Identifying authority structures in mathematics classroom discourse: a case of a teacher’s early experience in a new context

We explore a conceptual frame for analyzing mathematics classroom discourse to understand the way authority is at work. This case study of a teacher moving from a school where he is known to a new setting offers us the opportunity to explore the use of the conceptual frame as a tool for understanding how language practice and authority relate in a mathematics classroom. This case study illuminates the challenges of establishing disciplinary authority in a new context while also developing the students’ sense of authority within the discipline. To analyze the communication in the teacher’s grade 12 class in the first school and grade 9 class early in the year at the new school, we use the four categories of positioning drawn from our earlier analysis of pervasive language patterns in mathematics classrooms—personal authority, discourse as authority, discursive inevitability, and personal latitude.     

What students value in effective mathematics learning: a ‘Third Wave Project’ research study

Considerations of how mathematics can be effectively taught and learnt may be cognitive, affective or sociocultural in approach. ??What students value in effective mathematics learning?? is a research study of the Third Wave Project, an international consortium of research teams adopting a sociocultural approach to investigate the harnessing of relevant values to optimise school mathematics teaching and learning. This paper seeks to contextualise the study, as part of the study examines what the high-achieving East Asian mathematics students value. The study is framed by knowledge relating to the relative cultural influence on effectiveness in mathematics learning, Alan Bishop??s values in mathematics education and the role of interactions in education, in particular David Tripp??s idea of critical incidents as reflecting professional judgement (and, thus, underlying values). Features of the innovative qualitative research design are also presented, which include the facilitation of photo-voice, the argument for an international collaborative team and focus group interviews for all values research, and a two-stage data analysis process aimed at clarifying both etic and emic perspectives.     

The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching: practitioners’ views

This study investigates the different ways by which secondary school mathematics teachers view how advanced mathematics studies are relevant to expertise in classroom instruction. Data sources for this study included position papers and written notes from a group interview of 15 Israeli teachers who studied in a special master’s program, of which advanced mathematics courses comprise a sizeable share. Data analysis was iterative and comparative, aiming at identifying and characterizing teachers’ different perspectives. Overall, all participating teachers thought that the advanced mathematics studies in the program were relevant to their teaching of secondary school mathematics. Moreover, teachers specifically mentioned the importance of studying contemporary mathematics from research mathematicians. All teachers pointed out at least one specific feature that they viewed as relevant to their work: advanced mathematics courses (1) as a resource for teaching secondary school mathematics, (2) for improving understanding about what mathematics is, and (3) for reminding teachers what learning mathematics feels like.     

Shortcomings of Mathematics Education Reform in The Netherlands: A Paradigm Case?

This article offers a reflection on the findings of three PhD studies, in the domains of, respectively, subtraction under 100, fractions, and algebra, which independently of each other showed that Dutch students' proficiency fell short of what might be expected of reform in mathematics education aiming at conceptual understanding. In all three cases, the disappointing results appeared to be caused by a deviation from the original intentions of the reform, resulting from the textbooks' focus on individual tasks. It is suggested that this “task propensity”, together with a lack of attention for more advanced conceptual mathematical goals, constitutes a general barrier for mathematics education reform. This observation transcends the realm of textbooks, since more advanced conceptual mathematical understandings are underexposed as curriculum goals. It is argued that to foster successful reform, a conscious effort is needed to counteract task propensity and promote more advanced conceptual mathematical understandings as curriculum goals.     

The structural implications of measurement error in sociometry†

Measurement error, an inherent quality of any empirical data collection technique, is discussed in the context of sociometric data. These data have long been assumed to possess face validity and to be the data of choice in any study of the sentiment structure of small scale social systems. However, it is argued that while methods of sociometric analysis have become increasingly more sophisticated they have failed to yield unequivocal results because they do not distinguish structural complexity from measurement error. Through a discussion of increasingly more complex examples the distortion laden character of most sociometric data is illustrated. This distortion is introduced by the formalities of the sociometric test and it will not be removed by developing increasingly more sophisticated structural models or throwing out some of the data. Instead, when issues concerning the nature of specific relational networks are raised data of much higher quality than those which are commonly available are required. A technique for generating high quality sociometric data is briefly discussed. On the other hand, it is suggested that the extant body of sociometric data ought to be adequate when sizeable aggregations are examined for evidence of statistical tendencies in structure.     

The normalizer of γ1(N) in psl2(R)

We find the full normalizer of γ₁(N) in psl₂(R).     

Characterization of submartingales of a new class (Σr)

In this work, we shall consider a new class (Σ^r) of local submartingales of the form X_t = M_t + A_t, where (M_t)_{t ? 0} is a càdlàg (right continuous with left limits) local martingale, (A_t)_{t ? 0} is a càdlàg increasing process, and the measure (dA) is carried by the set {t: X_{t ?} = 0}.

The aim of the present paper is to study the positive and negative parts of processes of this class and establish some martingale characterizations. The formula of relative martingales is derived in terms of last passage time. Finally, by using balayage formula, we calculate predictable compensator.     

The estimation of the prevalence of delinquency: Two approaches and a correction of the literature†

The index “the proportion of a cohort that have become delinquent by a given age,” here called the “prevalence of delinquency,” is an important social indicator. In the present paper, we indicate methods by which this index can be estimated from data, and correct errors in previous sex‐ and race‐specific prevalence estimates published by Monahan (1960) for the city of Philadelphia. The difference between the sexes and between the races shown by these corrected prevalence estimates are of sufficient magnitude to render suspect any comparisons of prevalences of delinquency among cohorts which do not take account of the sex and race compositions of the cohorts to be compared.     

Place and use of new technology in the teaching of mathematics: ICMI activities in the past 25 years

Looking back at the place of technology in the past ICMEs during the last decades and on the two ICMI studies devoted to technology, it is obvious that the role and use of technology has given rise to a diversity of points of view and attitudes across the world. The ICMEs and the two studies are places where researchers, teacher educators and practitioners meet. To what extent do they reflect the evolution of the trends of research and/or of integration of technology into real practice? The study will develop a general analysis of the theoretical frameworks, issues and wishes related to the use of technology in mathematics teaching from the proceedings of past ICMEs and of the two ICMI studies. Both a quantitative and a qualitative point of view will be adopted. From the great diversity of questions and approaches, the study attempts to formulate the main trends and their evolution over time within the ICMI community, as well as some pertinent issues for the coming years.     

The accuracy of alternative stochastic models of participation in group discussion†

Four related models of participation in group discussion are presented and compared in the accuracy with which they predict proportional participation, mean run length, mean recurrence time and the variances of runs and recurrences. For some quantities, such as proportional participation and mean run length, the most restricted model does virtually as well as the least restricted model; for other quantities it does not. None of the models does a good job of predicting variances of runs or recurrences.     

Instructors’ use of technology in post-secondary undergraduate mathematics teaching: a local study

In this study, instructors of undergraduate mathematics from post-secondary institutions in Newfoundland were surveyed (N = 13) and interviewed (N = 8) about their use of, experiences with, and views on, technologically assisted teaching. It was found that the majority of them regularly use technologies for organizational and communication purposes. However, the use of math-specific technology such as computer algebra systems, or dynamic geometry software for instructional, exploratory, and creative activities with students takes place mostly on an individual basis, only occasionally, and is very much topic specific. This was even the case for those instructors who use technology proficiently in their research. The data also suggested that familiarity with and discussions of examples of technology implementation in teaching at regular and field-oriented professional development seminars within mathematics departments could potentially increase the use of math-specific technology by instructors.     

Post-primary students’ images of mathematics: findings from a survey of Irish ordinary level mathematics students

A questionnaire survey was carried out as part of a PhD research study to investigate the image of mathematics held by post-primary students in Ireland. The study focused on students in fifth year of post-primary education studying ordinary level mathematics for the Irish Leaving Certificate examination – the final examination for students in second-level or post-primary education. At the time this study was conducted, ordinary level mathematics students constituted approximately 72% of Leaving Certificate students. Students were aged between 15 and 18 years. A definition for ‘image of mathematics’ was adapted from Lim and Wilson, with image of mathematics hypothesized as comprising attitudes, beliefs, self-concept, motivation, emotions and past experiences of mathematics. A questionnaire was composed incorporating 84 fixed-response items chosen from eight pre-established scales by Aiken, Fennema and Sherman, Gourgey and Schoenfeld. This paper focuses on the findings from the questionnaire survey. Students’ images of mathematics are compared with regard to gender, type of post-primary school attended and prior mathematical achievement.     

The beliefs of ‘Tomorrow's Teachers’ about mathematics: precipitating change in beliefs as a result of participation in an Initial Teacher Education programme

Mathematics education research has given increasing attention to the role of affective factors in the learning process. While 'affect' is used to refer to a variety of aspects including feelings, emotions, beliefs, attitudes and conceptions, this paper focuses on 'beliefs' of elementary pre-service teachers. In particular, the study evaluates the effect of participation in a reform-based elementary pre-service teacher education (referred to as Initial Teacher Education (ITE)) programme on participants' 'beliefs about the nature of mathematics'. This was completed using two (sub)scales of the Aiken's Revised Mathematics Scale measuring Enjoyment of Mathematics (E) and belief in the Value of Mathematics (V). Both scales were administered before and after participants completed the mathematics education programme, which consisted of 5 compulsory and consecutive modules. This study reveals that entry-level pre-service teachers report generally positive beliefs about the value of and enjoyment in doing mathematics. The findings challenge previous research, which report the tendency of teachers' beliefs to be resistant to change while in teacher education and suggest that it is possible for ITE mathematics education programmes to stimulate improvement in beliefs and attitudes among participants. Particular programme features are identified as instrumental in this positive change in beliefs about mathematics.     

Pre-service science teachers’ perceptions of mathematics courses in a science teacher education programme1

Most science departments offer compulsory mathematics courses to their students with the expectation that students can apply their experience from the mathematics courses to other fields of study, including science. The current study first aims to investigate the views of pre-service science teachers of science-teaching preparation degrees and their expectations regarding the difficulty level of mathematics courses in science-teaching education programmes. Second, the study investigates changes and the reasons behind the changes in their interest regarding mathematics after completing these courses. Third, the current study seeks to reveal undergraduate science teachers’ opinions regarding the contribution of undergraduate mathematics courses to their professional development. Being qualitative in nature, this study was a case study. According to the results, almost all of the students considered that undergraduate mathematics courses were ‘difficult’ because of the complex and intensive content of the courses and their poor background mathematical knowledge. Moreover, the majority of science undergraduates mentioned that mathematics would contribute to their professional development as a science teacher. On the other hand, they declared a negative change in their attitude towards mathematics after completing the mathematics courses due to continuous failure at mathematics and their teachers’ lack of knowledge in terms of teaching mathematics.     

Some recent progress in applied mathematics(Ⅱ)

1　Unified information theoryOne characteristic of the 2 0 th century is its information revolution.With satellites,huge amounts of information are collected daily,proposing a challenge to our understandingof the information.Also,we realize that different ways of comprehension can be very mis-leading,since all information contains noise and our reasoning and sensing organs are limit-ed.To this end,Shannon and Weaver in 1949established the first theory of informationdealing with uncertainty on…