SNARC hunting: Examining number representation in deaf students

Learning and Individual Differences 15 (2005) 223 – 236 www.elsevier.com/locate/lindif

SNARC hunting: Examining number representation in deaf students Rebecca Bulla,T, Marc Marscharka,b, Gary Blatto-Valleeb a

School of Psychology, William Guild Building, University of Aberdeen, Aberdeen, AB24 2UB, Scotland, UK b National Technical Institute for the Deaf, Rochester Institute of Technology, Rochester, USA Received 12 November 2004; received in revised form 27 January 2005; accepted 31 January 2005

Abstract Many deaf children and adults show lags in mathematical abilities. The current study examines the basic number representations that allow individuals to perform higher-level arithmetical procedures. These representations are normally present in the earliest stages of development, but they may be affected by cultural, developmental, and educational factors. Deaf and hearing participants were asked to perform two number comparison tasks. Analysis of response times revealed that all participants showed effects normally associated with representation of magnitude on a visual-analog mental number line: SNARC, distance, and size effects. However, deaf participants were slower overall in making comparative judgements, suggesting that whilst their numerical representation does not differ from that of hearing individuals, the efficiency with which they process basic numerical information is lower. The results are discussed in terms of interactions between biologically determined numerical representations and cultural and schooling factors that differentially affect deaf and hearing individuals. D 2005 Elsevier Inc. All rights reserved. Keywords: Magnitude; SNARC; Deafness; Mathematics; Number

1. Introduction There are numerous reports showing that the academic performance of deaf children and adults often lags behind that of their hearing counterparts (Allen, 1986; Lang, 2003; Traxler, 2000). A range of T Corresponding author. E-mail address: [email protected] (R. Bull). 1041-6080/$ - see front matter D 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.lindif.2005.01.004

224

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

factors has been suggested to explain this academic lag including access to equal educational opportunities, motivation, teaching and learning styles, and the effectiveness of classroom communication (Marschark, Lang, & Albertini, 2002). At present, however, these are primarily speculation. What is needed, above all else, is informed research of the cognitive abilities underlying learning by deaf children (Marschark, Convertino, & LaRock, in press; see Detterman & Thompson, 1997). It has been well established that children with significant hearing losses lag behind hearing children on mathematical achievement tests by roughly three years, despite displaying normal IQ’s (Traxler, 2000; Wood, Wood, Kingsmill, French, & Howarth, 1984). It is somewhat unclear when this lag emerges; however, it can be seen as early as 8 years of age according to the (US) national norming of the Stanford Achievement Test, ninth edition (SAT9, Traxler, 2000). From 8 years onward, the lag remains relatively constant at bbelow basicQ levels, and deaf students’ scores also appear to asymptote at about age 13 or 14. Delays also have been demonstrated in particular domains of mathematical ability, such as in the understanding of fractions (Titus, 1995). Importantly, however, the constant nature of the lag in SAT scores and similar data patterns in mathematics-related studies (see below) suggest that deaf and hearing children’s mathematical development follows the same developmental course. Deaf children’s delays in mathematics skills have been demonstrated in mathematical research spanning the past five decades. In 1957, the National Council of Teachers of the Deaf using the Schonell Arithmetic Test reported an average difference of 2.5 years between deaf children and standardized norms in schools throughout England, although that test might have been biased by the need for English language skills. More recently, Wood et al. (1984) carried out a similar study, using the Vernon & Miller Graded Arithmetic Test, which contained little written instruction. Deaf participants still performed around 3.4 years behind their hearing counterparts. This finding of poor performance on mathematical achievement tests appears consistent worldwide, with similar developmental lags being found in studies carried out with Japanese, Norwegian, and American populations (Frostad & Ahlberg, 1999; Phelps & Branyan, 1990). The main purpose of the current study was to begin an exploration of the cognitive underpinnings of mathematical skills in deaf adults, starting at the most basic levels of numerical understanding, in the hope that this would guide future research examining more complex aspects of mathematical cognition and performance in educational settings. Researchers argue that although deaf individuals may process information differently from hearing individuals, they are not deficient in processing information (Marschark, 2003; Schick, 2005; Tharpe, Ashmead, & Rothpletz, 2002). Indeed, in some aspects of cognitive processing, deaf individuals show distinct advantages, for example, in speed of shifting visual attention and visual scanning (Rettenback, Diller, & Sireteaunu, 1999), peripheral detection of motion (Bavelier et al., 2000; Corina, Kritchevsky, & Bellugi, 1992; Neville & Lawson, 1987; Proksch & Bavelier, 2002; Swisher, 1993), and in the generation and manipulation of mental images (Chamberlain & Mayberry, 1994; Emmorey & Kosslyn, 1996; Emmorey, Kosslyn, & Bellugi, 1993; Talbot & Haude, 1993). However, it remains unclear whether such findings obtained in carefully-controlled laboratory settings have any implications for realworld learning and memory. At least one recent study suggests not. Marschark et al. (submitted for publication) examined visual gaze allocation (via eye-tracking) and classroom learning by deaf and hearing students. Deaf students were either skilled signers or new signers, whilst hearing students knew no sign language. Although deaf and hearing students clearly differed in their gaze allocation patterns, students varying in their sign language skills did not, nor did they differ significantly in the amount learned from the lectures.

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

225

Beyond differences in visuo-spatial skills between deaf and hearing individuals, studies of long-term or semantic memory have revealed several important differences in their organisation and utilisation of information. Studies using word association tasks, for example, have revealed that deaf individuals tend to have weaker strengths of association between concepts, asymmetrical category-exemplar relations, smaller set sizes, and much more variable associative structures relative to hearing peers (Marschark, Convertino, McEvoy, & Masteller, 2004; McEvoy, Marschark, & Nelson, 1999). Deaf adults and children also have been shown to tend toward item-specific processing, focusing on individual item information rather than relations among items (Marschark, DeBeni, Palazzo, & Cornoldi, 1993; Ottem, 1980; Richardson, McLeod-Gallinger, McKee, & Long, 1999; see Marschark, 2003, for a review). Although the extent to which such differences might affect the acquisition of mathematical skills by deaf students is yet to be determined, the understanding of mutual relations between quantities is a key aspect of basic number processing. Holding any kind of mental representation of numerical relations necessarily means holding some understanding of the cardinal and ordinal relation between numbers. Understanding relations between numbers, in turn, is thought to help in the development of more efficient counting strategies (such as counting-on from the largest addend). A less clear representation or a tendency to process information on an item-specific rather than relational basis may result in more difficulty establishing these early efficient counting procedures and hence a delay in the establishment of strong representations of arithmetic number facts in long-term memory. It has been proposed that number representations and mathematical thinking depend in part on a sense of approximate numerical magnitudes, or bnumber senseQ (Dehaene, 1997; Gallistel & Gelman, 1992). A large body of evidence now exists suggesting that individuals represent the relation between numerical magnitudes in a spatial format, with small magnitudes represented in the left side of space, and larger magnitudes in the right side of space. This association between space and numerical magnitude is referred to as the Spatial Numerical Association of Response Codes (SNARC). SNARC effects are assumed to arise when magnitude information is automatically activated from a long-term visual-spatial representation. In a seminal series of studies, Dehaene, Bossini, and Giraux (1993) asked participants to indicate the parity of digits with bimanual (left vs. right hand) responses. Assignment of odd and even responses to the right and left hand was switched halfway through the task. Despite the fact that magnitude was irrelevant to the judgements required, response times to large numbers were faster with the right hand, whilst those to smaller numbers were faster with the left hand. Berch, Foley, Hill, and Ryan (1999) found evidence of a SNARC effect in children as young as 9 years of age, again in a situation where magnitude was irrelevant to the task (parity judgement). Studies using Stroop-like paradigms with participants being asked to make physical size decisions, have also revealed that automatization in number processing is achieved gradually as numerical skills progress (Girelli, Lucangeli, & Butterworth, 2000; Rubinsten, Henik, Berger, & Shahar-Shalev, 2002). Additional evidence for this form of magnitude representation comes from studies showing distance and size effects in the comparison of numbers (e.g., Dehaene & Akhavein, 1995; Dehaene & Changeux, 1993; Deheane, Deheane-Lambertz, & Cohen, 1998; Deheane, Dupoux, & Mehler, 1990; Moyer & Landauer, 1967; Tzelgov, Meyer, & Henik, 1992). Where numbers are in close proximity on the mental number line (e.g., 4 and 5), the time to decide which is the larger number will be greater relative to numbers that are more distant on the number line (e.g., 1 and 5). Moreover, analyses of the time to compare numbers that are equally distanced, e.g., 3 and 7 vs. 43 and 47, reveal that participants are slower to make the comparison of the larger numbers. This finding is taken as evidence of a logarithmic representation of number, meaning that the psychological distance between larger numbers will be

226

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

smaller, despite having the same numerical distance (Brysbaert, 1995; Dehaene, 2003). Again, distance and size effects occur regardless of whether a magnitude judgement is relevant to the task. There is evidence for phylogenetic precursors and early ontogeny of the magnitude system (Feigenson, Deheane, & Spelke, 2004; Gallistel & Gelman, 1992), suggesting that the way in which we represent magnitude information is not determined by how we are educated. Therefore, without damage to the neurological systems that support magnitude representation, there is no a priori reason to expect that hearing and deaf participants would represent magnitude information in different ways. However, Dehaene (1997) also argues that this biologically determined number system interacts with cultural factors such as language acquisition and schooling to yield our ability for higher-level mathematics. Experience with the formal number system in a variety of numerical contexts may result in refinement of the number representation or to the formation of multiple representations (Siegler & Booth, 2004; Siegler & Opfer, 2003). For example, differences in number representation might be particularly apparent in native signers (i.e., those who acquire sign language from their deaf parents) relative to other deaf and hearing individuals. In signed face-to-face communication, the signer typically produces ordered sequences from left-to-right, so the observer perceives those signs with a right-to-left movement. This may influence the orientation of the mental number line (Iverson, Nuerk, & Willmes, 2004). More generally, given the differing educational experiences of deaf individuals and their preference for processing information in a different way, it is not unreasonable to suggest that the form of visual-spatial representation they formulate or wish to use could be somewhat different from that typically seen in hearing adults (Marschark, 1993, Chapter 8). Epstein, Hillegeist, and Grafman (1995) tested deaf students between the ages of 17 and 25 on a magnitude comparison task in which they were asked to determine which of two numbers presented (ranging from 1–99) was the largest and to press a spatially appropriate (left or right) key to indicate their choice of number. Accuracy of both deaf and hearing participants was very high. However, the deaf participants were found to have slower response times overall compared to hearing participants, although post hoc analyses showed no difference when comparisons involved single digits. Unfortunately, in this study, no assessments were made of speed to complete other comparison tasks. As such, it is not possible to conclude whether this slowness in magnitude judgement is a specific slowness in the processing or accessing numerical information, or representative of slowness in more general speed of processing. A similar limitation applies to a study by Marschark, Blatto-Vallee, Bull, and Cornoldi (2003), who found significant differences between deaf and hearing individuals in response times for both numerosity and length judgements. Studies of children with arithmetic learning difficulties (ALD, e.g., Koontz & Berch, 1996), have found that whilst normally achieving children automatically retrieved magnitude information that subsequently interfered with physical size judgements, children with ALD did not show the same interference suggesting less automatic activation of semantic numerical information. We would expect that where information about numerical magnitude is not retrieved so automatically from long-term memory, we also might find less robust distance and SNARC effects as well as slower comparison times overall (Epstein et al., 1995). The current study aimed to examine the strength and nature of numerical representation in deaf adults. If deaf adults have less automatic activation of numerical information, we would predict slower times to make comparative decisions, overall, and potentially, a less marked SNARC effect. Furthermore, if deaf adults process numerical information on an item-specific basis, then again, we might anticipate a less marked SNARC effect, as the relations to other magnitudes will be processed more slowly (i.e., less

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

227

automatically) or to a lesser extent. Two magnitude judgement experiments were conducted. Experiment 1 examined the automaticity issue by comparing deaf and hearing students under conditions expected to produce SNARC effects. Experiment 2 examined the bshapeQ of the internal number line by examining magnitude judgements within and across deciles.

2. Method 2.1. Participants Twenty deaf college students (14 males) and 20 hearing college students (11 males), ranging in age from 18 to 28 years, participated in the experiment as paid volunteers. All were enrolled as undergraduates at Rochester Institute of Technology and were recruited via posters and contacts with student groups. Deaf students’ hearing thresholds ranged from 65 dB to 120 dB in the better ear, with a mean of 96 dB. All of the deaf students used sign language, sometimes accompanied by speech, as the primary means of communication; eight had deaf parents (in one of those cases, only a single parent was deaf). Institutional records showed that the age of hearing loss onset for all but one of the students was 1 year. Although demographic and academic data were not available for the hearing students and there were some missing data for deaf students due to incomplete institutional records, deaf students’ scores on the American College Test (ACT) entrance examinations (n=16) indicated them to have an average mathematics subtest score of 15.56 (S.D.=3.65) of 36 possible. This mean corresponds to the 15th percentile relative to national norms for all students taking the test. The mean compares with an average mathematics subtest score of 17.33 (S.D.=4.12; 34th percentile) for the 984 deaf RIT students enrolled during the academic year for whom ACT scores were available. At face value, these scores are consistent with those of deaf students on the SAT9 (given in grades 1 through 12) in suggesting that deaf students, on average, lag behind hearing peers in mathematics skills (Traxler, 2000). 2.2. Experiment 1 2.2.1. Materials and procedure The experiments were administered using E-Prime laboratory software running on a laptop computer, which had two mouse buttons centered below the keypad. Participants kept their hands placed on the two buttons during test blocks. On each trial, participants saw a fixation point (+) for 500 ms presented in the centre of the computer screen, which was replaced by a single digit (1–9). Stimuli were 1.5 cm high on a 39 cm diagonal screen and remained in view until the participant responded. In one block of trials, they were instructed that if the number was less than five (1–4) they should press the left mouse key and if the number was greater than five (6–9) they should press the right mouse key. In a second (counterbalanced) block of trials, the instructions were reversed: they were instructed that if the number was less than five (1–4) they should press the right mouse key and if the number was greater than five (6–9) they should press the left mouse key. They were encouraged to respond as quickly as possible without making mistakes. After two practice trials, they received 64 test trials, four trials with each of eight digits, 32 in each block. All trials included feedback.

228

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

Instructions specific to each task were presented on the computer, in print. Deaf participants were instructed about the experiment in general and any questions were answered using American Sign Language (ASL) or simultaneous communication (sign language and spoken language), depending on their preference. The experimenter (GBV) was a hearing, native user of ASL. None of the participants indicated any difficulty in understanding the tasks. 2.3. Experiment 2 2.3.1. Materials and procedure The equipment and basic procedure was the same as in Experiment 1. On each trial, participants saw a fixation point (+) in the middle of the screen, which was replaced by two numbers that remained in view until the participant responded. They were instructed to press the mouse key on the side of the larger number and were encouraged to respond as quickly as possible without making mistakes. After two practice trials, they received 48 test trials (with feedback), 16 trials with numbers from each of three deciles, 10–19, 40–49, and 70–79. Stimuli on all trials had a difference of one, yielding eight comparisons for each decile, which were presented once with the larger number on the right and once with the larger number on the left. Although there is no evidence that deaf individuals, in general, have slower response times than hearing age-mates and, indeed, are faster than hearing individuals on some tasks (e.g., Emmorey & Kosslyn, 1996; Emmorey et al., 1993), a control task was administered here to ensure that the deaf and hearing students did not differ in this respect. In the bi-manual decision task, students saw two (1.5 cm1.5 cm) squares, one red and one blue, spaced 1.5 cm apart. The presentation was identical to that described for the number trials, except that stimuli were preceded by the (centered) word bredQ or bblueQ (for 1 sec). The squares remained on the screen until participants pressed one of the two keys to indicate the side on which that color square appeared. There were 20 trials, with the blue square appearing on each side on half of the trials. This condition was administered to assess general speed of responding in a bi-manual judgement task, and to rule out the possibility that any slowing in the magnitude decision task was a result of a general slowness in speed of processing information or initiating motor responses.

3. Results Mean response times for each number were calculated separately for the specified right and left responses. Only response times to correct trials were included in this calculation. Moreover, mean response times falling outside the 95th percentiles were also removed from the analysis to ensure that data were not skewed by a small number of exceedingly fast or slow responses. Because of the repeated measures nature of the data, only participants without missing data are included in the analysis (14 deaf participants and 17 hearing participants). 3.1. Experiment 1 Initial analyses examined response times of the deaf and hearing participants under conditions where the required response for each number was congruent with SNARC (low numbers, press left key; high

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

229

numbers press right key), and where the required response was incongruent with SNARC (low numbers, press right key; high numbers, press left key). A 2 (hearing status)2 (congruency)8 (number) mixed design ANOVA, with repeated measures on the last two factors revealed significant main effects of number, F(7, 203)=9.76, pb0.001, partial g 2=0.25; congruency, F(1, 29)=16.42, pb0.001, partial g 2=0.36; and hearing status, F(1, 29)=7.28, pb0.05, partial g 2=0.20. Response times were faster in the SNARC congruent condition, and hearing participants were faster overall than deaf participants (see Fig. 1). Post hoc analysis (Tukey HSD) of the main effect of number revealed that participants were slower to respond to the numbers 4 and 6 compared with 1, 2, 3, 7, 8, and 9, that is, the further the number was from the target number (5), the faster the comparison (i.e., typical distance effect). None of the two or three way interactions were found to be significant. The difference in the time to respond to each number with the right and left hand was then calculated (right hand RT left hand RT). We would predict that response times to low numbers would be faster with the left hand, resulting in a positive RT difference following this calculation. Response times to high value numbers should be faster with the right hand, resulting in a negative RT difference. An 8 (number)2 (hearing status) mixed design ANOVA with repeated measures on the first factor revealed a significant main effect of number, F(7, 203)=7.87, pb0.001, partial g 2=0.21, but no significant main effect of hearing status, F(1, 29)=1.80, pN0.05, and no interaction between number and hearing status, F(7, 203)b1. The nature of the SNARC effect was captured by regression analyses (Lorch & Meyers, 1990, Method 3; for a detailed discussion see Fias, Brysbaert, Geypens, & d’Ydewalle, 1996). The regression for RT difference as the criterion variable and number as the predictor variable confirmed that a SNARC effect was present with the regression weight for number significantly different from 0 for both deaf [t (7)= 3.21, pb0.05] and hearing participants [t (7)= 4.17, pb0.01]. Fig. 2 clearly shows that both deaf and hearing participants show very similar patterns of results, both showing the predicted SNARC effects of faster responses to low numbers with the left hand and higher value numbers with the right hand.

Response time (msecs)

800 750 700 650 600 550 Deaf congruent Deaf incongruent Hearing congruent Hearing incongruent

500 450 400 1

2

3

4

5

6

7

8

9

Number Fig. 1. Response times for each number under SNARC congruent (low numbers left, high numbers right), and SNARC incongruent (low numbers right, high numbers left) conditions, for deaf and hearing participants.

230

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

RT difference (right - left)

150 100 Deaf Hearing Linear (Deaf) Linear (Hearing)

50 0

Deaf: y = -17.268x + 81.233 R2 = 0.63

-50

Hearing: y = -21.136x + 96.936 2 R = 0.74

-100 -150 0

2

4

6

8

10

Number position Fig. 2. Response time difference (ms) between the right and left hand for deaf and hearing participants (bars show standard error).

Very few errors were made in the task (proportion accurate for deaf participants=0.93 and 0.97; hearing participants, = 0.98, 0.97, for congruent and incongruent conditions, respectively). Analysis of the proportion of correct responses (collapsed across number) in the SNARC congruent and incongruent conditions for deaf and hearing participants, revealed no significant main effects of congruency, F(1, 38)b1 or hearing status, F(1, 38)=2.28, pN0.05, and no significant interaction, F(1, 38)=2.06, pN0.05. 3.2. Experiment 2 If higher value magnitude information is represented on a logarithmic scale, then making a decision about such numbers should take longer, as the spatial distance between the numbers is much less, and hence magnitude discrimination more difficult. Therefore, we would expect that response times increase within each decile and across the deciles 10–19, 40–49, and 70–79. Alternatively, if magnitude information was presented on a linear scale, then the time taken to compare the magnitudes 47 and 48 should take no longer than the comparison of 42 to 43; both represent an equivalent distance on the number line. Finally, if participants were making the magnitude decision by only attending to the unit digit and ignoring the10s digit, we would predict no increase in response times across the decile ranges. A 3 (digit number size: small [2–4], medium [5–6], and large [7–9])3 (decile:10s, 40s, 70s)2 (hearing status) mixed design ANOVA with repeated measures on the first two factors revealed significant main effects of decile, F(2, 68)=68.87, pb0.001, partial g 2=0.67; digit size, F(2, 68)=15.94, pb0.001, partial g 2=0.32; and hearing status, F(1, 34)=4.07, p=0.05, partial g 2=0.11. None of the interactions were found to be significant. Fig. 3 shows that response times increase as the size of the second digit increases within each decile range, and that response times to the teen numbers are faster than the 40s and 70s numbers, which do not differ from each other. Deaf participants were significantly slower overall than hearing participants. Analysis of response times on the control color judgement task revealed no difference between the deaf and hearing participants, t (38)=0.54, pN0.05 (response times of 550.07 and 528.62 for the deaf and

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

231

950 Deaf 10

Response time

900

Hear 10

850

Deaf 40 Hear 40

800

Deaf 70 Hear 70

750 700 650 600 2 to 4

5 to 6

7 to 9

Number Fig. 3. Response times across the deciles ranges (10s, 40s, and 70s) for small (2–4), medium (5–6), and large (7–9) unit digits (bars show standard error).

hearing groups, respectively). Therefore, the differences found on the number judgement tasks cannot be attributed to differences in baseline processing or motor response speed.

4. Discussion The motivation behind this study was to begin to understand numerical processing in deaf students and, hopefully, gain some insight into the possible locus of their lags in their mathematical abilities relative to hearing age-mates. Clearly, deaf individuals do not represent a completely homogenous population, having experienced differing modes of communication and differing educational experiences. As such, we cannot claim that findings or implications discussed in the current study apply to all deaf individuals, but the findings do provide an insight into the skills shown by adults representative of a large deaf student population. Given phylogenetic evidence for the number line, along with early ontogenetic evidence for such a representation (Feigenson et al., 2004; Gallistel & Gelman, 1992), our expectations were that deaf adults should not show a difference in terms of SNARC and distance effects when compared with their hearing counterparts. If this is the case, then we are in a position to begin an examination of more complex numerical processing skills as the basis for the later mathematical difficulties. Alternatively, given evidence of slower magnitude processing by deaf adults (Epstein et al., 1995; Marschark et al., 2003), and less automatic activation of numerical information in individuals with arithmetic learning difficulties (Koontz & Berch, 1996), one potential outcome was the lack of typical SNARC and distance effects if number information is not being automatically activated. The current data suggest that deaf college students have a reduced efficiency with which they can retrieve magnitude information, similar to results seen with children with poor arithmetic skills. The speed with which deaf participants made magnitude decisions in the SNARC congruent condition was similar to the speed with which hearing participants made decisions in the SNARC incongruent condition. At this point, it is not possible to pinpoint whether this is due to slower encoding or identification of the Arabic symbols (Bull & Johnston, 1997) or slow retrieval of magnitude information.

232

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

However, the nature of the visual-spatial representation of number in deaf participants shows the typical pattern of SNARC and distance effects, and did not differ from the pattern of results displayed by the hearing participants. All participants showed evidence of distance effects with numbers below 10, and evidence of logarithmic representation of higher magnitude numbers. Furthermore, Marschark et al. (2003) found that whilst deaf adults were slower to make estimations of dot numbers and line lengths, they were just as accurate as hearing peers. Since collecting data for the current studies, research findings from Zarfaty, Nunes, and Bryant (2004) have also revealed that preschool deaf children’s basic number representation is at least as advanced as that of hearing children (note that number representation was assessed using different methodology than that presented here). They concluded that deaf children’s difficulties with mathematical learning were not a consequence of a delay in non-linguistic number representation (see also Iverson et al., 2004, for evidence of SNARC effects in deaf adult signers). However, evidence indicating comparable forms of mental representation in deaf and hearing individuals, both here and in the Zarfaty et al. (2004) study, does not speak to the efficiency with which that information is retrieved and/or applied. In the task used in the current study, we asked participants to perform a relational task where they either had to compare two numbers or classify a number as being above or below a certain number. As such, the processing of magnitude information is intentional for this task. Many other studies ask participants to make judgements (e.g., parity, physical size) where the retrieval of semantically related magnitude information is irrelevant to the task. Future studies examining the representation and retrieval of numerical information in deaf populations should address whether processing of the related semantic magnitude information is undertaken under such conditions (i.e., unintentional processing). Bebko (1998) suggested that delays in the automatization of a second language and cognitive skills in deaf children result in the need for their using of greater cognitive capacity in dealing with more complex tasks. Marschark (submitted for publication) similarly questioned the extent to which some of the cognitive differences observed between deaf and hearing individuals, and a tendency toward itemspecific processing in particular, are a function of the impoverished early communication environments of most deaf children, the ways in which they are educated, or some fundamental aspect of dealing with the world primarily through the visuo-spatial mode. He suggested that the latter alternative, which may be orthogonal, rather than mutually exclusive to the other two, could be particularly important for understanding learning and cognitive functioning of deaf adults and children. Because most deaf individuals (as opposed to those who have mild to moderate hearing losses) have significantly degraded auditory input, at best, they are unable to take full advantage of dual visual and auditory coding in the way that hearing individuals do in many contexts (Paivio, 1986). To the extent that they rely on interpreters for communication of linguistic information, they thus must engage more in consecutive rather than simultaneous information processing. Either a failure to apply relational processing strategies or their less frequent use by deaf individuals thus seems likely to influence learning and development, complicating attempts to fully characterize intellectual functioning in deaf adults and children across a variety of domains. Walsh (2003) argued that that concepts of quantity, space, and time share similar properties in that they are part of a generalized magnitude system. Gevers, Reynvoet, and Fias (2003, 2004) also provided evidence showing that other ordinal sequences, such as letters, months, and days were spatially organised. It would be interesting to see whether deaf participants show difficulties in these concepts given ad hoc reports of difficulties in sequencing historical events within a time frame. Certainly, there is anecdotal evidence in this regard from history instructors, even at the university level (D. LaRock,

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

233

personal communication, September 20, 2004); and deaf children have been shown to have difficulty, relative to hearing peers, in temporal/sequential ordering across several domains (see Marschark, 1993, Chapter 7–8 for related research concerning memory and IQ). Given evidence suggesting that deaf individuals tend to process information on an item-specific level–possibly a cause or an effect of weaker associative connections among concepts–tasks which require relational information to order temporal events may prove to be particularly difficult (e.g., in serial recall, Krakow & Hanson, 1985). So, where do deaf individual’s difficulties with mathematics come from? One possibility is that problems begin with the interaction between the innately determined core representation of number and cultural factors such as schooling and language acquisition. Clearly, the education and language education of deaf children varies considerably from hearing children (Marschark et al., 2002). Whilst formal schooling does provide the necessary tools (e.g., symbolic notation, number words, and so on) to carry out more complex numerical tasks, these skills need to be co-ordinated with existing representations of number. A lack of co-ordination between linguistic, symbolic, and analog forms of number representation is likely to lead to difficulties in the acquisition of arithmetic. We have linguistic as well as semantic representations of number, and each representation will be weighted differently according to the task. For example, Lee and Kang (2002) argued that whilst multiplication relies on an auditory verbal code, subtraction makes use of the visual-analog magnitude code (Dehaene, 1997). Therefore, if deaf participants do have access to a visual-analog magnitude code, as the current results would suggest, but have less efficient access to the auditory verbal code, we would predict selective problems in only certain areas of arithmetic and mathematics. Evidence might also be apparent for a lack of associative strength of magnitude information in semantic memory. When asked to solve arithmetic problems, adults typically show a range of errors (e.g., table errors, associative errors, e.g., responding 12 to the questions 3+4), because facts are thought to be stored in associative networks (see Ashcraft, 1992, for a review). Again, it would be interesting to see if the same pattern of errors is apparent in deaf individuals. Alternatively, there may be more a basic difference between deaf and hearing individuals at play here. Whereas hearing children learn about the world using bcorrelationsQ between auditory and visual information (e.g., between an object and its name), deaf children need to attend to and relate two visual information sources to accomplish the same thing (Marschark et al., submitted for publication). Given that approximately 95% of deaf children have hearing parents–most of whom lack effective communication with their young children–it is perhaps not surprising that information processing among deaf children proceeds somewhat differently than it does among hearing children with hearing parents. In their research involving hearing college students, Mayer and Morena (1998) demonstrated that individuals with less content knowledge benefit more from combined verbal and visual information. Sequential presentation of verbal and visual materials, the situation functionally faced by deaf children, increases cognitive load (Iding, 2000). Deaf children thus may become caught in a spiral of having less information, lesser automaticity in activating conceptual knowledge necessary in various situations, and hence being less able to apply knowledge they do have. The present results indicate the need to alter instructional strategies for deaf students (admittedly difficult in mainstream settings) in order to better match their cognitive skills and learning styles. The apparent inefficiency of deaf students’ retrieval of magnitude information suggests that classroom drills, assignments, and games (at least in lower grades) which emphasize retrieval of numerical quantities may improve mathematics skills. In fact, the disruption caused by attention-splitting between verbal descriptions and their visual referents has long been recognized as problematic for deaf children, due to their dependence on visual

234

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

language (Harris, Jones, Brookes, & Grant, 1986). Meanwhile, Todman and Seedhouse (1994) found that visual information presented successively was significantly more difficult for deaf children to integrate and retain than information presented simultaneously. Yet, the issue has not been addressed with regard to pedagogy beyond cautions to teachers against talking whilst facing the blackboard and the need to give deaf students more time to read print materials—cautions that are often forgotten or ignored (Marschark et al., 2002). A full understanding of deaf students’ knowledge and learning of mathematics (as well as other subjects) therefore requires greater elaboration of the cognitive basis of their learning. Whilst it might appear culturally sensitive to suggest that deaf and hearing individuals are fully comparable and learn in similar ways, the results presented here and a variety of prior findings clearly indicate that this is not the case. Beyond improving the current unacceptable situation with regard to deaf students’ mathematics abilities, research of this sort also would provide valuable insights into modes and consequences of information processing beyond those normally demonstrated for hearing students of all ages. References Allen, T. E. (1986). Patterns of academic achievement among hearing impaired students: 1974–1983. In A. N. Schildroth, & M. A. Karchmer (Eds.), Deaf children in america (pp. 161 – 206). San Diego, CA7 College-Hill Press. Ashcraft, M. H. (1992). Cognitive arithmetic: A review of data and theory. Cognition, 44, 75 – 106. Bavelier, D., Tomann, A., Hutton, C., Mitchell, T., Corina, D., Liu, G., et al. (2000). Visual attention to the periphery is enhanced in congenitally deaf individuals. Journal of Neuroscience, 20, 1 – 6. Bebko, J. (1998). Learning, language, memory, and reading: The role of language automatization and its impact on complex cognitive activities. Journal of Deaf Studies and Deaf Education, 3, 4 – 14. Berch, D. B., Foley, E. J., Hill, R. J., & Ryan, P. M. (1999). Extracting parity and magnitude from arabic numerals: Developmental changes in number processing and mental representation. Journal of Experimental Child Psychology, 74, 286 – 308. Brysbaert, M. (1995). Arabic number reading: On the nature of the numerical scale and the origin of phonological recoding. Journal of Experimental Psychology. General, 124, 434 – 452. Bull, R., & Johnston, R. S. (1997). Children’s arithmetical difficulties: Contributions from processing speed, item identification, and short-term memory. Journal of Experimental Child Psychology, 65, 1 – 24. Chamberlain, C., Mayberry, R. I. (1994). Do the Deaf bseeQ Better? Effects of Deafness on Visual-Spatial Skills. Poster presented at TENNET V meetings Montreal, May. Corina, D. P., Kritchevsky, M., Bellugi, U. (1992). Linguistic permeability of unilateral neglect: evidence from American sign language. Proceedings of the cognitive science conference (pp. 384–389). Hillside, NJ: Erlbaum. Dehaene, S. (1997). The Number Sense: How the Mind Creates Mathematics. London7 Penguin. Dehaene, S. (2003). The neural basis of the Weber–Fechner law: A logarithmic mental number line. Trends in Cognitive Sciences, 7, 145 – 147. Dehaene, S., & Akhavein, R. (1995). Attention, automaticity and levels of representation in number processing. Journal of Experimental Psychology. Learning Memory, and Cognition, 21, 314 – 326. Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology. General, 122, 371 – 396. Dehaene, S., & Changeux, J. (1993). Development of elementary numerical abilities: A neuronal model. Journal of Cognitive Neuroscience, 5, 390 – 407. Deheane, S., Deheane-Lambertz, G., & Cohen, L. (1998). Abstract representation of number in the animal and human brain. Nature Neuroscience, 21, 355 – 361. Deheane, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison digital: Analogical and symbolic distance effects in two-digit number comparison. Journal of Experimental Psychology. Human Perception and Performance, 16, 626 – 641.

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

235

Detterman, D. K., & Thompson, L. A. (1997). What is so special about special education? American Psychologist, 52, 1082 – 1090. Emmorey, K., & Kosslyn, S. (1996). Enhanced image generation abilities in deaf signers. A right hemisphere effect. Brain and Cognition, 32, 28 – 44. Emmorey, K., Kosslyn, S., & Bellugi, U. (1993). Visual imagery and visual-spatial language: Enhanced imagery abilities in deaf and hearing ASL signers. Cognition, 46, 139 – 181. Epstein, K. I., Hillegeist, E. G., & Grafman, J. (1995). Number processing in deaf college students. American Annals of the Deaf, 139, 336 – 347. Feigenson, L., Deheane, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8, 307 – 314. Fias, W., Brysbaert, M., Geypens, F., & d’Ydewalle, G. (1996). The importance of magnitude information in numerical processing: Evidence from the SNARC effect. Mathematical Cognition, 2, 95 – 110. Frostad, P., & Ahlberg, A. (1999). Solving story based arithmetic problems: Achievement of children with hearing-impairment and their interpretation of meaning. Journal of Deaf Studies and Deaf Education, 4, 283 – 293. Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44, 43 – 74. Gevers, W., Reynvoet, B., & Fias, W. (2003). The mental representation of ordinal sequences is spatially organised. Cognition, 87, B87 – B95. Gevers, W., Reynvoet, B., & Fias, W. (2004). The mental representation of ordinal sequences is spatially organised: Evidence from days of the week. Cortex, 40, 171 – 172. Girelli, L., Lucangeli, D., & Butterworth, B. (2000). The developmental of automaticity is accessing number magnitude. Journal of Experimental Child Psychology, 76, 104 – 122. Harris, M., Jones, D., Brookes, S., & Grant, J. (1986). Relations between the non-verbal context of maternal speech and rate of language development. British Journal of Developmental Psychology, 4, 261 – 268. Iding, M. K. (2000). Is seeing believing? Features of effective multimedia for learning science. International Journal of Instructional Media, 27, 403 – 415. Iverson, W., Nuerk, H. C., & Willmes, K. (2004). Do signers think differently? The processing of number parity in deaf participants. Cortex, 40, 176 – 178. Koontz, K. L., & Berch, D. B. (1996). Identifying simple numerical stimuli: Processing inefficiencies exhibited by arithmetic learning disabled children. Mathematical Cognition, 2, 1 – 23. Krakow, R. A., & Hanson, V. L. (1985). Deaf signers and serial recall in the visual modality: Memory for signs, fingerspelling, and print. Memory & Cognition, 13, 265 – 272. Lang, H. G. (2003). Perspectives on the history of deaf education. In M. Marschark, & P. E. Spencer (Eds.), Oxford handbook of deaf studies, language, and education (pp. 9 – 20). New York7 Oxford University Press. Lee, K. M., & Kang, S. Y. (2002). Arithmetic operation and working memory: Differential suppression in dual tasks. Cognition, 83, B63 – B68. Lorch, R. F., & Meyers, J. L. (1990). Regression analyses of repeated measures data in cognition research. Journal of Experimental Psychology. Learning Memory, and Cognition, 16, 149 – 157. Marschark, M. (1993). Psychological development of deaf children. New York7 Oxford University Press. Marschark, M. (2003). Cognitive functioning in deaf adults and children. In M. Marschark, & P. E. Spencer (Eds.), Oxford Handbook of Deaf studies, language, and education (pp. 464 – 477). New York7 Oxford University Press. Marschark, M. (submitted for publication). Intellectual functioning of deaf adults and children: Answers and questions. European Journal of Cognitive Psychology. Marschark, M., Blatto-Vallee, G., Bull, R., Cornoldi C. (2003). Relative Magnitude Judgments by Deaf and Hearing Individuals: Surprising Similarities and Surprising Differences. Paper presented at the annual meeting of the Psychonomic Society, Vancouver. Marschark, M., Convertino, C., McEvoy, C., & Masteller, A. (2004). Organization and use of the mental lexicon by deaf and hearing individuals. American Annals of the Deaf, 149, 51 – 61. Marschark, M., Convertino, C. M., & LaRock, D. (in press). Understanding and optimizing academic performance of deaf students: Access, opportunities, and outcomes. In D. F. Moores & D. S. Martin (Eds.), Deaf learners: new developments in curriculum and instruction. Washington, DC: Gallaudet University Press. Marschark, M., De Beni, R., Polazzo, M. G., & Cornoldi, C. (1993). Deaf and hearing-impaired adolescents’ memory for concrete and abstract prose: Effects of relational and distinctive information. American Annals of the Deaf, 138, 31 – 39.

236

R. Bull et al. / Learning and Individual Differences 15 (2005) 223–236

Marschark, M., Lang, H., & Albertini, J. (2002). Educating deaf students: From research to practice. New York7 Oxford University Press. Marschark, M., Pelz, J., Convertino, C., Sapere, P., Arndt, M. E., & Seewagen, R. (submitted for publication). Effects of threedimensional cues, interpreter feedback, and gaze allocation in learning via sign language interpreting. Mayer, R. E., & Morena, R. (1998). A split-attention effect in multimedia learning: Evidence for dual processing systems in working memory. Journal of Educational Psychology, 90, 312 – 320. McEvoy, C., Marschark, M., & Nelson, D. L. (1999). Comparing the mental lexicons of deaf and hearing individuals. Journal of Educational Psychology, 91, 1 – 9. Moyer, R. S., & Landauer, T. K. (1967). The time required for judgments of numerical inequality. Nature, 215, 1519 – 1520. Neville, H. J., & Lawson, D. (1987). Attention to central and peripheral visual space in a movement detection task: An eventrelated potential and behavioral study: II. Congenitally deaf adults. Brain Research, 405, 268 – 283. Ottem, E. (1980). An analysis of the cognitive studies with deaf subjects. American Annals of the Deaf, 125, 575 – 654. Paivio, A. (1986). Mental representations: A dual coding approach. Oxford7 Oxford University Press. Phelps, L., & Branyan, B. J. (1990). Academic achievement and nonverbal intelligence in public school hearing-impaired children. Psychology in the Schools, 27, 210 – 217. Proksch, J., & Bavelier, D. (2002). Changes in the spatial distribution of visual attention after early deafness. Journal of Cognitive Neuroscience, 14, 687 – 701. Rettenback, R., Diller, G., & Sireteaunu, R. (1999). Do deaf people see better? Texture segmentation and visual search compensate in adult but not in juvenile subjects. Journal of Cognitive Neuroscience, 11, 560 – 583. Richardson, J. T. E., McLeod-Gallinger, J., McKenn, B. G., & Long, G. L. (1999). Approaches to studying in deaf and hearing students in higher education. Journal of Deaf Studies and Deaf Education, 5, 156 – 173. Rubinsten, O., Henik, A., Berger, A., & Shahar-Shalev, S. (2002). The development of internal representations of magnitude and their association with Arabic numerals. Journal of Experimental Child Psychology, 81, 74 – 92. Schick, B. (2005). How might learning through an educational interpreter influence cognitive development? In E. Winston (Ed.), Educational interpreting: The questions we should be asking (pp. 73 – 87). Washington, DC7 Gallaudet University Press. Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75, 428 – 444. Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14, 237 – 243. Swisher, M. V. (1993). Perceptual and cognitive aspects of recognition of signs in peripheral vision. In M. Marschark, & M. D. Clark (Eds.), Psychological perspectives of deafness (pp. 229 – 265). Hillside, NJ7 Lawrence Erlbaum Associates. Talbot, K. F., & Haude, R. H. (1993). The relationship between sign language skill and spatial visualizations ability: Mental rotation of three-dimensional objects. Perceptual and Motor Skills, 77, 1387 – 1391. Tharpe, A., Ashmead, D., & Rothpletz, A. (2002). Visual attention in children with normal hearing, children with hearing aids, and children with cochlear implants. Journal of Speech, Language, and Hearing Research, 45, 403 – 413. Titus, J. C. (1995). The concept of fractional number among deaf and hard of hearing students. American Annals of the Deaf, 140, 255 – 263. Todman, J., & Seedhouse, E. (1994). Visual-action code processing by deaf and hearing children. Language and Cognitive Processes, 9, 129 – 141. Traxler, C. B. (2000). Measuring up to performance standards in reading and mathematics: Achievement of selected deaf and hard-of-hearing students in the national norming of the 9th Edition Stanford Achievement Test. Journal of Deaf Studies and Deaf Education, 5, 337 – 348. Tzelgov, J., Meyer, J., & Henik, A. (1992). Automatic and intentional processing of numerical information. Journal of Experimental Psychology. Learning, Memory, and Cognition, 18, 166 – 179. Walsh, V. (2003). A theory of magnitude: Common cortical metrics of time, space, and quantity. Trends in Cognitive Sciences, 7, 483 – 488. Wood, H., Wood, D., Kingsmill, M. C., French, J. R. W., & Howarth, S. P. (1984). The mathematical achievements of deaf children from different educational environments. British Journal of Educational Psychology, 54, 254 – 264. Zarfaty, Y., Nunes, T., & Bryant, P. (2004). The performance of young deaf children in spatial and temporal number tasks. Journal of Deaf Studies and Deaf Education, 9, 315 – 326.