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The topic of similarity plays an essential role in developing students’ deductive reasoning. However, knowing how to teach similarity and understanding how to incorporate deductive reasoning and proof along with plane geometry remain a challenge to both school curriculum creators and teachers. This study identified the problems and characteristics regarding how similarity is treated in secondary mathematics textbooks in Hong Kong in the past half century. The content analysis method was used to analyze six secondary mathematics textbook series published in different periods. From the epistemological perspective of the textbook contents, our analysis shows the historical context and learning trajectories of how similarity was treated in school curriculum. The natural axiomatic geometry paradigm is not emphasized too much at different stages and most of the textbooks did not provide formal proofs of similarity. The intuitive idea was gradually consolidated into a formal definition of similarity. Furthermore, the way that rigorous geometric deduction can be performed from intuitive concepts and experimental geometry to the idea of proofs and formal proofs is also discussed.

This paper discusses some of the recurrent issues which the authors have noticed in educational research. These might be concerned with the nature of social science research, such as its representation of the reality of student learning, the role for replication studies, and the tolerance for disagreements. There are also issues related to the researching process, including the scope of prior research that should be reviewed, the purpose of triangulation, the need to be data-sensitive, the value in the complementarity of quantitative and qualitative information, relationships between theory and data, researchers' statistical literacy levels, and over-dependence on software-generated results. The fluid multi-cultural context in which education and pedagogy are situated has also meant that accessibility to knowledge is language-mediated and social reality, ever-evolving.

It has become increasingly recognized that what teachers and students value affect teaching and learning in general and in the area of mathematics in particular. Yet, the extent to which this is so varies across cultural regions. In recent years, how the ethnic Chinese teach and learn mathematics has attracted much attention worldwide. It is precisely the purpose of the present study to investigate the value structures of three Chinese regions. Using a recently developed and validated questionnaire, students’ values in mathematics learning in the Chinese Mainland, Hong Kong and Taiwan are delineated. In the first place, the results reveal that there are six dimensions in the students’ value structure, namely achievement, relevance, practice, communication, information and communications technology as well as feedback. However, in each of the six value components derived from the principal components analysis, statistically significant differences between the regions were found.

Beliefs about mathematics education and their influences on teaching practices have been widely investigated in recent decades. There have been numerous empirical studies on the influences of religions on teachers’ and students’ beliefs about subjects such as sciences and language. However, the influences of worldviews in general and religions in particular, as one of the major sources of beliefs in relation to mathematics education, are under-researched. The current study is a first step to unpacking the relationship between teachers’ religions and their beliefs about mathematics teaching and learning. By means of semi-structured interviews with mathematics teachers of different religious backgrounds, teachers’ perceptions on the connection between their personal religious beliefs and their beliefs about teaching and learning are investigated. In-depth analyses of the perceptions of three mathematics teachers reveal the complex relationship between teachers’ religious beliefs and their teaching beliefs. First, there are some common values shared by different religions, which influence the beliefs about mathematics teaching and learning as well as education in general. Second, religion is a rich belief system, and the teachers appear to apply only a portion of their religious beliefs to guide their teaching. It is also possible that a teacher is influenced by more than one religion or cultural tradition. Despite its subtleties, our study provides evidence to support the alignment between teachers’ personal religious beliefs and their beliefs about mathematics teaching and learning.

The achievement in mathematics among the Chinese has aroused the interest of educationalists, sociologists and psychologists worldwide. The coining of the term Confucian Heritage Culture learner’s phenomenon mistakenly led researchers to look for direct attributions from Confucianism, without realizing that Confucianism is just one Chinese school of thought, and it is over-simplistic to draw causal relationships between schools of thought and social phenomena. This paper begins by introducing three major Chinese schools of thought—Confucianism, Daoism and Buddhism—and their views on education. The discussion will then focus on how these schools of thought might impact education in general and mathematics education in particular.

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. [Author abstract]

The spiral bianshi curriculum, an improvement on bianshi teaching developed by Gu (2000) and in line with Marton’s theory of variation (Marton & Booth, 1997), was tried out in a primary school in Hong Kong. This improved theoretical framework for the spiral bianshi curriculum comprises four types of bianshi problems—the inductive bianshi, the broadening bianshi, the deepening bianshi, and the applicative bianshi. Based on this framework, the research team developed a set of teaching materials on the three topics of division of fraction, speed, and volume. The materials were tried out in 21 Primary 6 classes (a total of 686 students) in a school. The effect was compared with a reference group using standard textbook materials in Hong Kong. A series of instruments, pre-tests, and post-tests were administered to gauge the effects on students’ performance in solving routine and non-routine problems, as well as the affective outcomes including self-concept, attitude towards learning mathematics, approaches to learning, and conceptions of mathematics. The intervention effects of the experimental design were examined by hierarchical regression analysis. The research reveals that students using spiral bianshi teaching materials performed significantly better than their counterparts using standard textbook materials. However, no significant differences were identified among affective learning outcome variables despite the positive results on cognitive learning outcomes. The findings indicate that spiral bianshi curriculum has high potential in enhancing students’ learning effectiveness. However, further studies are needed to map its strengths in detail.

Education for Sustainable Development (ESD) has been recently recognized as an important area in the new Chinese educational reform. As teachers play a pivotal role, knowing and developing an effective and easy-to-use instrument for tapping teachers' beliefs is essential. This article reports an attempt to develop an instrument with mixed methods. The finalized instruments comprise two subscales with satisfactory reliability indices obtained. Sustainability values (VSD) consists of four dimensions: respect and care for the community of life; ecological integrity; social and economic justice; and democracy, nonviolence, and peace. Teaching beliefs of ESD (TESD) consists of three dimensions: relevance to daily life; students' need in the future; and integrated teaching. With these validated instruments, future research and potential problems will be less strenuous.

Twelve experienced mathematics teachers in Hong Kong were invited to face-to-face semi-structured interviews to express their views about mathematics, about mathematics learning and about the teacher and teaching. Mathematics was generally regarded as a subject that is practical, logical, useful and involves thinking. In view of the abstract nature of the subject, the teachers took abstract thinking as the goal of mathematics learning. They reflected that it is not just a matter of “how” and “when”, but one should build a path so that students can proceed from the concrete to the abstract. Their conceptions of mathematics understanding were tapped. Furthermore, the roles of memorisation, practices and concrete experiences were discussed, in relation with understanding. Teaching for understanding is unanimously supported and along this line, the characteristics of an effective mathematics lesson and of an effective mathematics teacher were discussed. Though many of the participants realize that there is no fixed rule for good practices, some of the indicators were put forth. To arrive at an effective mathematics lesson, good preparation, basic teaching skills and good relationship with the students are prerequisite.