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Psychology of Mathematical Thinking

  • Roland W. Scholz
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Part of the Mathematics Education Library book series (MELI, volume 13)

Keywords

Mathematics Education Student Teacher Mathematical Concept Concept Formation Infinite Series 
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References

  1. Freeman, F. N. (1910). Untersuchungen über den Aufmerksamkeitsumfang und die Zahlauffassung bei Kindern und Erwachsenen. Leipzig: Veröffentlichungen des Instituts für experimentelle Pädagogok und Psychologie des Leipziger Lehrervereins.Google Scholar
  2. Goldberg, J. G. (1978). Psychological research into mathematics learning and teaching in the U.S.S.R. and Eastern Europe. In F. Swetz (Ed.), Socialist mathematics education. Southhampton, PA: Burgundy Press.Google Scholar
  3. Goldin, G. A. (1992). On developing a unified model for the psychology of mathematical learning and problem solving. In W. Geeflin & K. Graham (Eds.), Proceedings of the 16th PME Conference (Vol. 3, pp. 235–261). Durham, NH: University of New HampshireGoogle Scholar
  4. Glaserfeld, E. von (1991). Introduction. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. xiii–xx). Dordrecht: Kluwer.Google Scholar
  5. Katz, D. (1913). Psychologie und mathematischer Unterricht. Leipzig: Teubner.Google Scholar
  6. Piaget, J. (1968). The child’s conception of space. London: Routledge & Kegan Paul. [Original work published in 1948]Google Scholar
  7. Poincaré, H. (1910). Der Wert der Wissenschaft. Leipzig: Teubner.Google Scholar
  8. Poincaré, H. (1914). Wissenschaft und Methode. Leipzig: Teubner.Google Scholar
  9. Polya, G. (1954). How to solve it. Princeton, NJ: Princeton University Press.Google Scholar
  10. Rubinstein, S. L. (1958). Grundlagen der allgemeinen Psychologie. Berlin: Volk und Wissen.Google Scholar
  11. Siegler, R. S. (1986). Children’s thinking. Englewood Cliffs. NJ: Prentice-Hall.Google Scholar
  12. Vergnaud, G. (1990). Epistemology and psychology of mathematics education. In P. Necher & J. Kilpatrick (Eds.), Mathematics and cognition. Cambridge: Cambridge University Press.Google Scholar
  13. Vygotsky, L. S. (1978). Mind and society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.Google Scholar

References

  1. Boyer, C. B., & Merzbach, U. C. (1989). A history of mathematics. New York: Wiley.Google Scholar
  2. Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153–165), Dordrecht, Netherlands: Kluwer.Google Scholar
  3. Courant, R., & Robbins, H. (1978). What is mathematics? An elementary approach to ideas and methods. Oxford: Oxford University Press.Google Scholar
  4. Fischbein, E. (1963). Conceptele Figurale [in Roumanian]. Bucuresti: Editura Academiei, R.S.R.Google Scholar
  5. Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal of Research in Mathematics Education, 16(1), 3–17.Google Scholar
  6. Fischbein, E., Tirosh, D., & Melamed, U. (1981). Is it possible to measure the intuitive acceptance of a mathematical statement? Educational Studies in Mathematics, 12, 491–512.CrossRefGoogle Scholar
  7. Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317–326.CrossRefGoogle Scholar
  8. Linchevski, L., & Vinner, Sh. (1988). The naive concept of sets in elementary teachers. Proceedings of the Twelth International Conference, Psychology of Mathematics Education(Vol. 2.) Vezprem, Hungary.Google Scholar
  9. Mariotti, M. A. (1992). Imagini e concetti in geometria. L’Insegnamento della Matematica e delle Scienze Integrate, 15(9), 863–885.Google Scholar
  10. Maurer, S. B. (1987). New knowledge about errors and new views about learners: What they mean to educators and what more educators would like to know. In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 165–188). Hillsdale, NJ: Erlbaum.Google Scholar
  11. Resnik, L. B. (1983). Procédure et compréhension en arithmétique élémentaire. Seminaire de Didactique de Mathématiqe 1982–1983. Grenoble: IMAG.Google Scholar
  12. Tirosh, D., Graeber, A. D., & Glover, R. M. (1986). Preservice teachers’ choice of operation for multiplication and division word problems. Proceedings of the Tenth International Conference, Psychology of Mathematics Education (pp. 57–62). London: University of London Institute of EducationGoogle Scholar
  13. Vinner, Sh, (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall(Ed.), Advanced mathematical thinking (pp. 65–79). Dordrecht, Netherlands: Kluwer.Google Scholar

References

  1. Aebli, H. (1963). Über die geistige Entwicklung des Kindes. Stuttgart Klett.Google Scholar
  2. Aebli, H. (1978). Von Piagets Entw icklungspsychologie zur Theorie der kognitiven Sozialisation. In G. Steiner (Ed.), Piaget und die Folgen. Die Psychologie des 20. Jahrhunderts (Vol. 7, pp. 604–627). München: Kindler.Google Scholar
  3. Aebli, H. (1987). Development as construction: Nature and psychological and social context of genetic constructions. In B. Inhelder, D. de Caprona, & A. Cornu-Wells (Eds.), Piaget today (pp. 217–232). Hillsdale, NJ: Erlbaum.Google Scholar
  4. Bandura, A. (1982). Self-efficacy mechanisms in human agency. American Psychologist, 37(2), 122–147.Google Scholar
  5. Bruner, J. S. (1957). Beyond the information given. In J. M. Anglin (Ed.), Jerome S. Bruner: Beyond the information given. Studies in the psychology of knowing (pp. 218–238). New York: Norten. [Reprint]Google Scholar
  6. Bruner, J. S. (1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press.Google Scholar
  7. Bruner, J. S. (1973). Beyond the information given. New York: Norton.Google Scholar
  8. Case, R. (1978). Piaget and beyond: Toward a developmentally based theory and technology of instruction. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 1, pp. 167–228). Hillsdale, NJ: ErlbaumGoogle Scholar
  9. Case, R. (1985). Intellectual development: Birth to adulthood. Orlando, FL: Academic Press.Google Scholar
  10. Chandler, P., & Sweller, J. (1991). Cognitive load theory and the format of instruction. Cognition and Instruction, 8(4), 293–332.Google Scholar
  11. Glaser, R. (1984). Education and thinking. American Psychologist, 39(2), 93–104.Google Scholar
  12. Heckhausen, H. (1969). Förderung der Lernmotivierung und der intellektuellen Tüchtigkeiten. In H. Roth (Ed.), Begabung und Lernen (pp. 193–228). Stuttgart: Klett.Google Scholar
  13. Inhelder, B., & Piaget, J. (1955). De la logique de l’enfant à la logique de l’adolescent. Essai sur la construction des structures opératoires formelles. Paris: Presses Universitaires de France.Google Scholar
  14. Lindsay, P. H., & Norman, D. A. (1972). Human information processing. New York: Academic Press.Google Scholar
  15. Pascual-Leone, J. A. (1970). A mathematical model for the transition rule in Piaget’s developmental stages. Acta Psychologica, 63, 301–345.Google Scholar
  16. Pascual-Leone, J. A. (1976). A view of cognition from a formalist’s perspective. In K. Riegel & J. Meacham (Eds.), The developing individual in a changing world. Hague: Mouton.Google Scholar
  17. Piaget, J. (1947). Traité de logique. Essai sur la logique opératoire. Paris: Colin.Google Scholar
  18. Rumelhart, D. E. (1978). Schemata: The building blocks of cognition. In R. Spiro, B. Bruce, & W. Brewer (Eds.), Theoretical issues in reading comprehension. Hillsdale, NJ: Erlbaum.Google Scholar
  19. Rumelhart, D. E., & Norman, D. A. (1973). Active semantic networks as a model of human memory. Proceedings of the Third International Joint Conference on Artificial Intelligence. Stanford, California.Google Scholar
  20. Rumelhart, D. E., & Norman, D. A. (1976). Accretion, tuning, and restructuring: Three modes of learning. San Diego, CA: University of California, Technical Report # 63.Google Scholar
  21. Scardamalia, M., & Bereiter, C. (1985). Fostering the development of self-regulating in children’s knowledge processing. In S. F. Chipman, J. W. Segal, & R. Glaser (Eds.), Thinking and learning skills. Vol. 2: Research and open questions (pp. 563–577). Hillsdale, NJ: Erlbaum.Google Scholar
  22. Steiner, G. (1974a). On the psychological reality of cognitive structures: A tentative synthesis of Piaget’s and Bruner’s theories. Child Development, 45, 891–899.Google Scholar
  23. Steiner, G. (1974b). Kind und Mathematik. Zeitschrift für Pädagogik, 20(5), 677–702.Google Scholar
  24. Steiner, G. (1983). Number learning and constructing coherent networks by using Piaget-derived operative principles. In M. Zweng, T. Green, J. Kilpatrick, H. Pollak, & M. Sydam (Eds.), Proceedings of the Fourth International Congress on Mathematical Education. Boston: Birkhauser.Google Scholar
  25. Steiner, G. (1988). Lernen. Zwanzig Szenarien aus dem Alltag. Bern: Huber.Google Scholar
  26. Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12, 257–285.CrossRefGoogle Scholar
  27. Swelter, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition & Instruction, 2, 58–89.Google Scholar
  28. Thaeler, J. S. (1988). Input-output modification to basic graphs: A method of graphing functions. In A. F. Coxford & A. P. Shulte (Eds.), The ideas of algebra, K-12. Yearbook of the National Council of Teachers of Mathematics (pp. 229–241). Reston, VA: The Council.Google Scholar
  29. Zhu, X., & Simon, H. A. (1987). Learning mathematics from examples and by doing. Cognition & Instruction, 4, 137–166.Google Scholar

References

  1. Abulchanowa-Slawskaja, K. A., & Bruschlinski, A. W. (1989). Filosofsko-psichologiceskaja koncepcija S. L. Rubinsteina [S. L. Rubinstein’s philosophical-psychological conception]. Moskva: Mysl.Google Scholar
  2. Bol, E., Haenen, J. P. P., & Wolters, M. (Eds.). (1985). Education for cognitive development. Den Haag: SVO.Google Scholar
  3. Davydov, V. V. (Ed.). (1969). Psichologiceskie vozmoznosti mladsich skolnikov v usvoenii matematiki [The mental potentials of younger school students in the acquisition of mathematics]. Moskva: Prosvescenie.Google Scholar
  4. Davydov, V. V. (1977). Arten der Verallgemeinerung im Unterricht. Berlin: Volk und Wissen.Google Scholar
  5. Davydov, V. V. (1982). The psychological characteristics of the formation of elementary mathematical operations in children. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 234–238). Hillsdale, NJ: Erlbaum.Google Scholar
  6. Davydov, V. V. (1986). Problemy razvivajuscego obucenija [Problems of developmental teaching]. Moskva: Pedagogika.Google Scholar
  7. Davydov, V. V. (1988a). Problems of developmental teaching. The experience of theoretical and empirical psychological research. Excerpts. Soviet Education, 30(8), 15–97; 30(9), 3–83; 30(10), 3–77.Google Scholar
  8. Davydov, V. V. (1988b). Learning activity: The main problems needing further research. Multidisciplinary newsletter for activity theory, 1, 29–36.Google Scholar
  9. Davidov, V. V., Lompscher, J., & Markowa, A. K. (Eds.). (1982). Ausbildung der Lerntätigkeit bei Schülern. Berlin: Volk und Wissen.Google Scholar
  10. Elkonin, D. B. (1960). Detskaja psichologija [Child psychology]. Moskva: Ucpedgis.Google Scholar
  11. Elkonin, D. B. (1980). Psychologie des Spiels. Berlin: Volk und Wissen.Google Scholar
  12. Elkonin, D. B. (1989). Izzbrannye psichologiceskie trudy [Selected psychological studies]. Moskva: Pedagogika.Google Scholar
  13. Elkonin, D. B., & Davidov, V. V. (Eds.). (1962). Voprosy psichologii ucebnoj dejatelnosti mladsich skolnikov [Issues in the psychology of the learning activity of younger school students]. Moskva: Izd. APN.Google Scholar
  14. Elkonin, D. B., & Davidov, V. V. (Eds.). (1966). Vozrastnye vozmoznosti usvoenija znanij [Age-stage-related potentials in the acquisition of knowledge]. Moskva: Prosvescenie.Google Scholar
  15. Engested, N., Hedegaard, M., Karpatschof, B., & Mortensen, H. (Eds.). (1993). The societal subject. Aarhus: Aarhus University Press.Google Scholar
  16. Engeström, Y.(1987). Learning by expanding. Helsinki: Orienta-Konsultit Oy.Google Scholar
  17. Engeström, Y. (1990). Learning, working and imaging. Helsinki: Orienta-Konsultit Oy.Google Scholar
  18. Galperin, P. (1967). On the notion of internalization. Soviet Psychology, 5(3), 28–33.Google Scholar
  19. Galperin, P. (1969). Stages in the development of mental acts. In M. Cole & I. Maltzman(Eds.), A handbook of contemporary Soviet psychology (pp. 249–273). New York: Basic Books.Google Scholar
  20. Galperin, P. (1980). Zu Grundfragen der Psychologie. Berlin: Volk und Wissen.Google Scholar
  21. Galperin, P. (1982). Intellectual capabilities among older preschool children: On the problem of training and mental development. In W. W. Hartup (Ed.), Review of child development research (Vol. 6, pp. 526–546). Chicago, IL: The University of Chicago Press.Google Scholar
  22. Galperin, P., & Talysina, N. F. (Eds.). (1968). Formirovanie znanij i umenij na osnove teorii poetapnogo usvoenija umstvennych dejstvij [Formation of knowledge and skills on the basis of the step-wise acquisition of mental actions]. Moskva: Izd. Moskovskogo universiteta.Google Scholar
  23. Galperin, P., & Talysina, N. F. (Eds.). (1972). Upravlenie poznavatelnoj dejatelnostju ucascichsja. [Guiding the cognitive activity of students] Moskva: Izd. Moskovskogo universiteta.Google Scholar
  24. Hedegaard, M., Hakkarainen, P., & Engeström, Y. (Eds.). (1984). Learning and teaching on a scientific basis. Aarhus, Denmark: Aarhus universitet.Google Scholar
  25. Hildebrand-Nilshon, M., & Rückriem, G. (Eds.). (1988). Proceedings of the 1st international congress on activity theory. Berlin: System Druck.Google Scholar
  26. Jantzen, W. (1986). Abbild und Tätigkeit. Studien zur Entwicklung des Psychischen. Solms-Oberbiel: Jarick Oberbiel.Google Scholar
  27. Jantzen, W. (1991). Psychologischer Materialismus, Tätigkeitstheorie, Marxistische Anthropologie. Hamburg: Argument.Google Scholar
  28. Lektorski, W. P. (Ed.). (1990). Dejatelnost’: Teorii, metodologija, problemy [Tätigkeit: Theorien, Methodologie, Probleme]. Moskva: Politizdat.Google Scholar
  29. Leont’ev, A. N. (1964). Probleme der Entwicklung des Psychischen. Berlin: Volk und Wissen.Google Scholar
  30. Leont’ev, A. N. (1966/1978). Predislovie [Preface] In P. Fraisse & J. Piaget (Eds.), Traité de psychologie experimentale (Vols. 1–6). Moskva: Progress.Google Scholar
  31. Leont’ev, A. (1979). Tätigkeit, Bewußtsein, Persönlichkeit. Berlin: Volk und Wissen.Google Scholar
  32. Leont’ev, A. N., & Tichomirow, O. K. (1963). Posleslovie [Postscript to Russian translation of] Inhelder, B., & Piaget, J.: La genese des structures logiques elementaires. Moskva: Izd. inostrannoj literatury.Google Scholar
  33. Lompscher, J. (Ed.). (1967). Probleme der Lerntheorie. Berlin: Volk und Wissen.Google Scholar
  34. Lompscher, J. (Ed.). (1973). Sowjetische Beiträge zur Lerntheorie. Die Schule P. Galperins. Köln: Pahl-Rugenstein.Google Scholar
  35. Lompscher, J. (Ed.). (1975). Theoretische und experimentelle Untersuchungen zur Entwicklung geistiger Fähigkeiten. Berlin: Volk und Wissen.Google Scholar
  36. Lompscher, J. (Ed.). (1988). Persönlichkeitsentwicklung in der Lerntätigkeit. Berlin: Volk und Wissen.Google Scholar
  37. Lompscher, J. (Ed.). (1989a). Psychologische Analysen der Lerntätigkeit. Berlin: Volk und Wissen.Google Scholar
  38. Lompscher, J. (1989b). Formation of learning activity in pupils. In H. Mandl, E. De Corte, N. Bennett, & H. F. Friedrich (Eds.), Learning and instruction. European research in an international context (Vol. 2.2, pp. 47–66). Oxford: Pergamon Press.Google Scholar
  39. Lompscher, J. (1990). Aufsteigen vom Abstrakten zum Konkreten im Unterricht — Versuche zu einer alternativen Lehrstrategie. Berlin: Akademie der Pädagogischen Wissenschaften.Google Scholar
  40. Lompscher, J. (in press). Learning activity. In The international encyclopedia of education (2nd ed.). Oxford: Pergamon Press.Google Scholar
  41. Luria, A. (1982). Sprache und Bewußtsein. Berlin: Volk und Wissen.Google Scholar
  42. Luria, A. R. (1987). Die historische Bedingtheit individueller Erkenntnisprozesse. Berlin: Deutscher Verlag der Wissenschaften.Google Scholar
  43. Moll, L. C. (Ed.). (1990). Vygotsky and education. Instructional implications and applications of sociohistorical psychology. Cambridge: Cambridge University Press.Google Scholar
  44. Nikola, G., & Talysina, N. F. (1972). Formirovanie obscich priemov resenija arifmeticeskich zadac [Formation of general strategies for arithmethical problem solving]. In P. Galperin & N. F. Talysina (Eds.), Upravlenie poznavatelnoj dejatelnostiju ucascichsja [Guiding the cognitive activity of students] (pp. 209–261). Moskva: Izd. Moskovskogo universiteta.Google Scholar
  45. Obuchowa, L. F. (1972). Etapy razvitija detskogo myslenija [Developmental stages of childhood thought]. Moskva: Izd. Moskovskogo universiteta.Google Scholar
  46. Obuchowa, L. F. (1981). Koncepcija Zana Piaget: za i protiv [For and against the conception of Jean Piaget]. Moskva: Izd. Moskovskogo universiteta.Google Scholar
  47. Podolski, A. I. (1987). Stanovlenie poznavatelnogo dejstvija: naucnaja abstrakcija i realnost’ [The etiology of cognitive activity: Scientific abstraction and reality]. Moskva: Izd. Moskovskogo univeriteta.Google Scholar
  48. Rubinstein, S. L. (1958). Grundlagen der allgemeinen Psychologie. Berlin: Volk und Wissen.Google Scholar
  49. Rubinstein, S. L. (1962). Sein und Bewußtsein. Berlin: Akademie-Verlag.Google Scholar
  50. Rubinstein, S. L. (1963). Prinzipien und Wege der Entwicklung der Psychologie. Berlin: Akademie-Verlag.Google Scholar
  51. Salmina, N. G. (1981). Vidy i funkcii materializacii v obucenii [Types and functions of materialization in teaching]. Moskva: Izd. Moskovskogo universiteta.Google Scholar
  52. Salmina, N. G. (1988). Znak i simvol v obucenii [Signs and symbols in teaching]. Moskva: Izd. Moskovskogo universiteta.Google Scholar
  53. Saporoshez, A. W. (1986). Izbrannye psichologiceskie trudy [Selected psychological studies]. Moskva: PedagogikaGoogle Scholar
  54. Saporoshez, A. W., Sintschenko, W. P., Owtschinnikowa, O. W., & Tichomirow, O. K. (Eds.). (1983). A. N. Leont’ev i sovremennaja psichologija [A. N. Leont’ev and modern psychology]. Moskva: Izd. Moskovskogo universiteta.Google Scholar
  55. Säljö, R. (Ed.). (1991). Culture and learning. Learning and Instruction, 1(3).Google Scholar
  56. Seeger, F. (1989). Davydov’s theory of generalization: Theoretical thinking and representation in learning and teaching algebra. IDM Occasional Paper 117. Bielefeld: Universität Bielefeld.Google Scholar
  57. Talysina, N. F. (1969). Teoreticeskie problemy programmirovannogo obucenija [Theoretical problems of programmed teaching], Moskva: Izd. Moskovskogo universiteta.Google Scholar
  58. Talysina, N. F. (1975). Upravlenie processom usvoenija znanij [Guiding the process of knowledge acquisition]. Moskva: Izd. Moskovskogo universiteta.Google Scholar
  59. Talysina, N. F. (1988). Formirovanie poznavatelnoj dejatelnosti mladsich skolnikov [Training the cognitive ability of younger students], Moskva: Prosvescenie.Google Scholar
  60. Van Oers, B. (1990). The development of mathematical thinking in school: A comparison of the action-psychological and information-processing approaches. International Journal of Educational Research, 14(1), 51–66.CrossRefGoogle Scholar
  61. Vygotsky, L. S. (1964). Denken und Sprechen. Berlin: Akademie-Verlag.Google Scholar
  62. Vygotsky, L. (1985/1987). Ausgewählte Schriften (Vols. 1–2). Berlin: Volk und Wissen.Google Scholar
  63. Wertsch, J. V. (Ed.). (1985). Vygotsky and the social formation of mind. Cambridge, MA: Harvard University Press.Google Scholar
  64. Wertsch, J. V. (Ed.). (1985). Culture, communication and cognition: Vygotskyan perspectives. Cambridge: Cambridge University Press.Google Scholar
  65. Zaporozec, A. V. (1990). Entstehung und Aufbau der Motorik. Eine tätigkeitspsychologische Studie. Berlin: Deutscher Verlag der Wissenschaften.Google Scholar

References

  1. Carpenter, T., Fennema, E., & Romberg, T. (1993). Rational numbers: An integration of research. Hillsdale, NJ: Erlbaum.Google Scholar
  2. Gadamer, H-G. (1975). Truth and method. (G. Barden & J. Cumming, Trans, and Eds.). New York: Seabury Press.Google Scholar
  3. Gadamer, H-G. (1976). Philosophical hermeneutics. (D. E. Linge, Trans.). Berkeley, CA: University of California Press.Google Scholar
  4. Heidegger, M. (1962). Being and time. (J. Macquarrie & E. Robinson, Trans.). New York: Harper and Row.Google Scholar
  5. Lesh, R. (1983). Conceptual analyses of problem solving performance. In E. Silver (Ed.), Teaching and learning mathematical problem solving (pp. 309–329). Hillsdale, NJ: Erlbaum.Google Scholar
  6. Lesh, R., Hoover, M., & Kelly, A. E. (1993). Equity, assessment, and thinking mathematically: Principles for the design of model-eliciting activities. In I. Wirszup & R. Streit (Eds.), Developments in school mathematics education around the world (Vol. 3). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  7. Lesh, R., & Kelly, A. E. (1991). Human simulation of computer tutors: Lessons learned in a ten-week study of twenty human mathematics tutors. Paper presented at the International Psychology of Mathematics Education (PME) Conference, Assisi, Italy.Google Scholar
  8. Lesh, R., & Lamon, S. (1992). Assessing authentic performance in school mathematics. Washington, DC: AAAS.Google Scholar
  9. Lesh, R., Post, T., & Behr, M. (1989) Proportional reasoning. In M. Behr & J. Hiebert (Eds.), Number concepts and operations in the middle grades. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  10. Maher, C. A., Davis, R. B., & Alston, A. (1991). Brian’s representation and development of mathematical knowledge: A four-year study. The Journal of Mathematical Behavior, 102), 163–210.Google Scholar

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© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Roland W. Scholz
    • 1
  1. 1.Bielefeld/Zürich

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