Abstract
Mathematics education practitioners and researchers have long debated best pedagogical practices for introducing to students new concepts. We report on results from analyzing the behaviors of 25 Grade 4–6 students who participated individually in tutorial activities designed to compare the pedagogical effect of manipulating objects that are either generic (non-representational, not signifying specific contexts, e.g., a circle) or situated (representational, signifying specific contexts, e.g., a hot-air balloon). The situated objects gave rise to richer stories than the generic objects, presumably because the students could bring to bear their everyday knowledge of these objects’ properties, scenarios, and typical behaviors. However, in so doing, the students treated the objects’ only as framed by those particular stories rather than considering other possible interpretations. Consequently, these students did not experience key struggles and insights that the designers believe to be pivotal to their conceptual development in this particular content (proportionality). Drawing on enactivist theory, we analyze several case studies qualitatively to explicate how rich situativity filters out critical opportunities for conceptually pivotal sensorimotor engagement . We caution that designers and teachers should be aware of the double-edged sword of rich situativity: Familiar objects are perhaps more engaging but can also limit the scope of learning. We advocate for our instructional methodology of entering mathematical concepts through the action level.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abrahamson, D. (2004). Embodied spatial articulation: A gesture perspective on student negotiation between kinesthetic schemas and epistemic forms in learning mathematics. In D. E. McDougall & J. A. Ross (Eds.), Proceedings of the Twenty Sixth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 791–797). Toronto, Ontario: Preney.
Abrahamson, D. (2006). What’s a situation in situated cognition? (Symposium). In S. Barab, K. Hay, & D. Hickey (Eds.), Proceedings of the 7th International Conference of the Learning Sciences (Vol. 2, pp. 1015–1021). Bloomington, IN: ICLS.
Abrahamson, D. (2009). Embodied design: Constructing means for constructing meaning. Educational Studies in Mathematics, 70(1), 27–47.
Abrahamson, D. (2014). Building educational activities for understanding: An elaboration on the embodied-design framework and its epistemic grounds. International Journal of Child-Computer Interaction, 2(1), 1–16.
Abrahamson, D. (2015). The monster in the machine, or why educational technology needs embodied design. In V. R. Lee (Ed.), Learning technologies and the body: Integration and implementation (pp. 21–38). New York: Routledge.
Abrahamson, D., & Bakker, A. (2016). Making sense of movement in embodied design for mathematics learning. In N. Newcombe & S. Weisberg (Eds.), Embodied cognition and STEM learning [Special issue]. Cognitive Research: Principles and Implications, 1(1), 1–13. https://doi.org/10.1186/s41235-016-0034-3.
Abrahamson, D., & Kapur, M. (2018). Reinventing discovery learning: A field-wide research program. In D. Abrahamson & M. Kapur (Eds.), Practicing discovery-based learning: Evaluating new horizons [Special issue]. Instructional Science, 46(1), 1–10.
Abrahamson, D., Lee, R. G., Negrete, A. G., & Gutiérrez, J. F. (2014). Coordinating visualizations of polysemous action: Values added for grounding proportion. ZDM Mathematics Education, 46(1), 79–93.
Abrahamson, D., & Lindgren, R. (2014). Embodiment and embodied design. In R. K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (2nd ed.). Cambridge: Cambridge University Press.
Abrahamson, D., & Sánchez-García, R. (2016). Learning is moving in new ways: The ecological dynamics of mathematics education. Journal of the Learning Sciences, 25(2), 203–239.
Abrahamson, D., Shayan, S., Bakker, A., & van der Schaaf, M. (2016). Eye-tracking Piaget: Capturing the emergence of attentional anchors in the coordination of proportional motor action. Human Development, 58(4–5), 218–244.
Allen, J. W. P., & Bickhard, M. H. (2013). Stepping off the pendulum: Why only an action-based approach can transcend the nativist-empiricist debate. Cognitive Development, 28(2), 96–133.
Araújo, D., & Davids, K. (2004). Embodied cognition and emergent decision-making in dynamical movement systems. Junctures: The Journal for Thematic Dialogue, 2, 45–57.
Arsalidou, M., & Pascual-Leone, J. (2016). Constructivist developmental theory is needed in developmental neuroscience. Npj Science of Learning, 1, 16016.
Barab, S., Zuiker, S., Warren, S., Hickey, D., Ingram-Goble, A., Kwon, E. J., et al. (2007). Situationally embodied curriculum. Science Education, 91, 750–782.
Barsalou, L. W. (2010). Grounded cognition: Past, present, and future. Topics in Cognitive Science, 2(4), 716–724.
Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. D. English, M. G. Bartolini Bussi, G. A. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education, 2nd revised edition (pp. 720–749). Mahwah, NJ: Lawrence Erlbaum Associates.
Bruner, J. (1986). Actual minds, possible worlds. Cambridge: Harvard University Press.
Burton, L. (1999). The implications of a narrative approach to the learning of mathematics. In L. Burton (Ed.), Learning mathematics: From hierarchies to networks (pp. 21–35). London: Falmer Press.
Campbell, S. R. (2003). Reconnecting mind and world: Enacting a (new) way of life. In S. J. Lamon, W. A. Parker, & S. K. Houston (Eds.), Mathematical modeling: A way of life (pp. 245–256). Chichester, UK: Horwood Publishing.
Chemero, A. (2009). Radical embodied cognitive science. Cambridge, MA: MIT Press.
Clark, A. (2013). Whatever next? Predictive brains, situated agents, and the future of cognitive science. Behavioral and Brain Sciences, 36, 181–253.
Day, S. B., Motz, B. A., & Goldstone, R. L. (2015). The cognitive costs of context: The effects of concreteness and immersiveness in instructional examples. Frontiers in Psychology, 6.
de Freitas, E., & Sinclair, N. (2012). Diagram, gesture, agency: Theorizing embodiment in the mathematics classroom. Educational Studies in Mathematics, 80(1–2), 133–152.
Duijzer, A. C. G., Shayan, S., Bakker, A., Van der Schaaf, M. F., & Abrahamson, D. (2017). Touchscreen tablets: Coordinating action and perception for mathematical cognition. Frontiers in Psychology, 8(144).
Fillmore, C. J. (1968). The case for case. In E. Bach & R. Harms (Eds.), Universals in linguistic theory (pp. 1–88). New York, NY: Holt Rinehart and Winston.
Fillmore, C. J., & Atkins, B. T. (1992). Toward a frame-based lexicon: The semantics of RISK and its neighbors. In A. Lehrer & E. Kittay (Eds.), Frames, fields, and contrasts (pp. 75–102). Hillsdale, NJ: LEA.
Fuson, K. C., & Abrahamson, D. (2005). Understanding ratio and proportion as an example of the apprehending zone and conceptual-phase problem-solving models. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 213–234). New York: Psychology Press.
Gibson, J. J. (1977). The theory of affordances. In R. Shaw & J. Bransford (Eds.), Perceiving, acting and knowing: Toward an ecological psychology (pp. 67–82). Hillsdale, NJ: Lawrence Erlbaum Associates.
Goldstone, R. L., Landy, D., & Son, J. Y. (2008). A well-grounded education. In M. DeVega, A. M. Glenberg, & A. C. Graesser (Eds.), Symbols and embodiment (pp. 327–355). Oxford, UK: Oxford University Press.
Goldstone, R. L., & Sakamoto, Y. (2003). The transfer of abstract principles governing complex adaptive systems. Cognitive Psychology, 46, 414–466.
Goldstone, R. L., & Son, J. Y. (2005). The transfer of scientific principles using concrete and idealized simulations. Journal of the Learning Sciences, 14, 69–110.
Gravemeijer, K. P. E. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177.
Gray, E., & Tall, D. (1994). Duality, ambiguity, and flexibility: A “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116–140.
Greeno, J. G. (1994). Gibson’s affordances. Psychological Review, 101(2), 336–342.
Healy, L., & Sinclair, N. (2007). If this is our mathematics, what are our stories? International Journal of Computers for Mathematical Learning, 12(1), 3–21.
Howison, M., Trninic, D., Reinholz, D., & Abrahamson, D. (2011). The mathematical imagery trainer: From embodied interaction to conceptual learning. In G. Fitzpatrick, C. Gutwin, B. Begole, W. A. Kellogg, & D. Tan (Eds.), Proceedings of the annual meeting of The Association for Computer Machinery Special Interest Group on Computer Human Interaction: “Human Factors in Computing Systems” (CHI 2011) (Vol. “Full Papers,” pp. 1989–1998). New York: ACM Press.
Hutto, D. D., Kirchhoff, M. D., & Abrahamson, D. (2015). The enactive roots of STEM: Rethinking educational design in mathematics. In P. Chandler & A. Tricot (Eds.), Human movement, physical and mental health, and learning [Special issue]. Educational Psychology Review, 27(3), 371–389.
Hutto, D. D., & Sánchez-García, R. (2015). Choking RECtified: Embodied expertise beyond Dreyfus. Phenomenology and the Cognitive Sciences, 14(2), 309–331.
Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2008). The advantage of abstract examples in learning math. Science, 320, 454–455.
Kelso, J. A. S., & Engstrøm, D. A. (2006). The complementary nature. Cambridge, MA: M.I.T. Press.
Kim, M., Roth, W.-M., & Thom, J. S. (2011). Children’s gestures and the embodied knowledge of geometry. International Journal of Science and Mathematics Education, 9(1), 207–238.
Kirsh, D. (2013). Embodied cognition and the magical future of interaction design. In P. Marshall, A. N. Antle, E. V.D. Hoven, & Y. Rogers (Eds.), The theory and practice of embodied interaction in HCI and interaction design [Special issue]. ACM Transactions on Human–Computer Interaction, vol. 20, no. 1, 3, pp. 1–30.
Landy, D., & Goldstone, R. L. (2007). How abstract is symbolic thought? Journal of Experimental Psychology. Learning, Memory, and Cognition, 33(4), 720–733.
Lindgren, R., & Johnson-Glenberg, M. (2013). Emboldened by embodiment: Six precepts for research on embodied learning and mixed reality. Educational Researcher, 42, 445–452.
Mariotti, M. A. (2009). Artifacts and signs after a Vygotskian perspective: The role of the teacher. ZDM—The international Journal on Mathematics Education, 41, 427–440.
McNeil, N. M., Uttal, D. H., Jarvin, L., & Sternberg, R. J. (2009). Should you show me the money? Concrete objects both hurt and help performance on mathematics problems. Learning and Instruction, 19, 171–184.
Nathan, M. J. (2012). Rethinking formalisms in formal education. Educational Psychologist, 47(2), 125–148.
Negrete, A. G., Lee, R. G., & Abrahamson, D. (2013). Facilitating discovery learning in the tablet era: rethinking activity sequences vis-à-vis digital practices. In M. Martinez & A. Castro Superfine (Eds.), “Broadening Perspectives on Mathematics Thinking and Learning”—Proceedings of the 35th Annual Meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 35) (Vol. 10: “Technology” p. 1205). Chicago, IL: University of Illinois at Chicago.
Nemirovsky, R. (2003). Three conjectures concerning the relationship between body activity and understanding mathematics. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th Annual Meeting of the Int. Group for the Psychology of Mathematics Education (Vol. 1, pp. 105–109). Columbus, OH: Eric Clearinghouse for Science, Mathematics, and Environmental Education.
Nemirovsky, R., Kelton, M. L., & Rhodehamel, B. (2013). Playing mathematical instruments: Emerging perceptuomotor integration with an interactive mathematics exhibit. Journal for Research in Mathematics Education, 44(2), 372–415.
Newman, D., Griffin, P., & Cole, M. (1989). The construction zone: Working for cognitive change in school. New York: Cambridge University Press.
Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings: Learning cultures and computers. Dordrecht: Kluwer.
Ottmar, E., & Landy, D. (2017). Concreteness fading of algebraic instruction: Effects on learning. Journal of the Learning Sciences, 26(1), 51–78.
Palatnik, A., & Abrahamson, D. (2017). Taking measures to coordinate movements: Unitizing emerges as a method of building event structures for enacting proportion. In E. Galindo & J. Newton (Eds.), “Synergy at the crossroads”—Proceedings of the 39th Annual Conference of the North-American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 13 [Theory and research methods], pp. 1439–1442). Indianapolis, IN: Hoosier Association of Mathematics Teacher Educators.
Palatnik, A., & Abrahamson, D. (under review). Rhythmic movement as a tacit enactment goal mobilizing the emergence of mathematical structures. Educational Studies in Mathematics.
Piaget, J. (1968). Genetic epistemology (E. Duckworth, Trans.). New York: Columbia University Press.
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.
Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education. Child Development Perspectives, 3(3), 145–150.
Sfard, A. (2002). The interplay of intimations and implementations: Generating new discourse with new symbolic tools. Journal of the Learning Sciences, 11(2, 3), 319–357.
Sloutsky, V. M., Kaminski, J. A., & Heckler, A. F. (2005). The advantage of simple symbols for learning and transfer. Psychonomic Bulletin and Review, 12(3), 508–513.
Steffe, L. P., & Kieren, T. (1994). Radical constructivism and mathematics education. Journal for Research in Mathematics Education, 25(6), 711–733.
Stokes, D. E. (1997). Pasteur’s quadrant: Basic science and technological innovation. DC: Brookings.
Tahta, D. (1998). Counting counts. Mathematics Teaching, 163, 4–11.
Thompson, P. W. (2013). In the absence of meaning …. In K. Leatham (Ed.), Vital directions for mathematics education research (pp. 57–94). New York: Springer.
Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18, 37–54.
Varela, F. J. (1999). Ethical know-how: Action, wisdom, and cognition. Stanford, CA: Stanford University Press.
Varela, F. J., Thompson, E., & Rosch, E. (1991). The embodied mind. Cambridge, MA: M.I.T. Press.
Vérillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10(1), 77–101.
von Glasersfeld, E. (1983). Learning as constructive activity. In J. C. Bergeron & N. Herscovics (Eds.), Proceedings of the 5th Annual Meeting of the North American Group for the Psychology of Mathematics Education (Vol. 1, pp. 41–69). Montreal: PME-NA.
Vygotsky, L. S. (1997). Educational psychology (R. H. Silverman, Trans.). Boca Raton, FL: CRC Press LLC (Work originally published in 1926).
Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematics education. In I. Harel & S. Papert (Eds.), Constructionism (pp. 193–204). Norwood, NJ: Ablex Publishing Corporation.
Worsley, M., Abrahamson, D., Blikstein, P., Bumbacher, E., Grover, S., Schneider, B., et al. (2016). Workshop: Situating multimodal learning analytics. In C.-K. Looi, J. L. Polman, U. Cress, & P. Reimann (Eds.), “Transforming learning, empowering learners,” Proceedings of the International Conference of the Learning Sciences (ICLS 2016) (Vol. 2, pp. 1346–1349). Singapore: International Society of the Learning Sciences.
Acknowledgements
The research reported herein as well the writing of this chapter were supported by an REU (Rosen) under NSF IIS Cyberlearning EXP award 1321042.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Rosen, D., Palatnik, A., Abrahamson, D. (2018). A Better Story: An Embodied-Design Argument for Generic Manipulatives. In: Calder, N., Larkin, K., Sinclair, N. (eds) Using Mobile Technologies in the Teaching and Learning of Mathematics. Mathematics Education in the Digital Era, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-90179-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-90179-4_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-90178-7
Online ISBN: 978-3-319-90179-4
eBook Packages: EducationEducation (R0)