology, 138, 103-110.

Starkey, P. (1987, April). Early arithmetic competencies. Paper presented at the biennial meeting of the Society for Research in Child Development, Baltimore, MD.

Starkey, P., & Cooper, R. G. (1980). Perception of numbers by human infants. Science, 210, 1033-1035.

Starkey, P., & Gelman, R. (1982). The development of addition and subtraction abilities prior to formal schooling in arithmetic. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective. (pp. 99-116). Hillsdale, NJ: Lawrence Erlbaum.

Starkey, P., Spelke, E., & Gelman, R. (1983). Detection of intermodel numerical correspondences by human infants. Science, 222, 179.

Steffe, L. P., Cobb, P., & von Glasersfeld, E. (in press). Construction of arithmetical meanings and strategies. New York: Springer-Verlag.

Steffe, L. P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children's counting types: Philosophy, theory, and application. New York: Praeger Scientific.

Steinberg, R. M. (1984). A teaching experiment of the learning of addition and subtraction facts (Doctoral dissertation. University of Wisconsin-Madison, 1983). Dissertation Abstracts International, 44, 3313A.

Steinberg, R. M., (1985). Instruction on derived facts strategies in addition and subtraction. Journal for Research in Mathematics Education, 16, 337-355.

Stigler, J. W., Fuson, K. c., Ham, M., & Kim, M. S. (1986). An analysis of addition and subtraction word problems in American and Soviet elementary mathematics textbooks. Cognition and Instruction, 3, 153-17l.

Strauss, M. S., & Curtis, L. E. (1984). Development of numerical concepts in infancy. In C. Sophian (Ed.), Origins of cognitive skills (pp. 131-155). Hillsdale, NJ: Lawrence Erlbaum.

Tamburino, J. L. (1980). An analysis of the modelling processes used by kindergarten children in solving simple addition and subtraction story problems. Unpublished master's thesis, University of Pittsburgh.

Thomas, R. K., Fowlkes, D., & Vickery, J. D. (1980). Conceptual numerousness judgments by squirrel monkeys. American Journal of Psychology, 93, 247-257.

Thornton, C. A. (1978). Emphasizing thinking strategies in basic fact instruction. Journal for Research in Mathematics Education, 9, 214~227.

Underhill, R. (1984). External retention and transfer effects of special place value

References 431

curriculum activities. Focus on Learning Problems in Mathematics, 1 & 2, 108-130. Van Oeffelen, M. P., & Vos, P. G. (1982). Configuration effects on the enumeration

of dots: Counting by groups. Memory and Cognition, 10, 396-404. von Glasersfeld, E. (1982). Subitizing: The role of figural patterns in the development

of numerical concepts. Archives de Psychologie, 50, 191-218. Vygotsky, L. S. (1962). Thought and language. E. Hanfmann, & G. Vakar (Eds. and

Trans.) Cambridge, MA: MIT Press. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological pro­

cesses. M. Cole, V. John-Steiner, S. Scribner, & E. Souberman (Eds.). Cambridge, MA: Harvard University Press.

Vygotsky, L. S. (1986). Thought and language. A. Kozulin (Ed.). Cambridge, MA: MIT Press.

Wagner, S., & Walters, J. A. (1982). A longitudinal analysis of early number concepts: From numbers to number. In G. Forman (Ed.), Action and thought (pp. 137-161). New York: Academic Press.

Wallach, L. (1969). On the bases of conservation. In D. Elkind & J. H. Flavell (Eds.). Studies in cognitive development: Essays in honor of Jean Piaget (pp. 191-219). New York: Oxford University Press.

Walters, J. A., & Wagner, S. (1981, April). The earliest numbers. Paper presented at the biennial meeting of the Society for Research in Child Development, Boston.

Weiner, S. L. (1974). On the development of more and less. Journal of Experimental Child Psychology, 17, 271-287.

Whitehead, A. N., & Russell, B. (1910-13). Principia mathematica. Vols. 1-3. Cambridge, England: Cambridge University Press.

Whiteman, M., & Peisach, E. (1970). Perceptual and sensorimotor supports for con­servation tasks. Developmental Psychology, 2, 247-256.

Wilkinson, A. C. (1984). Children's partial knowledge of the cognitive skills of counting. Cognitive Psychology, 16, 28-64.

Willis, G. B., & Fuson, K. C. (1985). Teaching representational schemes for the more difficult addition and subtraction verbal problems. In S. K. Damarin & M. Shelton (Eds.), Proceedings of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 288-293). Columbus, Ohio.

Willis, G. B., & Fuson, K. C. (1986). Teaching schematic drawings for the solution of addition and subtraction word problems. Unpublished manuscript, Northwestern University.

Winer, G. A. (1974). Conservation of different quantities among preschool children. Child Development, 45, 839-842.

Winer, G. A. (1975). Analysis of the relation between conservation of large and small quantities. Psychological Reports, 36, 379-382.

Woods, S. S., Resnick, L. B., & Groen, G. J. (1975). An experimental test of five process models for subtraction. Journal of Educational Psychology, 67, 17-21.

Young, A. W., & McPherson, J. (1976). Ways of making number judgments and children's understanding of quantity relations. British Journal of Educational Psy­chology, 46, 328-332.

Zivin, G. (1979). The development of self-regulation through private speech. New York: John Wiley & Sons, Inc.

Author Index

A Acredolo, e., 343, 350 Andronov, v.P., 56, 60 Ashcraft, M.H., 285 Avesar, e., 338, 339, 349

B Baillargeon, R., 227, 251, 374 Baron, I., 337, 349 Baroody, A.J., 42, 43, 253, 266, 271, 272,

276,277, 278, 285, 295, 349, 360, 361, 387, 389, 390, 391, 414

Bartsch, K., 57, 291, 416 Bebout, H.e., 295 Beckwith, M., 68, 95, 123, 180 Behr, M., 266, 295 Beilin, H., 30, 342, 347 Bell, M.S., 37, 289, 291 Bergeron, A., 344, 347, 359, 360 Bergeron, I.e., 344, 347, 359, 360 Bessot, A., 349 Blevins, B., 345, 346 Blevins-Knabe, B., 346 Botvin, GJ., 351

Bradford, S.A., 28 Brainerd, C.I., 301, 337, 343, 344, 350,

354,361,414,415 Braisby, N., 226, 236, 243, 360, 386, 393,

401 Briars, D.I., 11, 12,34,35,37,38,40,

41,42,43,45, 49, 50, 51, 52, 53, 54, 57,59,75,78,86,90,109,211,224, 250, 253, 254, 256, 269, 270, 272, 273,275,280,351,360,361,380, 386, 387, 388, 391, 392, 393

Briga, I., 351 Bruner, I.S., 344 Brush, L.R., 345 Bryant, P.E., 337, 350 Burns, I., 37

C Campbell, I., 285, 294 Carpenter, T.P., 29, 250, 253, 254, 256,

261,264,272,273,274,278,280, 283, 284, 285, 293, 295, 342

Case, R., 56, 59, 209, 400, 415 Chi, M.T.H., 336

434 Author Index

Clements, D.H., 360 Cobb, P., 8,11,54,55,181,248,250,

253, 261, 270, 272, 275, 276, 278, 280,285,352,401,406

Comiti, C, 349 Cooper, R.G., 10,27,336,342,345,346,

347,351 Cowan, R., 339, 343, 347, 348, 349, 358 Crowther, R.D., 15, 17 Cuneo, D.o., 337, 338, 340, 349 Curtis, L.E., 10, 27, 336

D Daneman, M., 415 Darling, CL., 111, 120 Davis, H., 28 Davydov, v.v., 56, 60 De Corte, E., 249, 254, 281, 287 De Win, L., 254 Dilckerson, D.J., 338, 339, 349 Di.enes, Z.P., 289 Dineen, J.T., 285 Donaldson, M., 348 Durkin, K., 15, 17

E Eggleston, V.H., 68, 99, 206, 207, 208,

209,219,242 Elkind, D., 343, 344, 350 Erlwanger, S., 266, 295

F Fierman, B.A., 285 Fowlkes, D., 28 French, L., 241, 244 Frye, D., 226, 236, 243, 342, 348, 360,

386, 393, 401 Fuson, K.C, 11, 12,29,34,35,37,38,

40,41,42,43,45,49,50, 51, 52, 53, 54,57,59,65,68,74,75,86,89, 109,111,120,195,208,209,210, 211, 212, 216, 224, 225, 226, 228, 234, 238, 240, 243, 251, 253, 264, 265,270,271,272, 275, 276, 277, 278, 280, 284, 286, 287, 288, 289, 290,291,293,294,314,336,346, 347,349,351,352,355,360,361,

G

374, 380, 383, 386, 387, 388, 392, 405,415,416

Gallistel, CR., 27, 31, 35, 39, 68, 80, 81, 82, 90, 97, 116, 123, 132, 142, 157, 158,170,180, 195, 196, 197,206, 207,209,210,211,223,225,243, 244, 351, 360, 372, 373, 374, 375, 377, 380, 384, 385, 386, 387, 389, 397, 398, 399, 401, 402

Gearhart, M., 16, 37, 50, 116, 389 Gelman, R., 10,27,31,35,39,51,68,

80,81 , 82,90,97,116,123,132, 142, 157, 158, 170, 180, 195, 196, 197,206,207,208,209,210,211, 223,225,227,235,242,243, 244, 250,251,347, 348, 351, 355, 360, 372, 373, 374, 375, 377, 380, 384, 385,386,387,388,389,390,391, 392,393,397,398,399,401,402

Ginsburg, H.P., 17,42,68,86,224,245, 253,266,271,276,285,295, 360

Goldsmith, L.T., 10 Goldstein, A.G., 212, 217, 239, 303 Goodman, R.A., 285 Greco, P., 359 Greenfield, P.M., 344 Greeno, J.G., 51, 97, 227, 242, 250, 253,

254,255,256, 264, 273, 281, 293, 342, 372, 373, 374, 388, 390, 398, 399,402

Groen, G.J., 285 Guberman, S.R., 16,37,50,116,389 Gullen, G.E., 349 Guthrie, K.H., 342

H Hall, J.w., 29, 53, 208, 209, 210, 211,

224, 225, 226, 234, 238, 240, 243, 251,270,271,336,346,347,349, 352, 355, 374, 383, 415

Hall, V.C, 285 Ham, M., 253, 293, 294 Hamann, M.S., 285 Hatano, G., 57 Heldmeyer, K.H., 347

Author Index 435

Heller, 1. I. , 250, 253, 254, 255, 256, 264, 273,281,293,342

Heller, K.A., 347, 348, 349 Hennessey, A., 353 Herscovics, N., 344, 347, 359, 360 Hiebert, 1., 264 Hodges, R., 241, 244 Hooper, F.H., 343 Houlihan, D.M., 253, 285 Hudson, M., 342 Hudson., T., 339

J Jensen, E.M., 336

K Kamii, C.K., 295, 353 Kaplan, R., 86 Katz, H., 347 Kaufman, E.L., 10 Kaye, D.B., 285 Kim, M.S., 253, 293, 294 Kingma, 1.,350,414 Kintsch, w., 281 Klahr, D., 10,207,336,340,350,351 Kurland, M., 415 Kwon, Y., 68, 346, 349, 352

L LaPointe, K., 337, 338, 347 Larkin, 1.H., 250, 253, 254, 256, 269,

273, 280 Lawson, G., 337, 349 Leitner, E., 345, 346 Lesh, R.A., 289, 291 Levy, E.I., 68, 94 Lieberthal, E.M., 279 Lord, M.W., 10 Lowe, J., 226, 236, 243, 360, 386, 393,

401 Luria, A.R., 68, 74, 89, 195 Lyons, B.G., 216, 224, 225, 226, 228, 234,

238, 240, 346, 349, 352, 374, 383

M Mace, P.G., 345, 346 Mackay, C.K., 347, 349

Madson, C.R., 10 Mandler, G., 10, 336 Markman, E.M., 131, 146,216,227,346 Maroudas, c., 226, 236, 243, 360, 386,

393, 401 Matsuzawa, T., 28 McCloskey, M., 285 McGarrigle, l, 348 McLaughlin, lA., 337 McPherson, 1., 328, 332, 347, 351 Meck, E., 90, 235, 242, 245, 372, 373,

374,387,388,391,392,393,397, 399,402

Melnick, G.I., 349 Memmott, J., 28 Merkin, S., 235, 397, 402 Michie, S., 338, 339, 358, 414 Mierkiewicz, D., 41, 65 Millar, c., 347, 349 Miller, K., 29,39,41,44,355 Miller, P.H., 347, 348, 349 Miller, S.A., 343, 347 Miura, I., 291 Modgil, c., 301 Modgil, S., 301 Moore, C., 236, 342, 348 Moser, 1.M., 250, 253, 254, 256, 261,

264, 272, 273, 274, 278, 280, 283, 284, 285, 293, 295, 342

Murray, F.B., 347, 351

N Neches, R., 276 Nesher, P., 249 Nicholls, 1., 226, 236, 243, 360, 386, 393,

401 Nichols, E., 266, 295

o O'Donnell, J.P., 337, 338, 347 Olver, R.R., 344

p

Pariselle, c., 349 Parkman, lM., 285 Peisach, E., 347 Pepperberg, I., 28, 29

436 Author Index

Pergament, G.G., 209, 212, 216, 224, 225, 226, 228, 234, 238, 240, 346, 349, 352, 374, 383

Perusse, R., 28 Piaget, J., 300, 344, 347, 353, 361, 362,

363, 367, 414 Pollio, H.R., 39 Post, T.A., 285 Potter, M.e., 68, 94 Price, J., 43, 387, 390, 391 Pringle, R., 353 Pufall, P.B., 338

R Rathmell, E.C., 285, 295 Rees, R., 349 Reese, E.P., 336 Reese, TW., 10, 336 Resnick, L.B., 276, 285, 287, 288, 411 Restle, F., 68, 95,123, 180 Richards, J., 11, 12,34,35,37,38,40,

41,42,43,45,49,50,51, 52, 53, 54, 55,57,59,75,181,211,224,248, 253,261, 270, 272, 275, 276, 278, 351,352,361 , 380, 386,387, 388, 392, 401, 406

Riem, R., 15, 17 Riley, M.S., 51, 97, 227, 242, 250, 253,

254,255,256,264,273,281, 293, 342, 372, 373, 374, 388, 390, 398, 399,402

Robinson, M., 39, 41, 42, 317, 392 Roelinga, u., 414 Rogoff, B., 16 Rose, A.P., 353 Russac, R.1. , 360 Russell, B., 415 Russell, R.L., 17,68,86,224,245 Rutter, D.R., 15, 17

S Sadeghpour, M. , 387, 391 Sandieson, R., 415 Santilli, N.R., 342 Saxe, G.B., 16,37,50,56,60,68,77,86,

116, 119, 340, 347, 353, 387, 389, 391,414

Schaeffer, B., 68, 99, 206, 207, 208, 209, 219,242

Schonfeld, I.S., 347 Scott, J.L., 68, 99, 206, 207, 208, 209,

219, 242 Secada, w.G., 44, 52, 116,270,271,272,

276,278 , 287, 291, 303, 314, 336, 347, 415, 416

Shaw, R.E. , 338 Shebo, B.J. , 10, 336 Shire, B., 15, 17 Shoenfeld, E., 344 Shrager, J., 253 Sicilian, S., 387, 391 Siegel, A.w., 10 Siegel, L.S. , 212, 217, 239, 301, 303, 337,

338, 349, 350 Siegler, R.S., 39, 41, 42, 78, 90, 253, 285,

294,317 , 346,347,349,351,391, 392, 393

Silverman, 1.W., 351, 353 Smiley, S.S., 349 Smither, S.J., 349 Sokol, S.M., 285 Spelke, E., 27 Starkey, P., 10,27,250, 345,346,347 Stazyk, E.H., 285 Steffe, L.P., 8, 11,54,55, 181,248,253,

261, 263, 270, 275, 276, 278, 352, 401,406

Steinberg, R.M ., 278, 285 Stigler, J.w., 39, 41, 44, 57, 253, 254,

291, 293 , 294, 416 Strauss, M.S., 10, 27, 336

T Tamburino, J.L., 264 Thomas, R.K., 28 Thornton, C.A., 285

U Underhill, R., 291

V Van Oeffelen, M.P. , 336 Verschaffel, L., 249, 254, 281, 287

Author Index 437

Vickery, J.D., 28 Volkmann, J., 10 vonGlasersfeld, E., 8,10,11,54,55,

181, 248, 253, 261, 270, 275, 276, 278,352,401,406

Vos, P.G., 336 Vygotsky, L.S., 16,26,65,86,90, 193

W Wagner, S., 15, 50, 68, 139, 188, 347,

360,386 Wallace, J.G. ;10, 207, 336, 340, 350, 351 Wallach, L., 347, 353 Walters, J.A., 15,50,68, 139, 188,347,

360,386 Waxman, B., 285 Weiner, S.L., 342

Wertsch, J.Y., 16 Whitacre, J.D., 39 Whitehead, A.N., 415 Whiteman, M., 347 Wilkinson, A.C., 68,111,209,210,224,

240, 243, 385 Willis, G.B., 264, 278, 280, 284, 286, 287,

416 Winer, G.A., 349 Woods, S.S., 285

y Young, A.W., 347, 351

Z Zivin, G., 86, 195

Subject Index

A Addition

counting strategies for (see Counting, strategies; Sequence words, counting strategies)

facts, 23, 285-287, 296, 298, 416 multidigit algorithm, 248, 260, 278,

288-296, 298, 405, 416 school instruction and children's under­

standing of, 248, 271, 278-279, 293-295, 416

word problems computer models of, 254, 256, 269,

273, 280-282 representation of, 31, 247-250,

253-266, 269, 273-274, 280-284, 286-287,291-298,407, 410, 416

solution procedures for, 31, 247-250, 253-266,269-298,407,410

use of reversibility to solve, 259, 280-281,292,298

use of subset equivalence to solve, 259, 280-281, 292, 298

Apprehension of number (see Subitizing) Attention (see Counting, correspondence

errors in, effects of effort on)

B Base-ten system of numeration, 20, 22-23,

25, 34-35, 44, 55-56, 248, 260, 288-293, 355, 358, 389, 404-405, 414-416

C Cardinality principle (see Counting, princi­

ples, cardinality; Last-word respond­ing)

Cardinality rule (see Last-word responding) Cardinal number, 5, 248-249, 259-260,

275, 282-287, 298, 302, 335, 361-364, 367-368, 406-415, 417

Cardinal words and counting

cardinal-to-count transition, 22, 247, 251-253,257,261-264,268-271, 296-297,411-412

count-to-cardinal transition, 11, 19-21, 25, 205, 208-210, 228-239, 241, 243-247,250-253,257, 261-264, 266-271,296-297, 355-356,407, 411-412

440 Subject Index

Cardinal words (cont.)

and counting (cont.)

early relationships, 10-11, 205-246, 371-386, 396-402, 404-408, 411-412

embedded situations, 247-248, 269-282, 296-298, 407-408, 411-412

and representation of word problems, 253-264,269-287,292-298

single-set situations, 247, 250-266, 296,407-408,411-412

and solution of word problems, 253-264,269-287,292-298

strategies (see Counting, strategies; Sequence words, counting strategies)

subset situations, 247, 266-269, 296-297, 363

truly numerical counting (see Equiva­lence, truly numerical counting to establish)

use of transitions in addition and sub­traction, 247, 253-269, 407, 411-412

integration, 8, 11, 208, 246, 249, 256, 260, 262-263, 269-272, 275-277, 282,288,296,407-409,411, 413-414

operations (see Addition; Subtraction) and ordinal words, 363, 412-415 relations

equivalence (see Equivalence) order (see Order relations)

and sequence words, 362-364, 367-368 (see also Sequence words, cardi­nal/count transitions; Sequence words, and cardinality and counting correspondence; Sequence words, counting strategies)

Class versus collection terms (see Count­ing, correspondence errors in, effects of verbal label (class versus collec­tion) on)

Collection versus class terms (see Count­ing, correspondence errors in, effects of verbal label (class versus collec­tion) on)

Compensation (see Conservation of num­ber, Piagetian justifications for)

Conservation of number comparison strategies in, 332-335,

343-362, 365-368 correspondence in (see Conservation of

number, matching use in) counting use in (see Equivalence, use of

counting to establish, in dynamic sit­uation within a static compare situa­tion)

development of, 299-302, 334-335, 343-344, 347-368

identity conservation, 334-335, 343-344, 349-352, 354, 357, 359, 365-366

induction in, 335, 351-352, 354, 357, 366, 409

irrelevant strategies for, 334, 347 matching use in (see Equivalence, use of

matching to establish, in a dynamic situation within a static compare sit­uation)

perceptual strategies, 332, 334-335, 343-345, 347-350, 366

Piagetian justifications for, 335, 351-354, 366-367, 410

and transitive inference, 335, 343, 350, 352,361,366

transitivity in (see Conservation of num­ber, and transitive inference)

Correspondence in equivalence (see Equivalence, use of

matching to establish) errors (see Counting, correspondence

errors in) Countables, definition of (see Unit items,

perceptual) Counting

and addition (see Counting, strategies; Sequence words, counting strategies)

and cardinal words (see Cardinal words, and counting)

conceptual competence in, 76, 90, 200, 372, 374, 388-389, 393-395, 399-402

correspondence errors in dual errors, 63, 65, 67, 74, 76-85,

87-88,90,99, 102-104, 107-108, 117, 121, 126, 135-138, 141-144, 147, 150-157, 161-164, 167-170,

Subject Index 441

172-175, 180-181, 183-186, 191-197, 199

effects of age on, 65, 72-90, 102-105, 107, 109-113, 116, 118, 121, 124-125,129-131,133-158, 160-176,178-179,184-200

effects of array shape on (see Count­ing, correspondence errors in, effects of object arrangement on)

effects of distance between objects on (see Counting, correspondence errors in, effects of proximity of objects on)

effects of effort on, 63, 65, 69, 71, 82-83,85,89-90, 175, 189, 191-192, 195

effects of homogeneity of color on, 129-145,153-154,165-167, 170-173,178, 185, 187-188, 190-191

effects of homogeneity of objects on, 129-131, 145-158, 166-167, 171-173, 190, 192, 194-195

effects of location in row on, 63, 65, 68-69, 79-82, 89-90, 130-143, 147-154,156-158,160-171,173, 175,178-179, 185-189, 191, 194-197

effects of number of objects on, 63, 65, 69-70,83,85,90,129-174, 178-179,181,183,185-192, 194-195,200

effects of object arrangement on, 93-127,178-179, 188-189, 191-192, 197-199

effects of proximity of objects on, 129-131,158-174, 175, 178, 185, 187-191, 194

effects of sex of counter on, 63, 82, 175,193

effects of verbal label (class versus col­lection) on, 129, 131, 145-158, 175, 178, 195

local, 94, 98,122-126, 177, 180-181, 199

point-object errors, 63, 65-66, 72-80, 83-85,87-90,99, 102-104, 106-108,110,112-113, 115, 118-119,124-126,135-138,

140-144, 147, 149-156, 160-175, 180-200

recount errors, 63, 65, 67, 78-79, 81, 88, 99, 102-103, 108-116, 120, 124-126,136-137,144,162-164, 175, 181, 183, 192-193, 197-199

word-point errors, 63, 65-66, 72-80, 83-85,87-90,99,102-105, 107-108,110,119,121,124,126, 135-144, 147-156, 160, 162-175, 180, 182-188, 190-198,200

indicating acts used in moving objects, 94-99, 116-120,

122-126, 175-176, 178-181, 198-199,201,393

pointing, 64-67, 87, 93-99, 116-117, 120-127, 176-183, 187,200,393, 406 (see also Counting, correspon­dence errors in, dual errors; Count­ing, correspondence errors in, point-object errors; Counting, cor­respondence errors in, word-point errors)

internalization of, 63-65, 85-86, 90, 94, 116, 122, 176, 193

objects in circular arrays, 93-94, 110-116,

124-125, 175, 198 in disorganized arrays, 93-110,

122-124 in rows, 63-110,120-124, 126-127,

129-198,200-201,299,302-331, 354-360,361-368,371-386, 400-401

perceptual unit items (see Unit items, percept ual)

pointing (see Counting, correspondence errors in, dual errors; Counting, cor­respondence errors in, point-object errors; Counting, correspondence errors in, word-point errors; Count­ing, indicating acts used in, pointing)

principles abstraction, 395-396, 398, 401 (see

also Unit items, perceptual) cardinality, 19, 45-46, 205-246,

371-386, 388-389, 396-402 one-one, 371-386, 392-402 order irrelevance, 360, 397-398, 402

442 Subject Index

Counting (cont.)

principles (cont.)

principles about, 371, 397-400 stable-order, 35-36, 39-43, 371-392,

397-398, 400-401 procedural competence in, 76, 90, 200,

372, 374, 393-394, 399-402 Iremembering the objects already used in,

65, 93-100, 102-103, 106, 109-117,119-120,122-126, 177-180, 183, 192, 197-199

and sequence words (see Counting, strategies; Sequence words, and cardinality and counting correspon­dence; Sequence words, counting strategies; Unit items, sequence)

solution strategies (see Counting, strate­gies; Sequence words, counting strategies)

strategies add on up to s, 47, 257, 260, 264,

273, 296 count all, 20, 22, 47, 257, 260-261,

273, 296, 415 count down a with objects, 48, 260,

262-263, 270, 272, 274-275, 293, 297, 407

count down to a with objects, 48, 260, 270, 272-275, 297

count on a with objects, 47, 250, 260, 262-263, 270-272, 274-275, 284, 297,407,411-412,415-416

count up to s with objects, 48, 260, 270, 272-275, 297, 407

separate to a, 47, 257, 260, 264, 269, 296

take-away a, 47, 257, 260-261, 264, 269,272,277,280-281,284,293, 296

utilization competence in, 372, 374, 399, 402

Cross-cultural, 41, 44, 50, 52-53, 56-57, 59-60,253,291,293-294, 349, 359,414,416

D Developmental sequences

addition and subtraction solution proce­dures, 20, 22, 47-49, 250, 257,

E

260-264,269-281,283-285, 291-294,296-297,404,407, 411-412, 415-416

conservation solution strategies (see Con­servation of number, development of)

correspondence errors (see Counting, correspondence errors in, effects of age on)

identity conservation and equivalence conservation (see Conservation of number, identity conservation)

relationships between counting and cardinality

cardinality rule (see Cardinal words, and counting, count-to-cardinal tran­sition; Cardinal words, and count­ing, early relationships; Counting, principles, cardinality)

later relationships, 19-22,31,47-49, 55, 247-253, 257, 260-264, 266-285, 288-294, 296-298, 302, 361-364, 367-368, 404, 407, 409-417

representations of addition and subtrac­tion word problems, 31, 247-250, 253-266, 269, 273-274, 280-284, 286-287, 291-298, 407, 410, 416

subitizing and counting, 205, 207, 209-210,217-223,242

Effort and counting (see Counting, cor­respondence errors in, effects of effort on)

Enumeration (see Counting) Equivalence

and cardinal number, 6,8,29,31, 299-368, 403-404, 408-414

comparison strategies to establish, 299, 332-341, 343-362, 365-368

conservation (see Conservation of num­ber)

correspondence and (see Equivalence, use of matching to establish)

counting for (see Equivalence, use of counting to establish)

effects of change transformations on, 299, 341-342, 345-347, 359, 366

Subject Index 443

effects of displacement transformations on, 299-302, 342-345, 347-368

matching for (see Equivalence, use of matching to establish)

and Piaget, 31, 300-302, 333, 344, 347, 353-354, 361-364, 367-368, 409-414,417 (see also Conservation of number)

situations dynamic, 299-300, 332, 341-344,

365-366 dynamic within a static compare,

299-302, 332, 334-335, 344-368, 409-414,417

static compare, 299-341,355-361, 365-366, 408-410

strategies used to establish (see Equiva­lence, comparison strategies to establish; Equivalence, truly numeri­cal counting to establish; Equiva­lence, use of counting to establish; Equivalence, use of matching to establish)

truly numerical counting to establish, 31, 302, 335, 361-364, 367-368, 409-414, 417

use of counting to establish in a dynamic situation, 342-344 in a dynamic situation within a static

compare situation, 31, 300-302, 332-333, 343-345, 347-360, 362-368,408-414,417

in a static compare situation, 301-341, 408-410

use of matching to establish in a dynamic situation, 343-344 in a dynamic situation within a static

compare situation, 31, 300-301, 332-335, 343-345, 347-354, 360-361, 366-367, 410

in a static compare situation, 301-341, 410,414

Estimating, 10, 334-335, 340

F Fewer than (see Order relations)

G Greater than (see Order relations)

H Homogeneity versus heterogeneity of

counted objects (see Counting, cor­respondence errors in, effects of homogeneity of color on; Counting, correspondence errors in, effects of homogeneity of objects on)

How-to-count principles (see Counting, principles, cardinality; Counting, principles, one-one; Counting, prin­ciples, stable order)

I Identity conservation (see Conservation

of number, identity conserva­tion)

Idiosyncratic count lists (see Sequence words, levels in; Sequence words, stable conventional portions in)

Induction in conservation of number (see Conservation of number, induction in)

Instruction addition and subtraction sums to 18,

248,271,278-279,291,293-295, 415-416

multidigit addition and subtraction, 248, 278, 288-291, 405, 416

Irrelevant strategies (see Conservation of number, irrelevant strategies for)

L Last-word responding

and accurate counting, 205, 207-209, 211,221-222,226-227,242-243

and cardinal reference, 205, 208-211, 213,217,228-236,243-246,407

effects of performance demands on, 205, 208,210-211,214-215,227-228, 242-243

effects of set size on, 205, 207-213, 217-219, 223-226, 242-243, 378-381

learning, 206-210, 236-239, 241-246 memory and, 206, 209-212, 214, 224,

239-244 rule-learning and, 205, 208-211,

228-241,243-246

444 Subject Index

Last-word responding (cont.) and subitizing, 205, 207, 209-211, 213,

217-221,223,242 theoretical positions on, 207-210,

217-246 Less than (see Order relations) Linear arrays (see Conservation of number;

Counting, objects, in rows)

M Measure words, 3, 5-16, 19-24, 28-30,

46-47,51,388,391,397,403-404, 406,411

Memory

N

and the cardinality rule (see Last-word responding, memory and)

and counting (see Counting, remember­ing the objects already used in)

and counting for equivalence, 313-319

Nonnumerical number-word uses, 3,5, 7, 13-15, 403-404

Number words early experiences with, 3, 15-27,

403-405, 416 learning (see Sequence words, learning) meanings of

cardinal words (see Cardinal words) counting words (see Counting) measure words (see Measure words) non numerical words (see Nonnumeri-

cal word uses) ordinal words (see Ordinal words) sequence words (see Sequence words)

situations (see Number words, meanings of)

symbols for, 3-5, 7, 12-15,20,22, 24-27, 247, 265-266, 278, 284-291, 293-295, 389, 403-406, 416

uses of (see Number words, meanings of) Number-word situations (see Number

words, meanings of) Numeration system (see Base-ten system of

numeration)

o One-one counting errors (see Counting,

correspondence errors in) One-one counting principle (see Counting,

principles, one-one) One-to-one correspondence errors (see

Counting, correspondence errors in) Operations (see Addition; Subtraction) Order irrelevance counting principle (see

Counting, principles, order irrelevance)

Ordering, 3-8, 11-12, 14-15,95-99, 106, 110,123-124,126,177,180,189, 193,397,402,414

Order relations After/Before, 6,11-12,47,51-54,59,

355,358 fewer than (see Order relations, less

than) greater than (more than), 6, 8, 24,

53-54, 59, 299-368, 404, 408, 410 Just After/Just Before, 6, 11-12, 20,

47-48, 51-54, 59, 404, 413 less than, 6, 8, 53-54,.59, 299-368,

404, 408, 410 Ordinal words, 3-16, 19,22,24,46-47,

51,363,388,391,397,402-405, 412-415

p

Partitioning (see Counting, correspondence errors in)

Peano axioms, 415 Piagetian theory

conservation of number (see Conserva­tion of number)

truly numerical counting (see Equiva­lence, truly numerical counting to establish)

Pointing (see Counting, correspondence errors in; Counting, indicating acts used in, pointing)

Principles of counting (see Counting, prin­ciples)

Problem representation (see Addition, word problems, representation of;

Subject Index 445

Subtraction, word problems, representation of)

Procedural competence (see Counting, procedural competence in)

Processing space, 56, 59, 99, 105, 209-210, 240, 243, 400, 414

Q Quantification operators (see Counting;

Estimating; Subitizing) Quantitative invariance (see Conservation

of number; Equivalence)

R Relative magnitude determination (see

Order relations) Reversibility (see Conservation of number,

Piagetian justifications for)

S Schema

part-whole, 281-283, 287, 295 reversible unary, 259, 286-287, 296, 298 static, 259, 286-287, 296, 298 ADDEND-ADDEND-SUM, 259, 282,

286-287, 295-296, 298 Sequence words

backward sequences, 20-21, 45, 48-49, 52,55-57, 59, 260, 263, 275, 277-279, 284, 293, 297, 404, 407

cardinal/count transitions, 263, 276-278, 282,297

and cardinality and counting correspon­dence, 371-386, 400-402

and cardinal words, 362-364, 367-368 (see also Sequence words, cardi­nal/count transitions; Sequence words, and cardinality and counting correspondence; Sequence words, counting strategies)

counting strategies count all, 48 , 55, 276 count down a, 49, 55, 260, 263, 275,

277-279, 284, 293, 297, 404, 407

count down to a, 49, 55, 260, 275, 277-278, 297

count on a, 48, 55, 260, 263, 273, 275-279, 284-285, 291, 297, 404, 407,416

count up to s, 49, 55, 260, 274-275, 277-280,283-285,291-292, 294, 297,404,407,416

keeping-track processes used in, 47-49,260-261,263,275-279, 281-282,287,297, 407, 409, 412

elaboration in (see Sequence words, levels in)

errors in, 18-21,23,33-44,57-58, 371-382, 384-392, 397-398, 400-401

learning, 33-44, 57-58, 384-392, 397-398,400-401,404-405,416

levels in advanced, 33, 55-56, 292 bidirectional, 33,45,49, .55, 59, 407,

409-410,412-413 breakable chain, 33, 45, 47-48,

51-54,56,59,407,411-412 numerable chain, 33, 45, 48-49, 52,

54-57,59,275,407,409,411-412 (see also Sequence words, counting strategies)

string, 33, 45-46, 50, 58, 406, 408, 411-412

unbreakable list, 33, 45-47, 50-52, 58-59,406,408,411-412

nonstable sequence portions in, 36-37, 43-44, 58

sequence unit items in (see Unit items, sequence)

stable conventional portions in, 35-37, 39-43, 58

structure of English, 33-35, 41, 44, 52-53,56, 58-60,291,389,392, 401

Spew (see Sequence words, nonstable sequence portions in)

Stable-order principle (see Counting, prin­ciples, stable-order)

Strategies comparison (see Equivalence, compari­

son strategies to establish)

446 Subject Index

Strategies (cont.) counting (see Counting, strategies;

Sequence words, counting strategies) Subitizing, 10,28,205,207-211,

213-214,217-223,226,234, 240-245, 250, 332, 336-337, 343, 347, 349, 350-352

Subset situations (see Cardinal words, and counting, subset situations)

Subtraction counting strategies for (see Counting,

strategies; Sequence words, counting strategies)

facts, 285-287, 296, 298, 416 multidigit algorithm, 248, 260, 288-296,

298, 416 school instruction and children's under­

standing of, 248, 293-295, 416 word problems

computer models of, 254, 256, 269, 273, 280-282

representation of, 31, 247-250, 253-266, 269, 273-274, 280-284, 286-287, 291-298, 407, 410, 416

solution procedures for, 31, 247-250, 253-266, 269-298, 407, 410

use of reversibility to solve, 259, 280-281, 292, 298

use of subset equivalence to solve, 259, 280-281,292,298

Subvocal counting (see Counting, internali­zation of)

Symbols

T

meaning of =, 265-266, 295 meaning of -, 265, 293-294 use in situations (see Number words,

symbols for)

Tagging (see Counting) Transformations (see Conservation of num­

ber) Transitivity (see Conservation of number,

and transitive inference)

U Unit items

ideal, 260, 283-285, 288, 298, 335, 361-364, 367-368, 407, 409-414, 417

perceptual, 6-8,10,11,13,29,69, 86-88, 181, 192, 208, 247-249, 253-276, 283, 287-288, 291-292, 296-297, 335, 352-354, 357, 363, 367-368, 395-396, 401, 406-408, 410-413,416 (see also Counting, strategies)

sequence, 45, 48-49, 54-57, 59, 249, 259-260, 263, 274-283, 287-288, 292,296-298,407,409,411-412, 415-416 (see also Sequence, count­ing strategies)

simultaneous representation, 248, 258-260, 262-263, 266-282, 287, 296-298, 335, 352-354, 357, 362-363, 367, 407-412

single representation, 247-249, 253-266, 296-297, 352-354, 408, 412

Utilization competence (see Counting, utilization competence in)

V Videotaped counting study, 65-67, 69,

71-72,87,98-110, 124, 180

W Word problems (see Addition, word

problems; Subtraction, word problems)

Working memory (see Processing space)

Z Zero, 19-21,34,291