Theses abstracts (2)
No abstract supplied. (MEd)
The purpose of this research was to examine the possible associations between the perceived classroom environment of high school students, the level of mathematics anxiety that they possess, and their attitudes towards mathematics. This marks the first time that these three fields of research have been simultaneously combined. Data were gathered from 745 high school mathematics students in 34 classes in high schools in the Southern California area using three instruments: the What is Happening In this Class? (WIHIC) learning environment survey created by Fraser, McRobbie, and Fisher (1996), an updated version of Plake and Parker's (1982) Revised Mathematics Anxiety Ratings Scale (RMARS), and a mathematics version of selected scales from Fraser's (1981) Test of Science-Related Attitudes (TOSRA). This revised attitude instrument was called the Test of Mathematics-Related Attitudes (TOMRA). Using statistical methods, the three instruments were checked for internal consistency reliability, factor structure, and discriminant validity. The RMARS and WIHIC were both found to exhibit good reliability and factorial validity in mathematics classrooms in Southern California, while the TOMRA yielded two scales of the four a priori scales, Enjoyment of Mathematics Lessons and Normality of Mathematicians, which met reliability and factorial validity standards. Within-class gender differences were analysed using paired t-tests combined with a modified Bonferroni procedure and effect sizes. Between-student gender differences were investigated using MANOVA. Simple correlation and multiple regression analyses were performed to identify possible associations between the learning environment and anxiety/attitudes scales. Qualitative data were collected from interviews and inductive analysis was performed in order to refute or corroborate the quantitative findings. Significant within-class gender differences were found in four areas of the learning environment (Student Cohesiveness, Task Orientation, Cooperation, and Equity), but no gender differences in attitudes were found. All four learning environment areas were perceived in a more favourable light by females than by males. Individual gender differences were similar, with a significant difference also being found in Teacher Support, as well as both types of mathematics anxiety, namely, Learning Mathematics Anxiety and Mathematics Evaluation Anxiety. In order to carefully identify the relationships between the classroom learning environment and mathematics anxiety, analyses were conducted for both factors of mathematics anxiety. While no association between the learning environment and Mathematics Evaluation Anxiety was found, there were significant associations between Learning Mathematics Anxiety and three areas of the learning environment: Student Cohesiveness, Task Orientation, and Investigation. Significant associations between the Normality of Mathematicians attitude scale and the learning environment scales Equity and Involvement were identified, while three areas of the learning environment (Investigation, Task Orientation, and Cooperation) had a significant relationship with Enjoyment of Mathematics Lessons. Qualitative data analyses confirmed relationships between anxiety, attitudes, and classroom learning environments. The data also suggest that the structure of the mathematical content is linked with the level of anxiety that high school students feel.
This longitudinal study explored the extent to which mathematics education lecturers' constructivist beliefs and aligned practices were communicated to students in a New Zealand primary pre-service teacher education degree program. An integral part of this exploration was the identification of particular aspects of lecturers' practice that had a significant impact in enhancing the adoption of constructivist ideas on learning and constructivist-aligned teaching practices by student teachers. This goal had a practical focus on more effective course teaching within the chosen philosophical framework of constructivism. At a more theoretical level, there was a focus on the development of a constructivist approach to teacher education for teacher educators through the medium of mathematics education. A potential outcome of the development and widespread adoption of such a constructivist-aligned pedagogy within teacher education could be the significant furthering of a 'reform' (or transformative) agenda in school education with its potential for enhanced learning by children. The methodology comprised both quantitative (survey) and qualitative (interview) techniques to collect information which allowed the capture of different but complementary data, so building a 'rich' data set. The surveys were conducted using two learning environment instruments underpinned by particular constructivist perspectives: one focusing on the overall nature of the learning environment at an individual level from a critical constructivist perspective, and the other focusing on the nature of interactions between teacher and student teachers at a classroom level from a socio-cultural constructivist perspective. Surveys were conducted with the lecturers at the beginning and toward the end of the study, while the student teachers in these lecturers' classes were surveyed over a three year period. The interviews were semi-structured following an interpretative (evolving) research approach, with the 'results' of ongoing data analysis being fed into later interviews. The interview data were analysed for personal perceptions and understandings rather than for generalisation and prediction with the intention of focusing on the identification of emergent themes. Interviews were conducted with lecturers at the beginning of the study and again toward its conclusion while student teachers were interviewed at the end of the study. The lecturers claimed constructivism as their underlying philosophical belief system and the initial surveys established baseline data on the actual nature of the lecturers' beliefs and how these were perceived by the student teachers. Similarly, the initial interviews explored the espoused beliefs and congruent practices of lecturers and student teachers. These two sets of data were compared to establish their congruence or otherwise. Further interviews with the lecturers focused on the survey data and my reconstruction of what the lecturers had said previously when interviewed. Later survey and interview data were also examined against the baseline data for evidence of change over the four years of the study. The data demonstrated that the student teachers perceived the existence of moderate to strong socio-cultural constructivist-aligned classroom environments when considered at a class (group) level, and a moderate alignment with critical perspectives at the individual (personal) level. There was a high degree of consistency between staff and student teacher views, and the student teachers' views were consistent across the year groups (first, second and third years) and throughout the four years of the study. Lecturer practice(s) congruent with constructivism were the basis for student teacher change toward understanding and their adoption of constructivist ideas and aligned practices. Specific lecturer practices were identified as particularly effective in achieving such change. These effective lecturer practices may assist in establishing the foundations of a constructivist-aligned pedagogy for teacher education. The lecturers' modelling of the practices they were promoting for student teachers' practice was identified as a key element in promoting change. Indeed, the tension between traditional and transformative approaches was exacerbated in situations where lecturers' promotion of a preferred practice was different from that which they enacted. The continuing existence of such situations and associated tensions has the potential to limit the extent of any change.
This PhD thesis is concerned with the social psychology of cooperative learning and its effects in cognitive and social-emotional domains. It comprises two main studies and two exploratory studies undertaken during two 10-day, 16-hour learning intervention programs for Maths Word Problem-Solving (MWPS), respectively for 285 and 451 Grade-5 students in Singapore. Study 1 used a quasi-experimental design to investigate the outcomes of task-structures in an Individual Learning condition and three dyadic Cooperative Learning conditions that varied in the key elements: positive interdependence, individual accountability and group goals. The results indicated that a Cooperative Learning condition with a high level of positive interdependence in combination with a low level of individual accountability resulted in significantly lower MWPS academic achievement and peer–self-concept outcomes than the other conditions; whereas the other Cooperative conditions with lower levels of positive interdependence did not differ significantly from the Individual Learning condition in MWPS academic outcomes but produced better peer–self-concept outcomes. The discussion theorises how task-structured positive interdependence in cooperative conditions can potentially be so rigid that it limits individual control in overcoming a dyadic partner's error. In turn, this increases the likelihood that members of dyads would 'sink together' (rather than 'swim together') - which appears to produce relatively worse MWPS academic outcomes as well as being detrimental to peer self-concept outcomes. Therefore, optimal cooperative learning conditions for mathematics should allow interaction amongst student partners but not preclude individual control over any stage of the learning task. Study 2 comprised three interrelated investigations of the effects of rewarding learning behaviours and the effects of ability-structures on Individual, Equals (homogeneous) and Mixed (heterogeneous) dyads. All children were eligible to be rewarded for their own MWPS academic mastery achievements, but comparison groups in each of the ability-structures were either eligible or not eligible to be rewarded for displaying target learning behaviours (LB-Rewards or No-LB-Rewards). The academic program was based on Polya's problem-solving strategies of understanding the problem, devising a plan, carrying out the plan, and checking the results. Children in all learning conditions were instructed to use these problem-solving strategies and, according to their differently assigned learning conditions, to use learning behaviours (LB's) either 'for helping oneself' in Individual conditions or 'helping one's partner' in Equals and Mixed conditions. In 'LB-Rewards' conditions, teachers rewarded the children's displays of the assigned behaviours for learning alone or learning together, whereas in 'LB-No-Rewards' conditions they did not. The investigation in Study 2a encompassed the same dependent variables as Study 1. The results indicated that for maths (MWPS), Learning Behaviour rewards were detrimental to Individual Learning conditions with significantly lower MWPS gains when the rewards were used compared to when they were not, whereas the opposite pattern was found for Equals where the effects of Learning Behaviour significantly enhanced MWPS outcomes. For peer self-concept, effects varied across the Cooperative conditions' Learning Behaviour rewards conditions. An exploratory analysis of High-, Low- and Medium-ability revealed patterns of the inter-relationships between ability-structures and effects of rewarding. Study 2b is exploratory and involved traversing the traditional theoretical dichotomy of individual vs social learning, to develop a measure combining them both in 'self-efficacy for learning maths together and learning maths alone'. The effects of the various experimental conditions on factors in this measure were explored, allowing detailed insight into the complex, multi-dimensional and dynamic inter-relationships amongst all the variables. The findings have been developed into a theory of Incentive-values-Exchange in Individual- and Cooperative-learning, arguing that there are four main cooperative learning dimensions - 'individual cognitive endeavour', 'companionate positive influence', 'individualistic attitudes development' and 'social-emotional endeavour'. The argument is that students' motivation to learn cooperatively is the product of perceived equalisation of reward-outcomes in relation to each dyadic member's contributions to learning-goals on these dimensions. Hence, motivation varies across ability-structures and reward-structures in a complex manner. A further proposition of the theory is that social-emotional tendencies and biases form a dynamic system that tends to maintain dyadic partners' achievement levels relative to their ability-positioning. Study 2c is exploratory and extends Study 2b by illustrating its Incentive-values-Exchange theory. Samples of children's written descriptive reflections of their experiences in cooperative dyads are provided to illustrate the point made about the children's relationships and effects on each other for each of the factors on the individual- and cooperative-learning scales. As such, this section of the thesis offers a parsimonious explanation of cooperative learning and the effects of various learning conditions on the integrated cognitive, social and emotional domains. Practical implications in light of the study's findings of optimal conditions include the possibility of practitioners more closely tailoring cooperative learning conditions to meet the academic or social-emotional needs of learners at specific ability levels. Future directions for research include testing some of the learning dimensions and proposed theoretical configurations for them using controls identified by the statistical analyses together with qualitative observations, and further developing new methodologies for investigating the social-psychological causes and consequences of learning motivation.
This is a study of undergraduate and postgraduate students' participation in an induction to mathematics at an Australian university, asking why women were still poorly represented in the discipline, in new times, and when student numbers were dropping. At first, the author viewed both women's and men's participation through their engagement with the discipline, their ideas about the nature of mathematics and orientations on its knowledge production. The author looked at the sorts of awareness they brought to their decisions about becoming mathematical, through their justifications and concerns in doing so. The author have asked does a consciousness of modernity invade either their thinking or their department of mathematics, and if so how? In many respects these justifications and concerns were similar for women and men. From this understanding, the author found that to begin to ask the appropriate questions and see any change-in-process with respect to women's participation, the author had to learn to step aside from the natural sciences first-order description of participation and the philosophical intent of its projects and to learn what that casts out. The author turned to the post-modern and feminist theorists to look at students' conditions of participation and deeper social structuring of social space that might exclude women. The author looked at the communicative practices in students' accounts for the ways these might shape awareness and position them to participate. The author analysed students' compliance or resistance to what the author inferred were the social controls of membership. Some students indicated that there was a lack of dialogue within the induction on the nature of mathematics and being mathematical. Some read widely. Any boundary pushing by women and those supporting them was understated: Women were continuing with postdoctoral work, but not wanting 'to become mathematicians'. This was change in-process, outside rather than within the current collective practice of the department.
The nature of creative mathematical thinking undertaken by students in classroom settings was studied through analysis of the autonomy and spontaneity associated with these processes. The theoretical lens developed enabled simultaneous analysis of cognitive, social, and affective elements of the creative process, and student responses to successes and failures during their exploratory activity (resilience or optimism). Collective case study was employed, with each case progressively informing the analysis of subsequent cases. The classrooms of teachers who were seen by their school communities to display 'good teaching practice' were selected for study. It was anticipated that such classrooms would provide more opportunity to study creative thinking than classrooms chosen at random. During the research period, each student anticipated individually in post-lesson interviews that were stimulated by lesson video P material. To generate data to study student thinking, and the social and personal influences upon it, students were asked to identify parts of the lesson that were important to them, and discuss what was happening, and what they were thinking and feeling. Through this process, students who explored mathematical complexities to generate new mathematical knowledge were identified. Only 5 out of 86 students were identified undertaking thinking more complex than analysis, and 3 of these students undertook such activity more than once. In all except one case, the students worked alone. These five students, who varied in their mathematical performances, cultural backgrounds, socio-economic status, and the classes they attended, each engaged in the same six activities during their creation of new knowledge (a-f below). They manoeuvred Space to Think in classes where, in all but one case, creative thinking was not the teacher's explicit intention. In this Space to Think, students: (a) enacted optimism; (b) spontaneously identified mathematical complexities; (c) manoeuvred cognitive autonomy; (d) autonomously accessed mathematics; (e) spontaneously pursued their exploration of the identified complexity; and (f) posed questions to structure future exploratory activity. All displayed indicators of optimism in their interviews, and enacted optimism in their explorations. The optimistic activity of perceiving 'failure as specific' (looking into the problem to see what to change) was to be crucial to exploratory activity. It was found that students not only accessed mathematics by Recognizing cognitive artefacts, they also 'Looked-in' on dynamic visual lays generated by others, and idiosyncratically extracted relevant mathematics that licit within them. During spontaneous pursuit of their exploration, the cognitive activities, Synthetic-analysis and Evaluative-analysis (part of novel Building-with), were found to 'close the gap' between Analysis and Synthesis (during Constructing resulting in insight). These explicitly cognitive activities informed the identification of pedagogical moves that could promote and sustain creative thinking. e findings of this research should inform teachers, teacher educators, and policy makers, about pedagogical possibilities that could transform the learning and teaching of mathematics. They also suggest the need for further research into the theorised symbiotic relationship between resilience and mathematical exploration that is supported by empirical evidence in this study.
As mathematics and school mathematics are often misconceived as culture- and value-free, the classroom teaching of mathematics can be erroneously considered as easily transferable between cultural sites. In the context of an increasingly multi-cultural teaching profession in Victoria, Australia, this study examines the professional socialisation experiences of immigrant teachers of secondary mathematics. Adopting a socio-cultural perspective, the study aims to explore the nature of perceived value differences relating to mathematics, pedagogy and education; how these differences are negotiated; and the contextual factors through which such negotiations take place. The research design is mainly qualitative in nature. The study was structured through a state-wide survey of 159 secondary schools and a questionnaire survey of 35 immigrant teachers of secondary mathematics from these schools. Case studies were then conducted with 8 of the immigrant teachers, involving discussions, lesson observations, interviews, and content analyses (of teacher marking). Each of the inhabited continents has been represented by at least one of these eight teachers. A total of 34 perceived value differences were reported by the 8 teachers, whose teaching experience in Victoria ranged from 1 to 27 years. The value differences were found to be mathematical, mathematics educational, general educational, and organisational in nature. Opportunities for the perception of value differences were not found to be related to length of service in Victoria, and neither was it related to 'cultural distance'. Other than possibly responding with a sense of helplessness, the ways in which value differences were negotiated can be classified into six responsive approaches. These reflect different teacher assumptions and dispositions, including a need to espouse the home culture, a desire to portray the host culture, and a disposition to embody the essence of both the home and Australian values. That each teacher had responded to similar perceived value differences with different approaches demonstrates how sense-making of difference is contextualised within socio-cultural factors at the individual, interactional, institutional, and societal levels. Thus, beyond the celebration of teacher diversity, the findings demonstrate the complexities with which reality is co-constructed by both the immigrant teacher and her socio-cultural environment. Theoretically, the conceptualisation of personal value schema in this study helps to explain how values underpinning contextual factors interact to guide choice of responsive approaches. The study shows how 'competing', 'overriding' and 'second-hand' values can thus be regarded as part of a teacher's schema rather than as isolated, independent values. In this light, the negotiation of perceived value differences represents the reestablishment of harmony and equilibrium amongst the constituent values of teachers' personal value schemas. The valuing of numeracy and technology stood out strikingly from the data as being features of the Victorian mathematics curriculum. The potential for this to inform the ongoing Victorian curriculum reform is discussed. Implications for more effective induction and in-service professional development programs for immigrant mathematics teachers, for other teachers in transition, and for educational leaders, are also identified. Other implications for practice and suggestions for further research are also made, highlighting the potential of this thesis to contribute to both theory and practice.
The first aim of this study was to identify important aspects of mathematical thinking, and to investigate the relationships between the different aspects of mathematical thinking and mathematics achievement. The second aim was to examine possible gender and school location (urban, suburban, and rural) differences related to aspects of mathematical thinking and mathematics achievement. Two assessments were developed that were suitable for students in the Year 11 scientific stream in Jordan. One test was for aspects of mathematical thinking and the other for mathematics achievement, the latter being consistent with typical school achievement tests for these students in Jordan. The researcher chose and developed items to test mathematical thinking and mathematics achievement from the Third International Mathematics and Science Study (TIMSS), the internet, research literature, specialist books in mathematics and his own experience. The data were collected in the 2003-2004 academic year from over 500 Year 11 scientific stream students (both male and female) at 20 randomly selected schools from six directorates in the Irbid Governorate, Jordan. In addition, 13 teachers were individually interviewed, and four groups of students were interviewed in focus groups to obtain information about their opinions and about different methods of thinking in mathematics. The teacher interviews were used to identify consistencies and inconsistencies between the test results and the respondents' opinions of difficulty and importance. In addition, information was obtained about the classroom time teachers devoted to the different aspects of mathematical thinking and the teaching strategies they employed. Six aspects of mathematical thinking were identified by the study: Generalisation, Induction, Deduction, Use of Symbols, Logical thinking and Mathematical proof. Mathematical proof was also the most difficult aspect, while Logical thinking was the least difficult. Female students had significantly higher mean scores than males on three of the six aspects of mathematical thinking and on the total test scores. Students attending suburban schools had significantly higher mean scores than students at urban and rural schools on four aspects, and on the total scores. Using multiple regression analysis, all six aspects were found to be important for mathematics achievement. Mathematical proof and Generalisation were the most important aspects, Use of symbols and Logical thinking were next in importance, and Deduction and Induction were the least important aspects. Approximately 70 per cent of the variance in mathematics achievement was explained by the six aspects of mathematical thinking, gender, and school location. There was a high level of consistency between teacher opinions of the relative importance of aspects of mathematical thinking and the test results. However, there were some inconsistencies between the teacher opinions and test results with respect to relative difficulty levels of the six aspects. By clarifying the importance for mathematics achievement of the six aspects of mathematical thinking identified, this study has relevance for the teaching of mathematics to Year 11, scientific stream students in Jordan.
Mathematics has been seen to act as a 'critical filter’ in the social, economic and professional development of individuals. The Island of Mauritius relies to a great extent on its human resource power to meet the challenges of recent technological developments, and a substantial core of mathematics is needed to prepare students for their involvements in these challenges. After an analysis of the School Certificate examination results for the past ten years in Mauritius, it was found that boys were out-performing girls in mathematics at that level. This study aimed to examine this gender difference in mathematics performance at the secondary level by exploring factors affecting mathematics teaching and learning, and by identifying and implementing strategies to enhance positive factors. The study was conducted using a mixed quantitative and qualitative methodology in three phases. A survey approach was used in the Phase One of the study to analyse the performance of selected students from seventeen schools across Mauritius in a specially designed mathematics test. The attitudes of these students were also analysed through administration of the Modified Fennema-Sherman Mathematics Attitude Scale questionnaire. In Phase Two a case study method was employed, involving selected students from four Mauritian secondary schools. After the administration of the two instruments used in Phase One to these selected students, qualitative techniques were introduced. These included classroom observations and interviews of students, teachers, parents and key informants. Data from these interviews assisted in analysing and interpreting the influence of these individuals on students, and the influence of the students’ own attitudes towards mathematics on their learning of mathematics. The results of Phases One and Two provided further evidence that boys were outperforming girls in mathematics at the secondary level in Mauritius. It was noted that students rated teachers highly in influencing their learning of mathematics. However, the teaching methods usually employed in the mathematics classrooms were found to be teacher-centered, and it was apparent that there existed a lack of opportunity for students to be involved in their own learning. It was also determined that parents and peers played a significant role in students’ learning of mathematics. After having analysed the difficulties students encountered in their mathematical studies, a package was designed with a view to enhance the teaching and learning of the subject at the secondary level. The package was designed to promote student-centred practices, where students would be actively involved in their own learning, and to foster appropriate use of collaborative learning. It was anticipated that the package would motivate students towards learning mathematics and would enhance their conceptual understanding of the subject. The efficacy of the package was examined in Phase Three of the study when students from a number of Mauritian secondary schools engaged with the package over a period of three months. Pre- and post-tests were used to measure students’ achievement gains. The What Is Happening in This Class (WIHIC) questionnaire also was used to analyse issues related to the affective domains of the students. An overall appreciation of the approaches used in the teaching and learning package was provided by students in the form of self-reports. The outcomes of the Third Phase demonstrated an improvement in the achievement of students in the areas of mathematics which were tested. The students’ perceptions of the classroom learning environment were also found to be positive. Through their self-reports, students demonstrated an appreciation for the package’s strategies used in motivating them to learn mathematics and in helping them gain a better understanding of the mathematical concepts introduced.
This study investigated the relationship between Colombian mathematics teachers' conceptions of beginning algebra and their conceptions of their own teaching practices. The teachers' understandings of their teaching practices were explored with a view to unravelling their conceptions of change in their teaching. Focusing on the perspectives of teachers afforded opportunities that exposed the powerful role that the teachers' conceptions of social/institutional factors of teaching played in their conceptions of their practices. The degree to which they attributed these (external) factors as crucial determinants of their teaching provided the basis for a categorisation of (the) teachers' conceptions of their practices into four types of teachers, that goes from the (fully) 'external attributions' teacher to the (fully) 'internal attributions' teacher. From the findings, implications were identified for the creation of possibilities of change in the teaching of beginning algebra (and mathematics in general) in Colombia, as well as for developing the research on teachers' conceptions of mathematics and its teaching. A two-phase case study research design was chosen for this study. In Phase 1, which aimed to identify a variety of conceptions from an initial group of teachers in order to select case studies, data were collected from a group of 13 mathematics teachers, who taught at six different (state and private) schools in Bogota. The teachers varied in ages and teaching experience and were teaching in Grade 8. In Phase 2, a multi-case study with the participation of nine selected teachers was carried out. In Phase 1, data were collected through the use of two questionnaires and an interview, taking place in the following sequence: Questionnaire 1, Questionnaire 1 follow-up interview, and Questionnaire 2. Descriptors used in the questionnaires were developed assuming that teachers' conceptions of the nature of mathematics might be very different, and might range from a traditional 'instrumentalist' perspective of a collection of unrelated facts, rules and skills to one in which mathematics is a continually expanding field of human inquiry, where problem solving and understanding are central in mathematical activity. In Phase 2, data were collected through classroom observation, interviews, examination of curricular materials and a focus group session. The teachers dedicated part of the interviews to the construction of a concept map of the determinants of their practices. Data analysis was conducted in the language (Spanish) of the data collection. Data collected through the different sources were reviewed and classified in order to identify i) the teachers' conceptions of beginning algebra, and ii) the teachers' conceptions of their own teaching practices. In identifying the teachers' conceptions of beginning algebra, the focus of the analysis was placed on data related to the fundamental components of teaching (i.e., answers to the why, what and how of the teaching of beginning algebra). In identifying the teachers' conceptions of their own teaching, the focus was placed, at one point, on why the teachers taught Grade 8-algebra in the way they did and, at another point, on why they would (or would not) be willing to consider a different approach in their teaching of Grade 8-algebra. A significant contribution of this study is a typology of teachers' conceptions of their own practices which provides key insights to inform the provision of professional development and teacher education programs in Colombia. This contribution is particularly relevant to our understanding of the stability of mathematics teaching approaches in the Colombian context but has likely implications for a range of international education contexts. Another significant contribution of the study is represented by its theoretical implications for the development of the research into teachers' conceptions of mathematics and its teaching. The model of mathematics teachers' thought structures that emerged includes teachers' social 'knowledge, beliefs and attitudes' as an integral dimension of teachers' thinking.
This thesis records a study of major change. The study was designed to reveal and address the implications for teachers of primary mathematics, of moving from test-based assessment to a base built upon a balanced blend of norm-referenced and criteria-based assessments. In developing embedded authentic assessment through a process portfolio model, the teachers looked to change from the assessment of learning to assessment for learning. Consequently, through the efforts of the teachers involved, their students and those students' parents, the study traced a substantial pedagogical restructure. Based on an interpretative methodology, this study of significant assessment restructure used mainly qualitative approaches to data collection and analysis, supplemented by limited quantitative data. Interviews, participant observer interactions, surveys and joint teacher discussion and planning sessions were effective in mapping the change. Through frequent interaction, participating teachers shared their emerging understandings, along with difficulties and successes in the evolution and implementation of an effective, flexible process portfolio. From the beginning of the evolution, teachers working together to bring about improvements that would lead to students perceiving mathematics as meaningful, engendered a strong feeling of excitement, curiosity and 'team'. As the change progressed the team identified and met a range of challenges, not the least of which was gaining an understanding of the nature and function of a process portfolio strategy as against the product portfolio which was in use at that time in the study school. The resultant change was not implemented without barriers. Of prime concern across the group of teachers involved was the perennial problem of finding development time in what were already busy teaching days. However, for the change to be meaningful and lasting, it was imperative that the teachers invested considerable time in assuming ownership through genuine engagement in the evolution of the new concept. The engagement saw teachers experience first-hand the application of constructivist learning theory. It was an approach to learning that was largely unfamiliar to them and one they needed to understand in developing a successful process portfolio model. The study of that learning and the resultant change illustrated that a well-designed process portfolio structure offers widely diverse opportunities for teachers and students to work meaningfully with authentic mathematics. The enthusiastic prolonged engagement on the part of the students, with notable parental support, was deemed by the participant teachers to be suitable reward for the time and effort that they invested over the two years of the study. Following the teachers' prolonged commitment, the emergent portfolio was shared through an in-house booklet written to encourage other teachers to adopt authentic assessment, Process Portfolios in Primary Mathematics: A Guide. Within the booklet, explanation and illustration of the rationale, form and function of the unique process portfolio model offers starting points for others, should they embark on a similar course of assessment change in search of real student engagement in understanding mathematics. Subsequent sharing of the results of the study with the wider profession through journal articles and conference workshops is to be based on the contents of the guide booklet.
This study analyses the differences between upper-stream, lower-stream and mixed-ability mathematics classes in terms of student perceptions of their classroom learning environment. Both quantitative and qualitative data has been collected from students while qualitative data only was collected from pre-service teachers, practising teachers and parents. The sample for the quantitative data collection was comprised of 581 Year 9 and 10 students in 36 different classes taught by 28 different teachers in 7 schools covering 4 states of Australia. All of the schools are private schools and part of the Seventh-day Adventist school system. The questionnaire used an actual and preferred form of the 56 item version of the What is Happening in the Classroom? (WIHIC) survey along with 10 questions from the Test of Science Related Attitudes (TOSRA) modified for mathematics classrooms. For the qualitative data collection 40 interviews and 8 focus groups were conducted. Apart from comparing upper and lower-streams, other variables examined were: actual and preferred perceptions of the classroom learning environment, Year 9 with Year 10, males with females, English speakers with second language students, and attitudes with perceptions of learning environments. The most significant finding of the study was not only that lower-stream students have a more negative perception of their classroom learning environment, but that they seek less change. This negative perception is seen to be worse in Year 10 than Year 9, particularly in the areas of teacher support and task orientation. This study found a positive correlation between attitude and perceptions of classroom learning environment. This study also found a tacit acceptance of streaming as a practice by most participants in the study.
Mathematical modelling problems are embedded in written, representational, and graphic text. For students to actively engage in the mathematical-modelling process, they require literacy. Of critical importance is the comprehension of the problems' text information, data, and goals. This design-research study investigated the application of top-level structuring; a literary, organisational, structuring strategy, to mathematical-modelling problems. The research documents how students' mathematical modelling was changed when two classes of Year 4 students were shown, through a series of lessons, how to apply top-level structure to two scientifically-based, mathematical-modelling problems. The methodology used a design-based research approach, which included five phases. During Phase One, consultations took place with the principal and participant teachers. As well, information on student numeracy and literacy skills was gathered from the Queensland Year 3 'Aspects of Numeracy' and 'Aspects of Literacy' tests. Phase Two was the initial implementation of top-level structure with one class of students. In Phase Three, the first mathematical-modelling problem was implemented with the two Year 4 classes. Data was collected through video and audio taping, student work samples, teacher and researcher observations, and student presentations. During Phase Four, the top-level structure strategy was implemented with the second Year 4 class. In Phase Five, the second mathematical-modelling problem was investigated by both classes, and data was again collected through video and audio taping, student work samples, teacher and researcher observations, and student presentations. The key finding was that top-level structure had a positive impact on students' mathematical modelling. Students were more focussed on mathematising, acquired key mathematical knowledge, and used high-level, mathematically-based peer questioning and responses after top-level structure instruction. This research is timely and pertinent to the needs of mathematics education today because of its recognition of the need for mathematical literacy. It reflects international concerns on the need for more research in problem solving. It is applicable to real-world problem solving because mathematical-modelling problems are focussed in real-world situations. Finally, it investigates the role literacy plays in the problem-solving process.
Middle school initiatives (including heterogeneous classes and an integrated, flexible curriculum together with promotion of student input) have been implemented in schools in Western Australia in response to a perceived need to align schools more closely with a more student-centred approach to learning, in the expectation of meeting more students' needs and thereby reducing student dissatisfaction and increasing the possibility of students pursuing life long learning. Specific goals underlying the initiative include the development of independent learning and student responsibility for learning through a series of strategies such as self-paced learning, student involvement in negotiating their own learning, and a strong emphasis on respecting and valuing student input into the implementation of curricula. However, owing to the way that the curricula for Middle and Upper secondary school mathematics are currently structured, problems might arise for students in the transition from 'a relaxed to a highly discipline-based organization of content' (as described by Venville, Wallace, Rennie, Malone). Students accustomed to the current approaches implemented in Middle schools (Years 8 to 10) may be disadvantaged in the transition to Upper secondary school courses (Years 11 and 12) compared with those students who have been exposed to a more discipline-based organization of content throughout early adolescence and prior to entry into courses leading to tertiary entrance (T.E.E. courses). The aim of this project was to investigate the possible effects of Middle school initiatives in a group of students from three Middle schools in Western Australia in one subject area – mathematics – on the perceptions of self-efficacy and preparation in mathematics once the students encounter Year 11 Upper school courses. A survey containing Likert-type rating scales pertinent to four areas of interest – Self-efficacy in mathematics; Self-Directed Regulation; Views on current teaching; and Views on prior teaching were administered to students transferring from three 'feeder' Middle schools to Year 11 (Upper secondary school) classes in one Senior College in Western Australia for each of 4 consecutive years. Students were also asked for their comments regarding preparation for the challenges of their chosen courses in mathematics. In addition, their levels of performance in a range of mathematical skills were assessed using a teacher-developed test. The perceptions of their Middle and Senior School teachers were also sought. As the survey was administered to all students as a routine part of action research within the mathematics faculty at the Senior College, only the results of those students who subsequently agreed to be participants in the study are reported in this dissertation. Results indicated that a mismatch existed in approaches and skills between Middle School and Senior College Mathematics. The reliance on students making suitable choices for themselves, the absence of specialist teachers of mathematics in middle schools, mixed ability classes in which specialist teachers of mathematics find it difficult to operate successfully and a curriculum that was so flexible that teachers omitted key elements required for later studies were the main factors that resulted in a significant number of students making the transition from middle to senior school with insufficient preparation. Implications for the teaching of mathematics in these three Middle schools and the Upper school are discussed.
This study establishes a framework for the practice and the acquisition of mathematical knowledge. The natures of mathematics and rituals/ritual-like activities are examined compared and contrasted. Using a four-fold typology of core features, surface features, content features and functions of mathematics it is established that the nature of mathematics, its practice and the acquisition is typologically similar to that of rituals/ ritual-like activities. The practice of mathematics and its acquisition can hence be metaphorically compared to that of rituals/ritual-like activities and be enriched by the latter. A case study was conducted using the ritual metaphor at two levels to introduce and teach a topic within the current year eleven West Australian Geometry and Trigonometry course. In the first level, instructional materials were written using a ritual-like mentor-exemplar, exposition, replicate and extrapolate model (through the use of specially organised examples and exercises) based on the approaches of several mathematics text book authors as they attempted to introduce a topic new to the West Australian mathematics curriculum. In the second level, the classroom instruction was organised using a ritual-like pattern with direct exemplar mentoring and exposition by the teacher followed by replication and extrapolation from the students. Embedded within this ritual-like process was the personal (and communal) engagement with each student vis-a-vis the establishment of the relationships between the referent concepts, procedures and skills. This resulted in the emergence of solution behaviours appropriate to specific tasks imitating and extrapolating the mentored solution behaviours of the teacher. In determining the extent to which the instruction, mentoring and acquisition was successful, each student's solution 'behaviour was compared 'topographically' with the expected solution behaviour for the task at various critical points to determine the degree of congruence. Marks were allocated for congruence (or removed for incongruence), hence a percentage of congruence was established. The ritual-like model for the teaching and acquisition of mathematical knowledge required agreement with all stake-holders as to the purpose of the activity, expert knowledge on the part of the teacher, and within a classroom context requires students to possess similar levels of prerequisite mathematical knowledge. This agreement and the presence of an expert practitioner, provides the affirmation and security that is inherent in the practice of rituals. The study concluded that there is evidence to suggest that some aspects of mathematical ability are wired into the cognitive structures of human beings providing support to the hypothesis that some aspects of mathematics are discovered rather than created. The physical origin of mathematical abilities and activities was one of the factors used in this study to establish an isomorphism between the nature and practice of mathematics with that of rituals. This isomorphism provides the teaching and learning of mathematics with a more robust framework that is more attuned to the social nature of human beings. The ritual metaphor for the teaching and learning of mathematics can then be used as a framework to determine the relative adequacies of mathematics curricula, mathematics textbooks and teaching approaches.
Abstract not received.
There has been widespread interest in the potential impact of the graphics calculator on system wide 'high stakes' end of secondary school mathematics examinations. This thesis has focused on one aspect, the way in which examiners have gone about writing examination questions in a graphics calculator assumed environment. Two aspects of this issue have been investigated. The first concerns the types of questions that can be asked in a graphics calculator assumed environment and their frequency of use. The second addresses the level of skills assessed and whether with the introduction of the graphics calculator has been associated with an increase in difficulty as has been frequently suggested. A descriptive case study methodology was used with three examination boards, the Danish Ministry of Education, Victorian Curriculum and Assessment Authority and the International Baccalaureate Organization. Four distinct categories of questions were identified which differed according to the potential for the graphics calculator to contribute to the solution of the question and the freedom the student was then given to make use of this potential. While all examination boards made use of the full range of questions, the tendency was to under use questions in which required the use of the calculator for their solution. In respect to the level of skills assessed, it was found that both prior to and after the introduction of the graphics calculator, all three examination boards used question types that primarily tested the use of lower level mathematical skills. With exceptions, where graphics calculator active questions have been used, the tendency has been to continue to ask routine mechanistic questions. In this regard, there is no evidence of the introduction of the graphics calculator being associated with either lowering or raising of the level of the mathematical skills assessed. For all cases studied, the graphics calculator was introduced with minimal change to the curriculum and examination policies. The role of the graphics calculator in the enacted curriculum was left implicit. The resulting examinations were consistent with the stated policies. However, the inexperience of some examiners and a general policy of containment or minimal change enabled examiners to minimise the impact of the introduction of the graphics calculators on assessment.
The implementation of the Curriculum Framework into Western Australia in 1998 has changed the nature of mathematics education in the state. A central premise of the Framework is student engagement with education. The Curriculum Framework describes engagement in terms of student attitude, motivation and beliefs. The aim of the research reported in this thesis is to develop an understanding of mathematics teachers' perspectives on how they engage lower secondary state school students in ways commensurate with the mathematics learning outcomes of the Curriculum Framework. Three schools were investigated as case studies. In each school, three teachers were interviewed as to their understanding of what engagement with mathematics is and how they engage their students with the learning area. The schools were from a variety of socioeconomic and geographic parts of Perth, Western Australia. From the analytical findings of the case studies, three propositions Were developed. The first of these is that engagement with mathematics requires students to demonstrate a combination of doing mathematics and thinking about mathematics. The second proposition is that student centred learning approaches encourage students to engage with mathematics. The third proposition is that student success is a key ingredient to fostering student engagement with mathematics.
This study explores how an expert learning process may be beneficial in developing number sense in Year 6 students. The expert learning process was a set of reflective prompts designed to include questions that expert learners consider. The study focuses on the development of number sense due to its centrality in today's mathematics curriculum; and as it was a more specific topic, it made the task more manageable for this relatively small study. A case study method was used as this was an exploratory study and the author was interested in gaining in- depth, rich material about one setting. The setting was a Year 6 class from a primary school in the Perth metropolitan area, and the study lasted for two school terms. The data collection method used a mixed paradigm designed in order to strengthen internal validity, although the predominant approach was qualitative. Using qualitative data collection it was found that by engaging in the expert learning process, the students were prompted to reflect on their learning at various times throughout the number lessons, thus beginning to behave as self-regulated learners. The expert learning process also made the students more motivated in their attitudes towards the mathematics lessons, and after reflecting on the prompts provided, the students were at times directed towards more effective learning strategies. The class teacher's teaching style was found to be extremely important in affecting the outcome of this study. Her transmissive teaching style meant that she did not perceive the students' reflections as important and therefore the learning environment was not influenced very much by these reflections. The students were powerless in this situation and had to accept the fact that their learning activities did not change greatly. The study was designed to find out if the number sense performance of the Year 6 students improved through using the expert learning process. To investigate this the author took a quantitative approach using a number sense test, which had been used in an international study, and focused on analysing if there was any increased number sense performance. The results showed that there was no significant improvement in average number sense performance. It is important to note that the reason that number sense was not developed may have been due somewhat to the teacher's beliefs about mathematics. In this study the teacher did not believe that development of number sense should be the focus of her teaching. This again, reveals the importance of the teacher's role in guiding students to learn effectively.
This thesis explores, from the perspective of the software engineer, the application of computers to the teaching of secondary school mathematics. It identifies ways in which the uptake of computers in mathematics teaching can be encouraged, and show how the software engineer, working in partnership with the teacher, can play a pivotal role in enhancing teachers' use of technology. By conducting a number of interviews and undertaking several case studies with practising secondary school mathematics teachers, the research identifies the key factors influencing the integration of computers into the classroom. These factors range from the adequacy of teacher training through to the design of appropriate didactic software. The emphasis throughout is on the role the software engineer can play in the creation of software that will be used successfully by the teacher and will make a significant contribution to the overall teaching of mathematics. Cooperative teacher and software engineer partnerships are trialled in depth through the case studies. The outcome is the development of a software architecture aimed at creating educational software products that are adaptable to the pedagogical and epistemological orientations, and consequently the teaching practices, of individual teachers. The study also explores the various views mathematics teachers have of integrating technology; model the major factors influencing the integration of technology by mathematics teachers, and explore how these factors interrelate; explore processes of co-developing educational software with mathematics teachers; suggest how teacher training can be modified to more effectively encourage and assist mathematics teachers to integrate computers; categorize the various types of mathematics educational software; categorize the various educational tasks found in mathematics software; and develop an underlying reference architecture to create mathematics educational software that can be readily adapted to meet teachers' individual needs.
This study is an investigation of assessment and learning in the curriculum area of mathematics based on the current practices of a State government primary school. A case study approach provided a rich source of information on a means of measuring change over four years of schooling (Year 3 to Year 6). The learning was focussed on student knowledge, in relation to the mathematical tests selected as measurement tools in this study. A learning continuum developed was for students involved in the investigation and the opinions of their parents and teachers and selves sought via surveys. Designs selected that prevented disruption to the normal operation of the school at the centre of the investigation. The literature of Izard, Black and William, Forster and Broadfoot guided the educational context of the investigation into formal assessment practices for formative purposes. Modern Rasch analysis computer developments enabled the exploration of Rasch analysis techniques offering the researcher access to statistical information previously confined to statisticians and contributing to the investigation of effect size and the work of Cohen, Glass and Coe in measuring the magnitude of change. Sampling and analysis methods relating to validity and equating procedures supporting comparison issues required investigation and application. The disadvantages of decisions made outside and without consideration of teacher and students investigated with particular reference to the impact these decisions can make in relation to student learning.
This thesis reports an investigation of collaborative learning in senior mathematics classrooms. Collaborative learning in this study refers to forms of classroom organization in which students of approximately equal expertise work in small groups on relatively open- ended tasks, and are encouraged to share their thinking as they work together. In some mathematics classrooms, a goal of collaborative learning is to shift the locus of intellectual authority informally from the teacher to the students. Participants in this research were three teachers who, within the constraints of the school and examination systems, were constructing their classrooms as communities of mathematical inquiry. When students are working together optimally, their collaboration is 'rich in mutual discovery, reciprocal feedback and frequent sharing of ideas' and their interactions are characterised by indicators of shared understanding. Groups, however, sometimes fail to collaborate effectively. The author's particular interest was in the interplay between gendered behaviour and participation in collaborative activities, so she concentrated on student-student interactions during collaborative work. Her objective was to gain a better understanding of factors that influence the dynamics of interactions within collaborative groups, in the belief that this could provide guidance for teachers in planning collaborative activities, and increase equity for all students learning mathematics in the senior years of schooling. Data generation involved videotaping lessons and interviewing students and teachers. The emergent design of the study centred on identifying patterns of participation, and only then using gender as an interpretive frame. The major part of the analysis centred on studying the interactions among students during small-group discussions from the perspective of Positioning Theory, showing how students were differently positioned, at various times during an interaction. Analysis began with an intensive study of one group in one lesson. Themes that emerged from that analysis were carried forward to other lessons from all three schools, and eventually to the entire database of 46 lessons. The identification of the range of positions available to students during collaborative work in mathematics is a major outcome of the study. The cluster of positions typically occupied y a student was called that student's pattern of participation. By describing common Patterns of participation in collaborative activities, the research has taken an important step towards developing a better understanding of factors that promote or inhibit effective collaboration. The final stage of the analysis looked at the patterns from the perspective of gender. While some patterns observed were in line with expectations generated from previous research, others crossed traditional gender boundaries. A major strength of the study is the methodology that deferred considering gender until the final stage. As a consequence, the author was able to show the possibility of almost every identified position and many patterns of participation being occupied by either gender. These findings will help teachers to become more aware of the types of interactions taking place in their classrooms, and to plan ways of improving them.
Mathematics plays a key role in bolstering a country's knowledge economy. Australia's knowledge economy is negatively affected by the underachievement of Australian school students in geometry. Research indicates a continuing decline in student performance in geometry and a distinct lack of geometrical knowledge and understanding on the part of students and teachers. To address this issue a theory of success in geometry that focussed on background variables and attitude, was developed. In the theory it was hypothesised that success in geometry can be understood in terms of predictor variables and that attitude mediates the effects of the variables on success in geometry. A model of success in geometry was developed to systematically determine the relationships of the variables. Trainee teachers from the University of Western Sydney (n = 224) participated in the survey. Using Confirmatory Factor Analysis the use of one or two attitude scales was determined as were the items in the scales. Using Structural Equation Modeling (SEM) the relationships between the background factors (age, education, gender, left/right brain preference) on success in geometry (van Hiele level) mediated by attitude were determined. The evidence, however, suggests that attitude is not only correlated with the measures of success in geometry (van Hiele levels) but that it may also be a predictor of success in geometry. It was also hypothesised that attitude was composed of three analytically distinct factors (affective, cognitive and behavioural). The evidence suggests that this hypothesis cannot be rejected. This is an important finding as previous research has not been empirically able to distinguish these factors. In order to improve the success of Australian school students in geometry and assist teachers to succeed and consequently improve Australia's knowledge economy, the present research indicates that: all trainee teachers should have their van Hiele level of geometry understanding determined; appropriate geometry courses should be a mandatory part of the curriculum for all pre-service teachers whose van Hiele level is less than three; all trainee teachers should have a van Hiele level of three or four before they commence teaching; appropriate changes to the curriculum of trainee teachers should be made so that their stored general evaluative process produces a positive attitude to geometry, especially in female students; school students who intend to pursue a teaching career should complete mathematics courses with a geometry content.
This thesis reports on research involving a class of Year 8 students while they undertook four technology-based projects integrating the curriculum areas of mathematics, science and technology and enterprise. The study examines the structural features and pedagogical characteristics of each project and focuses on three key aspects of student learning that are argued to be enhanced by integration - student motivation, learning outcomes, and the capacity to transfer knowledge. The study was conducted over one school year and took place in a government high school in Western Australia. The participating class was a heterogenous group under the direction of the same two teachers for each project. One teacher was responsible for the mathematics and science learning areas while the other taught technology and enterprise. Each project was conducted over a full school term, providing the opportunity for 'prolonged engagement' and 'persistent observation' in order to describe in detail the characteristics of each setting. Each project has been described as a separate case study, detailing the events and outcomes individually, but the findings and conclusions of the study are drawn from the combined data provided by the case studies. Several complementary monitoring tools were employed including direct observation of performance tasks, field notes, interviews, concept mapping, audio taping, photographs, learning journals and analysis of artefacts such as student workbooks and teaching notes. The focus was to utilise research and assessment techniques that were primarily based on actual classroom tasks and to provide sufficient data to ensure a 'rich description' of the events that unfolded. A further dimension to the data was provided by using the What Is Happening In This Class? (WIHIC) questionnaire to draw on students' perceptions of the classroom environment. The results of the study revealed that success cannot be assumed with an integrated curriculum. Factors such as students' readiness and preparedness to learn in an integrated context have to be taken into account. The students may have to acquire the skills to learn in an integrated context, and they may also have to learn how to recognise and apply knowledge within and across curriculum areas. Although integrating the curriculum blurs the boundaries between subject areas, the learning purposes must be clear to both teachers and students. Purposeful learning was found to enhance motivation, but it was unclear whether students learned more in the integrated context than if their learning had been in a traditional, subject-based format. The results of the study revealed that success cannot be assumed with an integrated curriculum. Factors such as students' readiness and preparedness to learn in an integrated context have to be taken into account. The students may have to acquire the skills to learn in an integrated context, and they may also have to learn how to recognise and apply knowledge within and across curriculum areas. Although integrating the curriculum blurs the boundaries between subject areas, the learning purposes must be clear to both teachers and students. Purposeful learning was found to enhance motivation, but it was unclear whether students learned more in the integrated context than if their learning had been in a traditional, subject-based format.
No abstract available.
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. Departing from the traditional technical approach to teaching mathematics, the author was motivated to create the awareness in students that mathematics is more than just skills and drills, as it is often perceived. S/he incorporated a variety of word problems into the curriculum, and explored the use of discussions, peer-group activities, and reflection. S/he also examined how specific values can be imparted through mathematical concepts and practices. Since the ultimate purpose of this research was to change and improve the educational environment through collaborative effort from the participants, his/her methods were oriented towards the critical paradigm. Action research was used to explore the possibility of collegial participation in a traditional and hierarchical institution. The methods included the use of journal writing, open-ended questionnaires, conversational interviews and narrative observations. Constraints experienced included the short semesters, a packed curriculum, expectations of the participants and the institution, and a lack of autonomy for teachers to make radical changes to the curriculum. The impact of various socio-cultural and political challenges also affected the development and direction of the research. Language issues were significant because the majority of the students had a trilingual background of learning mathematics, that is, Chinese at primary level, Malay at secondary level and English at the present college level. Other socio-cultural aspects that were influential include the effects of the students' Chinese background, the examination-oriented curriculum, and certain political and economic challenges faced by Chinese Malaysians. All these issues were explored gradually through seven cycles of action research with the author's own students and a small group of colleagues. The research examined the resistance, challenges and constraints arising out of using (a) word problems in mathematics education and (b) action research as a research method, both of which are Western approaches. This thesis records the effects of using these techniques in a hierarchical and traditional Malaysian educational environment with students from a Chinese background.
This study investigated educational software design elements that encourage the development and use of higher-level thinking skills. This was carried out through an interpretive qualitative methodology within a constructivist conceptual framework. That is to say that the software used, the learning environment, and the methodology were based on the principles of constructivism. In addition, the study sought to determine the attributes of a constructivist learning environment that are particularly effective in supporting the software in its role in stimulating higher-level thinking. The setting for the research was a secondary school classroom in Western Australia and a participant- observational method was utilised. The learning environment was purposively established through discussions with the classroom teacher regarding the principles of constructivist learning environments. The software selected for use was the Geometry Inventor from LOGAL Software, principally because it had been designed in keeping with the tenets of constructivism. The research was carried out by initially formulating an operational definition of higher-level thinking. Specifically, the roles of cognitive tools, question types, format of student answers, manipulable objects and navigational issues were examined. Further, an investigation of the aspects of the learning environment that supported the occurrences of higher-level thinking was carried out. Data gathered through the use of the database and the participant observational method were enriched and supported with pre and post surveys, computer answer files from the software, student work samples, and interviews with students and the class teacher. The outcome of this study indicates that higher-level thinking is encouraged by the provision of a rich set of tools, and the presence of manipulable objects that allow students to generate exemplars of a new concept. In terms of the learning environment, cooperative learning and student responsibility and initiative are identified as key elements supporting the software in stimulating higher-level thinking. These issues have implications for teaching and learning. Specifically, the roles of cognitive tools, question types, format of student answers, manipulable objects and navigational issues were examined. Further, an investigation of the aspects of the learning environment that supported the occurrences of higher-level thinking was carried out. Data gathered through the use of the database and the participant observational method were enriched and supported with pre and post surveys, computer answer files from the software, student work samples, and interviews with students and the class teacher. The outcome of this study indicates that higher-level thinking is encouraged by the provision of a rich set of tools, and the presence of manipulable objects that allow students to generate exemplars of a new concept. In terms of the learning environment, cooperative learning and student responsibility and initiative are identified as key elements supporting the software in stimulating higher-level thinking. These issues have implications for teaching and learning.
This thesis uses a theoretical framework derived from activity theory to investigate the introduction of computer algebra systems (CAS) in first year university mathematics subjects. Both qualitative and quantitative data relevant to a case study of a group of approximately one hundred students, and two academics, were collected and analysed using a range of methods. The major question for this study was: What are the socio-cultural dynamics of learning with a new tool? More specifically, there are three questions: first, how do students in a particular context respond to their initial experience with a CAS as part of their first year mathematics subjects? Second, what relationships exist between aspects of students' personal histories, their goals for mathematical learning, and the range of experiences they report concerning using a CAS for the first time? Third, in a particular case in a particular setting, how do academics see their role in the introduction of computer algebra systems into mathematics teaching? The main findings include the identification of the critical nature of purpose, or multiple motivating 'objects' in activity systems. Personal identity as a learner of mathematics is constructed through choosing to engage at surface or deep levels, alone or with others. Students with a low level of computing background who had a high level of engagement and sense of purpose in their mathematical learning reported that they appropriated the new tool for their own personal use. Students with a high level of computing experience who were unable to form goals congruent with the learning tasks were less likely to appropriate the tool. In a similar way, lecturers with different purposes and different epistemological views of mathematics, in responding to contexts and personal goals, planned different teaching and learning experiences for their classes. The significance of this study is that it demonstrates how activity theory can be used successfully as a framework for an investigation that takes an expansive view of learning as a socio-cultural activity. Personal socio- cultural histories and motivations and social contexts influence students and academics as they form and reform their goals for engaging in learning and teaching activities. The study also highlights the gap between high school experiences of learning mathematics, dominated by rule following and the replication of pen and paper algorithms, and the more creative and challenging possibilities for making mathematics opened up by new technologies such as computer algebra systems.
Full thesis available from http://ro.uow.edu.au/theses/198
This study investigated students' understanding of rates of change in calculus. The study was designed to examine the possible difficulties that students in a foundation/bridging year program, MAZE, experienced as they worked on calculus tasks. MAZE is a full year pre-degree program in mathematics and science with emphasis on reading, writing and practical skills on these subjects in preparation for a possible science based degree programs. The specific aims to the study were to determine (i) the processes through which students develop an understanding of rates of change concepts, (ii) how and why these processes operate, and (iii) the implications of the findings for the teaching and learning of calculus in South Africa. The study involved those students registered for a one-year mathematics and science foundation year program at a university in South Africa. These are the students who have matriculated from secondary schools, but whose performances do not meet the admission criteria to study for a science based degree. A case study approach was adopted and qualitative methods were used to explore students' thinking and the learning environments in which thinking took place. Students' understandings of rates of change were examined by using their written responses to a class task and a test, conversations of related tasks from the learners' materials, and participant observations. Also, students' conversations were used to determine the social and sociomathematical norms that characterise the learning environments in which understanding occurred. Using excerpts of these conversations, the author examined the classroom environments in which learning took place and studied vignettes from students' responses to a class task and test to reach conclusions regarding the following four research questions that provided a focus for the study: (i) what understandings do MAZE students ascribe to tasks in the calculus learners' materials on rates of change? (ii) how do MAZE students ascribe understanding to these tasks? (iii) why are these understandings ascribed to the tasks? and (iv)what are the implications of the findings for the teaching and learning of rates of change? The study demonstrated that social and sociomathematical norms impacted on the classroom environment and that these norms differed from group to group of students within the same classroom. There were groups that could function well without the intervention of a facilitator, and those that would have benefited from such interventions. The investigation of students' thinking revealed that they were confused by numerical computations of rates of change, the roles that tangents and chords play in rates of change, and that they misinterpreted multiple representations of these rates. These misunderstandings occurred because of conceptually shallow conversations held and a language-of-mathematics barrier that frustrated attempts to develop mathematically viable concept images of rates. Students' prior knowledge, the quality of their discussions, and learners' materials were important factors contributing to the understandings that were made. Outcomes of the study suggest that: (i) procedural discourse should be contrasted with conceptual discourse in which reasons for calculating in particular ways become explicit in a conversation; (ii) more attention be given to the design of items leading to a solid image of the concept of average rates of change, and that (iii) writing to learn be considered as a pedagogical tool.
The major purpose of this study was to address the instructional needs of proof-type geometry problem solving. It was designed to address two research questions: (Q1). What are the predictive indicators of successful proof-type geometry problem solving? (Q2). Based on needs with an emphasis on formative evaluation, what is one design solution to support students solving proof-type problems in geometry? The overall study focused on a learning need assessment in the first phase of the study (Study 1) and a development process to translate instructional needs identified into a supportive instructional environment for proof-type geometry problem solving in the second phase (Study 2). The review of literature revealed that proof-type geometry problems have different learning requirements compared to other mathematical problems types. The solution process for proof-type geometry problems demands the adoption of a non-algorithmic approach in which students could activate problem-solving strategies that are domain specific. These strategies include heuristics such as using auxiliaries (parallel lines, bisectors and perpendiculars), alternative proving methods (indirect proof, reductio ad absurdum, method of contradiction). Equally important are the role of domain-general strategies during the solution of proof-type geometry problems such as working backward and logical inferencing. The literature review suggested that geometry content knowledge, general processes, and mathematical reasoning could be potential predictive indicators of successful proof-type geometry problem solving. However, the relative importance of these variables during the construction of geometry proofs had not been subjected to an empirical evaluation. Study 1 takes up the above issue by determining the relative importance of these variables in proof type geometry problem solving. Data were collected from 166 Sri Lankan students on three independent variables: Geometry Content Knowledge (GCK), General Problem-Solving processes (GPS) and Mathematical Reasoning Skills (MRS); and a dependent variable Proof- Type Geometry problem-solving (PTG). The relationship among these variables was examined through a multiple linear regression analysis procedure. This analysis showed that geometry content knowledge, general problem-solving processes, and mathematical reasoning are predictive indicators of successful proof-type geometry problem solving. Among these variables, geometry content knowledge was found to be the most influential one followed by general problem-solving processes and mathematical reasoning. Three experts participated in a series of meetings to translate the above findings into a support framework for helping students learn to solve proof-type geometry problems in Study 2. This development process resulted in a conceptual model consisting of three major components: Remedial, Instructional and Problem Solving. The Remedial Component was suggested to address the learning needs related to geometric reasoning development, the Instructional Component focused on the development of content knowledge related to Euclidian deductive system, and the Problem-Solving Component was designed to facilitate proof-type geometry problem- solving skills among students who have the prerequisite geometric content knowledge and reasoning skills. An iterative development process of design, development, review and revision was used to translate the Problem-Solving component into a Web-based, prototype learning environment in Part I of Study 2. This prototype, titled ANGEL (A Non-linear Geometry Environment for Learning), contained problem sets, process guidance, worked examples, diagram support and embedded content knowledge as core structural elements. Hyperlinked metacognitive supports were incorporated to facilitate the problem- solving process through guidance provided by general problem-solving processes such as analysis, representation, planning and use of knowledge retrieval by accessing embedded content. Although technology driven learning environments are mainly for student-technology interactions, ANGEL has additional advantages as it was designed for classroom use with teacher intervention to enhance social interactions: teacher-student and student-student that promote learning and construction. The usability of ANGEL was tested in a constructivist collaborative learning environment. Six students selected from an Australian high school solved a series of proof-type geometry problems in pairs in a two-hour problem-solving session with the help of ANGEL. During their problem-solving attempts, data were collected in the form of student verbalization of the solution process, observation of problem-solving attempts, and written workings in the workbook. Having completed the problem-solving session, the students were interviewed to collect data on how they perceived ANGEL as a learning tool. The qualitative data analysis showed that the target group of students accepted ANGEL as a learning tool and that students enjoyed using ANGEL in problem solving. These patterns of results suggest that ANGEL works as designed and assists students to construct knowledge related to proof-type geometry problem solving.
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This study researched the effect of money as a context on school students' mental computational performance and strategy choices across a range of ages. This study adds to existing research, which has compared students' mental computational methods with their written methods, by the provision of the single common context of money. The content topics of whole and other rational numbers (simple fractions, decimals, and some percentages) were covered. Forty-eight primary school students plus sixteen secondary school students were involved in this study, with equal numbers of both genders from the two primary schools and one secondary school in the Perth metropolitan area. The method followed was both quantitative—by scoring test results—and qualitative—through tape-recorded interviews. Students' prior experiences with money were documented and performance data were collected on students' mental computation ability for the two sets of mathematically identical items presented in a money-context and context- free. Student strategy choices were also documented. The semi- structured interviews consisted of nine money experience questions such as, How often do you get pocket money or an allowance? In addition, 10, 12, 13, and 13 pairs of mental computation items for Years 3, 5, 7 and 9 respectively. Where possible, common items were used across two or more year levels to ascertain growth in mental computation skills. Overall, results found that the context presentation did not make a difference to student performance and there was no correlation found between performance and student preference for one presentation or the other. No performance differences were found for gender. Year 3 recorded the lowest process scores, while Year 7 recorded the highest process scores although all the items used at both Year 7 and Year 9 were identical. The greatest growth in mental computation performance was found to occur from Year 3 to Year 5 and from Year 5 to Year 7. Further, for Year 3, results found that the context presentation had a negative impact on student performance. Some students were found to be using written methods mentally. Analysis of individual items revealed that context had a positive influence in some cases. However, despite the emphasis in modern curricula on the use of context, it appears that such an approach may have little value if used in contrived rather than real situations. Recommendations for teaching practice include promoting real experiences at school by linking students' out-of-school experiences to classroom learning, such as exploring students' pocket- money purchasing power or promoting mental computation for a variety of context tasks. It is considered likely that mental computation in classrooms tends to be non-contextual and it is recommended that teachers should make more use of context. It is further recommended that teachers use money as a context, with mental computation items presented as part of real shopping tasks. Oral presentation would remove typical school method cues—a 'sheet' and pencil—with the only visual stimuli being the goods and price labelling. Class shops could use simulations for the junior grades, while older grades could organise real money exchange experiences integrated with other curriculum areas such as raising money for charitable causes. Research on the effect of other common contexts such as food, time, and other measurement topics should also utilise real activities, with examples of such being readily found in the media. The provision of a variety of contexts is important for students as what constitutes a meaningful context may vary from individual to individual.
Research has shown that high school students' ability to solve projectile motion problems is limited by a variety of misconceptions. This study sought to confirm and analyse student and teacher misconceptions, and an experimental program was designed to teach projectile motion in Year 12 by confronting students' misconceptions explicitly and by linking practical experiments to the equations of motion. One teacher in each of two independent girls' schools in the Sydney metropolitan area agreed to teach the experimental projectile motion unit. These two teachers undertook a training program to show them the misconceptions they and their students may have. One other teacher in each school agreed to teach the topic as they had taught it for a number of years. The classes at one school were graded and the classes at the other school were not graded. Forty-seven students participated. All four teachers were female, each had been teaching for between ten and twenty years, and they all had excellent reputations within their schools and in the mathematics community. In order to determine whether the students possessed misconceptions about projectile motion, and then to determine the effectiveness of the experimental teaching program, pretests and posttests were conducted. The interview schedule included questions that had been used in other studies, and asked the same questions in a number of contexts to see if the students were consistent in their answers. The questions in the schedule covered fired objects, objects dropped from a moving carrier, and forces. The experimental teaching program used a constructivist approach; several activities were used so that students achieved a qualitative understanding of projectile motion before the equations of motion were introduced. Using the abstraction theory put forward by Mitchelmore and White (2000a; 2000b), the students became familiar with projectile motion in a number of contexts, and discussed the similarities. During the quantitative part of the program, the teachers linked the earlier qualitative activities to the equations. The teachers took every opportunity to discuss the misconceptions with the students. The two traditional teachers were asked to teach the course just as they had done in previous years. In essence, this approach involves the derivation of the equations of motion both horizontally and vertically, and then the solution of as many questions from the textbook as can be fitted in the allocated lessons. Projectile motion misconceptions were not discussed with the two comparison class teachers until after they had taught the unit and been interviewed as to their misconceptions. The results indicate that the classical misconceptions in the literature are 'alive and well' in both the mathematics and physics classes. The experimental program was successful in one class in helping students deal with their misconceptions about objects dropped from a moving carrier. The students in the other class were less successful. Both classes still had problems with fired objects. During the training program for the experimental lessons, both teachers exhibited misconceptions about objects dropped from a moving carrier. During their lessons and in the interview each teacher also showed an inclination towards the impetus idea that an object possesses a force that keeps it going until gravity takes over and the object then falls down. One teacher was more successful in dealing with her misconceptions than the other and the class results show that she was able to help her students deal with the dropped object misconception. The implications of the study are important in the teaching of projectile motion in both mathematics and physics. In the first instance, teachers in both subjects should work together to help their students deal with their misconceptions. It would be helpful to both subjects if the writers of the two syllabi could also see the similarities between the topics. This study serves as a step towards greater understanding of student misconceptions. It supports the notion that students develop their own informal ideas about projectile motion, and that these ideas are extremely difficult to change. The study shows that not only the students possessed the misconceptions but also their teachers. While the teaching program was successful in helping students deal with their misconceptions about objects dropped from a moving carrier, it was less successful in dealing with the impetus misconception. The most significant factor determining whether student misconceptions were eliminated was found to be the teachers' ability to deal with their own misconceptions.
This study determines primary level prospective teachers' procedural and conceptual knowledge about the concept of division. The sample included 17 primary grade prospective teachers in South Australia. The research process consisted at three phases. In first phase the focus of interest was whether the prospective teachers were able to represent and solve problems related to division of fractions. In the second phase the researcher assessed the participants' multiplicative thinking. In the last phase the researcher introduced an alternative model i.e. rate or ratio to represent and solve division involving whole numbers and afterward the participants were assessed whether they would be able to apply this model for representing and solving division of fractions. Data analysis involved the use of descriptive statistics. The findings revealed that overall the prospective teachers' knowledge about fraction deeply based on intuitive model of fair sharing and to some extent they used repeated subtraction model in their representation. Consequently they successfully represented division of whole numbers. However they could not represent division of fractions. Particularly they were not able to pose word problems for the mathematical expressions where the divisor was a fraction. A considerable improvement was observed in participants' performance to represent division of fraction after going through instruction intervention. However the prospective teachers possessed insufficient and inappropriate multiplicative thinking that is why they used the rate or ratio model mistakenly where the situations were not appropriate for the model. The research findings suggest that prospective teachers need to develop their multiplicative thinking to use the rate or ratio model flawlessly.
This research developed a framework describing students' developing understanding of function. The research started with the problem: How might typical learning paths of secondary school students' developing understanding of function be described and assessed? The following principles and research questions guided the development of the framework. Principle 1. The framework should be research-based. Principle 2. The framework should include key aspects of the function concept. Principle 3. The framework should be in a form that would enable teachers to assess and monitor students' developing understanding of this concept. Principle 4. The framework should reflect students' big ideas or growth points which describe students' key cognitive strategies, knowledge and skills in working with function tasks. Principle 5. The framework should reflect typical learning trajectories or a general trend of the growth points in students' developing understanding of function. The following questions guided the development of the framework of growth points: 1. What are the growth points in students' developing understanding of function? 2. What information on students' understanding of function is revealed in the course of developing the framework of growth points that would be potentially useful for teachers? The framework considered four key domains of the function concept: Graphs, Equations, Linking Representations and Equivalent Functions. Students' understanding of function in each of these domains was described in terms of growth points. Growth points are descriptions of students' 'big ideas'. The description of each growth point highlights students' developing conceptual understanding rather than merely procedural understanding of a mathematical concept. For example, growth points in students' understanding of function under Equations were: 1) interpretations based on individual points; 2) interpretations based on holistic analysis of relationships; 3) interpretations based on local properties; and, 4) manipulations and transformations of functions (in equation form) as objects. The growth points in each domain are more or less ordered according to the likelihood that these 'big ideas' would emerge. To identify and describe these growth points, Year 8, 9 and 10 students in Australia and the Philippines were given tasks involving function that would highlight thinking in terms of the process-object conception and the property-oriented conception of function. Students' performance on these tasks and their strategies served as bases for the identification and description of the growth points. The research approach was interpretive and exploratory during the initial stages of analysis. The research then moved to a quantitative approach to identify typical patterns across the growth points, before returning to an interpretive phase in refining the growth points in the light of these data. The main data were collected from students in the Philippines largely through two written tests. Interviews with a sample of students also provided insights into students' strategies and interpretations of tasks. The research outputs, the research-based framework and the assessment tasks, have the potential to provide teachers with a structure through which they can assess and develop students' growth in the understanding of function, and their own understanding of the function concept.
Problem solving has received a great deal of attention in the past two decades. Much work has been carried out in the fields of cognitive psychology and education on the topic of problem solving. Psychologists mainly focused on the cognitive processes underlying problem-solving activities, while educators mainly focused on practical strategies to improve students' problem-solving skills. The research studies carried out in these two fields were somewhat separate with different theoretical underpinnings. This thesis brings together the fields of cognitive psychology and education through the methodological advances of educational measurement developed to measure psychological constructs. More specifically, this thesis develops a theoretical framework for measuring problem-solving proficiencies, and applies item response theory to analyse students' responses to the test items. The development of a framework for the construction of test instruments for measuring problem-solving proficiencies requires a solid theoretical basis drawn from the published works of both psychologists and educators in problem-solving research. A detailed review of problem- solving literature is given to provide the groundwork for the development of the problem-solving framework. Empirical evidence has also been collected through previous years' problem-solving tests and trial tests to build a rationale for the identification of four dimensions of problem solving for this study. In this way, the framework developed has both the theoretical basis of psychological research as well as the practical utility for the assessment of students in the classrooms. A review of item response theory (IRT) is given to describe the mathematical models, the philosophy and the properties of a number of measurement models. In particular, multi- dimensional IRT models are deemed suitable for modelling the complex skills demands of problem-solving tasks. Model fit and item fit statistics are also discussed to demonstrate how these can be used as measures of the goodness-of-fit of the data to the model. The test development process focuses on the development of items that not only can tap into the problem-solving dimensions defined in the framework, but can also reveal the sources of error in the problem-solving process. The test items and the corresponding coding schemes of responses aim at capturing as much information as possible to increase the accuracy of the measurement of students' problem-solving proficiencies. Factor-analytic approaches to analysing item response data are compared to IRT analyses. There are a number of challenges in applying exploratory factor analyses. IRT approaches are more successful in modelling different problem-solving dimensions through a confirmatory approach. IRT model comparisons are carried out to assess the relative fit of the data to the models. The within-item multi- dimensional IRT model provides better fit than both the between-item multi-dimensional model and the uni-dimensional IRT model. This demonstrates that the test instrument developed with a sound theoretical basis indeed measures the underlying skills demands as the test designer intended, as the confirmatory approach of the multi- dimensional IRT analyses show. Overall, this thesis provides a methodology to demonstrate that complex problem-solving tasks can be modelled successfully through IRT analyses.
This study aimed to identify those beliefs of a sample of secondary mathematics teachers that impact significantly upon their practices. It represents an important addition to the limited amount of research concerning the beliefs of experienced secondary mathematics teachers and was designed cognisant of relevant understandings of the structure of belief systems. The design of the study also allowed for the identification of contextual variables at a class and individual teacher level, which were relevant to the translation of beliefs into practice. While the highly contextual nature of teachers' beliefs is fully acknowledged, the aim of the study was to determine those beliefs that were relevant across a relatively broad range of classroom contexts, and that would thus be more readily generalisable to other secondary mathematics teachers. In relation to teachers' practice, it was recognised that a sufficient level of generality needed to be considered in order to detect any correspondence with relatively decontextualised beliefs, hence the focus was on the extent to which the teachers' classroom environments could be characterised as constructivist. The study employed both quantitative and qualitative methods. These teachers participated in a semi-structured interview, and a number of their mathematics lessons were observed. Informal interviews and discussions with these teachers also occurred throughout, and subsequent to, the observation period. Significant associations were found between the teachers' beliefs and the extent to which their classroom environments could be described as constructivist. Specifically, teachers with a problem-solving view of mathematics and student-focused views of mathematics teaching and learning were more likely than other teachers to have classroom environments that were consistent with constructivist principles. Various associations with the teachers' personal data were also identified. The case studies revealed a number of particular beliefs that were associated with the creation of classroom environments that could be characterised as constructivist. These beliefs related to the teachers' views concerning the nature of mathematics, mathematics learning, the teachers' role and responsibilities in relation to both students' learning and their own professional learning, and most significantly, the ability and inclination of their students' to learn mathematics. These teachers participated in a semi-structured interview, and a number of their mathematics lessons were observed. Informal interviews and discussions with these teachers also occurred throughout, and subsequent to, the observation period. Significant associations were found between the teachers' beliefs and the extent to which their classroom environments could be described as constructivist. Specifically, teachers with a problem-solving view of mathematics and student-focused views of mathematics teaching and learning were more likely than other teachers to have classroom environments that were consistent with constructivist principles. Various associations with the teachers' personal data were also identified. The case studies revealed a number of particular beliefs that were associated with the creation of classroom environments that could be characterised as constructivist. These beliefs related to the teachers' views concerning the nature of mathematics, mathematics learning, the teachers' role and responsibilities in relation to both students' learning and their own professional learning, and most significantly, the ability and inclination of their students' to learn mathematics.
This researcher's job as a professional development instructor in mathematics and science throughout Miami-Dade County, involves working with both veteran and novice teachers on a daily basis. During inservice instruction in mathematics, the researcher has noticed a rather poor understanding of algebraic thinking concepts among the participants. The author began an informal survey of why so many of the participants were having so many problems understanding algebra concepts, and in most cases he was informed that algebra was a subject that they had never understood, and one that caused much anxiety even at the mere mention of the word algebra. Through further questioning, the researcher began to see a pattern exhibited among those with whom he spoke. This pattern seemed to indicate that most of the teachers surveyed stated that learning algebra had been relegated to lecture- driven, rule-oriented, memorisation and application of rules. Further questioning also indicated that many of the workshop participants chose to take no higher-level mathematics courses once they had completed algebra. Additionally, many participants reported that even though algebraic thinking was part of the curriculum for the county that they often skipped it, or paid little attention to it. As a result, the researcher began to question how these teachers' attitudes affected their students, and how the teachers' knowledge of algebra impacted their students. Based on these concerns, and the recognition of the importance of algebra in determining the types of mathematics courses taken, and ultimately the proficiency that students develop in mathematics. In order to answer each question completely, both qualitative and quantitative data were collected and analysed. The study concluded that teachers in the study group did improve their scores on the Algebra Thinking Test after the additional three days of inservice at a statistically significant level (p< 0.05) but the posttest scores of the study group and the comparison group were not statistically significantly different. Students in the classes of both the study and the comparison teachers improved their scores (p< 0.01) on the Algebraic Content Knowledge Thinking Test but the gain scores of both students groups was not statistically significant. Students in the classes of the study and comparison group teachers did not perceive statistically significant differences in their learning environment, based on their preposttest scores on the What is Happening in this Class? (WIHIC) instrument. Students in the classes of both groups of teachers experienced less enjoyment of mathematics lessons based on the pre-posttest scores on the scale Enjoyment of Mathematics Lessons (p<0.05) from the Test of Math Related Attitudes (TOMRA). Students in the classes of the study group teachers exhibited greater inquiry of mathematics lessons based on the pre-posttest scores on the scale Inquiry of Mathematics Lesson (p<0.05) from the Test of Math Related Attitudes (TOMRA). There was no statistical difference between the pre- posttest scores on the scale Inquiry of Mathematics Lesson (p<0.05) from the Test of Math Related Attitudes (TOMRA) for the comparison group teachers. Based on the multiple correlations students data, responses to the Inquiry in Mathematics Lessons scale was positive In order to answer each question completely, both qualitative and quantitative data were collected and analysed. The study concluded that teachers in the study group did improve their scores on the Algebra Thinking Test after the additional three days of inservice at a statistically significant level (p< 0.05) but the posttest scores of the study group and the comparison group were not statistically significantly different. Students in the classes of both the study and the comparison teachers improved their scores (p< 0.01) on the Algebraic Content Knowledge Thinking Test but the gain scores of both students groups was not statistically significant. Students in the classes of the study and comparison group teachers did not perceive statistically significant differences in their learning environment, based on their preposttest scores on the What is Happening in this Class? (WIHIC) instrument. Students in the classes of both groups of teachers experienced less enjoyment of mathematics lessons based on the pre-posttest scores on the scale Enjoyment of Mathematics Lessons (p<0.05) from the Test of Math Related Attitudes (TOMRA). Students in the classes of the study group teachers exhibited greater inquiry of mathematics lessons based on the pre-posttest scores on the scale Inquiry of Mathematics Lesson (p<0.05) from the Test of Math Related Attitudes (TOMRA). There was no statistical difference between the pre- posttest scores on the scale Inquiry of Mathematics Lesson (p<0.05) from the Test of Math Related Attitudes (TOMRA) for the comparison group teachers. Based on the multiple correlations students data, responses to the Inquiry in Mathematics Lessons scale was positive. The qualitative data suggested that the study group teachers who had participated in the three day inservice on algebraic thinking had developed a more positive attitude about algebra and additionally they seemed more relaxed and positive while teaching the algebraic thinking unit. The comparison group teachers on the other hand appeared not to be as positive about teaching the unit to their students, nor were they as positive about their own comfort with algebraic thinking. The implications of this study tend to agree with many other researchers who have concluded that elementary school teachers do not possess sufficient content knowledge in algebra. Furthermore, this study also indicated that inservice activities, in order to be effective, should be of sufficient breadth and depth in order to produce significant results. The qualitative data suggested that the study group teachers who had participated in the three day inservice on algebraic thinking had developed a more positive attitude about algebra and additionally they seemed more relaxed and positive while teaching the algebraic thinking unit. The comparison group teachers on the other hand appeared not to be as positive about teaching the unit to their students, nor were they as positive about their own comfort with algebraic thinking. The implications of this study tend to agree with many other researchers who have concluded that elementary school teachers do not possess sufficient content knowledge in algebra. Furthermore, this study also indicated that inservice activities, in order to be effective, should be of sufficient breadth and depth in order to produce significant results.
This study analyses the effect of introducing student portfolios as a means of assessing the learning of mathematics. It examines the intended and the unforeseen outcomes in terms of the students, the caregivers, and the teachers involved, using quantitative data to match classroom environments with the response to the innovation. A major focus of the qualitative aspect of the study is the decision-making process that was associated with the implementation of change. For this study, all the junior students in a New Zealand secondary school were asked to compile portfolios of their mathematical work. The portfolios were graded by the teachers, the marks contributing to the students' assessments for the year's work. At the outset, the plan was to survey the 510 students involved to determine their attitude towards mathematics, survey them again once the innovation was in place to quantify the classroom environment, then repeat the first survey. Analysis was expected to reveal whether classroom environments that approximated a 'portfolio culture' contributed to an improved attitude towards mathematics. This quantitative approach was supplemented with taped interviews of students and teachers, ongoing records of less formal interactions, review of examination marks and school reports, and questionnaires mailed to the homes of a sample of the students. As the study progressed, it emerged that the major impact was on the teachers, and the focus shifted to them. For four years, follow-up surveys were conducted with teachers, including those who had transferred to other schools. The study found that all students can benefit from portfolios, both in terms of skills and attitude towards mathematics. Portfolios legitimated the involvement of caregivers, a positive change that provided greater links between classroom activity and the world of employment. The professional practice of teachers was affected by portfolios, prompting development of new classroom resources and techniques, increased collegial cooperation, and well- informed reflection on teaching and assessment. Teachers maintain great influence on classroom culture, and for many of those involved in the study, portfolios prompted a renewed interest in the process undertaken by students as they develop mathematical ideas, and a change in the relationship between teacher and students. The 'portfolio culture' resulted in students improving in their appreciation of mathematics, and a changed role for the student within the social environment of the classroom.
This study investigated if there were any differences in the ability of students to generate their own mathematical generalisations when undertaking the same activity, presented under three different conditions: as a physical activity, as a pen and paper activity, and as a computer simulation. The problem used was an adaptation of a problem which was intended to enrich mathematics for gifted and talented mathematics students. This problem involved induction of a rule and was phrased in terms of racing cars driving around a circular track. Solutions could be generated by using simple methods such as counting or using more sophisticated mathematical principles involving factors or ratios. The focus of the study was to test the proposition that the physical or game activities would lead to better induction of generalisations than the pen and paper classroom activity. The study was designed to answer the following questions: Is there a school effect on performance? Is there a gender effect on performance? Are there differences in student enjoyment that arise from use of one of the three procedures? Are there differences in the number of students from each group that generate generalisations? Are there differences in the complexity of mathematical relationships identified by students from each group? The analysis of performance showed that there were no significant differences between conditions in complexity of generalisations or in number of generalisations. There was no significant school effect on performance. There was no significant gender effect on performance. The ratings of enjoyment level by students undertaking the activities showed very little difference between the three forms of presentation. The level of enjoyment was high for each form of the activity. The results of this study lend support to the view that claims for learning strategies need to be supported with careful research. The proposition that the physical or game activities would lead to better induction of generalisations than the pen and paper classroom activity was not supported in this study.
Accurate calculation of drug dosages and intravenous drip rates is a crucial aspect of nursing practice. The consequences of calculation error and the threat they pose to patient safety is a continuing cause of concern in the health care industry and has led to a focus in research on the adequacy of nurses' mathematical skills. Nurses need to possess the necessary mathematical and problem solving skills for dosage calculation. The study compared two different methods for teaching drug calculation and involved a small sample of first-year nursing students from a Sydney university. The first teaching method, used with students in the `formula group', was modelled on the formula- based approach commonly used to teach drug calculation in tertiary nursing programs. The second method, used with students in the `problem- solving group', focused on accessing students' existing mathematical problem-solving skills, further developing them, and applying them in the drug calculation context. Students in the problem-solving group were encouraged to select their preferred method from a number of problem-solving methods they had trialed in class. Students in both groups were taught essential arithmetic skills for drug calculation, and the importance of units of measurement in drug calculation was stressed. Both quantitative and qualitative measures were used to assess the effectiveness of the two teaching approaches. Research instruments included tests, surveys, including an attitude to mathematics survey, and an interview. These were used to obtain demographic data from students as well as feedback about their reactions to the teaching program they attended. Two key findings emerged. The major finding related to the outcome of comparing the two teaching approaches. The second related to the fact that students in both groups benefited greatly from learning specific mathematics skills required for drug calculation, such as arithmetic skills, estimation and checking strategies, and skills relating to units of measurement and conversion of metric units. In respect of the first finding, formula-based methods appeared to have a greater appeal to students and be more effective over the short duration of the teaching program. However, there was evidence that students in the problem-solving group benefited from having the freedom to select their preferred method for performing calculations. At the end of the program, compared to students in the formula group, they were more confident in their ability to perform calculations and had a higher expectation that their answers would be accurate. Further, for students who selected their method of choice for dosage calculations, learning was more likely to involve conceptual understanding, rather than the procedural learning associated with use of formula methods. By contrast, amongst students exposed to formula-based methods the prevailing perception was that having a formula to use was a way of compensating for students' own inadequacies and mathematical deficit; they saw it as a crutch. Their learning was more likely to be of a procedural nature, the type of understanding commonly associated with rote memorisation of formulae with little understanding of the underlying mathematical concepts. The findings of the study have important implications for students' learning to accurately calculate drug dosages and for the long-term retention of their drug calculation skills. Positive attitudes and high levels of confidence have been linked with achievement in mathematics and students' ability to persist when tasks become difficult. Conceptual understanding, rather than procedural understanding, has been found to be linked with long-term retention of skills and the ability to transfer skills to novel situations-an important skill for nurses in clinical practice. A number of limitations associated with the study, particularly the limited duration of the teaching intervention and the sample size, mean that the results should not be regarded as conclusive. Rather, they should be viewed as providing tentative conclusions that need to be further investigated in future research.
This study addresses the need to make informed curriculum decisions by providing information on instructional approaches and learning outcomes in secondary school mathematics, particularly in the topic of function, that offered possibilities for the use of the graphing calculator. The study investigated the implementation of a more active, inductive approach, including the integration of the graphing calculator, with the algebraic topic of function with Thai mathematics students. The teaching approach was designed in response to the requirements of the Thai National Education Act 1999 and contrasts with the traditional transmission approach currently seen in Thai classroom. In order to move from the traditional teaching and learning strategies, learning processes under the Thai National Education Act 1999 focus on the principle that students are most important in the learning processes, and are capable of learning and self-development. The teaching and learning process should enable the students to develop themselves at their own pace and to the best of their potentiality. The study investigated students' responses to a new teaching approach, their use of the graphing calculator and their developing understanding of the function concept. The teaching experiment included two classes of Year 10 Thai mathematics students who studied in a public school located in the north eastern part of Thailand. Twenty-four students volunteered to participate in these two classes, twelve in each. Three students from each class volunteered to take the roles of key informants. The two classes were taught, Class 2 commencing their program two weeks after Class 1. Both classes were taught by the researcher (called teacher- researcher in the report) assisted by one of the regular teachers at the school. The teacher-researcher is a Thai secondary mathematics teacher with more than twenty years experience. The influence of instruction was monitored through analysis of classroom observations, the teacher-researcher's and classroom teacher's field notes, teacher- researcher and classroom teacher discussions, students' diaries and the key informants' interviews after some teaching episodes. This enabled reflection on each teaching episode with Class 1 and some modifications to the materials for Class 2. At the completion of the teaching program, a function concept questionnaire and function test were administered to the six key informants. The questionnaire and the test were designed to investigate students' concept definition and concept image of function, and to assess each of three aspects of the conceptual knowledge of function, including interpreting a function, modelling a functional situation, and translating between different functional representations and within the same representation. Based on a sequence of student cognitive development and the research theoretical framework (Action-Process-Object perspective), all six key informants' responses to the questionnaire and the test were classified into four categories, namely an action, a process, a pre-object and an object conception of function. Results indicated that the six key informants were able to use action, process and object conceptions of function as required in various situations. They were mostly able to perform actions on functions whenever there were required. In some situations, they were able to link the necessary processes and properties to solve a problem, but in other situations they were not successful. Although the students had experienced all necessary processes and properties, they had not sufficient time to build the quality links that would enable them to solve problems more reliably. The inductive nature of the teaching program was more limited than originally envisaged. The nature of material, based on the Thai mathematics syllabus, made the use of a more inductive approach with the integrated use of the graphing calculator difficult to achieve. When teaching the classes, the teacher-researcher also had difficulties in moving away from his familiar strategies. Both students and teachers needed a greater time to adapt to these changes. The attitude questionnaire was administered with all students in order to investigate students' confidence in using the graphing calculator and their attitudes toward the collaborative use of the graphing calculator. Findings indicated positive attitudes toward integrating the graphing calculator in the teaching and learning of mathematics. They also indicated that they preferred to work using the graphing calculator on their own first, and then discuss and share results shown on its screen with each other, as well as asking for help when facing difficulties. Data from classroom observations and the key informants' interviews after the lessons were used to determine how and why the graphing calculators were employed in the problem contexts and learning situations. The study indicated that the graphing calculator could be used by the students as a tool to promote their conceptions of function in all three views (an action, a process and an object view of function). The study classified three roles of the graphing calculator used by the students, including a conceptual action representation tool (CART), a conceptual process representation tool (CPRT) and a conceptual object representation tool (CORT). However, students in the study usually attempted to use an algebraic or arithmetic strategy to solve a problem, rather than apply the graphing calculator. The study also identified three stages of the students learning to use the graphing calculator, namely 'awareness', 'learning the process' and 'application of the process'. In the stage of the application of the process, the study also identified three roles of the students during the period of using the graphing calculators: student as receiver, an assistant and partner. Results from the study were used to formulate a set of recommendations for Thai educational authorities aimed at assisting teachers and students to move towards meeting the requirements of the 1999 Education Act. The data of this study was collected entirely in Thailand. All lessons and interviews were conducted in the Thai language and teaching materials were also written in the Thai language. The materials presented in the appendices are translations of the printed materials used in the study. Excerpts of dialogue from the lessons and interviews have also been translated into English. The translations are as near as possible to the original statements in Thai.
Theories of self-efficacy, self-concept, self-worth and causal attributions have formed the framework for a large number of studies in educational fields. The reflections children make in learning contexts have rarely been linked to these concepts. Just as rare has been the qualitative classroom-based research on these concepts. This study seeks to add qualitative insights to the quantitative information and analysis of 8 to 12 year old children's capability beliefs, self-worth and reflections in the subject of mathematics. The similarities and differences between self-concept and self-efficacy at the domain level are also measured and discussed. A further contribution is the value of an intervention/learning program trialled with a class of eight to nine year olds taught by the researcher. The investigator and children attended a medium sized primary school in a middle socio-economic area of metropolitan Melbourne, in Victoria, Australia. The range of data is examined in two sections: one for the complete sample of 154 students, another for the above-mentioned class of eight to nine year olds. Findings of the study indicate children tend to maintain positive maths self-beliefs in a year, self-concept effects performance more than other researched beliefs, and, while there is a substantial relationship between various capability beliefs, they also differentiate within particular contexts. Some quantitative results confirm prior research; some are contrary to expectations, for example, specific maths achievement was predicted more highly by maths self- concept than by specific self-efficacy. The learning / intervention program trialled with one class led to improvement in children's subject value, self-beliefs and task behaviour, though alternative explanations for any program effect are reasonable.
An important aspect of conceptual development of multiplicative thinking in primary students is the development and use of models and representations. The study presented here explored the understanding of multiplication held by a group of Year 5 students in an inner Melbourne State School. The study was undertaken in response to perceived problems in Years 5 and 6, where students approached multiplication in an almost exclusively procedural way. Preliminary studies revealed that students' understanding and knowledge of the range of real-world situations, solved using multiplication had not developed. This study examined the role of representations such as equal groups, arrays, region models and real-world stories in conceptual development of multiplication. It explored the ways a group of Year 5 primary school students used models to extend primitive or simple conceptions of multiplication to more complex, sophisticated ideas leading to wore effective transfer to examples involving decimal fractions. The questions addressed by the study were: what do Year 5 students understand about multiplication and what models or representations do they have; does this help or hinder their movement from whole numbers to decimals; to what extent can their understandings be classified according to levels of understanding documented in the literature; and to what extent can a specifically devised intervention program impact existing understandings. The study classified the students' responses according to their use of models and representations, and categorised their explanations of symbols and equations as predominantly procedural or conceptual. The study was conducted as a teaching experiment. An interview protocol was derived from literature and used to collect data at the beginning of the study. The first interview was followed by an intervention program focussing on models and representations, particularly arrays and region models. A second interview was conducted following the intervention program. Some months later, the interviews were conducted again to explore retention rates. Results indicated a noticeable shift after intervention towards the use of more sophisticated models and representations of multiplication and corresponding increase in performance on the decimals tasks, and explanations of multiplication by decimals indicating greater conceptual understanding. This study has implications for teaching that could be addressed immediately. Providing opportunities to experience the wide range of multiplicative situations, and experience with a variety of models and representations are important aspects of a more sophisticated understanding of multiplication. A rich understanding has meaning and with this meaning comes an ability to assess the reasonableness of a solution. Teaching decimals with understanding, and explicitly addressing the different effects of multiplying with whole numbers compared with multiplying with decimals less than one, is an aspect of teaching often ignored. Further studies are indicated in the area of procedural and conceptual understanding, and research into the effects of early versus later introduction of algorithms for multiplication. Studies are also indicated in relation to the role of manipulatives, models, representations and images in every aspect of concept development and mathematical reasoning from early to upper primary school. In this study both procedural and conceptual understanding increased after teaching sessions focussed on meaning making.
While the amount of research into difficulties in mathematics has increased markedly over recent years there continues to be a need for more research into mathematics in the middle years of Primary School. The present study examined the extent to which performance on various maths related processing tasks (e.g. reading numbers, reading number statements, mental arithmetic) and measures of maths understanding (e.g. numeration and counting) predicted maths computation ability as determined by performance on typical Year 3-5 un-timed pen and paper arithmetic tasks. Analysis consisted of a stepwise regression for each of the three year levels. Some of these tasks were found to be highly predictive of achievement in arithmetic. The multiple regression was not only significant at each of the three year levels but accounted for a substantial proportion of achievement criterion variance: Year 3: 61%, Year 4: 59.8% and Year 5: 61.5%. Achievement in arithmetic was best predicted by a combination of factors at each year level with some similarities occurring across levels. The most striking of these is- Mental Arithmetic: multiplication which was found to be a predictive factor at all three levels. Other significant predictive factors included Mental Arithmetic: subtraction (Year 3), Numeration: tens of thousands (Years 3 and 4), Processing of 4-digit numerals (Years 4 and 5), and Mental Arithmetic: addition (Year 5).
This study focused on successful performance in school algebra. It sought to determine what high achievers in Year 10 algebra knew and felt about algebra that enabled them to succeed while their peers were less successful. The study aimed to identify essential cognitive and affective variables which associate significantly with successful performance in algebra and to develop a model to explain how these variables interact to facilitate that success. Implications for instruction were then drawn and recommendations made relating to curriculum design, teacher education and further associated research. The literature study indicated that three basic concepts, which are representation, generalisation, and functionality, appear to underlie algebraic thinking. A conceptual understanding of a subject together with a range of problem solving abilities distinguish expert from novice intellectual behaviour generally. In addition a range of affective variables including belief in the value of the task, self- concept and self-efficacy beliefs, together with attitudinal variables, impact significantly on successful performance generally. A two-stage research design was chosen for the study, involving a gender-balanced sample of 54 Year 10 extension (top stream) mathematics students - approximately half from a state school, and the remainder from a private school - from a semi-rural area of Australia. In Stage One all subjects were given a written test of algebra attainment and a series of written tests of the identified basic concepts, that is representation, generalisation and functionality. A questionnaire survey was conducted to determine their beliefs about and attitudes towards, themselves, algebra and mathematics in general. The questionnaire also asked about classroom experiences, perceived scholastic abilities, parental, and perceived teacher and peer group influences on the subjects' learning of algebra. The data were analysed quantitatively and qualitatively and comparisons made between the responses of a group of ten high and a group of ten low achievers in algebra. In Stage Two gender and school-balanced sub-samples from the high and low achievers groups of Stage One were selected for interview. These groups participated one-on-one in a one-hour audio/video recorded think-aloud mode algebra problem-solving interview conducted by the researcher. The subjects articulated their thinking procedures and feelings as they solved a range of routine to novel algebra problems. The data gathered from this interview were analysed using both quantitative and qualitative techniques and triangulated with that of Stage One. The Stage One findings confirmed that high achievers had a significant command of the concepts identified as basic to algebraic thinking. The questionnaire data showed high achievers held positive beliefs and attitudes about algebra and about their own capabilities. These beliefs and attitudes were significantly different from those of their less successful peers. High achievers were generally positive and confident about algebra. They valued the subject for aesthetic and practical reasons. They reported enjoying algebra and believed it enhanced their general thinking and problem solving capabilities. High achievers were self-aware and held strong self-efficacy beliefs. They knew they worked hard and were respected by their teachers and peers and were supported by their parents. Subjects who were less successful in algebra also reported parental support and generally good teacher relations, but reported not being inspired by algebra and not being confident that they could succeed in the subject. No gender or school effects (private or state) were found. The data gathered from algebra problem solving interviews conducted in Stage Two confirmed the high and low achiever ratings on the identified basic algebra concepts. This was particularly evident with generalisation and with functionality. Noticeably, however, even high achievers at this Year 10 level had difficulties with the function concept. The qualitative data analysis of the interview transcripts was aided by a grounded theory methodology which identified three stages in the algebra problem solving process, these are, information gathering, information processing and information reporting. The analysis identified eight cognitive and eight affective variables which associated significantly with successful performance in Year 10 algebra. Based on these findings an algebra learning success model was developed. The model postulates that success in algebra results from the reciprocal interaction of positive beliefs and attitudes towards algebra, a sound knowledge of the identified basic concepts of representation, generalisation, and functionality from and the application of a range of problem solving abilities. These abilities are developed, it is postulated, through the confidence building and optimistic experiences gained from early and sustained comprehension of and success with algebra. Implications for curriculum design and teacher education have been identified, presented and discussed, together with suggestions for further research relating to the algebra learning success model.
Computer Algebra Systems (CAS) are being increasingly used in mathematics education at both the secondary and tertiary level, either in hand-held supercalculator form or as desktop computer software. This research project has investigated the effectiveness of different ways of incorporating the use of the CAS Maple into undergraduate mathematics 'service' courses and the impact that the use of Maple has had on the students' development of mathematical understanding. A literature review was conducted of previous research relevant to the use of CAS in mathematics education. The first task was to discover the current thinking on how mathematical understanding develops, particularly at the advanced level. Secondly, issues relating to the use of technology, including CAS, in mathematics education, were drawn out. Thirdly, the review focussed specifically on research relating to the development of mathematical understanding when using CAS and the relationship between pencil and paper and CAS work. Finally, examples of good practice internationally were sought. The overall research design for the study was an action research or developmental research methodology whereby the development of CAS teaching materials was undertaken in a development, implementation and evaluation cycle. Over a two-year period, Maple resources were developed, implemented in teaching experiments and evaluated by means of student feedback responses. The students' responses to the Maple materials were ascertained through the use of questionnaires, structured interviews, journals and observations. Further information was provided through the analysis of students' Maple assignments and written work. The results from each teaching experiment were incorporated into the design of the next set of materials. The theoretical framework of the research is constructivist. Students are recognised as having individual knowledge schema that are constructed through the accommodation and assimilation of experience, in the Piagetian sense. The nature of mathematical understanding as being comprised of procedural and conceptual aspects is used to explore the development of understanding when using CAS. The notion of processes and objects, as put forward by Dubinsky et al and Sfard and Linchevski, and the idea of 'proceptuar thinking as proposed by Gray and Tall (1994), are also used in considering how the use of CAS impacts on the students' learning. The research here has concurred with existing evidence that CAS can be used in mathematics education as a black box, allowing students to focus on higher order skills; as a scaffold, carrying out processes where a students' procedural competence is weak; as an aid in the visualisation of concepts and as a means of linking multiple representations of mathematical objects. It was found that students reacted positively to the CAS components of their course when the CAS tasks were integrated throughout the course and when there was the opportunity for regular CAS use. In considering the idea of 'instrumental genesis' and the relation of the CAS work to pencil and paper work, it was found that linking pencil and paper tasks with the CAS assignments helped the students to relate their CAS work to their existing knowledge.
International comparative studies on mathematics achievement indicate that students from East Asian countries outperform their Western counterparts. This study compares mathematics curriculum and assessment policy and practice between Australian and Chinese primary schools, and investigates factors accounting for this achievement gap. Document analyses and case studies were used to examine the key differences in mathematics curriculum and assessment policy and practice between Australian and Chinese primary schools. The document analyses focused on the intended mathematics curriculum and assessment as presented in official documents both in China and Australia. Using a case study approach for in-depth study, three government primary schools were selected (one from Guangzhou, China and two from Sydney, Australia). Classroom teaching and assessment practices were observed, teachers' and parents' views of assessment practices were obtained through semi- structured interviews, and students' work samples and examination papers were analysed. Case studies analysed implemented curriculum and assessment practices. The findings of this study considered there was a gap in mathematics achievement between Australian and Chinese primary students, but the achievement gap was not the main focus of this study. It found that the Australian school curriculum and assessment differed markedly from the Chinese system. High mathematics achievement of Chinese students both at Australian and Chinese schools could not only to be attributed to higher standards of intended mathematics curriculum and assessment, or teacher knowledge or classroom practice. The mathematics achievement gap between Australian and Chinese students is better explained by cultural factors such as motivation to achieve, attributing success to effort, the influence of parental help and extra mathematics curricula (out-of-school). The success of Chinese students was based on the cultural values and beliefs of their family: 'to succeed is to honour one's family.' This study has contributed to our understanding of the mathematics achievement gap between East Asian and Western countries. In particular, it highlights the need to integrate case study with large-scale study in international comparative studies. In order to bridge the gap between intended and implemented curriculum, explicit professional training in assessment is essential for both Australian and Chinese teachers. To implement reform in Chinese education, the impact of cultural influence must be considered more seriously. In order to improve Australian students' mathematics achievement, parental support and achievement motivation must be addressed.