Strategies for written additions in adults

Studies about strategies used by adults to solve multi-digit written additions are very scarce. However, as advocated here, the specificity and characteristics of written calculations are of undeniable interest. The originality of our approach lies in part in the presentation of two-digit addition problems on a graphics tablet, which allowed us to precisely follow and analyse individuals’ solving process. Not only classic solution times and accuracy measures were recorded but also initiation times and starting positions of the calculations. Our results show that adults largely prefer the fixed columnar strategy taught at school rather than more flexible mental strategies. Moreover, the columnar strategy is executed faster and as accurately as other strategies, which suggests that individuals’ choice is usually well adapted. This result contradicts past educational intuitions that the use of rigid algorithms might be detrimental to performance. We also demonstrate that a minority of adults can modulate their strategy choice as a function of the characteristics of the problems. Tie problems and additions without carry were indeed solved less frequently through the columnar strategy than non-tie problems and additions with a carry. We conclude that the working memory demand of the arithmetic operation influences strategy selection in written problem-solving.     

Better elementary number processing in higher skill arithmetic problem solvers: Evidence from the encoding step

This paper addresses the relationship between basic numerical processes and higher level numerical abilities in normal achieving adults. In the first experiment we inferred the elementary numerical abilities of university students from the time they needed to encode numerical information involved in complex additions and subtractions. We interpreted the shorter encoding times in good arithmetic problem solvers as revealing clearer or more accessible representations of numbers. The second experiment shows that these results cannot be due to the fact that lower skilled individuals experience more maths anxiety or put more cognitive efforts into calculations than do higher skilled individuals. Moreover, the third experiment involving non-numerical information supports the hypothesis that these interindividual differences are specific to number processing. The possible causal relationships between basic and higher level numerical abilities are discussed.     

Mental representations of arithmetic facts: Evidence from eye movement recordings supports the preferred operand-order-specific representation hypothesis

There are three main hypotheses about mental representations of arithmetic facts: the independent representation hypothesis, the operand-order-free single-representation hypothesis, and the operand-order-specific single-representation hypothesis. The current study used electrical recordings of eye movements to examine the organization of arithmetic facts in long-term memory. Subjects were presented single-digit addition and multiplication problems and were asked to report the solutions. Analyses of the horizontal electrooculograph (HEOG) showed an operand order effect for multiplication in the time windows 150–300 ms (larger negative potentials for smaller operand first problems than for larger operand first ones). The operand order effect was reversed in the time windows from 400 to 1,000 ms (i.e., larger operand first problems had larger negative potentials than smaller operand first problems). For addition, larger operand first problems had larger negative potentials than smaller operand first in the series of time windows from 300 to 1,000 ms, but the effect was smaller than that for multiplication. These results confirmed the dissociated representation of addition and multiplication facts and were consistent with the prediction of the preferred operand-order-specific representation hypothesis.     

Negative numbers in simple arithmetic

Are negative numbers processed differently from positive numbers in arithmetic problems? In two experiments, adults (N?=?66) solved standard addition and subtraction problems such as 3?+?4 and 7 – 4 and recasted versions that included explicit negative signs—that is, 3 – (–4), 7?+?(–4), and (–4)?+?7. Solution times on the recasted problems were slower than those on standard problems, but the effect was much larger for addition than subtraction. The negative sign may prime subtraction in both kinds of recasted problem. Problem size effects were the same or smaller in recasted than in standard problems, suggesting that the recasted formats did not interfere with mental calculation. These results suggest that the underlying conceptual structure of the problem (i.e., addition vs. subtraction) is more important for solution processes than the presence of negative numbers.     

Improving arithmetic performance with number sense training: An investigation of underlying mechanism

A nonverbal primitive number sense allows approximate estimation and mental manipulations on numerical quantities without the use of numerical symbols. In a recent randomized controlled intervention study in adults, we demonstrated that repeated training on a non-symbolic approximate arithmetic task resulted in improved exact symbolic arithmetic performance, suggesting a causal relationship between the primitive number sense and arithmetic competence. Here, we investigate the potential mechanisms underlying this causal relationship. We constructed multiple training conditions designed to isolate distinct cognitive components of the approximate arithmetic task. We then assessed the effectiveness of these training conditions in improving exact symbolic arithmetic in adults. We found that training on approximate arithmetic, but not on numerical comparison, numerical matching, or visuo-spatial short-term memory, improves symbolic arithmetic performance. In addition, a second experiment revealed that our approximate arithmetic task does not require verbal encoding of number, ruling out an alternative explanation that participants use exact symbolic strategies during approximate arithmetic training. Based on these results, we propose that nonverbal numerical quantity manipulation is one key factor that drives the link between the primitive number sense and symbolic arithmetic competence. Future work should investigate whether training young children on approximate arithmetic tasks even before they solidify their symbolic number understanding is fruitful for improving readiness for math education.     

The role of fingers in the development of counting and arithmetic skills

Interactions between fingers and numbers have been reported in the existing literature on numerical cognition. The aim of the present research was to test whether hand interference movements might have an impact on children performance in counting and basic arithmetic problem solving. In Experiment 1, 5-year-old children had to perform both a one-target and a two-target counting task in three different conditions: with no constraints, while making interfering hand movements or while making interfering foot movements. In Experiment 2, first and fourth graders were required to perform addition problems under the same control and sensori-motor interfering conditions. In both tasks, the hand movements caused more disruption than the foot movements, suggesting that finger-counting plays a functional role in the development of counting and arithmetic.     

The roles of the central executive and visuospatial storage in mental arithmetic: A comparison across strategies

Previous research has demonstrated that working memory plays an important role in arithmetic. Different arithmetical strategies rely on working memory to different extents—for example, verbal working memory has been found to be more important for procedural strategies, such as counting and decomposition, than for retrieval strategies. Surprisingly, given the close connection between spatial and mathematical skills, the role of visuospatial working memory has received less attention and is poorly understood. This study used a dual-task methodology to investigate the impact of a dynamic spatial n-back task (Experiment 1) and tasks loading the visuospatial sketchpad and central executive (Experiment 2) on adults' use of counting, decomposition, and direct retrieval strategies for addition. While Experiment 1 suggested that visuospatial working memory plays an important role in arithmetic, especially when counting, the results of Experiment 2 suggested this was primarily due to the domain-general executive demands of the n-back task. Taken together, these results suggest that maintaining visuospatial information in mind is required when adults solve addition arithmetic problems by any strategy but the role of domain-general executive resources is much greater than that of the visuospatial sketchpad.     

Calibrating the mental number line

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line.     

Proximity and precedence in arithmetic

How does the physical structure of an arithmetic expression affect the computational processes engaged in by reasoners? In handwritten arithmetic expressions containing both multiplications and additions, terms that are multiplied are often placed physically closer together than terms that are added. Three experiments evaluate the role such physical factors play in how reasoners construct solutions to simple compound arithmetic expressions (such as “2?+?3?×?4”). Two kinds of influence are found: First, reasoners incorporate the physical size of the expression into numerical responses, tending to give larger responses to more widely spaced problems. Second, reasoners use spatial information as a cue to hierarchical expression structure: More narrowly spaced subproblems within an expression tend to be solved first and tend to be multiplied. Although spatial relationships besides order are entirely formally irrelevant to expression semantics, reasoners systematically use these relationships to support their success with various formal properties.     

Representation and working memory in early arithmetic

Working memory has been implicated in the early acquisition of arithmetic skill, but the relations among different components of working memory, performance on different types of arithmetic problems, and development have not been explored. Preschool and Grade 1 children completed measures of phonological, visual-spatial, and central executive working memory, as well as nonverbal and verbal arithmetic problems, some of which included irrelevant information. For preschool children, accuracy was higher on nonverbal problems than on verbal problems, and the best and only unique predictor of performance on the standard nonverbal problems was visual-spatial working memory. This finding is consistent with the view that most preschoolers use a mental model for arithmetic that requires visual-spatial working memory. For Grade 1 children, performance was equivalent on nonverbal and verbal problems, and phonological working memory was the best predictor of performance on standard verbal problems. For both age groups, problems with added irrelevant information were substantially more difficult than standard problems, and in some cases measures of the central executive predicted performance. Assessing performance on different components of working memory in conjunction with different types of arithmetic problems provided new insights into the developing relations between working memory and how children do arithmetic.     

Numerical ordering ability mediates the relation between number-sense and arithmetic competence

What predicts human mathematical competence? While detailed models of number representation in the brain have been developed, it remains to be seen exactly how basic number representations link to higher math abilities. We propose that representation of ordinal associations between numerical symbols is one important factor that underpins this link. We show that individual variability in symbolic number-ordering ability strongly predicts performance on complex mental-arithmetic tasks even when controlling for several competing factors, including approximate number acuity. Crucially, symbolic number-ordering ability fully mediates the previously reported relation between approximate number acuity and more complex mathematical skills, suggesting that symbolic number-ordering may be a stepping stone from approximate number representation to mathematical competence. These results are important for understanding how evolution has interacted with culture to generate complex representations of abstract numerical relationships. Moreover, the finding that symbolic number-ordering ability links approximate number acuity and complex math skills carries implications for designing math-education curricula and identifying reliable markers of math performance during schooling.     

Children's understanding of the arithmetic concepts of inversion and associativity

Previous studies have shown that even preschoolers can solve inversion problems of the form a+b-b by using the knowledge that addition and subtraction are inverse operations. In this study, a new type of inversion problem of the form d x e/e was also examined. Grade 6 and 8 students solved inversion problems of both types as well as standard problems of the form a+b-c and d x e/f. Students in both grades used the inversion concept on both types of inversion problems, although older students used inversion more frequently and inversion was used most frequently on the addition/subtraction problems. No transfer effects were found from one type of inversion problem to the other. Students who used the concept of associativity on the addition/subtraction standard problems (e.g., a+b-c=[b-c]+a) were more likely to use the concept of inversion on the inversion problems, although overall implementation of the associativity concept was infrequent. The findings suggest that further study of inversion and associativity is important for understanding conceptual development in arithmetic.     

Brief non-symbolic,approximate number practice enhances subsequent exact symbolic arithmetic in children

Recent research reveals a link between individual differences in mathematics achievement and performance on tasks that activate the approximate number system (ANS): a primitive cognitive system shared by diverse animal species and by humans of all ages. Here we used a brief experimental paradigm to test one causal hypothesis suggested by this relationship: activation of the ANS may enhance children’s performance of symbolic arithmetic. Over 2 experiments, children who briefly practiced tasks that engaged primitive approximate numerical quantities performed better on subsequent exact, symbolic arithmetic problems than did children given other tasks involving comparison and manipulation of non-numerical magnitudes (brightness and length). The practice effect appeared specific to mathematics, as no differences between groups were observed on a comparable sentence completion task. These results move beyond correlational research and provide evidence that the exercise of non-symbolic numerical processes can enhance children’s performance of symbolic mathematics.     

Children’s understanding of addition and subtraction concepts

After the onset of formal schooling, little is known about the development of children’s understanding of the arithmetic concepts of inversion and associativity. On problems of the form a + b − b (e.g., 3 + 26 − 26), if children understand the inversion concept (i.e., that addition and subtraction are inverse operations), then no calculations are needed to solve the problem. On problems of the form a + b − c (e.g., 3 + 27 − 23), if children understand the associativity concept (i.e., that the addition and subtraction can be solved in any order), then the second part of the problem can be solved first. Children in Grades 2, 3, and 4 solved both types of problems and then were given a demonstration of how to apply both concepts. Approval of each concept and preference of a conceptual approach versus an algorithmic approach were measured. Few grade differences were found on either task. Conceptual understanding was greater for inversion than for associativity on both tasks. Clusters of participants in all grades showed that some had strong understanding of both concepts, some had strong understanding of the inversion concept only, and others had weak understanding of both concepts. The findings highlight the lack of developmental increases and the large individual differences in conceptual understanding on two arithmetic concepts during the early school years.     

An evaluation of sex and cultural differences in arithmetic retrieval-induced forgetting

Retrieval practice of arithmetic facts (e.g. 2?×?3) can interfere with retrieval of other, closely related arithmetic facts (e.g. 2?+?3), increasing response time (RT) and errors for these problems. Here we examined potential sex and culture-related differences in arithmetic retrieval-induced forgetting (RIF). This was motivated by re-analyses of several published arithmetic RIF data sets that appeared to show that the effect occurred for women but not men. Experiment 1 (n?=?72) tested for possible sex differences in a diverse but predominantly Canadian university sample. Experiment 2 (n?=?48) examined potential sex differences in native Chinese participants, which previous research indicated may not be susceptible to the RIF effect for a particular subset of small addition problems (sum?≤?10). In Experiment 1, we found no evidence that the addition RIF effect differed between male and female adults. In Experiment 2, the Chinese adults showed RIF for tie problems (e.g. 2?+?2, 3?+?3, etc.) regardless of sex, but neither sex presented RIF for small non-tie addition problems. The results indicated that the RIF effect is not gender specific, and there might not be strong memory retrieval competition between addition and multiplication facts for non-tie problems in Chinese adults.     

An association between understanding cardinality and analog magnitude representations in preschoolers

The preschool years are a time of great advances in children’s numerical thinking, most notably as they master verbal counting. The present research assessed the relation between analog magnitude representations and cardinal number knowledge in preschool-aged children to ask two questions: (1) Is there a relationship between acuity in the analog magnitude system and cardinality proficiency? (2) Can evidence of the analog magnitude system be found within mappings of number words children have not successfully mastered? To address the first question, Study 1 asked three- to five-year-old children to discriminate side-by-side dot arrays with varying differences in numerical ratio, as well as to complete an assessment of cardinality. Consistent with the analog magnitude system, children became less accurate at discriminating dot arrays as the ratio between the two numbers approached one. Further, contrary to prior work with preschoolers, a significant correlation was found between cardinal number knowledge and non-symbolic numerical discrimination. Study 2 aimed to look for evidence of the analog magnitude system in mappings to the words in preschoolers’ verbal counting list. Based on a modified give-a-number task ( [Wynn, 1990] and [Wynn, 1992] ), three- to five-year-old children were asked to give quantities between 1 and 10 as many times as possible in order to assess analog magnitude variability within their developing cardinality understanding. In this task, even children who have not yet induced the cardinality principle showed signs of analog representations in their understanding of the verbal count list. Implications for the contribution of analog magnitude representations towards mastery of the verbal count list are discussed in light of the present work.     

Relationship between levels of oral and written language in beginning readers

The validity of a test battery, organized by a theoretical framework of levels of language processing and production, was evaluated at the end of kindergarten and theend of first grade. At the end of kindergarten two levels of oral language, phonemic and lexical, and at the end of first grade three levels of oral language, phonemic, lexical, and text, were correlated with word decoding and reading comprehension. At the end of first grade, the combination of phonemic and lexical skills accounted for more variance in both word decoding and reading comprehension than either phonemic or lexical skills alone. The strength of the relationship between specific levels of oral language and specific component reading skills changed after formal reading instruction was introduced. Functional relationships were found between improvement in phonemic skills or lexical skills and improvement in word decoding. Partial correlations between two levels of oral language with a third partialed out (receptive or expressive task requirements held constant) provided evidence for three semiindependent levels of oral language—phonemic, lexical, and text. Because the battery has concurrent and construct validity, school psychologists can use it to monitor beginning readers in order to prevent reading disabilities due to subtle language dysfunctions.     

Relationship between mental toughness and physical endurance

This study tested the criterion validity of the inventory, Mental Toughness 48, by assessing the correlation between mental toughness and physical endurance for 41 male undergraduate sports students. A significant correlation of .34 was found between scores for overall mental toughness and the time a relative weight could be held suspended. Results support the criterion-related validity of the Mental Toughness 48.