A USER-INTERFACE ENVIRONMENT AS A SUPPORT IN
MATHS TEACHING FOR DEAF CHILDREN
Maici Duarte Leite, Laura Sánchez García, Andrey R. Pimentel, Marcos S. Sunye
Marcos A. Castilho, Luis C. Bona and Fabiano Silva
Computer Science Departament of the Federal University of Paraná, UFPR, Curitiba, PR, Brazil
Keywords: Interface design, Educational interfaces, Accessibility, Cognitive theories, Deaf users.
Abstract: The use of theories stemming from different areas of knowledge has contributed both to knowledge
acquisition – since these theories help perceive individuals more effectively – and to the development of
educational software. This is the case, for instance, of cognitive theories, which in turn assist in terms of
both learning support and the clarification of possible needs in the acquisition of a given concept. In the
present paper, we propose a discussion about the help function (Leite, Borba and Gomes, 2008) provided by
educational software, focusing deaf students. This paper also presents an enhancement for the interface
design of the help function, based on the Conceptual Fields Theory and the Theory of Multiple External
Representations. This interface design enhancement is created by carrying out a new analysis regarding the
display of a help function through diagrams. In short, in the present paper we derive important contributions
from these two theoretical frameworks so as to provide deaf students with support in the acquisition of
mathematical concepts, considering that these deaf students are put in an inclusive context.
1 INTRODUCTION
In recent years, the use of technology as a
supporting resource to cognitive development has
become increasingly sophisticated, leading to a
number of interdisciplinary efforts involving areas
such as Psychology, Pedagogy and Design, amongst
others. Furthermore, these technological tools have
been taking increasingly more notice of the users’
specificities.
Psychology has been paramount to
understanding both user-interaction and knowledge
acquisition, particularly when it comes to designing
educational software.
A number of new educational projects point
towards the need to develop interfaces that fulfil the
demands of users with countless communication
needs. Indeed, deaf users are a good example of
these users.
Different studies have revealed that deaf students
tend to have even more difficulty with Mathematics
than hearing students. In fact, research (such as
Austin, 1975; Kelly, Lang, Pagliaro, 2003; Kelly,
Lang, Mousley, Davis, 2003; Nunes, Moreno, 2002;
Traxler, 2002) has shown that deaf students today
have great cognitive gaps when compared to hearing
students. The reasons for such gaps are of linguistic
and experiential nature, since hearing students and
deaf students acquire and develop language
differently (Bull, Marschark, Blatto-Vallee, 2005 ;
Zarfaty, Nunes, Bryant, 2004). This way, deaf
students have different social experiences, and this
fact moves the two groups, i.e. deaf students and
hearing students, further apart.
The different perspectives underlying the
education of deaf students, particularly in terms of
the way language is handled, have led to the
development of specific educational philosophies,
which in turn resulted in a significant educational
lag (Kelly, Lang, Pagliaro, 2003). Indeed, oralism –
the practice of teaching hearing-impaired and deaf
students to communicate through spoken language
and bilingualism – the practice of teaching hearing-
impaired and deaf students to communicate by
respecting their natural language and associating it
to the native language of the country in question
make up completely different philosophies. This
means that a change in paradigm would entail
drastic changes in the actual teaching process.
The range of products developed specially for
the deaf remains rather limited, though in certain
79
Duarte Leite M., García L., R. Pimentel A., Sunye M., Castilho M., Bona L. and Silva F. (2010).
A USER-INTERFACE ENVIRONMENT AS A SUPPORT IN MATHS TEACHING FOR DEAF CHILDREN.
In Proceedings of the 12th International Conference on Enterprise Information Systems - Human-Computer Interaction, pages 79-85
DOI: 10.5220/0002973500790085
Copyright
c
SciTePress
areas of knowledge one can find a large number of
translators or translation-related products for
Portuguese and Brazilian Sign Language. Products
such as these may even fulfil the needs of the deaf as
far as communication is concerned, but they
certainly do not make the inclusion of deaf children
into regular or mixed classrooms any easier, nor do
they fulfil the demands posed by specific areas of
knowledge, such as Mathematics, for instance.
The study of Leite, Borba and Gomes (2008)
proposed the design of an human-computer interface
for an educational software wich presented the help
function interaction based on the Conceptual Field
Theory (Vergnaud, 1986). The present work
enhances the first proposing contributions by the
application of the Multiple External Representation
Theory (MRE) (Ainsworth, 2006).
The present article is organised in the following
way: initially we carry out a brief discussion of the
difficulties faced by deaf children learning
Mathematics. After that, we present both the main
presuppositions of the Conceptual Fields Theory and
the contributions of the Theory of Multiple External
Representations. Next, we describe the contributions
for the interface design using Multiple External
Representations. Finally, in the last two sections we
carry out a discussion of the results obtained and
present our conclusions.
2 DEAF CHILDREN AND THEIR
LEARNING DIFFICULTIES IN
MATHEMATICS
The need to coordinate knowledge acquired inside
and outside school with teaching-learning activities
has been emphasised in the case of hearing children
(Carraher, Carraher, Schliemann, 1988), but it is also
valid for deaf children, particularly when one bears
in mind that these children tend to lag behind in
terms of age group/level of education (Traxler,
2000). If the previous knowledge acquired by deaf
children is not taken into account, their cognitive
development may be compromised.
Informal learning also has some impact on the
acquisition of mathematical knowledge. Therefore,
knowledge stored implicitly prior to school age
might need to be reviewed. One example of that is
additive composition, whereby any given number
may be perceived as the sum of other numbers,
which in turn is a concept that can be learned before
school age with money or other items. Another
element that may lead to difficulty is inferences
about time. Deaf children have far more difficulty
than hearing children with activities involving time
in a succession of events, and this is a key ability in
the learning of the inverse relation between addition
and subtraction (Nunes e Moreno, 2002).
Different authors (Bull, Marschark, Blatto-Vallee,
2005) assert that the fact that deaf students tend to
lag behind in terms of mathematical abilities is
intimately related to cultural factors of language
acquisition. Indeed, the lack of coordination between
linguistic, symbolic and analogical means of
representation of numbers may lead to difficulty in
the acquisition of arithmetical concepts. On the one
hand, hearing children learn certain correlations by
using visual and auditory information – e.g. when
they correlate objects to the oral expressions that
represent them. On the other hand, deaf children
need to watch and correlate two kinds of visual
information in order to carry out the same activity.
Therefore, even though both hearing and deaf
students have difficulties in learning Mathematics,
there certainly are peculiarities in the way deaf
students learn it.
Poor overall linguistic knowledge and competence
may be the main hindrances for deaf students to
learn Mathematics. Both cognitive development and
experiences lived through inside and outside the
classroom are directly related to linguistic issues.
Low linguistic development affects the relationships
and experiences of deaf children, potentially
harming their cognitive development (Zarfaty,
Nunes, Bryant, 2004).
Many of the inherent learning difficulties –
Mathematics being an example of it – faced by the
deaf have a lot to do with late language acquisition.
This calls for the adoption of special teaching
practices rather than traditional ones.
3 THE THEORY OF
CONCEPTUAL FIELDS
The main theoretical framework of Conceptual
Fields Theory (CFT), in which Leite, Borba and
Gomes (2008) based their study, was created by
Gérard Vergnaud’s (1986). According to it, a
conceptual field consist of a set of situations whose
domain requires the knowledge of numerous other
concepts of different natures. As Vergnaud (1986)
explains, a concept is like the base of a tripod, as
follows: Situations that provide concepts with
meaning (S), Invariable Relations and Properties of
the concept (I), and Symbolic Representations used
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80
in the presentation, description and
operationalisation of the concept (R). The analysis
of word problem, together with the study of the
processes and symbolic representations used by
students when discussing and solving word
problems, is the key element of the above-mentioned
theory (Vergnaud, 1991).
Vergnaud (1986, 1991) asserts that, in order to solve
a word problem, one must employ a great number of
theorems, i.e. knowledge equivalent to the properties
of a concept. When doing so, one resorts to abilities
and knowledge that Vergnaud (1986, 1991) names
theorems-in-action. In short, theorems-in-action
refer to representations of the relevant aspects of the
action in question. Indeed, these representations
reveal only the essential aspects of the action.
In more practical terms, one could speak of concrete
elements (such as sticks, marks on paper, fingers) as
the invariables employed. Since the response offered
would certainly not be the quantity of material used,
but rather the quantity about which the question asks
through the representation, the objects used are not
relevant; instead, what matters is the result.
Although word problems are worked on all through
elementary and secondary school, their diversity is
not fully explored as that would require the use of
numerous situations involving countless invariables.
What one normally does use is a limited range of
additions, subtractions and properties of these
operations. Similarly, the number of forms of
symbolic representations of these operations is also
limited (e.g. Charlie has 6 cars and Paul has 8. How
many cars do they have altogether?). This kind of
limited work makes students employ a reduced
amount of specific knowledge and present
difficulties when solving other problems which, in
turn, require the use of different meanings,
properties and representation forms.
An instance of word problem characterised by
change would be the following: “Mary used to have
14 letter papers. Her mother gave her 8 more. How
many letter papers does she have now?”, whereby a
change happens to an initial number, resulting in a
new number. In other words, an initial number goes
through a direct or indirect transformation, causing
this initial number to grow either bigger or smaller.
As we can see, the idea of time is implicit in a word
problem like this one, making it an invariable that
needs to be taken into account in order for one to
find the solution.
Vergnaud (1986) divided addition and subtraction
problems – isolated or combined over natural,
integer or real numbers sets – into six categories.
Carpenter and Moser (1982), as they explore natural
numbers only, divided the problems on four
categories: combination, comparison, change and
equalisation. These 4 categories give rise to 16
different situations, depending on where the
unknown number is located.
The Carpenter and Moser (1982) classification was
adopted by Leite, Borba and Gomes (2008), which
proposed a help-function using a different diagram
form for each one of the four problem categories.
4 VIRTUAL ENVIRONMENTS
AND THEIR MULTIPLE
REPRESENTATIONS:
SUPPORT TO TEACHING AND
LEARNING ONCLUSIONS
The Theory of Multiple External Representations
(Ainsworth, 2006) is a cognitive theory that
advocates the use of specific techniques to represent,
organise and present knowledge. MERs have the
following three key functions: complementary roles,
constrain interpretation and construct deeper
understanding.
The first function, “complementary roles”,
explores representations that, for being of different
types, complement each other and offer support to
the cognitive process. The main objective of the
“constrain interpretation” function, on the other
hand, is to use representations that helps the learner
to avoid misintrepretations about the concepts in
question. Finally, the “construct deeper
understanding” function uses MERs as a tool for
helping the learner to buid abstractions about the
concept and organize it in a higher level.
According to Ainsworth (2006, 2008),
representations may appear simultaneously or
alternately. Either way, learners must be able to
understand the form of representation, its relation to
domain, how to select an apropriate form of
representation and how to create an appropriate one
(Ainsworth, Wood, Bibby, 1996). This way, one can
construct new representations and access other
representation options which, in turn, help expand
the conceptual field in question.
Having a wide range of representations may
result in more effective learning conditions,
particularly when these representations provide
learners with a more in-depth view of the concept in
question, or when they suit the user’s cognitive
model. Furthermore, making use of more than one
representation type may help grab the students’
attention for longer (Ainsworth, 1999).
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81
Nevertheless, in order for the system to be truly
advantageous, i.e. in order for these functions to
actually be fulfilled, the design of a Multiple
External Representations environment must follow
certain rules, presented at the framework Detf
(Ainsworth, 2006).
As Ainsworth (1999) explains, MERs may be
used in numerous computer contexts and in
combination with users’ profiles and their own
representation preferences. In this case, we would
have both the communication specificities and the
mathematical invariables needed to employ a
concept described within CFT.
The main contribution made by the MERs to the
present study lies in the “complementary roles”
function. Indeed, as already pointed out above, this
function sugests that the use of different
representations stimulate learning further when
compared to single representations (Ainsworth,
2006) – which, by the way, is what CFT advocates
as well (Leite, Borba, Gomes, 2008). Another great
advantage of MERs is that fact that they are research
objects in learning, cognitive sciences and
constructivist theories.
5 COGNITIVE THEORIES:
CONTRIBUTIONS
Educational software does not always take into
account how able users may or may be not to
understand a concept, or the use of trial and error.
The interface design proposed by Leite, Borba,
Gomes (2008), for example, uses CFT because it
offered the necessary framework to solve problems
across different categories (combination,
comparison, change and equalisation).
At Leite, Borba, Gomes (2008), the use of help
through diagrams consists of presenting an image
that helps users to start building the concept in
question. To develop it, they used Vergnaud’s
invariables for each kind of problem. The authors
intention was to help learners to employ the correct
invariable, as well as to enable them to complement
the knowledge that may be lacking or incomplete.
Indeed, the use of diagrams to assist in the
resolution of problems is largely explored in the first
school years, when certain concepts are gradually
being formalised. Even though images may be
revealing and clarifying, formalising concepts is
necessary for further generalisations. In any case,
learners, which have difficulties, should choose the
option that suits them better. The study of Leite,
Borba, Gomes (2008) showed that the more choosed
option was the diagram form.
Let us take a problem, presented in Leite, Borba,
Gomes (2008), in which combination is involved:
“Peter has bought 15 oranges, and Helen has bought
6 oranges. How many oranges have they bought in
total?”. In this problem, one must employ the
invariables of a static relationship between two
quantities and their parts. In this light, we have come
up with a diagram (Figure 1) exploring two separate
sets of objects (parts) and then the whole set formed
by their addition. In this specific case, we had a
character and a certain number and another character
with another number, with a thin line separating
them. By using a circle surrounding both characters,
we expressed the addition of the elements.
Therefore, our intention was to show how a certain
whole is made up of parts, and one can determine
the whole by knowing the parts – or determine a part
by knowing the whole and one of the parts.
At the referred study, the use of diagrams was its
major contribution. The use of a diagram allow to
more easily explore the perceptual process, once it
groups the informatiom in a more cohesive form
(Ainsworth, 2008).
Figure 1: Combination Diagram (Leite, Borba, Gomes,
2008).
As Ainsworth (2008) asserts, sets of information
that are typically bound to different dimensions will
dificult its comprehension if they are put all together
in one representation. The use of different
representations, one for each set will make the
comprehension more confortable to the learner
Let us now turn to a problem of comparison,
presented in Leite, Borba, Gomes (2008), namely
“Peter has bought 10 oranges, and Helen has bought
6 oranges more than Peter. How many oranges has
Helen bought?”. When it comes to static numbers,
diagrams (Figure 2) allow one to identify one
number by adding another number to a pre-existing
static relation. In this specific case, we used the
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metaphor of two baskets initially containing the
same number of fruits, and then one of them being
added fruits, thus revealing the difference between
the numbers of fruits in the two baskets.
Figure 2: Comparison Diagram (Leite, Borba, Gomes
2008).
In the case of a word problem of change, we have
the following example: “Helen had 8 oranges in her
basket. She took 5 out of her basket and put them
into Peter’s basket. How many oranges does Helen
have now?”. The diagram (Figure 3) shows a
dynamic relation, i.e. the initial number went
through changes because of a direct or indirect
action, causing it to either grow bigger or smaller. In
this case, the variation involved an unknown number
and a situation of decrease. In order to express this
decrease in the original number through the diagram,
we used a contrast between the fruit colours all
through until the final state.
Figure 3: Change Diagram (Leite, Borba, Gomes (2008).
The use of temporality in this dynamic relation
diagram requires the presentation to depict action
through a time axis, thus allowing users to have the
same kind of perception. In the case of Figure 3, first
Helen appears with her apple basket, followed by
Peter with his. Finally, the colour contrast reinforces
the dynamic representation of the animation.
As for the equalisation category, we have the
following example: “In Peter’s basket there are 9
oranges, and in Helen’s basket there are 6 oranges.
How many oranges does Peter have to take out of
his basket in order to have the same number of
oranges as Helen?”. In this case, the problem
explores the invariable of a dynamic relation through
which numbers are equalised from a comparison
situation. For this particular example, we make use
of a weighing scale, as shown in Figure 4, to suggest
the invariable employed – i.e. the dynamic relation
of comparison between the numbers and their
equalisation.
Figure 4: Equalisation Diagram (Leite, Borba, Gomes
2008).
Like change word problems, equalisation word
problems should include a time axis so as to fulfil
the demands of the dynamic animation. In this case,
we achieve this by showing an unbalanced scale,
heavier on the side where there are more oranges. In
addition to that, we have a dotted line indicating the
state of balance/equalisation between both sides.
As for the messages about the kind of mathematical
error, our intention was to indicate possible wrong
procedures concerning the use of the algorithm. In
this light, we have used objective and clear
messages, so that users can understand them easily –
such as “add them again” or “subtract them again”.
In these examples, users had already chosen the
correct operation to find a solution, but needed to
repeat the operation itself, i.e. check the number
obtained. This is an indirect way of telling users to
pay attention to which aspect has not been
successful in their operation.
The four diagrams presented in Leite, Borba and
Gomes (2008) demanded dynamic and static
representations. Although the Conceptual Fields
Theory has made clear through the description of
invariants of each category and the referred study
has run the experiment using temporality at the
actions when they demanded statis representations,
the Theory of Multiple External Representations
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83
emphasizes the need of dynamic and statical form of
presenting as required by each problem.
The problems involving the categories of
Combination and Comparation exploited phenomena
with an statical representation. Although the study of
Leite, Borba and Gomes (2008) has validated only
um problem for each classification, the
generalization of statical representation extends to
the variations of them. The Multiple External
Representations emphasizes that the use of statical
diagrams is far more complex, once it is demanded
to communicate all event in only one moment
(Ainsworth, 2008)
The problems of Change and Igualization categories
the use of dynamic representations was needed,
since they involved invariants according to the
Conceptual Fields Theory. The use of dynamic
representations reduces the cognitive load, allowing
learners to focus on their actions on the
representations and its consequences in other
representations. One sugention would be present the
dinamic aspects in a image sequence.
6 CONCLUSIONS
For being based on the development of the
individual, cognitive theories tend to contribute
greatly to interface design.
While it is true that technology is present in the
most varied areas of Education, and therefore could
not be excluded from Special Education, on the
other hand it is also true that the use of obsolete
educational practices may distort and even
compromise the design of an interface.
Another aspect one must take into account
concerns the inclusion of deaf students in regular
schools. Indeed, neither the students themselves –
who probably used to attend so-called special
schools – nor the teachers – who certainly were not
trained to deal with students with special needs –
were prepared to deal with this new situation.
Although the study conducted by Leite (2007)
has already called attention to the need for an
inclusive interface that respects all users’
communication needs, in the present work we have
stressed the relevance of help functions through
diagrams.
Indeed, not only does this kind of symbolic
representation through diagrams allow for a better
understanding of the structure of the word problem,
but it also helps reveal the meaning of the operations
involved.
Even though it is often said that mathematical
difficulties (particularly concerning the acquisition
of concepts surrounding the notion of addition) have
a lot to do with the organisation of algorithms (i.e.
numeric calculations), the strategies adopted and the
invariables employed by the students are also crucial
elements that have a great impact on the acquisition
of these concepts.
By using representations that employ the same
concept, yet in a different, untraditional way (which
in turn can be chosen by users according to their
preferences), we have expanded the mathematical
concepts in question. Indeed, by broadening the
students’ experiences of these concepts and the
theorems-in-action, the mathematical concepts
acquired become more mature and solid.
In this light, we are convinced that our study has
brought about a new perspective in terms of what
mathematical software should include. This
contribution, we feel, have to do with both
respecting all students’ – hearing and hearing-
impaired – preferences and their learning time to
acquire a new concept. More importantly than that,
our contribution is based on a theory that focuses on
the acquisition of concepts.
As far as theories are concerned, another relevant
aspect of the present study is the fact that the two
theoretical frameworks chosen advocate the use of
representations to complement the acquisition of a
concept.
Indeed, the use of representations that
complement each other benefit users greatly,
because they help bridge the students’ knowledge
gaps in an interactive, fun way.
Furthermore, when it comes to the interpretations
of images with a theoretical intent, these
representations minimise the communication
difficulties faced by deaf students.
In conclusion, then, the present study raised
hypothesis about the contribution of cognitive
theories to the help function of mathematical
software. In this light, the next step would be to
compare and contrast the representations currently
used in the classroom to those recommended by the
two theoretical frameworks used here.
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