Can Dyscalculics Estimate the Results of Arithmetic Problems?

Profiles of mathematical deficits in children with dyslexia

Despite a high rate of concurrent mathematical difficulties among children with dyslexia, we still have limited information regarding the prevalence and severity of mathematical deficits in this population. To address this gap, we developed a comprehensive battery of cognitive tests, known as the UCSF Mathematical Cognition Battery (MCB), with the aim of identifying deficits in four distinct mathematical domains: number processing, arithmetical procedures, arithmetic facts retrieval, and geometrical abilities. The mathematical abilities of a cohort of 75 children referred to the UCSF Dyslexia Center with a diagnosis of dyslexia, along with 18 typically developing controls aged 7 to 16, were initially evaluated using a behavioral neurology approach. A team of professional clinicians classified the 75 children with dyslexia into five groups, based on parents' and teachers' reported symptoms and clinical history. These groups included children with no mathematical deficits and children with mathematical deficits in number processing, arithmetical procedures, arithmetic facts retrieval, or geometrical abilities. Subsequently, the children underwent evaluation using the MCB to determine concordance with the clinicians' impressions. Additionally, neuropsychological and cognitive standardized tests were administered. Our study reveals that within a cohort of children with dyslexia, 66% exhibit mathematical deficits, and among those with mathematical deficits, there is heterogeneity in the nature of these deficits. If these findings are confirmed in larger samples, they can potentially pave the way for new diagnostic approaches, consistent subtype classification, and, ultimately personalized interventions.

The interplay between math performances, spatial abilities, and affective factors: The role of task

Many studies have suggested that cognitive and affective abilities (such as math anxiety- MA and math self-efficacy) explain individual differences in math.

PURPOSE

The present study explores the interplay between MA, math self-efficacy, spatial anxiety and spatial abilities in explaining individual differences on two complex math tasks.

PROCEDURES

Ninety-three college students took part in the experiment and completed 3 emotional questionnaires, in addition to 2 math tasks and a mental rotation task.

FINDINGS

The interplay between math performances and cognitive and affective factors is related to task demand. MA and spatial abilities affected math performances directly, regardless of task. Spatial anxiety had only an indirect effect on math performances via MA, regardless of task.

CONCLUSIONS

These finding suggest that for math performances, contrary to MA, real spatial abilities rather than perceived spatial anxiety play a significant role in explaining individual differences.  Hence, the present result dissociates cognitive and emotional factors.

Neural Correlates of Numerical Estimation: The Role of Strategy Use

Introduction: Computation estimation is the ability to provide an approximate answer to a complex arithmetic problem without calculating it exactly. Despite its importance in daily life, the neuronal network underlying computation estimation is largely unknown. Methods: We looked at the neuronal correlates of two computational estimation strategies: approximated calculation and sense of magnitude (SOM)–intuitive representation of magnitude, without calculation. During an fMRI scan, thirty-one college students judged whether the result of a two-digit multiplication problem was larger or smaller than a given reference number. In two different blocks, they were asked to use a specific strategy (AC or SOM). Results: The two strategies activated brain regions related to calculation, numerical cognition, decision-making, and working memory. AC more than SOM elicited activations in multiple, domain-specific brain regions in the parietal lobule, including the left SMG (BA 40), the bilateral superior parietal lobule (BA 7), and the right inferior parietal lobule (BA 7). The activation level of the IFG was positively correlated to individual accuracy, indicating that the IFG has an essential role in both strategies. Conclusions: These finding suggest that the analogic code of magnitude is more involved in the AC than the SOM strategy.

Damage to the Intraparietal Sulcus Impairs Magnitude Representations of Results of Complex Arithmetic Problems

Past research investigating the role of the intraparietal sulcus (IPS) in numerical processes focused mainly on quantity and numerical comparisons as well on single digit arithmetic. The present study investigates the involvement of the IPS in estimating the results of multi-digit multiplication problems. For this purpose, the performance a 24-year-old female (JD) with brain damage in the left IPS was compared to an age-matched control group in the computation estimation task. When required to estimate whether the results of multi-digit multiplication problems are smaller or larger than given reference numbers, JD, in contrast to controls, did not show the common patterns of distance and size effects. Her strategy use was also atypical. Most control participants used both the approximated calculation strategy that involves rounding and calculation procedures and the sense of magnitude strategy that relies on an intuitive approximated magnitude representation of the results. In contrast, JD used only the former but not the latter strategy. Together, these findings suggest that the damage to the IPS impaired JD's representations of magnitude that play an important role in this computation estimation task.

Do Exact Calculation and Computation Estimation Reflect the Same Skills? Developmental and Individual Differences Perspectives

Groups of children in 4th, 5th, and 6th grades and college students performed exact calculation and computation estimation tasks with two-digit multiplication problems. In the former they calculated the exact answer for each problem, and in the latter they estimated whether the result of each problem was larger or smaller than a given reference number. The analyses of speed and accuracy both showed different developmental patterns of the two tasks. While the accuracy of exact calculation increased with age in childhood, the accuracy of the estimation task reached its maximum level already in 4th grade and did not change with age. The reaction time of the exact calculation task was longer than that of the estimation task. The reaction time for both tasks remained constant in childhood and decreased in adulthood, with the improvement in speed larger for the exact calculation task. Similarly, within group variability in accuracy was larger in the exact calculation task than in the computation estimation task. Finally, low correlation was found between the accuracy of the two tasks. Together, these findings suggest that exact calculation and computation estimation reflect at least in part different skills.

Approximation processes in arithmetic in old adulthood

Young and old adults estimated the results of multidigit multiplication problems relative to a reference number. Old adults were slower but slightly more accurate than young adults. They were less affected by the distance between the reference number and the exact answer than the young adults. The same strategies reported by past research–the approximated calculation strategy and the sense of magnitude strategy—were found here. The old adults showed a stronger preference toward the approximated calculation strategy than the young ones, and this probably led to the reduced effect of distance. These patterns are interpreted as reflecting two factors. The first is the extensive experience of the old adults with mental calculation, and the second is the decline in processing speed and in working memory resources with adulthood. The former is responsible for the more frequent use of the approximated calculation strategy and for the higher accuracy of the old adults, while the latter is responsible for their slower responses.

ADHD and math - The differential effect on calculation and estimation

Adults with ADHD were compared to controls when solving multiplication problems exactly and when estimating the results of multidigit multiplication problems relative to reference numbers. The ADHD participants were slower than controls in the exact calculation and in the estimation tasks, but not less accurate. The ADHD participants were similar to controls in showing enhanced accuracy and speed for smaller problem sizes, for trials in which the reference numbers were smaller (vs. larger) than the exact answers and for reference numbers that were far (vs. close) from the exact answer. The two groups similarly used the approximated calculation and the sense of magnitude strategies. They differed however in strategy execution, mainly of the approximated calculation strategy, which requires working memory resources. The increase in reaction time associated with using the approximated calculation strategy was larger for the ADHD compared to the control participants. Thus, ADHD seems to selectively impair calculation processes in estimation tasks that rely on working memory, but it does not hamper estimation skills that are based on sense of magnitude. The educational implications of these findings are discussed.

Solving Math Problems Approximately: A Developmental Perspective

Although solving arithmetic problems approximately is an important skill in everyday life, little is known about the development of this skill. Past research has shown that when children are asked to solve multi-digit multiplication problems approximately, they provide estimates that are often very far from the exact answer. This is unfortunate as computation estimation is needed in many circumstances in daily life. The present study examined 4^th graders, 6^th graders and adults’ ability to estimate the results of arithmetic problems relative to a reference number. A developmental pattern was observed in accuracy, speed and strategy use. With age there was a general increase in speed, and an increase in accuracy mainly for trials in which the reference number was close to the exact answer. The children tended to use the sense of magnitude strategy, which does not involve any calculation but relies mainly on an intuitive coarse sense of magnitude, while the adults used the approximated calculation strategy which involves rounding and multiplication procedures, and relies to a greater extent on calculation skills and working memory resources. Importantly, the children were less accurate than the adults, but were well above chance level. In all age groups performance was enhanced when the reference number was smaller (vs. larger) than the exact answer and when it was far (vs. close) from it, suggesting the involvement of an approximate number system. The results suggest the existence of an intuitive sense of magnitude for the results of arithmetic problems that might help children and even adults with difficulties in math. The present findings are discussed in the context of past research reporting poor estimation skills among children, and the conditions that might allow using children estimation skills in an effective manner.

Tracking practice effects in computation estimation

The present study investigated college students' ability to estimate the results of multi-digit multiplication problems and the extent to which this ability improves with practice. Participants judged whether the results of multiplication problems composed of two-digit numbers were larger or smaller than a given reference number. The reference numbers were either close or far from the exact answer. The effects of practice, size, and distance of the reference number from the exact answer were examined using four measures of performance: speed, accuracy, eye movements, and strategy use. The results show that together with enhanced speed and accuracy with practice, participants also changed the pattern of eye movements and the strategies they used. The eye movement analysis showed longer dwell time and more frequent first fixations toward the reference number with practice, suggesting that participants relied more on the reference number to solve the task with practice. The strategy analysis revealed that with practice participants reduced their use of the approximate calculation strategy, which involves multiplying the rounded operands and comparing the product to the reference number, and increased their reliance on the sense of magnitude strategy which does not involve any calculation, but is grounded in the ANS. This was done especially for trials in which the reference number was far from the exact answer, thus exhibiting enhanced adaptivity in strategy choice with practice.