Using Geometrical Representations as Cognitive Technologies

in Journal of Cognition and Culture
Restricted Access
Get Access to Full Text
Rent on DeepDyve

Have an Access Token?



Enter your access token to activate and access content online.

Please login and go to your personal user account to enter your access token.



Help

Have Institutional Access?



Access content through your institution. Any other coaching guidance?



Connect

In this article we provide a treatment of geometric diagrams as a semiotic ‘technology’ in mathematics education. To this end, we review two classroom observations and one experimental study that illuminate the potential for communicative breakdown in the use of this technology, as well as possibilities for mitigating the breakdown. In the classroom observations, we show that rather than ‘seeing through’ the diagrams to the idealized mathematical objects (a pervasive semiotic practice in academic mathematics), upper-elementary grade students may treat the misleading appearances of the diagrams as mathematically significant. We argue that one source of such communicative breakdowns is that the ‘rules’ for using the semiotic technology are rarely made explicit in classrooms. In the experimental study, we review findings showing that when students’ have access to mathematical definitions that distinguish between the diagram and the idealized mathematical object (e.g., a diagram of a mathematical point is the size of a small dot, but a true mathematical point has no size), they are less likely to rely on the misleading appearances of the diagrams; we also found that this effect increases over grade level. The issues explored in this article suggest that upper elementary students could benefit from explicit treatment of definitional practices in academic mathematics. At a broader theoretical level, the findings serve to emphasize the potential for any semiotic technology to have multiple meanings and the role of active sense making in communications with and about technologies.

Using Geometrical Representations as Cognitive Technologies

in Journal of Cognition and Culture

Sections

References

de KirbyK.SaxeG. B. Idealized objects and material diagrams: Definition use in early geometrical problem solving 2013 UC Berkeley, Berkeley CA Unpublished manuscript

DörflerW. CobbP.YackelE.McClainK. Means for meaning Symbolizing and communicating in mathematics classrooms; Perspectives on discourse tools and instructional design 2000 Mahwah NJ. Lawrence Erlbaum Associates 99 131

FischbeinE. The theory of figural concepts Educational Studies in Mathematics 1993 24 139 162

FontV.ContrerasÁ. The problem of the particular and its relation to the general in mathematics education Educational Studies in Mathematics 2008 69 33 52

GoodnowJ. J.MillerP. J.KesselF. Cultural practices as contexts for development 1995 San Francisco CA. Jossey-Bass

LaveJ.WengerE. Situated learning 1991 New York NY. Cambridge University Press

MesquitaA. L. On conceptual obstacles linked with external representation in geometry The Journal of Mathematical Behavior 1998 17 183 195

PeirceC. S. On the algebra of logic: A contribution to the philosophy of notation American Journal of Mathematics 1885 7 180 196

RadfordL. The seen, the spoken and the written: A semiotic approach to the problem of objectification of mathematical knowledge For the Learning of Mathematics 2002 22 14 23

RogoffB. Apprenticeship in thinking: Cognitive development in social context 1990 New York NY. Oxford University Press

RogoffB. The cultural nature of human development 2003 Oxford Oxford University Press

SaxeG. B. Culture and cognitive development: Studies in mathematical understanding 1991 Hillsdale NJ. Lawrence Erlbaum Associates

SaxeG. B. DemetriouA.RaftopoulosA. Practices of quantification from a socio-cultural perspective Cognitive Developmental Change: Theories Models and Measurement 2004 New York NY. Cambridge University Press 241 263

SaxeG. B. Cultural development of mathematical ideas: Papua New Guinea studies 2012 New York NY. Cambridge University Press

SaxeG. B.GubermanS. R.GearhartM. Social processes in early number development Monographs of the Society for Research in Child Development 1987 52 Ann Arbor MI. Society for Research in Child Development Serial No. 162

ScribnerS.ColeM. The psychology of literacy 1981 Cambridge MA. Harvard University Press

SfardA. CobbP.YackelE.McClainK. Symbolizing mathematical reality into being – or how mathematical discourse and mathematical objects create each other Symbolizing and communicating in mathematics classrooms: Perspectives on discourse tools and instructional design 2000 Mahwah NJ. Lawrence Erlbaum Associates 37 98

Figures

  • View in gallery
    The circles (cookies) that the teacher has drawn on the board.
  • View in gallery
    The teacher has partitioned one of the two drawn circles with perpendicular lines.
  • View in gallery
    (a) A number line with points for 0 and 1000 labeled. (b) −1000 and −1001 placed on the number line.
  • View in gallery
    (a) The teacher’s placement of –1001 on the number line; (b) the teacher’s erasure of –1001 and her query to the class about the number that should be placed at the indicated tick mark.
  • View in gallery
    (a) Jasmine’s placement of –1001; (b) Jasmine’s solution of –1006 or –1007.
  • View in gallery
    Task used by Fischbein (1993).
  • View in gallery
    Information sheet (with definitions) used in the experimental condition.
  • View in gallery
    Proportion of idealized responses for students in both treatment groups.

Index Card

Content Metrics

Content Metrics

All Time Past Year Past 30 Days
Abstract Views 29 29 6
Full Text Views 3 3 3
PDF Downloads 0 0 0
EPUB Downloads 0 0 0