^{*}

^{a}

^{a}

Current theoretical approaches point to the importance of several cognitive skills not specific to mathematics for the etiology of mathematics disorders (MD). In the current study, we examined the role of many of these skills, specifically: rapid automatized naming, attention, reading, and visual perception, on mathematics performance among a large group of college students (N = 1,322) with a wide range of arithmetic proficiency. Using factor analysis, we discovered that our data clustered to four latent variables 1) mathematics, 2) perception speed, 3) attention and 4) reading. In subsequent structural equation modeling, we found that the latent variable perception speed had a strong and meaningful effect on mathematics performance. Moreover, sustained attention, independent from the effect of the latent variable perception speed, had a meaningful, direct effect on arithmetic fact retrieval and procedural knowledge. The latent variable reading had a modest effect on mathematics performance. Specifically, reading comprehension, independent from the effect of the latent variable reading, had a meaningful direct effect on mathematics, and particularly on number line knowledge. Attention, tested by the attention network test, had no effect on mathematics, reading or perception speed. These results indicate that multiple factors can affect mathematics performance supporting a heterogeneous approach to mathematics. These results have meaningful implications for the diagnosis and intervention of pure and comorbid learning disorders.

Mathematics disorder (MD) is generally defined as poor mathematical abilities related to a specific brain function impairment (

The prevalence rate of MD is between 3-6% among children (

These reported prevalence rates suggest that comorbidity between learning disorders is very common. Despite this, most MD research does not take into account the role of comorbidity (

However, the cognitive processes underlying reading, mathematics and attention are not discrete. Reading and mathematics both involve verbal working memory, cognitive control, and the representation and retrieval of symbolic information (

Reading and mathematics both involve executive functions, entangling the relationship between RD, MD and ADHD further due to the central role of executive functions in ADHD (

Another complication in understanding the etiology of MD is that mathematics itself is heterogeneous. Many different tasks are labeled under the umbrella mathematics, such as counting, arithmetic, and geometry, and each of these tasks can involve different combinations of cognitive skills (

The goal of the current study was to examine the relationships between mathematics, reading and attention, in order to gain further understanding of the underlying cognitive mechanism of mathematics. To this end we analyzed the task performance of a large dataset of university students (

First, we examined the heterogeneity of the number and arithmetic task performance. Second, we looked at the role of phonological processing, verbal abilities, attention and perception speed in explaining individual differences in numerical and arithmetic abilities. We predicted that linguistic skills would positively relate to all of the mathematics tasks; however, different components of linguistic skills would affect each of the tasks differently. Specifically, phonological processing would relate to fact retrieval (

The participants were university students who came to the Student Support and Diagnostic Center for Learning Disabilities and ADHD for assessment to receive testing accommodations and benefits. The participants signed a consent form allowing their assessment results to be accessed for future research. Our dataset included 1,322 students that came to 3 test centers in Israel between the years 2007 and 2014. All the participants performed the entire diagnostic battery of the

Characteristic | Value |
---|---|

Age (in years) | |

Female | 52% |

Birth country Israel | 85% |

Hebrew mother tongue | 91% |

The

The goal of the task was to measure retrieval of arithmetic facts. The task included 80 simple arithmetic equations (e.g. 2 x 3 = 6) that were presented sequentially on the computer screen, the participant had to answer if the equation was correct or incorrect by keypress. The equations divided equally into addition, subtraction, multiplication and division problems (20 per category). The equations included 1-digit numbers and the possible solutions did not exceed 25. The equations were either correct (2 + 2 = 4, 40% of the trials), incorrect but with an associative distractor (2 + 6 = 12, 30% of the trials), or incorrect (3 * 4 = 10, 30% of the trials).

For the practice trials, an equation would appear for 10 seconds, while test trials were presented for 4 seconds, or until the participant selected an answer. Percent of correct answers and average reaction time (RT) per trial were measured.

The task was designed to measure procedural knowledge. It comprised of 96 equations that were presented sequentially on the computer screen, and participants needed to ascertain if the equation was correct or incorrect by keypress. The equations included numbers that ranged from 1-4 integers. All the equations required logarithmic, simple calculations. The equations divided equally into addition, subtraction, multiplication and division, and evenly into correct and incorrect solutions within each category (i.e. 12 addition problems with the correct solution). For the practice trials, an equation would appear for 10 seconds, while test trials were presented for 6 seconds or until the participant selected an answer. The entire task was about 10 minutes. Percent of correct answers and average RT per trial were measured.

This task measured understanding of the mental number line. It included 58 number lines that were presented sequentially on the computer screen. For each trial, different values appeared at the anchors of the number line, in black font, below the line (see

Example of trial from the Number Line Knowledge Task.

A 132-word text appeared on the computer screen. The participant was instructed to read the text aloud as accurately and quickly as possible without follow up questions. The results were voice-recorded via the computer and the examiner noted errors. Accuracy was measured as the percentage of words read correctly and pace as number of words per minute.

The task included 25 pseudowords that were presented sequentially on the computer screen. The words were displayed in pointed font. Hebrew letters are similar to consonants in English, while points are analogous to English vowels. After reading acquisition, Hebrew is easily read without points, however, they are required in order to read new words or pseudowords. All the words were made up of 2 or 3 syllables and followed grammar rules. None of the words looked or sounded like actual Hebrew words. The only way the words could be read was by grapheme-phoneme encoding. The participant was instructed to read the words aloud as accurately and quickly as possible. The participants’ results were voice recorded and the examiner marked errors on line. The measures taken were: accuracy, measured as the percentage of words read correctly, and RT, average time per item.

This task involved both grapheme-phoneme encoding and lexical retrieval. It included 40 pairs of pseudowords that were presented sequentially on the screen, similar to those from the previous task, however, they could sound like a word in Hebrew. The pairs of words differed by one consonant or one vowel (e.g. luv- buv). The participant had to decide by keypress which word sounded like a real word in Hebrew. Each pair was presented until the participant answered (maximum presentation time was 10 seconds) and the entire task was approximately 8 minutes. Accuracy, measured as percentage of correct answers, and average RT per trial, were measured.

This task measured phonological awareness. The participant heard recordings of 24 pseudowords via the computer. Each trial, the participant heard a pseudoword and was instructed to repeat it aloud. The pseudoword was played again and the participant was instructed to repeat the word without a specific sound. The maximum time allotted each trial was 10 seconds, and the entire task was approximately 5 minutes. The measures were accuracy, measured as percentage of correct answers, and average RT per trial.

The task included three texts, each text was followed by 9-11 questions (there was a total of 30 questions). The texts were ordered in level of difficulty, ranging from easy to difficult. The task was presented on the computer screen, and the students could click to go back and forth between the text and the questions. The questions were multiple choice with four possible answers per question. The questions included: basic understanding and fact finding, relationships between different parts of the text, and text interpretation. Performance was measured as the percent of correct answers from all of the questions.

The CPT measured sustained attention and impulsivity based on the principles of the attention tasks created by

The task used in the

In addition, the trials had varying types of cues. Some trials had no cue, others had center cues in the same location as the fixation point, double cues that appeared above and below simultaneously, or the spatial cue that appeared above or below the fixation point. The spatial cue could appear in the same location as the target stimulus, or the opposite location as the target stimulus. The target stimulus was presented for 1.7 seconds, while the whole trial lasted 4 seconds. The task lasted approximately 30 minutes.

There were several measures taken from this task. 1) Executive attention, the average RT for incongruent trials minus the average RT for congruent trials; 2) Alerting/arousal, average RT for un-cued trials minus the average RT for double cue trials; 3) Orienting of attention, the average RT of trials with central cues minus the average RT for spatial cue trials.

The task was based on (

This task had two conditions:

1) Parallel: this measured visual-spatial perception using a spatial frequency discrimination task. In each trial two horizontal gratings appeared simultaneously on the top and bottom halves of the computer screen. The two stimuli differed in spatial frequency. The participants had to ascertain whether the top or bottom was denser by pressing the up or down arrow keys on the keyboard. Each trial the reference grating appeared in either location along with a test grating. The task began with a contrast of 20% between the reference and test stimuli, the contrast adapted to the participant’s performance. Specifically, the reference and the test stimuli become more similar after every two correct answers or became less similar after every incorrect answer. The task ended when the participant reached his/her just noticeable difference or 80 trials. The just noticeable difference was the maximum density that a participant could correctly identify (defined by 18 replacements from increase in the density of the test stimuli to decreased or from decrease in the density of the test stimuli to increased).

2) Sequential: Similar to the other condition. While in the parallel task the two gratings appeared simultaneously, in the sequential task the gratings were presented sequentially and filled the entire computer screen (

To eliminate multicollinearity in future analysis, we examined the correlations between all of the tasks of the ^{i}. The cutoff of

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | - | ||||||||||||||

2 | .426^{**a} |
- | |||||||||||||

3 | .558^{**a} |
.407^{**a} |
- | ||||||||||||

4 | .316^{**a} |
.385^{**a} |
.295^{**a} |
- | |||||||||||

5 | .302^{**} |
.291^{**} |
.265^{**} |
.310^{**} |
- | ||||||||||

6 | .349^{**} |
.299^{**} |
.278^{**} |
.298^{**} |
.592^{**} |
-- | |||||||||

7 | .273^{**} |
.101^{**} |
.227^{**} |
.322^{**} |
.413^{**} |
.403^{**} |
- | ||||||||

8 | .178^{**} |
0.095 | .134^{**} |
.204^{**} |
.304^{**} |
.249^{**} |
.440^{**b} |
- | |||||||

9 | .037 | .005 | .000 | .029 | .031 | .029 | -.020^{b} |
-.001^{b} |
- | ||||||

10 | .024 | -.011 | .011 | -.010 | .015 | .026 | .006^{b} |
-.002^{b} |
.498^{**b} |
- | |||||

11 | -.023 | -.019 | -.033 | -.005 | .019 | .007 | -.005^{b} |
.008^{b} |
.321^{**b} |
.360^{**b} |
- | ||||

12 | .333^{**} |
.219^{**} |
.309^{**} |
.389^{**} |
.380^{**} |
.354^{**} |
.487^{**} |
.329^{**} |
-.019 | -.002 | -.025^{c} |
- | |||

13 | .313^{**} |
.163^{**} |
.327^{**} |
.355^{**} |
.291^{**} |
.274^{**} |
.264^{**} |
.171^{**} |
.013 | .036 | -.018^{c} |
.718^{**c} |
- | ||

14 | .146^{**} |
.108^{**} |
.281^{**} |
.300^{**} |
.223^{**} |
.186^{**} |
.283^{**} |
.257^{**} |
.007 | .016 | -.034^{c} |
.504^{**c} |
.587^{**c} |
- | |

15 | .269^{**} |
.136^{**} |
.304^{**} |
.283^{**} |
.386^{**} |
.365^{**} |
.492^{**} |
.330^{**} |
-.017 | -.006 | .005 | .405^{**} |
.366^{**} |
.348^{**} |
- |

16 | .290^{**} |
.193^{**} |
.318^{**} |
.336^{**} |
.422^{**} |
.396^{**} |
.526^{**} |
.417^{**} |
.016 | .032 | .017 | .446^{**} |
.398^{**} |
.400^{**} |
.685^{**} |

^{a}Correlations between reading tasks. ^{b}Correlations between attention tasks. ^{c}Correlations between math tasks.

**Correlation is significant at the Bonferroni adjusted alpha

As can be seen in

For the attention tasks, there were correlations between CPT omission and commission,

Following these modifications, we performed a principal component analysis with promax Kaiser Normalization rotation on the entire data set (

Rotate component matrix promax with Kaiser Normalization |
||||
---|---|---|---|---|

Measure | Factor 1: Cognitive processes (18.9%) | Factor 2: Math(16.3%) | Factor 3: Reading(7.7%) | Factor 4: Attention (6.3%) |

CPT omission | 0.81 | |||

VP serial | 0.77 | |||

VP parallel | 0.71 | |||

CPT commission | 0.63 | |||

RAN objects | 0.52 | |||

RAN letter/number | 0.47 | |||

Procedural knowledge | 1.11 | |||

Calculation automaticity | 0.62 | |||

Number line knowledge | 0.54 | |||

Pseudoword reading and identification | 0.78 | |||

Phoneme omission | 0.72 | |||

Text reading | 0.69 | |||

Reading comprehension | 0.38 | |||

Alerting | 0.74 | |||

Orienting of attention | 0.67 | |||

Executive attention | 0.49 |

We termed the first factor ‘‘perception speed” that included the following tasks: RAN (object and number/letter), CPT (omission and commission), and visual perception (serial and parallel) because all of the tasks required fast responses to a visual stimulus. The second factor we titled “mathematics” that included efficiency scores for calculation automaticity and procedural knowledge, and number line knowledge. The third factor we titled “reading” that included reading tasks: pseudoword reading and identification, reading comprehension, phoneme omission and text reading. The last factor we titled “attention” that included the 3 measures of the ANT task: alerting, executive attention and orienting of attention.

Model 1 of the SEM was designed based on the results of the principal component analysis. This was a statistical-fit led model, data driven from the previous analysis. We tested an independent three-factor model: perception speed, reading, and attention as independent influences of mathematics ability (see ^{2}^{2}^{2} change = 173.3, ^{2} change = 69.3, ^{2} change = 28, ^{2}^{2}^{2}

Visual processing P = visual processing parallel; Visual processing S = visual processing serial; CPT O = continuous performance task omission; CPT C = continuous performance task commission; RAN O = rapid automatized naming objects; RAN LN = rapid automatized naming letter/number; ANT orienting = attention network test orientation; ANT Alerting = attention network test alertness; ANT executive = attention network test executive.

Visual processing P = visual processing parallel; CPT O = continuous performance task omission; CPT C = continuous performance task commission; RAN LN = rapid automatized naming letter/number; ANT orienting = attention network test orientation; ANT Alerting = attention network test alertness; ANT executive = attention network test executive.

The previous model indicated that attention tested by ANT did not play a significant role in explaining mathematics task performance. However, a possible explanation is that attention was predictive of perception speed and reading that in turn predicted mathematics, ^{2}^{2} (χ^{2} change = .008, ^{2} change = 0.154, ^{2} change = .062,

Visual processing P = visual processing parallel. CPT O = continuous performance task omission. CPT C = continuous performance task commission. RAN LN = rapid automatized naming letter numbers. ANT orienting = attention network test orientation. ANT Alerting = attention network test alertness. ANT executive = attention network test executive.

The goal of the current study was to examine how performance in a comprehensive learning disorder assessment relates to mathematics performance, in a large group of adults that represented a wide range of arithmetic proficiency. First, in the factor analysis, we found that the mathematics subtests of the

Second, perception speed and reading, but not attention, shared unique variance with mathematics ability. We discovered that performance in the latent variable perception speed strongly predicted the latent variable mathematics. Moreover, the latent variable reading also had a direct influence on the latent variable mathematics. Importantly, the task reading comprehension had a strong and direct effect. We found no direct or indirect influence of the latent variable attention, tested by the ANT, on the latent variable mathematics, reading or perception speed.

Different types of arithmetic and number tasks can require different combinations of cognitive skills. Dehaene’s triple code model postulated that different skills are recruited based on the task demands (

According to Dehaene et al.’s model (2003) number representation in the semantic code is based on the mental number line. Hence, classic approaches (see

Both of these studies emphasize that improvement in the number line task is based on verbal knowledge of the symbolic number system, which includes understanding the relationship between hundreds, decades and units and of place value that is based on lexical rules. Accordingly, a cross-cultural study found that Italian speaking 1^{st} graders had higher accuracy in number line estimation task performance compared to German speaking 1^{st} graders. The difference between the two languages is the unique inversion property of German multi-digit numbers (e.g., 48 → “eight-and-forty”) (

Performance in the arithmetic tasks, procedural knowledge and calculation automaticity, highly correlated and were directly affected by sustained attention. CPT omission directly related to sustained attention, which is typically measured during long and boring tasks. Procedural knowledge and calculation automaticity are both easy and automatic tasks for adults, in comparison to children who calculate the answer each trial. Due to the involvement of retrieval and automatic nature of the tasks for adults, beyond calculation they also require sustained attention. Each trial the participant needs to attend to the arithmetic symbols and place value of the numbers, requiring constant vigilance for the duration of the task. These findings support the heterogeneous perspective (

We found that reading and phonological awareness predicted arithmetic abilities. The factor analysis results grouped reading comprehension, text reading and phonological awareness under one latent variable we named “reading”. However, the SEM results clearly differentiated between the effect of the latent variable reading, and variance related uniquely to reading comprehension. The contribution of reading comprehension was slightly larger than the contribution of the latent variable reading overall.

The present study is not the first to suggest that phonological awareness can explain individual differences in mathematics performance, several studies point to the critical role of phonological awareness on mathematics abilities in children (

The latent variable perception speed explained a large proportion of the variability in mathematics abilities. The latent variable perception speed was composed of tasks that required rapid responses to visual stimuli. One of the main tasks that test fast response is RAN.

It is important to recognize that others have found that poor RAN performance is a unique characteristic of RD and not MD (

Studies investigating the role of sustained attention in explaining individual differences in mathematics abilities is inconsistent, some studies demonstrated a relationship (

There are some indications that participants with learning disorders have deficits in some of the attentional networks tested by the ANT. Specifically, ADHD participants have presented poor executive attention and orienting of attention (

Our analysis was based on a population that came for learning disability assessment, and therefore, inferences from our findings can be drawn only to a limited extent for the whole population. Moreover, the tasks examined in the present study were drawn directly from the

We examined performance in mathematics, reading, attention and cognitive tasks in a large data set of college students (

We found different relationships between cognitive skills and the different mathematics tasks. Retrieval of arithmetic facts and procedural knowledge were influenced directly by sustained attention, while number line knowledge was directly affected by reading comprehension, demonstrating the heterogeneity of arithmetic and number tasks. These results have meaningful outcomes for the diagnosis and intervention of pure and comorbid learning disorders.

This work was funded by the Marie Curie CIG grant. Grant number 631731.

The authors declare no competing financial interests.

The authors would like to warmly thank Dr. Anat Ben Simon who explained the MATAL battery, examined the method section and gave us access to the data for the present manuscript. Additionally, we would like to thank the National Institute for Testing and Evaluation in Israel for collecting the data for the present manuscript.

Please note that for theoretical reasons we did not average 1) RAN numbers, letters and objects. 2) Visual processing serial and parallel. 3) Procedural knowledge and calculation automaticity. 4) Phonological omission and pseudoword reading and identification even though the correlations between each of these pairs was > .6.