Skip to main content
Thinking as Communicating
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 234
  • Cited by
    This book has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Kanbir, Sinan Clements, M. A. and Ellerton, Nerida F. 2018. Using Design Research and History to Tackle a Fundamental Problem with School Algebra. p. 87.

    Stouraitis, Konstantinos Potari, Despina and Skott, Jeppe 2017. Contradictions, dialectical oppositions and shifts in teaching mathematics. Educational Studies in Mathematics, Vol. 95, Issue. 2, p. 203.

    Ioannou, Marios 2017. Commognitive analysis of undergraduate mathematics students’ first encounter with the subgroup test. Mathematics Education Research Journal,

    Jackson, Kara and Nieman, Hannah 2017. Discourse and Education. p. 211.

    Stahl, Gerry 2017. Group practices: a new way of viewing CSCL. International Journal of Computer-Supported Collaborative Learning, Vol. 12, Issue. 1, p. 113.

    Tiedemann, Kerstin 2017. Mathematiklernen in der Familie. Journal für Mathematik-Didaktik, Vol. 38, Issue. 1, p. 1.

    Schacht, Florian 2017. Between the Conceptual and the Signified: How Language Changes when Using Dynamic Geometry Software for Construction Tasks. Digital Experiences in Mathematics Education,

    Skott, Jeppe 2017. Dealing with Conceptualisations of Learning. p. 133.

    Palatnik, Alik and Koichu, Boris 2017. Sense making in the context of algebraic activities. Educational Studies in Mathematics, Vol. 95, Issue. 3, p. 245.

    Tall, David 2017. And the Rest is Just Algebra. p. 43.

    Cromley, Jennifer G. Booth, Julie L. Wills, Theodore W. Chang, Briana L. Tran, Nhi Madeja, Michael Shipley, Thomas F. and Zahner, William 2017. Relation of Spatial Skills to Calculus Proficiency: A Brief Report. Mathematical Thinking and Learning, Vol. 19, Issue. 1, p. 55.

    Weitz, Marié and Venkat, Hamsa 2017. Improving Primary Mathematics Education, Teaching and Learning. p. 27.

    Radford, Luis 2017. Encyclopedia of Educational Philosophy and Theory. p. 1409.

    Branchetti, Laura and Morselli, Francesca 2017. Teaching and Learning in Maths Classrooms. p. 197.

    Heyd-Metzuyanim, Einat Munter, Charles and Greeno, James 2017. Conflicting frames: a case of misalignment between professional development efforts and a teacher’s practice in a high school mathematics classroom. Educational Studies in Mathematics,

    Graven, Mellony 2017. Research for educational change. Pythagoras, Vol. 38, Issue. 1,

    Ronda, Erlina and Adler, Jill 2017. Mining Mathematics in Textbook Lessons. International Journal of Science and Mathematics Education, Vol. 15, Issue. 6, p. 1097.

    Heyd-Metzuyanim, Einat and Schwarz, Baruch B. 2017. Conceptual change within dyadic interactions: the dance of conceptual and material agency. Instructional Science, Vol. 45, Issue. 5, p. 645.

    Kontorovich, Igor’ 2017. Why Johnny struggles when familiar concepts are taken to a new mathematical domain: towards a polysemous approach. Educational Studies in Mathematics,

    North, Marc 2017. In Pursuit of an Orientation for Life-preparation: A Case Study of the Subject Mathematical Literacy in South Africa. African Journal of Research in Mathematics, Science and Technology Education, p. 1.

  • Export citation
  • Recommend to librarian
  • Recommend this book

    Email your librarian or administrator to recommend adding this book to your organisation's collection.

    Thinking as Communicating
    • Online ISBN: 9780511499944
    • Book DOI:
    Please enter your name
    Please enter a valid email address
    Who would you like to send this to? *
  • Buy the print book

Book description

This book is an attempt to change our thinking about thinking. Anna Sfard undertakes this task convinced that many long-standing, seemingly irresolvable quandaries regarding human development originate in ambiguities of the existing discourses on thinking. Standing on the shoulders of Vygotsky and Wittgenstein, the author defines thinking as a form of communication. The disappearance of the time-honoured thinking-communicating dichotomy is epitomised by Sfard's term, commognition, which combines communication with cognition. The commognitive tenet implies that verbal communication with its distinctive property of recursive self-reference may be the primary source of humans' unique ability to accumulate the complexity of their action from one generation to another. The explanatory power of the commognitive framework and the manner in which it contributes to our understanding of human development is illustrated through commognitive analysis of mathematical discourse accompanied by vignettes from mathematics classrooms.


'Sfard has provided us with one of the most impressive, unified, homogenous theories of learning …'

Source: Computer-Supported Collaborative Learning

Refine List
Actions for selected content:
Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Send to Kindle
  • Send to Dropbox
  • Send to Google Drive
  • Send content to

    To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to .

    To send content to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

    Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

    Find out more about the Kindle Personal Document Service.

    Please be advised that item(s) you selected are not available.
    You are about to send:

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Albers D. J., & Alexanderson G. L. (1985). Mathematical people – profiles and interviews. Chicago: Contemporary Books.
The American heritage second college dictionary. (1985). Boston: Houghton Mifflin.
Anderson J. R., Reder L. M., & Simon H. A. (1996). Situated learning and education. Educational Researcher, 25(4), 5–11.
Austin J. L. (1962). How to do things with words (2nd ed.). Cambridge, MA: Harvard University Press.
Bakhtin M. (1981). The dialogic imagination: Four essays (M. Holquist & C. Emerson, Trans.). Austin: University of Texas Press.
Bakhtin M. (1986). Speech genres and other late essays (V. W. McGee, Trans.). Austin: University of Texas Press.
Bakhtin, M. (1999). The problem of speech genres. In Jaworski A. & Coupland N. (Eds.), The discourse reader (pp. 121–132). London: Routledge.
Bateson G. (1973). Steps to the ecology of mind. Frogmore, St. Albans, Herts, UK: Paladin Books.
Bauersfeld, H. (1988). Interaction, construction, and knowledge: Alternative perspectives for mathematics education. In Cooney T. J. & Grouws D. A. (Eds.), Effective mathematics teaching (pp. 27–46). Reston, VA: National Council of Teachers of Mathematics and Lawrence Erlbaum Associates.
Bauersfeld, H. (1995). “Language games” in mathematics classroom: Their function and their effects. In Cobb P. & Bauersfeld H. (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 271–292). Hillsdale, NJ: Lawrence Erlbaum Associates.
Bauman, Z. (1996). From Pilgrim to tourist – or a short history of identity. In Hall S. & Gay P. du (Eds.), Questions of cultural identity (pp. 18–36). London: Sage Publications.
Bauman Z. (2001). Identity in the globalized world. Social Anthropology, 9(2), 121–129.
Beach K. (1995). Activity as a mediator of sociocultural change and individual development: The case of school-work transition in Nepal. Mind, Culture, and Activity, 2(4), 285–302.
Ben-Yehuda M., Lavy I., Linchevski L., & Sfard A. (2005). Doing wrong with words: What bars students' access to arithmetical discourses. Journal for Research in Mathematics Education, 36(3), 176–247.
Bereiter C. (1985). Towards a solution of the learning paradox. Review of Educational Research, 55, 201–226.
Berlinski D. (2005). Infinite ascent: A short history of mathematics. New York: Modern Library.
Bills, C. (2002). Linguistic pointers in young children's description of mental calculations. In Cockburn A. & Nardi E. (Eds.), Proceedings of 26th Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 97–104). Norwich, England: School of Educational and Professional Development, University of East Anglia.
Blumer H. (1969). Symbolic interactionism: Perspective and method. Englewood Cliffs, NJ: Prentice-Hall.
Borges J. L. (1962/1964). Labyrinths: Selected stories and other writings (D. A. Yates & J. E. Irby, Eds.). New York: New Directions.
Borges J. L. (1998). Argumentum ornitologicum (A. Hurley, Trans.). In Collected fictions. New York: Viking. (Original work published 1960.)
Bourdieu, P. (1999). Structures, habits, practices. In Elliot A. (Ed.), The Blackwell reader in contemporary social theory (pp. 107–118). Oxford: Blackwell.
Boyer C. B., & Merzbach U. C. (1989). A history of mathematics (2nd ed.). New York: Wiley.
Bransford, J. D., Barron, B., Pea, R. D., Meltzoff, A., Kuhl, P., Bell, P., et al. (2006). Foundations and opportunities for an interdisciplinary science of learning. In Sawyer R. K. (Ed.), The Cambridge handbook of learning sciences (pp. 19–34). Cambridge: Cambridge University Press.
Brousseau G. (1997). Theory of didactical situations. Dordrecht, The Netherlands: Kluwer Academic.
Brown G., & Yule G. (1983). Discourse analysis. Cambridge: Cambridge University Press.
Brown J. S., & Burton R. R. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2, 155–192.
Brown J. S., Collins A., & Duguid P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32–42.
Brownell, W. A. (1935). Psychological considerations in the learning and teaching of arithmetic. In Reeve W. D. (Ed.), The teaching of arithmetic: Tenth yearbook of the National Council of Teachers of Mathematics (pp. 1–31). New York: Columbia University, Teachers College.
Bruner J. S. (1986). Actual minds, possible worlds. Cambridge, MA: Harvard University Press.
Bruner, J. S. (1990). The proper study of man: Four lectures on mind and culture. In Bruner J. (Ed.), Acts of meaning (pp. 1–32). Cambridge, MA: Harvard University Press.
Burkhard H.-D., & Schoenfeld A. H. (2003). Improving educational research: Toward a more useful, more influential, and better-funded enterprise. Educational Researcher, 32(9), 3–14.
Caduri G. (2005). The development of discourse on infinity. Unpublished master's thesis, University of Haifa, Haifa, Israel. (In Hebrew.)
Carroll L. (1998). Alice's adventures in Wonderland; And, Through the looking-glass and what Alice found there. Oxford, England: Oxford University Press.
Carruthers P., & Boucher J. (Eds.). (1998). Language and thought: Interdisciplinary themes. Cambridge: Cambridge University Press.
Cavaillès J. (1962). Philosophie mathematique. Paris: Hermann.
Cazden C. B. (2001). Classroom discourse: The language of teaching and learning (2nd ed.). Portsmouth, NH: Heinemann.
Chalouh, L., & Herscovics, N. (1988). Teaching algebraic expressions in a meaningful way. In Coxford A. F. & Shulte A. P. (Eds.), The ideas of algebra, K–12. (1988 Yearbook) (pp. 33–42). Reston, VA: National Council of Teachers of Mathematics.
Changeaux J.-P., & Connes A. (1995). Conversations on mind, matter, and mathematics (M. B. DeBevoise, Ed. and Trans.). Princeton, NJ: Princeton University Press.
Chevallard Y. (1985). Transposition didactique du savoir savant au savoir enseigne. Grenoble, France: La Pensee Sauvage Editions.
Chevallard Y. (1990). On mathematics education and culture: Critical afterthoughts. Educational Studies in Mathematics, 21(1), 3–28.
Chinn S. J. (1996). What to do when you can't learn the times tables. Aalborg, Denmark: Marko.
Chomsky N. (1957). Syntactic structures. ‘s-Gravenhage, The Netherlands: Mouton.
Clark A. (1997). Being there: Putting brain, body, and world together again. Cambridge, MA: MIT Press.
Cobb, P. (1996). Accounting for mathematics learning in the social context of the classroom. In Alsina C., Alvarez J., Hodgson B., Laborde C., & Perez A. (Eds.), Eighth International Congress on Mathematical Education: Selected Lectures (pp. 85–100). Seville, Spain: S. A. E. M. ‘THALES.’
Cobb P. (2002). A relational perspective on issues of cultural diversity and equity as they play out in the mathematics classroom. Mathematical Thinking and Learning, 4(2–3), 249–284.
Cobb P., & Bowers J. (1999). Cognitive and situated learning perspectives in theory and in practice. Educational Researcher, 25(4), 4–15.
Cobb, P., Gravemeijer, K. E. P., Yackel, E., McClain, K., & Whitenack, J. (1997). Situated cognition. In Kirshner D. & Whitson J. A. (Eds.), Mathematizing and symbolizing (pp. 151–233). Mahwah, NJ: Lawrence Erlbaum Associates.
Cobb, P., Wood, T. L., & Yackel, E. (1993). Discourse, mathematical thinking, and classroom practice. In Forman E., Minick N., & Stone A. (Eds.), Contexts for learning: Sociocultural dynamics in children's development (pp. 91–119). New York: Oxford University Press.
Cobb P., Yackel E., & Wood T. L. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2–33.
Cole M. (1996). Cultural psychology: A once and future discipline. Cambridge, MA: The Belknap Press of Harvard University Press.
Cole M., Gay J., Glick J. A., & Sharp D. W. (1971). The cultural context of learning and thinking: An exploration in experimental anthropology. New York: Basic Books.
Confrey, J. (1990). A review of the research on student conceptions in mathematics, science, and programming. In Cazden C. B. (Ed.), Review of research in education (Vol. 16, pp. 3–56). Washington, DC: American Educational Research Association.
Damasio A. R. (1999a). The feeling of what happens: Body and emotion in the making of consciousness. New York: Harcourt.
Damasio A. R. (1999b, December). How the brain creates the mind. Scientific American, 281, 112–117.
Davis P. J., & Hersh R. (1981). The mathematical experience. London: Penguin Books.
Davis R. (1988). The interplay of algebra, geometry, and logic. Journal of Mathematical Behavior, 7, 9–28.
Abreu G. (2000). Relationship between macro and micro socio-cultural context: Implications for the study of interactions in mathematics classroom. Educational Studies in Mathematics, 41(1), 1–29.
Dehaene S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press.
Dekel L. (2003). Colloquial and literate arithmetic discourse among children with learning difficulties and learning disabilitys in mathematics (in Hebrew). Unpublished masters thesis. The University of Haifa, Haifa, Israel.
Dennett D. C. (1996). Kinds of minds: Towards an understanding of consciousness. New York: Basic Books.
Descartes R. (1968). Discourse on method and the meditations (F. E. Sutcliffe, Trans.). Harmondsworth, England: Penguin Books.
Donald M. (1993). Origins of modern mind. Cambridge, MA: Harvard University Press.
Donmoyer R. (1996). This issue: A focus of learning. Educational Researcher, 25(4), 4.
Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. In Furinghetti F. (Ed.), 15th Annual Conference of the International Group of the Psychology of Mathematics Education (Vol. 1, pp. 32–48). Assisi, Italy.
Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In Tall D. (Ed.), Advanced mathematical thinking (pp. 95–125). Dordrecht, The Netherlands: Kluwer Academic.
Edwards D. (1993). But what do children really think? Discourse analysis and conceptual content in children's talk. Cognition and Instruction, 11(3–4), 207–225.
Edwards D. (1997). Discourse and cognition. London: Sage.
Edwards D., & Potter J. (1992). Discursive psychology. Newbury Park, CA: Sage.
Encyclopedia Britannica. (1998). Britannica CD 98, Multimedia edition. (International version).
Encyclopedia Britannica. (n.d.). Encylclopedia Britannica online. Retrieved from
Engeström Y. (1987). Learning by expanding: An activity-theoretical approach to developmental research. Helsinki: Orienta-Konsultit.
Erlwanger S. H. (1973). Benny's conception of rules and answers in IPI mathematics. Journal of Children's Mathematical Behavior, 1(2), 7–26.
Ernest P. (1993). Conversation as a metaphor for mathematics and learning. Paper presented at the British Society for Research into Learning Mathematics Day Conference, Manchester Metropolitan University.
Ernest P. (1994). Mathematics, education and philosophy: An international perspective. London: Routledge Falmer.
Filloy E., & Rojano T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.
Fischbein E. (1987). Intuition in science and mathematics. Dordrecht, The Netherlands: Reidel.
Fischbein E. (1989). Tactic models and mathematical reasoning. For the Learning of Mathematics, 9(2), 9–14.
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 for Research in Mathematics Education, 16, 3–17.
Fisher W. R. (1984). Narration as human communication paradigm: The case of public moral argument. Communication Monographs, 51, 1–22.
Fodor J. A. (1975). The language of thought. New York: Crowell.
Fodor J. A. (1983). The modularity of mind: An essay on faculty psychology. Cambridge, MA: MIT Press.
Forman, E. (1996). Forms of participation in classroom practice: Implications for learning mathematics. In Nesher P., Steffe L., Cobb P., Goldin G., & Greer B. (Eds.), Theories of mathematical learning (pp. 115–130). Hillsdale, NJ: Lawrence Erlbaum Associates.
Forman E. A., & Ansell E. (2002). Orchestrating the multiple voices and inscriptions of a mathematics classroom. Journal of the Learning Sciences, 11, 251–274.
Foucault M. (1972). The archaeology of knowledge; And, The discourse on language. New York: Pantheon Books.
Fouts R. (1998). Next of kin: My conversations with chimpanzees. New York: Harper Paperbacks.
Gadamer H.-G. (1975). Truth and method. New York: Seabury Press.
Garfinkel H. (1967). Studies in ethnomethodology. Englewood Cliffs, NJ: Prentice-Hall.
Garnett K. (1992). Developing fluency with basic number facts: Intervention for students with learning disabilities. Learning Disabilities Research and Practice, 7, 210–216.
Gauker, C. (n.d.). Languge and thought. Retrieved from
Geary D. C., Hoard M. K., & Hamson C. O. (1999). Numerical and arithmetical cognition: Patterns of functions and deficits in children at risk for a mathematical disability. Journal of Experimental Child Psychology, 74(3), 213–240.
Gee J. (1989). Literacy, discourse, and linguistics: Introduction. Journal of Education, 171(1), 6–17.
Gee, J. P. (1997). Thinking, learning, and reading: The situated sociocultural mind. In Kirshner D. & Whitson J. A. (Eds.), Situated cognition: Social, semiotic, and psychological perspective (pp. 235–260). Mahwah, NJ: Lawrence Erlbaum Associates.
Gee J. P. (2001). Identity as an analytic lens for research in education. Review of Research in Education, 25, 99–125.
Geertz C. (1973). The interpretation of cultures. New York: Basic Books.
Gelman R., & Gallistel C. R. (1978). The child's understanding of number. New York: Cambridge University Press.
Ginsburg H. P. (1997). Entering the child's mind: The clinical interview in psychological research and practice. New York: Cambridge University Press.
Goffman E. (1959). The presentation of self in everyday life. Garden City, NY: Doubleday.
Goffman E. (1967). Interaction ritual: Essays on face-to-face behavior. New York: Anchor Books.
Goldman S. R., Pelligrino J. W., & Mertz D. L. (1988). Extended practice of basic addition facts: Strategy changes in learning disabled students. Cognition and Instruction, 5, 223–265.
Goody J. (1977). The domestication of the savage mind. Cambridge: Cambridge University Press.
Greeno J. G. (1991). Number sense as a situated knowing in conceptual domain. Journal for Research in Mathematics Education, 22(3), 170–218.
Greeno J. G. (1997). On claims that answer the wrong question. Educational Researcher, 26(1), 5–17.
Grice, H. (1975). Logic and conversation. In Cole P. & Morgan J. L. (Eds.), Syntax and symantics: Speech acts (Vol. 3, pp. 41–58). New York: Academic Press.
Grice H. P. (1957). Meaning. Philosophical Review, 66, 377–388.
Groome D. (1999). An introduction to cognitive psychology: Processes and disorders. London: Routledge.
Gurnik M., & Saban A. (2003). Solving mathematical word problems with real-life content in mathematics and language lessons.Haifa, Israel: University of Haifa. Unpublished manuscript.
Hadamard J. (1954). An essay on the psychology of invention in the mathematical field. New York: Dover.
Hall S., & Gay du P. (1996). Questions of cultural identity. London: Sage Publications.
Halliday M. A. K. (1978). Language as social semiotics. London: Edward Arnold.
Halliday M. A. K. (2003). On language and linguistics. London: Continuum.
Halliday M. A. K., & Martin J. R. (1993). Writing science. Pittsburgh: University of Pittsburgh Press.
Halliday M. A. K., & Matthiessen C. M. I. M. (2004). An introduction to functional grammar (3rd ed.). London: Arnold.
Halmos P. R. (1985). I want to be a mathematician – an automathography in three parts. Washington, DC: Mathematical Association of America.
Hanks P. (Ed.). (1986). Collins dictionary of the English language.London: Collins.
Harel, G., Behr, M., Post, T., & Lesh, R. (1989). Fishbein's theory: A further consideration. In Vergnaud G., Rogalski J., & Artigue M. (Eds.), Proceedings of the 13th Annual Conference of the Psychology of Mathematics Education (pp. 52–59): Paris: University of Paris.
Harré R., & Gillett G. (1995). The discursive mind. Thousand Oaks, CA: Sage Publications.
Hauser M. D., Chomsky N., & Fitch W. T. (2002). The language faculty: What is it, who has it, and how did it evolve?Science, 298, 1569–1579.
Heider F. (1958). The psychology of interpersonal relations. New York: Wiley.
Herder, J. G. (1771/1967). Abhandlung über Ursprung der Sprache. In Suphon B. (Ed.), Herder, J. G., Sämtliche Werke V. George Olms.
Hershkowitz R. (1989). Visualisation in geometry – two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–75.
Hesse M. B. (1966). Models and analogies in science. Notre Dame, IN: University of Notre Dame Press.
Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In Grouws D. A. (Ed.), The handbook of research on mathematics teaching and learning (pp. 65–100). New York: Macmillan.
Holland D., Lachicotte W. J., Skinner D., & Cain C. (1998). Identity and agency in cultural worlds. Cambridge, MA: Harvard University Press.
Holquist M. (1990). Dialogism: Bakhtin and his world. London: Routledge.
Horn L. R., & Ward G. (Eds.). (2004). Handbook of pragmatics. Malden, MA: Blackwell.
Hoyles C., Noss R., & Pozzi S. (2001). Proportional reasoning in nursing practices. Journal for Research in Mathematics Education, 32(1), 4–27.
Hutchins E. (1995). Cognition in the wild. Cambridge, MA: MIT Press.
Jacoby S., & Ochs E. (1995). Co-construction: An introduction. Research on Language and Social Interaction, 28(3), 171–183.
Johnson M. (1987). The body in the mind: The bodily basis of meaning, imagination, and reason. Chicago: University of Chicago Press.
Jourdain, P. E. B. (1956). The nature of mathematics. In Newman J. P. (Ed.), The world of mathematics. New York: Simon & Schuster.
Kavale, K. A., & Forness, S. R. (1997). Defining learning disabilities: Consonance and dissonance. In Lloyd J. W., Kame'euni E. J., & Chard D. J. (Eds.), Issues in educating students with disabilities (pp. 3–25). Mahwah, NJ: Lawrence Erlbaum Associates.
Kieran C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematocs, 12(3), 317–326.
Kieran C., & Sfard A. (1999). Seeing through symbols: The case of equivalent expressions. Focus on Learning Problems in Mathematics, 21(1), 1–17.
Kilpatrick J., Swafford J., & Findell B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
Kipling R. (1894). The jungle book. Retrieved July 12, 2007, from
Kleiner I. (1989). Evolution of the function concept: A brief survey. The College Mathematics Journal, 20(4), 282–300.
Kline M. (1980). Mathematics: The loss of certainty. New York: Oxford University Press.
Kosc L. (1974). Developmental dyscalculia. Journal of Learning Disabilities, 7, 46–89.
Kozulin A. (1990). Vygotsky's psychology: A biography of ideas. New York: Harvester Wheatsheaf.
Kristal L. (2005). The development of discourse on derivative of lower placement 11th grade mathematics students. Unpublished master's thesis, University of Haifa, Haifa, Israel.
Krummheuer, G. (1995). The ethnography of argumentation. In Cobb P. & Bauersfeld H. (Eds.), The emergence of mathematical meaning: Interactions in classroom culture (pp. 229–269). Hillsdale, NJ: Lawrence Erlbaum Associates.
Kuhn T. (1962). The structure of scientific revolutions (2nd ed.). Chicago: University of Chicago Press.
Labov, W., & Waletzky, J. (1967). Narrative analysis: Oral versions of personal experience. In Helm J. (Ed.), Essays on the verbal and visual arts: Proceedings of the 1966 Annual Spring meeting of the American Ethnological Society (pp. 12–44). Seattle: University of Washington Press.
Lacan J. (1966). Ecrits 1. Paris: Editions du Seuil.
Lakatos I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.
Lakoff G. (1987). Women, fire and dangerous things: What categories reveal about the mind. Chicago: University of Chicago Press.
Lakoff, G. (1993). The contemporary theory of metaphor. In Ortony A. (Ed.), Metaphor and thought (pp. 202–250). Cambridge: Cambridge University Press.
Lakoff G., & Johnson M. (1980). The metaphors we live by. Chicago: University of Chicago Press.
Latour B. (1987). Science in action. Cambridge, MA: Harvard University Press.
Lave J. (1988). Cognition in practice. Cambridge: Cambridge University Press.
Lave J., & Wenger E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press.
Lemke J. L. (1993). Talking science: Language, learning, and values. Norwood, NJ: Ablex.
Lemke J. L. (2000). Across the scales of time: Artifacts, activities, and meanings in ecosocial systems. Mind, Culture, and Activity, 7(4), 273–290.
Leont'ev A. N. (1930). Studies in the cultural development of the child: 2. The development of voluntary attention in the child. Journal of Genetic Psychology, 37, 52–81.
Leont'ev A. N. (1981). Psychology and the language learning process. Oxford, England: Pergamon Press.
Lerman, S. (1998). A moment in the zoom of a lens: Towards a discursive psychology of mathematics teaching and learning. In Olivier A. & Newstead K. (Eds.), Proceedings of the 22nd Annual Meeting of the International Group for the Psychology of Mathematics Education. Stellenbosch, South Africa.
Lerman, S. (1999). Culturally situated knowledge and the problem of transfer in the learning of mathematics. In Burton L. (Ed.), Learning mathematics: From hierarchies to networks. London: Falmer Press.
Levinson S. (1983). Pragmatics. Cambridge: Cambridge University Press.
Lucy J. A. (1997). Linguistic relativity. Annual Review of Anthropology, 26, 291–313.
Lucy J. A., & Shweder R. (1979). Whorf and his critics: Linguistic and nonlinguistic influences on color memory. American Anthropologist, 81, 581–615.
Malik M. A. (1980). Historical and pedagogical aspects of definition of function. International Journal of Math Science and Technology, 1(4), 489–492.
Markova I. (2003). Dialogicality and social representations: The dynamics of mind. Cambridge: Cambridge University Press.
Markovits Z., Eylon B., & Bruckheimer M. (1986). Functions today and yesterday. For the Learning of Mathematics, 6(2), 18–24.
Maturana H. R. and Varela F. J. (1987). The Three of Knowledge. Boston: Shambhala.
Mayer R. E. (1983). Thinking, problem solving, cognition. New York: W. H. Freeman.
McDermott, R. P. (1993). The acquisition of a child by a learning disability. In Chaiklin S. & Lave J. (Eds.), Understanding practice. Cambridge: Cambridge University Press.
McGarrigle J., & Donaldson M. (1974). Conservation accidents. Cognition, 3, 341–350.
Mead, G. H. (1934). Mind, self, and society from the standpoint of a social behaviorist (Morris C. W., Ed.). Chicago: University of Chicago Press.
Mehan H. (1979). Learning lessons: Social organization in the classroom. Cambridge, MA: Harvard University Press.
Mehan, H. (1996). The construction of an LD student: A case study in the politics of representation. In Silverstein M. & Urban G. (Eds.), Natural histories of discourse (pp. 253–276). Chicago: University of Chicago Press.
Mehler J., & Bever T. G. (1967). Cognitive capacity of very young children. Science, 158, 141–142.
Minick, N. (1987). Development of Vygotsky's thought: An introduction. In Rieber R. W. & Carton A. S. (Eds.), The collected works of L. S. Vgotsiy (Vol. 1, pp. 17–38). New York: Plenum Press.
Minsky M. (1985). The society of mind. New York: Simon and Schuster.
Nardi B. (Ed.). (1996). Context and consciousness: Activity theory and human-computer interaction. Cambridge, MA: MIT Press.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Neisser U. (Ed.). (1987). Concepts and conceptual development: Ecological and intellectual factors in categorization. Cambridge: Cambridge University Press.
Nunes T., & Bryant P. (1996). Children doing mathematics. Oxford: Blackwell.
Nunes T., Schliemann A., & Carraher D. (1993). Street mathematics and school mathematics. Cambridge: Cambridge University Press.
O'Connor, M. C. (1998). Language socialization in the mathematics classroom: Discourse practices in group discussions. In Lampert M. & Blunk M. (Eds.), Talking mathematics: Studies of teaching and learning in school (pp. 17–55). Cambridge: Cambridge University Press.
O'Connor, M. C., & Michaels, S. (1996). Shifting participant frameworks: Orchestrating thinking practices in group discussions. In Hicks D. (Ed.), Discourse, learning, and schooling (pp. 63–103). Cambridge: Cambridge University Press.
Olby R. C. (1974). The path to the double helix. Seattle: University of Washington Press.
Ortony A. (1993). Metaphor and thought. Cambridge: Cambridge University Press.
Overbye D. (2006, August 15). An elusive proof and its elusive prover. New York Times, Section F, p. 1.
Peirce C. (1931–1935). Collected papers of Charles Sanders Peirce (Vol. 1–6). Cambridge, MA: Harvard University Press.
Peirce C. (1955). Philosophical writings of Peirce. New York: Dover.
Perkins D. (2001). Eureka effect: The art and logic of breakthrough thinking. New York: W. W. Norton.
Piaget J. (1952). The origins of intelligence of the child. London: Routledge and Kegan Paul.
Piaget J. & Garcia R. (1989). Psychogenesis and the history of science. New York: Columbia University Press.
Pinker S. (1994). The language instinct. New York: HarperCollins.
Pinker S. (2003). Blank slate: The modern denial of human nature. London: Penguin.
Plato . (1949). Meno. New York: Liberal Arts Press.
Poincaré H. (1952). Science and method. New York: Dover.
Pullum G. K. (1991). The great Eskimo hoax, and other irreverent essays on the study of language. Chicago: University of Chicago Press.
Reddy, M. J. (1979). The conduit metaphor: A case of frame conflict in our language about language. In Ortony A. (Ed.), Metaphor and thought. Cambridge: Cambridge University Press.
Reed H. J., & Lave J. (1979). Arithmetic as a tool for investigating the relations between culture and cognition. American Ethnologist, 6, 568–582.
Richter D. J. (n.d.). Ludwig Wittgenstein (1889–1951). In The Internet encyclopedia of philosophy. Retrieved from
Ricoeur P. (1977). The rule of metaphor: Multidisciplinary studies of the creation of meaning in language. Toronto: University of Toronto Press.
Rogoff B. (1990). Apprenticeship in thinking: Cognitive development in social context. Oxford: Oxford University Press.
Rogoff, B. (1995). Observing sociocultural activity on three planes: Participatory appropriation, guided participation, and apprenticeship. In Wertsch J. V., Rio P. Del, & Alvarez A. (Eds.), Sociocultural studies of mind. Cambridge: Cambridge University Press.
Rorty R. (1979). Philosophy and the mirror of nature. Princeton, NJ: Princeton University Press.
Rorty R. (1989). Contingency, irony, solidarity. Cambridge: Cambridge University Press.
Rosch, E. (1978). Principles of categorization. In Rosch E. & Lloyd B. B. (Eds.), Cognition and categorization. Hillsdale, NJ: Lawrence Erlbaum Associates.
Russell B. (1904). Recent works on the principles of mathematics. International Monthly, 4, 84.
Russell B. (1956). Portraits from memory, and other essays. New York: Simon and Schuster.
Rutherford, E. (n.d.). Retrieved in June 1999 from
Ruthing D. (1984). Some definitions of the concept of function from John Bernoulli to N. Bourbaki. Mathematical Intelligence, 6(4), 72–77.
Ryle G. (1949/2000). The concept of mind. Chicago: University of Chicago Press.
Sacks S. (Ed.). (1978). On metaphor. Chicago: University of Chicago Press. Sacks H. (1992). Lectures on conversation. Oxford: Blackwell.
Säljö, R., & Wyndhamn, J. (1993). Solving everyday problems in the formal setting: An empirical study of the school as context for thought. In Chaiklin S. & Lave J. (Eds.), Understanding practice: Perspectives on activity and context (pp. 327–342). Cambridge: Cambridge University Press.
Sapir E. (1949). Selected writings in language, culture, and personality. Berkeley: University of California Press.
Saxe G. B., & Esmonde I. (2005). Studying cognition in flux: A historical treatment of “Fu” in the shifting structure of Oksapmin mathematics. Mind, Culture, and Activity, 13(2), 171–225.
Scheffler, I. (1991). Educational metaphors. In Scheffler I. (Ed.), In praise of the cognitive emotions and other essays in the philosophy of education (pp. 45–55). New York: Routledge.
Schnotz W., Vosniadou S., & Carretero M. (Eds.). (1999). New perspectives on conceptual change. Oxford: Pergamon.
Schoenfeld, A. H. (1998). Making mathematics and making pasta: From cookbook procedures to really cooking. In Greeno J. G. & Goldman S. V. (Eds.), Thinking practices in mathematics and science learning (pp. 299–319). Mahwah, NJ: Lawrence Erlbaum Associates.
Schutz A. (1967). Collected papers: The problem of social reality. The Hague, The Netherlands: Martinus Nijhoff.
Schutz A., & Luckmann T. (1973). The structures of the life world. Evanston, IL: Northwestern University Press.
Scribner, S. (1997). Mind in action: A functional approach to thinking. In Cole M., Engstrom Y., & Vasquez O. (Eds.), Mind, culture, and activity: Seminal papers from the Laboratory of Comparative Human Cognition (pp. 354–368). Cambridge: Cambridge University Press. (Original work published in 1983.)
Scribner S., & Cole M. (1981). The psychology of literacy. Cambridge, MA: Harvard University Press.
Searle J. R. (2002). Consciousness and language. Cambridge: Cambridge University Press.
Searle J. R. (2004). Mind: A brief introduction. New York: Oxford University Press.
Sebeok, T. A. (2001). Nonverbal communication. In Cobley P. (Ed.), The Routledge companion to semiotics and linguistics. London: Routledge.
Sfard A. (1987). Teaching the theory of algorithms in high school. Unpublished doctoral dissertation, Hebrew University of Jerusalem, Jerusalem, Israel. (In Hebrew.)
Sfard A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.
Sfard, A. (1992). Operational origin of mathematical objects and the quandary of reification – the case of function. In Dubinsky E. & Harel G. (Eds.), The Concept of function: Aspects of epistemology and pedagogy (MAA Notes No. 25) (pp. 59–84). Washington, DC: Mathematical Association of America.
Sfard A. (1994). Reification as a birth of a metaphor. For the Learning of Mathematics, 14(1), 44–55.
Sfard A. (1995). The development of algebra: Confronting historical and psychological perspectives. Journal of Mathematic Behavior, 14, 15–39.
Sfard, A. (1997). Commentary: On metaphorical roots of conceptual growth. In English L. D. (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 339–371). Mahwah, NJ: Lawrence Erlbaum Associates.
Sfard A. (1998). Two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4–13.
Sfard, A. (2003). Balancing the unbalanceable: The NCTM Standards in the light of theories of learning mathematics. In Kilpatrick J., Martin W. G., & Schifter D. (Eds.), A research companion to principles and standards for school mathematics (pp. 353–392). Reston, VA: National Council of Teachers of Mathematics.
Sfard A. (2005). What could be more practical than good research? On mutual relations between research and practice of mathematics education. Educational Studies in Mathematics, 58(3), 393–413.
Sfard A. (2007). When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a cognitive standpoint. Journal for Learning Sciences.
Sfard A., & Kieran C. (2001a). Cognition as communication: Rethinking learning-by-talking through multi-faceted analysis of students' mathematical interactions. Mind, Culture, and Activity, 8(1), 42–76.
Sfard, A., & Kieran, C. (2001b). Preparing teachers for handling students' mathematical communication: Gathering knowledge and building tools. In Lin F. L. & Cooney T. J. (Eds.), Making sense of mathematics teacher education. Dordrecht, The Netherlands: Kluwer Academic.
Sfard A., & Lavie I. (2005). Why cannot children see as the same what grown-ups cannot see as different? Early numerical thinking revisited. Cognition and Instruction 23(2), 237–309.
Sfard A., & Linchevski L. (1994). The gains and the pitfalls of reification: The case of algebra. Educational Studies in Mathematics, 26, 191–228.
Sfard A., & Prusak A. (2005). Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity. Educational Researcher, 34(4), 14–22.
Shannon C. E., & Weaver W. (1949). The mathematical theory of communication. Urbana: University of Illinois Press.
Shaywitz S., Escobar M., Shaywitz B., Fletchers J., & Makuch R. (1992). Evidence that dyslexia may represent the lower tail of a normal distribution of reading ability. New England Journal of Medicine, 326(3), 145–150.
Simon T., Hespos S. J., & Rochat P. (1994). Do infants understand simple arithmetic? A replication of Wynn (1992). Cognitive Development, 10(2), 253–269.
Sinclair J. M., & Coulthard M. (1975). Toward an analysis of discourse: The English used by teachers and pupils. London: Oxford University Press.
Smith, H. (n.d.). Quotation retrieved from
Smith J. P., diSessa A. A., & Rochelle J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3(2), 115–163.
Sperber D., & Wilson D. (1986). Relevance: Communication and cognition. Malden, MA: Blackwell.
SperberWilson D.D. (1995). Relevance: Communication and cognition. Cambridge, MA: Blackwell.
Starkey P., & Cooper R. (1980). Perception of numbers by human infants. Science, 210(28), 1033–1034.
Steeves K. J., & Tomey H. A. (1998). Mathematics and dyslexia: The individual who learns differently may still be successful in math. Unpublished manuscript.
Stern D. G. (1995). Wittgenstein on mind and language. New York: Oxford University Press.
Suchman L. (2007). Human-machine reconfigurations: Plans and situated actions. Cambridge: Cambridge University Press.
Tall D., & Schwartzenberger R. (1978). Conflicts in the learning of real numbers and limits. Mathematics Teaching, 82, 44–49.
Tall D., & Vinner S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.
Tannen, D., & Wallat, C. (1999). Interactive frames and knowledge schemas in interactions. In Jaworski A. & Coupland N. (Eds.), The discourse reader (pp. 346–366). New York: Routledge.
Thom R. (1971). Modern mathematics: An educational and philosophical error?American Scientist, 59, 695–699.
Thurston W. P. (1990). Mathematical education. Notices of the American Mathematical Society, 37(7), 844–850.
Thurston W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30, 161–177.
Tomasello M. (1999). The cultural origins of human cognition. Cambridge, MA: Harvard University Press.
Toulmin S. (1972). Human understanding. Vol. 1.General Introduction and Part 1. Oxford: Calderon Press.
Turing A. M. (1950). Computing machinery and intelligence. Mind, 59, 443–460.
Dooren W., DeBock D., Janssens D., & Verschaffel L. (2005). Students' overreliance on linearity: An effect of school-like word problems?Paper presented at the 29th Conference of the International Group for the Psychology of Mathematics Education, Melbourne, Australia.
van Hiele, P. M. (1985). A child's thought and geometry. In Fuys D., Geddes D., & Tischler R. (Eds.), English translation of selected writing of Dina van Hiele-Geldorf and Pierre M. van Hiele (pp. 243–252). Brooklyn, NY: Brooklyn College, School of Education.
van Hiele, P. M. (2004/1959). A child's thought and geometry. In Carpenter T. P., Dossey J. A., & Koelher J. L. (Eds.), Classics in mathematics education research (pp. 60–67). Reston, VA: National Council of Teachers of Mathematics.
Varela F. J., Thompson E., & Rosch E. (1991). The embodied mind: Cognitive science and human experience. Cambridge, MA: MIT Press.
Varenne H., & McDermott R. P. (1998). Successful failure: The school America builds. Boulder, CO: Westview Press.
Vinner S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293–305.
Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In Tall D. (Ed.), Advanced mathematical thinking. Dordrecht, The Netherlands: Kluwer Academic.
Vinner S., & Dreyfus T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356–366.
Voigt J. (1985). Patterns and routines in classroom interaction. Recherches en Didatique des Mathématiques, 6(1), 69–118.
Voigt J. (1994). Negotiation of mathematical meaning and learning mathematics. Educational Studies in Mathematics, 26, 275–298.
Voigt, J. (1995). Thematic patterns of interaction and sociomathematical norms. In Cobb P. & Bauersfeld H. (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 163–201). Hillsdale, NJ: Lawrence Erlbaum Associates.
Voigt, J. (1996). Negotiation of mathematical meaning in classroom processes: Social interaction and learning mathematics. In Steffe L., Nesher P., Cobb P., Goldin G., & Greer B. (Eds.), Theories of mathematical learning (pp. 21–50). Mahwah, NJ: Kluwer Academic.
Vosniadou S. (1994). Capturing and modeling the process of conceptual change. Learning and Instruction, 4, 45–69.
Vosniadou S., Baltas A., & Vamvakoussi X. (Eds.). (2007). Reframing the conceptual change approach in learnign and instruction. Amsterdam: Elsevier.
Vygotsky L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.
Vygotsky, L. S. (1982). Consciousness as a problem in the psychology of behavior. In Vygotsky L. S. (Ed.), Collected works: Problems of the theory and history of psychology. Moscow: Pedagogica. (In Russian.)
Vygotsky L. S. (1986). Thought and language (A. Kozulin, Trans.). Cambridge, MA: MIT Press.
Vygotsky, L. S. (1987). Thinking and speech. In Rieber R. W., & Carton A. C. (Eds.), The collected works of L. S. Vygotsky. New York: Plenum Press.
Walkerdine V. (1988). The mastery of reason. London: Routledge.
Watson J. D. (2001). The double helix: A personal account of the discovery of the structure of DNA. New York: Touchstone.
Wenger E. (1998). Communities of practice: Learning, meaning, and identity. New York: Cambridge University Press.
Whorf B. L. (1940). Science and linguistics. Technology Review, 42(6), 229–231, 247–228.
Wikipedia. (n.d.). Language. Retrieved from
Wikipedia. (n.d.). Occam's razor. Retrieved from's_Razor.
William of Ockham. (1974). Ockham's theory of terms: Part I of the Summa logicae (M. J. Loux, Trans.). Notre Dame, IN: University of Notre Dame Press.
William of Ockham. (1984). Venerabilis inceptoris Guillelmi de Ockham brevis summa libri physicorum; Summula philosophiae naturalis; et Quaestiones in libros physicorum Aristotelis (Brown S., Trans.). St. Bonaventure, NY: St. Bonaventure University.
William of Ockham. (1990). Philosophical writings: A selection (Boehner P., Trans.). Indianapolis, IN: Hackett.
Wittgenstein L. (1953/2003). Philosophical investigations: The German text, with a revised English translation (3rd ed., G. E. M. Anscombe, Trans.). Malden, MA: Blackwell.
Wittgenstein L. (1961). Tractatus logico-philosophicus (D. F. Pears & B. F. McGuiness, Trans.). London: Routledge & Paul.
Wittgenstein, L. (1969). On certainty (Anscombe G. E. M. & Wright G. H., Eds.) (D. Paul & G. E. M. Anscombe, Trans.). Oxford: Blackwell.
Wittgenstein L. (1978). Remarks on the foundations of mathematics. Oxford: Blackwell.
Wittgenstein, L. (1980). Remarks on the philosophy of psychology (Vol. 2, Wright G. H. & Nyman H., Eds.) (C. G. Luckhardt & M. A. E. Aue, Trans.). Oxford: Blackwell.
Wittgenstein L. (1981). Zettel (2nd ed., Anscombe G. E. M. & Wright G. H. V., Eds.). Oxford: Blackwell.
Woodfield, A. (1993). Do your concepts develop? In Hookway C. & Peterson D. (Eds.), Philosophy and cognitive science (pp. 41–67). Cambridge: Cambridge University Press.
Wu H. (1997). The mathematics education reform: Why you should be concerned and what you can do. American Mathematical Monthly, 104, 954–962.
Wynn K. (1992). Addition and subtraction by human infants. Nature, 358, 749–750.
Wynn K. (1995). Origins of numerical knowledge. Psychological Science, 7, 164–169.
Xu F., & Carey S. (1996). Infants' metaphysics: The case of numerical identity. Cognitive Psychology, 30, 111–153.
Yackel E., & Cobb P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 58–477.


Full text views

Total number of HTML views: 0
Total number of PDF views: 532 *
Loading metrics...

Book summary page views

Total views: 572 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 18th November 2017. This data will be updated every 24 hours.