Instruction to support mental computation development in young children of diverse ability

Fostering young children's mental computation capacity is essential to support their numeracy development. Debate continues as to whether young children should be explicitly taught strategies for mental computation, or be afforded the freedom to develop their own. This paper reports on teaching experiments with two groups of students in their first year of schooling: those considered 'at-risk', and those deemed mathematically advanced. Both groups made considerable learning gains as a result of instruction. Importantly, the gains of the at-risk group are likely to renew both their own, and their teacher's confidence in their ability to learn. In this paper, the instructional programs are documented, highlighting the influence of instruction upon the children's development.

Cálculo mental com números racionais : um estudo com alunos do 6º ano de escolariedade

Tese de doutoramento, Educação (Didática da Matemática), Universidade de Lisboa, Instituto de Educação, 2016

Experimental Support of Argument-based Syntactic Computation

Linguistic theory, cognitive, information, and mathematical modeling are all useful while we attempt to achieve a better understanding of the Language Faculty (LF). This cross-disciplinary approach will eventually lead to the identification of the key principles applicable in the systems of Natural Language Processing. The present work concentrates on the syntax-semantics interface. We start from recursive definitions and application of optimization principles, and gradually develop a formal model of syntactic operations. The result – a Fibonacci- like syntactic tree – is in fact an argument-based variant of the natural language syntax. This representation (argument-centered model, ACM) is derived by a recursive calculus that generates a mode which connects arguments and expresses relations between them. The reiterative operation assigns primary role to entities as the key components of syntactic structure. We provide experimental evidence in support of the argument-based model. We also show that mental computation of syntax is influenced by the inter-conceptual relations between the images of entities in a semantic space.

Teacher change and professional development in mathematics education

It is generally agreed that if authentic teacher change is to occur then the tacit knowledge about how and why they act in certain ways in the classroom be accessed and reflected upon. While critical reflection can and often is an individual experience there is evidence to suggest that teachers are more likely to engage in the process when it is approached in a collegial manner; that is, when other teachers are involved in and engaged with the same process. Teachers do not enact their profession in isolation but rather exist within a wider community of teachers. An outside facilitator can also play an active and important role in achieving lasting teacher change. According to Stein and Brown (1997) “an important ingredient in socially based learning is that graduations of expertise and experience exist when teachers collaborate with each other or outside experts” (p. 155). To assist in the effective professional development of teachers, outside facilitators, when used, need to provide “a dynamic energy producing interactive experience in which participants examine and explore the complex components of teaching” (Bolster, 1995, p. 193). They also need to establish rapport with the participating teachers that is built on trust and competence (Hyde, Ormiston, & Hyde, 1994). For this to occur, professional development involving teachers and outside facilitators or researchers should not be a one-off event but an ongoing process of engagement that enables both the energy and trust required to develop. Successful professional development activities are therefore collaborative, relevant and provide individual, specialised attention to the teachers concerned. The project reported here aimed to provide professional development to two Year 3 teachers to enhance their teaching of a new mathematics content area, mental computation. This was achieved through the teachers collaborating with a researcher to design an instructional program for mental computation that drew on theory and research in the field.

Professional growth through working together : a study of reciprocal benefits for teacher and education advisor through classroom based professional development

Teacher professional development provided by education advisors as one-off, centrally offered sessions does not always result in change in teacher knowledge, beliefs, attitudes or practice in the classroom. As the mathematics education advisor in this study, I set out to investigate a particular method of professional development so as to influence change in a practising classroom teacher’s knowledge and practices. The particular method of professional development utilised in this study was based on several principles of effective teacher professional development and saw me working regularly in a classroom with the classroom teacher as well as providing ongoing support for her for a full school year. The intention was to document the effects of this particular method of professional development in terms of the classroom teacher’s and my professional growth to provide insights for others working as education advisors. The professional development for the classroom teacher consisted of two components. The first was the co-operative development and implementation of a mental computation instructional program for the Year 3 class. The second component was the provision of ongoing support for the classroom teacher by the education advisor. The design of the professional development and the mental computation instructional program were progressively refined throughout the year. The education advisor fulfilled multiple roles in the study as teacher in the classroom, teacher educator working with the classroom teacher and researcher. Examples of the professional growth of the classroom teacher and the education advisor which occurred as sequences of changes (growth networks, Hollingsworth, 1999) in the domains of the professional world of the classroom teacher and education advisor were drawn from the large body of data collected through regular face-to-face and email communications between the classroom teacher and the education advisor as well as from transcripts of a structured interview. The Interconnected Model of Professional Growth (Clarke & Hollingsworth, 2002; Hollingsworth, 1999) was used to summarise and represent each example of the classroom teacher’s professional growth. A modified version of this model was used to summarise and represent the professional growth of the education advisor. This study confirmed that the method of professional development utilised could lead to significant teacher professional growth related directly to her work in the classroom. Using the Interconnected Model of Professional Growth to summarise and represent the classroom teacher’s professional growth and the modified version for my professional growth assisted with the recognition of examples of how we both changed. This model has potential to be used more widely by education advisors when preparing, implementing, evaluating and following-up on planned teacher professional development activities. The mental computation instructional program developed and trialled in the study was shown to be a successful way of sequencing and managing the teaching of mental computation strategies and related number sense understandings to Year 3 students. This study was conducted in one classroom, with one teacher in one school. The strength of this study was the depth of teacher support provided made possible by the particular method of the professional development, and the depth of analysis of the process. In another school, or with another teacher, this might not have been as successful. While I set out to change my practice as an education advisor I did not expect the depth of learning I experienced in terms of my knowledge, beliefs, attitudes and practices as an educator of teachers. This study has changed the way in which I plan to work as an education advisor in the future.

Calculators in the Classroom: Help or Hindrance?

In a world where technology is ever present and ever changing, is too much technology at too young of an age detrimental to a child’s educational success? The purpose of this paper is to share the results of a four-month study that focused on the use of calculators in grade eight. This study was conducted in an eighth grade class, in a small kindergarten through twelfth grade school. This paper will share the findings of a study of a classroom in which calculator use was limited and mental computation was emphasized. The main focus of this study was whether or not there would be any improvement in the computation skills of my students and how, or if, their problem solving would be affected. As a result of this research project, I plan to permanently limit calculator use in grades seven and eight, as well as to implement a computational review that will be conducted yearly with all of my classes.

Analogies to semantic and syntactic mental labour in the Feist decision

The Supreme Court of the United States in Feist v. Rural (Feist, 1991) specified that compilations or databases, and other works, must have a minimal degree of creativity to be copyrightable. The significance and global diffusion of the decision is only matched by the difficulties it has posed for interpretation. The judgment does not specify what is to be understood by creativity, although it does give a full account of the negative of creativity, as ‘so mechanical or routine as to require no creativity whatsoever’ (Feist, 1991, p.362). The negative of creativity as highly mechanical has particularly diffused globally.

A recent interpretation has correlated ‘so mechanical’ (Feist, 1991) with an automatic mechanical procedure or computational process, using a rigorous exegesis fully to correlate the two uses of mechanical. The negative of creativity is then understood as an automatic computation and as a highly routine process. Creativity is itself is conversely understood as non-computational activity, above a certain level of routinicity (Warner, 2013).

The distinction between the negative of creativity and creativity is strongly analogous to an independently developed distinction between forms of mental labour, between semantic and syntactic labour. Semantic labour is understood as human labour motivated by considerations of meaning and syntactic labour as concerned solely with patterns. Semantic labour is distinctively human while syntactic labour can be directly humanly conducted or delegated to machine, as an automatic computational process (Warner, 2005; 2010, pp.33-41).

The value of the analogy is to greatly increase the intersubjective scope of the distinction between semantic and syntactic mental labour. The global diffusion of the standard for extreme absence of copyrightability embodied in the judgment also indicates the possibility that the distinction fully captures the current transformation in the distribution of mental labour, where syntactic tasks which were previously humanly performed are now increasingly conducted by machine.

The paper has substantive and methodological relevance to the conference themes. Substantively, it is concerned with human creativity, with rationality as not reducible to computation, and has relevance to the language myth, through its indirect endorsement of a non-computable or not mechanical semantics. These themes are supported by the underlying idea of technology as a human construction. Methodologically, it is rooted in the humanities and conducts critical thinking through exegesis and empirically tested theoretical development

References

Feist. (1991). Feist Publications, Inc. v. Rural Tel. Service Co., Inc. 499 U.S. 340.

Warner, J. (2005). Labor in information systems. Annual Review of Information Science and Technology. 39, 2005, pp.551-573.

Warner, J. (2010). Human Information Retrieval (History and Foundations of Information Science Series). Cambridge, MA: MIT Press.

Warner, J. (2013). Creativity for Feist. Journal of the American Society for Information Science and Technology. 64, 6, 2013, pp.1173-1192.

A Restriction-Centered Theory of Reasoning and Computation to materialize a Web 3.0

Researchers suggest that personalization on the Semantic Web adds up to a Web 3.0 eventually. In this Web, personalized agents process and thus generate the biggest share of information rather than humans. In the sense of emergent semantics, which supplements traditional formal semantics of the Semantic Web, this is well conceivable. An emergent Semantic Web underlying fuzzy grassroots ontology can be accomplished through inducing knowledge from users' common parlance in mutual Web 2.0 interactions [1]. These ontologies can also be matched against existing Semantic Web ontologies, to create comprehensive top-level ontologies. On the Web, if augmented with information in the form of restrictions andassociated reliability (Z-numbers) [2], this collection of fuzzy ontologies constitutes an important basis for an implementation of Zadeh's restriction-centered theory of reasoning and computation (RRC) [3]. By considering real world's fuzziness, RRC differs from traditional approaches because it can handle restrictions described in natural language. A restriction is an answer to a question of the value of a variable such as the duration of an appointment. In addition to mathematically well-defined answers, RRC can likewise deal with unprecisiated answers as "about one hour." Inspired by mental functions, it constitutes an important basis to leverage present-day Web efforts to a natural Web 3.0. Based on natural language information, RRC may be accomplished with Z-number calculation to achieve a personalized Web reasoning and computation. Finally, through Web agents' understanding of natural language, they can react to humans more intuitively and thus generate and process information.

Knowledge engineering for mental-health risk assessment and decision support

Mental-health risk assessment practice in the UK is mainly paper-based, with little standardisation in the tools that are used across the Services. The tools that are available tend to rely on minimal sets of items and unsophisticated scoring methods to identify at-risk individuals. This means the reasoning by which an outcome has been determined remains uncertain. Consequently, there is little provision for: including the patient as an active party in the assessment process, identifying underlying causes of risk, and eecting shared decision-making. This thesis develops a tool-chain for the formulation and deployment of a computerised clinical decision support system for mental-health risk assessment. The resultant tool, GRiST, will be based on consensual domain expert knowledge that will be validated as part of the research, and will incorporate a proven psychological model of classication for risk computation. GRiST will have an ambitious remit of being a platform that can be used over the Internet, by both the clinician and the layperson, in multiple settings, and in the assessment of patients with varying demographics. Flexibility will therefore be a guiding principle in the development of the platform, to the extent that GRiST will present an assessment environment that is tailored to the circumstances in which it nds itself. XML and XSLT will be the key technologies that help deliver this exibility.