An Intervention Program for Promoting Deaf Pupils ... · explicitly taught todeaf pupils in...

Our past research identified two aspects of deaf children’sfunctioning that places them at risk for underachievement inmathematics. The first is their reduced opportunities for in-cidental learning, and the second is their difficulty in makinginferences involving time sequences. This article examinesthe effectiveness of an intervention program to promote deafchildren’s numeracy that was designed to deal with these twofactors. The design involved a comparison of 23 deaf pupilsparticipating in the project with a baseline group formed by65 deaf pupils attending the same schools in the previousyear. The project pupils were tested before and after the in-tervention on the NFER-Nelson Age Appropriate Mathe-matics Achievement Test. The intervention was delivered bythe teachers during the time normally scheduled for mathe-matics lessons. The project pupils did not differ from thebaseline group at pretest but performed significantly betterat posttest. They also performed at posttest better than ex-pected on the basis of their pretest scores, according to normsprovided by the NFER-Nelson Age Appropriate Mathemat-ics Test for assessing the progress of hearing pupils. We con-clude that the program was effective in promoting deaf pu-pils’ achievement in numeracy.

We describe in this article an intervention project de-signed to raise the achievement of deaf pupils in math-ematics. It is well established that deaf pupils lag be-hind hearing pupils in mathematics. The National

Council of Teachers of the Deaf (1957) in England car-ried out a study with a large sample of deaf pupils andreported that deaf pupils were on average 2.5 years be-hind in mathematics achievement tests. About a decadelater, Wollman (1965) reported similar results in a sur-vey that included a third of the pupils from 13 schoolsfor the deaf in the United Kingdom. Wood, Wood, andHowarth (1983) found that no improvement in this sit-uation could be documented two decades later: in theirstudy deaf pupils were approximately 3.4 years behindin mathematics achievement when compared to theirhearing counterparts. In the first section of this article,we argue that it is possible to raise the mathematicsachievement of deaf pupils. In the second section, wedescribe the intervention program we carried out andthe results of its assessment. In the last section, we dis-cuss the implications for future research.

Is It Possible to Raise the MathematicsAchievement of Deaf Pupils?

We (Nunes & Moreno, 1998) have argued that hearingloss cannot be treated as a cause of difficulties in math-ematics but as a risk factor. Several findings in the liter-ature suggest that hearing loss is not a direct cause ofdifficulties in mathematics. First, not all deaf pupils areweaker in math than their hearing counterparts: ap-proximately 15% of the profoundly deaf pupils per-form at average or above average levels in standardizedtests (Wood et al., 1983). If hearing loss were a directcause of difficulties in mathematics, there should be no

We thank The Nuffield Foundation for the support received throughgrant number HQ/313[EDU] without which this work would not havebeen possible. We also thank the schools, teachers, and children who en-thusiastically participated in this project. Correspondence should be sentto Terezinha Nunes, Department of Psychology, Oxford Brookes Univer-sity, Gipsy Lane, Oxford OX3 0BP, United Kingdom (e-mail: [email protected]).

� 2002 Oxford University Press

An Intervention Program for Promoting Deaf Pupils’

Achievement in Mathematics

Terezinha NunesOxford Brookes University

Constanza MorenoRoyal Holloway, University of London

ditive composition informally, probably through theirexperiences with money, among other things. If a childis asked to pay for a sweet that costs 8 cents, for ex-ample, a child who understands additive compositionhas no difficulty in using one nickel and three penniesto pay. About 60% of 6-year-old and virtually all 7-year-old hearing children succeed in this task (Nunes &Bryant, 1996). In contrast, many deaf children, includ-ing some as old as 10 and 11 years (Nunes & Moreno,1998), cannot combine coins of different values into asingle amount.

A second difficulty of deaf children, often reportedby parents (see Gregory, 1995), is related to communi-cation about time. Research has shown (Moreno, 2000)that deaf children have significantly more difficultythan hearing children in tasks that require them to pro-cess a sequence of events over time, keeping in mind agap for an unknown element in the sequence. Moreno’stask required the children to listen to a short descrip-tion of a sequence of events—for example, “there weresome people waiting at the bus stop; the first personwho got to the bus stop was a girl; the second one, Idon’t remember; the third one was a lady.” The childwas asked to show which one of four pictures could bethe correct picture for this story. Only one alternativehad a girl in the first position and a lady in the thirdposition in the bus line. Moreno also used a controltask, where all the three elements in the sequence in thestory were mentioned and there was no need to think ofholding an empty space in the line for a person not re-called by the experimenter. The deaf children’s perfor-mance in the control tasks did not differ from that ofhearing children of the same age. In contrast, the hear-ing children performed significantly better when therewas a gap in the sequence. Moreno (2000) furthershowed in a short-term longitudinal study, by means ofa series of regression analyses, that the deaf children’sperformance in this task was a strong predictor of theirscores on a standardized mathematics test (the NFER-Nelson) 7 months later, even after controlling for ageand IQ. Thus, deaf children may need support whencommunicating and reasoning about time, particularlyif they need to consider a gap in a story sequence.

The ability to think of time and consider gaps in adescription is often required in the mathematics class-room when we teach pupils about the inverse relation

An Intervention Program for Mathematics 121

deaf pupils displaying achievements adequate for theirage level.

Second, most studies have found either no correla-tion or only a very small correlation (Nunes & Moreno,1998; Wood et al., 1983) between the level of hearingloss and mathematics attainment. This result suggeststhat hearing loss is not a direct cause of difficulties inmathematics.

Third, the development of deaf pupils’ counting(Secada, 1984) and computation skills (Hitch, Ar-nold, & Philips, 1983) and of their problem-solvingabilities (Nunes & Moreno, 1997) seems to follow thesame pattern as that of hearing pupils, albeit at a slowerpace. This suggests that there is no deviation but onlya delay in deaf pupils’ mathematics learning.

We proposed the alternative hypothesis that hear-ing loss places children at risk for difficulties in learn-ing mathematics. Unlike a causal hypothesis, a risk fac-tor hypothesis supposes that it is possible to preventa risk factor from leading to negative outcomes if thenecessary steps are taken.

Our past research has identified two specific diffi-culties of deaf pupils that can explain, at least in part,why they are at risk for low achievement in mathematics.

First, deaf children have fewer opportunities for in-cidental learning as a consequence of their hearing loss.This hypothesis has already been proposed by Furth(1966) and Rapin (1986), who suggested that deafyoungsters’ poor results in reasoning tasks and educa-tional assessments can be explained by an “informationdeprivation” (Rapin, 1986, p. 214). Deaf youngsterslack access to many sources of information (e.g., radio,conversations around the dinner table), and their inci-dental learning may suffer from this lack of opportu-nity. Consequently, some concepts that hearing chil-dren learn incidentally in everyday life may have to beexplicitly taught to deaf pupils in school. We (Nunes &Moreno, 1998) identified one mathematical concept—additive composition—crucial to progress in mathemat-ics, that is often mastered by children before they enterschool or quite early on in their school lives, and thatseems to create a significant obstacle for deaf children.Additive composition refers to the understanding thatany number can be seen as the sum of other numbers(Nunes & Bryant, 1996; Resnick & Omanson, 1987).There is evidence that hearing children learn about ad-

between addition and subtraction. Consider the prob-lem “Mary had some sweets; her grandmother gave herthree and now she has eight. How many did she haveto begin with?” If pupils have difficulty in making in-ferences in connection with time sequences, it will bevery difficult for them to work with problems that re-quire them to think about inversion and time. In a pre-vious study (Nunes & Moreno, 1997), we showed thatdeaf pupils were not behind hearing pupils when theysolved problems that did not involve inversion but wereconsiderably behind when the problems required suchinverse inferences. Later, Moreno found that deaf chil-dren’s performance in tasks that require making infer-ences about events over time is significantly behind thatof hearing pupils of the same age level and predictstheir mathematics achievement scores after an intervalof 7 months.

To promote the development of deaf pupils inmathematics, we developed a program with two aims.The first was to give the deaf pupils opportunities tolearn core mathematical concepts that many hearingpupils may learn informally outside school and to pro-mote connections between these informal concepts andmathematical representations used in school. The sec-ond aim was to promote deaf pupils’ access to informa-tion about word problems related to transformationsover time by representing the problems through draw-ings and diagrams and reducing the need to retain in-formation about sequences of events in memory.

Method

Participants

The project involved two groups of deaf pupils: thebaseline group and the project pupils, all drawn fromthe same six schools for the deaf or schools with unitsfor deaf pupils in London, United Kingdom.

The baseline group consisted of 65 pupils who wereattending the same schools as the project pupils andwere tested on the NFER-Nelson age-appropriatemathematics test approximately 1 year before the be-ginning of the project. They were drawn from a largersample of 82 pupils according to the criterion of beingin the same year groups (2–5) as the project pupils and

122 Journal of Deaf Studies and Deaf Education 7:2 Spring 2002

having taken the same age-appropriate NFER-Nelsontests.

Project pupils were 23 children in six differentclasses whose teachers agreed to implement the pro-gram and participate in the meetings with the re-searchers during the period of the project.1 The distri-bution of pupils per age appropriate test taken ispresented in Table 1.

Design

Before the implementation of the program, the projectpupils were assessed in the NFER-Nelson age-appropriate mathematics tests (beginning autumn1998). The program was administered by the teachersat their own pace, making use of some of the time (ap-proximately 1 hour per week) normally scheduled formathematics lessons over two terms (end of autumn1998, spring and beginning summer 1999). The pro-gram did not take up the whole time dedicated to theteaching of mathematics so that the teachers continuedwith their regular teaching of the curriculum in the re-mainder of the available time. At the end of the pro-gram, the pupils were assessed again in the NFER age-appropriate mathematics test (autumn 1999). Table 1shows that some of the children were tested on thesame age-appropriate test at the end of the programand others had moved on in age to the following testinglevel; 17 of the 23 children took the next age-level testat the end of the program.

The baseline group was assessed only once(autumn–spring 1996). Thus, most of the project chil-

Table 1 Number of pupils in the project and baselinegroups by age-appropriate test taken at pre- and posttest

7 years 8 years 9 years 10 years 11 years

Project pupilsat pretest 10 4 5 4 0(n � 23)Project pupilsat posttest 3 8 3 7 2(n � 23)Baselinepupils 22 19 19 5 0(n � 65)

makes the program more similar to an off-the-shelfbook than to a recipe to be strictly followed by teachersin detail.

Observations and teachers’ reports made it clearthat there was much variation in the implementation.Some teachers promoted pupil interaction more thanothers. Some teachers allowed the children to identifymistakes themselves by comparing their work, whereasother teachers provided feedback themselves. Teachersof older children tended to use materials less often thanthose of younger children when introducing problems.The materials chosen were sometimes simply blocks tobe used as representations of other objects described inthe problems and sometimes cut-out shapes (for ex-ample, little shorts and shirts) that offered more figu-rative representations of the objects mentioned in theproblems. In spite of these variations, all the teachersworked from the booklets and reported that the chil-dren enjoyed the work.

The program was not envisaged as a replacementto the mathematics curriculum taught in the class-room. Its aim was to bring the deaf children’s informalmathematical understanding to a solid basis for learn-ing the curriculum that they are taught in school.

The Program

The concepts covered in the program were:

1. additive composition and its application tonumber and measurement;

2. additive reasoning (that is, reasoning about therelations between addition and subtraction as inverseof each other);

3. multiplicative reasoning (that is, reasoningabout the relations between multiplication and divi-sion); and

4. ratio and fractions.

Each concept was explored by means of a series oftasks. In each series, the items were ordered accordingto their expected level of difficulty. The most difficultitems in one series were often more difficult than theeasier items in the following series. This gave the chil-dren a sense of progress in one series and also the feel-ing that some easier tasks would be coming later on.


dren are compared with different baseline groups atpre- and posttest because on each occasion they arecompared with the children who took the same age-appropriate test.

Procedure

The program was designed with the involvement ofnine teachers of the deaf who attended monthly meet-ings with the researchers over five school terms. In thespring and summer terms of 1998, the researchers pre-pared basic materials relevant to the teaching of eachof the key concepts and discussed each set of materialswith the teachers. The teachers tried out the materialsand reported back on the results, positive features, andthe difficulties that the pupils had experienced. Thematerials were revised on the basis of the teachers’feedback.

The revised program was organized in the book-lets, which were used in the implementation of theprogram in the classroom (examples are presented inFigures 1 through 8). The pupil booklets containedpictures to support the presentation of the situationsand no text. The teachers received a teacher book thatcontained the same pictures along with instructions ex-plaining the question. Teachers were asked to give theinstructions to their pupils in the pupils’ language ofinstruction (British Sign Language [BSL], Sign-Supported English [SSE], or English). They were freeto adapt the instructions to the pupils’ languageknowledge.

All teachers were asked to attempt all the itemswith the pupils, even if they expected some of the con-cepts to be either too easy or too difficult for their pu-pils. The teachers were encouraged to introduce newitems with practical materials if they felt this wasneeded to ensure that the children understood the situ-ations. They were also encouraged to use discussionsamong the pupils as the items were solved. They pro-vided their own explanations to the pupils. The pro-gram booklets worked as a starting point for them towork on everyday mathematical concepts. The teacherswere encouraged to use the program in their own way,as long as they gave the pupils all the tasks in the book-lets and followed the instructional sequence. This

The work with each concept is briefly described in thefollowing sections.

The aims of the section of the program on additivecomposition, number, and measurement were (1) tostrengthen the pupils’ understanding of additive com-position; (2) to strengthen their understanding of hownumbers are used to measure, thereby expanding theuse of additive composition; and (3) to introduce thenumber line as a working tool for representing andsolving problems.

Items on additive composition. Two examples are includedin Figure 1. The aim of the items was to use pupils’informal knowledge of money and strengthen their un-derstanding of additive composition. We discussedwith the teachers the difficulties of each type of itemand different ways of promoting children’s understand-ing of the crucial concepts. For example, pupils whohave difficulty with additive composition benefit fromrepresenting the value of nonunitary coins (nickel ordime) with their fingers before counting the total sumof money. They also benefit from working with valueswhere the number words facilitate the task (e.g., com-bining a 20p coin with 1p coins) before working withvalues where the number words do not help (e.g., com-


bining 10p and 1p coins). When pupils succeed in thesesimpler tasks, they can go on to more difficult itemsand be asked to think about how they solved the previ-ous items.

Items on measurement. We saw two advantages of includ-ing items on measurement in this series of tasks relatedto number concepts. First, it is important that pupilsthink about the quantities represented by numbers inorder to strengthen their understanding of number.Our previous work indicated that pupils’ understand-ing of measurement is often incomplete. When mea-suring, many are not sure where to start measuringfrom, the edge of the ruler, zero, or one. This suggeststhat they do not fully understand what the readingobtained from a ruler indicates. Second, rulers can beused as a number line. Thus, they are useful instrengthening pupils’ familiarity with mathematicalconventions. Some of the items involved measurementwith a broken ruler. Measuring with a broken ruler re-inforces pupils’ understanding that it is possible towork with number lines that start at any point. Someitems from this section are presented in Figure 2.

Our discussions with the teachers revealed thateven some of their older pupils were not aware of the

Figure 1 Two examples of additive composition items.

ings and diagrams, representing time through spatialrelations; and (3) to use the number line for calculationand for the demonstration of different solutions to thesame problems.

Problems involving addition and subtraction withdifferent number meanings and different levels of com-plexity (invisible addends, start unknown, comparison)were included to provide the pupils with the opportu-nity to explore additive reasoning broadly. Teacherswere encouraged to use the pupils’ records for the dis-cussion of different ways of solving the same problems.Some examples of items are presented in Figures 4and 5.

The classroom observations showed that someteachers used concrete materials to help the youngerchildren reason about problems that they found moredifficult. Once the pupils had solved some problems us-ing objects, they were able to work with the bookletwithout difficulty. The connection between the way inwhich they had counted the objects and counting onthe number line was easily made.

Problems with start unknown (Mary had somesweets; her friend gave her two; now she had eight; howmany did she have to begin with?) and with transfor-mation unknown (A boy had five cakes; he ate some;now he has three; how many did he eat?) are quite


precise meaning of the numbers obtained from readinga ruler. The examples in Figure 2 show how a pupilcounts the “longer lines” on the ruler as an indicationof the number of centimeters and the shorter lines asan indication of the number of half centimeters. Thepupil counted the lines, not the units of length.

Items for introducing work with the number line. Work withthe number line was included in the program for tworeasons. First, it is a form of conventional mathematicalrepresentation. Second, the number line offers a visualrepresentation of number sequences. It is known fromprevious work that deaf adults process informationpresented visually more efficiently than informationpresented orally. Thus, we expect the number line tobe a useful tool for deaf children when they calculate ordiscuss numerical information in the classroom. Twoexamples of tasks used to introduce the number line arepresented in Figure 3.

The teachers reported that the children did notfind it difficult to use the number line to represent ananswer. This facilitated the transition to using it as aninstrument in problem solving.

The aims of the section on additive reasoning were(1) to promote the coordination of addition and sub-traction as inverse of each other; (2) to work with draw-

Figure 2 Two items from the measurement tasks.

difficult for deaf pupils, as they involve making time-related inferences. One example is included in Figure 5.

Comparison problems are the most difficult of theadditive problems with natural numbers for hearingand deaf pupils alike. Our program introduced com-parison problems by initially connecting the compari-


son to additive transformations. Our previous research(Nunes & Bryant, 1996) has shown that this is aneffective way to make the solution of comparison prob-lems accessible to 6-year-olds. One example is shownin Figure 5. Several were included in the program.Teachers reported that the number line was a particu-

Figure 3 Two items introducing the number line.

Figure 4 Two ways of reasoning leading to the same answer. A discussion between the pupils can show theco-ordination between addition and subtraction.

schema and the concept of multiplication. Tables areused as a way of representing the connection betweenthe variables. The bottom items illustrate how the con-nection between multiplication and division was intro-duced. The text describing the problem situation wasused to guide the teachers’ explanation: each teacherwas asked to present it to the pupils in the languageused in the classroom, adapting or explaining the prob-lem in a different way, without supplying further infor-mation. The problem on the bottom left describes atypical multiplication situation with a missing factor(number of children) and provides information aboutthe product (total number of flowers). The pupils typi-cally used a division strategy to solve this problem:they circled the flowers in groups of two: each groupcorresponds to one child in the class. The problem onthe right presents an incomplete table where the ratioof flowers per vase has to be identified by division.

The items used to introduce graphs and their con-nection to multiplication were designed to help the pu-pils make a transition from more concrete drawings tomore abstract diagrams. The graphs were always pre-sented as representations of problems. In the initialproblems, the pupils represented their answers in the


larly useful instrument when discussing the logic ofcomparisons.

The aims of the section on multiplicative reasoningwere (1) to work from pupils’ intuitive understandingof correspondence as the basis of multiplicative reason-ing; (2) to work with drawings and diagrams that repre-sent two variables in the representation of multiplica-tive concepts; and (3) to introduce tables and graphs asmathematical representations for multiplicative rela-tions. Some examples are presented in Figures 6 and 7.

Much research has shown that pupils’ misconcep-tions in the domain of multiplicative reasoning arerooted in the idea that multiplication is simply re-peated addition (for a review, see Greer, 1992). We(Park & Nunes, 2000) have carried out an interventionstudy with hearing children ages 6 and 7 years, compar-ing instruction on multiplication as repeated additionand as an operation linked to the schema of correspon-dence. Children instructed through the correspon-dence schema performed significantly better in theposttest than those instructed through repeated addi-tion. Some examples of multiplicative reasoning prob-lems are included in Figure 6. The two items on topexemplify the connection between the correspondence

Figure 5 A problem with a missing transformation and a comparison problem.

graphs. Later, they were asked to obtain informationfrom the graph. Two sample problems are presented inFigure 7.

An analysis of the pupils’ productions showed thateven the younger pupils were able to have some successin reasoning about and representing multiplicativeproblems with tables and graphs.

The aims of the section on fractions were (1) towork with sharing and division as an intuitive startingpoint for reasoning about fractions; (2) to promote aconnection between pupils’ understanding of fraction


and division (1⁄4 seen as one divided into 4), and be-tween fraction and ratio (one out of four means thesame as a 1:3 ratio). Figure 8 presents two examples,one of an item that connects sharing with fractions andthe second that connects ratio and fraction.

The concept of fraction as traditionally taught(areas of a figure) was not stressed in the program. Al-though it is often used in the classroom, research sug-gests that children’s intuitions about sharing and divi-sion can provide a solid start for understandingfractions from about age 8 (Streefland, 1997). Children

Two rabbits live in eachhouse. We need one pellet offood for each rabbit. Howmany houses are there? Howmany rabbits? How manypellets of food do we need?

We are having a party. Eachchild that comes will get 3balloons. Fill in the table toshow how many balloons weneed if 2, 3 and 4 children cometo the party.

The teacher had birthday.Each child in the classbrought him two flowers. Theteacher received 24 flowers.How many children are in theclass?

We are fixing the flowers for aparty. Each vase will have thesame number of flowers. Fill inthe table with the number offlowers corresponding to thenumber of vases.

Figure 6 Four examples of multiplication and division items illustrating theuse of tables.

greater difficulty for the pupils but acknowledged thatthe easier items were accessible even to the youngestpupils.

Results

The effect of the intervention was assessed in two ways.The first analysis was a comparison between the scaledscores of the project pupils and those of the baselinegroup. Because different items and different numbersof items are used in the different age-level tests, a com-parison that includes tests administered at different agelevels must be based on scaled scores. Scaled scorestake into account these variations and are obtained by


at this age already realize that one pie shared amongtwo people will give larger pieces than one pie sharedamong four people. This helps them understand that1⁄2 is a larger number than 1⁄3. A common misconcep-tion about fractions documented in the literature is tothink that 1⁄3 is larger because 3 is more than 2.

The teachers found the fraction items difficult towork with, as they are not used to thinking about theconnection between ratio and fraction. It should bepointed out, though, that the National Numeracy Strat-egy in England includes as a specific aim of teach-ing about fractions to establish a connection betweenthe concepts of fraction and ratio. They also reported

Figure 7 Two examples of items illustrating the introduction of graphs atthe initial stages of the program (top) and the use of graphs at a later stage(bottom).

consulting a table for the appropriate test. Scaledscores in the NFER-Nelson vary between 0 and 76.

We carried out two comparisons, one between thepretest scores of the project pupils and the baselinegroup and the second between the posttest scores ofthe project pupils and the baseline. We had no reasonto expect a significant difference between the projectand the baseline pupils in the pretest because they hadbeen drawn from the same schools. If the interventionwere successful, the project pupils should perform sig-nificantly better than the baseline pupils in the posttest.

Figure 9 shows the mean scaled score for the proj-ect pupils at pre- and posttest and the baseline groupfor each age-appropriate test administered. The num-ber of pupils for the different data points is specified inTable 1.

To test whether the differences observed were sig-nificant, we used two analyses of covariance with thelevel of test taken as the covariate, the group member-ship (project pupils vs. baseline group) as the indepen-


dent variable, and the score in the pre- and in the post-test as the dependent variable, respectively in the firstand second analysis.

The comparison between the project pupils and thebaseline group did not produce significant results inthe pretest: F(1,84) � .651, ns. In contrast, the projectpupils differed significantly from the baseline group atposttest, F(1,85) � 7.296, p � .008. Thus, we concludethat the project pupils improved significantly in theirmathematics achievement during the time they wereengaged in the program.

The second analysis was a comparison between theproject pupils’ observed progress on the standardizedmathematics assessment and their expected progress.If the project were successful, their performance in theposttest should be better than the performance ex-pected from them on the basis of their pretest perfor-mance. The NFER norms for the age-appropriatemathematics tests include a prediction of performanceon tests administered at a subsequent age, taking into

Figure 8 Two examples of problems dealing with ratio and fractions.

formance at posttest was thus significantly better thanit would have been if their progress had been equiva-lent to the average amount of progress expected forhearing pupils during one school year.

Discussion

These quantitative analyses suggest that the interven-tion was effective in increasing the deaf pupils’ accessto the mathematics curriculum. The effects we docu-mented cannot be explained solely in terms of theteaching the children would have normally received inthe classroom because the baseline pupils had been ex-posed to the same curriculum in the same schools inthe previous year. No major policy changes in theteaching of mathematics were introduced during thisyear in the schools.2 The teachers did not increase theamount of time dedicated to teaching mathematics be-cause the project was implemented during the timenormally scheduled for mathematics lessons. No extraclassroom helpers were made available during the proj-ect. Thus, there is no other reason beyond their partici-pation in the project to expect that the pupils would


account performance on a first testing occasion. Theseguidelines are produced for hearing rather than deafchildren. Pupils whose observed performance is at alower level than the predicted score are said to havemade less progress than expected. Those pupils whoseobserved and predicted performances coincide are saidto have made average progress. Finally, pupils whoseobserved score is superior to the predicted score aresaid to have made more progress than expected. By us-ing the NFER guidelines, we obtained predicted scoresfor each pupil and compared these predicted scores tothe observed scores. We observed that 31.8% of the pu-pils made less progress than predicted and 68.2% hadhigher observed than predicted scores. Because the dis-tribution of the differences between observed and pre-dicted scores was skewed, a nonparametric statistic, theWilcoxon Signed Ranks Test for dependent samples,was used to compare the predicted with the observedscores. A significant difference indicates that the ob-served scores are significantly higher than the pre-dicted scores. This analysis showed that the mean rankfor the predicted scores was 7.86 and for the observedscores was 13.20 (z � 2.31, p � .02). The pupils’ per-

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Figure 9 Mean scaled scores in the NFER-Nelson for the baselinegroup and the project group in the pre- and posttests by age-appropriatetest taken.

perform significantly better than the baseline groupnor that they would make more progress than pre-dicted for hearing children during the year.

As in any other major intervention, where pupilsparticipate in a large number of activities, it is not pos-sible to say exactly what in the program led to the sig-nificantly greater level of progress. The program wasdesigned with deaf children’s needs in mind: the needfor visual support in communication in the mathemat-ics classroom and the need for the systematic teachingof concepts that hearing children might learn infor-mally. The activities we proposed were scrutinized bythe participating teachers of the deaf, who made sug-gestions for the visual and linguistic presentation. Thetasks were organized in blocks that tackled the sameconcept from different perspectives and thus createdfor the children the opportunity of thinking about thesame concept in different ways. The teachers encour-aged the children to use drawings and diagrams to ex-plain their answers, but the children did not have toinvent these diagrams for themselves: the programprovided them with examples of drawings, diagrams,tables, and graphs that could be used as tools for think-ing and communicating about mathematics. The pro-gram consisted exclusively of activities that involvereasoning: no time was dedicated to the teaching of al-gorithms. This was in line with the decision to create aconceptual basis that would provide access to the math-ematics curriculum instead of replacing it. All of theseaspects of the program were novel when compared tothe mathematics instruction at the time. Mathematicslessons, even for deaf pupils, seem to rely on languageto a large degree. Calculations are taught through therecall of verbally represented number bonds and rulesfor how to proceed. When there are misunderstandingsin the teacher’s explanations, there are few resources tosupport a discussion of the mathematical ideas, whichcould show what had been lost in the communication.Visual means of representing relations between vari-ables, such as tables and graphs, are typically not intro-duced in the context of problem solving, as was the casein this intervention; at this age level, the curriculumincludes only bar graphs and the pupils are asked toread frequencies, without a consideration of other usesof graphs than the display of information. It is likelythat all these differences jointly contributed to the pos-


itive impact of the program. It is not possible to teaseout the effects of the different aspects of the experienceprovided by the program. Although the question ofspecific effects should be addressed in future research,it would have been outside the aims of this interven-tion, which were to maximize the learning opportuni-ties for the deaf pupils. We believe that many factorschange when a program such as this is introduced inthe classroom and that experimental interventions out-side the classroom are needed to separate the effects ofdifferent aspects of an educational program.

Likely, in this case, the effects should be attributedboth to cognitive and motivational factors. The cogni-tive effects are likely to be based both on the specific de-sign for teaching the core concepts included in the pro-gram and on the use of drawings and diagrams inteaching. The instruction program was carefully de-signed to use children’s mathematical intuitions and tohelp them confront conceptual difficulties identified inprevious research. The program also provided the pu-pils with tools for representing the core concepts in amathematically adequate form. The use of drawingsand diagrams was chosen for its potential in addressingthe communication needs of deaf pupils. The pupilsseem to have found drawings and diagrams useful as ameans of representing their ideas and working towardsolutions. The teachers reported that their pupils, afterstarting to work with the project booklets, had begunto use drawings and diagrams at other moments in themathematics lessons.

Likely there were also motivational effects operatingbeyond these cognitive factors. Our own observationsand the teachers’ reports indicate that the pupils en-joyed working with the booklets. In one class they actu-ally celebrated the moments when the teacher askedthem to bring out the booklets: they clapped and showedgreat enthusiasm verbally. In another class the pupils didnot want to interrupt their work at lunch time becausethey were too engaged in a discussion about measuringwith a broken ruler, an event very unusual in the teach-er’s experience of this group of pupils.

Further research analyzing specific aspects of thisintervention is necessary to identify the most crucialcognitive elements in the project. It would be importantto know whether the teaching of concepts normallylearned informally by hearing children is necessary or

Moreno, C. (2000). Predicting numeracy in hearing impairedchildren. Doctoral dissertation, London, Institute of Edu-cation.

National Council of Teachers of the Deaf Research Committee.(1957). The teaching of arithmetic in schools for the deaf.Teacher of the Deaf, 152, 165–172.

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Received September 26, 2000; revisions received June 18, 2001;accepted July 5, 2001


whether the use of visual means of communicationalone can produce such positive results. If the teachingof informal concepts is necessary, and these have al-ready been mastered by hearing children, schoolswould have to consider the need to offer extra teachingto the deaf children outside regular lessons. If the useof visual means of representations is sufficient to ac-complish the desired results, these could well be intro-duced in the regular classroom where deaf and hearingchildren might be working together. Many hearingchildren likely would benefit from this more visual in-struction too.

Notes

1. One of the teachers went on maternity leave during theproject. The second author replaced her during this test, imple-menting the program 1 hour per week during the period sched-uled for mathematics lessons.

2. After the conclusion of this project, English schools havestarted to implement a program known as The Numeracy Hour.There was no overlap between the implementation of the inter-vention described here and the Numeracy Hour.

References

Furth, H. G. (1966). Thinking without language: Psychological im-plications of deafness. New York: Free Press.

Greer, B. (1992). Multiplication and division as models of situa-tions. In D. Grouws (Ed.), Handbook of research on mathe-matics teaching and learning. New York: Macmillan.

Gregory, S. (1995). Young deaf children and their families. Cam-bridge: Cambridge University Press.

Hitch, G. J., Arnold, P., & Philips, L. J. (1983). Counting pro-cesses in deaf children’s arithmetic. British Journal of Psy-chology, 74, 429–437.

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