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.

Expert Consensus on Nail Procedures and Selection of CPT Codes.

BACKGROUND: Dermatologists specialize in treating conditions of the skin, hair, and nails; however, it is our experience that the field of nail diseases is the least discussed facet of dermatology. Even less acknowledged is the complexity of nail procedures and how best to accurately code for these procedures. OBJECTIVE: To convene a panel of experts in nail disease to reach consensus on the most accurate and appropriate Current Procedural Terminology (CPT) codes associated with the most commonly performed nail procedures. METHODS: A questionnaire including 9 of the most commonly performed nail procedures and potential CPT codes was sent to experts in the treatment of nail disease, defined as those clinicians running a nail subspecialty clinic and performing nail procedures with regularity. A conference call was convened to discuss survey results. RESULTS: Unanimous consensus was reached on the appropriate CPT codes associated with all discussed procedures. LIMITATIONS: Although this article details the most commonly performed nail procedures, many were excluded and billing for these procedures continues to be largely subjective. This article is meant to serve as a guide for clinicians but should not be impervious to interpretation in specific clinical situations. CONCLUSION: Billing of nail procedures remains a practice gap within our field. The authors hope that the expert consensus on the most appropriate CPT codes associated with commonly performed nail procedures will aid clinicians as they diagnose and treat disorders of the nail unit and encourage accurate and complete billing practices.

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.

Acquired Idiopathic True Transverse Leukonychia.

A 34-year-old man from Djibouti presented with a 14-year history of relapsing and remitting transverse white bands on the fingernails with sparing of the toenails. Examination revealed several transverse, white bands following the contour of the lunula on seven of his fingernails that did not fade upon compression of the digits (Figure). There was no onycholysis. No other skin lesions were noted. The patient reported having lived for 4 years (2000-2004) in a house that had well water as its primary water supply. This 4-year period was a stressful point in our patient's life. During that time, he had been a student at university. He had had no reported occupational exposure to arsenic. He reported being a cigarette smoker since 1996 but denied any illicit drug use or alcohol consumption. His past medical history was significant for hepatitis A infection, but he denied any history of systemic illness, including renal disease, heart disease, and lung disease. He denied any family history of leukonychia. He denied any trauma or participation in activities that require excessive use of his hands, and also denied manipulation of the cuticles. The patient's liver function tests, lipid panel, complete blood count, and urinalysis were all within normal limits. A blood test revealed normal arsenic levels. Histologic examination of the nail plate showed segmental parakeratosis, with no evidence of fungal organisms upon PAS staining.

The Automatic Mapping of Magnitude to Temporal Order Is Space-Dependent.

Past research has shown that performance in ordinal magnitude tasks is enhanced when stimuli are presented in ascending order, suggesting that magnitude is mapped to temporal order, with small magnitude associated with early and large with late presentation. The present study addresses the automaticity of this effect and its limitations. We used the "same/different" task for numbers (Experiment 1) and physical sizes of shapes (Experiment 2) as well as identity of shapes (Experiment 3). The advantage for stimuli in ascending order was found for both numbers and physical sizes of shapes. However, it was limited to specific conditions - when magnitude processing was required for the task and when a "different" response was mapped to the right hand side. Thus, it seems that the automatic mapping of magnitude to temporal order is dependent on the mapping of magnitude to space.

Can Dyscalculics Estimate the Results of Arithmetic Problems?

The present study is the first to examine the computation estimation skills of dyscalculics versus controls using the estimation comparison task. In this task, participants judged whether an estimated answer to a multidigit multiplication problem was larger or smaller than a given reference number. While dyscalculics were less accurate than controls, their performance was well above chance level. The performance of controls but not of those with developmental dyscalculia (DD) improved consistently for smaller problem sizes. The performance of both groups was superior when the reference number was smaller (vs. larger) than the exact answer and when it was far (vs. close) from it, both of which are considered to be the markers of the approximate number system (ANS). Strategy analysis distinguished between an approximated calculation strategy and a sense of magnitude strategy, which does not involve any calculation but relies entirely on the ANS. Dyscalculics used the latter more often than controls. The present results suggest that there is little, if any, impairment in the ANS of adults with DD and that their main deficiency is with performing operations on magnitudes rather than with the representations of the magnitudes themselves.

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 4th graders, 6th 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.

Exploring the Boundaries of the Number-Temporal Order Association.

Past research has shown that numbers are associated with order in time such that performance in a numerical comparison task is enhanced when number pairs appear in ascending order, when the larger number follows the smaller one. This was found in the past for the integers 1-9 ( Ben-Meir, Ganor-Stern, & Tzelgov, 2013 ; Müller & Schwarz, 2008 ). In the present study we explored whether the advantage for processing numbers in ascending order exists also for fractions and negative numbers. The results demonstrate this advantage for fraction pairs and for integer-fraction pairs. However, the opposite advantage for descending order was found for negative numbers and for positive-negative number pairs. These findings are interpreted in the context of embodied cognition approaches and current theories on the mental representation of fractions and negative numbers.

When you don't have to be exact: investigating computational estimation skills with a comparison task.

The present study is the first systematic investigation of computational estimation skills of multi-digit multiplication problems using an estimation comparison task. In two experiments, participants judged whether an estimated answer to a multi-digit multiplication problem was larger or smaller than a given reference number. Performance was superior in terms of speed and accuracy for smaller problem sizes, for trials in which the reference numbers were smaller vs. larger than the exact answers (consistent with the size effect) and for trials in which the reference numbers were numerically far compared to close to the exact answers (consistent with the distance effect). Strategy analysis showed that two main strategies were used to solve this task-approximate calculation and sense of magnitude. Most participants reported using the two strategies. Strategy choice was influenced by the distance between the reference number and exact answer, and by the interaction of problem size and reference number size. Theoretical implications as to the nature of numerical representations in the ANS (approximate number system) and to the estimation processes are suggested.

Are all changes equal? comparing early and late changes in sequence learning.

Although it is known that a change in a learned motor sequence slows performance down, it is yet unknown if this impairment varies depending on whether the changed element is early or late in the sequence. In Experiment 1, we showed greater impairment in performance when changing the third vs. the sixth element in a 7-element sequence. The impairment was greater for the deviant and the following elements than for the preceding ones. In Experiment 2, we replicated the results of Experiment 1 and expanded them by showing that a change in the third element of a 4-element sequence produced similar results to those of the late change condition in the long 7-element sequence. It is proposed that during practice, associative relations between the sequence elements are formed together with the representation of the whole chunk. Following the change in sequence, the chunk representation is impaired and performance mainly reflects the associative links between the elements. An early change hampers these associative relations to a greater extent than a late change, and as a consequence slows performance down more than a late change does. The implications and advantages of such a mechanism are discussed.

Are 1/2 and 0.5 represented in the same way?

Adults' processing of unit and decimal fractions was investigated using the numerical comparison task. When unit fractions were compared to integers, the pattern of distance effect found suggests that they were perceived to be on the same mental number line as integers; however, their representation was undifferentiated, as they were perceived to have the same magnitude. This was found both with simultaneous and with sequential presentation. When decimal fractions were compared to integers, the pattern of results suggests that they were also represented on the same mental number line with integers, but their representation was differentiated. Possible explanations for the different patterns found for unit and decimal fractions are discussed. Moreover, compatibility between the magnitude of the whole fraction and that of its components relative to the compared integer affected performance in the case of decimal fractions and unit fractions presented simultaneously, but not in the case of unit fractions presented sequentially. This suggests that sequential processing reduces the components representation of fractions and the whole number bias.

"On the money" - monetary and numerical judgments of currency.

Numerical and monetary judgments of currency were examined using two tasks-a monetary value task (which coin has a higher monetary value), and a numerical value task (which coin has a higher numerical value). In Experiment 1 participants were presented with pictures of coins of the Israeli currency. The Israeli currency is the shekel, which is composed of 100 agorot (equivalent to a dollar composed of 100 cents). Higher discriminability between shekels compared to agorot due to importance in everyday life was reflected in faster monetary comparisons of shekel pairs compared to agorot pairs. Automatic processing of numerical value was demonstrated for monetary judgments. When presented with pairs composed of one coin from each monetary category, responses were faster to pairs that were monetary-numerical congruent (e.g., 10 shekels vs. 5 agorot) compared to incongruent (e.g., 5 shekels vs. 10 agorot). Numerical value judgments were unaffected by such congruency. There was evidence for the automatic activation of physical size mainly in the numerical task. A similar picture was obtained in Experiment 2 where instead of pictures of coins we used verbal descriptions of currency, demonstrating the generality of our results.

Numerical and physical magnitudes are mapped into time.

In two experiments we investigated mapping of numerical and physical magnitudes with temporal order. Pairs of digits were presented sequentially for a size comparison task. An advantage for numbers presented in ascending order was found when participants were comparing the numbers' physical and numerical magnitudes. The effect was more robust for comparisons of physical size, as it was found using both select larger and select smaller instructions, while for numerical comparisons it was found only for select larger instructions. Varying both the digits' numerical and physical sizes resulted in a size congruity effect, indicating automatic processing of the irrelevant magnitude dimension. Temporal order and the congruency between numerical and physical magnitudes affected comparisons in an additive manner, thus suggesting that they affect different stages of the comparison process.

Estimating linear effects in ANOVA designs: the easy way.

Research in cognitive science has documented numerous phenomena that are approximated by linear relationships. In the domain of numerical cognition, the use of linear regression for estimating linear effects (e.g., distance and SNARC effects) became common following Fias, Brysbaert, Geypens, and d'Ydewalle's (1996) study on the SNARC effect. While their work has become the model for analyzing linear effects in the field, it requires statistical analysis of individual participants and does not provide measures of the proportions of variability accounted for (cf. Lorch & Myers, 1990). In the present methodological note, using both the distance and SNARC effects as examples, we demonstrate how linear effects can be estimated in a simple way within the framework of repeated measures analysis of variance. This method allows for estimating effect sizes in terms of both slope and proportions of variability accounted for. Finally, we show that our method can easily be extended to estimate linear interaction effects, not just linear effects calculated as main effects.

Fractions but not negative numbers are represented on the mental number line.

The present study is the first to directly compare numerical representations of positive numbers, negative numbers and unit fractions. The results show that negative numbers and unit fractions were not represented in the same way. Distance effects were found when positive numbers were compared with fractions but not when they were compared with negative numbers, thus suggesting that unit fractions but not negative numbers were represented on the number line with positive numbers. As indicated by the semantic congruity effect, negative numbers were perceived to be small, positive numbers were perceived as large, while unit fractions were perceived neither as large nor small. Comparisons between negative numbers were faster than between unit fractions, possibly due to the smaller differences between the holistic magnitudes of the unit fractions. Finally, comparing unit fractions to 1 was faster than comparing them to 0, consistent with the idea that unit fractions are perceived as entities smaller than 1 (Kallai & Tzelgov, 2009). The results are consistent with the idea of a mental division between numbers that represent a quantity (positive numbers and unit fractions) and those that do not (negative numbers).