Barham, J., & Bishop, A. (1991) Mathematics and deaf students. In K. Durkin& B. Shire ... Milton. Keynes: Open University Press. Clements, M. A., & Ellerton, N. F. (1992). Over-emphasising ... 145-152). Durham, NC: Program. Committee.
Mathematics Education Research Journal
2001, Vol. 13, No.3, 204-218
Language, Arithmetic Word Problems, and Deaf Students: Linguistic Strategies Used to Solve Tasks Robyn Zevenbergen, Merv Hyde, & Des Power Gold Coast Campus, Griffith University There has been limited examination of the intersection between language and arithmetic in the performance of deaf students, although some previous research has shown that deaf and hearing-impaired} students are delayed in both their language acquisition and arithmetic performance. This paper examines the performance of deaf and hearing-impaired students in South-East Queensland, Australia, in solving arithmetic word problems. It was found that the subjects' solutions of word problems confirmed trends for hearing students, but that their performance was delayed in comparison. The results confirm other studies where deaf and hearing-impaired students are delayed in their language acquisition and this impacts on their capacity to successfully undertake the resolution of word problems. © 2002 MERGA Inc.
There is a general consensus that deaf and hearing-impaired students' competence in mathematics is inferior to that of their hearing counterparts (Frostad, 1996). Studies of mathematical achievement and understanding have generally concluded that there is no significant cognitive basis for major differences in mathematical performance between deaf and hearing students. Achievement differences that are observed are the result of a combination of linguistic and experiential delays for the deaf students (Serrano Pau, 1995; Titus, 1995). What is less clear are the reasons why this is the case, and the synergies between the poor performance of deaf and hearing-impaired students and that of hearing students who experience difficulties in mathematics. Some authors have relied on comparison of cognitive strategies (Titus, 1995; Zwiebel & Allen, 1988), while others have focused on the effects of sign language on the development of cognitive strategies (Frostad, 1999). What are less well researched are the effects of language. development and use on performance for deaf and hearing-impaired students. The effect of language on mathematics performance for disadvantaged students has been noted (Zevenbergen, 2000), and this suggests that students who have hearing loss experience difficulties similar to their hearing peers. In part, these delays may be due to the The terms deaf and hearing-impaired are used in accordance with most current descriptions in the Special Education literature. Tl1at is, the term deaf refers to hearing loss that prevents hearing of speech, even if the person with a. hearing loss is using amplification. Hearing-impaired refers to a hearing loss that still permits effective reception and interpretation of speech through amplified means. The term deaf may be used to describe certain cultural, linguistic, and social features of groups of deaf people and some hearing-impaired people in the same way as the term Greek Australians may be used to describe that cultural group. 1
Linguistic Strategies Used by Deaf Students to Solve Word Problems
205
development of an alternative form of language, that of sign language (Frostad, 1999), that impacts on the types of understandings that students develop with regard to mathematical concepts. Therefore this study explores the ways in which deaf and hearing-impaired students attempt to make sense of written arithmetic word problems and what strategies they use to respond to these word problems. In examining and theorising the effects of language on the resolution of mathematical tasks, Wood, Wood, Griffith, and Howarth (1986) have argued that the wordiness of mathematical problems create difficulties for hearing students as well as those with hearing loss. They contend that both deaf and hearing students are often able to do arithmetic questions such as "How many minutes between 10.40 a.m. and 1.20 p.m.?", but experience difficulties in transforming the words of the problem into a workable mathematical format. They suggest that the problems experienced by both groups of students indicate that the linguistic aspects of mathematics can create difficulties for them. This aspect of mathematics has been noted by many mathematics educators but it has had limited exploration in the area of hearing-impaired and deaf students. Wood et al. (1986) conclude that the role of language is critical to understanding the delays in performance of deaf students as opposed to earlier deficit models in which deafness in, and of itself, causes a cognitive deficit that accounts for an inability to solve problems. They suggest that the delay may be due to language factors rather than to some inherent cognitive deficit in the students. The effects of language are often demonstrated in studies of cognitive strategies used by deaf students. For example, Stone (1980) noted that deaf and hearing-impaired students had difficulties when undertaking sequencing tasks. Stone proposed that the capacity to sequence requires a need for understanding greater than and less than, but it was found that hearing-impaired students undertaking a-level (16+) and A-level (18+) examinations experienced difficulty in writing things in order. Studies with hearing students from sociallydisadvantaged backgrounds have shown similar difficulties in using binary oppositional terms such as more and less because less is not an integral part of the linguistic register of working-class students (Walkerdine, 1988). For such students, the use of these terms in mathematics becomes problematic - not due to lack of ability, but due to differences in language use and language understanding. Rittenhouse and Kenyon (1991) support and extend Stone's conclusion about deaf students by illustrating that differences in performance on sequencing and conservation tasks are not due to inferior cognitive abilities but rather to a lack of experience and language. In considering the specific aspects of mathematical language, Barham and Bishop (1991) have noted that the linguistic complexity of mathematical terminology can be confusing for deaf students. Distinguishing terms such as tens and tenths or sixty and sixteen is difficult for students who are lip-reading. Other difficulties are centred on the ways in which words are used in mathematics classrooms that are very different from their non-school use. For example, the use of up in the non-school context and then add up in the mathematics classroom can be confusing. Kidd and Lamb (1993) also comment
206
Zevenbergen, Hyde, & Power'
on words with double meanings such as interest and table. Such multiplicity of meanings is very confusing for deaf students. Furthermore, Kidd and Lamb argue that technical vocabulary (such as reciprocal, premium) and variations of form (year/yearly, base/basic) can result in confusion. Also, there are numerous ways of expressing the same idea or concept. Thus, for deaf students who have difficulty in understanding a concept one way, there is greater difficulty when the same concept is expressed in a different manner. In addition, logical connectives such as if and because are rarely explicitly taught to hearing students; they learn them through interactions. Hearing-impaired students do riot have as much opportunity to learn the meaning of such words, so they have difficulty with this area of mathematics. Extending beyond word meanings, word problems involving the use of linguistic forms applied to arithmetic concepts and strategies have been found to cause difficulties (Daniele, 1993; Serrano Pau, 1995; Wood et al., 1986). The exact relationship between language and mathematics for deaf students has not been clearly established, with some researchers finding that there is some relationship with the levels of literacy in English language (Zwiebel & Allen, 1988), while others suggest that it is more closely related to the level of hearing loss (Wood et al., 1986). The relationship between language and mathematics has further emerged through the study of word problems. Using Heller and Greeno's (1978) original jclassification of word problems and research reported by Del Campo and Clements (1987), it is clear that the wording of arithmetic tasks has a significant effect on the successful completion of tasks. Even where the arithmetic is simple and involves addition and subtraction of two numbers where the sum is less than ten, many students in the upper primary years can experience difficulty solving word problems (Le'an, Clements, & Del Campo, 1990). Serrano Pau (1995) studied deaf students attempting change, compare, and combine problems in order to examine their problem-solving abilities in comparison with their reading comprehension levels. While Serrano Pau acknowledged that the relationship between reading comprehension and problem-solving ability was important, he also acknowledged that ineffective problem-solving strategies were also relevant in poor performance. In his study of deaf students solving word problems, Frostad (1999) found that the skills learnt for counting using sign language were not successful when transferred to word problems, thereby creating an even more complex interplay between language and arithmetic for deaf students. Luckner and McNeill (1994) identified three learner capabilities relevant for deaf students when considering linguistic aspects of mathematics. Th~se were: (a) intellectual skills, including a knowledge of the concepts and rule structures relating to the problem solving task; (b) organised information in the form of appropriate schemata to enable an understanding of the problem; and (c) cognitive strategies that allow the learner to select the relevant information and strategies necessary to the problem's solution. Luckner and McNeill confirmed that deaf students have difficulties in relation to arithmetic problem-solving and that it is important to identify ways to assist deaf students to develop organisational
Linguistic Strategies Used by Deaf Students to Solve Word Problems
207
artd procedural skills. In summary, researchers have claimed that while deaf students consistently show delays in comparison with their hearing peers in arithmetic problemsolving, there is no simple or direct relationship established between these delays and the students' linguistic and experiential deficits or their degree of hearing loss (Wood et aL, 1986). Indeed, Wood et aL go so far as to suggest that what is clear from the research is that deaf and hearing~impairedstudents as a group have a greater variability in their ,performance on mathematical tasks than the general student population. They suggest that it is critical to recognise the different demands of linguistic tasks. Where tasks are embedded in everyday contexts using everyday terms, deaf students may perform well, but where the language becomes difficult, then there is greater scope for error. .The present research reports on a project investigating deaf students' performance on arithmetic word problems. Previous studies have considered aspects of the strategies used by deaf students within a cognitive perspective. This project specifically examined linguistic aspects of the strategies used. It investigated the following questions: 1. How do deaf and hearing-impaired students compare with their hearing peers when solving arithmetic word problems? 2. What aspects of arithmetic word problems create difficulties for deaf and hearing-impaired students when solving such problems? Using the research instrument developed by Lean, Clements, and Del Campo (1990) and their data as a comparison base, this project investigated linguistic strategics used by Queensland deaf and hearing-impaired students. l
Word Problems The effects of syntactic structure on students' capacity to solve arithmetic word problems has been well documented (De Corte & Verschaffel, 1991; Lean et aL, 1990). This approach represents a significant shift from earlier work on surface structure and demonstrates that even when a problem can be solved with the same arithmetic operation, the levels of difficulty can vary considerably depending on the type of problem and level of complexity of a given type of problem. Riley, Greeno, and Heller (1983) identified three problem types based on the types of action undertaken in the resolution of the problem. These problem types are: • Change questions which involve a process whereby there is an event that alters the value of the quantity, for example" Peter had 3 oranges. Michelle gave him 2 more oranges; how many oranges does Peter have now? • Combine questions that relate to static situations where there are two amounts. These are considered either as separate entities or in relation to each other, for example, Sarah has 4 oranges; Michelle has two oranges; how many oranges do they have altogether? • Compare questions that involve the comparison of two amounts and the difference between them, for example, Ben has 5 oranges; Alice has 2more oranges than Ben; how many oranges does Alice have?
208
Zevenbergen, Hyde, & Power
The questions can then be divided into distinct subcategories depending on the position of the unknown quantity within the problem. Further distinctions in the questions need to be noted. For change questions, the direction of change causes distinct characteristics in the questions, as change may be increasing or decreasing, and hence require different actions. For compare questions, the relationship between the sets (more or less) is a further variable affecting the question. As a result, De Corte and Verschaffel (1991) identify 14 distinct categories of word problems. The questions range in levels of difficulty and produce substantially different response rates depending on the question type and the variations within those types (De Corte & Verschaffel, 1991; Lean et al., 1990).
Method All moderately to profoundly deaf and hearing-impaired students in SouthEast Queensland were tested and a number of these students were subsequently interviewed. Students from both mainstream and special education settings were included in the study. The students had been ascertained to require special educational assistance because of hearing loss according to Education Queensland's guidelines for Ascertainment of Hearing-Impaired Students. All students had mastered basic number facts (in this context this means basic addition and subtraction facts with numbers with totals up to 20) and had basic pnglish competency skills as determined from school records and .teacher judgements. Comparable numbers of boys and girls participated in the study and students did not have significant or uncorrected impairments other than their deafness. In several grade levels the number of students who met the study criteria was low. Deaf students usually begin school by the age of five and undertake 12 years of schooling. For those deaf or hearing-impaired students in the study who were placed in regular school classes, an actual grade level could be recorded. However, for those students who were placed in Special Education Units, the concept of grade level is less clear. As such, the Grade levels described hereafter for the Queensland deaf and hearing-impaired students in this study are an indication only and cannot be exactly mapped to the (Victorian) grade levels described by Lean et al. in their study of hearing students. While this limits statistical comparison in inferential terms, it does permit examination of the deaf students' development of mathematics problem-solving ability, and the identification of any substantial delays in achievement between the deaf and hearing student populations. It also permits examination of the strategies used by the deaf and hearing-impaired students in attempting the problems. Table 1 shows the number of deaf and hearing-impaired students across 12 years of schooling. Only data from the grades 1 to 7 are presented hereafter so as to allow some comparison to the Lean et al. (1990) data.
Linguistic Strategies Used by Deaf Students to Solve Word Problems
209
Table 1
Numbers of Students from Each Grade Level Grade
1
2
3
4
5
6
7
8
9
10
11
12
Number
3
6
'7
4
11
12
6
15
7
3
o
3
The arithmetic word problems were those originally used in Australia and Papua New Guinea with hearing students by Lean, Clements, and Dei Campo (1990). The three types of questions - change, combine and compare - were used in the study. The study involved two key phases. Phase One consisted of a test instrument which students across South-East Queensland completed. This consisted of the 24 word problems developed by Lean et aL (1990) using a pencil-and-paper format (only the 16 questions that relate to Grades 1 to 7 and to the previous study by Lean et aL are presented in this paper). Equipment suchas toys, blocks, and pencils was made available to the students should they choose to use concrete representations of the word problem. The test was administered to the students in the classropm setting, with the teacher and/ or researcher present to help read and clarify questions if needed. A trained research assistant (a teacher trained in mathematics and hearing-impaired education) supervised the implementation of the survey in. conjunction with the classroom teacher. This phase provided baseline data to compare deaf and hearing-impaired students' performance against that of the Lean et aL study. From the data collected through the test, target students were identified and up to six students from each grade level were interviewed to provide in-depth data on the strategies used to solve the word problems. Phase Two consisted of case studies of these target students. Students were encouraged to use a "think aloud" strategy while solving the tasks. The interviews were videorecorded and were undertaken by a research assistant using sign language. Students were able to use their pr~ference for signed English or oral communication. All interviews were transcribed, and analysis undertaken of both the mathematical and non.. mathematical language used in the task resolution.. The analysis of the text transcripts was conducted using a content analysis procedure outlined by Silverman (1993). This procedure identifies the major lexical items used across syntactic forms, and examines and confirms these across successive transcripts. For example, most of the students associaled the phrase less than as take away (subtraction) in their transcripts and the verb gave in a similar manner. Some problems had greater syntactic complexity and used less familiar terms, for example, two sentences described the participants in the problem and their mathematical relationship, followed by a question referring to one of the participants: "How many did she have at the start?" These produced more varied lexical and related mathematical responses. The responses provided indicators of the problem-solving strategies, the cognitive strategies and the mathematical reasoning being used, and displayed features of the language of the students.
[ Zevenbergen, Hyde, & Power
210
Results The data presented here (Tables 2 and 3) represent the percentage of correct responses offered by students to each problem type. These sets of data corne from the pencil-and-paper instrument. In the Discussion section, we outline some of the linguistic demands of the tasks and discuss some of the difficulties encountered and strategies used by the students in this study. This latter set of data is qualitative and comes from the in-depth interviews conducted on a sample of students. Table 2
Percentage Correct of Compare Questions Across Grades Grades Compare Questions
1
2
3
4
5
6
7
John has 2 buckets. Eric has 6 buckets. How many more buckets than John does Eric have?
0
0
0
0
18
8
33
38
53
60
78
85
95
94
0
0
0
0
27
33
33
34
42
61
72
88
93
96
Gina has some boxes. Ken has 3 boxes. Gina has 1 more box than Ken. How 0 many boxes does Gina have?
50
86
50
55
83
83
59
66
74
85
84
93
94
0
0
0
50
36
17
33
41
57
49
66
80
88
92
0
17
0
25
46
25
50
41
42
54
69
77
82
84
0
33
33
0
0
50
33
17
43
43
56
64
69·
67
. Lean et al. Hearing data Nick has 2 cups. Sarah has 7 cups. How many cups less than Sarah does Nick have?
Lean et al. Hearing data
Lean et al. Hearing data Jo has some dolls. Pat has 5 dolls. Jo has 2 dolls less than Pat. How many dolls does Jo have?
Lean et al. Hearing data Bill has some trucks. Tina has 5 trucks. Tina has 2 trucks more than Bill. How many trucks does Bill have?
Lean et al. Hearing data Sam has some marbles. Sarah has 6 marbles. Sarah has 2 marbles less than Sam. How many marbles does Sam have?
Lean et al. Hearing data
As the sample size for some grade levels was small, it was not possible to conduct any inferential analysis that could be statistically significant, even if the data from the present study and the data from the Lean et a1. (1990) study could be compared directly. However, what can be observed are the trends that
II
Linguistic Strategies Used by Deaf Students to Solve Word Problems
211
emerged from the responses. Overall, the trends confirm other research in that the successful performance of deaf and hearing-impaired students is delayed in comparison with their hearing peers, and that the complexity of the word problems creates similar demands for both sets of students. In the tables, data are shown for the same sets of questions used in the Lean et a1. (1990) study. The Lean et a1. study was conducted with Australian and Papua New Guinea students. However, only the data for the Australian students is used for comparison in this study, since it can be assumed that the students would have greater cultural and schooling similarities to the research group than the Papua New Guinea sample. The data used in the Lean et a1. study were for the seven years of primary school in Victoria. Table 3
Percentage Correct of Change Questions Across Grades Grades Change Questions
1
2
3
4
5
6
7
Barbara had 2 eggs. Dan gave Barbara 1 more egg. How many eggs does 67 Barbara have now?
67
86
75
100
67
100
Lean et al. Hearing data
90
89
90
95
97
97
97
Jack ha's 4 pens. Dianne took 3 of Jack's pens. How many pens did Jack have then?
33
83
86
75
91
67
100
Lean et al. Hearing data
90
86
89
95
97
99
99
33
17
43
25
18
42
50
48
63
73
80
86
93
98
33
50
71
75
55
67
67
76
78
81
85
94
94
94
a
a
29
25
36
58
50
Jeff had 3 bananas. Carmel gave Jeff some more bananas. Jeff then had 5 bananas. How many bananas did Carmel give Jeff?
Lean et aI. Hearing data Anna had five books. Tom took some of Anna's books. Then Anna only had 2 books left. How many of Anna's books did Tom take?
Lean et aI: Hearing Paul had some pencils. His father gave him 2 more pencils. Then he had 5 pencils. How many pencils did Paul have at the start?
Lean et aI: Hearing Sally has some pictures. She lost 2 of her pictures. Then she had 3 pictures. How many pictures did she have at the start?
Lean et al. Hearing data
. 38
71
66
76
84
93
93
0
0
43
0
36
50
50
48
57
65
76
85
89
94
212
Zevenbergen, Hyde, & Power
The responses of these deaf and hearing-impaired students confirm the trends that have been noted in similar studies of word problems involving hearing-impaired students, that is, performance of deaf and hearing-impaired . students is delayed in arithmetic and mathematics. The delay can be observed in this study where, in each grade level, lower proportions of shidents were able to correctly respond to the same questions compared with the hearing students in the Lean et al. (1990) study. As hearing-impaired and deaf students age, and have greater experience with school and mathematics, they seem better able to respond correctly to the questions. Students respond to the complexity of the word problem as it varies according to the syntax of th~ question and the operation to be undertaken either addition or subtraction. In Tables 2 and 3, the complexity of the word problem generally increases from first to last.
Summary The overall data could not be analysed for statistical significance due to the small numbers in each year level cohort and the difficulty of direct comparison (for example, in relation to grade descriptors) with the Lean et al. (1990) data. While there were inconsistencies among some sets of data, the overall patterns of the responses confirm similar patterns of difficulty experienced for hearing and deaf and hearing-impaired subjects. As noted by other authors researching aspects of the development of numeracy, mathematical and arithmetic concepts for deaf and hearing-impaired students, the data in this project further confirm the delay in the capacity of deaf and hearing-impaired students to respond appropriately to tasks such as those commonly found in mathematics classrooms. When the language of the question conforms to the usual procedures used within the mathematics classroom, there is less delay in performance. For example, in two Combine questions, the posing of the problem IIDavid has 2 dogs and Jim has 4 dogs. How many dogs to they have _~ClJJ()getherr conforms to the general practice of addition of two numbers. The use of the signifier altogether provides a linguistic cue for addition. The data presented suggest that there is little delay in this type of question between our cohort and the subjects reported by Lean et al. (1990). In contrast, the problem IIHelen has 3 ribbons. Lyn also has some ribbons. Lyn and Helen have 7 ribbons altogether. How many ribbons does Lyn have?'! is more complex syntactically. Although the word altogether is used and the sum is provided (7 ribbons), there is an unknown set associated with one of the people (as opposed to the summative set), and the syntactic and semantic demands are significantly greater. Our data suggest that this type of problem causes greater difficulties for deaf and hearing-impaired students. Similar trends can be observed in the Change questions where the first two forms are commonly used in the classroom and there is considerably less delay for the students in this cohort, whereas there appears to be greater delay in the less commonly used questions, that is, Change Questions 3-6 and the Compare questions.
Linguistic Strategies Used by Deaf Students to Solve Word Problems
213
Discussion In this section, we discuss the strategies used by hearing-impaired and deaf students when they attempted arithmetic word problems. As has been noted in Newman's research into the taxonomy of responding to word problems (Clements & Ellerton, 1992; Ellerton & Clements, 1992), the role of language is critical. Mathematics is a very particularised discourse with highly specific and concise forms of language use. To be effective in comprehending and interpreting the task posed, students must be able to understand the highly specific register of mathematics. Deaf and hearing-impaired students use strategies that indicate a restricted understanding of the intricacies of mathematical language. For many of these students the subtle intricacies of the mathematical register may be elusive. In the sense of this paper, restricted is not meant to infer deficit, but rather that the specific nuances of the mathematical register are frequently different from everyday language so that strategies used for deconstructing meaning in the latter context may not be as effective in the mathematical context Where students have been able to make sense of words in the everyday context, this meaning is transferred to the mathematical context. Often such strategies may work, but in other cases they may not. The linguistic transcripts of the student interviews indicate that there are a number of strategies that are used by the students in this study - with more or less effect. For many students-hearing and hearing-impaired-there is a reliance on trigger words that may offer students cues as to the meaning of the problem. For example, the trigger word more is often seen to call for an additive operation. In part, this could be due to the widespread practice of teachers introducing the addition operation in the context of word problems relevant to the students' non-school experiences. Common practices consist of teachers posing questions such as "1 had three apples, then I bought three more, how many did I have altogether?" Such tasks expose students to practices where they can extrapolate the meaning of more to imply addition. The study provides evidence that students whose access to the lexical density and complexity of the mathematics register is restricted tend to develop strategies that rely on taking key words and using these to make sense of a problem. This strategy may work well when the problems are those that are commonly used and conform to the parameters of general use. However, where the problems become more complex due to varied syntactical arrangements, such strategies are not as successful.
I
Decontextualised Tasks Two types of questions were used in the study. We use the term decontexualised to connote what would have once been referred to as pure mathematics. That is, the problems have no specific application and are presented as a 'straight' arithmetic problem, for example the items in Table 4. We use the term contextualised to connote those questions where the mathematics has been embedded in a context. Most of the problems in this study are of this form, for example the items in Table 5.
!
214
Zevenbergen, Hyde, & Power
Table 4
Percentage Correct of Pure Number Comparison Questions Grades Pure Number Questions
1
2
3
4
5
6
7
Which number is 3 more than 4?
0
0 ·14
0
18
8
33
Lean et al. Hearing data Which number is 2 less than 3?
Lean et al. Hearing data 5 is 1 more than which number?
Lean et al. Hearing data 6 is 2 less than which number?
Lean et al. Hearing data
48
55
58
65
60
71
67
0
0
29
0
18
25
67
34
43
58
76
73
85
88
0
17
0
0
18
0
17
45
51
61
71
77
78
82
0
17
14
0
0
0
0
18
34
29
35
43
45
54
Table 5
Percentage Correct of Combine Questions Across Grades Grades Combine Questions
1
2
3
4
5
6
7
100
91
100
100
David has 2 dogs and Jim has 4 dogs. How many dogs do they have altogether?
67
83 100
Lean et al. Hearing data
79
91
93
98
99
99
100
0
0
0
25
27
25
67
45
63
69
71
75
85
82
Helen has 3 ribbons. Lyn also has some ribbons. Helen and Lyn have 7 ribbons altogether. How many ribbons does Lyn have?
Lean et al. Hearing data
As noted above, common trigger words in mathematics are the terms more and less, often associated with the operations of addition and subtraction respectively. The patterns of responses confirmed those in the Lean et al. (1990) study. At the decontextualised level where the binary oppositional terms of more and less are used, there were fewer students able to respond correctly to Item 4 (Table 4; "6 is 2 less than which number?") where the relational term less was integral to the item. Students misconstrued items using the term less in a number of different ways. In the c,?mparative tasks such as Item 2 (Table 4) where the task asks "Which number is 2 less than 3?", some students perceived the use of less to connote subtraction and were able to respond correctly, in spite of not understanding the task. However, most students saw the task as a comparative one in which they had to identify which was the lesser in value of 2 or 3. The wording of Item 1 (Table 4) "Which number is 3 more than 4?" was interpreted similarly-the students saw it as asking which is the greater (or lesser) of the two
I
Linguistic Strategies Used by Deaf Students to Solve Word Problems
215
numbers, rather than as addition or subtraction equations. The signifier than seemed to be redundant wording as many of the students did not know the word or its role in the sentence, resulting in the (mis)interpretation of the task as being I/which is more, 3 or 4?". The use of propositions has been found to cause difficulties for students whose first language is not English. McGregor (1991), working in the multiculturalclassroom, found that non-English-speaking background students found the use of propositions particularly difficult. She cites the example where the statements are made: "The temperature fell by 10 degrees, '" to 10 degrees, .... from 10 degrees" and note~that difficulties are caused by the use of prepositional terms. Similar studies with deaf students, such as that of Rudner (a 1978 study, as cited in Barham & Bishop, 1991, p. 182), noted a difficulty for deaf students who struggled with conditionals (if, when), comparatives (greater than, less than), negatives (without, not), and inferentials (should, could). As noted in the previous section, the use of than in the decontextualised tasks created difficulties for problem solution. Similar difficulties were noted with the use of is in Items 3 and 4 in Table 4: 1/5 is 1 more than which number" and 1/6 is 2 less than which number?" where students interpreted the question as being 1/5 and 1 more is which nUJJ:,lber?" Again the students tended to ignore the key terms and rely on commonly-used classroom examples. The reliance on trigger words was common in this study, as was the redundancy of prepositional terms such as than. The case studies revealed these two aspects of language that were often not apparent in the quantitative data. This adds another dimension to the study of deaf and hearing-impaired students undertaking mathematics.
Contextualised Word Problems Where the problems are stated simply and in order of operation, there was a greater likelihood of the students being able to offer correct responses (e.g., Item I, Table 3: "Barbara had 2 eggs. Dan gave Barbara 1 more egg. How many eggs does Barbara have now?"). The actions involved in the Change tasks (giving and taking) provided strong contextual cues for the students as to the appropriate operation. Difficulty arose in the other four Change tasks when the indefinite some entered the word problems. Not only was the amount missing, but it also produced confusion as to the appropriate action (operation) to be undertaken. Where the non-stated amount came in the first line of the task, the students experienced great difficulty in making sense of the task. For example, in Item 5, Table 3 ("Paul had some pencils. His father gave him two more pencils. Then he had 5 pencils. How many pencils did Paul have at the start?"), the lack of a definitive amount confused the students as to the requirements of the task. The students searched for key words (such as lost) to offer some cues as to what they might need to do. In the case of Item 6, Table 3, where something was lost students took this as a cue for subtraction. Where the first line was a statement that obviously did not add to the mathematical information in the task, the students ignored it and saw it as having no relevance to the task. They
· 216
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then proceeded to" operate on the numbers given in the other sentences using such cues as they could extract from the task. The comparative tasks (e.g., Item 1, Table 2: "John has 2 buckets. Eric has 6 buckets. How many more buckets than John does Eric have?"), proved to be difficult due to the structure of the final sentence requiring a comparison to be made and a difference noted. A~ has been noted with the decontextualised tasks in the previous sections, students were at risk of interpreting the task as one of determining which is the bigger or smaller quantity and naming that quantity. These answers tended to be of the form, "Eric: because he has got 6 buckets." Alternatively, students would interpret the more or less in the wording to signify addition or subtraction. Answers of this form tended to be "8 buckets". Other students made some information redundant and interpreted the question to be asking: "How many buckets does Eric have?" and would respond with "6".
Conclusion The sample of students in this study is small so it is not possible to make any generalised claims about the development of deaf and hearing-impaired students' performance on" arithmetic word problems. However, the data collected confirmed previous studies in which deaf students were found to be somewhat delayed in their performance in mathematics. The delay seems to be exacerbated by the difference between the questions commonly posed in I classrooms and those that are novel. Where questions conformed to common meanings, such as the trigger word more equating with addition, the students were more successful in responding to the tasks. However, where the word problems were different in expectations, or more complex syntactically, they posed greater difficulty for the students. The data collected in this project support the notion that the specificity of language used in arithmetic word problems creates difficulties for deaf students. Many of the deaf students relied on a key-word approach to comprehending the tasks. Recognising that they could not understand everything, students use their understanding of key words and use this knowledge base to attempt to construct meaning for the tasks. From the responses offered in these tasks, this appears to be "a very common strategy used by this cohort of students. The overgeneralisation of the meaning of key words resulted in many incorrect responses. With the restricted understanding of syntax, deaf and hearingimpaired students are compelled to rely on fragmented sentences and need to make sense of that to which they can gain access. The complex syntactic and lexical density of the word problems, along with the lack of redundant and supportive information found in such word problems, hinders the capacity of deaf students to make sense of the tasks, and hence their capacity to offer correct responses. As has been shown in other studies (De Corte & Verschaffel, 1991), where the order and structure of word problems is reconstituted so as to make greater synergies between spoken language and the action of the problem, then students have greater success. However, progress through mathematics is often associated with understanding the highly specific register and structure of
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mathematical language, thus creating a dilemma in mathematics education for these students. By re-organizing word problems so as to make them more accessible, students are not gaining access to the highly specific register of the discipline.
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Authors Robyn Zevenbergen, School of Education and Professional Studies, Faculty of Education, Gold Coast Campus, Griffith University, P.M.B. 50, GCMC, Bundall, QLD 9726, Australia. Email: Merv Hyde, Center for Deafness Studies and Research, Faculty of Education, Gold Coast Campus, Griffith University, P.M.B. 50, GCMC, Bundall, QLD 9726, Australia. Email: Des Power, Center for Deafness Studies and Research, Faculty of Education, Gold Coast Campus, Griffith University, P.M.B. 50, GCMC, Bundall, QLD 9726, Australia. Email:
