On fractions and non-standard representations: Pre-service teachers' concepts

The present study examined the reasoning strategies and arguments given by pre-service school teachers as they solved two problems regarding fractions in different symbolic representations. In the first problem, the pre-service school teachers were asked to compare between two different fractions having the same numerical representation, and in the second problem, they were asked to compare between different notational representations of the same fraction. Numeration systems in bases other than ten were used to generate various representations of fractions. All students were asked to provide justifications to their responses. Strategies and arguments relative to pre-service teachers' concepts of fractions and place value were identified and analyzed based on results of 38 individual clinical interviews, and written responses of 124 students. It was found that the majority of students believe that fractions change their numerical value under different symbolic representations.     

Relations of Preschoolers' Social Acceptance to Peer Ratings and Self-Perceptions

To examine the relations of preschoolers' social acceptance to peer ratings and self-perceptions, 53 preschoolers were asked to rate how much they liked or disliked their peers and to justify these ratings. Preschoolers also rated their peers' aggressive, prosocial, and sociable behavior. Finally, they completed a pictorial self-perception scale that assessed their views of their physical competence and their relationships with mother and with peers. Children who were better liked by peers were rated as more prosocial, more sociable, and less aggressive than less liked children. Preschoolers often reported liking certain peers because they perceived that those peers liked them; they often reported disliking certain peers because they perceived those peers as aggressive. In contrast to findings with older children, preschoolers' social acceptance was not significantly related to any aspect of their self-perceptions. The results provide evidence for the validity of peer ratings by preschool-age children and bring up issues related to the development and assessment of self-perceptions among preschoolers.     

A second chance for higher education: Academic preparatory programs in Israel

The Israeli academic preparatory programs (APPs) challenge the sorting regime of regular education by offering a genuine second chance for higher education to youngsters who have failed to acquire the necessary prerequisites via the mainstream. The study analyzes the success of the APPs to produce changes in their students in light of the failure of a different second-chance framework, the community college, to do so. The analysis of a sample of APP graduates indicates that background characteristics and high school history, powerful sources of differentiation upon admittance to the APPs, lose their discriminating power after graduation. All graduates join any postsecondary educational framework, regardless of their initial disadvantages. The success of the APPs in eliminating initial gaps among the students is assigned mainly to their nonconformity to the selection criteria of regular education. The purpose of this research was to assess whether second-chance educational programs that challenge the sorting criteria of the mainstream succeed in enhancing social equality in education. Outcomes of the challenging second-chance academic preparatory programs in Israel are analyzed in light of the reported failure of community colleges to moderate inequalities in higher education.     

Interweaving mathematics and pedagogy in task design: a tale of one task

In this article we introduce a usage-goal framework within which task design can be guided and analyzed. We tell a tale of one task, the Pentomino Problem, and its evolution through predictive analysis, trial, reflective analysis, and adjustment. In describing several iterations of the task implementation, we focus on mathematical affordances embedded in the design and also briefly touch upon pedagogical affordances.     

On learning fundamental concepts of group theory

The research reported in this paper explores the nature of student knowledge about group theory, and how an individual may develop an understanding of certain topics in this domain. As part of a long-term research and development project in learning and teaching undergraduate mathematics, this report is one of a series of papers on the abstract algebra component of that project.The observations discussed here were collected during a six-week summer workshop where 24 high school teachers took a course in Abstract Algebra as part of their work. By comparing written samples, and student interviews with our own theoretical analysis, we attempt to describe ways in which these individuals seemed to be approaching the concepts of group, subgroup, coset, normality, and quotient group. The general pattern of learning that we infer here illustrates an action-process-object-schema framework for addressing these specific group theory issues. We make here only some quite general observations about learning these specific topics, the complex nature of understanding, and the role of errors and misconceptions in light of an action-process-schema framework. Seen as research questions for further exploration, we expect these observations to inform our continuing investigations and those of other researchers.We end the paper with a brief discussion of some pedagogical suggestions arising out of our considerations. We defer, however, a full consideration of instructional strategies and their effects on learning these topics to some future time when more extensive research can provide a more solid foundation for the design of specific pedagogies.Work on this project has been partially supported by the National Science Foundation.     

Relative and absolute thinking in visual estimation processes

This study has two main goals: (1) to investigate the processes involved in visual estimation (part I of the study), and (2) to investigate the processes of judgment in visual estimation situations, which mostly involved proportional reasoning (part II). The study was conducted with 9-year old children in the third grade. Four strategies were expressed by the children in visual estimation situations. Exposure to a unit in the Agam project, designed to enhance visual estimation capabilities resulted in changes in the children's strategies. These changes reflected the processes by which children overcame their limited ability to process visual information. The development of proportional reasoning was investigated through a series of judgment situations. Although, as was expected, most of the children showed an additive behavior, these situations stimulated some children towards qualitative proportional reasoning, where easy/difficult considerations played an important role.     

Teachers' concerns about inclusive education in Ahmedabad,India

This study was undertaken to determine the concerns of primary school teachers about the inclusion of students with disabilities in Ahmedabad, India. A total of 560 teachers, working in government‐run schools, returned the completed survey. A two‐part questionnaire was used in this study. Part 1 gathered information relating to personal and professional characteristics of the teachers. Part 2 was a 21‐item Likert scale titled Concerns about Inclusive Education – Gujarati. The major finding of the study was that the teachers in Ahmedabad were moderately concerned about including students with disabilities in their classrooms. The teachers were most concerned about lack of infrastructural resources and least concerned about lack of social acceptance of students with disabilities in inclusive education classrooms. Significant differences existed in teacher concerns based on the following background variables: gender, qualifications in special education, teaching experience and number of students with disabilities in class. A number of implications are discussed to address teacher concerns for inclusive education in India.