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

Neural Representation Precision of Distance Predicts Children's Arithmetic Performance

Focusing on the distance between magnitudes as the starting point to investigate the mechanism of relation detection and its contribution to mathematical thinking, this study explores the precision of neural representations of numerical distance and their impact on children's arithmetic performance. By employing neural decoding techniques and representational similarity analysis, the present study investigates how accurately the brain represents numerical distances and how this precision relates to arithmetic skills. Twenty-nine school-aged children participated, completing a dot number comparison task during fMRI scanning and an arithmetic fluency test. Results indicated that neural activation patterns in the intra-parietal sulcus decoded the distance between the presented pair of dots, and higher precision in neural distance representation correlates with better arithmetic performance. These findings suggest that the accuracy of neural decoding can serve as an index of neural representation precision and that the ability to precisely encode numerical distances in the brain is a key factor in mathematical abilities. This provides new insights into the neural basis of mathematical cognition and learning.

Investigating frontoparietal networks and activation in children with mathematics learning difficulties: Cases with different deficit profiles

Approximately 15%-20% of school-aged children suffer from mathematics learning difficulties (MLD). Most children with developmental dyscalculia (DD) or MLD also have comorbid cognitive deficits. Recent literature suggests that research should focus on uncovering the neural underpinnings of MLD across more inclusive samples, rather than limiting studies to pure cases of DD or MLD with highly stringent inclusion criteria. Therefore, this study aims to identify neural aberrancies that may be common across multiple MLD cases with different deficit profiles. Nine MLD cases and 45 typically developing (TD) children, all around 7 years old (27 boys), were recruited. Using functional near-infrared spectroscopy (fNIRS), brain data were collected during an approximate resting state and a mathematical computation task (addition). Graph theory was then applied to assess global and nodal network indicators of brain function. When comparing the network indicators and brain activation of the MLD cases to those of TD children, no unified neural aberrancy was found across all cases. However, three MLD cases did show distinct neural aberrancies compared to TD children. The study discusses the implications of these findings, considering both the neural aberrancies in the three MLD cases and the neural similarities found in the other six cases, which were comparable to those of the TD children. This raises important questions about the presence and nature of aberrant neural indicators in MLD across large cohorts and highlights the need for further research in this area.

Anodal High-definition Transcranial Direct Current Stimulation Over the Left (but not Right) Parietal Cortex Facilitates Mental Arithmetic

Non-invasive brain stimulation (NIBS) techniques such as transcranial direct current stimulation (tDCS) allow for investigating the functional involvement of specific brain areas in mental arithmetic. In this study, we employed for the first time high-definition (HD)-tDCS, which offers enhanced spatial precision, to explore the functional roles of the left and right intraparietal sulcus (IPS) in mental arithmetic. A total of 25 participants underwent anodal left IPS, anodal right IPS, and sham stimulation in separate sessions in counterbalanced order while solving single- and multi-step addition and subtraction problems. We found that stimulation of the left IPS, but not the right IPS or sham stimulation, improved arithmetic performance speed. These results provide further evidence for the functional involvement of the left IPS in a broad range of arithmetic tasks and highlight the potential of NIBS for cognitive enhancement.

Supplementary Information

The online version contains supplementary material available at 10.1007/s41465-024-00314-0.

Predicting individual differences in preschoolers' numeracy and geometry knowledge: The role of understanding abstract relations between objects and quantities

Preschoolers' mathematics knowledge develops early and varies substantially. The current study focused on two ontogenetically early emerging cognitive skills that may be important predictors of later math skills (i.e., geometry and numeracy): children's understanding of abstract relations between objects and quantities as evidenced by their patterning skills and the approximate number system (ANS). Children's patterning skills, the ANS, numeracy, geometry, nonverbal intelligence (IQ), and executive functioning (EF) skills were assessed at age 4 years, and their numeracy and geometry knowledge was assessed again a year later at age 5 (N = 113). Above and beyond children's initial knowledge in numeracy and geometry, as well as IQ and EF, patterning skills and the ANS at age 4 uniquely predicted children's geometry knowledge at age 5, but only age 4 patterning uniquely predicted age 5 numeracy. Thus, although patterning and the ANS are related, they differentially explain variation in later geometry and numeracy knowledge. Results are discussed in terms of implications for early mathematics theory and research.

Are they equivalent? An examination of task variants of approximate number comparison

Nonverbal numerical ability supports individuals' numerical information processing in everyday life and is also correlated with their learning of mathematics. This ability is typically measured with an approximate number comparison paradigm, in which participants are presented with two sets of objects and instructed to choose the numerically larger set. This paradigm has multiple task variants, where the two sets are presented in different ways (e.g., two sets are presented either simultaneously or sequentially, or two sets are presented either intermixed or separately). Despite the fact that different task variants have often been used interchangeably, it remains unclear whether these variants measure the same aspects of nonverbal numerical ability. Using a latent variable modeling approach with 270 participants (Mage = 20.75 years, SDage = 2.03, 94 males), this study examined the degree to which three commonly used task variants tapped into the same construct. The results showed that a bi-factor model corresponding to the hypothesis that task variants had both commonalities and uniqueness was a better fit for the data than a single-factor model, corresponding to the hypothesis that task variants were construct equivalent. These findings suggested that task variants of approximate number comparison did not measure the same construct and cannot be used interchangeably. This study also quantified the extent to which general cognitive abilities were involved in both common and unique parts of these task variants.

A brief tablet-based intervention benefits linguistic and communicative abilities in toddlers and preschoolers

Young children's linguistic and communicative abilities are foundational for their academic achievement and overall well-being. We present the positive outcomes of a brief tablet-based intervention aimed at teaching toddlers and preschoolers new word-object and letter-sound associations. We conducted two experiments, one involving toddlers ( ~ 24 months old, n = 101) and the other with preschoolers ( ~ 42 months old, n = 152). Using a pre-post equivalent group design, we measured the children's improvements in language and communication skills resulting from the intervention. Our results showed that the intervention benefited toddlers' verbal communication and preschoolers' speech comprehension. Additionally, it encouraged vocalizations in preschoolers and enhanced long-term memory for the associations taught in the study for all participants. In summary, our study demonstrates that the use of a ludic tablet-based intervention for teaching new vocabulary and pre-reading skills can improve young children's linguistic and communicative abilities, which are essential for future development.

Numeracy, gist, literal thinking and the value of nothing in decision making

The onus on the average person is greater than ever before to make sense of large amounts of readily accessible quantitative information, but the ability and confidence to do so are frequently lacking. Many people lack practical mathematical skills that are essential for evaluating risks, probabilities and numerical outcomes such as survival rates for medical treatments, income from retirement savings plans or monetary damages in civil trials. In this Review, we integrate research on objective and subjective numeracy, focusing on cognitive and metacognitive factors that distort human perceptions and foment systematic biases in judgement and decision making. Paradoxically, an important implication of this research is that a literal focus on objective numbers and mechanical number crunching is misguided. Numbers can be a matter of life and death but a person who uses rote strategies (verbatim representations) cannot take advantage of the information contained in the numbers because 'rote' strategies are, by definition, processing without meaning. Verbatim representations (verbatim is only surface form, not meaning) treat numbers as data as opposed to information. We highlight a contrasting approach of gist extraction: organizing numbers meaningfully, interpreting them qualitatively and making meaningful inferences about them. Efforts to improve numerical cognition and its practical applications can benefit from emphasizing the qualitative meaning of numbers in context - the gist - building on the strengths of humans as intuitive mathematicians. Thus, we conclude by reviewing evidence that gist training facilitates transfer to new contexts and, because it is more durable, longer-lasting improvements in decision making.

The developmental relationship between nonsymbolic and symbolic fraction abilities

A fundamental research question in quantitative cognition concerns the developmental relationship between nonsymbolic and symbolic quantitative abilities. This study examined this developmental relationship in abilities to process nonsymbolic and symbolic fractions. There were 99 6th graders (M_age = 11.86 years), 101 10th graders (M_age = 15.71 years), and 102 undergraduate and graduate students (M_age = 21.97 years) participating in this study, and their nonsymbolic and symbolic fraction abilities were measured with nonsymbolic and symbolic fraction comparison tasks, respectively. Nonsymbolic and symbolic fraction abilities were significantly correlated in all age groups even after controlling for the ability to process nonsymbolic absolute quantity and general cognitive abilities, including working memory and inhibitory control. Moreover, the strength of nonsymbolic-symbolic correlations was higher in 6th graders than in 10th graders and adults. These findings suggest a weakened association between nonsymbolic and symbolic fraction abilities during development, and this developmental pattern may be related with participants' increasing proficiency in symbolic fractions.

Directionality in the interrelations between approximate number, verbal number, and mathematics in preschool-aged children

A fundamental question in numerical development concerns the directional relation between an early-emerging non-verbal approximate number system (ANS) and culturally acquired verbal number and mathematics knowledge. Using path models on longitudinal data collected in preschool children (M_age  = 3.86 years; N = 216; 99 males; 80.8% White; 10.8% Multiracial, 3.8% Latino; 1.9% Black; collected 2013-2017) over 1 year, this study showed that earlier verbal number knowledge was associated with later ANS precision (average β = .32), even after controlling for baseline differences in numerical, general cognitive, and language abilities. In contrast, earlier ANS precision was not associated with later verbal number knowledge (β = -.07) or mathematics abilities (average β = .10). These results suggest that learning about verbal numbers is associated with a sharpening of pre-existing non-verbal numerical abilities.

How many seconds was that? Teaching children about time does not refine their ability to track durations

Over development, children acquire symbols to represent abstract concepts such as time and number. Despite the importance of quantity symbols, it is unknown how acquiring these symbols impacts one's ability to perceive quantities (i.e., nonsymbolic representations). While it has been proposed that learning symbols shapes nonsymbolic quantitative abilities (i.e., the refinement hypothesis), this hypothesis has been understudied, especially in the domain of time. Moreover, the majority of research in support of this hypothesis has been correlational in nature, and thus, experimental manipulations are critical for determining whether this relation is causal. In the present study, kindergarteners and first graders (N = 154) who have yet to learn about temporal symbols in school completed a temporal estimation task during which they were either (1) trained on temporal symbols and effective timing strategies ("2 s" and counting on the beat), (2) trained on temporal symbols only ("2 s"), or (3) participated in a control training. Children's nonsymbolic and symbolic timing abilities were assessed before and after training. Results revealed a correlation between children's nonsymbolic and symbolic timing abilities at pre-test (when controlling for age), indicating this relation exists prior to formal classroom instruction on temporal symbols. Notably, we found no support for the refinement hypothesis, as learning temporal symbols did not impact children's nonsymbolic timing abilities. Implications and future directions are discussed.

Metacognitive and Non-Metacognitive Processes in Arithmetic Performance: Can There Be More than One Meta-Level?

The nature of the development of arithmetic performance has long been intensively studied, and available scientific evidence can be evaluated and synthesized in light of Nelson and Narens’ model of metacognition. According to the Nelson–Narens model, human cognition can be split into two or more interrelated levels. Obviously, in the case of more than two levels, cognitive processes from at least one level can be described as both meta- and object-level processes. The question arises whether it is possible that the very same cognitive processes are both controlled and controlling. The feasibility of owning the same cognitive processes—which are considered the same from an external point of view of assessment—as both meta- and object-level processes within the same individual opens the possibility of investigating the transition from meta-level to object-level. Modeling cognitive development by means of a series of such transitions calls forth an understanding of possible developmental phases in a given domain of learning. The developmental phases of arithmetic performance are described as a series of transitions from arithmetical facts to strategies of arithmetic word problem solving. For school learning and instruction, the role of metacognitive scaffolding as a powerful educational approach is emphasized.

EXPRESS: Arithmetic operations without symbols are unimpaired in adults with math anxiety

This study characterizes a previously unstudied facet of a major causal model of math anxiety. The model posits that impaired "basic number abilities" can lead to math anxiety, but what constitutes a basic number ability remains underdefined. Previous work has raised the idea that our perceptual ability to represent quantities approximately without using symbols constitutes one of the basic number abilities. Indeed, several recent studies tested how participants with math anxiety estimate and compare non-symbolic quantities. However, little is known about how participants with math anxiety perform arithmetic operations (addition and subtraction) on non-symbolic quantities. This is an important question because poor arithmetic performance on symbolic numbers is one of the primary signatures of high math anxiety. To test the question, we recruited 92 participants and asked them to complete a math anxiety survey, two measures of working memory, a timed symbolic arithmetic test, and a non-symbolic "approximate arithmetic" task. We hypothesized that if impaired ability to perform operations was a potential causal factor to math anxiety, we should see relationships between math anxiety and both symbolic and approximate arithmetic. However, if math anxiety relates to precise or symbolic representation, only a relationship between math anxiety and symbolic arithmetic should appear. Our results show no relationship between math anxiety and the ability to perform operations with approximate quantities, suggesting that difficulties performing perceptually based arithmetic operations does not constitute a basic number ability linked to math anxiety.

Can training in the approximate number system improve the informal mathematics ability of preschoolers?

Recent studies show that comparison or arithmetic training in the approximate number system (ANS) can improve the early mathematics ability of preschool children. However, no studies have compared the training effects of ANS comparison training with those of ANS arithmetic training on the early mathematics ability of preschool children. The current study pseudorandomly assigned 87 children aged 4-5 years to one of three training groups (the ANS comparison, ANS arithmetic, and control groups) for 4 weeks of training. The results showed that compared with the control group, the ANS comparison training and ANS arithmetic training equally improved the ANS acuity and informal mathematics ability of preschool children. In addition, the study found that there may be a bidirectional causal relationship between ANS and mathematics in preschoolers, but this relationship needs to be further investigated using longitudinal studies. Taken together, these findings emphasize the importance of ANS-based training in improving preschoolers' ANS acuity and informal mathematics ability before formal school enrollment.

Teasing apart the unique contributions of cognitive and affective predictors of math performance

Math permeates everyday life, and math skills are linked to general educational attainment, income, career choice, likelihood of full-time employment, and health and financial decision making. Thus, researchers have attempted to understand factors predicting math performance in order to identify ways of supporting math development. Work examining individual differences in math performance typically focuses on either cognitive predictors, including inhibitory control and the approximate number system (ANS; a nonsymbolic numerical comparison system), or affective predictors, like math anxiety. Studies with children suggest that these factors are interrelated, warranting examination of whether and how each uniquely and independently contributes to math performance in adulthood. Here, we examined how inhibitory control, the ANS, and math anxiety predicted college students' math performance (n = 122, mean age = 19.70 years). Using structural equation modeling, we find that although inhibitory control and the ANS were closely related to each other, they did not predict math performance above and beyond the effects of the other while also controlling for math anxiety. Instead, math anxiety was the only unique predictor of math performance. These findings contradict previous results in children and reinforce the need to consider affective factors in our discussions and interventions for supporting math performance in college students.

Sharpening, focusing, and developing: A study of change in nonsymbolic number comparison skills and math achievement in 1st grade

Children's ability to discriminate nonsymbolic number (e.g., the number of items in a set) is a commonly studied predictor of later math skills. Number discrimination improves throughout development, but what drives this improvement is unclear. Competing theories suggest that it may be due to a sharpening numerical representation or an improved ability to pay attention to number and filter out non-numerical information. We investigate this issue by studying change in children's performance (N = 65) on a nonsymbolic number comparison task, where children decide which of two dot arrays has more dots, from the middle to the end of 1st grade (mean age at time 1 = 6.85 years old). In this task, visual properties of the dot arrays such as surface area are either congruent (the more numerous array has more surface area) or incongruent. Children rely more on executive functions during incongruent trials, so improvements in each congruency condition provide information about the underlying cognitive mechanisms. We found that accuracy rates increased similarly for both conditions, indicating a sharpening sense of numerical magnitude, not simply improved attention to the numerical task dimension. Symbolic number skills predicted change in congruent trials, but executive function did not predict change in either condition. No factor predicted change in math achievement. Together, these findings suggest that nonsymbolic number processing undergoes development related to existing symbolic number skills, development that appears not to be driving math gains during this period. Children's ability to discriminate nonsymbolic number improves throughout development. Competing theories suggest improvement due to sharpening magnitude representations or changes in attention and inhibition. The current study investigates change in nonsymbolic number comparison performance during first grade and whether symbolic number skills, math skills, or executive function predict change. Children's performance increased across visual control conditions (i.e., congruent or incongruent with number) suggesting an overall sharpening of number processing. Symbolic number skills predicted change in nonsymbolic number comparison performance.

Testing the role of symbols in preschool numeracy: An experimental computer-based intervention study

Numeracy is of critical importance for scholastic success and modern-day living, but the precise mechanisms that drive its development are poorly understood. Here we used novel experimental training methods to begin to investigate the role of symbols in the development of numeracy in preschool-aged children. We assigned pre-school children in the U.S. and Italy (N = 215; Mean age = 49.15 months) to play one of five versions of a computer-based numerical comparison game for two weeks. The different versions of the game were equated on basic features of gameplay and demands but systematically varied in numerical content. Critically, some versions included non-symbolic numerical comparisons only, while others combined non-symbolic numerical comparison with symbolic aids of various types. Before and after training we assessed four components of early numeracy: counting proficiency, non-symbolic numerical comparison, one-to-one correspondence, and arithmetic set transformation. We found that overall children showed improvement in most of these components after completing these short trainings. However, children trained on numerical comparisons with symbolic aids made larger gains on assessments of one-to-one correspondence and arithmetic transformation compared to children whose training involved non-symbolic numerical comparison only. Further exploratory analyses suggested that, although there were no major differences between children trained with verbal symbols (e.g., verbal counting) and non-verbal visuo-spatial symbols (i.e., abacus counting), the gains in one-to-one correspondence may have been driven by abacus training, while the gains in non-verbal arithmetic transformations may have been driven by verbal training. These results provide initial evidence that the introduction of symbols may contribute to the emergence of numeracy by enhancing the capacity for thinking about exact equality and the numerical effects of set transformations. More broadly, this study provides an empirical basis to motivate further focused study of the processes by which children’s mastery of symbols influences children’s developing mastery of numeracy.

Operational momentum during children's approximate arithmetic relates to symbolic math skills and space-magnitude association

Operational momentum (OM) refers to the behavioral tendency to overestimate or underestimate the results of addition or subtraction, respectively. The cognitive mechanism of the OM effect and how it is related to the development of symbolic math abilities are not well understood. The current study examined whether individual differences in the OM effect are related to symbolic arithmetic abilities, number line estimation performance, and the space-magnitude association effect in young children. In this study, first-grade elementary school children manifested the OM effect during approximate addition and subtraction. Individual differences in the OM effect were not correlated with number line estimation error. Interestingly, children who showed a greater degree of the OM effect performed not worse, but better on the symbolic arithmetic task. In addition, the OM effect was correlated with the space-magnitude association (size congruity) effect measured with the Numerical Stroop task. More specifically, the OM bias was correlated with the ability to inhibit interference from competing information on the incongruent trials of the Numerical Stroop task. Our results suggest that the inaccuracy of numerical magnitude representations is not the source of the OM effect. Given that children with better math ability showed a greater OM bias, a stronger OM effect may reflect better intuition in arithmetic operations. Altogether, we carefully interpret these findings as suggesting that a greater OM effect reflects superior intuition or fundamental knowledge of arithmetic operations and a more adult-like maturation of the reorienting component of the attentional system.

Developmental brain dynamics of numerical and arithmetic abilities

The development of numerical and arithmetic abilities constitutes a crucial cornerstone in our modern and educated societies. Difficulties to acquire these central skills can lead to severe consequences for an individual's well-being and nation's economy. In the present review, we describe our current broad understanding of the functional and structural brain organization that supports the development of numbers and arithmetic. The existing evidence points towards a complex interaction among multiple domain-specific (e.g., representation of quantities and number symbols) and domain-general (e.g., working memory, visual-spatial abilities) cognitive processes, as well as a dynamic integration of several brain regions into functional networks that support these processes. These networks are mainly, but not exclusively, located in regions of the frontal and parietal cortex, and the functional and structural dynamics of these networks differ as a function of age and performance level. Distinctive brain activation patterns have also been shown for children with dyscalculia, a specific learning disability in the domain of mathematics. Although our knowledge about the developmental brain dynamics of number and arithmetic has greatly improved over the past years, many questions about the interaction and the causal involvement of the abovementioned functional brain networks remain. This review provides a broad and critical overview of the known developmental processes and what is yet to be discovered.

Communicating personalized risks from COVID-19: guidelines from an empirical study

As increasing amounts of data accumulate on the effects of the novel coronavirus SARS-CoV-2 and the risk factors that lead to poor outcomes, it is possible to produce personalized estimates of the risks faced by groups of people with different characteristics. The challenge of how to communicate these then becomes apparent. Based on empirical work (total n = 5520, UK) supported by in-person interviews with the public and physicians, we make recommendations on the presentation of such information. These include: using predominantly percentages when communicating the absolute risk, but also providing, for balance, a format which conveys a contrasting (higher) perception of risk (expected frequency out of 10 000); using a visual linear scale cut at an appropriate point to illustrate the maximum risk, explained through an illustrative 'persona' who might face that highest level of risk; and providing context to the absolute risk through presenting a range of other 'personas' illustrating people who would face risks of a wide range of different levels. These 'personas' should have their major risk factors (age, existing health conditions) described. By contrast, giving people absolute likelihoods of other risks they face in an attempt to add context was considered less helpful. We note that observed effect sizes generally were small. However, even small effects are meaningful and relevant when scaled up to population levels.

First and Second Graders Successfully Reason About Ratios With Both Dot Arrays and Arabic Numerals

Children struggle with exact, symbolic ratio reasoning, but prior research demonstrates children show surprising intuition when making approximate, nonsymbolic ratio judgments. In the current experiment, eighty-five 6- to 8-year-old children made approximate ratio judgments with dot arrays and numerals. Children were adept at approximate ratio reasoning in both formats and improved with age. Children who engaged in the nonsymbolic task first performed better on the symbolic task compared to children tested in the reverse order, suggesting that nonsymbolic ratio reasoning may function as a scaffold for symbolic ratio reasoning. Nonsymbolic ratio reasoning mediated the relation between children's numerosity comparison performance and symbolic mathematics performance in the domain of probabilities, but numerosity comparison performance explained significant unique variance in general numeration skills.

The numerator bias exists in millions of real-world comparisons

Fractions are crucial, from math and science education to daily activities, but they are hard. A puzzling aspect of fractions is that people over-rely on the numerator when comparing a pair of fractions. Previous work has considered this numerator bias mostly as a reasoning mishap. Still, in a vast amount of pairwise comparisons, across many real-world domains, not just education textbooks, we report a high prior probability that the larger fraction has the larger numerator, and, for a relevant case, we provide formal arguments why. The existence of such a regularity suggests that the numerator bias may reflect a rational adaptation that detects and exploits likely events. In a pair of visual-proportion tasks (discrete and continuous fractions), we confirm that the numerator bias in participants adapts to experimented regularities. Even though weak education and math abilities play a role, adaptation to informative priors outside the classroom poses a challenge to educators, learners, and decision-makers.

Approximate arithmetic training does not improve symbolic math in third and fourth grade children

BACKGROUND

Prior studies reported that practice playing an approximate arithmetic game improved symbolic math performance relative to active control groups in adults and preschool children (e.g. Park & Brannon, 2013, 2014; Park et al., 2016; Szkudlarek & Brannon, 2018). However, Szkudlarek, Park and Brannon (2021) recently failed to replicate those findings in adults. Here we test whether approximate arithmetic training yields benefits in elementary school children who have intermediate knowledge of arithmetic.

METHOD

We conducted a randomized controlled trial with a pre and post-test design to compare the effects of approximate arithmetic training and visuo-spatial working memory training on standardized math performance in third and fourth grade children.

RESULTS

We found that approximate arithmetic training did not yield any significant gains on standardized measures of symbolic math performance.

CONCLUSION

A Bayesian analysis supports the conclusion that approximate arithmetic provides no benefits for symbolic math performance.

Effects of Action Video Game Play on Arithmetic Performance in Adults

This experimental study investigated the state (short-term) effects of action video game (AVG) training on arithmetic performance and their persistence over time. In addition, it examined group differences between experienced and novice AVGers. Twenty-nine college students without a prior AVG experience were randomly assigned to one of the two training groups: AVG and non-AVG. After 40 minutes of video game training, the arithmetic problem-solving speed and accuracy of non-AVG group increased, while the AVG group's arithmetic performance decreased, thus suggesting a possibility of state effects of a non-AVG training on arithmetic performance. The state effects did not persist over time; on a delayed posttest, both groups' arithmetic performance was similar to their pretraining scores. In addition, there were nonsignificant differences in arithmetic performance between experienced and novice AVGers. Implications for investigating the game mechanics and transfer mechanism between the game and transfer task are discussed.

Failure to replicate the benefit of approximate arithmetic training for symbolic arithmetic fluency in adults

Previous research reported that college students’ symbolic addition and subtraction fluency improved after training with non-symbolic, approximate addition and subtraction. These findings were widely interpreted as strong support for the hypothesis that the Approximate Number System (ANS) plays a causal role in symbolic mathematics, and that this relation holds into adulthood. Here we report four experiments that fail to find evidence for this causal relation. Experiment 1 examined whether the approximate arithmetic training effect exists within a shorter training period than originally reported (2 vs 6 days of training). Experiment 2 attempted to replicate and compare the approximate arithmetic training effect to a control training condition matched in working memory load. Experiments 3 and 4 replicated the original approximate arithmetic training experiments with a larger sample size. Across all four experiments (N = 318) approximate arithmetic training was no more effective at improving the arithmetic fluency of adults than training with control tasks. Results call into question any causal relationship between approximate, non-symbolic arithmetic and precise symbolic arithmetic.

Examining aging and numerosity using an integrated diffusion model

Two experiments are presented that use tasks common in research in numerical cognition with young adults and older adults as subjects. In these tasks, one or two arrays of dots are displayed, and subjects decide whether there are more or fewer dots of one kind than another. Results show that older adults, relative to young adults, tend to rely more on the perceptual feature, area, in making numerosity judgments when area is correlated with numerosity. Also, convex hull unexpectedly shows different effects depending on the task (being either correlated with numerosity or anticorrelated). Accuracy and response time (RT) data are interpreted with the integration of the diffusion decision model with models for the representation of numerosity. One model assumes that the representation of the difference depends on the difference between the numerosities and that standard deviations (SDs) increase linearly with numerosity, and the other model assumes a log representation with constant SDs. The representational models have coefficients that are applied to differences between two numerosities to produce drift rates and SDs in drift rates in the decision process. The two tasks produce qualitatively different patterns of RTs: One model fits results from one task, but the results are mixed for the other task. The effects of age on model parameters show a modest decrease in evidence driving the decision process, an increase in the duration of processes outside the decision process (nondecision time), and an increase in the amount of evidence needed to make a decision (boundary separation). (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Effects of Visual Training of Approximate Number Sense on Auditory Number Sense and School Math Ability

Research with children and adults suggests that people's math performance is predicted by individual differences in an evolutionarily ancient ability to estimate and compare numerical quantities without counting (the approximate number system or ANS). However, previous work has almost exclusively used visual stimuli to measure ANS precision, leaving open the possibility that the observed link might be driven by aspects of visuospatial competence, rather than the amodal ANS. We addressed this possibility in an ANS training study. Sixty-eight 6-year-old children participated in a 5-week study that either trained their visual ANS ability or their phonological awareness (an active control group). Immediately before and after training, we assessed children's visual and auditory ANS precision, as well as their symbolic math ability and phonological awareness. We found that, prior to training, children's precision in a visual ANS task related to their math performance - replicating recent studies. Importantly, precision in an auditory ANS task also related to math performance. Furthermore, we found that children who completed visual ANS training showed greater improvements in auditory ANS precision than children who completed phonological awareness training. Finally, children in the ANS training group showed significant improvements in math ability but not phonological awareness. These results suggest that the link between ANS precision and school math ability goes beyond visuospatial abilities and that the modality-independent ANS is causally linked to math ability in early childhood.

Preschoolers exhibit greater on-task behavior following physically active lessons on the approximate number system

OBJECTIVE

To determine how the dual-task nature of incorporating physical activity with instructional activities immediately impacts acuity of the approximate number system and on-task behavior in preschoolers.

METHODS

Using a randomized within-participants repeated-measures crossover design, 51 children completed an approximate number system task before and after either 20-min of physically active instruction corresponding to 38% heart rate reserve (HRR; light-to-moderate intensity) or conventional sedentary instruction at corresponding to 21% HRR (very light intensity).

RESULTS

Findings revealed that preschool-aged children exhibited similar learning and greater on-task behavior following a single bout of physically active instruction relative to conventional sedentary instruction. Overall, preschoolers accrued 931.3 ± 8.2 more steps and an additional 9 minutes at or above light-intensity activity during the physically active instruction.

CONCLUSION

Accordingly, these findings suggest that the dual-task nature of physically active learning does not compromise learning, reduces the need for redirecting off-task behavior, and ultimately allows children to avoid sedentary behavior in educational contexts.

Strengthening spatial reasoning: elucidating the attentional and neural mechanisms associated with mental rotation skill development

Spatial reasoning is a critical skill in many everyday tasks and in science, technology, engineering, and mathematics disciplines. The current study examined how training on mental rotation (a spatial reasoning task) impacts the completeness of an encoded representation and the ability to rotate the representation. We used a multisession, multimethod design with an active control group to determine how mental rotation ability impacts performance for a trained stimulus category and an untrained stimulus category. Participants in the experimental group (n = 18) showed greater improvement than the active control group (n = 18) on the mental rotation tasks. The number of saccades between objects decreased and saccade amplitude increased after training, suggesting that participants in the experimental group encoded more of the object and possibly had more complete mental representations after training. Functional magnetic resonance imaging data revealed distinct neural activation associated with mental rotation, notably in the right motor cortex and right lateral occipital cortex. These brain areas are often associated with rotation and encoding complete representations, respectively. Furthermore, logistic regression revealed that activation in these brain regions during the post-training scan significantly predicted training group assignment. Overall, the current study suggests that effective mental rotation training protocols should aim to improve the encoding and manipulation of mental representations.

Modeling the interaction of numerosity and perceptual variables with the diffusion model

Ratcliff and McKoon (2018) proposed integrated diffusion models for numerosity judgments in which a numerosity representation provides evidence used to drive the decision process. We extend this modeling framework to examine the interaction of non-numeric perceptual variables with numerosity by assuming that drift rate and non-decision time are functions of those variables. Four experiments were conducted with two different types of stimuli: a single array of intermingled blue and yellow dots in which both numerosity and dot area vary over trials and two side-by-side arrays of dots in which numerosity, dot area, and convex hull vary over trials. The tasks were to decide whether there were more blue or yellow dots (two experiments), more dots on which side, or which dots have a larger total area. Development of models started from the principled models in Ratcliff and McKoon (2018) and became somewhat ad hoc as we attempted to capture unexpected patterns induced by the conflict between numerosity and perceptual variables. In the three tasks involving numerosity judgments, the effects of the non-numeric variables were moderated by the number of dots. Under a high conflict, judgments were dominated by perceptual variables and produced an unexpected shift in the leading edge of the reaction time (RT) distributions. Although the resulting models were able to predict most of the accuracy and RT patterns, the models were not able to completely capture this shift in the RT distributions. However, when subjects judged area, numerosity affected perceptual judgments but there was no leading edge effect. Based on the results, it appears that the integrated diffusion models provide an effective framework to study the role of numerical and perceptual variables in numerosity tasks and their context-dependency.

On the relationship between mathematics and visuospatial processing in Turner syndrome

A common neurocognitive phenotype of Turner syndrome (TS) includes coincident deficits in math and visuospatial reasoning while overall IQ remains intact. However, research has highlighted disparities in the relationship between these properties in women with TS, suggesting that not all visuospatial domains are equally related to mathematics in this group. Here, we present findings from a longitudinal investigation of visuospatial processing and its relationship to math performance in adolescent girls with TS and age-matched healthy controls. Participants completed a standardized battery of math and visuospatial tests once a year for 4 years. Linear mixed effects modeling was used to examine the relationship between mathematics and each visuospatial domain over time. Our results indicate that math performance was related to visual tracking, visual-motor coordination, and figure-ground processing. Such visuospatial domains appear to be uniquely affected by TS and could contribute to their deficits in math performance. Furthermore, differences in math and visuospatial test performance between girls with TS and healthy controls remain stable over time. Our results have important implications for the role of visuospatial processing in early math performance and may inform the development of effective interventions aimed at improving math education in children with TS.

Training nonsymbolic proportional reasoning in children and its effects on their symbolic math abilities

Our understanding of proportions can be both symbolic, as when doing calculations in school mathematics, or intuitive, as when folding a bed sheet in half. While an understanding of symbolic proportions is crucial for school mathematics, the cognitive foundations of this ability remain unclear. Here we implemented a computerized training game to test a causal link from intuitive (nonsymbolic) to symbolic proportional reasoning and other math abilities in 4th grade children. An experimental group was trained in nonsymbolic proportional reasoning (PR) with continuous extents, and an active control group was trained on a remarkably similar nonsymbolic magnitude comparison. We found that the experimental group improved at nonsymbolic PR across training sessions, showed near transfer to a paper-and-pencil nonsymbolic PR test, transfer to symbolic proportions, and far transfer to geometry. The active control group showed only a predicted far transfer to geometry. In a second experiment, these results were replicated with an independent cohort of children. Overall this study extends previous correlational evidence, suggesting a functional link between nonsymbolic PR on one hand and symbolic PR and geometry on the other.

Anodal High-definition Transcranial Direct Current Stimulation over the Posterior Parietal Cortex Modulates Approximate Mental Arithmetic

The representation and processing of numerosity is a crucial cognitive capacity. Converging evidence points to the posterior parietal cortex (PPC) as primary "number" region. However, the exact role of the left and right PPC for different types of numerical and arithmetic tasks remains controversial. In this study, we used high-definition transcranial direct current stimulation (HD-tDCS) to further investigate the causal involvement of the PPC during approximative, nonsymbolic mental arithmetic. Eighteen healthy participants received three sessions of anodal HD-tDCS at 1-week intervals in counterbalanced order: left PPC, right PPC, and sham stimulation. Results showed an improved performance during online parietal HD-tDCS (vs. sham) for subtraction problems. Specifically, the general tendency to underestimate the results of subtraction problems (i.e., the "operational momentum effect") was reduced during online parietal HD-tDCS. There was no difference between left and right stimulation. This study thus provides new evidence for a causal involvement of the left and right PPC for approximate nonsymbolic arithmetic and advances the promising use of noninvasive brain stimulation in increasing cognitive functions.

Short-term numerosity training promotes symbolic arithmetic in children with developmental dyscalculia: The mediating role of visual form perception

Studies have shown that numerosity-based arithmetic training can promote arithmetic learning in typically developing children as well as children with developmental dyscalculia (DD), but the cognitive mechanism underlying this training effect remains unclear. The main aim of the current study was to examine the role of visual form perception in arithmetic improvement through an 8-day numerosity training for DD children. Eighty DD children were selected from four Chinese primary schools. They were randomly divided into the intervention and control groups. The intervention group received training on an apple-collecting game, whereas the control group received an English dictation task. Children's cognitive and arithmetic performances were assessed before and after training. The results showed that the intervention group showed a significant improvement in arithmetic performance, approximate number system (ANS) acuity, and visual form perception, but not in spatial processing and sentence comprehension. The control group showed no significant improvement in any cognitive ability. Mediation analysis further showed that training-related improvement in arithmetic performance was fully mediated by the improvement in visual form perception. The results suggest that short-term numerosity training enhances the arithmetic performance of DD children by improving their visual form perception.

Challenging the neurobiological link between number sense and symbolic numerical abilities

A significant body of research links individual differences in symbolic numerical abilities, such as arithmetic, to number sense, the neurobiological system used to approximate and manipulate quantities without language or symbols. However, recent findings from cognitive neuroscience challenge this influential theory. Our current review presents an overview of evidence for the number sense account of symbolic numerical abilities and then reviews recent studies that challenge this account, organized around the following four assertions. (1) There is no number sense as traditionally conceived. (2) Neural substrates of number sense are more widely distributed than common consensus asserts, complicating the neurobiological evidence linking number sense to numerical abilities. (3) The most common measures of number sense are confounded by other cognitive demands, which drive key correlations. (4) Number sense and symbolic number systems (Arabic digits, number words, and so on) rely on distinct neural mechanisms and follow independent developmental trajectories. The review follows each assertion with comments on future directions that may bring resolution to these issues.

Preschoolers and multi-digit numbers: A path to mathematics through the symbols themselves

Numerous studies from developmental psychology have suggested that human symbolic representation of numbers is built upon the evolutionally old capacity for representing quantities that is shared with other species. Substantial research from mathematics education also supports the idea that mathematical concepts are best learned through their corresponding physical representations. We argue for an independent pathway to learning "big" multi-digit symbolic numbers that focuses on the symbol system itself. Across five experiments using both between- and within-subject designs, we asked preschoolers to identify written multi-digit numbers with their spoken names in a two-alternative-choice-test or to indicate the larger quantity between two written numbers. Results showed that preschoolers could reliably map spoken number names to written forms and compare the magnitudes of two written multi-digit numbers. Importantly, these abilities were not related to their non-symbolic representation of quantities. These findings have important implications for numerical cognition, symbolic development, teaching, and education.

Number, time, and space are not singularly represented: Evidence against a common magnitude system beyond early childhood

Our ability to represent temporal, spatial, and numerical information is critical for understanding the world around us. Given the prominence of quantitative representations in the natural world, numerous cognitive, neurobiological, and developmental models have been proposed as a means of describing how we track quantity. One prominent theory posits that time, space, and number are represented by a common magnitude system, or a common neural locus (i.e., Bonn & Cantlon in Cognitive Neuropsychology, 29(1/2), 149-173, 2012; Cantlon, Platt, & Brannon in Trends in Cognitive Sciences, 13(2), 83-91, 2009; Meck & Church in Animal Behavior Processes, 9(3), 320, 1983; Walsh in Trends in Cognitive Sciences, 7(11), 483-488, 2003). Despite numerous similarities in representations of time, space, and number, an increasing body of literature reveals striking dissociations in how each quantity is processed, particularly later in development. These findings have led many researchers to consider the possibility that separate systems may be responsible for processing each quantity. This review will analyze evidence in favor of a common magnitude system, particularly in infancy, which will be tempered by counter evidence, the majority of which comes from experiments with children and adult participants. After reviewing the current data, we argue that although the common magnitude system may account for quantity representations in infancy, the data do not provide support for this system throughout the life span. We also identify future directions for the field and discuss the likelihood of the developmental divergence model of quantity representation, like that of Newcombe (Ecological Psychology, 2, 147-157, 2014), as a more plausible account of quantity development.

The Acuity and Manipulability of the ANS Have Separable Influences on Preschoolers' Symbolic Math Achievement

The approximate number system (ANS) is widely considered to be a foundation for the acquisition of uniquely human symbolic numerical capabilities. However, the mechanism by which the ANS may support symbolic number representations and mathematical thought remains poorly understood. In the present study, we investigated two pathways by which the ANS may influence early math abilities: variability in the acuity of the ANS representations, and children's' ability to manipulate ANS representations. We assessed the relation between 4-year-old children's performance on a non-symbolic numerical comparison task, a non-symbolic approximate addition task, and a standardized symbolic math assessment. Our results indicate that ANS acuity and ANS manipulability each contribute unique variance to preschooler's early math achievement, and this result holds after controlling for both IQ and executive functions. These findings suggest that there are multiple routes by which the ANS influences math achievement. Therefore, interventions that target both the precision and manipulability of the ANS may prove to be more beneficial for improving symbolic math skills compared to interventions that target only one of these factors.

"Approximate number system" training: A perceptual learning approach

Recent research suggests that humans perceive quantity using a non-symbolic "number sense." This sense is then thought to provide a foundation for understanding symbolic numbers in formal education. Given this link, there has been interest in the extent to which the approximate number system (ANS) can be improved via dedicated training, as this could provide a route to improving performance in symbolic mathematics. However, current evidence regarding the trainability of the ANS comes largely from studies that have used short training durations, leaving open the question of whether improvements occur over a longer time span. To address this limitation, we utilized a perceptual learning approach to investigate the extent to which long-term (8,000+ trials) training modifies the ANS. Consistent with the general methodological approach common in the domain of perceptual learning (where learning specificity is commonly observed), we also examined whether ANS training generalizes to: (a) untrained locations in the visual field; (b) an enumeration task; (c) a higher-level ratio comparison task; and (d) arithmetic ability. In contrast to previous short-term training studies showing that ANS learning quickly asymptotes, our long-term training approach revealed that performance continued to improve even after thousands of trials. We further found that the training generalized to untrained visual locations. At post-test there was non-significant evidence for generalization to a low-level enumeration task, but not to our high-level tasks, including ratio comparison, multi-object tracking, and arithmetic performance. These results demonstrate the potential utility of long-term psychophysical training, but also suggest that ANS training alone (even long-duration training) may be insufficient to modify higher-level math skills.

Shared and distinct neural circuitry for nonsymbolic and symbolic double-digit addition

Symbolic arithmetic is a complex, uniquely human ability that is acquired through direct instruction. In contrast, the capacity to mentally add and subtract nonsymbolic quantities such as dot arrays emerges without instruction and can be seen in human infants and nonhuman animals. One possibility is that the mental manipulation of nonsymbolic arrays provides a critical scaffold for developing symbolic arithmetic abilities. To explore this hypothesis, we examined whether there is a shared neural basis for nonsymbolic and symbolic double-digit addition. In parallel, we asked whether there are brain regions that are associated with nonsymbolic and symbolic addition independently. First, relative to visually matched control tasks, we found that both nonsymbolic and symbolic addition elicited greater neural signal in the bilateral intraparietal sulcus (IPS), bilateral inferior temporal gyrus, and the right superior parietal lobule. Subsequent representational similarity analyses revealed that the neural similarity between nonsymbolic and symbolic addition was stronger relative to the similarity between each addition condition and its visually matched control task, but only in the bilateral IPS. These findings suggest that the IPS is involved in arithmetic calculation independent of stimulus format.

Using Hierarchical Linear Models to Examine Approximate Number System Acuity: The Role of Trial-Level and Participant-Level Characteristics

The ability to intuitively and quickly compare the number of items in collections without counting is thought to rely on the Approximate Number System (ANS). To assess individual differences in the precision of peoples' ANS representations, researchers often use non-symbolic number comparison tasks in which participants quickly choose the numerically larger of two arrays of dots. However, some researchers debate whether this task actually measures the ability to discriminate approximate numbers or instead measures the ability to discriminate other continuous magnitude dimensions that are often confounded with number (e.g., the total surface area of the dots or the convex hull of the dot arrays). In this study, we used hierarchical linear models (HLMs) to predict 132 adults' accuracy on each trial of a non-symbolic number comparison task from a comprehensive set of trial-level characteristics (including numerosity ratio, surface area, convex hull, and temporal and spatial variations in presentation format) and participant-level controls (including cognitive abilities such as visual-short term memory, working memory, and math ability) in order to gain a more nuanced understanding of how individuals complete this task. Our results indicate that certain trial-level characteristics of the dot arrays contribute to our ability to compare numerosities, yet numerosity ratio, the critical marker of the ANS, remains a highly significant predictor of accuracy above and beyond trial-level characteristics and across individuals with varying levels of math ability and domain-general cognitive abilities.

Testing the Efficacy of Training Basic Numerical Cognition and Transfer Effects to Improvement in Children's Math Ability

The goals of the present study were to test whether (and which) basic numerical abilities can be improved with training and whether training effects transfer to improvement in children's math achievement. The literature is mixed with evidence that does or does not substantiate the efficacy of training basic numerical ability. In the present study, we developed a child-friendly software named "123 Bakery" which includes four training modules; non-symbolic numerosity comparison, non-symbolic numerosity estimation, approximate arithmetic, and symbol-to-numerosity mapping. Fifty-six first graders were randomly assigned to either the training or control group. The training group participated in 6 weeks of training (5 times a week, 30 minutes per day). All participants underwent pre- and post-training assessment of their basic numerical processing ability (including numerosity discrimination acuity, symbolic/non-symbolic magnitude estimation, approximate arithmetic, and symbol-to-numerosity mapping), overall math achievement and intelligence, 6 weeks apart. The acuity for numerosity discrimination (approximate number sense acuity; hereafter ANS acuity) significantly improved after training, but this training effect did not transfer to improvement in symbolic, exact calculation, or any other math ability. We conclude that basic numerical cognition training leads to improvement in ANS acuity, but whether this effect transfers to symbolic math ability remains to be further tested.

Cognitive and Neural Effects of a Brief Nonsymbolic Approximate Arithmetic Training in Healthy First Grade Children

Recent studies with children and adults have shown that the abilities of the Approximate Number System (ANS), which operates from early infancy and allows estimating the number of elements in a set without symbols, are trainable and transferable to symbolic arithmetic abilities. Here we investigated the brain correlates of these training effects, which are currently unknown. We trained two Groups of first grade children, one in performing nonsymbolic additions with dot arrays (Addition-Group) and another one in performing color comparisons of the same arrays (Color-Group). The training program was computerized, throughout seven sessions and had a pretest-posttest design. To evaluate cognitive gains, we measured math skills before and after the training. To measure the brain changes, we used electroencephalogram (EEG) recordings in the first and the last training sessions. We explored the changes in N1 and P2p, which are two electrophysiological components sensitive to nonsymbolic numeric computations. A passive Control-Group receiving no intervention also had their math skills evaluated. We found that the two training Groups had similarly gain in math skills, suggesting no specific transfer of the nonsymbolic addition training to math skills at the behavioral level. In contrast, at the brain level, we found that only in the Addition-Group the P2p amplitude significantly increased across sessions. Notably, the gain in P2p amplitude positively correlated with the gain in math abilities. Together, our results showed that first graders rapidly gained in math skills by different interventions. However, number-related brain networks seem to be particularly sensitive to nonsymbolic arithmetic training.

Improving number abilities in low achieving preschoolers: Symbolic versus non-symbolic training programs

BACKGROUND

Although previous correlational studies have shown that both symbolic and non-symbolic abilities relate to mathematical abilities, correlational studies cannot show the cause and effect of these abilities for mathematical success.

AIMS

The current study examined the effect of a non-symbolic training program, called PLUS and a symbolic training program, called DIGIT, to provide further insight into the causal nature of domain specific factors that contribute to mathematical abilities.

METHODS

and Procedures: Forty-nine preschool children who had low mathematical abilities were recruited and randomly allocated to the DIGIT and PLUS training programs. Performance on a number of mathematical tasks was compared to 20 preschoolers with no mathematical difficulties.

OUTCOMES AND RESULTS

Performance in both training programs improved on the Test of Early Mathematical Abilities as well as on a non-symbolic Approximate Number Sense task, counting tasks, and digit recognition tasks, immediately after five weeks of training and this improvement remained six months later.

CONCLUSIONS AND IMPLICATIONS

This study provides further evidence that symbolic and non-symbolic abilities bi-directionally impact on each other and that ordinality knowledge is an important factor of mathematical development.

Approximate Arithmetic Training Improves Informal Math Performance in Low Achieving Preschoolers

Recent studies suggest that practice with approximate and non-symbolic arithmetic problems improves the math performance of adults, school aged children, and preschoolers. However, the relative effectiveness of approximate arithmetic training compared to available educational games, and the type of math skills that approximate arithmetic targets are unknown. The present study was designed to (1) compare the effectiveness of approximate arithmetic training to two commercially available numeral and letter identification tablet applications and (2) to examine the specific type of math skills that benefit from approximate arithmetic training. Preschool children (n = 158) were pseudo-randomly assigned to one of three conditions: approximate arithmetic, letter identification, or numeral identification. All children were trained for 10 short sessions and given pre and post tests of informal and formal math, executive function, short term memory, vocabulary, alphabet knowledge, and number word knowledge. We found a significant interaction between initial math performance and training condition, such that children with low pretest math performance benefited from approximate arithmetic training, and children with high pretest math performance benefited from symbol identification training. This effect was restricted to informal, and not formal, math problems. There were also effects of gender, socio-economic status, and age on post-test informal math score after intervention. A median split on pretest math ability indicated that children in the low half of math scores in the approximate arithmetic training condition performed significantly better than children in the letter identification training condition on post-test informal math problems when controlling for pretest, age, gender, and socio-economic status. Our results support the conclusion that approximate arithmetic training may be especially effective for children with low math skills, and that approximate arithmetic training improves early informal, but not formal, math skills.

Innate or Acquired? - Disentangling Number Sense and Early Number Competencies

The clinical profile termed developmental dyscalculia (DD) is a fundamental disability affecting children already prior to arithmetic schooling, but the formal diagnosis is often only made during school years. The manifold associated deficits depend on age, education, developmental stage, and task requirements. Despite a large body of studies, the underlying mechanisms remain dubious. Conflicting findings have stimulated opposing theories, each presenting enough empirical support to remain a possible alternative. A so far unresolved question concerns the debate whether a putative innate number sense is required for successful arithmetic achievement as opposed to a pure reliance on domain-general cognitive factors. Here, we outline that the controversy arises due to ambiguous conceptualizations of the number sense. It is common practice to use early number competence as a proxy for innate magnitude processing, even though it requires knowledge of the number system. Therefore, such findings reflect the degree to which quantity is successfully transferred into symbols rather than informing about quantity representation per se. To solve this issue, we propose a three-factor account and incorporate it into the partly overlapping suggestions in the literature regarding the etiology of different DD profiles. The proposed view on DD is especially beneficial because it is applicable to more complex theories identifying a conglomerate of deficits as underlying cause of DD.