Personal and Ubiquitous Computing

, Volume 16, Issue 4, pp 405–419 | Cite as

Tangibles for learning: a representational analysis of physical manipulation

  • Andrew Manches
  • Claire O’Malley
Original Article


Manipulatives—physical learning materials such as cubes or tiles—are prevalent in educational settings across cultures and have generated substantial research into how actions with physical objects may support children’s learning. The ability to integrate digital technology into physical objects—so-called ‘digital manipulatives’—has generated excitement over the potential to create new educational materials. However, without a clear understanding of how actions with physical materials lead to learning, it is difficult to evaluate or inform designs in this area. This paper is intended to contribute to the development of effective tangible technologies for children’s learning by summarising key debates about the representational advantages of manipulatives under two key headings: offloading cognition—where manipulatives may help children by freeing up valuable cognitive resources during problem solving, and conceptual metaphors—where perceptual information or actions with objects have a structural correspondence with more symbolic concepts. The review also indicates possible limitations of physical objects—most importantly that their symbolic significance is only granted by the context in which they are used. These arguments are then discussed in light of tangible designs drawing upon the authors’ current research into tangibles and young children’s understanding of number.


Tangible technologies Physical manipulatives Mathematics learning Educational technology Virtual manipulatives 



This research was supported by an Economic Social Research Council Case PhD studentship awarded to Professors Claire O’Malley and Steve Benford, and co-sponsored by Futurelab (Grant no. PTA-033-2006-00025). We would also like to thank the schools and children who have taken part and contributed greatly to our work.


  1. 1.
    Blackwell A (2003) Cognitive dimensions of tangible programming languages. Paper presented at the Proceedings of the first joint conference of the empirical assessment in software engineering and psychology of programming interest groupsGoogle Scholar
  2. 2.
    Salomon G (1990) Cognitive effects with and of computer technology. Commun Res 17(1):26–44MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cox R (1999) Representation construction, externalised cognition and individual differences. Learn Instr 9(4):343–363CrossRefGoogle Scholar
  4. 4.
    Ainsworth S (2006) Deft: a conceptual framework for considering learning with multiple representations. Learn Instr 16(3):183–198MathSciNetCrossRefGoogle Scholar
  5. 5.
    Marshall P (2007) Do tangible interfaces enhance learning? In: Proceedings of the 1st international conference on tangible and embedded interaction, Baton Rouge, Louisiana, 2007. ACM Press. doi: 10.1145/1226969.1227004
  6. 6.
    O’Malley C, Stanton Fraser D (2004) Report 12: literature review in learning with tangible technologies. Futurelab series. Futurelab, BristolGoogle Scholar
  7. 7.
    Montessori M (1912) The montessori method. Frederick Stokes Co, New YorkzbMATHGoogle Scholar
  8. 8.
    Froebel F (1909) Pedagogics of the kindergarten. D.Appleton and Company, New YorkGoogle Scholar
  9. 9.
    McNeil NM, Jarvin L (2007) When theories don’t add up: disentangling the manipulatives debate. Theory Pract 46(4):309–316CrossRefGoogle Scholar
  10. 10.
    Dienes Z (1964) Building up mathematics, 2nd edn. Hutchinson Educational, LondonGoogle Scholar
  11. 11.
    Clements DH (1999) ‘Concrete’ manipulatives, concrete ideas. Contemp Issues Early Child 1(1):45–60MathSciNetCrossRefGoogle Scholar
  12. 12.
    Triona LM, Klahr D (2003) Point and click or grab and heft: comparing the influence of physical and virtual instructional materials on elementary school students’ ability to design experiments. Cogn Instr 21(2):149–173CrossRefGoogle Scholar
  13. 13.
    Zhang JJ (1997) The nature of external representations in problem solving. Cogn Sci 21(2):179–217CrossRefGoogle Scholar
  14. 14.
    Scaife M, Rogers Y (1996) External cognition: how do graphical representations work? Int J Hum-Comput Stud 45(2):185–213CrossRefGoogle Scholar
  15. 15.
    Rogers Y (2004) New theoretical approaches for human-computer interaction. Annu Rev Inf Sci Technol 38:87–143Google Scholar
  16. 16.
    Anderson ML (2003) Embodied cognition: a field guide. Artif Intell 149(1):91–130CrossRefGoogle Scholar
  17. 17.
    Mix KS (2009) Spatial tools for mathematical thought. In: Mix KS, Smith LB, Gasser M (eds) The spatial foundations of cognition and language. Oxford Scholarship Online Monographs, New York, pp 40–66Google Scholar
  18. 18.
    Hutchins E (2005) Material anchors for conceptual blends. J Pragmat 37(10):1555–1577CrossRefGoogle Scholar
  19. 19.
    Hurtienne J (2009) Sad is heavy and happy is light. Population stereotypes of tangible object attributes. In: Third international conference on tangibles and embedded interaction, Cambridge, pp 61–68Google Scholar
  20. 20.
    Antle AN (2007) The cti framework: Informing the design of tangible systems for children. Paper presented at the proceedings of the 1st international conference on Tangible and embedded interaction, Baton Rouge, LouisianaGoogle Scholar
  21. 21.
    Antle AN, Droumeva M, Corness G (2008) Playing with the sound maker: do embodied metaphors help children learn? In: Proceedings of the 7th international conference on interaction design and children, ACM, pp 178–185Google Scholar
  22. 22.
    Larkin JH, Simon HA (1987) Why a diagram is (sometimes) worth 10000 words. Cogn Sci 11(1):65–99CrossRefGoogle Scholar
  23. 23.
    Manches A, O’Malley C, Benford S (2010) The role of physical representations in solving number problems: a comparison of young children’s use of physical and virtual materials. Comput Educ 54(3):622–640CrossRefGoogle Scholar
  24. 24.
    Mandler G, Shebo BJ (1982) Subitizing: an analysis of its component processes. J Exp Psychol-Gen 111(1):1–22CrossRefGoogle Scholar
  25. 25.
    Riggs K, Ferrand L, Lancelin D, Fryziel L, Dumur G, Simpson A (2006) Subitizing in tactile perception. Psychol Sci 17(4):271–272CrossRefGoogle Scholar
  26. 26.
    Patten J, Ishii H (2000) A comparison of spatial organization strategies in graphical and tangible user interfaces. Paper presented at the Proceedings of DARE 2000 on designing augmented reality environments, Elsinore, DenmarkGoogle Scholar
  27. 27.
    Manches A, O’Malley C, Benford S (2009) Physical manipulation: evaluating the potential for tangible designs. In: Proceedings of the 3rd international conference on tangible and embedded interaction. ACM, Cambridge, UK, pp 77–84. doi: 10.1145/1517664.1517688
  28. 28.
    Spelke ES (1990) Principles of object perception. Cogn Sci 14(1):29–56CrossRefGoogle Scholar
  29. 29.
    Wilson M (2001) The case for sensorimotor coding in working memory. Psychon Bull Rev 8(1):44–57CrossRefGoogle Scholar
  30. 30.
    Goldin-Meadow S, Nusbaum H, Kelly SD, Wagner S (2001) Explaining math: gesturing lightens the load. Psychol Sci 12(6):516–522CrossRefGoogle Scholar
  31. 31.
    Roth W-M (2002) From action to discourse: the bridging function of gestures. Cogn Syst Res 3(3):535–554CrossRefGoogle Scholar
  32. 32.
    Kirsh D, Maglio P (1994) On distinguishing epistemic from pragmatic action. Cogn Sci 18(4):513–549CrossRefGoogle Scholar
  33. 33.
    Kaput J (1992) Technology and mathematics education. In: Grouws D (ed) Handbook of research on mathematics teaching and learning. Macmillan, New YorkGoogle Scholar
  34. 34.
    O’Hara KP, Payne S (1999) Planning and the user interface: The effects of lockout time and error recovery cost, vol 50. Academic Press, New York. First published. doi: 10.1006/ijhc.1998.0234
  35. 35.
    Ellis S, Siegler RS (1997) Planning as strategy choice: why don’t children plan when they should? In: Friedman S, Scholnick EK (eds) The developmental psychology of planning. Lawrence Erlbaum Associates, London, pp 183–208Google Scholar
  36. 36.
    Martin T, Schwartz D (2005) Physically distributed learning: adapting and reinterpreting physical environments in the development of fraction concepts. Cogn Sci 29:587–625CrossRefGoogle Scholar
  37. 37.
    Martin T (2007) Physically distributed learning with virtual manipulatives for elementary mathematics. In: Robinson D, Schraw G (eds) Recent innovations in educational technology that facilitate student learning. Information Age Publishing, CharlotteGoogle Scholar
  38. 38.
    Alibali MW, DiRusso AA (1999) The function of gesture in learning to count: more than keeping track. Cogn Dev 14(1):37–56CrossRefGoogle Scholar
  39. 39.
    Gentner D (1983) Structure-mapping: a theoretical framework for analogy. Cogn Sci 7(2):155–170CrossRefGoogle Scholar
  40. 40.
    Canobi KH, Reeve RA, Pattison PE (2002) Young children’s understanding of addition concepts. Educ Psychol 22(5):513–532CrossRefGoogle Scholar
  41. 41.
    Lakoff G, Núñez R (2000) Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books, New YorkzbMATHGoogle Scholar
  42. 42.
    Hatano G, Osawa K (1983) Digit memory of grand experts in abacus-derived mental calculation. Cognition 15(1–3):95–110CrossRefGoogle Scholar
  43. 43.
    Moretto G, di Pellegrino G (2008) Grasping numbers. Exp Brain Res 188(4):505–515CrossRefGoogle Scholar
  44. 44.
    Edwards L (2005) Metaphors and gestures in fraction talk. Paper presented at the 4th congress of the european society for research in mathematics education, Sant Feliu de Guixols, SpainGoogle Scholar
  45. 45.
    Goldin-Meadow S (2000) Beyond words: The importance of gesture to researchers and learners. Child Dev 71(1):231–239CrossRefGoogle Scholar
  46. 46.
    Valenzeno L, Alibali MW, Klatzky R (2003) Teachers’ gestures facilitate students’ learning: a lesson in symmetry. Contemp Educ Psychol 28(2):187–204CrossRefGoogle Scholar
  47. 47.
    Halford GS, Boulton-Lewis GM (1992) Value and limitations of analogs in teaching mathematics. In: Demetriou A, Efkliades A, Shayer M (eds) Modern theories of cognitive development go to school. Routledge and Kegan Paul, LondonGoogle Scholar
  48. 48.
    Uttal DH, Scudder KV, DeLoache JS (1997) Manipulatives as symbols: a new perspective on the use of concrete objects to teach mathematics. J Appl Dev Psychol 18(1):37–54CrossRefGoogle Scholar
  49. 49.
    Meira L (1998) Making sense of instructional devices: the emergence of transparency in mathematical activity. J Res Math Educ 29(2):121–142MathSciNetCrossRefGoogle Scholar
  50. 50.
    Fishkin KP (2004) A taxonomy for and analysis of tangible interfaces. Personal Ubiquitous Comput 8(5): 347–358. doi: 10.1007/s00779-004-0297-4
  51. 51.
    Girouard A, Solovey ET, Hirshfield LM, Ecott S, Shaer O, Jacob RJK (2007) Smart blocks: a tangible mathematical manipulative. Paper presented at the Proceedings of the 1st international conference on tangible and embedded interaction, Baton Rouge, LouisianaGoogle Scholar
  52. 52.
    Wyeth P, Wyeth G (2001) Electronic blocks: tangible programming elements for preschoolers. In: Hirose M (ed) Proceedings of the Eighth IFIP TC13 conference on human-computer interaction. IOS Press, Amsterdam, pp 496–503Google Scholar
  53. 53.
    Lackner TM, Dobson K, Rodenstein R, Weisman L (1999) Sensory puzzles. Paper presented at the CHI ‘99 extended abstracts on Human factors in computing systems, Pittsburgh, PennsylvaniaGoogle Scholar
  54. 54.
    Jones MG, Minogue J, Tretter TR, Negishi A, Taylor R (2006) Haptic augmentation of science instruction: does touch matter? Sci Educ 90(1):111–123CrossRefGoogle Scholar
  55. 55.
    Resnick M (1998) Technologies for lifelong kindergarten. Educ Tech Res Dev 46(4):43–55CrossRefGoogle Scholar
  56. 56.
    NLVM (2007) National library of virtual manipulatives.
  57. 57.
    Raffle HS, Parkes AJ, Ishii H (2004) Topobo: A constructive assembly system with kinetic memory. Paper presented at the Proceedings of the SIGCHI conference on Human factors in computing systems, Vienna, AustriaGoogle Scholar
  58. 58.
    Shaer O, Leland N, Calvillo-Gamez EH, Jacob RJK (2004) The tac paradigm: Specifying tangible user interfaces, vol 8. Springer-Verlag, Berlin. First published. doi: 10.1007/s00779-004-0298-3
  59. 59.
    Price.S, Pontual T, Sheridan J, Roussos G (2009) The effect of representation location on interaction in a tangible learning environment. Paper presented at the Proceedings of the 3rd International Conference on Tangible and Embedded Interaction, Cambridge, United KingdomGoogle Scholar
  60. 60.
    Wilkie K, Holland S, Mulholland P (2009) Evaluating musical software using conceptual metaphors. Br Comput Soc, pp 232–237Google Scholar
  61. 61.
    Khandelwal M, Mazalek A (2007) Teaching table: a tangible mentor for pre-k math education. In: Proceedings of the 1st international conference on tangible and embedded interaction (TEI ‘07). ACM Press, New York, pp 191–194. doi: 10.1145/1226969.1227009
  62. 62.
    Scarlatos T, Scarlatos L (2009) Tangible maths application. Accessed 12 Sept 2009

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.London Knowledge LabInstitute of EducationLondonUK
  2. 2.School of PsychologyUniversity of NottinghamNottinghamUK

Personalised recommendations