Students' difficulties, conceptions and attitudes towards learning algebra : an intervention study to improve teaching and learning

The skills necessary to identify and analyse errors and misconceptions made by students are needed by teachers of all levels especially at the lower secondary school level in Malaysia. If students are to be successful in tackling mathematical problems later in their schooling, the one prerequisite is the mastery of basic concepts in algebra. Despite the best efforts of the teachers, students still develop algebra misconceptions. Is it possible to reduce or eliminate these misconceptions? The research involved a survey of 14 year-old students in Form 2 (Grade 8) in the Penampang district of Sabah, East Malaysia. The focus of this study lies in students’ difficulties, conceptions and attitudes towards learning algebra in the framework of conceptual change. A possible way to help students overcome their learning difficulties and misconceptions is by implementing diagnostic teaching involving conflict to foster conceptual change. The study involved evaluating the efficacy of a conceptual change instructional programme involving cognitive conflict in (1) facilitating Form 2 students’ understanding of algebra concepts, and (2) assessing changes in students’ attitudes towards learning mathematics, in a mixed quantitativequalitative research design.A 24-item Algebra Diagnostic Test and a 20-item Test of Mathematics-Related Attitudes (TOMRA) questionnaire were administered as a pretest and a posttest to 39 students in each of a heterogeneous high-achieving class and a below-average achieving class. In addition 9 students were purposefully selected to participate in the interview.The results of the study indicated that students’ difficulties and misconceptions from both classes fell into five broad areas: (1) basic understanding of letters and their place in mathematics, (2) manipulation of these letters or variables, (3) use of rules of manipulation to solve equations, (4) use of knowledge of algebraic structure and syntax to form equations, and (5) generalisation of rule for repetitive patterns or sequences of shapes.The results also showed that there was significant improvement in students’ achievement in mathematics. Further, students’ attitude towards inquiry of mathematics lessons showed significant positive improvement. Enjoyment remained high even though enjoyment of mathematics lesson showed no change. Also, changes in students’ understanding (from unintelligible to intelligible, intelligible to plausible, plausible to fruitful) illustrated the extent of changes in their conceptions.Different pedagogies can affect how conceptual change and challenge of misconceptions occurs. Therefore, knowledge of the origin of different types of misconceptions can be useful in selecting more effective pedagogical techniques for challenging particular misconceptions. Also, for teachers to create an effective learning experience they should be aware of and acknowledge students’ prior knowledge acquired from academic settings and from everyday previous personal experiences. Since all learning involves transfer from prior knowledge and previous experiences, an awareness and understanding of a student’s initial conceptual framework and/or topic can be used to formulate more effective teaching strategies. If this idea is taken a step further, it could be said that, because misconceptions comprise part of a conceptual framework, then understanding origins of misconceptions would further facilitate development of effective teaching strategies.Further research is needed to help teachers to understand how students experience conflict, how students feel when they experience conflict, and how these experiences are related to their final responses because cognitive conflict has both constructive and destructive potential. Thus, by being able to interpret, recognise and manage cognitive conflict, a teacher can then successfully interpret his/her students’ cognitive conflict and be able to make conceptual change more likely or help students to have meaningful learning experiences in secondary school algebra

Constraint structure of O(3) nonlinear sigma model revisited

We study the constraint structure of the O(3) nonlinear sigma model in the framework of the Lagrangian, symplectic, Hamilton-Jacobi as well as the Batalin-Fradkin-Tyutin embedding procedure.Comment: 17 page

Field-enlarging transformations and chiral theories

A field-enlarging transformation in the chiral electrodynamics is performed. This introduces an additional gauge symmetry to the model that is unitary and anomaly-free and allows for comparison of different models discussed in the literature. The problem of superfluous degrees of freedom and their influence on quantization is discussed. Several "mysteries" are explained from this point of view.Comment: 14 pages, LaTeX-file, BI-TP 93/0

Symplectic quantization of self-dual master Lagrangian

We consider the master Lagrangian of Deser and Jackiw, interpolating between the self-dual and the Maxwell-Chern-Simons Lagrangian, and quantize it following the symplectic approach, as well as the traditional Dirac scheme. We demonstrate the equivalence of these procedures in the subspace of the second-class constraints. We then proceed to embed this mixed first- and second-class system into an extended first-class system within the framework of both approaches, and construct the corresponding generator for this extended gauge symmetry in both formulations.Comment: 27 page

New Technologies’ Promise to the Self and the Becoming of the Sacred: Insights from Georges Bataille’s Concept of Transgression

This article draws on Georges Bataille’s concept of transgression, a key element in Bataille’s theory of the sacred, to highlight structural implications of the way the self-empowerment ethos of new technologies suffuses the digital tracking culture. Pointing to the original conceptual stance of transgression, worked out against prohibition, I first argue that, beyond a critique of new technologies’ promise of self-empowerment as coming at the expense of an acknowledgement of the ultimate taboo—death—is the problem of the sanitizing of the tension between the crossing of the line of the symbolic taboo and prohibition; this undermines a “libidinal investment” towards the sacred, which is central in Bataille’s theory. Second, focussing on “eroticism”, since this embodies the emancipative potential of the Bataillean sacred, I argue that while a fear of eroticism marks out the digital technological realm, this is covered up by the blurring of boundaries between pleasure, fun and sex(iness) that currently governs our experience with technological devices

The Diversity of Religious Diversity. Using Census and NCS Methodology in Order to Map and Assess the Religious Diversity of a Whole Country

Religious diversity is often captured in “mapping studies” that use mostly qualitative methods in order to map and assess the religious communities in a given area. While these studies are useful, they often present weaknesses in that they treat only limited geographic regions, provide limited possibilities for comparing across religious groups and cannot test theories. In this article, we show how a census and a quantitative national congregations study (NCS) methodology can be combined in order to map and assess the religious diversity of a whole country (Switzerland), overcoming the problems mentioned above. We outline the methodological steps and selected results concerning organizational, geographic, structural, and cultural diversity

‘NOT A RELIGIOUS STATE’ A study of three Indonesian religious leaders on the relation of state and religion

This article explores the concept of a ‘secular state’ offered by three Indonesian religious leaders: a Catholic priest, Nicolaus Driyarkara (1913–1967), and two Muslim intellectuals who were also state officials, Mukti Ali (1923–2004) and Munawir Sjadzali (1925–2004). All three, who represented the immediate generation after the revolution for Indonesian independence from the Dutch (1945), defended the legitimacy of a secular state for Indonesia based on the state ideology Pancasila (Five Principles of Indonesia). In doing so, they argued that a religious state, for example an Islamic state, is incompatible with a plural nation that has diverse cultures, faiths, and ethnicities. The three also argued that the state should remain neutral about its citizens’ faith and should not be dominated by a single religion, i.e. Islam. Instead, the state is obliged to protect all religions embraced by Indonesians. This argument becomes a vital foundation in the establishment of Indonesia’s trajectory of unique ‘secularisation’. Whilst these three intellectuals opposed the idea of establishing a religious or Islamic state in Indonesia, it was not because they envisioned the decline of the role of religion in politics and the public domain but rather that they regarded religiosity in Indonesia as vital in nation building within a multi-religious society. In particular, the two Muslim leaders used religious legitimacy to sustain the New Order’s political stability, and harnessed state authority to modernise the Indonesian Islamic community