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The performance of 24 French-Quebec 8‒9-year-old children with Mathematical Learning Disability (MLD) in Arabic and spoken number recognition, comprehension and production tasks designed to assess symbolic number processing was compared to that of 37 typically developing children (TD). Children with MLD were less successful than TD children in every symbolic numerical task, including recognition of Arabic and spoken numbers. These results thus suggested that this deficit of symbolic number recognition could compromise symbolic number comprehension and production. Children with MLD also presented with general cognitive difficulties as reading difficulties. Taken together, our results clearly showed that children with MLD presented with a symbolic numerical processing deficit that could be largely attributed to their poorer written language skills.

According to the

The cognitive processing of numbers has been conceptualized in various theoretical models. In the triple-code model (

Other models have been proposed to account for the development of numerical abilities in children. According to the model of number acquisition (

According to both the triple-code model (

The functional cognitive origin of MLD remains controversial. Some of this uncertainty stems from significant differences in the methods and tasks used to measure numerical skills in children. In previous studies, researchers usually used comparison tasks to assess numerical abilities in children with MLD and mainly found that they presented deficits in processing Arabic numbers (

Finally, the functional origin of MLD also remains controversial. According to some researchers, MLD is a pure numerical deficit that can be observed in isolation in children (

To summarize, there is still no consensus regarding the cognitive and functional origin of MLD. The originality of the present study lies in the systematic exploration, in the same group of children with MLD, of processes devoted to: the recognition of Arabic and spoken numbers, the access to number sense from Arabic and spoken numbers, the production of symbolic numbers from analogic numerosities (analogic-to-symbolic) and vice versa (symbolic-to-analogic). Moreover, the possible link between number processing and cognitive abilities was specifically examined.

The present study was designed to answer the following research questions: 1) Do children with MLD present with a deficit affecting Arabic number processing and/or spoken number processing? 2) Do children with MLD present with a deficit affecting number sense access, number production, and/or number recognition? And 3) Is the numerical deficit observed in children with MLD in relation with a more general cognitive deficit?

The focus of the present study was to investigate the numerical deficit in children who presented mathematics difficulties at school, as identified by their teacher.

Seventy-six 8- or 9-year-old third grade French-speaking children were recruited for this study. They had no history of sensory, physical, neurological, language, intellectual and/or psychiatric illness. All these children were recruited from eleven French schools in Quebec City, Canada. Of these schools, one was classified as being in a poor socioeconomic environment, nine as average, and one as being in an advantaged environment, according to the socioeconomic index published by Québec’s

The initial research sample included 37 children with typical development and 39 children with mathematical difficulties, as identified by the child’s teacher or special education teacher or speech-language pathologist. MLD is rarely diagnosed in Quebec and, therefore, we adopted an enrolment method based on the judgment by parents and teachers of mathematics abilities of children. The high proportion of children identified with mathematical difficulties was the direct result of this enrolment method, and does not represent the actual prevalence of MLD. These 39 children with mathematical difficulties met the

In summary, there were 37 TD children (mean age in months = 107.3, ^{2}_{p} = .001). There were 21 girls (57%) and 16 boys (43%) in the TD group and 20 girls (83%) and 4 boys (17%) in the MLD group and the difference in gender distribution between the two groups was significant (χ^{2}(1, ^{2}(1, ^{2}(2,

A series of ANOVAs (Group: TD vs. MLD) on the total score and subtests of the Zareki-R were performed. The performance of the children with MLD was significantly poorer than that of the TD children with respect to general mathematical abilities (see

Measure | TD ( |
MLD ( |
ANOVAs (Group: TD vs. MLD) |
||
---|---|---|---|---|---|

η^{2}_{p} |
|||||

Mathematic abilities (/163) | 140.8 (9.3) | 112.7 (9.8) | 127.687 | < .001* | .684 |

Dot counting (/6) | 5.7 (0.5) | 5.4 (0.8) | 3.350 | .072 | .054 |

Backward oral counting (/4) | 3.5 (0.9) | 2.6 (1.2) | 11.838 | .001* | .167 |

Number dictation (/16) | 15.4 (1.0) | 13.4 (2.4) | 20.238 | < .001* | .255 |

Mental calculation (/44) | 37.1 (4.8) | 26.8 (5.9) | 55.762 | < .001* | .486 |

Number reading (/16) | 15.4 (0.9) | 14.2 (1.8) | 12.931 | .001* | .180 |

Number positioning on an analogic scale (/24) | 17.9 (3.3) | 14.2 (3.7) | 17.055 | < .001* | .224 |

Magnitude comparison on spoken numbers (/16) | 14.7 (1.3) | 12.2 (2.1) | 34.272 | < .001* | .367 |

Estimation of sets of dots (/5) | 4.2 (0.8) | 3.8 (1.2) | 3.409 | .070 | .055 |

Contextual estimation of quantities (/10) | 7.9 (2.5) | 5.4 (2.9) | 13.413 | .001* | .185 |

Spoken problem-solving (/12) | 9.1 (2.0) | 5.3 (2.3) | 46.907 | < .001* | .443 |

Magnitude comparison of Arabic numbers (/10) | 9.8 (0.5) | 9.5 (0.7) | 3.683 | .060 | .059 |

Neuropsychological abilities | |||||

Non-verbal reasoning (/36) | 28.4 (3.2) | 25.6 (3.0) | 10.461 | .002* | .151 |

Processing speed (/126) | 39.5 (7.4) | 34.9 (5.4) | 6.686 | .012* | .102 |

Visuospatial short-term memory (/16) | 7.0 (1.6) | 6.3 (1.3) | 3.344 | .072 | .054 |

Visuospatial working memory (/14) | 6.3 (2.0) | 5.4 (1.3) | 3.935 | .052 | .063 |

Verbal short-term memory (/12) | 9.1 (2.0) | 8.2 (2.5) | 2.614 | .111 | .042 |

Verbal working memory (/12) | 6.9 (2.1) | 5.2 (1.4) | 12.776 | .001* | .178 |

Linguistic abilities | |||||

Reading and spelling (/90) | 77.9 (6.8) | 66.0 (10.0) | 30.499 | < .001* | .341 |

Word comprehension and production (/80) | 53.2 (3.8) | 50.7 (2.9) | 7.367 | .009* | .111 |

Reaction time (in ms) | 513.9 (135.9) | 634.9 (203.3) | 10.022 | .002* | .145 |

*These ability scores were significantly lower than those of TD children (

All children first underwent a general neuropsychological assessment tapping the cognitive domains sensitive to number processing. They were tested individually in a quiet room at their school. Testing took place from February to June of the school year. Each child was tested in one session lasting between 45 and 60 minutes.

The assessment battery first included measures of non-verbal reasoning with Raven's Coloured Progressive Matrices (

Since the experimental tasks were administered using a computer to record response time, a reaction time task was administered to the children. This task was presented on a computer screen with DMDX (

The performance of the MLD group (a series of ANOVAs, Group: TD vs. MLD) was significantly lower than that of the TD group for non-verbal reasoning, processing speed, verbal working memory, word comprehension, word production, reading and spelling (see

A 2 (Group: TD vs. MLD) x 2 (Task: Right vs. Left reaction time) ANOVA was performed on response latencies. Children with MLD were slower than TD children (^{2}_{p} = .145). However, neither Task effect (^{2}_{p} < .001) nor interaction (^{2}_{p} = .001) were observed.

As presented in the introduction, the co-occurrence of MLD with other cognitive and linguistic disorders is very common (

TD children as well as those with MLD underwent an experimental numerical assessment designed to identify the locus of the deficit at the origin of MLD. The presentation order of the experimental tasks was randomized between children. All the stimuli were presented on a computer screen with DMDX (

The ability to access number sense from symbolic numbers was assessed with two tasks administered in the Arabic and spoken codes. They involved recognizing symbolic numbers and accessing their numerosities (in other words, number sense).

In the Arabic code, stimuli, displayed on the computer screen, consisted of two white squares (side = 10 cm) separated by a 2-cm space, containing a blue (left stimulus) and a yellow (right stimulus) Arabic number (Calibri, 22 millimeters) and children were asked to select the largest one. Magnitude comparisons were performed on 30 pairs in which numbers varied from 1 to 99. The size of numbers was controlled with regard to the ENS and the ANS, as well as to medium numerosities from 5 to 9, which had a particular position and overlapped the two previous systems. Indeed, these pairs varied along three numerical sizes: 8 small pairs from 1 to 4 were contrasted with 7 medium pairs from 5 to 9 and 15 large pairs from 10 to 99. The pairs also varied along three ratios between the numbers: 10 pairs with ratio 1/2, 10 pairs with ratio 2/3, and 10 pairs with ratio 3/4. The side of the correct response (i.e., the largest number) was counterbalanced: each pair appeared twice, once in ascending (e.g. 1-2) and once in descending (e.g. 2-1) order, for a total of 60 pairs. Pairs were presented in random order. Five practice trials were offered to allow the children to become familiar with the task.

Each trial started with the presentation of the pair, remaining on the screen until response or until 4000 milliseconds, and followed by a 500-millisecond delay. Blue and yellow stickers were respectively stuck on the left Alt key and right Alt key of the keyboard. As this task required choosing between two possible responses, one displayed on the left and one displayed on the right side, the children were asked to respond by pressing the button on the side of the correct response (i.e. the largest number). Responses and latencies were recorded by the computer, from the onset of the first number word uttered.

In the spoken code, two white squares containing blue and yellow pictures of earphones were displayed on the computer screen. The children listened to pairs of spoken numbers, separated by 1000 milliseconds, and were asked to select the largest one. They had to press the left blue button or right yellow button respectively if the largest number was the first or second number heard.

Two tasks in the Arabic and spoken codes were administered to assess the ability to access number sense from symbolic numbers. More particularly, these tasks investigated the direct association between a symbolic number and the corresponding numerosity. Two numbers were presented on the computer screen and children were asked to decide if they represented the same magnitude or not. Compared to the magnitude comparison, this task is cognitively more demanding. First, the concurrent processing of two different modalities within the same trial is complex, as pointed out by

In the Arabic code, a set of dots in random spatial arrangement and an Arabic number were presented on the computer screen and the children had to decide if the two numbers (set of dots and Arabic number) represented the same magnitude or not. In the spoken code, a set of dots and an earphone picture were presented on the computer screen; at the same time, the children listened to a spoken number and had to decide if the two numbers (set of dots and spoken number) represented the same magnitude or not.

Stimuli, separated by a 2-cm space, consisted of a black square (10-cm sides) containing a set of blue dots (left stimulus) and a white square containing a black Arabic number in the Arabic code (Calibri, 22 millimeters) or a yellow earphone picture in the spoken code (right stimulus).

Magnitude judgments were performed on pairs in which the numbers varied from 1 to 9. The 18 pairs varied along two numerical sizes: 9 small pairs with 1 to 4 dots, and 9 medium pairs with 5 to 9 dots. Different pairs also varied along three ratios of dots: 3 pairs with ratio 1/2, 3 pairs with ratio 2/3, and 3 pairs with ratio 3/4. The side on which the largest number was presented in pairs with two different numbers was counterbalanced: each pair appeared twice, once in ascending (e.g., 1-2) and once in descending (e.g., 2-1) order, for a total of 36 pairs. Pairs were presented in random order. Five practice trials were offered to allow the children to become familiar with the task.

Each trial started with the presentation of the pair, remaining on the screen until response or until 4000 milliseconds, and followed by a 500-millisecond delay. Green and red stickers were respectively stuck on the left Ctrl key and the right Ctrl key of the keyboard. As this task required choosing between two possible responses (“yes, the two numbers (set of dots and symbolic number) represent the same magnitude” or “no, the two numbers (set of dots and symbolic number) did not represent the same magnitude”), one displayed on the left and one displayed on the right side, the children were asked to respond by pressing the left Ctrl key button for the answer “yes” or the right Ctrl key for the answer “no”. Responses latencies were recorded by the computer from the onset of the number word uttered.

Two tasks were administered to assess symbolic number production from analogic numerosities. These tasks involved numerosity recognition as well as production of symbolic (Arabic or spoken) numbers. A set of dots in random spatial arrangement (black squares (10-cm sides) which contained a set of blue circles) was presented on the computer screen. Children were asked to estimate the numerosity represented by the dots by writing (without counting) the corresponding Arabic number on a sheet of paper (Arabic code) or by orally producing the corresponding spoken number (spoken code).

Number production was performed on sets in which the number of dots varied from 1 to 99. The 30-dot sets varied along three numerical sizes: 10 small numerosities with 1 to 4 dots (1, 1, 2, 2, 2, 3, 3, 3, 4, 4) were contrasted with 10 medium numerosities with 5 to 9 dots (5, 5, 6, 6, 7, 7, 8, 8, 9, 9) and with 10 large numerosities with 10 to 99 dots (12, 17, 26, 31, 46, 53, 64, 79, 85, and 98 for the Arabic code; 13, 18, 25, 32, 47, 51, 66, 74, 89, and 94 for the spoken code). Because of the nature of the task and the small size of dots sets, it was highly probable that children transcode small numerosities by counting dots. Therefore, in this task, we expected that children with MLD would be as successful as TD children in this condition of small numerosities.

They were presented in random order. Five practice trials were offered to allow the children to become familiar with the task.

Each trial started with the presentation of a set of dots, remaining on the screen until response or until 4000 milliseconds, and followed by a 500-millisecond delay. Because it was an estimation task and an exact response could not be expected, it was not possible to consider true or false responses in analyses. Therefore, a ratio (child’s response / target number) between child’s response and the target number was first calculated (for example, a ratio of 0.5 corresponded to a response of 6 for the target number 12). Second, the absolute distance (ADR) between the child’s calculated ratio and the perfect ratio 1 was calculated: ADR = 1 – child’s response / target number. Finally, response time was also recorded by the experimenter, who pressed a response key as soon as the child started his/her response. The response latencies were recorded from the onset of the number word uttered. This measure was used in the study by

Two tasks were administered to assess analogic number production from a symbolic number. These tasks involved symbolic number (Arabic or spoken) recognition, as well as production of numerosities in an analogic format. In the Arabic code, an Arabic number (Calibri, 22 millimeters) was presented on the computer screen. In the spoken code, an earphone picture was presented on the computer screen and the children listened to a spoken number. Then they were asked to quickly circle with a pencil (without counting) the corresponding number of dots on a sheet of paper on which a set of one hundred dots in random spatial arrangement was drawn. The participants were not informed that the response sheet comprised 100 dots, to avoid they resort to a proportion strategy. An illustration of the task in each code is presented in

Illustration of the Arabic-to-Analogic Number Transcoding Task (a) and the Spoken-to-Analogic Number Transcoding Task (b).

Arabic and spoken numbers varied from 1 to 99. The 30 numbers varied along three numerical sizes: 10 small numbers from 1 to 4 dots (1, 1, 2, 2, 2, 3, 3, 3, 4, 4) were contrasted with 10 medium numbers from 5 to 9 dots (5, 5, 6, 6, 7, 7, 8, 8, 9, 9) and with 10 large numbers from 10 to 99 dots (12, 17, 26, 31, 46, 53, 64, 79, 85, and 98 for the Arabic code; 11, 19, 24, 33, 48, 52, 65, 76, 87, and 95 for the spoken code). Stimuli were presented in random order. Five practice trials were offered to allow the children to become familiar with the task.

Each trial started with the presentation of symbolic numbers, remaining on the screen until response or until 7000 milliseconds, and followed by a 500-millisecond delay. With respect to the analyses of responses, the ADR was calculated.

Two lexical decision tasks were administered to assess the recognition of symbolic numbers. In the Arabic code, a symbol was presented on the computer screen. In the spoken code, an earphone picture was presented on the computer screen; at the same time, children listened to a spoken symbol. In both tasks, children were asked to decide if the symbol seen or heard was a number or not.

In the Arabic code, stimuli consisted of a white square containing digits (1, 2, 3, 4, 5, 6, 7, 8, 9 presented twice), letters (A, B, C, D, E, F, G, H, J), or mathematical signs (+, -, x, ÷, =, ≠, <, >, //), written in black, for a total of 36 stimuli. In the spoken code, stimuli were digits (1 (/œ̃/), 2 (/dœ/), 3 (/tʁwa/), 4 (/kɑtʁ/), 5 (/sɛ̃k/), 6 (/sis/), 7 (/sɛt/), 8 (/ɥit/), 9 (/nœf/), presented twice), monosyllabic words (en 'in' (/ɛ̃/), beau 'beautiful' (/bo/), bruit 'noise' (/bʁɥi/), pauvre 'poor' (/povʁ/), fonte 'melting' (/fɔ̃t/), fauve 'tawny' (/fov/), chaque 'each' (/ʃɑk/), hyène 'hyena' (/jɛn/), mousse 'foam'(/mus/)), or non-words (u (/y/), ti (/ti/), prui (/pʁɥi/), toupre (/tupʁ/), zonp (/zɔ̃p/), chup (/ʃup/), fat (/fɑt/), hiap (/jap/), mosse /mɔs/) for a total of 36 stimuli. Words and pseudo-words were selected to resemble digit names with respect to number and nature of phonemes as well as syllabic structure. Furthermore, words were controlled in frequency with regard to numbers from the basis Lexique 3 (

Stimuli were presented in random order for each participant. Five practice trials were offered to allow the children to familiarize themselves with the task. Each trial started with the presentation of the digit, remaining on the screen until response (in the Arabic code only) or until 4000 milliseconds, and followed by a 500-millisecond delay. Children were asked to respond by pressing the button on the side of the correct response (i.e., on the left green button to answer “yes, it is a digit”, or on the right red button to answer “no, it is not a digit”). Responses were recorded by the computer from the onset of the number word uttered.

A 2 (Group: TD vs. MLD) x 3 (Size: Small vs. Medium vs. Large) x 2 (Code: Arabic vs. Spoken) ANOVA was performed on the mean number of correct responses. The analysis revealed a significant Group effect (^{2}_{p} = .115), indicating that children with MLD were less successful than TD children (TD: Mean (^{2}_{p} = .063), no Size effect (^{2}_{p} = .003), as well as no interaction (^{2}_{p} = .001 for the Group x Code interaction; ^{2}_{p} = .004 for the Group x Size interaction; ^{2}_{p} = .007 for the Code x Size interaction; and ^{2}_{p} = .005 for the Group x Code x Size interaction). The Group effect disappeared with non-verbal reasoning as well as reading and spelling variables as covariates; it remained significant when each of the other cognitive variables as well as the socioeconomic environment variable were introduced as covariate.

A 2 (Group: TD vs. MLD) x 3 (Size: Small vs. Medium vs. Large) x 2 (Code: Arabic vs. Spoken) ANCOVA was performed on response latencies with reaction time as covariate. The analyses revealed no Group effect (^{2}_{p} = .031), a Size effect (^{2}_{p} = .284), a Code effect (^{2}_{p} = .812), but no interaction (^{2}_{p} = .041 for the Code X Size interaction; ^{2}_{p} = .045 for the Group X Code interaction; ^{2}_{p} = .032 for the Group X Size interaction; and ^{2}_{p} = .023 for the Group X Code X Size interaction). Children were faster for small than medium than large numerosities, as well for Arabic than spoken numbers. The mean response latency was 955 milliseconds for Arabic numbers and 3890 milliseconds for spoken numbers.

A 2 (Group: TD vs. MLD) x 2 (Size: Small vs. Medium) x 2 (Code: Arabic vs. Spoken) ANOVA was performed on the mean number of correct responses. The analyses revealed a significant Group effect (^{2}_{p} = .108), indicating that children with MLD were less successful than TD children (TD: Mean (^{2}_{p} = .737) was also observed, but no Code effect (^{2}_{p} = .033). A Code X Size interaction ^{2}_{p} = .068 but no other interaction was observed (^{2}_{p} = .014 for the Group x Code interaction; ^{2}_{p} = .007 for the Group x Size interaction; and ^{2}_{p} = .006 for the Group x Code x Size interaction). The Group effect disappeared with reading and spelling variables as covariates; it became marginal with non-verbal reasoning and verbal working memory variables as covariates; finally, it remained significant when the other cognitive variables and the socioeconomic environment variable were introduced as covariates.

A 2 (Group: TD vs. MLD) x 2 (Size: Small vs. Medium) x 2 (Code: Arabic vs. Spoken) ANCOVA was performed on response latencies with reaction time as covariate. The analyses revealed a Group effect (^{2}_{p} = .129), a Size effect (^{2}_{p} = .141), a Code effect (^{2}_{p} = .161), but no interaction (^{2}_{p} = .008 for the Group x Code interaction; ^{2}_{p} < .001 for the Group x Size interaction; (^{2}_{p} = .012) for the Code x Size interaction; and ^{2}_{p} = .031 for the Group x Code x Size interaction). Children were faster for small than medium numerosities, as well for Arabic than spoken numbers. The mean response latency was 1285 milliseconds for Arabic numbers and 2028 milliseconds for spoken numbers.

First, a 2-sample test for equality of proportions, with continuity correction using the prop.test R 3.2.2 (

A 2 (Group: TD vs. MLD) x 2 (Size: Medium vs. Large) x 2 (Code: Arabic vs. Spoken) ANOVA was performed on the ADR. The analysis revealed a significant Group effect (^{2}_{p} = .117), indicating that the ADR was larger in children with MLD than in TD children (TD: Mean (^{2}_{p} = .457), indicating that the children’s ADR was significantly larger for large than medium numerosities. A Code effect was also observed (^{2}_{p} = .106), indicating that the children’s ADR was smaller for spoken than Arabic numbers. No interaction was observed (^{2}_{p} = .011 for the Group x Code interaction; ^{2}_{p} = .008 for the Group x Size interaction; (^{2}_{p} = .029) for the Code x Size interaction; and ^{2}_{p} = .001 for the Group x Code x Size interaction). The Group effect disappeared with reading and spelling as covariates, but remained significant when each of the other cognitive variables and the socioeconomic environment variable were introduced as covariate.

A 2 (Group: TD vs. MLD) x 3 (Size: Small vs. Medium vs. Large) x 2 (Code: Arabic vs. Spoken) ANOVA was performed on response latencies. The analyses revealed no Group effect (^{2}_{p} = .020), a Size effect (^{2}_{p} = .911), a Code effect (^{2}_{p} = .775), a Code x Size interaction (^{2}_{p} = .104), a Group x Size interaction (^{2}_{p} = .061), but no other interaction (^{2}_{p} = .004 for the Group x Code interaction and ^{2}_{p} = .014 for the Group x Code x Size interaction). Children were faster for Arabic than spoken numbers. The mean response latency was 1237 milliseconds for Arabic numbers and 3704 milliseconds for spoken numbers. Post-hoc analysis showed that, in children with MLD, the response latencies were similar for medium and large numerosities (

First, a 2-sample test for equality of proportions with continuity correction using the prop.test, R 3.2.2 was performed in Arabic and spoken modalities for children whose ADR was perfect (i.e. equal to 0) for small numerosities. As expected, the proportion of MLD children who got a perfect ADR equal to 0 in Arabic (26/37, 70.3%) and spoken (20/37, 54.1%) codes was equivalent to that of children with MLD (Arabic numbers: 12/24, 50.0%;

A 2 (Group: TD vs. MLD) x 2 (Size: Medium vs. Large) x 2 (Code: Arabic vs. Spoken) ANOVA was performed on the ADR (see ^{2}_{p} = .267), indicating that the ADR was larger in children with MLD, than in TD children, and a significant Size effect (^{2}_{p} = .105), indicating that the children’s ADR was significantly larger for large than medium numerosities. No Code effect was observed (^{2}_{p} = .017), indicating that the children with MLD were as successful for Arabic as for spoken numbers. Furthermore, a significant Group x Size interaction was observed (^{2}_{p} = .070). More precisely, post-hoc analysis showed that, in children with MLD, the ADR was similar for medium and large numerosities (

TD and MLD performances on Symbolic-to-Analogic number transcoding tasks (error bars = standard deviation).

A 2 (Group: TD vs. MLD) x 3 (Size: Small vs. Medium vs. Large) x 2 (Code: Arabic vs. Spoken) ANOVA was performed on response latencies. The analyses revealed a marginal Group effect (^{2}_{p} = .058), a Size effect (^{2}_{p} = .684), a Code effect (^{2}_{p} = .460), a Code x Size interaction (^{2}_{p} = .074), but no other interaction (^{2}_{p} < .001 for the Group x Code interaction; ^{2}_{p} = .009 for the Group x Size interaction; and ^{2}_{p} = .017 for the Group x Code x Size interaction). Children were faster for small than medium than large numerosities, as well for Arabic than spoken numbers. The mean response latency was 2484 milliseconds for Arabic numbers and 5122 milliseconds for spoken numbers.

A 2 (Group: TD vs. MLD) x 2 (Code: Arabic vs. Spoken) ANOVA was performed on the mean number of correct responses. The analyses revealed a significant Group effect (^{2}_{p} = .072) indicating that children with MLD were less successful than TD children (TD: Mean (^{2}_{p} = .051) and no interaction (^{2}_{p} = .028) were observed. The Group effect disappeared with non-verbal reasoning, reading and spelling, and word comprehension and production as covariates; it became marginal with visuospatial short-term memory, as well verbal working memory variables as covariates; finally, it remained significant when processing speed, visuospatial working memory, verbal short term memory, and the socioeconomic environment variable were introduced as covariates.

A 2 (Group: TD vs. MLD) x 2 (Code: Arabic vs. Spoken) ANCOVA was performed on response latencies with reaction time as covariate. The analyses revealed no Group effect (^{2}_{p} = .054), a Code effect (^{2}_{p} = .548), and no interaction (^{2}_{p} < .001). Children were faster for Arabic than spoken numbers. The mean response latency was 793 milliseconds for Arabic numbers and 1659 milliseconds for spoken numbers.

In this study, we investigated the processing of symbolic numbers in children with MLD. In addition to cognitive, linguistic, and mathematic tests, the children were administered experimental tasks involving symbolic number recognition, comprehension, and production designed to answer the following research questions: 1) Do children with MLD present with a deficit affecting Arabic number processing and/or spoken number processing? 2) Do children with MLD present with a deficit affecting number sense access, number production, and/or number recognition? And 3) Is the numerical deficit observed in children with MLD in relation with a more general cognitive deficit?

In summary, results showed that children with MLD had poorer performances than TD children in all symbolic numerical tasks for which moderate and large effect size were found. With respect to the access to number sense from symbolic numbers, children with MLD had lower performances than TD children, regardless of the numerical code, the task, and the size of magnitude. Compared to TD children, children with MLD were also impaired in the production and estimation of medium and large symbolic numbers, regardless of the Arabic or spoken numerical code. In number recognition, they were impaired for Arabic and spoken numbers. This study is the first to show, in the same group of participants, that children with MLD not only presented with an Arabic number processing deficit but also with a spoken number processing deficit and a recognition number deficit. Altogether, results suggested that children with MLD had a general symbolic number processing impairment. Regarding the cognitive capacities, the MLD group was significantly poorer than the TD group for non-verbal reasoning, processing speed, verbal working memory, word comprehension, word production, and reading and spelling.

With regard to our first research question, our results clearly suggest that MLD is not limited to a specific code but affects both Arabic and spoken symbolic numbers. The present study was thus in line with studies showing that children with MLD had difficulty with Arabic (e.g.,

For the second research question, our results are the first to show that MLD affects number access, number production, and number recognition. First, with respect to number access, our results were in agreement with those reported by

With respect to number production, our results also confirmed those of studies showing that children with MLD were impaired in tasks consisting of transcoding numerosities in spoken numbers (for example,

Finally, with respect to number recognition, our results pointed to a deficit in recognizing Arabic and spoken numbers in children with MLD. Recognition is a prerequisite for all subsequent number processing abilities. Having difficulty recognizing numbers would thus lead to subsequent impairment in the automatic access to number sense. In other words, a deficit at this processing stage would compromise symbolic number comprehension and production. To our knowledge, this is the first study to show the presence of number recognition impairment in children with MLD.

For the third research question, our results showed that children with MLD also presented with general cognitive difficulties, in addition to mathematical and numerical difficulties. Compared to TD children, they showed lower performance in tests exploring non-verbal reasoning, processing speed, verbal working memory, word comprehension, word production, and reading and spelling. Compared to norms however, no children with MLD presented with a deficit in non-verbal reasoning and processing speed, while two of them presented with a deficit in verbal short-term memory and verbal working memory. Finally, compared to norms, half of the children with MLD presented with a reading and spelling deficit. Most of the Group effects became marginal when the non-verbal reasoning scores were entered as covariates in the analyses. Moreover, most of the Group effects disappeared when the reading and spelling performances were entered as covariates in the analyses. This is not surprising because a high comorbidity of MLD and other developmental deficits such as dyslexia (

In conclusion, the present study showed that children with MLD had poorer performances than those of TD children in all symbolic numerical tasks, regardless of the numerical codes. To our knowledge, this is the first study to show that children with MLD had a symbolic number deficit affecting number access and production abilities, highly suggestive of a deficit in the access to number sense from Arabic, and spoken symbolic numbers. Such a deficit is congruent with results recently reported by

The present study has some implications in guiding effective educational and therapeutic strategies for children with MLD. Teachers and clinicians should better take into account and try to improve symbolic numerical deficiencies in children. A first approach could be to develop exercises and games for the spoken and written counting, building on a constructive and multiple encoding (e.g., song with and without fingers for the spoken numbers; drawing and kinaesthetic encoding for the Arabic numbers). Furthermore, teachers and clinicians could strengthen the link between numerical codes (analogical, Arabic and spoken numbers) by systematically presenting numbers in various formats in exercises and games. External representation as concrete objects could be used to illustrate and manipulate symbolic numbers. Some studies (e.g.,

The present study has some limitations. First, children were allowed to provide a response within many seconds (4 seconds in Judgment tasks; 7 seconds in transcoding tasks), although they were explicitly asked to answer as quickly as possible. Several examples were also given to encourage children to proceed as fast as possible and to prevent them from counting. Moreover, children answered within 1749 ms for small numerosities and 2028 ms for medium numerosities in judgment task. Such response latency made it highly unlikely the use of a counting strategy. Such response latencies are congruent with

The present study begs various other research questions. Further studies should first examine how the development of Arabic numbers is independent of or linked to the development of spoken numbers. According to the developmental model of number acquisition (

We would like to thank the following Québec schools for their participation: La Mosaïque, St Fidèle, La Farandole, St Pierre et St Laurent de l’île d’Orléans, Sous bois, St Jean Baptiste, Cap Soleil et St-Pierre, L’Orée des bois, Trivent, and Institut Saint-Joseph.

The authors have no funding to report.

The authors have declared that no competing interests exist.

This study was approved by the Research Ethics Committee of the Research Centre of the Institut universitaire en santé mentale de Québec.