Psychology 91
2011. Vol.2, No.2, 91-97
Copyright © 2011 SciRes. DOI:10.4236/psych.2011.22015
Gesture as a Cognitive Support to Solve
Mathematical Problems
Mauro Francaviglia1, Rocco Servidio2
1Department of Mathematics, University of Turin, Turin, Italy;
2Department of Linguistics, University of Calabria, Cosenza, Italy.
Received October 7th, 2010; revised December 12th, 2010; accepted December 15th, 2010.
The aim of this study was to investigate the relationships between gestures and mathematical problem solving. It
concentrates on the idea that gestures can improve the student’s mathematical conceptual abilities. The educa-
tional aim of the current study was to understand whether Penelope sewing the cloth every day will be eventu-
ally able to finish it in 50 days, before Ulysses returns in his home-town. To analyse children’s gestures we ap-
plied the McNeill classification. The participants were five children aged between 9 and 10 years, attending the
fifth-grade class of a primary school in Turin, Italy. We used the observational method to analyse the children’s
gestures behaviour. At the end of the analysis, we collected a corpus of 538 gestures. Results show that children
use different gesture patterns to communicate their own mathematical ideas. Overall, these findings suggest that
gestures facilitate children’ learning of mathematical concepts and improve their cognitive strategies to the
problem solution.
Keywords: Gestures, Mathematical Problem Solving, Mathematical Learning, Cognitive Processing
In the last years many studies have attempted to better un-
derstand how the integration of gestures and spoken language
can be used not only to improve the student’s learning process
of Mathematics means, but also to assess problem solving
strategies (Alibali et al., 1999; Edwards, 2009; Radford, Ed-
wards, & Arzarello, 2009; Rasmussen, Stephan, & Allen, 2004).
Although some mathematical concepts are often difficult to
understand for many students, a growing body of research
shows how the semiotics perspective can be used to analyse the
construction of mathematical concepts (Arzarello et al., 2009;
Arzarello, Francaviglia, & Servidio, 2006; Bazzini, 2001; Lim
et al., 2009). Students are able to operate with gestures discov-
ering how perceptual knowledge is a useful strategy to create
into their own brain mental representations of mathematical
When students work in the classroom, gestures are prevalent
because they play an important role to support teaching and
learning activities. Focusing on gestures role in the educational
setting, researchers have examined the cognitive mechanisms of
the gesture production as mental strategies that help children to
reduce they cognitive effort when discusses about mathematical
problems (Cook & Goldin-Meadow, 2006; Goldin-Meadow et
al., 2001).
The purpose of this study was to investigate how children
use gestures to communicate mathematical ideas, improving the
sharing of knowledge within the group. We consider the ges-
tures as cognitive scaffolding that help children to solve
mathematical problems. Another concurrent aim of this study
was to contribute to the collection of a corpus of gestures
within the domain of Mathematics. A conceptual framework on
gestural mathematical behaviour represents an important chal-
lenge to understand the role of non-verbal behaviour as a didac-
tical strategy. In a typical cooperative work setting, gestural
behaviours take the place of the images to communicate ideas
about structured or abstract scientific topics.
To study the relationships between gestures and children’s
problem solving strategies, the Penelope’s history has been
selected as a didactical task for its appropriateness and appeal
to the children’s age. The problem was to understand whether
Penelope sewing the cloth every day would be able to finish it
in 50 day before Ulysses arrives. It is worthy that Penelope
every day finishes a span of cloth, but during the night, she
unpacks half of span. Knowing that the cloth must be 15 spans
long; does Penelope manage to await the return of Ulysses
without marrying? Finally, the interactions occurred while sub-
jects worked to understand whether Penelope during her
day/night work would be eventually able to complete success-
fully the spam before than Ulysses would come back to Ithaca.
In this current research, we adopt the observational method to
analyse children’s gestural behaviour intent to solve mathe-
matical problems. The observational method is a research tech-
nique suited to examine human and animal behaviour by inves-
tigating the body movements.
The paper is organized as follows. First, the theoretical back-
ground is briefly introduced. Next, Section 3 introduces and
explains the research methodology of the current study. Results
are described in the Section 4. Finally, in the last section con-
clusion and future direction that addressed the importance of
the relationship between gestures and Mathematics are intro-
Theoretical Framework
In the current work, the definition of gesture concerns the
movements of the hands, in order to achieve some communica-
tive aims performing meaningfully a didactical task. However,
in the last years many broader definitions of gestures have been
proposed. For an excellent overview of gesture research, see
Kendon (2004).
Gestures not only support the mental processes but they are
also used as mediators of a socio-cultural participation (Vygot-
skij, 1962). In this view, teachers and students share the same
language offering a productive way to understand mathematical
notions and then assuring them a good comprehension of the
scientific knowledge. Researchers in communication and con-
struction of mathematical meaning explored different ways to
represent the scientific information. In particular, gestures are
analysed not as an alternative approach to understand mathe-
matical concepts, but rather as an integrative strategy, since
they favour a natural communication mechanism able to easily
deliver abstract ideas that are normally considered too complex
to be understood. With respect to the gestures function they
involve familiars’ representational forms that can be used as
didactical scaffolding during the mathematical problem solving,
also improving the cognitive strategies of children.
The relationships between gestures and mathematical think-
ing have been analysed from different standpoints (Reynolds &
Reeve, 2002). Gestures and body movements have received
great attention as means to investigate both mathematical
thinking and communication strategies. This attention has in-
creased since when Mathematics was considered as an embod-
ied, socially and constructed human product (Bazzini, 2001). In
particular, the aim of these studies was to investigate how ges-
tures and bodily movements can influence the construction of
mathematical meaning. The embodied vision of the gestures
behaviour emphasize the importance to transform abstracts
concepts in a visual and concrete form, making easier for peo-
ple learning and using mathematical concepts during their edu-
cational activities (Gallese & Lakoff, 2005; Nemirovsky &
Ferrara, 2009). Gestures have a direct role in the speech pro-
duction process, where they facilitate the planning of it. When
people speak, gestures play a role to organize conceptual ideas
that are highly spatial or motoric (Kita, 2003). Other studies
have shown that gestures are more linked with words that are
spatial and concrete than with words that are non-spatial and
abstract (Goldin-Meadow, 2003; Lakoff & Núñez, 2000).
As well as based on this approach, gestures are a particular
modality of embodied cognition; they are connected to the
speech and they can serve as important bridges between inter-
nal imagery (or mental state) formal and symbolic expressions
of mathematical ideas (Bazzini, 2001; Kita, 2000). Gestures are
therefore an important source of information, since body
movements support the oral communication reducing language
ambiguity and improving the share of scientific concepts. In
addition, gestures are related to problem solving process rein-
forcing and improving the conceptual representation of mathe-
matical concepts. When all subjects share within the group the
same cognitive strategies, this mechanism reduces their mental
effort necessary to apply the problem solving methodologies.
The participants were selected from an age interval within
which previous studies had provided different examples of how
children use gestures to solve mathematical problems (Arza-
rello et al., 2006; Cook & Goldin-Meadow, 2006). The partici-
pants were five subjects (2 males and 3 females), aged between
9 and 10 years (M = 9.8; SD = 0.35), attending the fifth-grade
class of a primary school in Turin, Italy. For the purpose of this
study, the students remained in their classroom, and then
worked in-group to solve the mathematical problem. No pre-
liminary test was used to measure the participant’s cognitive
abilities. They participated to the research after having obtained
their consent from the school Director.
Participants worked in their school to grant them a familiar
environment. Each work session lasted approximately 40 min-
utes. The session began with the reading by children of their
task. While the participants worked to solve the problem, the
researchers remained in the classroom and videotaped their
activity. However, researchers did not give to participants any
tip or suggestion but just supported them by providing paper
and other materials. Using a digital video camera, which was
situated on a tripod during all the observational sessions, we
videotaped the participant behaviours. The video camera was
set, in order to record carefully all participants’ activities and
making easier the next codifier work.
After this initial phase, the second step was to analyse the
data analyze by applying the observational method. We re-
cruited one codifier with the intent to codify participant ges-
tures. The codifier, before starting to collect the behavioural
data, followed a training course on how to recognize the be-
haviours (or gestures) listed in the taxonomy. Thus, the codifi-
ers with the support of a researcher better learn in fact to recog-
nize all behaviours and after this training session the codifiers
began with the behavioural data.
The gestures were codified from the videotapes among par-
ticipants’ interactions in the school, using the taxonomy de-
signed according to the previous studies (Edwards, 2009; Rad-
ford, Edwards, & Arzarello, 2009). The taxonomy has been
designed by observing the participants behaviour while they
worked to solve the mathematical problem. The taxonomy aim
was twofold. The first aim was to record the behavioural ges-
tures linked with the mathematical concepts; the second aim
was to collect the behavioural actions that participants realized
during the workgroup session.
Then, we accomplished a taxonomy that included five main
gestures macro-categories: Deictic Gestures; Iconic Gestures;
Regulator Gestures; Adaptive Gestures; Problem Solving Ges-
tures. In order to define the participant’s activity, we have indi-
viduated for each behavioural macro-categories other specific
The gestures were initially identified by using the scheme of
McNeill’s (1996) classification. After this initial work, we have
individuated further categories. The most important gesture
types individuated by McNeill (1996) are: Iconic Gestures,
which “bear a close formal relationship to the semantic content
of speech”. Metaphoric Gestures means, “the pictorial content
presents an abstract idea rather than a concrete object or event”
(p. 14). Beat, a repetitive gesture that “indexes the word or
phrase it accompanies as being significant” (p. 15), and Deixis,
a “pointing movement [that] selects a part of the gesture space”
(p. 80). The gestures produced during the participant’s activities
were analysed by using the observational method. Table 1
shows an example of the gestures taxonomy used to analyse
participant’s behaviour.
Defining the Target Behaviour
The target gestures (or taxonomy) were the result of the par-
ticipant’s behavioural analysis while they were intent to solve
the mathematical problem. As we mentioned above, the re-
searcher analysing participant’s behaviour has identified and
analysed five main behavioural categories:
1) Deictic Gestures: we intend behaviours in which the sub-
ject uses them to point and serve to move the listener attention
from the speaker to the near region of the referent. Deictic
Gestures indicate also how to individuate things, directing-to
and positioning-for. These behaviours are used to determine
how the subjects interact not only with objects but also com-
municate ideas and conceptual aspects by gestures.
2) Iconic Gestures: all gestures used to illustrate spoken
words. Subjects use them to encode difficult messages, reduc-
ing the communicative effort of the speaker and then helping
the receiver to easily decode the messages. As a typical exam-
ple, illustrators emphasize actions that represent geometric
shapes. Among these behaviours, we include actions that illus-
trate certain aspects that connect spoken language and gestures.
In particular, these gestures provide emphasis to concurrent
speech, controlling the conversation between subjects.
3) Regulators Gestures: all the gestures that support both in-
teraction and communication between emitting and recipient.
These behaviours control the flow of conversation such as nod-
ding the head up and down to indicate agreement and as though
signalling the other to continue the conversation.
4) Adaptors Gestures: all the gestures that are not used inten-
tionally during the communication or interaction. These behav-
iours include actions that presuppose the subject’s ability to
equilibrate their own motivational internal state to continue
their activity. They include behaviours in which subjects touch
some body parts as cognitive strategies to improve the quality
of the conceptual analysis of the problem.
5) Problem Solving Gestures: we intended all gestures used
by subjects to help other colleagues while work together to
solve mathematical problems. In particular, we intend all be-
haviours that indicate the cognitive strategies that subject real-
ize to solve the problem.
A student coder, blinded to the main study hypothesis, was
trained both to use the taxonomy and to apply the rating system
methodology. The coder analysed several examples of the
videotapes collected during the participant’s problem solving
activities. At the end of each training session, we checked the
coder performance by calculating an inter-rate agreement. We
compute the percent agreement between coder and researcher
involved in the current study. According to the taxonomy be-
haviour, we have checked the rate achieved to identify correctly
the participants’ gestural behaviour. To determine the in-
ter-reliability during data coding, we analysed randomly some
participant’s behaviour selected from collected videotape. The
Table 1.
Example of regulators gesture s .
Code Description
Head movements Children move the head, neck, eyes
Hand gestures Children use hands to touch some body parts
Posture Body postures and other parts such as torso and arms
inter-reliability was calculated by using the Cohen Kappa sta-
tistic method, which agreement index obtained ranged from
0.70 to 0.80. Generally, a Kappa value of 0.70 is considered to
be satisfactory.
Coding Gestures
Each observational session was analysed for every behav-
ioural category, for each participant of all groups and for the
whole duration of the work session. We have adopted the direct
observation method to analyse participant behaviour. The coder
observed a single participant for the entire time, recording the
beginning and the end of each behavioural pattern (duration) of
all instances of the taxonomy behaviours. Besides, the coder
counted each single duration pattern obtaining the frequencies
number of the behaviours realized by participants. We have
used a checklist method to register participant’s behaviour,
which is an inventory of all target behaviours that the codifier
marked when the behaviour occurred.
The videotape analysis was segmented into meaningful ac-
tion units. In fact, we coded only behaviours in which the par-
ticipant performed gestures directed to solve the mathematical
problem. For example, moving in the room was not considered
as a relevant behaviour for our analysis, whereas asking the
support or explaining to other participants’ own ideas how to
solve the problem was instead coded as a relevant behaviour.
Both the frequencies of the relevant actions and the duration
they lasted were registered.
Statistical Analysis
We have used the chi-square statistical test to evaluate the
relationships between frequency and duration in comparison
with each gesture category. We needed to determine whether
the distribution of gesture behaviour for frequency was signifi-
cantly different from the distribution of the duration.
Quantita ti ve S u mmary
A corpus of 538 gestures was collected analysing the video-
tape session; there was a total of 20 minutes of videotape data.
The corpus does include a large number of Deictic Gestures
(e.g., use of the paper and other behaviours) that occurred while
the participants worked together to solve a written problem.
The participant’s gestures were produced while they discussed
to find the problem solution. The average rate of gestures pro-
duction while the participants discussed within the group was
3.32 gestures per minute. The total gestures produced by each
participant during the problem solving session were 39.03.
Table 2 shows the major gestures realized by participants dur-
ing the problem solving session.
Table 2.
Types of gestures by most sal ient dimensions.
Deictic Iconic Regulator Adaptive
solving Total
Frequency 74 72 139 109 144 538
Percentage 13.75 13.38 25.84 20.26 26.77 100
The participant gestural behaviours have also been analysed.
Chi-square result showed no significant differences among
gestures behaviour and the two variables measured (frequency
χ2 (19) = 4.60, p > .05 and duration χ2 (17) = 7.40, p > .05).
Statistical analysis revealed that gestural behaviours were dis-
tributed among all participants’ activities. For the educational
task proposed, we found that participants realized several ges-
tures that evolved during the different phases of the whole dis-
cussion process. When the participants’ comprehension of the
problem was better, then them realized more articulated gesture
configurations. This result confirms that gestures are important
for all phases of the problem solving process, and then was
associated with specific behavioural category. In the next sec-
tion, we give an overview of the principal gestures behaviours
that participants realized during their educational activities.
Qualitative Summary
In this section, we provide a qualitative analysis of the par-
ticipants’ gestural behaviour. They realized different type’s
gestures to communicate their own ideas to other people, in
order to infer the solution of the problem. The gestures sup-
ported the discussion among all participants; in particular, when
they are not yet able to individuate the correct solution of the
problem. We analysed the participants’ gestural behaviour
while they worked together to find the solution.
As we already said, the problem assigned was to understand
if Penelope sewing the cloth every day would be able to finish
it in 50 day before Ulysses arrives. To solve this problem, all
participants realized gestures and used structured objects such
as paper and pencil to represent their own concepts. Participants
needed to combine gestures and tools reinforcing their mental
representation of the mathematical concepts and then to use
them to explain and describe the problem.
The gestures shown in Figure 1(a) and in Figure 1(b) repre-
sent two kinds of iconic movements, which refer to concrete
actions or events (Edwards, 2009). Both participants described
an imaginary dimension of the span, displaying clearly the
physical process regarding the construction of it. After, see
Figure 1(b), the hand movement is similar, but they realized a
shape that includes a spatial dimension of the span. In virtue of
their shared socio-cultural background, participant interpreted
easily these gestures as referring to the action, which they mean
to represent the dimension of a span.
When gestures and speech are correctly associated, subjects
reproduced the same hand movement confirming their active
role of the attention in the learning process (see Figure 2). Par-
ticipants used behaviours compatible with their verbal language
and relied on paper writing to make more comprehensible their
mental representation of the concepts. Gestures reproduced by
other people represent thence a careful cognitive support to
stimulate the discussion within the group, improving the quality
of communication.
Figure 1.
Hand gesture that represen ts p r e c i s e s h a p e .
In Figure 3(a) and Figure 3(b) subjects combine a conceptual
integration between gestures and tools. In this context, the af-
fordances of the gesture hand and the physical objects allow
participant to produce a new conceptual analysis of the problem
solving. Participants, in turn, analysed the mathematical proce-
dure by creating a new shape or conceptual strategy to identify
the correct solution of the problem under investigation.
We needed to determine whether the distribution of gesture
behaviour for frequency was significantly different from the
distribution of the duration.
Examining participant’s behaviour, we found that gestures
reinforced the scientific communication (e.g., gestures and
speech converged on the same hand representation), so they
tried to find a solution bearing in mind the hand representation.
When the gestures were more structured and correct, partici-
pants reduced their spontaneous production focusing on spe-
cific aspects of the problem. In addition, these representations
improved the spatial relationship between speech and hand
movement as showed in Figure 1(a) and Figure 1(b). In fact, the
form of the gestures with palms perpendicular to the table and
facing each other, suggested that children attempted to realize a
(b) (c)
Figure 2.
Subject (a) reproduces the initial shape. Other subjects reproduce the
same shape ((b) and (c)).
(a) (b)
Figure 3.
Objects and tools as support sol vi n g mathematical problems.
mental model of the span conceptualizing it and therefore re-
ducing its abstract meaning (Arzarello et al., 2009; Arzarello et
al., 2006; Bazzini, 2001). Therefore, these gestures represent
also a metaphoric idea of the concept of span; ideal entities are
treated as objects. Participants tried to understand the span
properties represented it by using hand movements. These be-
haviours provided to the participants concrete entities that allow
them to express themselves through the gestures abstract means.
Based on obtained results, we propose a conceptual scheme
of gesture behaviour, which aim is to outline the gesture/per-
ception cycle as a dynamic mechanism that their production
involves (Figure 4). Initially, the subjects produced a gesture
that modifies the environmental context, by creating new stim-
uli. After, the subjects perceived this behavioural pattern and
then analysed it, in order to identify the conceptual properties
that gestures communicate.
An important aspect of the analysis phase concerns the con-
ceptualization process. In this phase, subjects plan the next
movement, showing how the cognitive processes elaborate the
problem aims and then they decide the action that it is more
suitable, in order to achieve the correct solution or to revise the
previous mental strategies.
According to Parrill and Sweetser (2004) even metaphoric
gestures have an iconic property, since by means of the hand
shapes and movement, they appeal to some visual or concrete
representation. These situations offer specific elements that
underlie an inferential thinking as support to the children un-
derstanding of the abstract concepts expressed by using ges-
tures movements. Specifically, a metaphoric gesture includes
two conceptual mapping sequences: one iconic between real
space and visual/concrete situation, and a second one between
the conceptual space and the intended abstract meaning (Ed-
wards, 2009). Gestures allow the subjects to manipulate
mathematical concepts by using only certain kind of actions
and configurations, which offer concrete and stable representa-
tions of the abstract concepts. These kinds of gestures display
some type of iconicity. We can see how the gestures refer to a
symbolic content rather than to a concrete object. This type of
reference is called “chain of signification” (Edwards, 2009).
Finally, gestures can be seen as simple illustrations of stu-
dents’ ideas, which incorporated hands-on materials into con-
ceptual mathematical aspects (see Figures 2(a), 2(b), 2(c)).
Gestures and tools may play an important role in students’
learning. In this context, the interaction between gestures and
tools acted as scaffold to support the development of a concep-
tual understanding. This process encouraged the discussion and
other appropriate cognitive strategies that supported the sub-
jects thinking. Besides, these are clear examples that a more
efficient cognitive strategy requires representational forms to
communicate own conceptual ideas.
In this paper, we have examined the role of gestures as fa-
cilitators to solve mathematical problems. We have argued that
gestures play an important role for the learning and that they
assist children in acquiring, restructuring and tuning their own
knowledge. Moreover, much of learner’s conceptual knowledge
is socially constructed and shared, and gestures should be used
to assist them in this endeavour. Our findings show that ges-
tures reveal important information about children’s reasoning
on mathematical problem solving strategy. Children realize
Figure 4.
Conceptual schem e to represent the gesture/perception cycle.
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
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.
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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