ELSA FOISACKMALMÖ STUDIES IN EDUCATIONAL SCIENCES No. 7, 2003

ELSA FOISACK

DÖVA BARNS BEGREPPSBILDNING I MATEMATIK

English abstract

Deaf children´s concept formation in mathematics

The question why deaf children have difficulties in learning mathematics is the basis of this study. The aim of the study is to illuminate deaf children's concept formation in mathematics by describing how some deaf children express themselves and act on their way towards understanding two basic concepts, the concept of multiplication with whole numbers and the concept of length.

Theories developed by Feuerstein are used in order to describe how deaf children develop concepts, and to investigate possibilities to help deaf children develop their cognitive potential in a more effective and adequate way. Concept maps illustrate steps and pathways taken by the pupils. The importance of language in concept formation, with focus on sign language is illuminated.

The children in this study were pupils in a School for the Deaf, a bilingual school with the languages Swedish Sign Language and Swedish. Seven 11-year-old pupils, all the pupils in one group in grade 4, were studied. Video recordings were made of pupil-teacher interactions in problem solving situations in sign language only, with paper and pencil, with learning materials and with real things.

 A large variety in the pupils ability to solve the problems was found depending on different factors identified by Feuerstein e.g. self-confidence, looking for meaning, search of challenge, intention to finish the work and use of known facts. No difference was found concerning the steps towards comprehension of the concepts for the deaf pupils in the study compared to those of hearing pupils. In accordance with earlier studies it was found that the deaf pupils needed more time to learn mathematics than hearing pupils normally do. As a consequence they may learn certain concepts at a later age and the pathways towards comprehension may vary compared to those of hearing pupils.

The structure of sign language and the lack of an established terminology in mathematics are also of importance. The bilingual situation for deaf pupils is a reason for developing methods of teaching mathematics to deaf pupils alternative to methods used today.

Summary

This study is based on the question why deaf children have difficulties in learning mathematics. International research (Frostad, 1998; Magne, 1991; Moores, 2000) shows that deaf pupils achieve much lower results on tests in mathematics than hearing pupils do. On the other hand there is no research available today showing that the cognitive potential of deaf pupils differs from that of hearing pupils (Martin, 1991). If it is a fact that deaf pupils do not use their cognitive potential to a full extent, it is of great importance to investigate why, and to find possible ways of improvement. Other ways of assessing than by ordinary tests might show other results than those referred to.

Mathematics in its broad sense is constantly becoming more and more important in our society. Mathematics is not only one out of several subjects in school, it is a way of thinking necessary in everyday life. Nunes and Bryant (1996) define this as numeracy, which they consider to be of equal importance as literacy. Numeracy has to be developed by every child in three main steps. First you need to be logical, second you need to learn conventional systems and third you need to use your mathematical thinking meaningfully and appropriately in situations.

The children in this study were pupils in a Special School for the Deaf, a bilingual school with the languages Swedish Sign Language and Swedish. Going to school in a Special School for the Deaf is in some ways different from going to a regular school. The sign language environment is crucial as well as the bilingual approach with the two languages Swedish Sign Language and Swedish in its written form as two separate subjects (Skolverket, 2000/2002). But the basic objectives are the same as in all schools in Sweden. In mathematics the deaf pupils are taught on the basis of the same curriculum as all other pupils in Sweden (Lpo94, 1994) syllabi and criteria for grades (Skolverket, 2000) are the same. In the curriculum for compulsory schools in Sweden the aim of learning mathematics is “to master basic mathematical thinking and be able to use it in everyday life”.

As to communication and particularly language as an important tool for developing thinking the importance of sign language for sign language users in learning mathematics is emphasized in this study. Deaf children learn sign language in a natural way if they are exposed to it in a sign language environment. That is the case for most deaf children in Sweden today. Swedish is regarded as a second language for deaf people. Most deaf children learn Swedish at the same time as they learn to read and many of them not until they start school. Thus it is self-evident that mathematics is taught in sign language from the beginning and that sign language is the language in which they develop basic concepts. By the time the ability to read and write is developed, the pupils can find new information on their own and express themselves in written language, but there is still a need for direct communication in sign language. In the syllabus for the schools for the deaf (Skolverket, 2000/2002) there is a general text on bilingualism from which the following quotation is taken:

Learning occurs through both languages. Bilingualism is therefore important in all school subjects, not only in the two subjects Swedish and Swedish Sign Language. Every subject has its own subject-specific concepts and a terminology that the pupils do not meet in other subjects. As a consequence it is every teacher’s obligation to make sure that the pupil masters these concepts in both languages.

Concerning mathematics there are a great amount of well defined subject- specific concepts and related terminology in spoken/written languages that pupils in school need to master. The quotation above shows that there is a need for an established terminology in Swedish Sign Language for mathematics. It does not seem fair that tests used in national assessments should be available in written Swedish only, the second language of deaf pupils. Most learning material is also in written Swedish.

Aim

The aim of the study is to illuminate deaf children’s concept formation in mathematics by describing how some deaf children express themselves and act on their way towards understanding two basic concepts, the concept of multiplication with whole numbers and the concept of length.

How deaf children achieve knowledge of basic mathematical concepts is studied in order to find ways to meet the needs of deaf children learning mathematics and to promote more effective and adequate teaching in general.

Questions of significance to the study:

  • How do deaf pupils, those who have already understood as well as those who are on their way to understanding the concept of multiplication with whole numbers and the concept of length, express themselves and act when confronted with a mathematical problem of this nature?
  • Does the way deaf pupils express themselves in sign language influence their concept formation?

From a mathematics education perspective the following questions are then raised:

  • What steps are needed for understanding the concept?
  • Are the steps the same for deaf pupils as for hearing pupils?

In order to follow the development of the pupils on their way towards understanding the concept learning situations are arranged on the basis of theories developed by Feuerstein concerning mediated learning.

Theoretical framework

Two separate perspectives were used in the study, cognitive education and mathematics education.

The cognitive education perspective is based on theories developed by Feuerstein (Feuerstein et al., 1991), Structural Cognitive Modifiability (SCM) and Mediated Learning Experience (MLE). Feuerstein has a constructivistic view of learning in contrast to a behaviouristic view. SCM is closely related to theories by Piaget, but the uniqueness in Feuerstein’s theory is the connection with the theory called Mediated Learning Experience (MLE). Feuerstein (1988) has been influenced by Vygotsky (1978) as to how learning is developed in a social context. The most important characteristics of MLE are mediation of intention and reciprocity, mediation of transcendency and mediation of meaning. MLE is used in the method for assessing children’s cognitive potential developed by Feuerstein, called Dynamic Assessment (Feuerstein et al., 1979) to reveal the learning potential of an individual. In dynamic assessment interaction is crucial. How much mediation is used and of what kind is registered for the purpose of the pupils to understand and solve problems on their own.

Mediation and dynamic assessment are used in this study to reveal the process of learning how to form concepts in mathematics. The theories developed by Feuerstein are used in order to describe how deaf children learn, and to investigate if it is possible to help deaf children to develop their cognitive potential in a more effective and adequate way than is usually done.

From the mathematics education perspective insightful learning, problem solving and communication are considered to be crucial in developing mathematical knowledge (Verschaffel & De Corte, 1996). Conceptual knowledge in interaction with procedural knowledge is of importance (Hiebert & Lefevre, 1987). Insight and understanding of how to solve a task must generally precede the training of the calculating skill. According to Ahlberg (1992) children need to learn to count and to solve problems. Automatized calculations are timesaving, but they do not develop the ability to formulate, understand and solve problems.

Steps children in general take on their pathways towards comprehension of the concepts were searched for and are described in concept maps (Novak, 1998). In the concept map the concepts are hierarchically ordered and are connected to each other in a network by link words to build statements. Concept maps were used in this study to give an overarching description of mathematical concepts and their connections included in this study. Concept maps illustrate steps and pathways taken by some of the pupils in the study in forming concepts.

The importance of language in concept formation, with focus on sign language, is illuminated in the study. Since the study is theoretically based on a constructivistic view with concepts being developed in a sociocultural context, a language for direct communication is crucial. Swedish Sign Language (Bergman, 1979) is the language developed by deaf people in Sweden and is used among people who do not hear.

Empirical study

The study is focusing on the process of learning basic mathematical concepts. By choosing to concentrate on the concept multiplication with whole numbers and the concept of length in two separate parts of the study, it was possible to look into both arithmetic and geometry, since they represent different aspects of mathematical knowledge and since both number and space are mentioned as basic concepts in the current syllabus (Skolverket, 2000).

The children in this study were pupils in grade 4 in a Special School for the Deaf. There were seven 11-year-old pupils in the study. They were all the pupils in one of two parallel groups in the fourth grade. All seven had been taught in the same school since the first grade. The level of knowledge was not considered when organizing the groups. None of the pupils in the study had deaf parents. As many as half the number of the parents had another language than Swedish as their first language.

Video recordings were made of pupil-teacher interactions in problem solving situations. Communication was brought about in sign language. The pupils met the teacher one by one several times.

Problems to be solved were generated out of a given situation. The solution of a problem presented was discussed in four different ways: in sign language with no material available, with paper and pencil, with learning materials i.e. Centimo and Cuisenaire-rods, and with real objects. In the assessment lesson it was important to start in the most abstract way and let the most concrete way be the last. To go from telling to acting was needed for the assessment of the pupil’s level of abstraction. Later on the pupils were free to choose ways to solve or to explain the problem.

In the multiplication study a problem was chosen and the difficulty of the problem was increased by changing one of two factors. The problem was to find out how many apples are needed if three/four children are to have three apples each, if seven children are to have three apples each or if one-hundred-and-three children are to have three apples each. The reason why I chose the numbers of children to be 3 or 4, 7 and 103, was that they were numbers with a certain meaning to the pupils. They were the number of children in groups they belonged to themselves. There were 7 pupils in their class, there were 3 boys and 4 girls in the class and there were 103 pupils in the school altogether at the time, a matter that had recently been focused on in a speech by the principle of the school.

In the length study the teacher and each of the pupils discussed the lengths of shelves to be put on the wall in different places. It was a problem they were familiar with from the subject of woodwork. Another reason was that we could put real shelves in front of us on the table. We could talk about lengths in a horizontal direction and the lengths could be kept shorter than one meter to focus on working with measurements in centimetres only. The pupils were challenged to estimate, measure and compare the lengths of different shelves. Four different shelves were used.

The first lesson with each pupil and in each of the two parts of the study was regarded as an assessment lesson. After the lesson a brief analysis was made from the video recordings and an assessment was made of the pupil’s ability to solve the problem presented and the level of understanding the concept in question. Assessment was also made of what else the pupil might need to know about how to learn and how to develop understanding of the concept. Dynamic assessment was used by mediation if needed for the pupil to solve the problem and to develop better understanding of the concept. How much mediation was needed and of what kind was registered. With pupils who solved the problem on their own, attention was concentrated on how the pupil described his or her way of thinking in order to find new ways of helping other pupils to solve the problem.

With pupils who needed help to solve the problem, analyses were made to find what was needed for the pupil to solve the problem and to develop better understanding of the concept. One or two lessons were consciously planned out of the needs of the pupil. The goal for the teaching was for each pupil to understand the concept to the extent it was presented in the study. The teacher interacted with the pupil using mediation if needed for the pupil to solve the problem and to develop understanding of the concept.

A test was then given to the pupils in the study containing the same kind of problems as in the lessons. The tests were presented in written Swedish and was given to the pupils individually. After the pupil stopped working on the test, the teacher initiated communication if needed for the pupil to understand the solution of the problem. Video recordings were made to be used in the analysis. For each pupil an assessment was made of what mathematical steps he or she mastered during the first lesson and in the test situation. The result from the two occasions were compared.

Conclusion

When analysing the data from the Cognitive Education perspective I found a large variety in the ability of the pupils to solve the problems depending on factors defined by Feuerstein (1980) e.g. self-confidence, looking for meaning and search of challenge, intention to finish the work, use of known facts. They are all factors of importance to communicative competence and to problem solving.

When analysing the data from the mathematics education perspective I found no difference in general concerning steps towards comprehension for the pupils in the study compared to those of hearing pupils as far as comparing spoken Swedish for hearing pupils and signed Swedish for deaf pupils. In the area of number sense, several pupils in the study did not master three-digit numbers, an ability usually automatized at an earlier age by hearing pupils. In accordance with earlier studies deaf pupils need more time to learn mathematics than hearing pupils. As a consequence they may learn certain concepts at a later age and the pathways towards comprehension may vary compared to those of hearing pupils.

How deaf pupils’ way of expressing themselves in sign language influenced their concept formation was studied. It was analysed out of the four characteristics of sign language iconicity, simultaneity, movement and spatiality. It was found that the structure of sign language could be of help but it could also be an obstacle in mathematics. It was helpful when showing what something looked like iconically and what happened. On the contrary arguments have been raised whether the visual aspects of sign language may hamper concept formation in mathematics. In this study the possibilities of using the pupils’ expressions in sign language to reveal their level of knowledge and to promote their concept development have been focused on. In constructing one’s own knowledge it is essential that all possibilities are taken advantage of.

The importance of teaching mathematics by problem solving and by communication to deaf pupils as well as to hearing pupils has been emphasized in this study, which is concerned with mathematics in its broad sense, including mastering basic mathematical thinking as well as ability to use it in everyday life. For deaf pupils, a more developed terminology in sign language would make learning subject-specific concepts of mathematics less dependent on competence in spoken/written language. The pupils could then be offered better conditions to reach a more abstract level at an earlier age. The bilingual situation of deaf pupils is a reason for developing methods for teaching mathematics to deaf pupils, approaches differing from or supplementing methods used today.