Arithmetic Problem Solving Research Articles

Evidence for children's error sensitivity during arithmetic word problem solving

Abstract Solving simple arithmetic word problems is often challenging for children. Recent research suggests that children often fail to solve certain of these problems because they fail to inhibit erroneous heuristic intuitions that bias their judgment. However it is unclear whether these errors result from an error monitoring or inhibition failure. Our study focuses on this critical error detection. Eight to eleven year-old schoolchildren were given problems in which an intuitively cued heuristic answer conflicted with the correct answer and a control version in which this conflict was not present. After solving each version children were asked to indicate their response confidence. Results showed that children showed a sharp confidence decrease after having failed to solve the conflict problems. This indicates that erring children have some minimal awareness of the questionable nature of their answer and underscores that they have more arithmetic understanding than their errors might seem to suggest.

Development of common neural representations for distinct numerical problems

How the brain develops representations for abstract cognitive problems is a major unaddressed question in neuroscience. Here we tackle this fundamental question using arithmetic problem solving, a cognitive domain important for the development of mathematical reasoning. We first examined whether adults demonstrate common neural representations for addition and subtraction problems, two complementary arithmetic operations that manipulate the same quantities. We then examined how the common neural representations for the two problem types change with development. Whole-brain multivoxel representational similarity (MRS) analysis was conducted to examine common coding of addition and subtraction problems in children and adults. We found that adults exhibited significant levels of MRS between the two problem types, not only in the intraparietal sulcus (IPS) region of the posterior parietal cortex (PPC), but also in ventral temporal-occipital, anterior temporal and dorsolateral prefrontal cortices. Relative to adults, children showed significantly reduced levels of MRS in these same regions. In contrast, no brain areas showed significantly greater MRS between problem types in children. Our findings provide novel evidence that the emergence of arithmetic problem solving skills from childhood to adulthood is characterized by maturation of common neural representations between distinct numerical operations, and involve distributed brain regions important for representing and manipulating numerical quantity. More broadly, our findings demonstrate that representational analysis provides a powerful approach for uncovering fundamental mechanisms by which children develop proficiencies that are a hallmark of human cognition.

Mathematics Learning Development: the Role of Long-Term Retrieval

This study assessed the relation between long-term memory retrieval and mathematics calculation and mathematics problem solving achievement among elementary, middle, and high school students in nationally representative sample of US students, when controlling for fluid and crystallized intelligence, short-term memory, and processing speed. As hypothesized, structural equation modeling comparing elementary school students and middle and high school students revealed that long-term retrieval skills became a better predictor of both mathematics calculation and mathematics problem solving as age and grade increased. Future research should focus on the effectiveness of interventions to improve long-term retrieval skills in general, and arithmetic facts retrieval and problem solving procedures in particular, at all grades, including high school.

Interference of lateralized distractors on arithmetic problem solving: a functional role for attention shifts in mental calculation.

Solving arithmetic problems has been shown to induce shifts of spatial attention in simple probe-detection tasks, subtractions orienting attention to the left side and additions to the right side of space. Whether these attentional shifts constitute epiphenomena or are critically linked to the calculation process is still unknown. In the present study, we investigate participants' performance on addition and subtraction solving while they have to detect central or lateralized targets. The results show that lateralized distractors presented in the hemifield congruent to the operation to be solved interfere with arithmetical solving: participants are slower to solve subtractions or additions when distractors are located on the left or on the right, respectively. These results converge with previous data to show that attentional shifts underlie not only number processing but also mental arithmetic. They extend them as they reveal the reverse effect of the one previously reported by showing that inducing attention shifts interferes with the solving of arithmetic problems. They also demonstrate that spatial attentional shifts are part of the calculation procedure of solving mentally arithmetic problems. Their functional role is to access, from the first operand, the representation of the result in a direction congruent to the operation.

Mathematical skills in children with dyslexia

The mathematical performance of 17 children with developmental dyslexia (DD) was assessed and compared to a control group to examine whether difficulties related to reading and phonological processing affect the development of mathematical skills. The DD group performed worse than the controls on number fact retrieval, multi-step arithmetic problem solving, and multi-digit calculation, whereas their scores on tasks tapping approximate arithmetic and conceptual understanding (i.e., place value, calculation principles) were equal to the controls. In view of the Triple-code model, the findings demonstrate that children with DD have problems with tasks depending on verbal–phonological number codes but have no problems with tasks depending on analogue magnitude representations or visual-Arabic number codes.

When problem size matters: differential effects of brain stimulation on arithmetic problem solving and neural oscillations.

The problem size effect is a well-established finding in arithmetic problem solving and is characterized by worse performance in problems with larger compared to smaller operand size. Solving small and large arithmetic problems has also been shown to involve different cognitive processes and distinct electroencephalography (EEG) oscillations over the left posterior parietal cortex (LPPC). In this study, we aimed to provide further evidence for these dissociations by using transcranial direct current stimulation (tDCS). Participants underwent anodal (30min, 1.5 mA, LPPC) and sham tDCS. After the stimulation, we recorded their neural activity using EEG while the participants solved small and large arithmetic problems. We found that the tDCS effects on performance and oscillatory activity critically depended on the problem size. While anodal tDCS improved response latencies in large arithmetic problems, it decreased solution rates in small arithmetic problems. Likewise, the lower-alpha desynchronization in large problems increased, whereas the theta synchronization in small problems decreased. These findings reveal that the LPPC is differentially involved in solving small and large arithmetic problems and demonstrate that the effects of brain stimulation strikingly differ depending on the involved neuro-cognitive processes.

Perceiving fingers in single-digit arithmetic problems.

In this study, we investigate in children the neural underpinnings of finger representation and finger movement involved in single-digit arithmetic problems. Evidence suggests that finger representation and finger-based strategies play an important role in learning and understanding arithmetic. Because different operations rely on different networks, we compared activation for subtraction and multiplication problems in independently localized finger somatosensory and motor areas and tested whether activation was related to skill. Brain activations from children between 8 and 13 years of age revealed that only subtraction problems significantly activated finger motor areas, suggesting reliance on finger-based strategies. In addition, larger subtraction problems yielded greater somatosensory activation than smaller problems, suggesting a greater reliance on finger representation for larger numerical values. Interestingly, better performance in subtraction problems was associated with lower activation in the finger somatosensory area. Our results support the importance of fine-grained finger representation in arithmetical skill and are the first neurological evidence for a functional role of the somatosensory finger area in proficient arithmetical problem solving, in particular for those problems requiring quantity manipulation. From an educational perspective, these results encourage investigating whether different finger-based strategies facilitate arithmetical understanding and encourage educational practices aiming at integrating finger representation and finger-based strategies as a tool for instilling stronger numerical sense.

Relationships Between Strategy Switching and Strategy Switch Costs in Young and Older Adults: A Study in Arithmetic Problem Solving

Background/Study Context: This study investigated age-related differences in within-item strategy switching (i.e., revising initial strategy choices to select a better strategy while solving a given problem) and in strategy switch costs (i.e., longer latencies when participants switch strategies than when they do not switch strategy during strategy execution).Methods: In a computational estimation task, participants had to give approximate products to two-digit multiplication problems (e.g., 41 × 67) while rounding up (i.e., do 50 × 70 for 41 × 67) or rounding down (i.e., do 40 × 60 for 41 × 67) operands to their nearest decades. After executing a cued strategy during 1000 ms, participants had the possibility to switch to another strategy (or repeat the same strategy) in a selection condition. In an execution condition, participants were forced to repeat the same strategy or to switch to another strategy.Results: It was found that (1) older adults were less able than young adults to switch strategy after starting to execute a cued strategy (36.1% vs. 45.8%); (2) older adults showed larger switch costs than young adults (422 vs. 223 ms); and (3) strategy switches and strategy switch costs correlated in older adults but not in young adults.Conclusion: These findings have important implications for our understanding of the mechanisms underlying within-item strategy switching and aging effects on these mechanisms as well as, more generally, of strategic variations during cognitive aging.

Neural correlates of mathematical problem solving.

This study explores electroencephalography (EEG) brain dynamics associated with mathematical problem solving. EEG and solution latencies (SLs) were recorded as 11 neurologically healthy volunteers worked on intellectually challenging math puzzles that involved combining four single-digit numbers through basic arithmetic operators (addition, subtraction, division, multiplication) to create an arithmetic expression equaling 24. Estimates of EEG spectral power were computed in three frequency bands - θ (4-7 Hz), α (8-13 Hz) and β (14-30 Hz) - over a widely distributed montage of scalp electrode sites. The magnitude of power estimates was found to change in a linear fashion with SLs - that is, relative to a base of power spectrum, theta power increased with longer SLs, while alpha and beta power tended to decrease. Further, the topographic distribution of spectral fluctuations was characterized by more pronounced asymmetries along the left-right and anterior-posterior axes for solutions that involved a longer search phase. These findings reveal for the first time the topography and dynamics of EEG spectral activities important for sustained solution search during arithmetical problem solving.

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.

Updating executive function and performance in reading comprehension and problem solving. [La función ejecutiva de actualización y el rendimiento en comprensión lectora y resolución de problemas

In this investigation, the capacity of the working memory (WM) updating executive function to predict individual differences in reading comprehension and problem solving was analyzed in 5th-graders of Prima- ry Education. In addition, we examined whether this relation is direct or mediated by domain-general or domain-specific variables. For this purpose, a series of tasks was administered to assess fluid intelligence, WM infor- mation updating, arithmetic abilities, arithmetic problem solving, lexical processing, and reading comprehension in 49 students aged between 10 and 11 years. The results support the idea that updating is an important predictor of reading comprehension, beyond the influence of domain- specific skills and fluid intelligence. In the case of problem solving, our findings confirm that updating plays an important role although, perhaps due to task content, the relation seems to be mediated by fluid intelligence at this developmental stage..

Arithmetic and algebraic problem solving and resource allocation: the distinct impact of fluid and numerical intelligence.

This study investigates cognitive resource allocation dependent on fluid and numerical intelligence in arithmetic/algebraic tasks varying in difficulty. Sixty-six 11th grade students participated in a mathematical verification paradigm, while pupil dilation as a measure of resource allocation was collected. Students with high fluid intelligence solved the tasks faster and more accurately than those with average fluid intelligence, as did students with high compared to average numerical intelligence. However, fluid intelligence sped up response times only in students with average but not high numerical intelligence. Further, high fluid but not numerical intelligence led to greater task-related pupil dilation. We assume that fluid intelligence serves as a domain-general resource that helps to tackle problems for which domain-specific knowledge (numerical intelligence) is missing. The allocation of this resource can be measured by pupil dilation.

Hippocampal-neocortical functional reorganization underlies children's cognitive development.

The importance of the hippocampal system for rapid learning and memory is well recognized, but its contributions to a cardinal feature of children's cognitive development – the transition from procedure-based to memory-based problem solving strategies – are unknown. Here we show that the hippocampal system is pivotal to this strategic transition. Longitudinal fMRI in children, ages 7 to 9, revealed that the transition from use of counting to memory-based retrieval parallels increased hippocampal and decreased prefrontal-parietal engagement during arithmetic problem solving. Critically, longitudinal improvements in retrieval strategy use were predicted by increased hippocampal-neocortical functional connectivity. Beyond childhood, retrieval strategy use continued to improve through adolescence into adulthood, and was associated with decreased activation but more stable inter-problem representations in the hippocampus. Our findings provide novel insights into the dynamic role of the hippocampus in the maturation of memory-based problem solving, and establish a critical link between hippocampal-neocortical reorganization and children's cognitive development.

Brain hyper-connectivity and operation-specific deficits during arithmetic problem solving in children with developmental dyscalculia.

Developmental dyscalculia (DD) is marked by specific deficits in processing numerical and mathematical information despite normal intelligence (IQ) and reading ability. We examined how brain circuits used by young children with DD to solve simple addition and subtraction problems differ from those used by typically developing (TD) children who were matched on age, IQ, reading ability, and working memory. Children with DD were slower and less accurate during problem solving than TD children, and were especially impaired on their ability to solve subtraction problems. Children with DD showed significantly greater activity in multiple parietal, occipito-temporal and prefrontal cortex regions while solving addition and subtraction problems. Despite poorer performance during subtraction, children with DD showed greater activity in multiple intra-parietal sulcus (IPS) and superior parietal lobule subdivisions in the dorsal posterior parietal cortex as well as fusiform gyrus in the ventral occipito-temporal cortex. Critically, effective connectivity analyses revealed hyper-connectivity, rather than reduced connectivity, between the IPS and multiple brain systems including the lateral fronto-parietal and default mode networks in children with DD during both addition and subtraction. These findings suggest the IPS and its functional circuits are a major locus of dysfunction during both addition and subtraction problem solving in DD, and that inappropriate task modulation and hyper-connectivity, rather than under-engagement and under-connectivity, are the neural mechanisms underlying problem solving difficulties in children with DD. We discuss our findings in the broader context of multiple levels of analysis and performance issues inherent in neuroimaging studies of typical and atypical development.

The functional anatomy of single-digit arithmetic in children with developmental dyslexia

Some arithmetic procedures, such as addition of small numbers, rely on fact retrieval mechanisms supported by left hemisphere perisylvian language areas, while others, such as subtraction, rely on procedural-based mechanisms subserved by bilateral parietal cortices. Previous work suggests that developmental dyslexia, a reading disability, is accompanied by subtle deficits in retrieval-based arithmetic, possibly because of compromised left hemisphere function. To test this prediction, we compared brain activity underlying arithmetic problem solving in children with and without dyslexia during addition and subtraction operations using a factorial design. The main effect of arithmetic operation (addition versus subtraction) for both groups combined revealed activity during addition in the left superior temporal gyrus and activity during subtraction in the bilateral intraparietal sulcus, the right supramarginal gyrus and the anterior cingulate, consistent with prior studies. For the main effect of diagnostic group (dyslexics versus controls), we found less activity in dyslexic children in the left supramarginal gyrus. Finally, the interaction analysis revealed that while the control group showed a strong response in the right supramarginal gyrus for subtraction but not for addition, the dyslexic group engaged this region for both operations. This provides physiological evidence in support of the theory that children with dyslexia, because of disruption to left hemisphere language areas, use a less optimal route for retrieval-based arithmetic, engaging right hemisphere parietal regions typically used by good readers for procedural-based arithmetic. Our results highlight the importance of language processing for mathematical processing and illustrate that children with dyslexia have impairments that extend beyond reading.

Causal role of spatial attention in arithmetic problem solving: Evidence from left unilateral neglect

Recent behavioural and brain imaging studies have provided evidence for rightward and leftward attention shifts while solving addition and subtraction problems respectively, suggesting that mental arithmetic makes use of mechanisms akin to those underlying spatial attention. However, this hypothesis mainly relies on correlative data and the causal relevance of spatial attention for mental arithmetic remains unclear. In order to test whether the mechanisms underlying spatial attention are necessary to perform arithmetic operations, we compared the performance of right brain-lesioned patients, with and without left unilateral neglect, and healthy controls in addition and subtraction of two-digit numbers. We predicted that patients with left unilateral neglect would be selectively impaired in the subtraction task while being unimpaired in the addition task. The results showed that neglect patients made more errors than the two other groups to subtract large numbers, whereas they were still able to solve large addition problems matched for difficulty and magnitude of the answer. This finding demonstrates a causal relationship between the ability to attend the left side of space and the solving of large subtraction problems. A plausible account is that attention shifts help localizing the position of the answer on a spatial continuum while subtracting large numbers.

Emulating Human Supervision in an Intelligent Tutoring System for Arithmetical Problem Solving

This paper presents an intelligent tutoring system (ITS) for the learning of arithmetical problem solving. This is based on an analysis of a) the cognitive processes that take place during problem solving; and b) the usual tasks performed by a human when supervising a student in a one-to-one tutoring situation. The ITS is able to identify the solving strategy that the student is following and offer adaptive feedback that takes into account both the problem's constraints and the decisions previously made by the user. An observational study shows the ITS's accuracy at emulating expert human supervision, and a randomized experiment reveals that the ITS significantly improves students' learning in arithmetical problem solving.

Working memory, worry, and algebraic ability

Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship.

An integrative view of Luria’s perspective on arithmetic problem solving: The two sides of environmental dependency

Introduction: A. R. Luria was the first author to hypothesize that executive dysfunction can lead to specific deficits in arithmetic problem solving, showing that patients’ performance depends on the structure of the tasks. Cummings (1995. Anatomic and behavioral aspects of frontal-subcortical circuits. Annals of the New York Academy of Sciences, 15, 1–13) proposed the term “environmental dependency” to define such behavioral disorders triggered by the characteristics of the test and pointed out also the role of executive impairments. Few studies compare executive functioning and problem solving in brain-damaged patients, and none have examined the question from this point of view. Thus, the main aim of the present paper was to study the relationship between environmental dependency and executive functions. Method: Fifty neurological patients with frontal, subcortical, and posterior brain lesions were compared to 45 matched healthy controls and were divided into two groups (dysexecutive/nondysexecutive) according to their performances on executive tasks. Then, we confronted the results of the two groups on an experimental protocol designed in accordance with Luria’s proposals. We made also comparisons between groups on the basis of lesion location. Results: Our findings indicate a high association between executive functions and environmental dependency, showing that dysexecutive patients’ performances were dependent on task demands. In addition, a specific frontal behavior not associated with executive functions and characterized by the solving of insoluble problems was highlighted. Conclusion: The discussion focused on the interest to take into account the methodological and clinical contributions of environmental dependency. Based on our findings and theoretical arguments, we highlight the need to fractionate this concept.

Mathematical learning difficulties subtypes classification

Mathematics is a complex subject including different domains such as arithmetic, arithmetic problem solving, geometry, algebra, probability, statistics, calculus, … that implies mobilizing a variety of basic abilities associated with the sense of quantity, symbols decoding, memory, visuospatial capacity, logics, to name a few. Students with difficulties in any of the these abilities or in their coordination, may experience mathematical learning difficulties. Understanding the cognitive nature of the various mathematical domains, as well as the mechanisms mediating cognitive development, has fascinated researchers from different fields: from mathematics education to developmental and cognitive psychology and neuroscience. The field of cognitive psychology has a long history in the studies of cognitive difficulties involved in developing the representation and learning general use of numbers in mathematics (e.g., Campbell, 2005). However, as Fletcher et al. (2007) note, there are “no consistent standards by which to judge the presence or absence of LDs [learning difficulties] in math” (p. 207), and there is still disagreement concerning the question of a definition, operational criteria, and prevalence (Lanfranchi et al., 2008; Mazzocco, 2008). In general, the term Mathematical Learning Difficulty (MLD) is used broadly to describe a wide variety of deficits in math skills, typically pertaining the domains of arithmetic and arithmetic problem solving. We will use MLDs to refer to learning difficulties in these domains as well as other mathematical domains like the ones mentioned above. Within the field of mathematics education, many frameworks and theories have been developed to analyze teaching and learning processes and difficulties involved with these and other mathematical tasks (e.g., Freudenthal, 1991; Schoenfeld, 1992, 2011; Bharath and English, 2010). Recently, the field has shown interest in perspectives from cognitive neuroscience (e.g., Grabner and Ansari, 2010). Although developmental and classification models in these fields have been developed (for example, Geary and Hoard, 2005; Desoete, 2007; von Aster and Shalev, 2007), to our knowledge, no single framework or model can be used for a comprehensive and fine interpretation of students' mathematical difficulties, not only for scientific purposes, but also for informing mathematics educators. As mathematics educators1, we believe that reaching a model that combines existing hypotheses on MLD, based on known cognitive processes and mechanisms, could be used to provide a mathematical profile for every student. Our aims with this contribution are to: (1) provide an overview of the most relevant hypotheses in the present day's literature regarding possible deficits that lead to MLD and of possible classifications of MLD subtypes; (2) and to build on such literature, using a multi-deficit neurocognitive approach, to propose a classification model for MLD describing four basic cognitive domains within which specific deficits may reside. In order to reach our first objective we will describe the current hypotheses on neurocognitive deficits that may lead to MLD specifically related to numbers, and then we will provide examples of the most relevant classifications of MLD, based on a possible deficit in basic cognitive functions.