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

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

 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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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).

A USER-INTERFACE ENVIRONMENT AS A SUPPORT IN MATHS TEACHING FOR DEAF CHILDREN

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

ICEIS 2010 - 12th International Conference on Enterprise Information Systems

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

A USER-INTERFACE ENVIRONMENT AS A SUPPORT IN MATHS TEACHING FOR DEAF CHILDREN

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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