Anna Sfard's homepage

picture of anna sfard

My research is devoted to study of human thinking in general and of mathematical thinking in particular. Inspired by the  critique of the traditional divide between thinking and speaking issued, among others, by Wittgenstein and Vygotsky, I define thinking as individualized form of interpersonal communication. The disappearance of the time-honored dichotomy is epitomized in the 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.

In my empirical research I follow the development of mathematical discourses, with the word development referring to the growth of these discourses both throughout individual lives and in the course of history. The  focus of many of my studies is on the issue of objectification, that is, on  those discursive mechanisms that bring about the emergence of new mathematical objects. In a series of studies in Israel, Canada, and US, most of them conducted together with PhD students, I have been investigating such topics as the development of algebraic discourse, the discourse on negative numbers, early numerical discourse, the mathematical discourse of students diagnosed with learning disabilities and those regarded as high-achievers, and the professional discourse of high-school mathematics teachers. Results of these studies have been published in a series of journal articles and book chapters (some of them in books I co-edited with others), and they have now been summarized in a book due to appear in the beginning of 2008..

Based at the University of Haifa, Israel, I am also affiliated with  Michigan State University in USA (as Lappan-Phillips-Fitzgerald Professor of Mathematics Education) and the Institute of Education, University of London, in UK.


JLS cover

Sfard, A. & McClain, K., Eds. (2002). Learning tools: Perspectives on the role of designed artifacts in mathematics learning. Mahwah, NJ: Laurence Erlbaum Associates [also publishes as special issue of Journal of Learning Sciences, 11(2,3).]

ESM cover

Kieran, C., Forman, E., and Sfard, A., Eds. (2003). Learning discourse: Bridging the individual and the social: discursive approaches to research in mathematics education. Dodrecht, TheNetherlands: Kluwer Academic Press [also published as the special issue of Educational Studies in Mathematics,  46(1-3)]

book cover

Sfard, A. (in press). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, UK: Cambridge University Press. (to appear: January 08 in US, March 08 in UK) [Table of contents + introduction ]   [more about the book]   [amazon]


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. [download]

Sfard, A. & Linchevski, L. (1994) . The gains and the pitfalls  of reification: The case of  algebra. Educational Studies in Mathematics , 26, 191-228. Reprinted in P. Cobb (ed.), Learning Mathematics -- Constructivist and Interactionist theories of mathematical development.(pp. 87-124). Dordrecht: Kluwer Academic  Publishers.

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 Mathematical Behavior,  14, 15-39. 

Sfard, A. (1997). On metaphorical roots of conceptual growth (A commentary). In English, L. (ed.), Mathematical reasoning: Analogies, metaphors, and images. pp. 339-371. London: Erlbaum.

Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), pp. 4-13. [English]  [Hebrew]

Sfard, A. (2000a). Symbolizing mathematical reality into being: How mathematical discourse and mathematical objects create each other. In P. Cobb,K. E. Yackel, & K. McClain (Eds), Symbolizing and communicating: perspectives on Mathematical Discourse, Tools, and Instructional Design (pp. 37-98). Mahwah, NJ: Erlbaum. [download]

Sfard, A. (2000b). On reform movement and the limits of mathematical discourse. Mathematical Thinking and Learning, 2(3), 157-189. [download]

Sfard, A. (2000 c). Steering (dis)course between metaphor and rigor: Using focal analysis to investigate the emergence of mathematical objects. Journal for Research in Mathematics Education, 31(3), 296-327

Sfard, A. & Kieran, C.  (2001). 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. (2001a). On the gains and dilemmas of calling different things the same name. Quarterly of Cognitive Science, 1(3/4), 359-388.

Sfard, A. ( 2001b). Learning mathematics as developing a discourse. In R. Speiser, C. Maher, C. Walter (Eds), Proceedings of 21st Conference of PME-NA (pp. 23-44). Columbus, Ohio: Clearing House for Science, mathematics, and Environmental Education. [download]

Sfard, A. (2001c). Communicating to learn or learning to communicate? Mathematics education in quest for new answers to old questions. (Book review). Zenralblatt fur Didiaktik Mathematik / International Reviews on Mathematical Education, 33(1), 1-9.

Sfard, A. (2001d). There is More to Discourse than Meets the Ears: Learning from mathematical communication things that we have not known before. Educational Studies in Mathematics, 46(1/3), 13-57.

Sfard, A., Forman, E. & Kieran, K. (2001). Learning discourse: Sociocultural approaches to research in mathematics education. Educational Studies in Mathematics. 46(1/3), 1-12.

Sfard, A. (2002) The interplay of intimations and implementations: Generating new discourse with new symbolic tools. The Journal of Learning Sciences, 11(2,3), 319-358.

Sfard, A., & Kay McClain, Eds. (2002). Analyzing tools: Perspectives on the Role of Designed Artifacts in Mathematics Learning. Journal for Learning Sciences 11(2&3), 153-161.

Sfard, A. (2003). Balancing the unbalanceable: The NCTM Standards in the light of theories of learning mathematics. In  J. Kilpatrick, Martin, G., & Schifter, D. (Eds.), A Research Companion for NCTM Standards (pp. 353-392).  Reston, VA: National Council for Teachers of Mathematics.

Ben-Yehuda, M.,  Lavy,  I., Linchevski, L., Sfard, A. (2005), Doing wrong with words or What bars students’ access to arithmetical discourses. The Journal for Research in Mathemtics Education, 36(3), 176 – 247. [download]

Sfard, A. & Lavi, I. (2005). Why cannot children see as the same what grownups cannot see as different? – early numerical thinking revisited. Cognition and Instruction, 23(2), 237-309. [download]

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 .[download the article]    [download Mary Juzwik's critique]  [download my rejoinder]

Sfard, A. & Prusak, A. (2005). Identity that makes a difference: Substantial learning as closing the gap between actual and designated identities. In H.L. Chick & J.L. Vincent (Eds.) Proceedings of the Twenty-ninth Meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 37-52), Department of Science and Mathematics Education, University of Melbourne, Victoria, Australia.

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.  (2005).  What changes when learning goes to school: The communicational version, the case of mathematics. European Journal of School Psychology, 3(1), 301-326 .

Sfard, A. (2005). Challenging discourse. A commentary. In A. Chronaki & I. Christiansen (Eds.), Challenging Perpsectives on Mathematics Classroom, 323 - 338.  Communication. London: Information Age Publishing.

Sfard, A. (2005). Discourse in flux - Commentary to Cognition in flux by Geoffrey Saxe. Mind, Culture, and Activity, 12(3&4), 233-250.

Sfard, A. (2005). Essential dialogue. Review essay on “Dialogicality and social representations” by Ivana Markova. Human Development, 48(6) , 363 - 370.

Sfard, A. (2007). When the rules of discourse change, but nobody tells you - making sense of mathematics learning from commognitive standpoint.  Journal of Learning Sciences,  16(4), 567–615.

Ben-Zvi, D., & Sfard, A. (2007). Ariadne’s Thread, Deadalus’ Wings, and the Learner’s Autonomy. Education and Didactics, 1(3), 123 - 141. [download]