How To Improve Mathematical Skills Pdf

Mathematical Skill

Mathematical skills are conceptualized as a separate area that includes verbal components (number knowledge, counting, computation, and reasoning) and nonverbal components (math notation, reasoning in time and space, and computation).

From: Encyclopedia of Applied Psychology , 2004

Systems Neuroscience of Mathematical Cognition and Learning

Teresa Iuculano , ... Vinod Menon , in Heterogeneity of Function in Numerical Cognition, 2018

Introduction

Mathematical skill acquisition is hierarchical in nature, and each iteration of increased proficiency builds on knowledge of a lower-level primitive. For example, learning to solve arithmetical operations such as "3 + 4" requires first an understanding of what numbers mean and represent (e.g., the symbol "3" refers to the quantity of three items, which derives from the ability to attend to discrete items in the environment). Thus, all forms of mathematical cognition, from basic to complex, require proficiency in a fundamental system of "number sense," including elemental properties of numbers, principles of cardinality, numerosity as abstract representations of sets, and the axiomatic rules by which numerical quantity is manipulated (Dantzig, 1930; Dehaene, 1997). The brain systems supporting mathematical cognition parallel these behavioral constructs and function as a set of (partly) hierarchically organized and dynamically interacting systems. Each brain system subserves specific perceptual and cognitive processes, including visual and auditory processing, quantity processing, working memory, declarative memory, attention, and cognitive control (Fig. 15.1). Importantly, the topology of brain systems engaged during mathematical cognition varies considerably not only across individuals but also with learning and development, as individuals gain proficiency in mathematical skills.

The basic building blocks of mathematical cognition, which are learned early in a child's development, include understanding numerical magnitude and the ability to manipulate symbolic and nonsymbolic quantity ("number sense"). Human imaging studies using electroencephalography and functional magnetic resonance imaging (fMRI) have shown that these building blocks are subserved by specific and interacting brain systems (Fig. 15.1). First, visual and auditory association cortices decode the visual form (e.g., visual shape of the symbol "3") or phonological feature (e.g., the word "three") of numerical stimuli. Next, the fusiform gyrus (FG) in the higher-level visual cortex (i.e., ventral temporal–occipital cortex) plays an important role in visual object recognition and in forming relevant representations of symbolic stimuli. As the child begins to learn the use of orthographic symbols (i.e., written symbols and number words), new representations for these visual stimuli are developed in the FG (Allison, McCarthy, Nobre, Puce, & Belger, 1994; Ansari, 2008; Dehaene, Molko, Cohen, & Wilson, 2004; Park, Hebrank, Polk, & Park, 2012; Shum et al., 2013). Simultaneously, the superior temporal sulcus (STS) and middle temporal gyrus (MTG) are involved in forming representations of number words (Thompson, Abbott, Wheaton, Syngeniotis, & Puce, 2004). The parietal attention system helps build semantic representations of quantity (Ansari, 2008) from multiple low-level primitives (Box 15.1), including the ability to attend to and individuate individual objects in space, and the ability to perceive the numerosity of such objects (i.e., "oneness," "threeness"). These low-level primitives are processed in the posterior parietal cortex (PPC), particularly in its intraparietal sulcus (IPS) subdivision and are established many years before a child learns to process numerical symbols and number words (Hyde, Boas, Blair, & Carey, 2010). Taken together, the perceptual and cognitive brain systems that underlie "number sense" come on line early in development, providing a critical foundation for the acquisition of later mathematical skills.

Box 15.1

Glossary

Systems Neuroscience: refers to a subdiscipline of neuroscience and systems biology, which studies the function of brain circuits and systems. It concerns the study of brain regions and their function during a task and their interaction with other—proximal and distal—brain regions and the derived neural systems/networks.

Primitives: refer to a core set of cognitive capacities—pertinent to the intended domain—that can be utilized and integrated to develop higher-order cognitive capacities.

Schema of knowledge: refers to a cognitive construct that helps to organize categories of information and the relationship between them.

Hub: refers to a central structure within a network.

Distance effect: refers to the measurable behavior—assessed by accuracy and reaction time—which instantiates that it is harder to discriminate two sets of items as their distance decreases. It is more difficult to compare 8 versus 9—i.e., distance of 1 unit—than it is to compare 8 versus 5—i.e., distance of 3 units.

Neural distance effect: refers to the neural correlate of the distance effect, such that functional activation in dedicated brain regions is greater when the numerosities to be compared have a smaller distance (i.e., 8 vs. 9), compared to a larger distance (i.e., 8 vs. 5).

Proficiency in mathematics includes learning not only how to perceive and process (i.e., represent) numerical information but also to manipulate it, which requires engagement of multiple neurocognitive networks in the brain (Fig. 15.1). First, frontoposterior parietal brain circuits underlying working memory processes support the online manipulation of discrete quantities (e.g., individual objects) by creating short-term representations of such quantities and to help solve more complex problems (Metcalfe, Ashkenazi, Rosenberg-Lee, & Menon, 2013). Second, the prefrontal cortex (PFC) helps guide and maintain attention in the service of goal-directed problem-solving and cognitive control. Third, the medial temporal lobe (MTL) memory system, anchored in the hippocampus, plays an important role in long-term memory formation, and consolidation of mathematical concepts (i.e., arithmetical facts) and generalizations beyond individual problem attributes within a broader schema of knowledge (Box 15.1) (Davachi, 2006; Davachi, Mitchell, & Wagner, 2003). Together, these systems provide necessary cognitive scaffolding and play varying roles in mathematical skill acquisition throughout learning and development.

Considerable heterogeneity of mathematical proficiency in both children and adults is well documented, with some individuals demonstrating remarkable abilities (Cowan & Carney, 2006; Cowan & Frith, 2009; Cowan, O'Connor, & Samella, 2003; Iuculano et al., 2014; Pesenti, Seron, Samson, & Duroux, 1999; Pesenti et al., 2001) and others showing marked deficits (Butterworth, 2005, pp. 455–467; Butterworth & Kovas, 2013; Butterworth & Reigosa-Crespo, 2007, pp. 65–81; Butterworth, Varma, & Laurillard, 2011; Iuculano, Tang, Hall, & Butterworth, 2008; Kucian & von Aster, 2015; Landerl, Bevan, & Butterworth, 2004; Rousselle & Noel, 2007). This heterogeneity in mathematical abilities is likely due, in part, to variability in learning and brain plasticity throughout development. Importantly, aberrant plasticity of the brain systems described above (Fig. 15.1) is thought to underlie the pathogenesis of learning disabilities specific to mathematics or otherwise referred to as mathematical learning disabilities (MLD) (DSM-V Association, 2013). Thus, the study of typical and atypical neurocognitive development and plasticity provides a unique opportunity to help characterize sources of heterogeneity in mathematical learning.

This chapter synthesizes the extant literature on functional brain circuits underlying "number sense" and scaffolding of mathematical cognition and learning in children and adults, with a focus on sources of heterogeneity in typical and atypical development. We take a systems neuroscience approach (Box 15.1) to elucidate sources of heterogeneity arising from multiple distributed neural circuits critical for number form identification, magnitude and quantity representations, working memory, declarative memory, attention, and cognitive control (Fig. 15.1) (Arsalidou & Taylor, 2011; Fias, Menon, & Szucs, 2013; Qin et al., 2014). We conclude by highlighting directions for future research.

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Foundations of Children's Numerical and Mathematical Skills

Ian M. Lyons , Daniel Ansari , in Advances in Child Development and Behavior, 2015

Abstract

Numerical and mathematical skills are critical predictors of academic success. The last three decades have seen a substantial growth in our understanding of how the human mind and brain represent and process numbers. In particular, research has shown that we share with animals the ability to represent numerical magnitude (the total number of items in a set) and that preverbal infants can process numerical magnitude. Further research has shown that similar processing signatures characterize numerical magnitude processing across species and developmental time. These findings suggest that an approximate system for nonsymbolic (e.g., dot arrays) numerical magnitude representation serves as the basis for the acquisition of cultural, symbolic (e.g., Arabic numerals) representations of numerical magnitude. This chapter explores this hypothesis by reviewing studies that have examined the relation between individual differences in nonsymbolic numerical magnitude processing and symbolic math abilities (e.g., arithmetic). Furthermore, we examine the extent to which the available literature provides strong evidence for a link between symbolic and nonsymbolic representations of numerical magnitude at the behavioral and neural levels of analysis. We conclude that claims that symbolic number abilities are grounded in the approximate system for the nonsymbolic representation of numerical magnitude are not strongly supported by the available evidence. Alternative models and future research directions are discussed.

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The Interplay Between Learning Arithmetic and Learning to Read: Insights From Developmental Cognitive Neuroscience

Jérôme Prado , in Heterogeneity of Function in Numerical Cognition, 2018

Abstract

The acquisition of mathematical skills is increasingly thought to rely on the co-option of nonlinguistic capacities. Yet, mathematical and linguistic skills appear to be associated across development. For example, scores on tests of arithmetic and reading correlate in children. Additionally, arithmetic and reading disabilities have a comorbidity of about 40%. Here I use the lens of developmental cognitive neuroscience to evaluate the most popular explanation of the link between arithmetic learning and reading acquisition in children: the idea that the phonological mechanisms of language might contribute to the development of arithmetic skills in children. I will argue that evidence for this hypothesis is limited in the literature. I will then turn to the more recent proposal that the relationship between arithmetic and reading may be mediated by a common reliance on procedural memory. Although this hypothesis requires further testing, it is a promising explanation of the interplay between learning to read and learning arithmetic in children.

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The Mathematical Brain Across the Lifespan

K. Vanbinst , B. De Smedt , in Progress in Brain Research, 2016

6 Conclusions and Future Directions

Well-developed mathematical skills are crucial to life success in modern Western society, and the ability to acquire and retrieve arithmetic facts is a major building block for the successful development of mathematical skills ( Kilpatrick et al., 2001). The goal of this chapter was to review the available evidence on the domain-specific and domain-general neurocognitive determinants of individual differences in children's arithmetic development, other than the factor nonsymbolic numerical magnitude processing, which might have been overemphasized as the primary determinant of individual differences in mathematics and of dyscalculia. We focused on the contributions of symbolic numerical magnitude processing, working memory, and phonological processing, as these cognitive determinants have received the most attention so far and their roles in arithmetic can be predicted against the background of existing brain imaging data.

It is evident from the studies reviewed earlier that symbolic numerical magnitude processing is a major determinant of individual differences in arithmetic, even across primary school. Working memory, particularly the central executive, plays a role in learning arithmetic, but its influence appears to be dependent on the learning stage and the experience of children, which need to be taken into account in future research on the associations between working memory and mathematics. The available evidence on phonological processing, although more limited in nature, suggests that it plays a more subtle role in children's acquisition of arithmetic facts.

The earlier reviewed studies also highlight that it is crucial to investigate the roles of domain-specific and domain-general cognitive factors in concert. This requires longitudinal studies in which each of these factors is investigated, in order to understand its relative contribution as well as the mediating and moderating roles of these cognitive factors in children's arithmetic development. It is highly likely that different pathways contribute to individual differences in children arithmetic (see LeFevre et al., 2010, for a similar rationale), which are to be determined in future work. Importantly, such research should carefully characterize the dependent variable, ie, the arithmetic strategies, skill level, and expertise. The same accounts for the independent variables under investigation. Such characterization is needed in order to determine potential pathways as well as time points during which a particular cognitive factor may exert its largest effects. On a related note, impairments in the earlier reviewed cognitive factors all constitute risk factors for developing deficits in arithmetic, and consequently dyscalculia. As has been argued by Pennington (2006) on the origins of developmental disorders, it is unlikely that one single deficit accounts for the emergence of such a disorder. This requires future studies on dyscalculia to consider the relative contribution of these risk factors.

We would like to emphasize that in addition to the determinants reviewed in this chapter, other cognitive factors have been related to individual differences, which were not discussed into detail. One such factor, that has received some recent interest, is inhibitory control (Cragg and Gilmore, 2014, for a review). For example, it has been suggested that difficulties in arithmetic fact development might be related to difficulties in suppressing irrelevant information during the fact retrieval process (eg, Barrouillet and Lépine, 2005; Geary et al., 2012c; Verguts and Fias, 2005), although not all studies have found consistent associations between inhibitory control and arithmetic fact retrieval (eg, Censabella and Noël, 2008). Another factor, recently proposed by De Visscher and Noël (2014a,b), is an individual's (hyper)sensitivity to interference in memory. Specifically, these authors argued that during the arithmetic fact learning process, the storage of problem–answer associations in long-term memory might be hindered through interference, provoked by feature overlap of the to be learned problem–answer associations or arithmetic facts. These authors showed that individual differences in the sensitivity to this interference are related to individual differences in arithmetic fact retrieval in children and adults (De Visscher and Noël, 2014a,b). Future studies should investigate the relative importance of these domain-general cognitive factors by also including domain-specific factors, such as symbolic numerical magnitude processing in their designs.

Another area in need of exploration is how noncognitive factors interact with the cognitive factors reviewed earlier in their prediction of individual differences in arithmetic. These factors include mathematics anxiety (eg, Ma, 1999; Maloney and Beilock, 2012; Ramirez et al., 2013), parental involvement (Fan, and Chen, 2001, for a meta-analysis), or attitudes toward mathematics (Ma and Kishor, 1997, for a meta-analysis). Mathematics anxiety is particularly relevant in this context, as it might affect children's strategic behavior in arithmetic: children with high math anxiety might feel not confident enough to retrieve arithmetic facts from memory and as a consequence, rely more on procedural strategies (ie, counting and decomposition), which are slower and more error prone and put a high demand on working memory resources (eg, Ashcraft and Krause, 2007).

It is also important to point out that the development of arithmetic does not occur in isolation but is dependent on the educational environment in general and the degree to which a mathematics curriculum emphasizes the acquisition of this skill and the use of specific strategies in particular. For example, cross-cultural studies have highlighted substantial differences between Chinese and US children (Campbell and Xue, 2001; Geary et al., 1996), which might be explained by the extent to which curricular programs place an emphasis on fact retrieval. It therefore would be interesting to explore how determinants of individual differences in arithmetic vary as a function of the type of mathematics education.

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Ordinal Instinct: A Neurocognitive Perspective and Methodological Issues

Orly Rubinsten , in Continuous Issues in Numerical Cognition, 2016

12.1 Scientific knowledge and developments

Numerical cognition is important because mathematical skills are essential for productive functioning in our progressively complex, technological society. In addition, numerical development has been a focus of the continuing theoretical debate concerning the origins of cognition and how it develops throughout life. Approximately 10% of the population has low numeracy skills, a condition called developmental dyscalculia (DD) or mathematical learning disability (MLD) ( Kaufmann et al., 2013; Rubinsten & Henik, 2009; Szűcs & Goswami, 2013). Numerical difficulties result in reduced educational and employment achievements, as well as increased costs to physical and mental health (eg, Parsons & Bynner, 2005; Reyna, Nelson, Han, & Dieckmann, 2009). Some argue that in Western societies, poor numeracy is a more significant handicap than poor literacy (eg, Estrada, Martin-Hryniewicz, Peek, Collins, & Byrd, 2004; Parsons & Bynner, 2005). Hence, mathematical skills may have an impact on social mobility and poverty levels. Despite its manifest importance, paradoxically, and compared to other cognitive abilities such as reading and attention, the neurocognitive development of numerical abilities has been neglected by both clinicians and researchers.

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Developmental Disorders and Interventions

Deborah M. Riby , Melanie A. Porter , in Advances in Child Development and Behavior, 2010

3 Numeracy

Despite frequent anecdotal reports of poor mathematical skills, there are a limited number of studies exploring mathematical ability within the syndrome. Limited evidence suggests that some aspects of mathematical skill are much poorer than predicted by overall intellectual abilities ( Bellugi, Marks, et al., 1988; Bellugi, Sabo, & Vaid, 1988; Howlin et al., 1998; Paterson, Girelli, Butterworth, & Karmiloff-Smith, 2006, Udwin, Davies, & Howlin, 1996). These deficits may relate to the visuospatial and the spatial construction deficits outlined above. Generally, mathematical skills (or quantitative reasoning) are a significant weakness within academic capabilities. Mirroring the pattern seen within other domains of cognition, some components of mathematics are relatively more impaired than others (see O'Hearn & Landau, 2007). For example, a selection of studies involving infants and toddlers suggest that numerical magnitude (number line) and approximate number estimates may be more impaired than number recognition, memory for mathematical facts (addition and multiplication), and verbal encoding of numbers at this early stage of development (Ansari et al., 2003; Krajcsi, Lukács, Igács, Racsmány, & Pléh, 2009, O'Hearn & Landau, 2007, and see O'Hearn & Luna, 2009 for a review). Moreover, individuals with WS appear to experience particular difficulties discriminating spatial arrays of larger numbers as opposed to smaller numbers (Van Herwegen, Ansari, Xu, & Karmiloff-Smith, 2008), perhaps at least partially reflecting (i) the mental capacity of the task, (ii) spatial attention deficits, or (iii) problems with visual tracking. Importantly, when considering a range of numerical and mathematical skills, it is important to remember that these skills will be affected not only by lower-level cognitive deficits including information processing, attention, and spatial skills, but also by higher-level (executive) functions such as working memory, impulsivity, nonverbal reasoning, and problem-solving deficits. It is therefore difficult to separate deficits of mathematical ability from other cognitive abilities or deficits. However, problems with number skills can be widespread and have implications for everyday functioning; specifically, individuals experience great difficulty telling the time and, as adults, the vast majority are unable to manage their own finances (Davies, Udwin, & Howlin, 1998; Udwin, 1990).

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The Mathematical Brain Across the Lifespan

E. Eger , in Progress in Brain Research, 2016

Abstract

The human species has developed complex mathematical skills which likely emerge from a combination of multiple foundational abilities. One of them seems to be a preverbal capacity to extract and manipulate the numerosity of sets of objects which is shared with other species and in humans is thought to be integrated with symbolic knowledge to result in a more abstract representation of numerical concepts. For what concerns the functional neuroanatomy of this capacity, neuropsychology and functional imaging have localized key substrates of numerical processing in parietal and frontal cortex. However, traditional fMRI mapping relying on a simple subtraction approach to compare numerical and nonnumerical conditions is limited to tackle with sufficient precision and detail the issue of the underlying code for number, a question which more easily lends itself to investigation by methods with higher spatial resolution, such as neurophysiology. In recent years, progress has been made through the introduction of approaches sensitive to within-category discrimination in combination with fMRI (adaptation and multivariate pattern recognition), and the present review summarizes what these have revealed so far about the neural coding of individual numbers in the human brain, the format of these representations and parallels between human and monkey neurophysiology findings.

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Neurocognitive Components of Mathematical Skills and Dyscalculia

Wim Fias , in Development of Mathematical Cognition, 2016

Abstract

The search for the cognitive determinants of mathematical skill has a long history. For some time it has been thought that mathematical proficiency is not determined by a single unique underlying cognitive factor but by multiple cognitive components such as memory, spatial processing, or executive function. Yet it remains unclear exactly what these cognitive components are and how it is that they have an impact on mathematical skills. I argue that specific neurocognitive explanatory models of cognitive components promise to increase our understanding of how these components play a role in numerical and mathematical tasks and determine performance. I outline how recent advances in the understanding of the neurocognitive mechanisms of sensory processing, working memory, and executive functions lead to meaningful hypotheses about their functional involvement in mathematical performance. I also touch upon how this might shed light on dyscalculia and its comorbidity with other learning deficits.

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Foreword: Mathematical Cognition, Language, and Culture: Understanding the Links

John Towse , in Language and Culture in Mathematical Cognition, 2018

In a number of ways, analysis of mathematical skills has been less well supported than analysis of reading skills as a topic of psychological study (see Towse, 2015, to see how these two compare as research topics). So it is good to remember that while these two domains differ in important ways and likely diverge at various developmental phases, there are some noteworthy connections and intersections. In an elegant program of work, Göbel shows how and when reading affects and molds number development and when it does not. This complements the more specific focus considered by Shaki and Fischer who reveal subtleties in how reading habits (for many, left to right but, for some, right to left) can shape symbolic number associations. There are not only long-term impacts but also short-term or transient effects that combine into a compelling account of how empirical research can make sense of heterogeneous findings. Donlan also complements the understanding of links between language and number by studying the trajectory of what has come to be known as the approximate number system in children with specific language impairments, further emphasizing the connections between skill domains and the potential for atypical development, to reveal important insights for cognition.

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Connectionism and Learning

T.R. Shultz , in International Encyclopedia of Education (Third Edition), 2010

Primality

Prime-number detection is a more advanced mathematical skill that has also been modeled with connectionist methods. The primality of an integer n can be determined by seeing if n is divisible by any integers between 2 and the integer part of $\sqrt{n}$ . It is efficient to test in this order because the smaller the prime divisor, the more composites it detects in a fixed range of integers.

A connectionist system called knowledge-based cascade-correlation (KBCC) learned this algorithm from examples by recruiting previously learned knowledge of divisibility. KBCC is based on a simpler connectionist algorithm called cascade-correlation (CC) that learns from examples by recruiting single hidden units. CC and KBCC have simulated a large number of phenomena in learning and cognitive development. KBCC has the added advantage that it can recruit its existing network knowledge as well as single hidden units. Both CC and KBCC are constructive learners that build their new learning on top of existing knowledge.

For primality, the pool of source knowledge contained networks that had previously learned whether an integer could be divided by each of a range of divisors, for example, a divide-by-2 network, a divide-by-3 network, etc., up to a divisor of 20 (see Figure 7 ). These source networks were trained on integers from 2 to 360. Twenty KBCC target networks trained on 306 randomly selected integers from 21 to 360 only recruited source networks involving prime divisors below the square root of the largest number they were trained on (360). Moreover, they recruited these sources in order from small to large, and avoided recruiting single hidden units, source networks with composite divisors, any divisors greater than square root of 360 even if prime, and divisor networks with randomized connection weights.

KBCC target networks never recruited a divide-by-2 source network; however, this was because they, instead, used the least significant digit of n, binary coded, to directly determine if n was odd or even. As with people who use the 1's digit of base-10 numbers to check for divisibility by 5 or 10, this is a shortcut to dividing by 2.

The KBCC target networks learned to classify their training integers 3 times faster than did knowledge-free networks, with fewer recruits on fewer network layers, and generalized almost perfectly to novel test integers. Networks without knowledge of divisibility did not generalize better than chance guessing.

As predicted by this simulation, university students testing the primality of integers also mainly used prime divisors below $\sqrt{n}$ and ordered their divisors from small to large. The recommendation for education is not only to use examples, but also to structure curricula so that later learning can build on existing knowledge.

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