Modelling arithmetic strategies

This thesis examines children's arithmetic strategies and their relation to the concepts of commutativity and associativity. Two complementary methods were used in this research: empirical studies and computational models.Empirical studies were carried out to identify the strategies children used for solving problems like 3 + 4, and 3 + 4 + 7, and the conceptual knowledge associated with them. Their understanding of subtraction problems where the minuend is less than the subtrahend (e.g. 6 - 8) was also considered. A study with 105 subjects revealed a variety of strategies and information about children's knowledge of commutativity and associativity. Four levels of performance of commutativity were also identified. A longitudinal study was carried out with 12 children in order to obtain details of children's changes in strategy, and to double check the results obtained in the main study. The strategies observed to be used by children over a 20 month period parallel those found in previous studies, which show a general transition to more efficient methods. However, the longitudinal study revealed that development of such arithmetic strategies is a slow process. Furthermore, the studies indicated that knowledge of commutativity is a prerequisite for associativity.Models of the observed strategies have been implemented in the form of production rules in a computer program called PALM. The process of implementation highlighted features of children's problem solving that had not been 'detected during the studies.In addition to models that describe the space of strategies, a model of learning has been implemented for the transition from procedural knowledge of commutativity to that of associativity. The model is capable of generalizing its inbuilt knowledge, for instance, its ability to solve 2-term arithmetic expressions, to allow it to solve more complex problems, such as 3-term arithmetic expressions. A further model has been constructed for learning arithmetic strategies that are more efficient than those already represented in the program. It learns specific rules by adding conditions for efficient problem solving to its previous general rules

Rules and principles in cognitive diagnoses

Cognitive simulation is concerned with constructing process models of human cognitive behavior. Our work on the ACM system (Automated Cognitive Modeler) is an attempt to automate this process. The basic assumption is that all goal-oriented cognitive behavior involves search through some problem space. Within this framework, the task of cognitive diagnosis is to identify the problem space in which the subject is operating, identify solution paths used by the subject, and find conditions on the operators that explain those solution paths and that predict the subject's behavior on new problems. The work presented in this paper uses techniques from machine learning to automate the tasks of finding solution paths and operator conditions. We apply this method to the domain of multi-column subtraction and present results that demonstrate ACM's ability to model incorrect subtraction strategies. Finally, we discuss the difference between procedural bugs and misconceptions, proposing that errors due to misconceptions can be viewed as violations of principles for the task domain

Modality-Independent Effects of Phonological Neighborhood Structure on Initial L2 Sign Language Learning

The goal of the present study was to characterize how neighborhood structure in sign language influences lexical sign acquisition in order to extend our understanding of how the lexicon influences lexical acquisition in both sign and spoken languages. A referent-matching lexical sign learning paradigm was administered to a group of 29 hearing sign language learners in order to create a sign lexicon. The lexicon was constructed based on exposures to signs that resided in either sparse or dense handshape and location neighborhoods. The results of the current study indicated that during the creation of the lexicon signs that resided in sparse neighborhoods were learned better than signs that resided in dense neighborhoods. This pattern of results is similar to what is seen in child first language acquisition of spoken language. Therefore, despite differences in child first language and adult second language acquisition, these results contribute to a growing body of literature that implicates the phonological features that structure of the lexicon is influential in initial stages of lexical acquisition for both spoken and sign languages. This is the first study that uses an innovated lexicon-construction methodology to explore interactions between phonology and the lexicon in L2 acquisition of sign language

Relationships between number skills and cognitive abilities in people with specific arithmetic difficulties and people with dyslexia

[Introduction]:Aims and rationale for the studiesThe overall aim of this thesis was to analyse the relationships between cognitive abilities and number skills in children and adults. Examining the links between number skills and cognitive abilities is important both to improve our theoretical knowledge and to inform practitioners who are assessing and teaching children who have number skills difficulties. One important theoretical debate that can be informed by this work is whether normally developing individuals solve problems involving numbers using distinct cognitive modules that are specialised for such work or whether they utilise more generalpurpose cognitive systems. If weaknesses in particular number skills are associated with particular cognitive deficits, it will support the hypothesis that people utilise their general cognitive architecture. Although research into the interactions between children's cognitive profiles and their responses to different teaching programmes is in the early stages, some studies have suggested that tailoring teaching to a child's cognitive profile can be effective. Therefore identifying groups of children with number skills difficulties that have homogeneous cognitive profiles may help in the design of future intervention strategies.ScopeThree main areas of investigation were conducted, all of which examined the links between cognitive abilities and number skills.• An examination of the relationships between three number skills (number fact recall, counting speed and place value understanding) and three cognitive abilities (non-verbal reasoning, auditory-verbal-sequential short-term memory and visual-spatial short-term memory) in normally developing children. • An examination of the cognitive and number skills profiles of children with specific arithmetic abilities (SAD). These children had poor arithmetic attainment, but much better reading attainment. The assessment of these children's cognitive and attainment profiles was comprehensive. The children's verbal, non-verbal and spatial abilities were assessed as well as their psychomotor, visual-spatial memory and auditory-verbal memory abilities. Particular attention was paid to the balance of verbal and spatial abilities in these children as previous research has indicated that children with specific arithmetic difficulties share a homogenous ability profile with poor spatial ability, but better verbal ability.• An examination of the number skills profiles of children and adults with dyslexia. A wealth of previous research has indicated that dyslexic individuals have working memory weaknesses (Hulme, 1981; Shankweiler, Liberman, Mark, Fowler & Fisher, 1979). Three number skills (number fact recall, counting speed and place value understanding) were assessed in dyslexic children, to determine whether a diagnosis of dyslexia was associated with a particular number skills profile. As children with dyslexia had a specific difficulty with number fact recall, the number fact recall of dyslexic adults was compared with non-dyslexic adults, to determine whether this difficulty persisted into adulthood.Structure of the thesisThe thesis is divided into nine chapters. Chapter 1 describes the aims of the thesis and gives an outline of its content. Chapter 2 describes and evaluates the two major models of normal adult numerical processing. Chapter 3 describes current knowledge about how children develop number skills; particular emphasis is placed on the interplay between conceptual understanding and procedural skills. Chapter 4 describes and evaluates previous research into the attainment, cognitive and psychosocial strengths and weaknesses of children with arithmetic difficulties. The limitations of the various research methodologies utilised in previous studies are examined. Chapter 5 provides an overview of how dyslexia is defined; current knowledge about the cognitive profiles of dyslexic individuals is also discussed. Research into the number skills of dyslexic children is described and evaluated. Chapter 6 describes and evaluates Study 1, which had three main aims: to produce norms for some new computerised tests of number skills; to examine how place value understanding, counting speed and number fact recall develop injunior age children; to examine the relationships between cognitive and number skills junior aged children. Chapter 7 reports the results of Studies 2 and 3. The aim of Study 2 was to examine the ability profiles of children with specific arithmetic abilities. The results indicated that children with large verbal/spatial ability discrepancies were over-represented in the group with specific arithmetic difficulties. The cognitive profiles of the children with specific arithmetic abilities were examined in Study 3. The children were divided into four groups: low general conceptual ability; non-verbal learning difficulty; low verbal reasoning; and specific memory weakness. An illustrative case study of a child in each group is provided. Chapter 8 describes and evaluates Study 4, in which the counting speed, number fact recall and place value understanding of children with SAD and children with dyslexia was compared to a randomly selected sample of children attending mainstream schools. The children with dyslexia showed weaknesses on the test of number fact recall and one test of counting speed, but they had unimpaired place value understanding. In contrast the children with specific arithmetic difficulties were impaired both on the tests of place value understanding and number fact recall. Chapter 9 describes and evaluates Study 5, in which the number fact recall of a group of dyslexic students was compared to a group of non-dyslexic students who were matched on intellectual ability. The adults with dyslexia were slower and less accurate at recalling number facts. Chapter 10 draws together the results of the five studies. The findings are discussed in reference to models of adult numerical processing and Rourke's non-verbal learning difficulty classification (Rourke & Del Dotto, 1994). A multiple-route model of arithmetic difficulties is proposed and methods that could be used to evaluate the model are described. Recommendations for the diagnostic assessment of children with arithmetic difficulties and for cognitively tailored teaching are made

Using a cognitive architecture to examine what develops

Different theories of development propose alternative mechanisms by which development occurs. Cognitive architectures can be used to examine the influence of each proposed mechanism of development while keeping all other mechanisms constant. An ACT-R computational model that matched adult behavior in solving a 21-block pyramid puzzle was created. The model was modified in three ways that corresponded to mechanisms of development proposed by developmental theories. The results showed that all the modifications (two of capacity and one of strategy choice) could approximate the behavior of 7-year-old children on the task. The strategy-choice modification provided the closest match on the two central measures of task behavior (time taken per layer, r = .99, and construction attempts per layer, r = .73). Modifying cognitive architectures is a fruitful way to compare and test potential developmental mechanisms, and can therefore help in specifying “what develops.

Inhibitory control and children's mathematical ability

Following recent research linking executive functioning to children 's skills, this thesis explores the relationship between children's inhibition effciency and mathematical ability. This relationship was initially explored using six Stroop task variants containing verbal, numerical or pictorial stimuli. The results indicated that, in the numerical variants only, children of lower mathematical abilty possess less effcient inhibition mechanisms, compared to children of higher mathematical ability. Thus, it is proposed that low-abilty mathematicians may possess a domain-specifc problem with the inhibition of numerical information. The increased interference scores of the lowability mathematicians, however, were only evident under those conditions which also required a degree of switching between temporary strategies. A series of experiments also examined children's ability to inhibit prepotent responses and switch between strategies whilst performing mental arithmetic. The aim of these experiments was to provide a more naturalistic and appropriate exploration of the hypothesized relationship between mathematical abilty and inhibition effciency. These results also indicated that low-ability mathematicians possess fewer executive resources to cope with increased inhibition demands. A further systematic manipulation of switching and inhibition demands revealed that the low-abilty mathematicians experienced a particular difculty when both types of inhibitory demands (i.e. inhibiting a prepotent response and inhibiting an established strategy)were present. This suggests that their reduction in inhibition effciency stems from the amount of demands, rather than the type of demands placed on the executive system. Furthermore, the results indicated that inhibition effciency may be a specifc element of mathematical ability rather than an element of intellectual ability in general. The final study involved a group of low-abilty mathematicians and examined the disturbing impact of irrelevant information on their arithmetic word problem solving abilty. This study revealed that irrelevant numerical (IN) information has a more detrimental impact on performance than irrelevant verbal (IV) information. It is proposed that it is more difcult to inhibit IN information, as it appears more relevant to intentions, and thus, enters WM with a higher level of activations. In sum, the results indicate that low-abilty mathematicians have a reduced domainspecific working memory capacity, characterized by ineffcient inhibition mechanisms

Developing use of strategy in childhood mental addition

The aim of this research was to look at the strategies used by children when doing mental addition problems of the varying levels of complexity. An authentic arithmetic task was designed for use in a school environment. The central aim was to study developing fluency in mental arithmetic as achieved through recruiting various strategies into solving more complex problems than those studied by existing research. The nature of mental addition strategies was inferred from children's solution times when doing sequences of sums. Three studies were carried out on 7–11 year-old children from two local schools. [Continues.

Exact Equality and Successor Function: Two Key Concepts on the Path Towards Understanding Exact Numbers

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers, as prerequisites to the concept of exact numbers: the fact that all numbers can be generated by a successor function, and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Mundurucu (an Amazonian language), and young western children (3-4 years old) understand these fundamental properties of numbers.Psycholog

Teaching addition and subtraction by the method of bidirectional translation: an empirical study

Bidirectional Translation, devised by the author, is a structured approach to the teaching of addition and subtraction which aims to give children greater understanding of arithmetical operations. The approach systematically involves both: the translation of numerical representations into hypothetical, real world contexts; and the extraction of the appropriate numerical operations from hypothetical, real world contexts. It is this emphasis on translation from and to both the numerical representation and realistic contexts which gives rise to the name, Bidirectional Translation. An experimental group of 90 primary one children were taught to add and subtract (within 10) by the method of Bidirectional Translation. Post-test comparison of the experimental subjects' performance with that of a control group showed significantly superior performance on the part of the experimental subjects in terms of the utilizability of addition, the evocability of addition, the utilizability of subtraction and the evocability of subtraction for five different classes of verbal context, namely: Part-Part Whole, Separating, Joining, Equalizing and Comparison contexts. In all instances the probability of the results being chance ones were less than 5% and in most, were less than 1%. In both the experimental and control groups, most children performed better when they were required to utilize concepts than when they were required to evoke concepts. Similarly they performed better when they were required to add than when they were required to subtract. The differences, however, were not always significant. It is suggested that the effectiveness of the methodology of Bidirectional Translation is rooted in a structure which allows the child to make his/her thinking explicit and which allows the teacher to monitor this