Individual differences in learning from worked examples by senior secondary mathematics students

The primary purpose of this research was to examine individual differences in learning from worked examples. By integrating cognitive style theory and cognitive load theory, it was hypothesised that an interaction existed between individual cognitive style and the structure and presentation of worked examples in their effect upon subsequent student problem solving. In particular, it was hypothesised that Analytic-Verbalisers, Analytic-Imagers, and Wholist-lmagers would perform better on a posttest after learning from structured-pictorial worked examples than after learning from unstructured worked examples. For Analytic-Verbalisers it was reasoned that the cognitive effort required to impose structure on unstructured worked examples would hinder learning. Alternatively, it was expected that Wholist-Verbalisers would display superior performances after learning from unstructured worked examples than after learning from structured-pictorial worked examples. The images of the structured-pictorial format, incongruent with the Wholist-Verbaliser style, would be expected to split attention between the text and the diagrams. The information contained in the images would also be a source of redundancy and not easily ignored in the integrated structured-pictorial format. Despite a number of authors having emphasised the need to include individual differences as a fundamental component of problem solving within domainspecific subjects such as mathematics, few studies have attempted to investigate a relationship between mathematical or science instructional method, cognitive style, and problem solving. Cognitive style theory proposes that the structure and presentation of learning material is likely to affect each of the four cognitive styles differently. No study could be found which has used Riding's (1997) model of cognitive style as a framework for examining the interaction between the structural presentation of worked examples and an individual's cognitive style. 269 Year 12 Mathematics B students from five urban and rural secondary schools in Queensland, Australia participated in the main study. A factorial (three treatments by four cognitive styles) between-subjects multivariate analysis of variance indicated a statistically significant interaction. As the difficulty of the posttest components increased, the empirical evidence supporting the research hypotheses became more pronounced. The rigour of the study's theoretical framework was further tested by the construction of a measure of instructional efficiency, based on an index of cognitive load, and the construction of a measure of problem-solving efficiency, based on problem-solving time. The consistent empirical evidence within this study that learning from worked examples is affected by an interaction of cognitive style and the structure and presentation of the worked examples emphasises the need to consider individual differences among senior secondary mathematics students to enhance educational opportunities. Implications for teaching and learning are discussed and recommendations for further research are outlined.

Accelerating junior-secondary mathematics

Unfortunately, in Australia there is a prevalence of mathematically underperforming junior-secondary students in low-socioeconomic status schools. This requires targeted intervention to develop the affected students’ requisite understanding in preparation for post-compulsory study and employment and, ultimately, to increase their life chances. To address this, the ongoing action research project presented in this paper is developing a curriculum of accelerated learning, informed by a lineage of cognitivist-based structural sequence theory building activity (e.g., Cooper & Warren, 2011). The project’s conceptual framework features three pillars: the vertically structured sequencing of concepts; pedagogy grounded in students’ reality and culture; and professional learning to support teachers’ implementation of the curriculum (Cooper, Nutchey, & Grant, 2013). Quantitative and qualitative data informs the ongoing refinement of the theory, the curriculum, and the teacher support.

50 mathematics lessons : rich and engaging ideas for secondary Mathematics.

Es una colección de cincuenta lecciones de matemáticas que requieren poco tiempo de preparación y que resultan idóneas para profesores de secundaria muy ocupados y con varias clases en el mismo día.

Teachers and computers for secondary mathematics

In the first stage of a three-year study in which the effects of using computers for the teaching and learning of mathematics are being explored, a questionnaire was developed and administered to teachers of students in grades 7–10 in a representative sample of co-educational post-primary schools in Victoria, Australia. Using open and closed response formats, the information sought included data on the teachers' professional backgrounds, computer ownership and use, and their beliefs and practices in using computers for the teaching of mathematics. In this article, findings related to ownership, professional development, perceptions of technological skills, beliefs about the efficacy of computer use in mathematics, and data on how teachers are using computers for teaching secondary mathematics are presented and discussed.

Primary and secondary mathematics practice: how different is it?

This paper describes the practices of ninety-nine teachers at the beginning of their involvement in a large-scale project investigating the influence of subject cultures on school and teacher change. Data collected from these teachers, using the IMYMS. Components of effective teaching and Learning mapping procedure, were analysed to investigate similarities and differences in primary and secondary teachers’ perceptions of their practice in mathematics. Results of the analysis show different patterns across a range of components of effective teaching and learning.

A first priority when working with teachers is to help them become aware of and make problematic aspects of their current practice …. Only then would they have reason to attempt to reform their instructional practices when working with us. (Yackel, 1994, p. 386)

Using everyday experiences in teaching secondary mathematics in Malawi : possibilities and constraints for change

Proponents of socially and culturally oriented mathematics education have argued that teaching approaches which value and connect with the learner's prior knowledge and everyday experience are more likely to promote active, meaningful, relevant and liberatory learning than approaches which rely on transmission and abstract presentation of mathematical content. In Malawi, proposals to reform the outdated secondary mathematics curriculum have been made with the aim of aligning mathematics instruction with the social and political changes in the current Malawian society. Using a case study approach, this study investigated the extent to which everyday experiences could be used as a vehicle for changing the learning and teaching of secondary mathematics in Malawi. The study was collaborative, taking place over a period of five months in severely overcrowded and poorly resourced classes in two schools. It involved three mathematics teachers in a cycle of planning and teaching mathematics lessons based on the use of everyday experiences, and observation of and reflection on these lessons, in order to document the effects of using everyday experiences on student learning and teachers' teaching practices. The data was collected through student questionnaires; classroom observations and fieldnotes; interviews and reflective meetings with teachers; and informal meetings with key education officials in Malawi. Mathematics examination results from students involved in this study and a corresponding group from the previous year were collected. A reflective and critical approach was adopted in the interpretation and discussion of the data. Teachers' participation in this study resulted in heightened awareness of their teaching roles and the value of linking school mathematics with everyday experience. The study also shows that students found mathematics interesting and important to learn despite their lack of success in it. In addition, the study documented a number of constraints to change in mathematics instruction such as teachers' focus on mathematics content and examination requirements, and students' resistance to inquiry learning. It also recorded possibilities and barriers to collaboration both between teachers and researchers, and teachers themselves. The findings of this study are timely since they could serve to inform the reform of the Malawian secondary mathematics curriculum currently being undertaken, which began without a critical examination of the classroom conditions necessary to accommodate a socio-politically relevant mathematics education.

Fostering meaningful learning by using word problems in post-secondary mathematics

This research is a case study of action research investigating the effects of using word problems with Chinese Malaysian post-secondary students studying mathematics as an enabling science. A variety of word problems, a discussion method, reflective and peer learning, specific values in mathematics, and relevant socio-cultural and language issues were explored.

Computers for secondary mathematics: who uses them and how?

In the first stage of a three year ARC-funded study in which the effects of using computers for the teaching and learning of mathematics are being explored, a questionnaire has been developed and is being administered to a large number of students in grades 7-10 in a representative sample of co-educational post-primary schools in Victoria. Using open and closed response formats, the information sought includes: background and biographical details; data on computer ownership and use =96 frequency and mode/s =96 for mathematics learning at home and in mathematics classes; and beliefs about mathematics and the use of computers for learning mathematics. In this paper, the focus will be on some of the preliminary findings related to ownership and use of computers for learning mathematics at home and at school. In previous research, factors such as gender, socio-economic background, and ethnicity have been identified as contributors to inequitable mathematics learning outcomes. The questionnaire data will be analysed to examine if these and other factors are related to any inequities identified in student's access to computers and in how they use computers in their learning of mathematics. The results will be presented and pertinent implications discussed.

Computers are taking mathematics into the next century : gender differences in attitudes of secondary mathematics students to the use of computers

Secondary students who participated in a computer enhanced mathematics program expressed positive attitudes about the use of computers. They viewed computers as a source of pleasure, success, relevance and/or power in mathematics. Girls were more likely than boys to qualify their support for the use of computers and more likely to view computers as a source of success in mathematics. Boys were more likely to claim that computers brought pleasure or relevance to mathematics learning.

Deepening the mathematical knowledge of secondary mathematics teachers who lack tertiary mathematics qualifications

A professional learning program for unqualified practising secondary mathematics teachers regarding senior secondary mathematics teaching is described in this paper. The VCE (Victorian Certificate of Education) mathematics professional learning program for senior secondary mathematics was designed for practising secondary teachers of mathematics who had no experience of teaching advanced senior secondary mathematics and who had not completed the recommended qualifications. Professional learning episodes, artefacts and reflections of three teachers who participated in the program are analysed to identify the development of these teachers' mathematical and pedagogical content knowledge (PCK). The PCK framework developed by Chick et al. was used to analyse teachers' knowledge, and the cases of teachers' knowledge presented in the paper illustrate the entwining of knowledge of mathematics and knowledge of teaching and learning The findings indicate that a program designed for senior secondary mathematics can enable practising teachers to deepen and broaden their understanding of junior secondary mathematical pedagogy.

Supporting "out-of-field" teachers of secondary mathematics

The key aspects of the recommendations made by the Council of Australian Governments in its National Numeracy Report (2008) to help reduce the shortage of teachers of secondary mathematics are discussed. The aims and objectives of the professional learning programs for 'out-of-field' mathematics teachers are highlighted.