Integrating concrete and virtual materials in an elementary mathematics classroom : a case study of success with fractions

This paper describes an approach to introducing fraction concepts using generic software tools such as Microsoft Office's PowerPoint to create "virtual" materials for mathematics teaching and learning. This approach replicates existing concrete materials and integrates virtual materials with current non-computer methods of teaching primary students about fractions. The paper reports a case study of a 12-year-old student, Frank, who had an extremely limited understanding of fractions. Frank also lacked motivation for learning mathematics in general and interacted with his peers in a negative way during mathematics lessons. In just one classroom session involving the seamless integration of off-computer and on-computer activities, Frank acquired a basic understanding of simple common equivalent fractions. Further, he was observed as the session progressed to be an enthusiastic learner who offered to share his learning with his peers. The study's "virtual replication" approach for fractions involves the manipulation of concrete materials (folding paper regions) alongside the manipulation of their virtual equivalent (shading screen regions). As researchers have pointed out, the emergence of new technologies does not mean old technologies become redundant. Learning technologies have not replaced print and oral language or basic mathematical understanding. Instead, they are modifying, reshaping, and blending the ways in which humankind speaks, reads, writes, and works mathematically. Constructivist theories of learning and teaching argue that mathematics understanding is developed from concrete to pictorial to abstract and that, ultimately, mathematics learning and teaching is about refinement and expression of ideas and concepts. Therefore, by seamlessly integrating the use of concrete materials and virtual materials generated by computer software applications, an opportunity arises to enhance the teaching and learning value of both materials.

Justifications in young children’s language in an early childhood classroom

This paper examines the interactional phenomenon of justification as it is produced in young children’s language. A justification provides a reason for one’s position and can be produced in children’s language at an early age. There are various pragmatic reasons for justifications. For example, justifications may be drawn upon by members to compensate for the disruption of the existing social order or to explain something that is possibly questionable. Justifications are also drawn upon to extend or close disputes. This study uses the analytical techniques of conversation analysis and membership categorisation to analyse video-recorded and transcribed interactions of young children (aged 4-6 years) in a preparatory classroom in a primary school in Australia. The focus is an episode that occurred within the block play area of the classroom that involved a dispute of ownership relating to a small, wooden plank. In analysing this dispute, justifications were frequent occurrences and the young participants drew upon justificatory devices in their everyday arguments. As the turns surrounding the justificatory language were examined, a pattern emerged: in each excerpt observed, a justification arose in response to a challenge. This pattern provided the basis for developing a model that helped to discern where, why and what type of justifications occurred in the interaction. To depict this interactional phenomenon, the model of ‘if x, then y’ was used, ‘x’ referring to the challenge or prompt, and ‘y’ referring to the justificatory response. Justifications related to the concepts of ownership and were used as devices by those engaged in disputes to support their positions and provide reasons for their actions. The children drew upon these child-constructed rules as resources to use in disputes with their peers, in order to construct and maintain the social order of the block area in the classroom.

Scaffolding techniques : a teacher training for cooperative learning in Thailand primary education

In general, the benefits of using cooperative learning include academic achievement, communication skills, problem-solving, social skills and student motivation. Yet cooperative learning as a Western educational concept may be ineffective in a different learning system. The study aims to investigate scaffolding techniques for cooperative learning in Thailand primary education. The program was designed to foster Thai primary school teachers’ cooperative learning implementation that includes the basic tenets of cooperative learning and socio-cognitive based learning. Two teachers were invited to participate in this experimental teacher training program for one and a half weeks. Then the teachers implemented a cooperative learning in their mathematics class for six weeks. The data from teacher interview and classroom observation indicated that the both teachers are able to utilise questions to scaffold their students’ engagement in cooperative learning. This initiative study showed that difficulty or failure of implementing cooperative learning in Thailand education may not be derived from cultural difference. The paper discussed the techniques the participant teachers applied with proactive scaffolding, reactive scaffolding and scaffolding questions that can be used to facilitate the implementation of cooperative learning in Thai school.

Teacher aides and their pedagogical contributions in the underachieving Indigenous mathematics classroom

This study examines the pedagogical contributions made by teacher aides in underperforming Indigenous mathematics secondary classrooms. Three teaching teams, each consisting of a teacher and their teacher aide, responded to semi-structured interviews. Their mathematics classrooms were observed for details of pedagogical contributions to the mathematics lessons. It was found that the pedagogical contributions of the teacher aides varied from co-teaching contributions, to the provision of menial support and behaviour management. The techniques used by the teacher aides to provide student feedback, to support behaviour management and to undertake questioning vary greatly, and this variance is also evident in the classroom atmosphere. Teacher aides are providing pedagogical contributions, and are engaged in instructional interactions, and are in a sense “teaching”.

Scaffolding techniques : a teacher training for cooperative learning in Thailand primary education

Using cooperative learning in classrooms promotes academic achievement, communication skills, problem-solving, social skills and student motivation. Yet it is reported that cooperative learning as a Western educational concept may be ineffective in Asian cultural contexts. The study aims to investigate the utilisation of scaffolding techniques for cooperative learning in Thailand primary mathematics classes. A teacher training program was designed to foster Thai primary school teachers’ cooperative learning implementation. Two teachers participated in this experimental program for one and a half weeks and then implemented cooperative learning strategies in their mathematics classes for six weeks. The data collected from teacher interviews and classroom observations indicates that the difficulty or failure of implementing cooperative learning in Thailand education may not be directly derived from cultural differences. Instead, it does indicate that Thai culture can be constructively merged with cooperative learning through a teacher training program and practices of scaffolding techniques.

Revisiting Polya's summation techniques using a spreadsheet : from addition tables to Bernoulli polynomials

Recurrence relations in mathematics form a very powerful and compact way of looking at a wide range of relationships. Traditionally, the concept of recurrence has often been a difficult one for the secondary teacher to convey to students. Closely related to the powerful proof technique of mathematical induction, recurrences are able to capture many relationships in formulas much simpler than so-called direct or closed formulas. In computer science, recursive coding often has a similar compactness property, and, perhaps not surprisingly, suffers from similar problems in the classroom as recurrences: the students often find both the basic concepts and practicalities elusive. Using models designed to illuminate the relevant principles for the students, we offer a range of examples which use the modern spreadsheet environment to powerfully illustrate the great expressive and computational power of recurrences.

Task profiling : A task-Based approach to measuring the integration of ICT in the classroom

The measurement of ICT (information and communication technology) integration is emerging as an area of research interest with such systems as Education Queensland including it in their recently released list of research priorities. Studies to trial differing integration measurement instruments have taken place within Australia in the last few years, particularly Western Australia (Trinidad, Clarkson, & Newhouse, 2004; Trinidad, Newhouse & Clarkson, 2005), Tasmania (Fitzallen 2005) and Queensland (Finger, Proctor, & Watson, 2005). This paper will add to these investigations by describing an alternate and original methodological approach which was trialled in a small-scale pilot study conducted jointly by Queensland Catholic Education Commission (QCEC) and the Centre of Learning Innovation, Queensland University of Technology (QUT) in late 2005. The methodology described is based on tasks which, through a process of profiling, can be seen to be artefacts which embody the internal and external factors enabling and constraining ICT integration.