M. FRANCAVIGLIA ET AL.
96
different kinds of gestures that demonstrate how the problem
solving processes include behavioural categories that enhance
the conceptualization phase. More specifically, gesture provides
a more complete representation of the problem and then of the
solution rather than just speech alone. In addition, in this paper
we have also identified the importance to use tools, which not
only support children’s reflection, but also provide a very pow-
erful predictor of the problem solution.
These gestures are an evidence of how children think about
mathematical concepts, once they have learned how to manipu-
late them symbolically. All children involved in the current
investigation produced gestures that evoked hands-on activities,
such as drawing and physical manipulation of the ideal struc-
tures. The drawing activities represent a further analysis of the
quality of gesture structure. In fact, many of these gestures were
precise with regard to the structural details of the agreement
cloth structure. Our findings are in line with other studies that
have investigated the role of gestural behaviour from an educa-
tional standpoint (Church, Ayman-Nolley, & Alibali, 2001;
Goldin-Meadow, 1999; Kelly et al., 2002; Singer & Goldin-
Meadow, 2005). In addition, these studies are significant also
from an educational perspective since they represent a new
analytic framework to analyse the children cognitive process
involved in learning and in problem solving activities. In par-
ticular, gestures can be successfully used in the didactical con-
text as a natural support without modifying the natural children
behaviour.
We have also shown here that gestures evolve during the dis-
cussion process. Initially, children produce gesture that is more
spontaneous. In this phase, the children’s ideas about the prob-
lem are not always clear. After this initial phase, children in-
creased their problem understanding by incorporating new in-
formation and then creating an articulated mental representation
of the assigned problem. This conceptualization was expressed
by using gestural movements. Therefore, understanding how
children represent problems will allow to better explaining how
they construct and identify their cognitive strategies to solve a
mathematical problem. Furthermore, this connection between
gestures, speech and drawing makes the children more aware of
the cognitive strategies available that they can apply to solve a
mathematical problem. For example, children sometimes rep-
resented the length of the span by using some gestures, showing
how the problem entities changed continuously. Such cases
represent important aspects that show how children’s thinking
evolves during the problem solving activities. During these
didactical activities, children construct in fact new strategies
(not always correct) that often modify other realized previously.
Further Direction
Our research provides a large scientific evidence of the rela-
tionship existing between gestures and mathematical problem
solving. A further investigation should be to examine also the
relationship between gesture and metacognition (Perfect &
Schwartz, 2002). Metacognition encompasses a set of proce-
dures that allow cognitive systems equipped with it to predict
or evaluate their ability to perform a given cognitive operation.
This problem is being currently addressed and will be the sub-
ject of forthcoming publication. It will allow to individuate, for
example, a possible distinction between cognitive and meta-
cognitive gestures.
Acknowledgements
We warmly thank Prof. Ferdinando Arzarello head of the
Department of Mathematics of University of Torino, Italy, for
having provided the educational material (video) on children
activities. We also thank the student (Eleonora Crudo) of the
University of Calabria (Cosenza, Italy) that has analysed the
children behaviours.
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