Skip to main content
SpringerLink
Log in

Discussion I of Part II

Representing and Meaning-Making: The Transformation of Transformation

  • Chapter
  • First Online:
Book cover Transformation - A Fundamental Idea of Mathematics Education

  • 1022 Accesses

Abstract

This commentary simultaneously offers a broad and a narrow perspective. It is narrow in that it does not attempt to synthesize or digest the chapters in this section. It can be called broad as it attempts to sketch some salient features of the future development of and learning geometry, of the transformation of transformation. Four such strands of reasoning are discussed: the paramount significance of meaning-making, the role of artefacts as socially and culturally embedded, embodiment and enactment, and, finally, emotions, meaning-making, and triangulation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In what follows, I will not discuss the specific details of embodied or enactment approaches. First, because these paradigms are still very much „under construction” and exhibit a great diversity; second, because space would not allow to go into the details. I will also not discuss the brain-focused approaches that sometimes label themselves as approaches to embodied cognition (for a short overview, see, e.g., Di Paolo and De Jaegher 2012). Interesting and fascinating as these approaches may be, they reduce the role of the body to excitatory patterns in the brain that since recently can be traced with brain-scanners. For an overview and discussion of cognitive neuroscience see Campbell (2010). Dehaene, e.g., has presented an interesting tripartite model of the development of number (1992), but his numerous attempts to show how the components of this model are processed by and localized in the brain have not been very conclusive (see, e.g., Dehaene 1997).

  2. 2.

    This issue would deserve an extended discussion because the relation of gestures to language is still very much in need of clarifications. Gestures are often seen as developmental precursors to language (see, e.g., Tomasello 2008). At the same time, they seem so important because gestures potentially express what cannot be expressed through language. For an interesting account of gestural language see Sacks (1989). Goldin-Meadow (2003) has discussed intensely the gesture-speech mismatches, and Sinclair (2010) has elaborated the idea of overt and covert forms of knowing in mathematics education, gestures indicating covert knowing.

  3. 3.

    For reasons of space, I will not present of Vygotsky’s ideas in more detail, also because this has been done extensively elsewhere over the past years (see, e.g., van der Veer and Valsiner 1991, 1994; Daniels et al. 2007).

  4. 4.

    A term coined already in a 1973 volume by Stone and others (Stone et al. 1973; see also Dornes 1993).

  5. 5.

    Already Hegel expressed this specific feature of triangulation in his “Quadratum est lex naturae, triangulum mentis” (Hegel 1801, p. 533).

  6. 6.

    Rachman is a leading figure in the Financial Times. Although his intention is to present an account of today’s global situation, it must be doubted if he really can leave his fixation on the investment and money-making perspective behind.

References

  • Bandura, A. (1997). Self-efficacy: The exercise of control. New York: Freeman.

    Google Scholar 

  • Bandura, A. (2001). Social cognitive theory: An agentic perspective. Annual Review of Psychology, 52, 1–26.

    Google Scholar 

  • Baron-Cohen, S., Bolton, P., Wheelwright, S., Scahill, V., Short, L., Mead, G., & Smith, A. (1998). Autism occurs more often in families of physicists, engineers, and mathematicians. Autism, 2, 296–301.

    Google Scholar 

  • Baron-Cohen, S., Wheelwright, S., Burtenshaw, A., & Hobson, E. (2007). Mathematical talent is linked to autism. Human Nature, 18, 125–131.

    Google Scholar 

  • Bullowa, M. (Ed.) (1979). Before speech: The beginnings of human communication. Cambridge: Cambridge University Press.

    Google Scholar 

  • Campbell, S. R. (2010). Embodied minds and dancing brains: New opportunities for research in mathematics education. In B. Sriraman & L. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 309–331). Heidelberg: Springer

    Google Scholar 

  • Cole, M. (1996). Cultural psychology. A once and future discipline. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Daniels, H., Cole, M., & Wertsch, J. V. (2007). The Cambridge companion to Vygotsky. New York: Cambridge University Press.

    Google Scholar 

  • Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44, 1–40.

    Google Scholar 

  • Dehaene, S. (1997). The number sense. How the mind creates mathematics. New York: Oxford University Press.

    Google Scholar 

  • Di Paolo, E., & De Jaegher, H. (2012). The interactive brain hypothesis. Frontiers in Human Neuroscience, 6, Article 163 (doi: 10.3389/fnhum.2012.00163)

    Google Scholar 

  • Dornes, M. (1993). Der kompetente Säugling. [The competent infant]. Frankfurt/Main. Fischer.

    Google Scholar 

  • Dornes, M. (2005). Die emotionalen Ursprünge des Denkens [The emotional origins of thinking]. West-End. Neue Zeitschrift für Sozialforschung, 2(1), 3–48

    Google Scholar 

  • Dyson, G. (2010). Kayaks vs. canoes http://www.edge.org/q2010/q10_2.html

  • Elias, N. (1994). The Civilizing process. Oxford: Blackwell (original 1934)

    Google Scholar 

  • Engeström, Y., & Escalante, V. (1995). Mundane tool or object of affection? The rise and fall of the postal buddy. In B. Nardi (Ed.), Context and consciousness: activity theory and human-computer interaction (pp. 325–373). Cambridge: The MIT Press.

    Google Scholar 

  • Fonagy, P,. Gergely, G., Jurist, E., & Target, M. (2002). Affect regulation, mentalization, and the development of the self. New York: Other Press.

    Google Scholar 

  • Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School engagement: Potential of the concept, state of the evidence. Review of Educational Research, 74(1), 59–109.

    Google Scholar 

  • Goldin, G. A., Epstein, Y. M., Schorr, R. Y., & Warner, L. B. (2011). Beliefs and engagement structures: Behind the affective dimension of mathematical learning. ZDM-The International Journal on Mathematics Education, 43, 547–560.

    Google Scholar 

  • Goldin-Meadow, S. (2003). Hearing gesture-How our hands help us think. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Hegel, G. W. F. (1801). „Habilitationsthesen“. In: Georg Willhelm Friedrich Hegel, Werke in Zwanzig Bänden. Band 2. Jenaer Schriften 1801-1807 (S. 533). Frankfurt am Main: Suhrkamp.

    Google Scholar 

  • Hobson, R. P. (1993). Autism and the development of mind. Hillsdale: Lawrence Erlbaum.

    Google Scholar 

  • Hobson, R. P. (2002). The cradle of thought. Exploring the origins of thinking. London: Macmillan.

    Google Scholar 

  • Holodynski, M. (2006). Die Entwicklung der Emotionen. Berlin: Springer.

    Google Scholar 

  • Honneth, A. (1995). The Struggle for recognition. The moral grammar of social conflicts. 1995. Cambridge: Polity Press.

    Google Scholar 

  • Hutchins, E. (2010). Enaction, imagination, and insight. In J. Stewart, O. Gapenne, & E. A. Di Paolo (Eds.), Enaction. Toward a new paradigm for cognitive science (pp. 425-450). Cambridge, MA: MIT Press.

    Google Scholar 

  • Latour, B. (1993). Aramis, or the love of technology. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Lave, J. (1988). Cognition in practice. Mind mathematics and culture in everyday life. Cambridge: Cambridge University Press.

    Google Scholar 

  • Lave, J., & Wenger, E. (1991). Situated learning - Legitimate peripheral participation. Cambridge: Cambridge University Press.

    Google Scholar 

  • Leder, G. C., Pehkonen, E., & Törner, G. (Eds.) (2002). Beliefs: A hidden variable in mathematics education? Dordrecht: Kluwer.

    Google Scholar 

  • Leont’ev, A. N. (1981). Problems of the development of the mind. Moscow: Progress Publishers.

    Google Scholar 

  • Leont’ev, A. N. (1978). Activity, consciousness, and personality. Englewood Cliffs, N.J.: Prentice-Hall

    Google Scholar 

  • Maschietto, M., & Bartolini Bussi, M. G. (2009). Working with artefacts: gestures, drawings and speech in the construction of the mathematical meaning of the visual pyramid. Educational Studies in Mathematics, 70, 143–157.

    Google Scholar 

  • Maass, J., & Schlöglmann, W. (Eds.) (2009). Beliefs and attitudes in mathematics education: New research results. Rotterdam: Sense.

    Google Scholar 

  • Meltzoff, A. N. (2002). Elements of a developmental theory of imitation. In A. N. Meltzoff &W. Prinz (Eds.), The imitative mind (pp. 19-41). Cambridge: Cambridge University Press.

    Google Scholar 

  • Meltzoff, A. N. (2007). ‘Like me’: A foundation for social cognition. Developmental Science, 10(1), 126–134.

    Google Scholar 

  • Meltzoff, A. N., Kuhl, P. K., Movellan, J., & Sejnowski, T. J. (2009). Foundations for a new science of learning. Science, 325, 284–288.

    Google Scholar 

  • Meltzoff, A. N., & Moore, M. K. (1977). Imitation of facial and manual gestures by human neonates. Science, 198, 75–78.

    Google Scholar 

  • Ogden, C. K., & Richards, I. A. (1923). The meaning of meaning: A study of the influence of language upon thought and of the science of symbolism. London: Routledge & Kegan Paul.

    Google Scholar 

  • Otte, M. F. (2011). Evolution, learning, and semiotics from a Peircean point of view. Educational Studies in Mathematics, 77, 313–329.

    Google Scholar 

  • Papousek, H., & Papousek, M. (1974). Mirror image and self-recognition in young infants. Developmental Psychology, 7, 149–157.

    Google Scholar 

  • Papousek, H., & Papousek, M. (1977). Cognitive aspects of preverbal social interaction between human infants and adults. In R. Porter & M. O’Connor (Eds.), Ciba Foundation Symposium 33: Parent-infant interaction (pp. 241–269). New York: Wiley

    Google Scholar 

  • Papousek, H., & Papousek, M. (1981). How human is the human newborn, and what else is to be done. In K. Bloom (Ed.), Prospective issues in infancy research (pp. 137–155). Hillsdale: Erlbaum.

    Google Scholar 

  • Rachman, G. (2011). Zero-sum future-American power in an age of anxiety. New York: Simon & Schuster.

    Google Scholar 

  • Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic‐cultural approach to students’ types of generalization. Mathematical Thinking and Learning 5(1), 37–70

    Google Scholar 

  • Radford, L., Bardini, C., Sabena, C., Diallo, P., & Simbagoye, A. (2005). On embodiment, artefacts, and signs: A semiotic-cultural perspective on mathematical thinking. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th International Group for the Psychology of Mathematics Education (vol. 4, pp. 113–122). Melbourne, Australia

    Google Scholar 

  • Robutti, O. (2006). Motion, technology, gesture in interpreting graphs. International Journal of Computer Algebra in Mathematics Education, 13, 117–126.

    Google Scholar 

  • Roth, W.-M. (2001). Gestures: Their role in teaching and learning. Review of Educational Research, 71(3), 365–392.

    Google Scholar 

  • Roth, W.-M., & Welzel, M. (2001). From activity to gestures and scientific language. Journal of Research in Science Teaching, 38(1), 103–136.

    Google Scholar 

  • Sabena, C. (2007). Body and signs: A multimodal semiotic approach to teaching-learning processes in early calculus. Unpublished Doctoral Dissertation, Università degli Studi di Torino, Torino

    Google Scholar 

  • Sacks, O. (1989). Seeing voices - A journey into the world of the deaf. Berkeley: University of California Press.

    Google Scholar 

  • Sinclair, N. (2010). Knowing more that we can tell. In B. Sriraman & L. English (Eds.), Theories of mathematical education: Seeking new frontiers (pp. 595–612). Heidelberg: Springer

    Google Scholar 

  • Sriraman, B., & English, L. (2010) (Eds.), Theories of mathematical education: Seeking new frontiers. Heidelberg: Springer

    Google Scholar 

  • Steinbring, H. (2005). Do mathematical symbols serve to describe or to construct reality? In M. Hoffmann, J. Lenhard, & F. Seeger (Eds.), Activity and sign (pp. 91–104). New York: Springer.

    Google Scholar 

  • Stern, D. (1971). A microanalysis of mother-infant interaction. Journal of the American Academy of Child Psychiatry, 10, 501–517

    Google Scholar 

  • Stern, D. (1985). The interpersonal world of the infant: A view from psychoanalysis and developmental psychology. New York: Basic Books.

    Google Scholar 

  • Stewart, J. (2010). Foundational issues in enaction as a paradigm for cognitive science: From the origin of life to consciousness and writing. In J. Stewart, O. Gapenne, & E. A. Di Paolo (Eds.), Enaction. Toward a New Paradigm for Cognitive Science (pp. 1-31). Cambridge: MIT Press.

    Google Scholar 

  • Stone, L. J., Smith, H. T., & Murphy, L. B. (Eds.) (1973). The Competent Infant. New York: Basic Books.

    Google Scholar 

  • Thom, A. (1973), Modern mathematics: Does it exist? In A. G. Howson (Ed.), Developments in mathematical education. Proceedings of the 2nd International Congress on Mathematical Education (pp.  194–210). Cambridge: Cambridge University Press

    Google Scholar 

  • Tomasello, M. (1999). The cultural origins of human cognition. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Tomasello, M. (2005). Constructing a language: A usage-based theory of language acquisition. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Tomasello, M. (2008). Origins of human communication. Cambridge, MA: MIT Press.

    Google Scholar 

  • Tomasello, M. (2006). Why don’t apes point? In N. J. Enfield & S. C. Levinson (Eds.), Roots of human sociality: Culture, cognition and interaction (pp. 506-524). Oxford: Berg.

    Google Scholar 

  • Trevarthen, C. (1979). Communication and cooperation in early infancy: A description of primary intersubjectivity. In M. Bullowa (Ed.), Before Speech: The Beginnings of human communication (pp. 321-347). Cambridge: Cambridge University Press.

    Google Scholar 

  • Trevarthen, C. (1980). The foundations of intersubjectivity: Development of interpersonal and cooperative understanding in infants. In D. Olson (Ed.), The social foundations of language and thought: Essays in honor of J. S. Bruner (pp. 316–342). New York: Norton.

    Google Scholar 

  • Trevarthen, C. (1994). Infant semiosis. In W. Nöth (Ed.), Origins of semiosis: Sign evolution in nature and culture (pp. 219–252). Berlin: Mouton de Gruyter.

    Google Scholar 

  • Trevarthen, C., & Hubley, P. (1978). Secondary intersubjectivity: Confidence, confiding and acts of meaningin the first year. In A. Lock (Ed.), Action, gesture, and symbol: The emergence of language (pp. 183–230). London: Academic.

    Google Scholar 

  • Van der Veer, R., & Valsiner, J. (1991). Understanding Vygotsky: A quest for synthesis. Oxford: Blackwell.

    Google Scholar 

  • Van der Veer, R., & Valsiner, J. (Eds.) (1994). The Vygotsky reader. Oxford: Blackwell.

    Google Scholar 

  • Varela, F. J., Thompson, E., & Rosch, E. (1991). The embodied mind: Cognitive science and human experience. Cambridge, MA: MIT.

    Google Scholar 

  • Von Holst, E., & Mittelstaedt, H. (1950).  Das Reafferenzprinzip. Wechselwirkungen zwischen Zentralnervensystem und Peripherie. Die Naturwissenschaften, 37, Heft 20, 464–476.

    Google Scholar 

  • Von Uexküll, J. (2010). A foray into the worlds of animals and humans; with a theory of meaning. Minneapolis: University of Minnesota Press (original work published 1934).

    Google Scholar 

  • Vygotsky, L.S. (1994). The problem of the environment. In van der Veer, R. & Valsiner, J. (Eds.), The Vygotsky reader (pp. 338–354). Oxford: Blackwell. (Original work 1934).

    Google Scholar 

  • Vygotsky, L. S. (1997). The history of the development of higher mental functions. In R. W. Rieber (Ed.), The collected works of L. S. Vygotsky, Vol. 4 New York: Plenum. (Original work 1931).

    Google Scholar 

  • Vygotsky, L. S. (1999). Tool and sign in the development of the child. In R. W. Rieber (Ed.), The collected works of L. S. Vygotsky, Vol. 6 (pp. 1–68). New York: Kluwer. (Original work 1930).

    Google Scholar 

  • Wing, L. (1996). The autistic spectrum: A guide for parents and professionals. London: Constable.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. the Institut für Didaktik der Mathematik, Universität Bielefeld, Steinhagen, Germany

    Falk Seeger (Retired Senior Lecturer)

Authors
  1. Falk Seeger
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. IEIM - Institut für Mathematik, Universität Paderborn, Paderborn, Germany

    Sebastian Rezat

  2. Fakultät für Mathematik - IDM, Universität Bielefeld, Bielefeld, Germany

    Mathias Hattermann

  3. Fakultät für Mathematik - IDM, Universität Bielefeld, Bielefeld, Germany

    Andrea Peter-Koop

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Seeger, F. (2014). Discussion I of Part II. In: Rezat, S., Hattermann, M., Peter-Koop, A. (eds) Transformation - A Fundamental Idea of Mathematics Education. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3489-4_19

Download citation

Publish with us

Policies and ethics

Search

Navigation