Diversions : Book mathematics - Part 1

The article presents several excerpts from the books including "The Geometry of Love," by John Cheever, "Mr. Midshipman Hornblower," by C.S. Forester, and "The Stinky Cheese Man and Other Fairly Stupid Stories," by Jon Scieszka and Lane Smith.

Designing open-ended problems to challenge preservice teachers' views on mathematics and pedagogy

This study examines preservice elementary teachers' reported experiences of posing open-ended mathematics problems. Responses of 33 students in a mathematics teacher education course were analysed for the strategies participants used, what they learned and the challenges encountered from an opportunity to collect digital images and pose open-ended problems related to those images. Results indicate that preservice teachers reported a shift in the ways they viewed mathematics and how it might be taught. The school curriculum both constrained and provided possibilities for preservice teachers in noticing mathematics beyond the textbook and mathematics classroom. This study adds to our understanding of teaching as a learning practice and the art of posing mathematical problems as a significant aspect of that practice.

Shut the decimal box - a mathematics dice game

The table-top game of Shut the Box is popular in Europe and England, but not well known in Australia. Sets can be purchased, but the game can be played with home-made equipment. A variant of the game which illustrates the standard rules, with a decimal twist to the scoring, is described. A list of the equipment required is also provided.

Picture books stimulate the learning of mathematics

In this article we describe our experiences using picture books to provide young children (five- to six-year-olds) with a learning environment where they can explore and extend preliminary notions of mathematics-related concepts, without being taught these concepts explicitly. We gained these experiences in the PICO-ma project, which aimed to generate more knowledge about the effect of picture books on young children's learning of mathematics. The project's goal is to investigate how picture books can contribute to the development of mathematical concepts in young children, and how the actions of the teacher can strengthen the characteristics of picture books that support learning. The reading sessions described in this article were not intended to be mathematics 'lessons'. Instead, the reading sessions were intended to tell the children a pleasant story and, at the same time, give them something to think about. Based on our research we provide reasons for using picture books to develop mathematical thinking, and include recommendations for practitioners interested in using picture books for mathematics learning.

Exemplary mathematics lessons : a view from the west

Cross-cultural comparative research provides a powerful means of achieving better understanding of one’s own practice. This article contrasts exemplary mathematics teaching, as highlighted in this special issue on its practice and development in East Asia, with the increasing emphasis on effective teaching in the West. It provides an overview of the papers in this issue and examines the similarities and differences between what is seen as exemplary practice and the ways in which its development is supported in the educational systems represented. It attempts to identify some of the cultural factors that influence our understanding of what constitutes exemplary practice and the possibilities for its development in the East and the West.

The nature of spontaneity in high quality mathematics learning experiences

Spontaneity has been linked to high quality learning experiences in mathematics (Csikszentmihalyi & Csikszentmihalyi, 1992; Williams, 2002).This paper shows how spontaneity can be identified by attending to the nature of social elements in the process of abstracting (Dreyfus, Hershkowitz, & Schwarz, 2001). This process is elaborated through an illustrative example—a Year 8 Australian male student who scaffolded his learning by attending to images in the classroom that were intended for other purposes. Leon’s cognitive processing was not ‘observable’ (Dreyfus et al., 2001) in classroom dialogue because Leon ‘thought alone’. Post-lesson videostimulated reconstructive interviews facilitated study of Leon’s thought processes and extended methodological techniques available to study thinking in classrooms.

Calculators, mathematics and young children : a study of six children using calculators as part of the mathematics curriculum during their first two years of school

The thesis investigates the role a calculator can play in the developing number knowledge of three girls and three boys as part of their mathematics program, during their first two years at primary school. Random sampling was used initially to select six girls and six boys from the twenty-four children entering a 1993 prep class. These twelve children were interviewed on entrance to school and based on the performance of the twelve children on the initial interview, a girl and a boy were chosen from the higher, middle and lower achievers to take part in the full study. The class teachers involved were previously participants in the ‘Calculators in Primary Mathematics’ research program and were committed to the use of calculators in their mathematics program. A case study approach using qualitative methods within the activity theory framework is used to collect relevant data and information, an analysis of five interviews with each child and observations of the children in forty-one classroom lessons provides comprehensive data on the children's developing number knowledge during the two years. The analysis questionnaires establishes each teacher's perceptions of the children's number learning at the beginning and end of each year, compares teacher expectations with children's actual performance for the year and compares curriculum expectations with children's actual performance. A teacher interview established reasons for changes in teaching style; teacher expectations; children's number learning; and was used to confirm my research findings. An activity theory framework provides an appropriate means of co-coordinating perspectives within this research to enable a description of the child's number learning within a social environment. This framework allows for highlighting the mediation offered by the calculator supporting the children's number learning in the classroom. Levels of children's developing number knowledge reached when working with a calculator and as a result of calculator use are mapped against the levels recommended in ‘Mathematics in the National Curriculum’ (National Curriculum Council, December 1988), and the Curriculum and Standards Framework: Mathematics (Board of Studies 2000). Findings from this comparison illustrate that the six children's performance in number was enhanced when using a calculator and indicate that on-going development and understanding of number concepts occurred at levels of performance at least two years in advance of curriculum recommendations for the first two years of school.

Applying outcomes of lifelong learning to organisational achievement

The focus of this doctoral research study is making the most what a person knows and can do, as an outcome from their lifelong learning, so as to better contribute to organisational achievement. This has been motivated by a perceived gap in the extensive literature linking knowledge with organisational achievement. Whereas there is a rich body of literature addressing the meta-philosophies giving rise to the emergence of learning organisations there is, as yet, scant attention paid to the detail of planning and implementing action which would reveal individual/organisational opportunities of mutual advantage and motivate, and sustain, participation at the day-to-day level of the individual. It is in this space that this dissertation seeks to contribute by offering a mechanism for bringing the, hindsight informed, response “but that’s obvious” into the abiding explicit realm at the level of the individual. In moving beyond the obvious which is prone to be overlooked, the emphasis on “better” in the introductory sentence, is very deliberately made and has a link to awakening latent individual, and hence organisational, capabilities that would otherwise languish. The evolved LCM Model – a purposeful integration valuing the outcomes from lifelong learning (the L) with nurturing a culture supporting this outcome (the C) and with responsiveness to potentially diverse motivations (the M) – is a reflective device for bringing otherwise tacit, and latent, logic into the explicit realm of action. In the course of the development of the model, a number of supplementary models included in this dissertation have evolved from the research. They form a suite of devices which inform action and lead to making the most of what an individual knows and can do within the formal requirements of a job and within the informal influences of a frequently invisible community of practice. The initial inquiry drew upon the views and experiences of water industry engineering personnel and training facilitators associated with the contract cleaning and waste management industries. However, the major research occurred as an Emergency Management Australia (EMA) project with the Country Fire Authority (CFA) as the host organisation. This EMA/CFA research project explored the influence of making the most of what a CFA volunteer knows and can do upon retention of that volunteer. In its aggregate, across the CFA volunteer body, retention is a critical community safety objective. A qualitative research, ethnographic in character, approach was adopted. Data was collected through interviews, workshops and outcomes from attempts at action research projects. Following an initial thirteen month scoping study including respondents other than from the CFA, the research study moved into an exploration of the efficacy of an indicative model with four contextual foci – i.e. the manner of welcoming new members to the CFA, embracing training, strengthening brigade sustainability and leadership. Interestingly, the research environment which forced a truncated implementation of action research projects was, in itself, an informing experience indicative of inhibitors to making the most of what people know and can do. Competition for interest, time and commitment were factors governing the manner in which CFA respondents could be called upon to explore the efficacy of the model, and were a harbinger of the influences shaping the more general environment of drawing upon what CFA volunteers know and can do. Subsequent to the development of the indicative model, a further 16 month period was utilised in the ethnographic exploration of the relevance of the model within the CFA as the host organisation. As a consequence, the model is a more fully developed tool (framework) to aid reflection, planning and action. Importantly, the later phase of the research study has, through application of the model to specific goals within the CFA, yielded operational insight into its effective use, and in which activity systems have an important place. The model – now confidently styled as the LCM Model – has three elements that when enmeshed strengthen the likelihood of organisational achievement ; and the degree of this meshing, as relevant to the target outcome, determines the strength of outcome. i.e. - • Valuing outcomes from learning: When a person recognises and values (appropriately to achievement by the organisation) what they know and can do, and associated others recognise and value what this person knows and can do, then there is increased likelihood of these outcomes from learning being applied to organisational achievement. • Valuing a culture that is conducive to learning: When a person, and associated others, are further developing and drawing upon what they know and can do within the context of a culture that is conducive to learning, then there is increased likelihood that outcomes from learning will be applied to organisational achievement. • Valuing motivation of the individual: When a person’s motivation to apply what they know and can do is valued by them, and associated others, as appropriate to organisational achievement then there is increased likelihood that appropriately drawing upon outcomes from learning will occur. Activity theory was employed as a device to scope and explore understanding of the issues as they emerged in the course of the research study. Viewing the data through the prism of activity theory led not only to the development of the LCM Model but also to an enhanced understanding of the role of leadership as a foundation for acting upon the model. Both formal and informal leadership were found to be germane in asserting influence on empowering engagement with learning and drawing upon its outcomes. It is apparent that a “leaderful organisation”, as postulated by Raelin (2003), is an environment which supports drawing upon the LCM model; and it may be the case that the act of drawing upon the model will move a narrowly leadership focused organisation toward leaderful attributes. As foreshadowed at the beginning of this synopsis, nurturing individual and organisational capability is the guiding mantra for this dissertation - “Capability embraces competence but is also forward-looking, concerned with the realisation of potential” (Stephenson 1998, p. 3). Although the inquiry focussed upon a need for CFA volunteer retention, it began with a broader investigation as part of the scoping foundation and the expanded usefulness of the LCM Model invites further investigation. The dissertation concludes with the encapsulating sentiment that “You have really got to want to”. With this predisposition in mind, this dissertation contributes to knowledge through the development and discussion of the LCM model as a reflective device informing transformative learning (Mezirow and Associates 1990). A leaderful environment (Raelin 2003) aids transformative learning – accruing to the individual and the organisation - through engendering and maintaining making the most of knowledge and skill – motivating and sustaining “the will”. The outcomes from this research study are a strong assertion that wanting to make the most of what is known and can be done is a hallmark of capability. Accordingly, this dissertation is a contribution to the “how” of strengthening the capability, and the commitment to applying that capability, of an individual and an organisation.

Investigation into the skill levels achieved by mathematics students in the V.C.E. and the H.S.C. mathematics courses

This study examines whether recent changes to the mathematics courses offered in the final year of secondary school (Year 12) in the state of Victoria, Australia have affected the learning outcomes of students in terms of then: skill levels in algebra, calculus and problem solving; and in terms of their preparation for a tertiary mathematics unit. The impact of these changes on the transition from secondary to tertiary mathematics is also considered. A comparison is made between students who attempted a first year mathematics unit at the University of Melbourne (U. of M.) having completed the new V.C.E. (Victorian Certificate of Education) mathematics courses and mathematics courses from the previous H.S.C. (Higher School Certificate) system. The comparison involves the use of tests administered upon entrance to a tertiary mathematics unit at the U. of M., and questionnaires. In 1991, V.C.E, students and H.S.C. students attempted the same mathematics test at the U. of M. and their results were compared. In 1992, the tests were attempted by V.C.E. students only. To compare new V.C.E. students and H.S.C. students, questions on the 1991 test were matched with similar questions on the 1992 tests and a panel of experts determined what the H.S.C. students who attempted the 1991 test would have been expected to average on these matched questions on the 1992 tests had they attempted them. These expected average scores were then compared with the actual scores of the new V.C.E. students. The scores of the groups were scaled when necessary. Questionnaires were administered to 1991 U. of M, mathematics students who were part of the V.C.E. pilot group in 1990, secondary mathematics educators, tertiary mathematics educators, and 1991 V.C.E. (1992 U. of M.) students. The mathematical misconceptions exhibited by new V.C.E. students are discussed and their frequencies stated. The research indicates that the new V.C.E. mathematics courses have provided the V.C.E. mathematics students in this study with significantly lower skill levels and a significantly poorer preparation for a tertiary mathematics unit than those which were previously provided by the H.S.C. mathematics courses.