Numerical Comparison Task Research Articles

The speed of magnitude processing and executive functions in controlled and automatic number comparison in children: an electro-encephalography study

BackgroundIn the numerical Stroop paradigm (NSP) participants decide whether a digit is numerically or physically larger than another simultaneously presented digit. This paradigm is frequently used to assess the automatic number processing abilities of children. Currently it is unclear whether an equally refined evaluation of numerical magnitude occurs in both controlled (the numerical comparison task of the NSP) and automatic (the physical comparison task of the NSP) numerical comparison in both children and adults. One of our objectives was to respond this question by measuring the speed of controlled and automatic magnitude processing in children and adults in the NSP. Another objective was to determine how the immature executive functions of children affect their cognitive functions relative to adults in numerical comparison.Methods and resultsThe speed of numerical comparison was determined by monitoring the electro-encephalographic (EEG) numerical distance effect: The amplitude of EEG measures is modulated as a function of numerical distance between the to-be-compared digits. EEG numerical distance effects occurred between 140–320 ms after stimulus presentation in both controlled and automatic numerical comparison in all age groups. Executive functions were assessed by analyzing facilitation and interference effects on the latency of the P3b event-related potential component and the lateralized readiness potential (LRP). Interference effects were more related to response than to stimulus processing in children as compared with adults. The LRP revealed that the difficulty to inhibit irrelevant response tendencies was a major factor behind interference in the numerical task in children.ConclusionThe timing of the EEG distance effect suggests that a refined evaluation of numerical magnitude happened at a similar speed in each age group during both controlled and automatic magnitude processing. The larger response interference in children than in adults suggests that despite the similar behavioural profile of children and adults, partially different cognitive processes underlie their performance in the NSP. Further, behavioural effects in the NSP depend on interactions between comparison, facilitation/interference and response-related processes. Our data suggest that caution is needed when using the NSP to compare behavioural markers of the numerical processing skills of children and adults.

Interference Effects in a Numerical Stroop Paradigm in 9- to 12-year-old Children with ADHD-C

Attention deficit hyperactivity disorder (ADHD) is characterized by deficient self-regulation, poor attentional control, and poor response inhibition. To date, however, the extent to which these deficits affect basic interference control remains a matter of controversy. Secondly, ADHD has been reported to be associated with arithmetic deficits. It remains unclear whether such deficits are a secondary consequence of the above-mentioned characteristics of ADHD or whether basic numerical magnitude representations are also affected. In the present study we attempted to investigate these issues using a basic numerical interference paradigm. Nine- to twelve-year-old children with ADHD-C (attention-deficit hyperactivity-disorder combined type) and control children without ADHD (each n = 16) were presented with two digits of possibly different physical sizes (e.g., 3 7). This numerical Stroop task requires subjects to make a magnitude classification concerning either the physical or the numerical stimulus dimension. The irrelevant dimension can be congruent (same response), incongruent (different response), or neutral (no response association). Children with ADHD-C performed worse than control children in most analyses. The most important finding was a significant interaction of congruity effects with group in the numerical comparison task. Children with ADHD-C tended to show larger congruity and interference effects than controls, and these were not attributable to a speed-accuracy trade-off. The results might reflect differential processing speeds, or a different degree of automatic activation of physical and numerical magnitudes in children with and without ADHD-C. Alternative explanations, such as insufficient inhibition of selective (domain-specific) attention are also discussed. The order of authorship is alphabetical, as the authors contributed equally to the study. We wish to thank the participating children, their parents, and their schools for contributing to this project. We are also grateful to Susanna Bloder and Maria Hoellwarth for data collection, and Elise Klein and Katharina Dressel for checking the references. In particular, we wish to thank Stuart Fellows for checking English style and grammar. This research was supported by funding of the European Community (Neuromath: HPRN-CT-2000-00076 and RTN NUMBRA proposal 504927), by grants of the RWTH Aachen (IZKF “BioMAT.”; VV N50, 51, 69c; START AZ 160/05) and the German Research Society (DFG: KFO110/TP2) supporting the research of Hans-Christoph Nuerk.

Principles underlying the design of "The Number Race", an adaptive computer game for remediation of dyscalculia

BackgroundAdaptive game software has been successful in remediation of dyslexia. Here we describe the cognitive and algorithmic principles underlying the development of similar software for dyscalculia. Our software is based on current understanding of the cerebral representation of number and the hypotheses that dyscalculia is due to a "core deficit" in number sense or in the link between number sense and symbolic number representations.Methods"The Number Race" software trains children on an entertaining numerical comparison task, by presenting problems adapted to the performance level of the individual child. We report full mathematical specifications of the algorithm used, which relies on an internal model of the child's knowledge in a multidimensional "learning space" consisting of three difficulty dimensions: numerical distance, response deadline, and conceptual complexity (from non-symbolic numerosity processing to increasingly complex symbolic operations).ResultsThe performance of the software was evaluated both by mathematical simulations and by five weeks of use by nine children with mathematical learning difficulties. The results indicate that the software adapts well to varying levels of initial knowledge and learning speeds. Feedback from children, parents and teachers was positive. A companion article [1] describes the evolution of number sense and arithmetic scores before and after training.ConclusionThe software, open-source and freely available online, is designed for learning disabled children aged 5–8, and may also be useful for general instruction of normal preschool children. The learning algorithm reported is highly general, and may be applied in other domains.

Flexible mental processes in numerical size judgments: The case of hebrew letters that are used to convey numbers

In addition to its primary linguistic function, the Hebrew alphabet is sometimes used as a means of number notation (i.e., the system of gematria). Hebrew letters, Arabic numerals, Hebrew number names, and Hebrew letter names were used in a numerical size comparison task, in which two visually presented symbols were compared for numerical value while irrelevant variations in their physical size had to be ignored. A size congruity effect, indicated by faster responses when differences in physical and numerical size were consistent, was larger for Arabic numerals than for number names. The effect for Hebrew letters was similar to that for Arabic numerals and was stronger than that observed for letter names. These results suggest flexible processing of Hebrew letters, so that they function as ideographic symbols in an arithmetic context. A distance effect, indicated by an inverse relationship between reaction time and numerical distance, was found for all notations but was particularly strong for Hebrew letters.

Neural correlates of distance and congruity effects in a numerical Stroop task: an event-related fMRI study

This study aimed at investigating the neural correlates of a number-size congruity task. Using an event-related fMRI design, we presented one-digit number pairs to 17 participants in a number-size interference task that required subjects to focus on one stimulus property (e.g., numerical size) and to ignore the other (physical size). In different blocks, participants were asked to decide which digit of a digit pair was numerically larger (numerical comparison task) or physically larger (physical comparison task). Stimuli were classified into three categories: (a) congruent: physical and numerical comparison leads to the same response; (b) incongruent: physical and numerical comparison leads to different responses; (c) neutral: the stimuli differ only with regard to the task-relevant stimulus property. Behavioral results reflect robust distance effects (quicker reaction times for long distances relative to short ones) and size congruity effects (longer reaction times for incongruent relative to congruent stimuli) in both tasks. Imaging results reveal that–compared to congruent trials–incongruent trials led to a stronger activation in the dorsolateral prefrontal cortex and the anterior cingulate cortex, areas associated with attentional control. The distance effect (neutral condition only) led to a stronger activation in bilateral parietal areas including the intraparietal sulcus (IPS).

Approximate quantities and exact number words: dissociable systems

Numerical abilities are thought to rest on the integration of two distinct systems, a verbal system of number words and a non-symbolic representation of approximate quantities. This view has lead to the classification of acalculias into two broad categories depending on whether the deficit affects the verbal or the quantity system. Here, we test the association of deficits predicted by this theory, and particularly the presence or absence of impairments in non-symbolic quantity processing. We describe two acalculic patients, one with a focal lesion of the left parietal lobe and Gerstmann’s syndrome and another with semantic dementia with predominantly left temporal hypometabolism. As predicted by a quantity deficit, the first patient was more impaired in subtraction than in multiplication, showed a severe slowness in approximation, and exhibited associated impairments in subitizing and numerical comparison tasks, both with Arabic digits and with arrays of dots. As predicted by a verbal deficit, the second patient was more impaired in multiplication than in subtraction, had intact approximation abilities, and showed preserved processing of non-symbolic numerosities.

Bilateral field interactions and hemispheric asymmetry in number comparison

We investigated bilateral field interactions and hemispheric asymmetry in number comparison. A numerical comparison task with three different stimulus notations (Arabic numerals, word numerals and bar graphs) was used. In all conditions, a target was displayed in one visual field, simultaneously with a distractor of the same format in the other visual field. Participants had to indicate manually whether the magnitude of the target was small or large, ignoring the distractor stimulus. Only in the condition with Arabic numerals did we obtain some evidence for a LVF advantage, which argues against a strong laterality of number magnitude representations. Significant interactions between target and distractor values were observed, indicating rich interhemispheric interactions. The interactions were mainly situated at the response stage, but the presence of a bilateral identity effect also points to interactions at the input level.

Understanding the concepts of proportion and ratio among grade nine students in Malaysia

The purpose of the study was to investigate the concepts of ratio and proportion constructed by grade nine students by investigating grade nine students proportional reasoning schemes and procedures on three types of tasks: missing value, numerical comparison and qualitative reasoning. Comparisons among the different categories were made and the strategies used in solving these problems were identified. The relationship between student grades on a national examination and their knowledge of proportional reasoning was determined. The results of the quantitative analysis indicated that students performed generally well on the missing value tasks but their scores on the numerical comparison and qualitative tasks were much lower. The results indicate that only a small percentage of students who did well on the national exams were able to solve complex proportional problems and the grades obtained were not indicative of their knowledge of ratio and proportion. The difficulty experienced by the ninth graders indicated that students frequently used additive reasoning, that is a comparison of two numbers by subtraction rather than division. It appears that students cannot begin to understand the functional and scalar relationship inherent in a proportion until they first develop multiplicative reasoning.

The Development of Automaticity in Accessing Number Magnitude

This study traces developmental changes in automatic and intentional processing of Arabic numerals using a numerical-Stroop paradigm. In Study 1, university students compared the numerical or physical size of Arabic numerals varying along both dimensions. In Study 2, first graders (mean age = 6 years 6 months), third graders (mean age = 8 years 4 months), and fifth graders (mean age = 10 years 3 months) were tested to examine developmental changes in numerical and physical comparisons. In the numerical comparison task, a size congruity effect was found at all ages (i.e., relative to a neutral control, congruent physical sizes facilitated, and incongruent sizes interfered with, the numerical comparison). The pattern of facilitation and interference, however, was modulated by age. In the physical comparison task, the incongruity between physical and numerical size affected only older children and adults. These findings strongly suggest that the automatization in number processing is achieved gradually as numerical skills progress.

Mental comparison

The symbolic distance effect was studied in a numerical comparison task by means of behavioural measures and event-related potentials (ERP). Two numbers in the range from 11 to 20 were presented simultaneously and the subjects had to indicate if the right side number was smaller or larger than the left side number. Reaction time, error rate and ERPs were selectively averaged according to the numerical differences between the two numbers. The symbolic distance effect was confirmed: Reaction times and error rates decreased significantly with increasing difference between the two numbers. A significant distance effect was also found for both P300 amplitude and latency. The larger the numerical difference the larger the P300 amplitude and the P300 latency tended to be shorter. This finding suggests that the numerical difference is effective already during stimulus encoding.