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.

Teaching and learning mathematics with robotics in middle-years of schooling

Engaging and motivating students in mathematics lessons can be challenging. The traditional approach of chalk and talk can sometimes be problematic. The new generation of educational robotics has the potential to not only motivate students but also enable teachers to demonstrate concepts in mathematics by connecting concepts with the real world. Robotics hardware and the software are becoming increasing more user-friendly and as a consequence they can be blended in with classroom activities with greater ease. Using robotics in suitably designed activities promotes a constructivist learning environment and enables students to engage in higher order thinking through hands-on problem solving. Teamwork and collaborative learning are also enhanced through the use of this technology. This paper discusses a model for teaching concepts in mathematics in middle year classrooms. It will also highlight some of the benefits and challenges of using robotics in the learning environment.

Primary students’ interpretation of maps : gesture use and mapping knowledge

Maps are used to represent three-dimensional space and are integral to a range of everyday experiences. They are increasingly used in mathematics, being prominent both in school curricula and as a form of assessing students understanding of mathematics ideas. In order to successfully interpret maps, students need to be able to understand that maps: represent space, have their own perspective and scale, and their own set of symbols and texts. Despite the fact that maps have an increased prevalence in society and school, there is evidence to suggest that students have difficulty interpreting maps. This study investigated 43 primary-aged students’ (aged 9-12 years) verbal and gestural behaviours as they engaged with and solved map tasks. Within a multiliteracies framework that focuses on spatial, visual, linguistic, and gestural elements, the study investigated how students interpret map tasks. Specifically, the study sought to understand students’ skills and approaches used to solving map tasks and the gestural behaviours they utilised as they engaged with map tasks. The investigation was undertaken using the Knowledge Discovery in Data (KDD) design. The design of this study capitalised on existing research data to carry out a more detailed analysis of students’ interpretation of map tasks. Video data from an existing data set was reorganised according to two distinct episodes—Task Solution and Task Explanation—and analysed within the multiliteracies framework. Content Analysis was used with these data and through anticipatory data reduction techniques, patterns of behaviour were identified in relation to each specific map task by looking at task solution, task correctness and gesture use. The findings of this study revealed that students had a relatively sound understanding of general mapping knowledge such as identifying landmarks, using keys, compass points and coordinates. However, their understanding of mathematical concepts pertinent to map tasks including location, direction, and movement were less developed. Successful students were able to interpret the map tasks and apply relevant mathematical understanding to navigate the spatial demands of the map tasks while the unsuccessful students were only able to interpret and understand basic map conventions. In terms of their gesture use, the more difficult the task, the more likely students were to exhibit gestural behaviours to solve the task. The most common form of gestural behaviour was deictic, that is a pointing gesture. Deictic gestures not only aided the students capacity to explain how they solved the map tasks but they were also a tool which assisted them to navigate and monitor their spatial movements when solving the tasks. There were a number of implications for theory, learning and teaching, and test and curriculum design arising from the study. From a theoretical perspective, the findings of the study suggest that gesturing is an important element of multimodal engagement in mapping tasks. In terms of teaching and learning, implications include the need for students to utilise gesturing techniques when first faced with new or novel map tasks. As students become more proficient in solving such tasks, they should be encouraged to move beyond a reliance on such gesture use in order to progress to more sophisticated understandings of map tasks. Additionally, teachers need to provide students with opportunities to interpret and attend to multiple modes of information when interpreting map tasks.

Embedding information literacy in a first year business undergraduate course

This article reports on a project to embed information literacy skills development in a first-year undergraduate business course at an Australian university. In accordance with prior research suggesting that first-year students are over-confident about their skills, the project used an optional online quiz to allow students to pre-test their information literacy skills. The students' lower than expected results subsequently encouraged greater skill development. However, not all students elected to undertake the first quiz. A final assessable information literacy quiz increased the levels of student engagement, suggesting that skill development activities need to be made assessable. We found that undertaking the information literacy quizzes resulted in a statistically significant improvement in students' information literacy skills from the pre-test to the post-test. This research therefore extends previous research by providing an effective means of delivering information literacy skill development to large cohorts of first-year students.

Direct instruction in mathematics : issues for schools with high Indigenous enrolments : a literature review

Direct instruction, an approach that is becoming familiar to Queensland schools that have high Aboriginal and Torres Strait Islander populations, has been gaining substantial political and popular support in the United States of America [USA], England and Australia. Recent examples include the No Child Left Behind policy in the USA, the British National Numeracy Strategy and in Australia, Effective Third Wave Intervention Strategies. Direct instruction, stems directly from the model created in the 1960s under a Project Follow Through grant. It has been defined as a comprehensive system of education involving all aspects of instruction. Now in its third decade of influencing curriculum, instruction and research, direct instruction is also into its third decade of controversy because of its focus on explicit and highly directed instruction for learning. Characteristics of direct instruction are critiqued and discussed to identify implications for teaching and learning for Indigenous students.

Re-engaging adolescent students in science : an experienced teachers' classroom

Adolescents are both aware of and have the impetuous to exploit aspects of Science, Technology, Engineering and Mathematics (STEM) within their personal lives. Whether they are surfing, cycling, skateboarding or shopping, STEM concepts impact their lives. However science, mathematics, engineering and technology are still treated in the classroom as separate fragmented entities in the educational environment where most classroom talk is seemingly incomprehensible to the adolescent senses. The aim of this study was to examine the experiences of young adolescents with the aim of transforming school learning at least of science into meaningful experiences that connected with their lives using a self-study approach. Over a 12-month period, the researcher, an experienced secondary-science teacher, designed, implemented and documented a range of pedagogical practices with his Year-7 secondary science class. Data for this case study included video recordings, journals, interviews and surveys of students. By setting an environment empathetic to adolescent needs and understandings, students were able to actively explore phenomena collaboratively through developmentally appropriate experiences. Providing a more contextually relevant environment fostered meta-cognitive practices, encouraged new learning through open dialogue, multi-modal representations and assessments that contributed to building upon, re-affirming, or challenging both the students' prior learning and the teacher’s pedagogical content knowledge. A significant outcome of this study was the transformative experiences of an insider, the teacher as researcher, whose reflections provided an authentic model for reforming pedagogy in STEM classes.

Coursework masters students' perspectives on career development learning

Coursework Masters’ students have been identified as a ‘forgotten’ cohort by the Queensland University of Technology (QUT) Careers and Employment (C&E) Service. Traditionally, these students have been included in undergraduate career development activities. However, coursework Masters’ students are arguably a specialised cohort due to the advanced learning implicit in many of their degrees. A brief survey was designed to better understand how to assist QUT coursework Masters students with career development activities. Students were emailed an invitation to complete a brief online survey asking for their assistance in planning appropriate career development activities for their cohort. Questions included reasons for undertaking a Masters degree and preferred topics and times for career development activities. 615 students completed the survey. This brief report will report the results of the survey and highlight suggestions for career development practitioners and services

Indigenous mathematics : creating an equitable learning environment

This chapter examines how a change in school leadership can successfully address competencies in complex situations and thus create a positive learning environment in which Indigenous students can excel in their learning rather than accept a culture that inhibits school improvement. Mathematics has long been an area that has failed to assist Indigenous students in improving their learning outcomes, as it is a Eurocentric subject (Rothbaum, Weisz, Pott, Miyake & Morelli, 2000, De Plevitz, 2007) and does not contextualize pedagogy with Indigenous culture and perspectives (Matthews, Cooper & Baturo, 2007). The chapter explores the work of a team of Indigenous and non-Indigenous academics from the YuMi Deadly Centre who are turning the tide on improving Indigenous mathematical outcomes in schools and in communities with high numbers of Aboriginal and Torres Strait Islander students.

Primary students' performance on map tasks : the role of context

This study investigated the longitudinal performance of 583 students on six map items that were represented in various graphic forms. Specifically, this study compared the performance of 7-9-year-olds (across Grades 2 and 3) from metropolitan and non-metropolitan locations. The results of the study revealed significant performance differences in favour of metropolitan students on two of six map tasks. Implications include the need for teachers in non-metropolitan locations to ensure that their students do not overly fixate on landmarks represented on maps but rather consider the arrangement of all elements encompassed within the graphic.

Co-constructing self-efficacy narratives : a study of four mature age university students

This thesis reports on a study in which research participants, four mature aged females starting an undergraduate degree at a regional Australian university, collaborated with the researcher in co-constructing a self-efficacy narrative. For the purpose of the study, self-efficacy was conceptualized as a means by which an individual initiates action to engage in a task or set of tasks, applies effort to perform the task or set of tasks, and persists in the face of obstacles encountered in order to achieve successful completion of the task or set of tasks. Qualitative interviews were conducted with the participants, initially investigating their respective life histories for an understanding of how they made the decision to embark on their respective academic program. Additional data were generated from a written exercise, prompting participants to furnish specific examples of self-efficacy. These data were incorporated into the individual's self-efficacy narrative, produced as the outcome of the "narrative analysis". Another aspect of the study entailed "analysis of narrative" in which analytic procedures were used to identify themes common to the self-efficacy narratives. Five main themes were identified: (a) participants' experience of schooling . for several participants their formative experience of school was not always positive, and yet their narratives demonstrated their agency in persevering and taking on university-level studies as mature aged persons; (b) recognition of family as an early influence . these influences were described as being both positive, in the sense of being supportive and encouraging, as well as posing obstacles that participants had to overcome in order to pursue their goals; (c) availability of supportive persons – the support of particular persons was acknowledged as a factor that enabled participants to persist in their respective endeavours; (d) luck or chance factors were recognised as placing participants at the right place at the right time, from which circumstances they applied considerable effort in order to convert the opportunity into a successful outcome; and (e) self-efficacy was identified as a major theme found in the narratives. The study included an evaluation of the research process by participants. A number of themes were identified in respect of the manner in which the research process was experienced as a helpful process. Participants commented that: (a) the research process was helpful in clarifying their respective career goals; (b) they appreciated opportunities provided by the research process to view their life from a different perspective and to better understand what motivated them, and what their preferred learning styles were; (c) their past successes in a range of different spheres were made more evident to them as they were guided in self-reflection, and their self-efficacious behaviour was affirmed; and (d) the opportunities provided by their participation in the research process to identify strengths of which they had not been consciously aware, to find confirmation of strengths they knew they possessed, and in some instances to rectify misconceptions they had held about aspects of their personality. The study made three important contributions to knowledge. Firstly, it provided a detailed explication of a qualitative narrative method in exploring self-efficacy, with the potential for application to other issues in educational, counselling and psychotherapy research. Secondly, it consolidated and illustrated social cognitive theory by proposing a dynamic model of self-efficacy, drawing on constructivist and interpretivist paradigms and extending extant theory and models. Finally, the study made a contribution to the debate concerning the nexus of qualitative research and counselling by providing guidelines for ethical practice in both endeavours for the practitioner-researcher.

Middle school students' perceptions of engineering

This paper focuses on implementing engineering education in middle school classrooms (grade levels 7-9). One of the aims of the study was to foster students’ and teachers’ knowledge and understanding of engineering in society. Given the increasing importance of engineering in shaping our daily lives, it is imperative that we foster in students an interest and drive to participate in engineering education, increase their awareness of engineering as a career path, and inform them of the links between engineering and the enabling subjects, mathematics, science, and technology. Data for the study are drawn from five classes across three schools. Grade 7 students’ responded to initial whole class discussions on what is an engineer, what is engineering, what characteristics engineers require, engineers (family/friends) that they know, and subjects that may facilitate an engineering career. Students generally viewed engineers as creative, future-oriented, and artistic problem finders and solvers; planners and designers; “seekers” and inventors; and builders of constructions. Students also viewed engineers as adventurous, decisive, community-minded, reliable, and “smart.” In addition to a range of mathematics and science topics, students identified business studies, ICT, graphics, art, and history as facilitating careers in engineering. Although students displayed a broadened awareness of engineering than the existing research suggests, there was limited knowledge of various engineering fields and a strong perception of engineering as large construction.

Recognising Torres Strait Islander women’s knowledges in their children’s mathematics education

This paper discusses women’s involvement in their children’s mathematics education. It does, where possible, focus Torres Strait Islander women who share the aspirations of Aborginal communities around Australia. That is, they are keen for their children to receive an education that provides them with opportunities for their present and future lives. They are also keen to have their cultures’ child learning practices recognised and respected within mainstream education. This recognition has some way to go with the language of instruction in schools written to English conventions, decontextualised and disconnected to the students’ culture, Community and home language.

The design and development of an undergraduate signal processing laboratory

The School of Electrical and Electronic Systems Engineering of Queensland University of Technology (like many other universities around the world) has recognised the importance of complementing the teaching of signal processing with computer based experiments. A laboratory has been developed to provide a "hands-on" approach to the teaching of signal processing techniques. The motivation for the development of this laboratory was the cliche "What I hear I remember but what I do I understand." The laboratory has been named as the "Signal Computing and Real-time DSP Laboratory" and provides practical training to approximately 150 final year undergraduate students each year. The paper describes the novel features of the laboratory, techniques used in the laboratory based teaching, interesting aspects of the experiments that have been developed and student evaluation of the teaching techniques

Assessing students in senior science: an analysis of questions in contextualised chemistry exams

This study explores the development of a coding system for analysing test questions in two context-based chemistry exams. We describe our unique analytical procedures before contrasting the data from both tests. Our findings indicate that when a new curriculum is developed such as a context-based curriculum, teachers are required to combine the previously separate domains of context and concept to develop contextualised assessment. We argue that constructing contextualised assessment items requires teachers to view concepts and context as interconnected rather than as separate entities that may polarise scientific endeavour. Implications for practice, curriculum and assessment-development in context-based courses are proposed.