Language and the complexity of the world

Nature is complex, exceedingly so. A repercussion of this “complex world constraint” is that it is, in practice, impossible to connect words to the world in a foolproof manner. In this paper I explore the ways in which the complex world constraint makes vagueness, or more generally imprecision, in language in practice unavoidable, illuminates what vagueness comes to, and guides us to a sensible way of thinking about truth. Along the way we see that the problem of ceteris paribus laws is exactly the problem of vagueness and susceptible to similar treatment

PISA 2012 Results: Creative Problem Solving

PISA is an ongoing programme that offers insights for education policy and practice, and that helps monitor trends in students’ acquisition of knowledge and skills across countries and economies and in different demographic subgroups within each country. PISA results reveal what is possible in education by showing what students in the highest-performing and most rapidly improving school systems can do. The findings allow policy makers around the world to gauge the knowledge and skills of students in their own countries in comparison with those in other countries, set policy targets against measurable goals achieved by other school systems, and learn from policies and practices applied elsewhere. While PISA cannot identify cause-and-effect relationships between policies/practices and student outcomes, it can show educators, policy makers and the interested public how education systems are similar and different – and what that means for students

Revisiting the 'unreasonable effectiveness' of mathematics

Although the phrase 'unreasonable effectiveness' of mathematics is widely used, it is not clear what it means. To understand this phrase critically, we first need to understand the meaning of mathematics and what it means to use it in the sciences. This paper begins by considering the different views on the nature of mathematics,the diversity of which points to the difficulty in understanding what mathematics really is, a difficulty which adds to the mysteriousness of the applicability of mathematics. It is also not clear as to what is applied when we apply mathematics. What is clear however is that mathematics cannot be applied to the world but only to some descriptions of the world. This description occurs through the medium of language and models, thus leading us to consider the role of mathematics as language. The use of a language like English to describe the world is itself unreasonably effective and the puzzle with mathematics is just one reflection of this larger mystery of the relation between language and the world. The concluding parts of this paper argue how the view of mathematics as language can help us understand the mechanisms for its effective applicability

For the Love of Poetry and Mathematics: January 6, 2012

Together we combine to become twice what 60 could ever be.The same ghost looked into my eyes.Also, the center of the circle is inside the circle.There is no way to draw a line, other than a border line,A language as precise as poetry to delineate universe and being,Deceptive as one over n as n goes to infinity, summed.The world is a complex systemnot defined by any voice factored out of my voicelessness,the hypotenuse of course is the man who came between usdoubting every figured guess,so, inductively, all of God’s natural numbers fail. The poem above comprises lines or phrases from several pieces that were presented at “A Reading of Poetry with Mathematics” on Friday, January 6, 2012, during the Joint Mathematics Meetings in Boston, MA. The sources of the individual lines and the event as a whole will be discussed in the following

Teachers’ pedagogical knowledge in integration of information communication technology and students’ performance in mathematics in public secondary schools in Makueni county, Kenya

Introduction: In many nations, the use of information and communication technology (ICTs) in and for education is increasingly growing, and it is now widely recognized as both a need and an incentive to develop and improve the education provided to people around the world. Purpose: The intend of this study was to establish the relationship between teachers’ pedagogical knowledge in ICT integration and students’ performance in Mathematics in public secondary schools in Makueni County, Kenya. Methodology: The study used a correlation research design. The target population was 251 principals, 407 mathematics teachers, and 14,752 Form three students, giving a total of 15,410 respondents. The study sample consisted of 25 principals, 42 mathematics teachers, and 375 students. The researcher used questionnaires for mathematics teachers, interview schedules for principals, and focus group discussions for students. Cronbach’s alpha was used to determine the reliability of the questionnaire, giving a correlation coefficient of 0.939. Quantitative data were analyzed using descriptive and inferential statistics, while qualitative data were categorized into themes and thematically analyzed. Results: The findings revealed a significant relationship between teachers’ pedagogical knowledge in ICT integration and students’ performance in Mathematics. Recommendations: The study recommends that mathematics teachers should choose appropriate technologies to incorporate in their teaching methodologies in the actual classroom to enrich what they teach, how they teach, and what students learn

On the untrustworthiness of axiomatic-founded science

The idea of science being the best – or the only – way to reach the truth about our cosmos has been a major belief of modern civilization. Yet, science has grown tall on fragile legs of clay. Every scientific theory uses axioms and assumptions that by definition cannot be proved. This poses a serious limitation to the use of science as a tool to find the truth. The only way to search for the latter is to redefine the former to its original glory. In the days well before Galileo and Newton, science and religion were not separated. They worked together to discover the truth and while the latter had God as its final destination, the former had God as its starting point. Science is based on the irrational (unproven) belief that the world is intelligible along many other assumptions. This poses a serious limitation to science that can only be overcome if we accept the irrationality of the cosmos. The motto “Credo quia absurdum” holds more truth than one can ever realize at first glance. There is nothing logical in logic, whereas there is deep wisdom in the irrational. For while the former tries to build castles on moving sand, the latter digs deep inside the depths of existence itself in order to build on the most concrete foundations that there can be: the cosmos itself. The only way forward is backwards. Backwards to a time when religion led the quest for knowledge by accepting what we cannot know, rather than trying to comprehend what we do not. Science was anyway based on that in the first place

Reflections on teaching a mathematics education course

Teaching and learning involve reflexive actions and should be chosen thoughtfully and deliberately, not because someone has decided “what works.” In this study, I examine how complex conversations might offer pedagogical and theoretical (re)considerations in a teacher education course on mathematics. The term “math methods” is a doubly weighted phrase, for both mathematics and methods connote particular ideologies prevalent in current educational rhetoric. In order to unpack the impact of these words, I engage in research based on inquiry, historical analysis, and personal reflections, all of which I use in an eclectic, thoughtful, and explorative manner. The two main research questions I will explore in this dissertation involve effort by “teacher” and “student” in which both are learners, knowers and participants. The first question is how can complex conversations—those involving multiple perspectives—aid pre-service teachers in becoming reflective practitioners, effective professionals, and inquiring pedagogues? Specifically, teaching mathematics as a relational activity—in which a hermeneutical perspective is crucial—brings forth epistemological questions and issues. The historical situatedness of teacher education and mathematics education become relevant with respect to current epistemological perspectives of teachers and researchers, and these influences are examined in the context of pre-service teachers’ positionalities. The second question involves an examination of how I am transformed as I experience and reflect on participation in these complex conversations. While engaged as an instructor, I am simultaneously influenced by research in complexity theory, curriculum theory, and teacher education. In complex conversations, we can find possibilities for teaching and learning, even potential ways of being that we do not yet know. Complex conversations encourage a different form of interaction, a different way of imagining the world—different from a Ramist method of hierarchies, different from a patriarchal positioning of supervisors over teachers and teachers over students, and different from mathematics as what is. In (re)imagining what mathematics can be, it is important to recognize how mathematics is currently construed. May mathematics education be(come) a field of study that allows for differences, multiple perspectives, and authentic questions, where ideas do not converge or diverge but co-emerge

Folly to the Greeks: Good Reasons to Give up Reason

A discussion of why a strong doctrine of 'reason' may not be worth sustaining in the face of modern scientific speculation, and the difficulties this poses for scientific rationality, together with comments on the social understanding of religion, and why we might wish to transcend common sense

World-class apprenticeship standards: Report and recommendations

The aim of this research was to identify world-class apprenticeship standards and to make suggestions as to how these could be applied to the English system. By ‘world class’ we mean that the standards described here are acknowledged to be among the best in the world. Thirteen indicators for world-class apprenticeship standards were identified through the research and these have been divided into four sub-sections: (1) training, (2) skills and expertise, (3) recognition and (4) progression. Each of the indicators is explained separately in this report but they have to be understood as being in close relationship to each other. As well as identifying world-class apprenticeship standards the indicators are also designed to compare these standards to apprenticeship standards that are of a good level but do not necessarily feature amongst the best in the world. Identifying and applying standards in apprenticeships is important because apprenticeship training, especially if delivered at a world-class standard, can raise the number of people in employment, increase individual and company productivity and enhance economic growth. The findings from our research suggest that world-class apprenticeship standards require: • extended apprenticeships of between three to four years; • broad and in-depth scientific and industrial skills and knowledge; • the presence of a ‘master’ in the company to train an apprentice; • high-quality knowledge-based education and training; • recognition through an occupational title on completion of the training; • apprentices to acquire all the skills and knowledge necessary to work effectively in an occupation; • apprentices to become skilled workers in an occupational area with a critical and creative approach; and • progression routes into employment as well as into further education and training. This report is based on interviews with seven experts from Australia, Denmark, England, Germany, Ireland, the Netherlands and South Korea on vocational education and training with a review of the literature

Designing Open and Distance Learning for Teacher Education in Sub-Saharan Africa: A toolkit for educators and planners

Everyone remembers a good teacher. Good teachers are the key to educational expansion and improvement. In many countries in Sub-Saharan Africa, there is an urgent need to expand the number of primary and secondary teachers. In all African countries, there is an equally important need to improve the quality of teaching. To achieve this, it is clear that new approaches to teacher education are essential. Existing institutions of teacher education will continue to play an important role, but, alone, they will not meet the goals of Education for All (EFA) by 2015. It is fortunate that, just as the twin needs to improve the quantity and quality of teachers become imperative, so new forms of education and training are becoming available. The world is witnessing a revolution in information and communication technologies (ICTs), which can offer training and support of a type and at a cost hitherto impossible to consider, and thus, must be fully explored given the scale and urgency of demand. In doing so, however, it will be necessary to build on existing and well-tested strategies, including the best models of open and distance learning. This toolkit is the third in a series of recent publications by the Africa Region Human Development Department of the World Bank to share knowledge and experience on how distance education and ICTs can support education in Sub-Saharan Africa. It emphasizes the rigorous process by which new forms of distance-education programs for teacher education can be planned and implemented. The best models of established programs are considered along with the potential for incorporating, as the means become available, new modes of communication. Most forms of teacher education, particularly those concerned with qualification upgrading and ongoing professional development, will have to be based in schools. The authors demonstrate how school-based programs, appropriately resourced and supported, have the potential not only to raise significantly the number and quality of teachers, but also to improve classroom practice and school organization, generally. The guidance and advice, which is drawn from many years of experience in design and implementation, and embraces a range of case studies from across the region, will be of considerable value to those preparing new policies and programs of teacher education and to those seeking to improve existing programs