Continuity as a computational effect

The original purpose of component-based development was to provide techniques to master complex software, through composition, reuse and parametrisation. However, such systems are rapidly moving towards a level in which software becomes prevalently intertwined with (continuous) physical processes. A possible way to accommodate the latter in component calculi relies on a suitable encoding of continuous behaviour as (yet another) computational effect. This paper introduces such an encoding through a monad which, in the compositional development of hybrid systems, may play a role similar to the one played by 1+, powerset, and distribution monads in the characterisation of partial, nondeterministic and probabilistic components, respectively. This monad and its Kleisli category provide a universe in which the effects of continuity over (different forms of) composition can be suitably studied.This work is financed by the ERDF - European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT - Fundacao para a Ciencia e a Tecnologia within project POCI-01-0145-FEDER-016692.The first author is also sponsored by FCT grant SFRH/BD/52234/2013, and the second one by FCT grant SFRH/BSAB/113890/2015. Moreover, D. Hofmann and M. Martins are supported by the EU FP7 Marie Curie PIRSES-GA-2012-318986 project GeTFun: Generalizing Truth-Functionality and FCT project UID/MAT/04106/2013 through CIDMA

The effect of money as a context on students\u27 mental computation performance in years 3, 5, 7 and 9

The purpose of this study was to research the effect of money as a context on school students’ mental computational performance and strategy choices across a range of ages. This study adds to existing research, which has compared students\u27 mental computational methods with their written methods, by the provision of the single common context of money. The content topics of whole and other rational numbers (simple fractions, decimals, and some percentages) were covered. Forty-eight primary school students plus sixteen secondary school students were involved in this study, with equal numbers of both genders from the two primary schools and one secondary school in the Perth metropolitan area. The method followed was both quantitative-by scoring test results-and qualitative-through tape-recorded interviews. Students\u27 prior experiences with money were documented and performance data were collected on students\u27 mental computation ability for the two sets of mathematically identical items presented in a money-context and context-free. Student strategy choices were also documented. The semi-structured interviews consisted of nine money experience questions such as, How often do you get pocket money or an allowance? In addition, 10, 12, 13, and 13 pairs of mental computation items for Years 3, S, 7 and 9 respectively. Where possible, common items were used across two or more year levels to ascertain growth in mental computation skills. Overall, results found that the context presentation did not make a difference to student performance and there was no correlation found between performance and student preference for one presentation or the other. No performance differences were found for gender. Year 3 recorded the lowest process scores, while Year 7 recorded the highest process scores although all the items used at both Year 7 and Year 9 were identical. The greatest growth in mental computation performance was found to occur from Year 3 to Year 5 and from Year 5 to Year 7. Further, for Year 3, results found that the context presentation had a negative impact on student performance. Some students were found to be using written methods mentally. Analysis of individual items revealed that context had a positive influence in some cases. However, despite the emphasis in modem curricula on the use of context, it appears that such an approach may have little value if used in contrived rather than real situations. Recommendations for teaching practice include promoting real experiences at school by linking students\u27 out-of-school experiences to classroom learning, such as exploring students\u27 pocket-money purchasing power or promoting mental computation for a variety of context tasks. It is considered likely that mental computation in classrooms tends to be non-contextual and it is recommended that teachers should make more use of context. It is further recommended that teachers use money as a context, with mental computation items presented as part of areal shopping tasks. Oral presentation would remove typical school method cues – a ‘sheet’ and pencil – with the only visual stimuli being the goods and price labelling. Class shops could use simulations for the junior grades, while older grades could organise real money exchange experiences integrated with other curriculum areas such as raising money for charitable causes. Research on the effect of other common contexts such as food, time, and other measurement topics should also utilise real activities, with examples of such being readily found in the media. The provision of a variety of contexts is important for students as what constitutes a meaningful context may vary from individual to individual

Experiences of Adults with Developmental Disability and a Teacher of Mathematics in the Money Club

In my experiences, students with Developmental Disability (DD) are routinely excluded from Algebra and other high-level mathematics courses. People with DD do not have the opportunity to learn Algebra, which may support the understanding and provide purpose for learning money and budgeting skills that, perhaps, could help them avoid financial difficulties as adults. The purpose of this study was to investigate the experiences of young adults with DD and me, their teacher researcher, about mathematics, finance, and self within a 6 week program of instruction titled the Money Club. This includes how adults with DD reason, apply, perceive, and solve applied mathematics problems in finance. Access to general education curriculum is not only a federally mandated law, but also a core interest to me as an educator. I want my students to learn what everyone else is learning, go places everyone else is going, and do things side by side with other people. Many people with DD do not learn algebra (even if the course title is Algebra). They will focus on basic skills, life skills, and non-skilled labor-intensive processes. I wanted people with DD to have access to higher mathematics and extend their life skills to include it. In high school, people with DD learn basic mathematics and life skills, which consist of adding, subtracting, multiplying, and dividing numbers; or as my peer teachers say, working on their mathematics skills.\u27 They are taught many of the things I offered in the Money Club at a topical level: saving money, overspending, balancing their personal budgets, and avoiding people who take advantage of them. The Money Club\u27s difference is that we used algebraic ways of thinking to accomplish these goals: Use of variables, function tables, graphs, matrices, and other higher order mathematics concepts and skills. We demonstrated exactly what happens mathematically when they do not balance their budget, spend too much, take out a bad loan, use a high interest credit card, go to the pawnshop, or invest their money. The Money Club provided a gateway to higher mathematics through an immediately interesting subject: Money.\u2

Pupils learning algebra with ICT in Key Stage 3 mathematics classrooms

This thesis is set in the context of learning early secondary algebra using ICT. It argues that the support offered by a collaborative interaction of elements (pupils, teachers, language and computers) in lessons, that is, the affective dimension of pupils' mathematical constructions, has not been emphasised enough in studies about the impact of ICT. Through a classroom-based intervention study, pupils’ conceptual understanding in algebra was monitored. The study examined the effect of introducing a technological tool on pupils’ interest in algebra, their involvement and engagement in mathematical learning, and the role of the teacher in ICT-enhanced settings. Research was conducted during secondary mathematics lessons in England and in Kenya. This intervention used ‘dialogic teaching’ and Grid Algebra as catalysts, with traditional resources alongside new technologies. The study sought to demonstrate that a ‘blended’ approach to learning can mediate the pupils’ access to algebraic concepts. The study employed a mixed-method strategy including: written work, observations, interviews and pupil questionnaires. Analysis of collected data underlines the value of formative feedback, clear expectations and developing mathematical language. The study concludes that appropriate use of computer software can have a significant impact on a whole year group. Additionally, emphasising affective aspects which reinforce ICT use in mathematics instruction can create an enabling environment for active learning. This thesis makes an original contribution to effective teacher development and delivery on the premise of a social model of learning mathematics with ICT

TEACHING AS CARING RELATIONS: A CASE STUDY OF AN EXEMPLARY SECONDARY SPECIAL EDUCATION TEACHER

Thesis (Ph.D.) - Indiana University, School of Education, 2004ABSTRACT Sandra Joy Washburn TEACHING AS CARING RELATIONS: A CASE STUDY OF AN EXEMPLARY SECONDARY SPECIAL EDUCATION TEACHER This interpretive case study examines the teaching practices and thoughts of a secondary special educator considered by many (teachers, administrators, students, university faculty, pre-service teachers) to be an exemplary teacher. During two years, the researcher spent an average of two days per week observing the public high school teacher, his students, and co-teaching partners in several general education math classes that included students with disabilities. During the second year of the study, the researcher also observed and worked with the teacher and his students in a Resource Class. The data included field notes, taped interviews, student course evaluations, teaching materials, and other artifacts. The original research questions focused on instructional practices, co-teaching, student perceptions, teacher caring and relationships, and the development of community. The researcher systematically categorized multiple data types and sources to arrive at her categories and conclusions. The study's findings show that caring teaching and pedagogical mastery are interdependent and that students perceive teacher competence at promoting students' academic engagement and success as demonstrable proof of caring teaching. The study also determines that this exemplary teacher perceived his role to extend beyond the teaching of subject matter content, that he felt obligated to promote the development of caring and responsible students who were knowledgeable about interpersonal relationships and dimensions of the self. By portraying a strong moral authority, displaying high levels of enthusiasm for teaching, students, and subject matter, forming close, caring, personal relationships with students, and consistently modeling expected behaviors, the teacher created a classroom climate that supported student engagement, effort, and success with academic and behavioral expectations. Finally, the findings show many benefits of the co-teaching model, and that students and teachers preferred co-teaching over solo-teaching. The study aims to provide substantive content to the discussion of how best to engage low-achieving students, including students with disabilities, at the secondary level. Implications for teacher practice and teacher education are discussed