Sarah Lisa Hopkins

Explaining the Acquisition of a Complex Skill: Methodological and Theoretical Considerations Uncovered in the Study of Simple Addition and the Moving-On Process

Research into the development of mathematical cognition has accelerated in recent times. In this literature review, findings from different approaches to studying simple addition performance are reviewed. A review of findings from studies incorporating observation, self-report or both revealed the complex process of change associated with developing retrieval strategies for simple addition. These findings are synthesized using a model of change labelled the Moving-On Process to represent the wave-like transitional pattern of strategy use preceding retrieval. A review of findings from studies based on the analysis of reaction times revealed a robust finding known as “the problem-size effect.” Different explanations of the problem-size effect, incorporating different theoretical accounts of the nature of retrieval, exact answers, and strategy choice, are compared and contrasted. A review of findings based on a combination of both approaches uncovered important methodological considerations associated with analyzing reaction times and offers direction for future research aimed at unravelling the convoluted construct of automaticity.

Retrieval of Simple Addition Facts: Complexities Involved in Addressing a Commonly Identified Mathematical Learning Difficulty

There exists a substantial number of studies that have identified a subset of low-achieving mathematics students who do not develop a reliance on retrieval for simple addition but who continue to use a counting strategy to solve these problems. There are few studies, however, that have focused on how retrieval of simple addition facts may be improved. This study employed a combined methodological approach to examine the effect extended practice had on increasing a reliance on retrieval for simple addition. An intervention aimed at improving the efficiency of extended practice was also piloted. Although most students improved with extended practice, the extent of improvement was not practical for all students and the intervention did not generally improve the effectiveness of extended practice. The findings emphasize the critical importance of continuing such research and draw attention to the complexities involved in addressing retrieval difficulties for simple addition.

Self-reported delinquency and reputational orientations of high school and incarcerated adolescent loners and nonloners

The self-reported delinquency and reputational orientations of loners and nonloners were examined in two related investigations. In the first, 1328 high school students and 132 institutionalized youth were classified as either loners or nonloners. Two multivariate analyses of variance revealed no significant main effect of loner status for delinquency, but there were main effects for gender and age. For reputational orientations, there were main effects of loner status, gender, and age. Of the 113 participants identified as loners (75 male and 38 female), 98 were age and gender matched with nonloners for further investigation. Two multivariate analyses of variance with follow-up univariates revealed a main effect of delinquency for loner status, with loners reporting higher levels of involvement in physical aggression, stealing offences, and vehicle-related offences. A significant main effect of loner status was also evident for reputational orientation. Overall, the reputational orientations of loners were suggestive of individuals who wish to be nonconforming but desire this to be of a private nature, which may be why they commit delinquent activities without the presence of an audience. These findings are discussed in the light of reputation enhancement theory.

Differential Profiles of Risk of Self-Harm among Clinically Referred Primary School Aged Children.

Risk of self-harm among clinic referred children aged 6- to 12-years-old was investigated using the recently developed Self-Harm Risk Assessment for Children (SHRAC) instrument which comprises six factors: Affect traits; verbalizing of self-harm; socialization; dissociation; self-directing; and self-appraisal. The SHRAC was completed by the mothers of 125 primary school aged children referred to a University based clinic for psychological assessment. These children were subsequently assigned to one of four groups (non clinical, externalizers, internalizers, and combined internalizers/externalizers). Multivariate analysis of variance with univariate F and Scheffé follow-ups revealed differences in risk of self-harm between the four groups. Overall, individuals with both internalizing (anxiety/withdrawn, anxiety/depressed and somatic complaints) and externalizing (rule-breaking behavior and aggressive behavior) characteristics were more at risk of self-harm. These individuals scored highest across each of the six domains. The findings demonstrate the SHRAC to be a reliable instrument for educational psychologists to use with children they suspect of being at risk of self-harm.

The Prevalence and Disadvantage of Min-Counting in Seventh Grade: Problems with Confidence as Well as Accuracy?.

ABSTRACT In this research, we examined how 200 students in seventh grade (around 12 years old) solved simple addition problems. A cluster approach revealed that less than half of the cohort displayed proficiency with simple addition: 35% predominantly used min-counting and were accurate, and 16% frequently made min-counting errors. Students who frequently used min-counting for simple addition, regardless of accuracy, showed lower achievement in mathematics compared to students who relied on accurate retrieval-based strategies. The findings indicate that for many students at this stage of schooling, performance is characterized by accurate min-counting, which is suggestive of a lack of confidence with retrieval. Further, this pattern of performance appears to be more prominent among girls. Confidence with retrieval may be impeding strategy development and hindering learning. The role of confidence in retrieval development needs to be better understood and possibilities for further research are discussed.

Preparing beginning teachers for inclusion: designing and assessing supplementary fieldwork experiences

ABSTRACT Inclusive education requires restructuring educational provision so that mainstream schools are able to provide for the needs of all students in their communities. To help realise this goal, initial teacher education programmes need to better prepare new graduates for teaching students with complex special education needs, including students with intellectual disability. Concerns about the capacity of current school-based placements to prepare new teachers for inclusive classrooms have led some teacher education institutions to develop supplementary fieldwork experiences. The current study involved an investigation into such an experience and looked at the benefits to pre-service teachers (PSTs) of tutoring a young adult with intellectual disability. The findings indicate that PSTs learned effective strategies for differentiating a programme of work and, in their first year of study, were developing a teacher identity. The importance of aligning experiences with coursework units is highlighted and the need for valid assessments of how well initial teacher education programmes are preparing beginning teachers for inclusion, and what these assessments may look like, is discussed.

Building Stronger Teacher Education Programmes to Prepare Inclusive Teachers

In response to concerns about the capacity of current school-based placements to prepare new teachers for inclusive classrooms, teacher education institutions are developing additional fieldwork programmes. We describe an innovative fieldwork programme that resulted from a partnership between the Faculty of Education at Monash University and Wallara (a disability-service provider), and we present findings from an investigation into the pedagogical decisions pre-service teachers made in this programme when tutoring young adults with intellectual disabilities. The findings suggest that well-structured fieldwork programmes that take place on campus can provide valuable opportunities for preparing new teachers for inclusive classrooms.

Teaching primary matheBmatics with challenging tasks: How should lessons be structured?

ABSTRACT Engaging students in a challenging (cognitively demanding) task and launching a mathematics lesson with a task before instruction are two characteristics of a reform-oriented approach to mathematics instruction often considered together. The authors systematically contrasted teaching with challenging tasks using a task-first lesson structure with that of a discussion-first lesson structure to three composite classes of first- and second-grade students (n = 73). Subsequent assessments of mathematical performance revealed that the discussion-first lesson structure was somewhat more efficacious in improving fluency performance but both structures similarly improved problem-solving performance. The findings suggest there is more than one way of incorporating challenging tasks into mathematics lessons to produce sizeable learning gains.

Capturing diversity in the classroom: uncovering patterns of difficulty with simple addition

In this chapter, we draw attention to a common problem encountered in mathematics classrooms, poor number fluency with simple addition. We believe poor number fluency restricts children’s ability to learn mathematics and that a different type of research is needed to inform teachers’ practice to improve number fluency. We used a variety of techniques to capture differences in how children performed simple addition at a time when curriculum documents suggest that proficiency is expected. The findings illustrate marked diversity in performance and reveal varied patterns of difficulty, including the use of highly inefficient counting strategies, inconsistent counting errors and low retrieval on key problem types. The research approach described here provides clear pointers for where teachers need to target instruction for individual children. We propose that inclusive teaching practices are those that are matched to children’s individual patterns of difficulty and suggest how this may be achieved.

Comparative Pairs Marking Supports Authentic Assessment of Practical Performance Within Constructivist Learning Environments

Organisation is a key factor for successful everyday adaptive functioning in schools. Although it is widely recognised that the school environment requires students to be organised if they are to be successful, particularly at the secondary school level, there is currently a lack of instruments available for measuring this construct. To address this, a new teacher report measure of student disorganisation behaviours in the secondary school environment was developed. Following a review of the literature and currently available instrumentation, three phases were implemented: focus-group interviews with primary and secondary teachers; a pilot test of the newly developed instrument to check face validity and item functioning; and an analysis of data from a large scale administration based on the Rasch measurement model. The outcome was a 23-item teacher report scale entitled the Student Disorganisation Scale.