# One is Not Born a Mathematician: In Conversation with Vasily Davydov

## Abstract

That mathematics education has been one of the central concerns of educational systems worldwide is no secret. It is also an established consensus that as far back as eighty years ago, Russian psychologists such as Vygotsky, Luria, Meshcheryakov, and Davydov have pioneered work that contributed to the understanding of teaching and learning and shifted the trajectory of schooling in the Western World. Looking back, we feel that there is more to this legacy that awaits to be further explained and extended to potentially address some of the pressing issues in mathematics education. To bring forth this legacy, the authors engage in an imaginary conversation with Vasily Davydov to tease out notions that include, inter alia, language and interaction, learning and teaching, and empirical and theoretical thinking. The utilisation of a conversation as a method of inquiry for the purpose of this paper was intentional as it not only encompasses the very method of teaching advocated by Davydov, thus conveying that the means is the message, but it is also conducive to the exploration of simultaneously surfacing ideas and phenomena. The questions asked, clarifications provided, and contradictions that still remain chart a map that displays theoretical vistas and empirical landscapes drawn and inspired by Davydov’s legacy. Specifically, Fellus and Biton bring forth citations from Davydov’s works that are used as signposts in the conversation that unfolds.## References

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Berteletti, I., & Booth, J. R. (2015). Finger Representation and Finger-Based Strategies in the Acquisition of Number Meaning and Arithmetic. Development of Mathematical Cognition: Neural Substrates and Genetic Influences, 2, 109.

Berteletti, I., & Booth, J. R. (2015). Perceiving fingers in single-digit arithmetic problems. Frontiers in psychology, 6, 226. doi:10.3389/fpsyg.2015.00226

Biton, Y., Fellus, O., & Hershkovitz, S. (2016, August 3-7). Border crossing: Bringing together pre-service teachers’ technological, pedagogical, and content knowledge through the use of digital textbooks in mathematics. The 2016 International Group for the Psychology of Mathematics Education: How to solve it? Szeged, Hungary.

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Boaler, J., Wiliam, D. & Brown, M. L. (2000). Students’ experiences of ability grouping—disaffection, polarisation and the construction of failure. British Educational Research Journal, 26(5), 631−648.

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Davydov, V. V. (1988). The concept of theoretical generalization and problems of educational psychology. Studies in Soviet Thought 36 (3), pp. 169−202.

Davydov, V. V. (1991). Psychological abilities of primary school children in learning mathematics. Soviet studies in mathematics education, Vol. 6. (J. Teller, Trans.) National Council of Teachers of Mathematics, Reston, Virginia.

Davydov, V. V. (1995). The influence of L. S. Vygotsky on education theory, research, and practice. Educational Researcher, 24(3), pp. 2−21.

Davydov, V. V. (1998a). The concept of developmental teaching. Journal of Russian and East European Psychology, 36(4), pp. 11−36.

Davydov, V. V. (1998b). What is formal learning activity? Journal of Russian and East European Psychology, 36(4), pp. 37−47.

Davydov, V. V. (1999). What is real learning activity? In M. Hedegaard & J. Lompscher (Eds.). Learning activity and development. (pp. 123−138). Aarhus: Aarhus University Press.

Davydov, V. V., & Andronov, V. P. (1981). Psychological conditions of the origination of ideal actions (Project Paper No. 81−2). Wisconsin University, Madison: Wisconsin Research and Development Center for Individualized Schooling.

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Mason, R. & McFeetors, J. (2007). Student trajectories in high school mathematics: Issues of choice, support, and identity-making. Canadian Journal of Science, Mathematics, and Technology Education, 7(4), 291−316.

McCrink, K., & Wynn, K. (2004). Large-number addition and subtraction by 9-month-old infants. Psychological Science, 15(11), 776−781.

McFeetors, J., & Mason, R. (2005). Voice and success in non-academic mathematics courses: (Re)Forming identity. For the Learning of Mathematics, 25(3), 16−23.

Ministry of Education and Training (Ontario). (2005). The Ontario curriculum grades 9 and 10: Mathematics. Toronto: Queen’s Printer.

Muijs, D., & Dunne, M. (2010). Setting by ability–or is it? A quantitative study of determinants of set placement in English secondary schools. Educational Research, 52(4), 391−407.

Newman, S. D. (2016). Does finger sense predict addition performance? Cognitive processing, 17(2), 139−146.

Noël, M. P. (2008). Finger gnosia: A predictor of numerical abilities in children? Child Neuropsychology, 11(5), 413−430.

Norris, J., Sawyer, R. D., & Lund, D. (Eds.). (2012). Duoethnography: Dialogic methods for social, health, and educational research, (Vol. 7). Walnut Creek, CA: Left Coast Press.

Oakes, J. (1990). Multiplying inequalities: The effects of race, social class, and tracking on opportunities to learn mathematics and science. Santa Monica, CA: The RAND Corporation.

Piaget, J. (1972). Mathematical structures and the operational structures of the intellect. In W. E. Lamon (Ed.). Learning and the nature of mathematics. Chicago: Science Research Associates.

Preckel, F., Götz, T., & Frenzel, A. (2010). Ability grouping of gifted students: Effects on academic self‐concept and boredom. British Journal of Educational Psychology, 80(3), 451−472.

Valsiner, J., & Van der Veer, R. (1993). The encoding of distance: The concept of the zone of proximal development and its interpretations. In R. R. Cocking & K. A. Renninger (Eds.), The development and meaning of psychological distance (pp. 35–62). Hillsdale, NJ: Erlbaum.

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, (Eds.). Cambridge, Massachusetts: Harvard University Press.

Vygotsky, L. S. (1986). Thought and language, revised edition. Cambridge, MA: The MIT Press.

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Wertsch, J. (1991). Voices of the mind: A sociocultural approach to mediated action. Cambridge, MA: Harvard University Press.

Wiliam, D., & Bartholomew, H. (2004). It’s not which school but which set you’re in that matters: The influence of ability grouping practices on student progress in mathematics. British Educational Research Journal, 30(2), 279−293.

Zazkis, R., & Koichu, B. (2015). A fictional dialogue on infinitude of primes: Introducing virtual duoethnography. Educational Studies in Mathematics, 88(2), pp. 163−181.

Badets, A., & Pesenti, M. (2010). Creating number semantics through finger movement perception. Cognition, 115(1), 46−53.

Bakhtin, M. M. (1981). The dialogic imagination: Four essays. M. Holquist, (Ed.). (C. Emerson & M. Holquist, Trans.). Texas: University of Texas Press, Texas.

Berteletti, I., & Booth, J. R. (2015). Finger Representation and Finger-Based Strategies in the Acquisition of Number Meaning and Arithmetic. Development of Mathematical Cognition: Neural Substrates and Genetic Influences, 2, 109.

Berteletti, I., & Booth, J. R. (2015). Perceiving fingers in single-digit arithmetic problems. Frontiers in psychology, 6, 226. doi:10.3389/fpsyg.2015.00226

Biton, Y., Fellus, O., & Hershkovitz, S. (2016, August 3-7). Border crossing: Bringing together pre-service teachers’ technological, pedagogical, and content knowledge through the use of digital textbooks in mathematics. The 2016 International Group for the Psychology of Mathematics Education: How to solve it? Szeged, Hungary.

Biton, Y., Fellus, O., Hershkovitz, S., & Hoch, M. (2015, April 15). Using the lens of complexity theory to examine a second chance at school mathematizing through Facebook. Paper presented at the 2015 annual meeting of the American Educational Research Association: Toward justice: Culture, language and heritage in educational research and praxis. Chicago, Illinois. Retrieved from the AERA Online Paper Repository.

Boaler, J. (1997). When even the winners are losers: Evaluating the experiences of ‘top set’ students, Journal of Curriculum Studies, 29(2), 165−182. http://dx.doi.org/10.1080/002202797184116

Boaler, J. (2005). The ‘Psychological Prison’ from which they never escaped: The role of ability grouping in reproducing social class inequalities, FORUM, 47(2−3), 135−144. http://dx.doi.org/10.2304/forum.2005.47.2.2

Boaler, J. (2013). Ability and mathematics: the mindset revolution that is reshaping education. Forum 55(1), pp. 143−152. Symposium Journals.

Boaler, J., Wiliam, D. & Brown, M. L. (2000). Students’ experiences of ability grouping—disaffection, polarisation and the construction of failure. British Educational Research Journal, 26(5), 631−648.

Bourdieu, P. (1986). The forms of capital. In: Richardson, J.G. (Ed.), Handbook of theory and research for the sociology of education. Greenwood Press, New York, pp. 241–258.

Davydov, V. V. (1986/2008). Problems of developmental instruction: A theoretical and experimental psychological study. New York: Nova Science Publishers.

Davydov, V. V. (1988). The concept of theoretical generalization and problems of educational psychology. Studies in Soviet Thought 36 (3), pp. 169−202.

Davydov, V. V. (1991). Psychological abilities of primary school children in learning mathematics. Soviet studies in mathematics education, Vol. 6. (J. Teller, Trans.) National Council of Teachers of Mathematics, Reston, Virginia.

Davydov, V. V. (1995). The influence of L. S. Vygotsky on education theory, research, and practice. Educational Researcher, 24(3), pp. 2−21.

Davydov, V. V. (1998a). The concept of developmental teaching. Journal of Russian and East European Psychology, 36(4), pp. 11−36.

Davydov, V. V. (1998b). What is formal learning activity? Journal of Russian and East European Psychology, 36(4), pp. 37−47.

Davydov, V. V. (1999). What is real learning activity? In M. Hedegaard & J. Lompscher (Eds.). Learning activity and development. (pp. 123−138). Aarhus: Aarhus University Press.

Davydov, V. V., & Andronov, V. P. (1981). Psychological conditions of the origination of ideal actions (Project Paper No. 81−2). Wisconsin University, Madison: Wisconsin Research and Development Center for Individualized Schooling.

Davydov, V. V., & Kudriavtsev, V. T. (1998). Developmental education: The theoretical foundations of continuity between the preschool and primary school stages. Russian Education & Society, 40(7), 37−64.

Degner, K. M. (2012). Demography as destiny: The role of parental connoisseurship and mathematics course taking patterns among high school students. PhD (Doctor of Philosophy) thesis, University of Iowa.

Devlin, K. (2005). The math instinct: Why you’re a mathematical Genius (Along with Lobsters, Birds, Cats and Dogs). Thunder Mouth Press, New York.

Ewers, R. M., & Didham, R. K. (2006). Confounding factors in the detection of species responses to habitat fragmentation. Biological Reviews, 81(01), 117−142.

Gerdes, P. (1988). On culture, geometrical thinking, and mathematics education. Educational Studies in Mathematics, 19(2), 137−162.

Gracia-Bafalluy, M., & Noël, M. P. (2008). Does finger training increase young children’s numerical performance?. Cortex, 44(4), 368−375.

Harré, R., and Gillett, G. (1994). The discursive mind. London: Sage.

Josselson, R., Lieblich, A., Sharabany, R., & Wiseman, H. (1997). Conversation as method: Analyzing the relational world of people who were raised communally. Thousand Oaks, CA: Sage Publications.

Kawai, M. (1963). On the newly-acquired behaviors of the natural troop of Japanese monkeys on Koshima Island. Primates, 4(1), 113−115.

Klein, E. (1966). A Comprehensive Etymological Dictionary of the English Language. (Vol. 1) Amsterdam: Elsevier Publishing Company.

Lave, J. (1997). The culture of acquisition and practice of understanding. In D. Kirshner & J. A. Whitson (Eds.). Situated cognition: Social, semiotic, and psychological perspectives. (pp. 63−82). Mahwah, NJ: Erlbaum.

Levitin, K., & Davydov, V. V. (1982). One is not born a personality: Profiles of Soviet education psychologists. Moscow: Progress Publishers.

Linchevski, L. & Kutscher, B. (1998). Tell me with whom you’re learning and I’ll tell you how much you’ve learned: Mixed-ability versus same-ability grouping in mathematics, Journal for Research in Mathematics Education, 29, 533−554. http://dx.doi.org/10.2307/749732

Lockhart, P. (2009). A mathematician’s lament: How school cheats us out of our most fascinating and imaginative art form. New York, NY: Belevue Literary Press.

Mason, R. & McFeetors, J. (2007). Student trajectories in high school mathematics: Issues of choice, support, and identity-making. Canadian Journal of Science, Mathematics, and Technology Education, 7(4), 291−316.

McCrink, K., & Wynn, K. (2004). Large-number addition and subtraction by 9-month-old infants. Psychological Science, 15(11), 776−781.

McFeetors, J., & Mason, R. (2005). Voice and success in non-academic mathematics courses: (Re)Forming identity. For the Learning of Mathematics, 25(3), 16−23.

Ministry of Education and Training (Ontario). (2005). The Ontario curriculum grades 9 and 10: Mathematics. Toronto: Queen’s Printer.

Muijs, D., & Dunne, M. (2010). Setting by ability–or is it? A quantitative study of determinants of set placement in English secondary schools. Educational Research, 52(4), 391−407.

Newman, S. D. (2016). Does finger sense predict addition performance? Cognitive processing, 17(2), 139−146.

Noël, M. P. (2008). Finger gnosia: A predictor of numerical abilities in children? Child Neuropsychology, 11(5), 413−430.

Norris, J., Sawyer, R. D., & Lund, D. (Eds.). (2012). Duoethnography: Dialogic methods for social, health, and educational research, (Vol. 7). Walnut Creek, CA: Left Coast Press.

Oakes, J. (1990). Multiplying inequalities: The effects of race, social class, and tracking on opportunities to learn mathematics and science. Santa Monica, CA: The RAND Corporation.

Piaget, J. (1972). Mathematical structures and the operational structures of the intellect. In W. E. Lamon (Ed.). Learning and the nature of mathematics. Chicago: Science Research Associates.

Preckel, F., Götz, T., & Frenzel, A. (2010). Ability grouping of gifted students: Effects on academic self‐concept and boredom. British Journal of Educational Psychology, 80(3), 451−472.

Valsiner, J., & Van der Veer, R. (1993). The encoding of distance: The concept of the zone of proximal development and its interpretations. In R. R. Cocking & K. A. Renninger (Eds.), The development and meaning of psychological distance (pp. 35–62). Hillsdale, NJ: Erlbaum.

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, (Eds.). Cambridge, Massachusetts: Harvard University Press.

Vygotsky, L. S. (1986). Thought and language, revised edition. Cambridge, MA: The MIT Press.

Vygotsky, L. S. (1987a). The collected works of L. S. Vygotsky. Vol. 1: Problems of general psychology. R. W. Reiber & A. S. Carton (Eds.). (N. Minick, Trans.). New York, NY: Plenum Press.

Vygotsky, L. S. (1987b). The collected works of L. S. Vygotsky. Vol. 3: Problems of general psychology. R. W. Reiber & A. S. Carton (Eds.). (N. Minick, Trans.). New York, NY: Plenum Press.

Wenger, E. (1998). Communities of practice: learning, meaning, and identity. Cambridge University Press.

Wertsch, J. (1991). Voices of the mind: A sociocultural approach to mediated action. Cambridge, MA: Harvard University Press.

Wiliam, D., & Bartholomew, H. (2004). It’s not which school but which set you’re in that matters: The influence of ability grouping practices on student progress in mathematics. British Educational Research Journal, 30(2), 279−293.

Zazkis, R., & Koichu, B. (2015). A fictional dialogue on infinitude of primes: Introducing virtual duoethnography. Educational Studies in Mathematics, 88(2), pp. 163−181.