Promoting student understanding through complex learning

The world’s increasing complexity, competitiveness, interconnectivity, and dependence on technology generate new challenges for nations and individuals that cannot be met by “continuing education as usual” (The National Academies, 2009). With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualization. These technologies have led to significant changes in the forms of mathematical thinking that are required beyond the classroom. This paper argues for the need to incorporate future-oriented understandings and competencies within the mathematics curriculum, through intellectually stimulating activities that draw upon multidisciplinary content and contexts. The paper also argues for greater recognition of children’s learning potential, as increasingly complex learners capable of dealing with cognitively demanding tasks.

iPed : pedagogy for digital text production

Reading and writing are being transformed by global changes in communication practices using new media technologies. This paper introduces iPed, a research-based pedagogy that enables teachers to navigate innovative digital text production in the literacy classroom. The pedagogy was generated in the context of a longitudinal digital literacy intervention in a school that services low-socioeconomic and ethnically diverse students. iPed synthesizes four key pedagogies that were salient in the analysis of over 180 hours of lesson observations – Link, Challenge, Co-Create, and Share. The strengths of the pedagogy include connecting to students’ home cultures, critical media literacy, collaborative and creative digital text production, and gaining cosmopolitan recognition within global communities.

Grade A Taipei junior high school principals’ self-reported leadership practices

The principal’s leadership and curriculum development are considered the core elements for creating a high performing junior high school. In Taiwan, mathematics curriculum reform has been an ongoing topic since 1994. The pedagogy, classroom interactions, and the underlying philosophy of mathematics education have varied with different versions of guidelines. These changes inevitably increase the requirement for principals’ leadership in order to effectively implement the curriculum reform. Principals’ leadership is essential to the success of the implementation in their school. This study aimed to explore and identify the leadership of junior high school principals whose schools had been judged by the Taipei City Government as Grade A junior high schools. Principals’ implementations of the reformed mathematics curriculum were used as examples to generate insights of their leadership. This study drew upon a multiple-case study approach. Data were collected from interviews, observations, and documentations. Bass and Avolio’s (1997) full range leadership theory provided a structure for gaining insight into these principals’ leadership practices. Five Grade A Taipei junior high school principals participated and shared their leadership concepts and experiences. Findings revealed that the leadership preferences of the five principles varied considerably. Management by exception-active, contingent reward, individualised consideration, and idealised influence were Grade A Taipei junior high school principals’ preferred leadership practices. In addition, principals’ leadership strategies associated with these practices were identified. These principals had adopted a range of leadership strategies according to the staff and school needs. Results of this study have implications for both Taiwanese principals and education departments. Principals can enhance their leadership by gaining more understanding about the Grade A principals’ leadership practices and strategies. Taiwanese education departments can improve school leadership training programs by focusing on these practices and strategies, which may also lead to more effective strategies for implementing national curriculum reform.

Primary students' success on the structured number line

Number lines are part of our everyday life (e.g., thermometers, kitchen scales) and are frequently used in primary mathematics as instructional aids, in texts and for assessment purposes on mathematics tests. There are two major types of number lines; structured number lines, which are the focus of this paper, and empty number lines. Structured number lines represent mathematical information by the placement of marks on a horizontal or vertical line which has been marked into proportional segments (Figure 1). Empty number lines are blank lines which students can use for calculations (Figure 2) and are not discussed further here (see van den Heuvel-Panhuizen, 2008, on the role of empty number lines). In this article, we will focus on how students’ knowledge of the structured number line develops and how they become successful users of this mathematical tool.

Assessing and learning in internships: Reflecting on pedagogy and progress

An elective internship unit as part of a work integrated learning program in a business faculty is presented as a case study. In the unit, students complete a minimum of 120 hours work placement over the course of a 13 week semester. The students are majoring in advertising, marketing, or public relations and are placed in corporations, government agencies, and not for profit organisations. To support and scaffold the students’ learning in the work environment, a range of classroom and online learning activities are part of the unit. Classroom activities include an introductory workshop to prepare students for placement, an industry panel, and interview workshop. These are delivered as three workshops across the semester. Prior to commencing their placement, students complete a suite of online learning modules. The Work Placement Preparation Program assists students in securing obtaining a placement and make a successful transition to the work environment. It provides an opportunity for students to source possible work placement sites, prepare competitive applications, develop and rehearse interview skills, deal with workplace issues, and use a student ePortfolio to reflect on their skills and achievements. Students contribute to a reflective blog throughout their placement, with feedback from academic supervisors throughout the placement. The completion of the online learning modules and contribution to a reflective blog are assessed as part of the unit. Other assessment tools include a internship plan and learning contract between the student, industry supervisor, and academic supervisor; job application including responses to selection criteria; and presentation to peers, academics and industry representatives at a poster session. The paper discusses the development of the internship unit over three years, particularly learning activities and assessment. The reflection and refinement of the unit is informed by a pedagogical framework, and the development of processes to best manage placement for all stakeholders. A model of best practice is proposed, that can be adapted to a variety of discipline areas.

Years 2 to 6 students' ability to generalise : models, representations and theory for teaching and learning

Over the last three years, in our Early Algebra Thinking Project, we have been studying Years 3 to 5 students’ ability to generalise in a variety of situations, namely, compensation principles in computation, the balance principle in equivalence and equations, change and inverse change rules with function machines, and pattern rules with growing patterns. In these studies, we have attempted to involve a variety of models and representations and to build students’ abilities to switch between them (in line with the theories of Dreyfus, 1991, and Duval, 1999). The results have shown the negative effect of closure on generalisation in symbolic representations, the predominance of single variance generalisation over covariant generalisation in tabular representations, and the reduced ability to readily identify commonalities and relationships in enactive and iconic representations. This chapter uses the results to explore the interrelation between generalisation and verbal and visual comprehension of context. The studies evidence the importance of understanding and communicating aspects of representational forms which allowed commonalities to be seen across or between representations. Finally the chapter explores the implications of the studies for a theory that describes a growth in integration of models and representations that leads to generalisation.

Three primary school students’ cognition about 3D rotation in a virtual reality learning environment

This paper reports on three primary school students’ explorations of 3D rotation in a virtual reality learning environment (VRLE) named VRMath. When asked to investigate if you would face the same direction when you turn right 45 degrees first then roll up 45 degrees, or when you roll up 45 degrees first then turn right 45 degrees, the students found that the different order of the two turns ended up with different directions in the VRLE. This was contrary to the students’ prior predictions based on using pen, paper and body movements. The findings of this study showed the difficulty young children have in perceiving and understanding the non-commutative nature of 3D rotation and the power of the computational VRLE in giving students experiences that they rarely have in real life with 3D manipulations and 3D mental movements.

Pedagogy and participation : literacy education for low-literate refugee students of African origin in a western school system

For ESL teachers working with low-literate adolescents the challenge is to provide instruction in basic literacy capabilities while also realising the benefits of interactive and dialogic pedagogies advocated for the students. In this article we look at literacy pedagogy for refugees of African origin in Australian classrooms. We report on an interview study conducted in an intensive English language school for new arrival adolescents and in three regular secondary schools. Brian Street’s ideological model is used. From this perspective, literacy entails not only technical skills, but also social and cultural ways of making meaning that are embedded within relations of power. The findings showed that teachers were strengthening control of instruction to enable mastery of technical capabilities in basic literacy and genre analysis. We suggest that this approach should be supplemented by a critical approach transforming relations of linguistic power that exclude, marginalise and humiliate the study students in the classroom.

Narrative on a doctoral narrative : reflections on postgraduate study and pedagogy

This paper explores the stages of one student’s intellectual journey through a Doctor of Philosophy program of study in an Australian university. It outlines the theoretical and methodological insights made as she came to understand that data was discourse, entailing a politics and position of power that ran contrary to the aims of the study that the student was undertaking in and on her own community. The article is a reflective narrative produced from the experience of having to come to terms with some of the personal and professional tensions and contradictions that postgraduate study can, and maybe should, engender if it is to be any of real value.

Early childhood teachers' mathematical content knowledge

The process of becoming numerate begins in the early years. According to Vygotskian theory (1978), teachers are More Knowledgeable Others who provide and support learning experiences that influence children’s mathematical learning. This paper reports on research that investigates three early childhood teachers mathematics content knowledge. An exploratory, single case study utilised data collected from interviews, and email correspondence to investigate the teachers’ mathematics content knowledge. The data was reviewed according to three analytical strategies: content analysis, pattern matching, and comparative analysis. Findings indicated there was variation in teachers’ content knowledge across the five mathematical strands and that teachers might not demonstrate the depth of content knowledge that is expected of four year specially trained early years’ teachers. A significant factor that appeared to influence these teachers’ content knowledge was their teaching experience. Therefore, an avenue for future research is the investigation of factors that influence teachers’ content numeracy knowledge.

Towards a classification of criterion referenced assessment models in mathematics courses : student and academic perspectives

We present the findings of a study into the implementation of explicitly criterion- referenced assessment in undergraduate courses in mathematics. We discuss students' concepts of criterion referencing and also the various interpretations that this concept has among mathematics educators. Our primary goal was to move towards a classification of criterion referencing models in quantitative courses. A secondary goal was to investigate whether explicitly presenting assessment criteria to students was useful to them and guided them in responding to assessment tasks. The data and feedback from students indicates that while students found the criteria easy to understand and useful in informing them as to how they would be graded, it did not alter the way the actually approached the assessment activity.

Teachers' work and pedagogy in an era of accountability

A great deal of educational policy proceeds as though teachers are malleable and ever-responsive to change. Some argue they are positioned as technicians who simply implement policy. However, how teachers go about their work and respond to reform agendas may be contingent upon many factors that are both biographical in nature and workplace related. In this paper we discuss the work of middle school teachers in low-socioeconomic communities from their perspectives. Referring to reflective interviews, meeting transcripts and an electronic reporting template, we examine how teacher participants in a school reform project describe their work - what they emphasise and what they down-play or omit. Using Foucaultian approaches to critical discourse analysis and insights from Dorothy Smith's (2005) Institutional Ethnography, we consider the 'discursive economy' (Carlson, 2005) in teachers' reported experiences of their everyday practices in northern suburbs schools in South Australia in which a democratic progressive discourse exists alongside corporate and disciplinary discourses.

Pedagogy of feedback on student academic writing : research supervisor practices for the end of the research degree candidature

All academic writing is advanced with the benefit of feedback about the writing. In the case of the academic writing genres of the research proposal and the dissertation, feedback is usually provided by the research supervisor. Given that academic writing development is a process, and in the case of the research proposal and dissertation, writing which develops over time, it seems likely that the nature of feedback on drafts written early in the candidature may be different from feedback provided by the research supervisor later in a student’s candidature. ----- ----- When a research supervisor has been reading a student’s writing over a period of time, their own familiarity with the writing generates a risk to their ability to provide critical and objective feedback. Particularly by the end of a student’s candidature, the research supervisor’s familiarity with the work may cause them to miss elements of writing improvement. ----- ----- The author, as a research supervisor, has developed a feedback grid to facilitate feedback on the final drafts of a dissertation. This feedback grid is generated by the embedded promises in the early sections of the dissertation, which are then used to audit the content of the final sections of the dissertation to ascertain whether promises made have been fulfilled. This provides a strategy for the research supervisor to step back from the work and read the dissertation with the agenda of a dissertation examiner. ----- ----- The grid is one strategy within a broader pedagogy of providing feedback on writing samples.