Skip to main content
SpringerLink
Log in

Shaping Perception: Designing for Participatory Facilitation of Collaborative Geometry

  • Published:
Digital Experiences in Mathematics Education Aims and scope Submit manuscript

Abstract

Geometris is a technologically enabled, body-scale, educational game, in which collaborating players recreate target polygons, prompted on a digital screen, by activating a set of sensors–vertices on a large physical mat interface. We report on an evaluation study that draws on theoretical frameworks from ecological dynamics, genetic epistemology, and socio-cultural semiotics. Micro-analysis of three adult–child groups at play implicates two design features supporting mediated development of geometry skills: (1) spatial distribution across two displays – the screen and the mat – poses cross-display figural mapping as a tactical problem whose perceptual solution constitutes the game’s learning objective; (2) a multi-sensor input interface – the mat’s ‘vertices’ – enables flexible divisions of group labor for scaffolding solution enactment. We put forth the construct of participatory facilitation – an emergent interaction pattern in groups with inter-personal differences in content-domain knowledge and sensorimotor co-ordination. We tentatively generalize principles for designing informal educational activity architecture that create opportunities for relative experts to enculturate content learning via participatory facilitation.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. Each Geometris level contains eight target shapes, each of which must be completed by simultaneously activating its vertices, regardless of order. Geometris also includes an untimed Practice level. See Durán-López et al. (2017) for details on the hardware and the software.

  2. Whereas perceptual affiliation of sensory stimuli is a Gestalt perception, highlighting a Gestalt in the context of mathematical activity marks it for learners as a culturally significant referent.

  3. Very young children occasionally try to interact with the projection screen, perhaps based on experience with touchscreens.

  4. We observed one teenaged player make a hexagon by himself using his head, knees, feet, and elbows. Such contortion is atypical within Geometris play.

  5. We describe these players as relative experts, because they, along with their partners, were equally new to the game. Nevertheless, their perceived expertise relative to their partners seemed to sanction their informal teaching behavior through facilitation of play, which is our phenomenon of interest.

  6. Because Geometris’ design includes figural mapping as part of the game strategy, performance-oriented facilitation could still expose relative motives to game states through which they could learn this ideal form, if incidentally.

  7. All names are pseudonyms.

  8. In repositioning, Mike occasionally activated and held unnecessary vertices.

  9. While this decrease could result from fatigue, there is no demonstrable decrease across each group’s first round of play.

  10. We can imagine other types of competence-oriented facilitation – such as increasing the distance between players, modeling form, or throwing certain types of passes – as compatible with normative play.

References

  • Abrahamson, D. (2002). When “the same” is the same as different differences: Aliya reconciles her perceptual judgment of proportional equivalence with her additive computation skills. In D. Mewborn, P. Sztajn, E. White, H. Wiegel, R. Bryant, & K. Nooney (Eds.), Proceedings of the twenty-fourth annual meeting of the north American chapter of the International Group for the Psychology of mathematics education (Vol. 4, pp. 1658–1661). Columbus: PME-NA.

    Google Scholar 

  • Abrahamson, D. (2009). Orchestrating semiotic leaps from tacit to cultural quantitative reasoning: The case of anticipating experimental outcomes of a quasi-binomial random generator. Cognition and Instruction, 27(3), 175–224.

    Google Scholar 

  • Abrahamson, D. (2012a). Discovery reconceived: Product before process. For the Learning of Mathematics, 32(1), 8–15.

    Google Scholar 

  • Abrahamson, D. (2012b). Rethinking intensive quantities via guided mediated abduction. Journal of the Learning Sciences, 21(4), 626–649.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Abrahamson, D., & Wilensky, U. (2007). Learning axes and bridging tools in a technology-based design for statistics. International Journal of Computers for Mathematical Learning, 12(1), 23–55.

    Google Scholar 

  • Abrahamson, D., Gutiérrez, J., Charoenying, T., Negrete, A., & Bumbacher, E. (2012). Fostering hooks and shifts: Tutorial tactics for guided mathematical discovery. Technology, Knowledge, and Learning, 17(1–2), 61–86.

    Google Scholar 

  • Barnes, B., Henry, J., & Bloor, D. (1996). Scientific knowledge: A sociological analysis. Chicago: University of Chicago Press.

    Google Scholar 

  • Bartolini Bussi, M., & Mariotti, M. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, B. Sriraman, & D. Tirosh (Eds.), Handbook of international research in mathematics education (2nd ed., pp. 746–783). New York, NY: Routledge.

    Google Scholar 

  • Broderbund. (1996). Logical journey of the Zoombinis. Novato: Broderbund Software (video game).

    Google Scholar 

  • Cazden, C. (1981). Performance before competence: Assistance to child discourse in the zone of proximal development. Quarterly Newsletter of the Laboratory of Comparative Human Cognition, 3(1), 5–8.

    Google Scholar 

  • CCSSI (2017). Standards for Mathematical Practice. (http://www.corestandards.org/Math/Practice/).

  • DeLiema, D., Enyedy, N., & Danish, J. (2019). Roles, rules, and keys: How different play configurations shape collaborative science inquiry. Journal of the Learning Sciences, 28(4–5), 513–555.

    Google Scholar 

  • Durán-López, E., Iyer, G., & Rosenbaum, L. (2017). Geometris: A collaborative embodied geometry game. In G. Mark, S. Fussel, F. Mueller, & J. Tanenbaum (Eds.), Proceedings of the 35th conference on human factors in computing systems (pp. 214–217). Denver: CHI.

    Google Scholar 

  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131.

    Google Scholar 

  • Enyedy, N., Danish, J., & DeLiema, D. (2015). Liminal blends: How students blend symbols, experiences, and their own bodies together in order to co-construct meaning in a collaborative augmented reality learning environment. International Journal of Computer-Supported Collaborative Learning, 10(1), 7–34.

    Google Scholar 

  • Flood, V. (2018). Multimodal revoicing as an interactional mechanism for connecting scientific and everyday concepts. Human Development, 61(3), 145–173.

    Google Scholar 

  • Fyfe, E., McNeil, N., Son, J., & Goldstone, R. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26(1), 9–25.

    Google Scholar 

  • Gilligan, K., Hodgkiss, A., Thomas, M., & Farran, E. (2019). The developmental relations between spatial cognition and mathematics in primary school children. Developmental Science, 22(4), #e12786.

    Google Scholar 

  • Gleason, M., & Schauble, L. (1999). Parents’assistance of their children’s scientific reasoning. Cognition and Instruction, 17(4), 343–378.

    Google Scholar 

  • Goldstone, R., Landy, D., & Son, J. (2009). The education of perception. Topics in Cognitive Science, 2(2), 265–284.

    Google Scholar 

  • Goodwin, C. (1994). Professional vision. American Anthropologist, 96(3), 603–633.

    Google Scholar 

  • Habgood, M., & Ainsworth, S. (2011). Motivating children to learn effectively: Exploring the value of intrinsic integration in educational games. The Journal of the Learning Sciences, 20(2), 169–206.

    Google Scholar 

  • Hall, R., Ma, J., & Nemirovsky, R. (2015). Rescaling bodies in/as representational instruments in GPS drawing. In V. Lee (Ed.), Learning technologies and the body (pp. 112–131). New York: NY: Routledge.

    Google Scholar 

  • Hegedus, S., & Penuel, W. (2008). Studying new forms of participation and identity in mathematics classrooms with integrated communication and representational infrastructures. Educational Studies in Mathematics, 68(2), 171–183.

    Google Scholar 

  • Holbert, N., & Wilensky, U. (2014). Constructible authentic representations: Designing video games that enable players to utilize knowledge developed in-game to reason about science. Technology, Knowledge and Learning, 19(1–2), 53–79.

    Google Scholar 

  • 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. Kellogg, & D. Tan (Eds.), Proceedings of the annual meeting of the Association for Computer Machinery Special Interest Group on computer–human interaction (pp. 1989–1998). New York: ACM Press.

    Google Scholar 

  • Kafai, Y. (1996). Learning design by making games: Children’s development of design strategies in the creation of a complex computational artifact. In Y. Kafai & M. Resnick (Eds.), Constructionism in practice: Designing, thinking and learning in a digital world (pp. 71–96). Mahwah: Lawrence Erlbaum Associates.

    Google Scholar 

  • Kelton, M. & Ma, J. (2020, On-line). Assembling a torus: Family mobilities in an immersive mathematics exhibition. Cognition and Instruction, (30).

  • Kiili, K., & Perttula, P. (2012). Exerbraining for schools: Combining body and brain training. Procedia Computer Science, 15, 163–173.

    Google Scholar 

  • Leung, A., Baccaglini-Frank, A., & Mariotti, M. (2013). Discernment of invariants in dynamic geometry environments. Educational Studies in Mathematics, 84(3), 439–460.

    Google Scholar 

  • Ma, J. (2017). Multi-party, whole-body interactions in mathematical activity. Cognition and Instruction, 35(2), 141–164.

    Google Scholar 

  • Malinverni, L., Ackermann, E. & Pares, N. (2016). Experience as an object to think with: From sensing-in-action to making-sense of action in full-body interaction learning environments. In Proceedings of the tenth international conference on tangible, embedded, and embodied interaction (pp. 332–339). Eindhoven, The Netherlands: ACM.

  • Mayer, R. (2005). Cognitive theory of multimedia learning. In R. Mayer (Ed.), The Cambridge handbook of multimedia learning (pp. 31–48). New York: Cambridge University Press.

    Google Scholar 

  • Nathan, M., & Walkington, C. (2017). Grounded and embodied mathematical cognition: Promoting mathematical insight and proof using action and language. Cognitive Research: Principles and Implications, 2(1), 20.

    Google Scholar 

  • NCTM. (2000). Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics.

    Google Scholar 

  • Nemirovsky, R., Kelton, M., & Rhodehamel, B. (2013). Playing mathematical instruments: Emerging perceptuomotor integration with an interactive mathematics exhibit. Journal for Research in Mathematics Education, 44(2), 372–415.

    Google Scholar 

  • Nemirovsky, R., Tierney, C., & Wright, T. (1998). Body motion and graphing. Cognition and Instruction, 16(2), 119–172.

    Google Scholar 

  • Newell, K. (1986). Constraints on the development of coordination. In M. Wade & H. Whiting (Eds.), Motor development in children: Aspects of co-ordination and control (pp. 341–361). Amsterdam: Martinus Nijhoff Publishers.

    Google Scholar 

  • Newell, K., & Ranganathan, R. (2010). Instructions as constraints in motor skill acquisition. In I. Renshaw, K. Davids, & G. Savelsbergh (Eds.), Motor learning in practice: A constraints-led approach (pp. 17–32). Florence: Routledge.

    Google Scholar 

  • Newman, D., Griffin, P., & Cole, M. (1989). The construction zone: Working for cognitive change in school. New York, NY: Cambridge University Press.

    Google Scholar 

  • Okamoto, Y., Weckbacher, L., & Hallowell, D. (2014). How is spatial reasoning related to mathematical thinking and how important is early exposure to spatial activities? In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the 38th Conference of the International Group of the Psychology of Mathematics Education (Vol. 1, pp. 177–179). Vancouver: PME.

    Google Scholar 

  • Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of geometry (E. Lunzer, trans.). New York: Basic Books.

    Google Scholar 

  • Price, S., & Duffy, S. (2018). Opportunities and challenges of bodily interaction for geometry learning to inform technology design. Multimodal Technologies and Interaction, 2(3), #41.

    Google Scholar 

  • Rau, M., & Schmidt, T. (2019). Disentangling conceptual and embodied mechanisms for learning with virtual and physical representations. In S. Isotani, A. Ogan, P. Hastings, B. McLaren, & R. Luckin (Eds.), Artificial intelligence in education (pp. 419–431). Cham: Springer.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Shvarts, A. & Abrahamson, D. (2019). Dual-eye-tracking Vygotsky: A microgenetic account of a teaching/learning collaboration in an embodied-interaction technological tutorial for mathematics. Learning, Culture and Social Interaction, 22, (#100316).

  • Shvarts, A., & Bakker, A. (2019). The early history of the scaffolding metaphor: Bernstein, Luria, Vygotsky, and before. Mind, Culture, and Activity, 26(1), 4–23.

    Google Scholar 

  • Stevens, R., & Hall, R. (1998). Disciplined perception: Learning to see in technoscience. In M. Lampert & M. Blunk (Eds.), Talking mathematics in school: Studies of teaching and learning (pp. 107–149). Cambridge: Cambridge University Press.

    Google Scholar 

  • Sweller, J., van Merrienboer, J., & Paas, F. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10(3), 251–296.

    Google Scholar 

  • Uttal, D., Meadow, N., Tipton, E., Hand, L., Alden, A., Warren, C., & Newcombe, N. (2013). The malleability of spatial skills: A meta-analysis of training studies. Psychological Bulletin, 139(2), 352–402.

    Google Scholar 

  • Vilar, L., Araújo, D., Davids, K., & Renshaw, I. (2012). The need for ‘representative task design’ in evaluating efficacy of skills tests in sport: A comment on Russell, Benton and Kingsley (2010). Journal of Sports Sciences, 30(16), 1727–1730.

    Google Scholar 

  • Vygotsky, L. (1934/2001). Lektsii po pedologii [Lectures on paedology]. Izhevsk, Russia: Izdatel’kii dom Udmurtskii universitet.

  • Vygotsky, L. (1930/1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press.

  • Wilensky, U., & Stroup, W. (2000). Networked gridlock: Students enacting complex dynamic phenomena with the HubNet architecture. In B. Fishman & S. O’Connor-Divelbiss (Eds.), The 4th International Conference of the Learning Sciences (pp. 282–289). Mahwah: Lawrence Erbaum Associates.

    Google Scholar 

  • Williams-Pierce, C. (2016). Provoking mathematical play through hidden deep structures. In C. Looi, J. Polman, U. Cress & P. Reimann (Eds), Proceedings of the 12th international conference of the learning sciences (vol. 2, pp. 1241–1242). Singapore: International Society of the Learning Sciences.

  • Wittgenstein, L. (1953). Philosophical investigations (G. Anscombe, trans.). Upper Saddle River: Prentice Hall.

    Google Scholar 

  • Wolfgang, C., Stannard, L., & Jones, I. (2003). Advanced constructional play with LEGOs among preschoolers as a predictor of later school achievement in mathematics. Early Child Development and Care, 173(5), 467–475.

    Google Scholar 

  • Wood, D., Bruner, J., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17(2), 89–100.

    Google Scholar 

Download references

Acknowledgements

The authors wish to thank the DEME Guest Editors and the 3 anonymous reviewers for their formative comments on this manuscript. We also wish to thank the exhibits staff, volunteers, and participating visitors at the Lawrence Hall of Science, University of California Berkeley, for making this study possible. The ideas presented in this article were workshopped and refined with members of the Embodied Design Research Laboratory at the University of California, Berkeley. Geometris was collaboratively designed and created by Elena Durán-López, Ganesh V. Iyer, and Leah F. Rosenbaum, with significant guidance from Professor Kimiko Ryokai and Dr. Noura Howell.

Author information

Authors and Affiliations

  1. Graduate School of Education, University of California, Berkeley, Berkeley Way West, Room 4110, 2121 Berkeley Way, Berkeley, CA, 94720-1670, USA

    Leah F. Rosenbaum, Japleen Kaur & Dor Abrahamson

Authors
  1. Leah F. Rosenbaum
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Japleen Kaur
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dor Abrahamson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leah F. Rosenbaum.

Ethics declarations

Conflict of Interest

On behalf of all authors, the corresponding author claims no conflicts of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rosenbaum, L.F., Kaur, J. & Abrahamson, D. Shaping Perception: Designing for Participatory Facilitation of Collaborative Geometry. Digit Exp Math Educ 6, 191–212 (2020). https://doi.org/10.1007/s40751-020-00068-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40751-020-00068-2

Keywords

Search

Navigation