Pedagogies of practice and opportunities to learn about classroom mathematics discussions

In this paper, we argue that to prepare pre-service teachers for doing complex work of teaching like leading classroom mathematics discussions requires an implementation of different pedagogies of teacher education in deliberate ways. In supporting our argument, we use two frameworks: one curricular and one pedagogical. The curricular framework is based on the work of Hammerness et al. (Preparing teachers for a changing world. What teachers should learn and be able to do. San Francisco, Jossey-Bass Educational Series, pp 358–388, 2005) outlining four main goals of teacher learning: a vision of practice, knowledge of students and content, dispositions for using this knowledge, and a repertoire of practices and tools. The pedagogical framework is based on the work of Grossman et al. (Teach Teach Theory Pract 15(2):273–289, 2009a; Teach Coll Record 111(9):2055–2100, 2009b) outlining three pedagogies of practice: representations, decompositions, and approximations of practice. We use the curricular framework to examine the opportunities for teacher learning that were afforded by these three different pedagogies of practice in a unit on leading classroom mathematics discussion in a secondary mathematics methods course. We use evidence from our analysis to show how the coordination of those pedagogies of practice is better than any one of them in addressing important goals for teacher learning about classroom discussions.     

Expanding opportunities to learn to support inclusive education through drama‐enhanced literacy practices

This study examines how the Early Years Educators at Play (EYEPlay) professional development (PD) programme supported inclusive learning settings for all children, including English language learners and students with disabilities. The EYEPlay PD model is a year‐long programme that integrates drama strategies into literacy practices within real‐classroom contexts. Inclusive education refers to ensuring equal opportunities to access and participation in learning activities for all students. Cultural‐historical activity theory was used to understand and unpack the drama practices. Twelve semi‐structured focus group interviews were conducted with 19 preschool teachers. The data were analysed via constant‐comparative and interpretive methods. The study findings showed that EYEPlay PD practices enhanced inclusive learning settings for diverse groups of students by increasing access and expanding opportunities to learn, and supporting a positive learning environment.     

让学生与数学思考同行

新课标将"数学思考"作为义务教育数学课程的总目标之一,可见"数学思考"在现阶段教育数学教学中的作用。"数学思考"的实现程度已成为衡量课堂有效性的一项重要指标,小学数学教学中更应关注学生的数学思考,让数学思考贯穿于整个教学过程中,从而提升学生学习活动的思维含量、构建高效的数学课堂。本文对目前的"数学思考"现状进行了简单的介绍,根据现状分析,重点阐述了培养"数学思考"的策略,主要通过创设问题情境、创见性提问、把握时机和注重细节等几个方面,对提升学生"数学思考"的方法进行了详细的分析。     

Digital technologies to teach and learn mathematics: Context and re-contextualization

The central assumption of this paper is that, especially in the field of digital technologies to teach and learn mathematics, the influence of the context in which research is carried out has not been given enough attention, so that research results are not really useful outside this context. We base our discussion on the work of a group of European teams carrying out research with a special methodology of “cross-studies” and carrying out “cross-analyses” of particular studies. A context for a research study is described as a dynamic construction by researchers, connecting relevant contextual characteristics in the settings (empirical and academic) where research activity takes place and helping to gain insight from the outcomes of the study. Analyzing the design of two “Didactical Digital Artefacts,” and the associated cross-studies involving teams of three countries, we identify more or less conscious influences of characteristics in the researchers' contexts upon research outcomes. Cross-studies and cross-analysis help to go further by making researchers more aware of their context and of its characteristics. It also helps researchers to “re-contextualize,” that is to say to identify new contextual characteristics in the settings they are acting in, to gain insight from research outcomes that emerged in other contexts.     

让学生在体验中学习数学

教学环境充分民主自由，数学内容走进学生生活，数学学习融于自主活动。     

Coach learning to help teachers learn to enact conceptually rich,student-focused mathematics lessons

The past decade has witnessed a strong, standards-based call for improving what mathematics is taught and how it is taught. In the USA, districts have hired instructional coaches to help teachers shift their teaching from algorithm-based instruction to instruction that is more student-centered and conceptually focused. The purpose of this study was to contribute to the field’s understanding of (a) the specific coaching practices that help teachers enact more conceptual-based forms of instruction; and (b) how coaches learn to enact those practices. Using a design-based implementation research approach, we trained coaches using a particular model for one-on-one coaching (Content-Focused Coaching); the coaches then worked with teachers to plan lessons aligned with the coaching model. Data consisted of videotapes of pre-lesson conferences that were transcribed and coded according to the model. Analyses of 32 coaches’ practice over a 2-year period suggest that each of the three components of our coaching model (attention to student thinking, pedagogy, and mathematics) demonstrated statistically significant improvement over time. An illustrative analysis of five coaching sessions of one coach revealed a progression over five sessions from planning discussions that stayed at the level of general strategies to more specific conversations about teaching a particular task and then to deeper discussions that integrate attention to mathematical concepts, student thinking, and pedagogical moves. We view this delineation of coach learning as an important first step in laying the groundwork for the design of future coach training.

     

Scaffolding practices that enhance mathematics learning

It is over 25 years since Wood, Bruner and Ross (1976, Journal of Child Psychology and Psychiatry, 17, 89–100) introduced the idea of ‘scaffolding’ to represent the way children’s learning can be supported. Despite problems, this metaphor has enduring attraction in the way it emphasises the intent to support a sound foundation with increasing independence for the learner as understanding becomes more secure. It has resonance with the widely accepted notion in teaching of construction and the constructivist paradigm for learning. The discussion that follows will characterise some teaching approaches that can be identified as scaffolding, revisiting some of the original classifications, and identifying further scaffolding strategies with particular reference to mathematics learning. Examples will be given from studies relating to geometry learning with four to 6 year olds and to arithmetic learning with older pupils.     

Motivating adults to learn mathematics in the workplace: a trade union approach

ABSTRACT

The aim of this qualitative study is to explore adults’ motivation to learn mathematics in the workplace and the role that the trade union education approach promoted in the United Kingdom plays in that motivation. The findings draw on data from 20 semi-structured in-depth interviews with adults learning mathematics, organised through their trade union representatives. Trade unions promote education which uses teaching and learning approaches based on collectivist and activist principles, which is different from mainstream education, so the research explores what might be learnt from this approach. The findings show that the trade union-led approach enables even long-held negative feelings towards mathematics to become positive. The research finds a strong link between supportive social networks, including Union Learning Representative, as well as positive social and emotional encounters in the classroom that develop adult learners’ confidence, increasing their motivation to both learn and use mathematics in their everyday lives. This change in feelings and motivation is termed an Affective Mathematical Journey. These findings, while taking place in a non-traditional context, nevertheless are relevant to practitioners working with adults in both traditional (school and college) and non-traditional (workplace and community) settings. There are also indications of the positive influence this learning has on the relationship between individual members and their trade union organisations.     

Using PIAAC data to learn more about the literacy practices of adults

ABSTRACT

The present article connects a secondary analysis of quantitative data from the Programme for the International Assessment of Adult Competencies (PIAAC) with the theoretical approach of ‘literacy practices’ and related research results from the so-called New Literacy Studies (NLS) tradition, which follows a cultural practices paradigm.

According to the literacy as social practice approach, the analysis of adults’ literacy and numeracy practices could provide relevant policy information about how to address target groups in adult literacy and basic education. Thus, a Latent Class Analysis was carried out with the German PIAAC dataset in order to differentiate the adult population by their uses of literacy, numeracy and ICT.

As a result of this procedure, three subgroups of adults can be distinguished by the frequency in which they use selected skill-related activities. Surprisingly, an adult’s individual literacy level does not clearly predict group membership. A further interesting result is that participants in one of the groups seem to compensate for the few chances they have to use their skills at work by using them more often in their everyday life. Both results contribute to the need to draw a more differentiated picture of adults with lower literacy skills.     

The role of family orientation in predicting Korean boys' and girls' achievement motivation to learn mathematics

Informed by achievement goal orientation and self-determination theories, we explored the role of cultural/contextual factors on Korean students' achievement motivation. Specifically, we examined the role of the Korean middle school students' family orientation as a mediator between their perceptions of parent goals or motivating styles and their achievement motivations in learning math, when their perceptions of classroom goal structures were controlled. We also investigated gender differences in the role of the students' family orientation and of their perceptions of their parents and classroom variables in predicting their own achievement motivations. Multi-sample path analyses indicated that both Korean boys' and girls' family orientation mediated between their perceptions of parent variables and their own achievement motivations. Korean students' family orientation predicted mainly controlled forms of motivation and ego-focused goals. In addition, the relationships between students' perceptions of parent variables and their achievement motivations statistically varied across gender.     

Learning opportunities and negotiating social norms in mathematics class discussion

In this paper, we describe our current understanding of the relationship between learning opportunities and the negotiation of social norms in mathematics class discussion. With supportive social norms, a student non-acceptance can lead to constructive discussion in which students compare, contrast, elaborate and refute mathematics assertions just as mathematicians do. Under this circumstance, opportunities for learning are ample because students are more likely to find a situation problematic during inquiry activities. However, incompatible interpretations often cause emotional responses. Negative emotional responses, when not dealt with appropriately, tend to discourage students from participating in inquiry activities. Therefore, teachers need to take an active role in helping students interpret such situations through negotiating social norms. The negotiation of social norms makes possible the negotiation of mathematical meaning.     

The role of high school racial composition and opportunities to learn in students' STEM college participation

We analyze longitudinal data from students who spent their academic careers in North Carolina (NC) public secondary schools and attended NC public universities to investigate the importance of high school racial composition and opportunities to learn in secondary school for choosing a science, technology, engineering, and mathematics (STEM) major. We consider school racial composition and opportunities to learn as contexts that shape students' decisions regarding college majors. Results of cross‐classified hierarchical logistic models indicate that attending schools with predominantly White students is negatively associated with declaring a STEM major and with graduating with a STEM major irrespective of students' own race. The finding suggests that for students in North Carolina, attending racially isolated White high schools is related to a decrease in adolescents' participation in STEM during college.     

Understanding practices in mathematics education: structure and text

The practices of mathematics education can be investigated at a wide variety of levels: from the actions of individual students or teachers through classroom interactions, school structures, curriculum specifications and materials, teacher development programmes and local, national or international systems of instruction and assessment. These levels are, however, inter-related. The study of a national curriculum gains significance as we see how it impacts upon and is interpreted by teachers and students. The study of an individual’s actions makes more sense when these are interpreted in light of the broader context within which the individual is situated. This article is a version of a plenary lecture presented at the Psychology of Mathematics Education conference in 2009, addressing the conference theme “In Search for Theories in Mathematics Education”. In it, I trace the development of my own search for theories to address my wish to understand practices in the field and suggest some tools from linguistics, critical discourse analysis and social theory that can help to make such connections between the various levels of investigation, illustrating with a range of examples.     

Effects of a classroom discourse intervention on teachers' practice and students' motivation to learn mathematics and science

Student interest and motivation in STEM subjects has dropped significantly throughout secondary education, for which teacher–student interactions are named as a central reason. This study investigated whether a video-based teacher professional development (TPD) intervention on productive classroom discourse improved students' learning motivation and interest development over the course of a school year. The teachers' intervention group (IG; n = 6) was compared with a control group (CG; n = 4) who participated in a traditional TPD programme on classroom discourse. The teachers showed a significant increase in constructive feedback and decrease in simple feedback as a function of the treatment. Pre- and post-tests revealed that students in the IG (n = 136) significantly increased their perceived autonomy, competence and intrinsic learning motivation as compared with those in the CG (n = 90). They also showed significantly greater interest changes in the subjects compared with their peers in the CG.     

Master teachers' responses to twenty literacy and science/mathematics practices in deaf education

Under a grant to improve outcomes for students who are deaf or hard of hearing awarded to the Association of College Educators--Deaf/Hard of Hearing, a team identified content that all teachers of students who are deaf and hard of hearing must understand and be able to teach. Also identified were 20 practices associated with content standards (10 each, literacy and science/mathematics). Thirty-seven master teachers identified by grant agents rated the practices on a Likert-type scale indicating the maximum benefit of each practice and maximum likelihood that they would use the practice, yielding a likelihood-impact analysis. The teachers showed strong agreement on the benefits and likelihood of use of the rated practices. Concerns about implementation of many of the practices related to time constraints and mixed-ability classrooms were themes of the reviews. Actions for teacher preparation programs were recommended.     

引入图形计算器,改变学生的数学学习思维方式

随着新一轮的课堂教学改革,图形计算器作为教与学的工具,进入了中学课堂。本文通过两个典型教学案例,探讨了图形计算器在教学过程中对改变学生的数学学习思维方式的作用。