Effectiveness of Cognitively Guided instruction Practices in Below Grade-Level Elementary Students

This study asked the question, What is the effect of Cognitively Guided Instruction (CGI) practices on the math beliefs and abilities of below grade-level second grade students? A mixed methods approach was used to measure the impact of a four-week CGI-based intervention on below grade-level second graders’ math proficiency, as well as their math affect, self-concept, and anxiety. Data was also collected on participants’ strategy sophistication, conceptual understanding, and ability to explain their thinking. Findings suggest that CGI practices led to improvement in specific math abilities in below grade-level students, including fluency and automaticity, Base 10 understanding, strategy sophistication, and conceptual understanding. They also led to higher observed levels of math affect and self-concept and reduced levels of math anxiety. These findings indicate that there is significant value in utilizing CGI-based practices with below grade-level elementary students

Perceptual load and enumeration: Distractor interference depends on subitizing capacity

Attention is limited, both in processing capacity (leading to phenomena of “inattentional blindness”) and in the capacity for selective focus (leading to distraction). Load theory (e.g., Lavie, 1995) accounts for both limitations by proposing that perceptual processing has limited capacity but proceeds automatically and in parallel on all stimuli within capacity. Here we tested these claims by applying load theory to the phenomenon of “subitizing”: the parallel detection and individuation of a limited number of items, established in enumeration research. We predicted that distractor interference will be found within but not beyond a person’s subitizing capacity (measured as the transition from parallel to serial slope). Participants reported the number of target shapes from brief displays while ignoring irrelevant cartoon-image distractors. As predicted, distractor cost on enumeration performance was found within subitizing capacity and eliminated in larger set sizes. Moreover, individual differences results demonstrated that distractor effects depended on an individual’s capacity (i.e., their serial-to-parallel transition point), rather than on set size per se. These results provide new evidence for the load theory hypotheses that perceptual processing is automatic and parallel within its limited capacity, while extending it to account for selective attention during enumeratio

An investigation into the structure of numerical cognition

This thesis reports work relating to theoretical frameworks in the area of numerical cognition that have been developed by McCloskey, Caramazza & Basili (1985), Clark & Campbell (1991), Dehaene (1992) and Noel & Seron (1992). The associations between numerical cognition and memory processes in relation to the working memory model of Baddeley (1986) were investigated. The first study used the factor analytic method to elucidate the factor structure of the processes that underlie numerical cognition, and to investigate the various components of the working memory model in relation to arithmetic. A battery of 21 tests was administered to 100 participants. The contribution of the factor analytic study to the structure of numerical cognition is discussed. An examination of the factors (labelled 'access to representations' and 'working memory') identified specific aspects of numerical cognition that were investigated further using experimental methods. The data on magnitude comparisons of numbers and animals that have been found to load onto Factor 1 were reanalysed. Similar patterns were found with the two types of stimuli in some cases. This suggested that Dehaene's notion of a 'number line' might not be specific to numbers. To build on the investigation of magnitude comparisons two experiments were carried out using the dual task paradigm. The results confirmed that magnitude judgements are represented at the level of semantic processing and may not be specific to numbers. The subitizing circles test was also found to load onto Factor 1. This raised a question about the common processes that may be involved both in this test and in other tests loading on that factor. A dual task experiment was used to investigate that possibility. It appeared from the results that the verbally presented tasks in the control and experimental groups produced interference with the s ubitizing task. This result lent support for the view that subitizing is an early pre-lexical perceptual process, possibly based on canonical representations ofthe stimuli. Complex addition and multiplication loaded onto Factor 2, 'working memory' and a further dual task experiment was conducted to investigate the speCUlative view held by Aschraft (1995), that the visuo-spatial sketchpad may playa role in arithmetic problem solving. The results lent support for the view held by Aschraft (1995) of the involvement of the visual-spatial component of working memory in the calculation of multi-digit addition problems. Thus the research reported in this thesis has used a range of investigative techniques and data analysis, with the aim of clarifying the scope and the limitations of major recent models of numerical cognition and the role of working memory in numerical processing. The results of the research programme supported those models which link numerical cognition with other forms of mental processing by identifying specific ways in which diverse numerical processes such as magnitude comparison, subitizing and the calculation of multi-digit problems draw on forms of processing associated with other types of stimuli

Processing of quantitative information, investigated with fMRI.

Ever since the discovery of the ‘number neurons’, the neural representation of quantity in the brain has been thought of as a number-selective coding system. In such a system, the neuron is activated by a specific quantity but numerically close quantities also activate the neuron. Recent fMRI studies also confirmed the existence of a number-selective system in humans. Several computational modelling studies predicted a number-sensitive coding stage as a necessary preceding stage to the number-selective neurons (Verguts & Fias, 2004). In this coding scheme, the coding is analogous to the number it represents. This can be implemented by neurons that respond monotonically to number (e.g., more strongly for larger numbers). Recently, the biological reality of such a system has been demonstrated by use of single-cell recording, in the lateral intraparietal area (LIP) of the macaque monkey. In this thesis, we searched for evidence of number-sensitive coding in humans. Using a priming paradigm, we found behavioural evidence for a number-sensitive system in humans for small non-symbolic numerosities (1 to 5). Using event-related fMRI, we showed number-sensitive activation in the human LIP area in the same number range. Remarkably, we could not extend these results for larger numerosities (2 to 64). Whereas the lack of results in the behavioural priming experiment could be due to an insensitivity of the method, this was not a plausible explanation in the fMRI experiment, as the activity measured in human LIP significantly decreased for numerosities larger than 8. We therefore concluded that the number-sensitive system is liable to a capacity limit for higher numerosities, which could be caused by the use of lateral inhibition. We further suggest that the implementation of this lateral inhibition is dependent on the particular task set, and that the capacity limit is not present (or less stringent) when numerosity is not behaviourally relevant. This could explain the finding of number-sensitive neurons for larger numerosities in monkeys. Finally, we suggest that a different mechanism is employed when numerical value of large numerosities is relevant. This leads to the conclusion that dot patterns in the small and large number range are processed differently

Visual Enumeration and Estimation: Brain mechanisms, Attentional demands and Number representations.

The work presented in this thesis explored the roles of attention and number awareness in visual enumeration and estimation through a variety of methods. First, a distinction was made between different attentional modes underlying estimation and enumeration in an in-depth single case study of a patient with simultagnosia. Subsequently I demonstrated that, in visual enumeration, subitizing and counting are dissociable processes and they rely on different brain structures. This was done through a neuropsychological single case study as well as through the first large sample neuropsychological group study using a voxel-based correlation method. Following this, behavioural methods were used to examine the relations between subitizing and estimation. I found that, under conditions encouraging estimation, subitizing is an automatic process and may lead to the exact representation of small numbers, which contrasts with approximate representations for larger numerosities. Finally, a functional MRI study was conducted to highlight the brain regions that are activated for subitizable numerosities, but not for larger numerosities under distributed attention conditions. The imaging study provided converging evidence for automatic subitizing leading to an exact number representation. The last chapter discusses the implications of the contrast between subitization and counting for understanding numerical processing

Dual Task Interference in Low-Level Abilities: The Role of Working Memory and Effects of Mathematics Anxiety

Mathematics anxiety is a negative affective reaction to situations involving mathematical thought and is commonly believed to reduce cognitive functioning by impairing the efficient use of working memory resources. The conventional theory describes that the processing disadvantage associated with high levels of math anxiety increasingly impairs performance as working memory demands increase in a math task. Despite this convention, recent reports demonstrate that the high math anxious disadvantage can be measured in tasks that are relatively free of working memory assistance (Maloney, Ansari, & Fugelang, 2011; Maloney, Risko, Ansari, & Fugelsang, 2010). The present study examines these relatively low level effects in college adults. A dual task paradigm was designed to test the engagement of different processing faculties in number comparison (Experiment 1) and enumeration (Experiment 2). The results of the present study mostly replicated the math anxiety effects reported in the literature; however, the dual task settings provide key insight into their interpretation. The results obtained are explained in the context of the Attentional Control Theory (Eysenck, Derakshan, Santos, & Calvo, 2007), and reasoning is provided for the extension of the math anxiety construct to include components related to attentional control. Finally, implications drawn from this extension are used to explore the interaction between math anxiety and achievement for future research

An Exploration of the Relationships Among Cattell-Horn-Carroll (CHC) Theory-Aligned Cognitive Abilities and Math Fluency

Math fluency, which refers to the ability to solve single digit arithmetic problems quickly and accurately, is a foundational mathematical skill. Recent research has examined the role of phonological processing, executive control, and number sense in explaining differences in math fluency performance in school-aged children. Identifying the links between these cognitive abilities and math fluency skills has important implications for screening, assessment, and intervention efforts in schools. As extant mathematics research in the context of Cattell-Horn-Carroll (CHC) theory has evaluated either broad mathematics performance or math calculation skills, little is known about the specific relationships between math fact fluency and broad and narrow cognitive abilities. The present study investigated the relationships among Math Fact Fluency performance and the CHC theory-aligned broad and narrow cognitive abilities using a child-age subset of the Woodcock Johnson IV standardization sample. Results of the path analyses indicated that General Intellectual Ability (GIA) exhibited significant direct and indirect effects on Math Fact Fluency performance. With regard to broad cognitive abilities, Processing Speed had the greatest direct effect on Math Fact Fluency. Likewise, in the narrow abilities model, Perceptual Speed was most related to Math Fact Fluency, after accounting for GIA. Contrary to initial hypotheses, Working Memory, Phonetic Coding, and Attentional Control did not significantly contribute to Math Fact Fluency. Finally, the inclusion of Math Problem Solving within the cognitive abilities model resulted in a moderate direct effect on Math Fact Fluency performance. These findings are discussed in terms of directions for future research as well as implications for clinicians and educators

The effects of musical engagement on numerical cognition in early childhood.

Music and mathematics have been associated since Pythagoras, but there is limited evidence on whether musical engagement promotes the acquisition of cognitive skills in young children that support later learning of mathematics. The goal of the present study was to test if numerical cognition and working memory was enhanced in pre-schoolers who were actively engaged in music. The performance of two groups of children (aged 3 to 5) in numeracy, numerosity, subitizing, and memory tasks were compared. The children in the music group (n = 28) had participated in weekly 30-minute music classes for at least 6 months prior to the study. The control group (n = 28) were children attending regular preschools without any additional music classes. Older children (>= 4 years) performed significantly better than younger children (< 4 years) on most measures. A series of ANCOVAs with music group and age as factors and socioeconomic deprivation (NZDep) as a continuous covariate showed that the music group performed significantly better on several measures related to numerical cognition, including numerosity discrimination and subitizing with canonical (symmetrical) displays. Overall, these results provide evidence that musical engagement helps preschool children develop numerical cognition skills