Theses abstracts (3)
New Zealand teachers’ use of equipment has increased as a result of their participation in the Numeracy Development Project. The purpose of this study was to discover how closely students’ reasons for their equipment choices matched their teachers’ reasons for including the same pieces of equipment in their numeracy programmes. In the teachers’ reasons for equipment choices, the surface features of equipment seemed equally important as the conceptual development the equipment can support. In contrast, the reasons given for equipment choices by the 34 Year 3 students who were interviewed were almost exclusively concerned with how the equipment might help them to solve the given problem. The students’ success rates at solving problems declined as the equipment became more structured; this paralleled the teachers’ equipment choices. The equipment choices of the four teachers interviewed in this study were not strongly consistent with the equipment use recommended in the NDP materials.
Learning the meaning of common fractions and how to operate with them is a traditionally difficult aspect of learning mathematics. The symbol system used to represent fractions, one whole number written above another whole number, is not transparent to the meaning of fractions. This difficulty of interpreting fractions contributes to traditional practice in teaching fractions emphasising the syntax of fractions over their semantics. Without a sense of the size of fractions students must rely solely on learning the fraction syntax, as there is no feasible way of checking the reasonableness of an answer.
To investigate how well students had developed a sense of the size of fractions, a large cross-section of students from Years 4–8 (over 300 students in each grade) completed 37 tasks designed to draw upon a quantitative sense of fractions. The students’ answers provided data for both a conceptual analysis and a related Rasch item analysis. The conceptual analysis found that although some students have a strong sense of the size of fractions, many students have developed psuedo fraction concepts. Rather than seeing fractions as relational numbers, more than 10% of the students responded as if fractions corresponded to a count of the number of parts in a fraction representation. Questions involving the fraction notation increased the variety of incorrect interpretations of fractions. The Rasch analysis confirmed that the items related to the same trait and provided an ordering of the difficulty of the questions. The Rasch item map provided backing for a description of how students develop a sense of the size of fractions.
The current methods of developing students’ understanding of fractions have resulted in a large number of pseudo concepts being formed. The basis of making meaning from models used to introduce fractions, needs to be the focus of teaching fractions. As well as using counter-examples to limit the number of unintended features of models students associate with fractions, comparison of length rather than area should be used to introduce fractions.
In his keynote address to the National Council of Teachers of Mathematics research presession, Finbarr Sloane (2006b) challenged mathematics education researchers to "quantify qualitative insights." This quasi-experimental study used blended methods to investigate the development of two-digit addition and subtraction strategies. Concurrent classroom teaching experiments were conducted in two in-tact first grade (6- to 8-year olds) classrooms (n = 41) in a mid-Atlantic American public school. From a pragmatic emergent perspective, design research (Gravemeijer & Cobb, 2006) was used to develop local instructional theory. An amplified theoretical framework for early base-ten strategies is explicated. Multilevel modeling for repeated measures was used to evaluate the impact of pedagogical decisions and tools and the association of particular pedagogical practices with the emergence of incrementing and decrementing by 10 (N10) or decomposition (1010) strategies (Beishuizen, Felix, & Beishuizen, 1990).
The two matched classes were not statistically different in terms of gender, poverty, race, preassessment performance, and special education services. After the first unit of instruction with differentiated pedagogical tools, there were highly significant (p< .001) differences between classes in 1010 use. Differences decreased over time with no statistical difference demonstrated during the postassessment. Students in both classes were significantly more likely to use N10 during the last structured interview than in the first (p< .0001). Furthermore, there was no statistical difference between the two classes in using an advanced strategy; however, all students were statistically more likely to use an advanced strategy at the conclusion of the study than they were after the first unit of study (p< .033). The order of emergence of 1010 and N10 was not statistically associated with the ability to develop both strategies, but there was a highly significant association (p < .001) between use of an advanced strategy and success on a district-mandated written assessment of two-digit addition and subtraction.
Two original instructional sequences of contextually-based investigations are presented. Protocols transcribed from videotaped lessons and dynamic assessment interviews are presented to illuminate specific constructs detected and to illustrate the pedagogical techniques. Recommendations for future studies, curricular changes, and the need of early intervention are discussed.
This research study used action research to investigate changes in a teacher’s practice which occurred with the implementation of teaching strategies aimed at increasing students’ autonomy. The study involved students in the Year 12 mathematics classes at a Catholic coeducational college in a Queensland provincial city. An initial study over one term was carried out in 1997. This informed the major study of 1998. The changed teaching approach emphasised collaborative learning, the use of self-regulatory learning strategies, making sense of rather than memorising mathematical concepts, and reflecting on learning behaviours and on understanding of concepts. The students were encouraged to be actively engaged in their learning and to accept responsibility for it. Action research allowed the students to contribute to the teacher’s planning and evaluation of the teaching and learning in each cycle. Data were collected from students’ journals, interviews, classroom observations, questionnaires, and the teacher’s fieldnotes. Narrative analysis centred on three themes: issues affecting the students, issues affecting the teacher, and issues associated with carrying out research into practice. The students perceived themselves as being more autonomous than they had been previously. Nearly all accepted the responsibility for their own learning, and were able to work together to improve the learning of all class members. They also believed that the changed approach had contributed to improved results. The teacher was able to act as a facilitator of the students’ learning, rather than feeling responsible for ensuring that all learnt. Action research proved a suitable methodology for a full-time teacher seeking to improve her practice.Students’ responses to the study support the need to explain the purposes of changes to the ways in which they are taught and expected to learn. It proved important to allow the students time to adapt to the changes, and to allow them to have input into decisions about their learning. They needed time to reflect on their learning for them to become aware of what they were doing and whether it was proving effective. The teacher realised the need to be flexible in her teaching approach, and to provide sufficient scaffolding for the students as not all students wished to become more autonomous learners.
This thesis reports on a longitudinal study of students’ understanding of decimal notation. Over 3000 students, from a volunteer sample of 12 schools in Victoria, Australia, completed nearly 10000 tests over a 4-year period. The number of tests completed by individual students varied from 1 to 7 and the average inter-test time was 8 months. The diagnostic test used in this study, (Decimal Comparison Test), was created by extending and refining tests in the literature to identify students with one of 12 misconceptions about decimal notation.
Particular longitudinal measures and definitions of the prevalence of misconceptions were adapted from the medical literature. These measures were further refined to overcome the effect of repeated testing (which resulted in a 10% improvement) as well as various sampling issues. Analysis was conducted at both the coarse level (4 behaviours) and fine level (12 ways of thinking).
Improved estimates of the prevalence of expertise as well as for the various misconceptions are provided. Only 30% of Grade 6 students and 70% of Grade 10 students demonstrate expertise on this test and about 25% of students between Grades 7 and 10 completed tests by choosing the decimal with the fewest digits as the largest number, a behaviour which results from several different ways of thinking. Despite its high prevalence, this particular behaviour is not well known amongst teachers. Three phenomena were investigated: persistence, hierarchy and regression. The misconceptions which are most persistent are those that involve the treatment of the decimal portion of a number as a whole number. A hierarchy of the misconceptions was determined by considering the relative rate to expertise on the next test: the hierarchy is different for primary and secondary students. About 20% of students were involved in regression, that is, they completed one test as an expert, but were unable to do so on a later test. This analysis provides additional evidence that many students are receiving teaching that covers over rather than overcomes their misconceptions. For example, some students appear to be following algorithms for comparing decimal numbers (such as rounding to two decimal places), but revert to a latent misconception when their incomplete algorithm fails. Furthermore, support is provided for the hypothesis that some misconceptions are due to the interference of new teaching.
Little is known about how integrating ICT into classroom programmes actually enhances the learning process. This study investigated the link between integrating spreadsheets into a primary mathematics programme and the development of numeracy.
An eclectic approach to methodology was used, reflecting both the quantitative and qualitative paradigms. Written and oral assessments, surveys, interviews and observation were utilized to gather data, and gain a clearer vision of how these children developed their understanding of numeracy. For the purpose of this research, the more specific view of numeracy was taken, namely, the understanding (number sense), usage (including the development of strategies), and application (including problem solving and communication), of number in a variety of contexts.
Analysis of the results, using a Wilcoxon signed ranks test, showed the class that engaged in the spreadsheet work made significant progress in their use of strategies and content knowledge, between assessments, whereas the control class did not. Consideration of the other evidence reveals it is probable that the spreadsheet’s propensity for the development of visual patterns; the interactive nature of learning it enables; the potential it offers for the learner to explore numerical, tabular and graphical representations simultaneously; the rapidity of response; and the motivation of learners (perhaps inherent to those aspects), have contributed to the facilitation of the students’ understanding.
Although a correlation between the use of spreadsheets in primary mathematics programmes and the development of numeracy is difficult to establish, significant development occurred and insights into the way spreadsheets facilitate that development were gained.
Problem-solving approaches to teaching mathematics have been recommended in curriculum documents for some time but there is evidence to suggest that there has been limited classroom implementation both in Australia as well as overseas. This investigation explored the level of implementation of mathematical problem solving in primary classrooms in NSW. Teachers’ beliefs about the role of problem solving in learning mathematics as well as their classroom practices were also investigated. To explore what teachers believe and what they do in relation to problem solving, this investigation examined primary school teachers’ reported beliefs about the role of problem solving in learning mathematics and their reported practices in classrooms. It aimed to discover how beliefs about mathematical problem solving influenced decision making in teachers’ classrooms and what factors promoted and hindered the implementation of problemsolving approaches. The results of this investigation could provide benefits to several different groups involved in mathematics education. Preservice teacher educators and presenters of inservice education courses could benefit from increased knowledge about the role of beliefs in determining practices as well as potential constraints on desirable practices. Associated with this may be the need to challenge teachers’ beliefs that might not support the development of practices that promote problem-solving approaches. For practising teachers, professional development could focus on examining their beliefs and providing the necessary support for teachers to realise the aim of assisting their pupils to achieve problem-solving competence. Curriculum developers may benefit from an increased awareness of the difficulties associated with implementing recommended approaches. Finally, participating teachers may benefit from their involvement in the investigation through opportunities to reflect on their practice while completing the survey instrument as well as during interview discussions with the researcher.
Data collection focused on teachers’ beliefs about mathematics, teaching mathematics and learning mathematics, as well as on their reported practices since all of these factors impact on what occurs in teachers’ classrooms. A combination of methods was used to collect data so that there was increased confidence in the research findings. In this way, the results of one method could be tested against another for consistency, thus enhancing trustworthiness and dependability. The data collection for this investigation was divided into two phases. The first phase involved the use of a questionnaire to seek data on teachers’ reported problem-solving beliefs and practices as little is known overall in this area, particularly within the context of Australian teachers. Responses were received from 162 primary school teachers currently teaching in NSW. The instrument was designed with reference to similar instruments that had been used by other researchers in the field and incorporated a combination of closed and open questions. The second phase of data collection incorporated interviews and observations that were conducted in the field. These methods were used to explore the problem-solving teaching approaches used by a small number of teachers in particular school settings. To collect data about teachers’ planning for instruction, and opportunities that support or constrain innovative practices, it was more appropriate to explore particular contexts that would provide a rich set of data. Analyses of data confirmed the spread of teachers’ beliefs, the diversity of their practice, and revealed issues that could hinder their problem-solving efforts in classrooms. A small group of surveyed teachers reported holding very traditional views that were quite distinct from another group who reported support for very contemporary views. These differences were also apparent in relation to reported classroom practices and appeared to be linked to the current teaching grade level of the respondents. This was confirmed during the interviews and observations as it seems easier for teachers of the lower primary grades to implement practices identified as supporting problem-solving approaches. For teachers of upper primary grades, parents’ and school expectations impinge on teachers’ practices and potentially constrain their problem-solving efforts. For the two teachers who participated in the classroom observations, considerable energy was required to resist constraints and implement problem-solving approaches. Recommendations for practice and future research include the need for an examination of constraints on practice, the role of reflective practice in implementing innovative practices, the viability of teaching through problem solving as a necessary and important teaching approach, and the use of a variety of problem types in preparing students to be successful problem solvers. In addition, teachers may need to be encouraged to continually reflect on practice and teacher educators may need to raise the awareness of preservice and inservice teachers to the issues involved in implementing problem-solving approaches in their classrooms.
Problem-based learning is an approach to learning that has evolved from the integration of post-empiricist epistemology and current learning propositions. It is spreading through higher education, especially in the fields of medicine, architecture and educational administration. Most of the research on the implementation of problem-based learning in education has stemmed from studies carried out with adult learners but the approach has also been implemented at the Centre for Problem Based- Learning at Illinois Mathematics and Science Academy (IMSA) with gifted secondary students and in several elementary schools in the USA. The main aim of this thesis is to examine through literature review and critical analysis the philosophical and learning principles underpinning problem-based learning and to ascertain if this approach may further enhance learning in primary education in Australia. Problem- based learning emanates from the seeking of tentative solutions to problems. It is based upon post-empiricist epistemology which has evolved principally from the work of Dewey, Popper and Quine who viewed knowledge as ever-developing, continually changing and tentative, and open to question and disproof. The writer considers that teachers' implicit views of knowledge influence their pedagogy. Therefore it is important for teachers to become aware of their views of knowledge and understand why they hold them. This thesis discusses how post- empiricism differs from the dominant rationalist and empiricist epistemologies of knowledge, which underlie and influence much current teacher practice. It shows how teachers would need to develop different understandings of knowledge and knowledge acquisition in order to implement problem-based learning. At the same time it shows the validity of such understandings. Many of the learning propositions and processes already in use in some primary schools appear to be consistent with those advocated by supporters of problem-based learning. It would be expected that this would facilitate the introduction of this approach. However such learning principles need to be explored more closely. This is a theoretical study based on the analysis of literature and points out reasons why problem-based learning has been introduced successfully. Although problem based learning has been implemented in some elementary schools in the USA, the use of such an approach in primary education in Australia has not been widely documented. The suitability of such an approach to learning in Australian primary schools is explored and ways in which it could be introduced to the profession are suggested. More research based on practice is needed in this area. A future potential study is described, the findings of which could be distributed to school networks and professional development courses to allow teachers to become more familiar with this challenging but rewarding approach to learning.
The primary purpose of this thesis is to gain a better understanding of the espoused beliefs and instructional practices of secondary mathematics teachers. It aimed to investigate mathematics teachers' beliefs regarding the nature of mathematics, mathematics teaching, learning and assessment reported in the literature, and to explore the various links between these beliefs and instructional practice. The present study aimed to extend the earlier work that has been carried out through the analysis of data from further cultural contexts, i.e. by investigating Greek secondary mathematics teachers' beliefs about mathematics, and mathematics teaching, learning and assessment, and to contribute to an understanding of the impact that the cultural context could have on teaching practice. The study was conducted in three parts: - A pilot study was carried out to examine the feasibility of the proposed method of data collection. - A survey was administered to explore the areas of interest and to produce data to examine the research questions statistically. - The survey was followed by a study focusing on two in-depth case studies Through extensive interviews, observation and video taping mathematics lessons, a detailed picture of the complex relationships between enacted beliefs and teaching practice has emerged. Close examination of a number of the lessons of the two teachers was carried out using detailed analytical criteria that clarified the terrain of everyday teaching practice. This analysis displayed the complexity and the pervasive nature of emotions, values and beliefs, and everyday practice experienced by these teachers. Replicating and broadening previous findings, two orientations that characteristic of secondary mathematics teachers' beliefs were identified: - A contemporary-constructivist orientation consisting of the following complementary views: The socio-constructivist view; The dynamic problem-driven view; The cooperative view, and a - Traditional- transmission-information processing orientation, consisting of the following (not mutually exclusive but complementary) views: The static view; The mechanistic view. The analysis of the case studies teachers' interviews demonstrated that the classroom is a complex site of political, historical, social and cultural influences. The two teachers in this case study demonstrated that their beliefs were not always consistent with their instructional practices, as it has been documented in previous studies. In both cases, the veteran secondary mathematics teachers' beliefs about mathematics were more traditional than their beliefs about mathematics teaching and learning. A significant finding of this study is however that, Greek veteran secondary mathematics teachers' beliefs about mathematics learning and teaching (including assessment) were less traditional than their actual teaching practice. Overall, the analysis of the case study teachers' interviews and classroom observations indicated that both teachers generally agreed that there is a strong relationship between teaching practice and teachers' beliefs about mathematics, and mathematics teaching and learning. It was also clear that in some instances the case studies teachers' beliefs were influenced by prior effective teaching practices. The major causes of inconsistencies between the espoused beliefs and their teaching practice suggested by the two veteran secondary mathematics teachers, were the classroom situation, prior experiences and social norms. Overall, the findings of this study indicate that the broad social and cultural climate within the classroom appears to impact significantly on teachers' espoused and enacted beliefs about mathematics, and mathematics learning, teaching and assessment.
This research is concerned with the integration of technology in the teaching and learning of mathematics in the secondary schools of Fiji. In particular, the focus of the thesis is on how technology, specifically, scientific calculators, graphic calculators and computers have been used in the past to teach mathematics, what are the current practices and what resources and changes will be required in the future to accommodate such technology in the teaching and learning of mathematics. The framework for gathering data was designed from the literature on educational change and technology integration in mathematics education. The four themes identified were policies and guidelines, syllabuses and curriculum, availability of resources and professional development, which formed the conceptual framework for the study. Principals, Heads of the Mathematics Departments and mathematics teachers in secondary schools were surveyed regarding such integration. The supporting role of the Ministry of Education's Curriculum Development Unit, Teachers' College and the University were also examined, since these were important agents for training mathematics teachers to facilitate such integration. Data collected from closed questions using the questionnaires were number coded and analysed using Rasch analysis. Data collected from open-ended questions, from the questionnaires and interviews, were entered into Microsoft Excel spreadsheet and analysed descriptively and inferentially. Additional information was gathered and analysed from relevant literature and documentation. The results of the research and the data analysis identified important issues that were used to make practical suggestions and develop recommendations which the Ministry, the school system, the Teachers' College and the University could adopt to support integration of technology in classroom-based mathematics teaching. The outcomes provided a picture of why policies should be formulated in order to improve mathematics teaching and learning using technology in Fiji and these are also likely to be relevant to other South Pacific nations. As the world is heading towards the Information Age, it is concluded that Fiji's Education System needs to do more in terms of policies, syllabuses modification, resources and teacher training to provide adequately for the integration of technology in the teaching and learning of mathematics. Recommendations for future research include a follow-up survey to determine the changed situation as the new draft Prescription for Forms 5 and 6 is being developed by the Curriculum Development Unit in Fiji. Research could also be done at primary and tertiary level to investigate current and desirable future practices in relation to the integration of technology in the teaching and learning of mathematics.
This study explored the relationship between estimation skill and computational ability for whole and rational numbers. The methods carried out were both quantitative as well as qualitative and data were collected from three primary schools along with their associated high school in the Perth area. The year levels chosen were 5, 7 and 9. There were two classes from each chosen primary school representing Year 5 and Year 7 and three classes of Year 9 from the high school. The total number of students involved was 91, 77 and 73 from the three respective year levels. Instruments used for collecting data were group- administered tests and interview. Two parallel tests with identical items, where one of the pair was estimation and the other written computation were administered to all the students in the chosen year levels. Interviews were conducted for the group of selected students based on the criteria: slightly above the average and slightly below the average. There were eighteen students with nine in each group. The results of the correlation shows that performance in estimation is positively correlated with written computation in all the year levels. Moreover, the t-test result reveals that there is no significant difference between the two tests except in Year 7. Hence, the findings indicate that a child who is good in estimation skill can also perform well in written computation. As such, the importance of achieving estimation skill in a child would be very helpful in solving computation problems with understanding. On the other hand, children's performance related to the development of estimation skill and computational ability seems to be in positive direction from Year 5 to Year 7. Whereas the Year 9's performance is lower than Year 7. Among the topics, the children fared better in whole numbers compared to other topics. Performance tends to follow in a descending order from whole number to ratios. The disparities between estimation skill and computational ability are also more towards the difficult topics like division and multiplication of fractions and decimals. At the same time, the feedback from the interviewees tended to show that, the children from slightly above the average are better at choosing their own sensible strategies for solving the problems, whereas the students from slightly below average are more prone to the rote-learned algorithms. Although, male students appeared to perform better than the female students, the differences in performances are not that high. Thus, the result depicts that there are no significant gender issues in the selected year levels and topics. Further research needs to be carried out in order to determine the relationship between estimation skill and computational ability with topics other than whole and rational numbers, especially in measurement topics. Moreover, such studies can be done involving larger samples, and in other countries as well. Doing so can highlight the importance of the integration of estimation skill in teaching and learning mathematics.
Student written responses to basic questions in geometry are often expressed in unclear language, are confused, or ambiguous. In discussing written responses with students, correct explanations and qualifying comments are often forthcoming. An investigation into such a contradictory situation is warranted. This study investigated the inconsistencies that emerged between students' written test responses and their verbal explanations offered in subsequent interview situations. The participants were pre-service primary-teacher education students; the objective of the testing was to provide van Hiele levels of understanding for each student. There were several components to the study: the initial Preliminary study, the Main Study (consisting of a written test and interview), and an investigation of Response Pattern Errors. The Preliminary Study was conducted in two phases. Phase 1 consisted of an introductory levels test, patterned on Usiskin's earlier multiple-choice test (1982), and was completed by 184 teacher- education students. Subsequent interviews were held with five students. Phase 2 took the form of a post-test, held after the conclusion of the university semester, and once again was accompanied by interviews, on this occasion with eight volunteers. Discrepancies emerged between students' geometric understanding revealed in response to sensitive questioning and probing by the researcher, and their responses provided in written test answers. Interview sessions as a component of a Preliminary Study revealed several inhibiting factors that had influenced students in many of their responses to written test questions. The intention of the testing was to provide students with a realistic view of their van Hiele levels of geometric understanding. In many cases, the influence of the inhibiting factors precluded this. The Main Study was designed to explore the existence and investigate the influence of these factors in greater detail. The Main Study was conducted with a new cohort of 192 students. The students sat for a geometry test entitled the Personal Geometry Profile (PGP) whose questions were selected from a written test designed as part of the doctoral study by Lawrie (1998). Six geometric concepts were tested, namely, square, right triangle, circle, congruence, isosceles triangle, and parallel lines. The questions addressed the first three van Hiele levels. Forty-two participants subsequently volunteered for in-depth interviews. These interviews confirmed and consolidated the list of factors that had been noted in the preliminary investigation. From an analysis of the results of the written test and from input provided during the interview sessions, the inhibiting influences were categorised under six major factors. The first and perhaps the most crucial, was the influence exerted by several aspects of language. The five other major inhibiting factors that prevented students from providing adequate written responses to PGP test questions were: failure to recognise question context, unusual question format (as far as some students were concerned), unfamiliarity with some common geometric conventions, and alternative conceptions. Furthermore, affective issues, often in concert with these other factors, exerted an important influence. included in the study was an investigation into the causes of Response Pattern Errors (RPEs). These inconsistencies in the assignation of levels had occurred in most research studies associated with van Hiele levels of geometric reasoning. This study was no exception, producing eighteen RPEs from among the interviewed students. Analysis of the responses generating the RPEs revealed that they each were the result of the interaction of the same inhibiting factors that were identified during the interviews. They were not caused by a lack of geometric content knowledge or low level of reasoning. These findings unequivocally uphold the notion of hierarchy among levels as espoused by van Hiele. Overall, this investigation provided evidence of, and insight into, significant factors influencing students' written responses to questions in geometry. As a consequence of the investigation, a conceptual framework was developed, illustrating the complex interaction among the student's intellect and experience, and the influential factors that determine the worth of a geometrical response.
This study explores the characteristics of high school teachers' beliefs and practices as they apply to the teaching of mathematics thematically in the 1996 Stage 5 Standard course. A questionnaire consisting of beliefs and instructional practice items was administered to a sample of 122 Standard course teachers representing 69 high schools in metropolitan Sydney. The independent variables were gender, faculty position, socioeconomic status of the teaching area, teaching experience and academic qualifications, while the dependent variables were the teachers' beliefs and instructional practices. With the exception of the independent variable teaching socio-economic area, significant differences were found in the other independent variables in responses to a number of individual questionnaire items. In general, teachers' mathematical beliefs were found to be a moderate predictor of instructional practice for the total sample and a number of instructional, curricular and organisational factors were identified as mediating between teachers' beliefs and their instructional practices. Teachers' beliefs were also found to reflect a constructivist orientation while teachers' practices reflected a behaviourist orientation. The separation of teaching profiles into either constructivist or behaviourist oriented styles of teaching mathematics thematically was necessary for the purposes of this study only, as it is unlikely that the teaching style of any one teacher fill solely/totally into either profile. Subsequent interviews with 10 teachers drawn from the same sample revealed that in general teachers valued the humanistic advantages of teaching mathematics thematically. However, most of the interviewees were convinced that the mastering of basic skills, instead of themes, was the most appropriate instructional strategy to teach students undertaking the Standard course in mathematics. These students were perceived by their teachers as generally lacking in basic numeracy and literacy skills and in the motivation to learn within a school context. Teachers also commented on the lack of structure in teaching mathematics thematically and were of the opinion that its inclusion in the Standard course should be optional rather than mandatory. In addition, there was the perception that students, parents, colleagues, head teachers and the recent neo- behaviourist educational agenda, favour teaching in topics rather than thematically. Teachers were also of the opinion that there is a degree of incompatibility in content and methodology between thematic instruction and the approach taken by the course performance descriptors and the new single School Certificate Test. The study suggests that, due to the aforementioned reasons, the teaching of mathematics thematically in the Standard course is not being thoroughly enacted.
As a result of two years working with the pre-service primary teachers in a College in Fiji the author became aware of the difficulty many of the students were having understanding the primary school mathematics they would be required to teach. During that time she had attempted to help them overcome the difficulties by using different teaching approaches and activities but was far from satisfied with her efforts. Hence she decided to make a concerted effort to help the students by planning, implementing and partially evaluating a mathematics education unit, known as the Teaching Program for the first semester of their course. This work formed the basis of this study. For the Teaching Program a constructivist teaching approach was chosen with number sense as the underlying theme. To examine the aspects of the Program the author's observations and those of the students (reported in their mathematics journals) were used. To evaluate the effectiveness of the Teaching Program quantitative data was collected and analysed from traditional testing of the class of forty students as well as data from case studies of six of the pre-service teachers in the class. Case studies were used as the main source of data to determine what features of the Teaching Program were linked to positive changes. The findings suggested that a significant development of the cognitive aspects of the students' number sense did occur during the time of the Teaching Program but not as much as was hoped for. As a result of the analysis of the data the author came to a greater realisation of the importance of the non-cognitive aspects of number sense and the necessity for a greater consideration of them in the development of a Program. A major development that occurred was in the author's understanding of the knowledge and learning of mathematics. Her ideas of a teaching paradigm of social constructivism had not guided her sufficiently to incorporate activities and procedures to develop the non-cognitive aspects. She suggests that a paradigm which extends the theory of social constructivism to give greater consideration of these aspects of learning in general, and hence numeracy and number sense in particular, was needed. As a result of this study, her introduction to the theory of enactivism appears to be giving her some direction in this search at this stage.
Laptop computers have been used in mathematics classrooms for approximately 10 years and mere has not been a comprehensive study into how laptop computers are used, teachers' attitudes towards laptop computer use, and perceived student benefits. This study focuses on: How laptop computers are used in the mathematics classroom, concentrating on the types of software used and the type and the activities conducted; Teachers' attitudes towards the use of laptop computers in the mathematics classroom, in particular how their attitudes and experiences may effect the use of laptop computers; Teachers' perceptions of student benefits in terms of understanding and performing mathematical tasks using laptop computers. This study shows that there is a wide variety of software used and a large number of activities completed in mathematics classrooms with laptop computers. Teachers favoured using spreadsheets above any other type of software and tended to use the laptop computers for computational, open-ended activities rather than conceptual tasks. Teachers had varying attitudes about how and when laptop computers should be used in the mathematics classroom. There appeared to be connections between a teacher's own use of the computer and the way me teacher used the laptop computer in the classroom. Some teachers used the laptop computer very frequently whilst others used them sparingly. The most valuable type of in-service about using computers came from the teacher's own faculty, through formal and informal discussions. Finally, not all teachers believed there were benefits for their students from using laptop computers. There was no conclusive evidence about whether teachers believed their students had an increased ability to understand mathematics due to using laptop computers, but there was evidence of increased student motivation.
Computer Algebra Systems (CAS), a powerful mathematical software currently available on hand held calculators, is becoming increasingly available to assist secondary students learn school mathematics. This study investigates how two teachers taught introductory differential calculus to their Year 11 classes using multiple representations in a CAS-supported curriculum. This thesis aims to explore the impact of the teaching on students' understanding of the concept derivative. Understanding of the concept of derivative was gauged using an innovative Differentiation Competency Framework that was developed to describe understanding of the concept of derivative. It consists of eighteen competencies for formulation and interpretation of derivatives with, and without, translation between different representations. It clarified the objectives of the curriculum, purpose for using particular CAS activities, and also guided the construction of individual test items on the Differentiation Competency Test that enabled individual and class learning about the concept of derivative to be identified. The Framework also helped identify each teacher's privileging characteristics and facilitated analysis of the learning in relation to the teaching. This study found that using multiple representations was important in developing understanding of the concept of derivative but that the graphical and the symbolic representations were the most useful and important to emphasize and link. Analysis of the teaching actions showed that the teachers used CAS in ways that were consistent with their teaching approach and preferred use of representations and that a conceptual teaching method and student-centred style supported understanding of the concept of derivative. Teaching is directly linked to learning and each class developed a different understanding of the concept of derivative that related to the combined effect of their teacher's privileging characteristics: calculus content, teaching approach, and use of CAS. This study also shows that if a CAS-supported curriculum is to be successfully implemented, it needs to acquire institutional status including a corresponding change in assessment to legitimise new teaching practices.
This study explored Years 4, 6 and 8 students' understanding of the area concept. In particular, it identified and explained students' intuitive judgement rules, that is, the method by which a person integrates information about perceived stimuli when comparing area of rectangular and near-rectangular regions when no length dimensions were given. The study was based on the work of Anderson and Cuneo and the method of Information Integration Theory (IIT). IIT is a technique that classifies area judgement rules as additive, Area = Height + Width, where salient dimensions are added, or multiplicative, Area = Height x Width where salient dimensions are added. Thirty-six students aged 9 to 13 years, twelve at the beginning, 12 at the middle, and twelve at the end of the area-formulae instructional sequence (Years 4, 6 and 8), were given three exercises. The first exercise involved the students estimating the sizes of rectangular regions on a graphic rating scale. The second and third exercises also involved estimation and the graphic rating scale, but used near-rectangular regions. An additional exercise containing area calculation tasks similar to those found in many textbooks and classrooms was also presented to the students. The results of the study found that approximately one half of the students perceived area as being related to the sum of the rectangle's linear dimensions. Such students perceived that doubling the lengths of the sides of a rectangle doubles the area and were referred to as predominantly 'additive thinkers'. In estimating the size of rectangular regions, these students showed a preference for three strategies: vertical alignment of the test piece, use of a ruler (to measure the test piece), and use of the index finger (to measure the test piece informally). They also did not attempt many of the area calculation tasks, and for the ones they did attempt, they concentrated on boundary counting. Boundary counting included either grid line counting around the four sides of the figure or a count of the spaces around the four sides of the figure. The proportion of additive students remained largely the same across the Years. That is, the students at the end of the area-formulae instructional sequence who had increased levels of instruction had not advanced beyond the students in the middle of the area-formulae instructional sequence. These students in turn did not appear to have advanced much beyond the capability exhibited by students at the start of the area-formulae instructional sequence. It was clear, particularly from the classroom tasks, that many of these students experienced confusion between area and perimeter irrespective of their level of area-formulae instruction. However, the proportion of students exhibiting additive thinking reduced for the near-rectangular regions. These regions appeared to divert some students' attention from the linear dimensions to the surface area of the regions. Students who think multiplicatively in terms of their judgements of area were equally likely to use a form of perimeter to calculate area as were students who think additively in terms of area. Students who used a predominantly multiplicative integration of the stimulus cues should have been capable of correctly using multiplication to calculate the area of rectangles from the dimensions of the rectangles. This was found not to be the case in the area calculation tasks closely resembling classroom textbook area formula exercises. Over 65% of the multiplicative students were found to add the salient dimensions. They seemed to employ an additive integration of the dimensions indicating confusion with perimeter. Most students who employed multiplicative judgement rules were found to be more likely to draw a diagram and be able to calculate the area than students who employed additive judgement rules. The majority of students experienced 'intra-individual' rule changes. In the case of the additive thinkers, for one of the three exercises they thought multiplicatively. Similarly for the multiplicative thinkers for one of the three exercises they thought additively. The area calculation task strategies for the additive thinkers and the multiplicative thinkers in this group were the same. They attempted all tasks and these attempts included calculations of perimeter, half perimeter as well as calculations of area through direct one to one counting of the congruent sub-regions. Computational errors also prevented these students from obtaining a correct area solution. For this group also, there was a lot of confusion between area and perimeter.
This research focuses on the relationships between affective constructs of mathematics beliefs, mathematics self-concept, mathematics teaching attitudes, mathematics teaching self-efficacy, cognitive constructs of surface and deep approaches to learning and achievement with respect to mathematics education for pre-service teachers in Hong Kong. A model of Affect-Process-Product-for-Learning is postulated to explain the relationships and the mechanism of the interaction of the affective and cognitive constructs. The design of the investigation consists of three stages, in which both qualitative and quantitative methods are utilised: the pilot study, main study and supplementary study. The pilot study investigates the relationships among the selected affective and cognitive constructs for a small sample of pre-service teachers. It also tests the hypothesised model of relationships among these constructs. The results of the pilot study shows that there exist statistically significant correlations among the constructs and a path model is therefore proposed to explain the influences of the affective constructs and learning approaches on achievement. The main study is a large-scale study of 410 pre-service teachers. Comprehensive models involving affective and cognitive constructs with respect to mathematics education were explored by structural equation modelling (SEM). By competing model strategy the results of the main study indicate how affective and learning-approach characteristics influence achievement in mathematics and mathematics education. The findings also confirm the mediation role of mathematics teaching self-efficacy linking between affective constructs and learning approaches that subsequently influences the achievement. The findings are consistent with Bandura's (1997) self-efficacy theory. Furthermore the supplementary study is a qualitative follow-up of the main study using qualitative interviews. The findings of this study help to illuminate affective characteristics and learning approaches that provide us with a deeper understanding of the nature of each construct for the pre- service teachers and how the affective and learning-approach characteristics influence mathematics and mathematics education achievements. In addition the results of the present study support self- enhancement theory (Hattie, 1992; Marsh and Yeung, 1997; Shavelson and Bolus, 1982) that claims that affective constructs are the determinants of academic achievement. Finally the thesis proposes an Affect Process- Product for learning model (APPLE) based on the quantitative SEM results, qualitative interview analysis and information processing theory to explain how the affective constructs influence the cognitive constructs and the interactions between the characteristics.
Advances in technology have transformed the way in which mathematics in general, and mathematical functions in particular, are taught and learnt. Secondary schools and, to a lesser extent post-secondary institutions have been keen to include technological tools into their learning programs. The basic assumption is that the integration of tools such as Computer Algebra Systems and graphics calculators has the potential to better assist students with their learning, and are superior to the more traditional methods of course delivery. Furthermore, the majority of studies that have reported these findings have been carried out with average to high-achieving students, and little research has been done with low achievers, particularly at the post-secondary level. The purpose of this longitudinal action research study was to explore the nature and extent to which mathematically weak undergraduate students used two technological tools — Computer Algebra Systems and graphics calculators — and what influenced their use of these tools. Three research questions were addressed: (1) How do low- achieving first year university students undertaking a bridging course in mathematics engage with and make use of technological tools? (2) What characterises the solution responses of these students as they engage with technological tools? (3) How do low-achieving students' beliefs and attitudes impact on their subsequent engagement and performance with technological tools? Three teaching experiments conducted over a period of three academic years are reported. The first teaching experiment, the Computer Algebra System Phase, was designed to provide background information concerning the preferred strategies employed by students using Derive when confronted by technology-rich, technology- neutral, and technology-absent assessment tasks. The findings that emerged from the Computer Algebra System Phase informed and provided the direction for the second teaching experiment: the Initial Graphics Calculator Phase. It included the quantitative and qualitative analyses of nine students' responses to technology-rich assessment tasks that utilized a graphics calculator. The third and final year of the study, the Final Graphics Calculator Phase, examined in detail ten students as they engaged with a graphics calculator. As with the Initial Graphics Calculator Phase both qualitative and, where appropriate, quantitative data were collected through assessment tasks, questionnaires, self-report journals, interviews and field notes. In- depth analysis of individual case studies (within-case analysis) was followed by pair- or group-wise comparisons (cross-case analysis) to explore patterns or characteristics that students shared across common backgrounds, such as school- and mature-age status, prior mathematics and tool-based experiences, and attitudes towards and beliefs about using a technological tool.
In the middle years of schooling, the topic of equation solving generally begins teaching students how to find solutions to linear equations using traditional by-hand techniques. This is a student's first formal introduction to algebraic equation solving and many master the techniques involved, however, few students acquire a conceptual understanding of the notion of a solution or the solution process. A graphics calculator is capable of plotting a function, generating a table of values and testing whether a number is a solution to an equation, all in a matter of moments. They are currently used in VCE mathematical studies and examinations. The aim of this investigation is to study the effects of a graphics calculator based approach on the learning and teaching of equation solving. This approach involved constant access, by an experimental group, to a TI-83 graphics calculator. A graphics calculator based approach to equation solving may assist students in developing a better understanding of the key concepts and solving techniques, and the application of these to the solution of problems. Constant access to graphics calculators in the classroom may allow students who lack pen-and-paper techniques the opportunity to reason mathematically about problems and mathematical ideas. To accomplish this aim a classroom-based study was conducted in a Melbourne metropolitan school. An experimental-control design was used with the four year eight classes of two teachers. Each teacher taught a control and experimental (graphics calculator) class. The study examined the achievement of the two treatment groups with pre and post-tests. Test performances of all students were compared and analysed to determine the benefits from the various instructional techniques. It was found that there were no significant differences between treatment groups and the only significant difference was between the. teachers of the study. It was also found that low ability students, from the graphics calculator environment, made the largest gains. Although no statistical differences were found, students in the experimental group were more successful with questions involving graphs and tables while the students in the control group were better at solving equations by hand.
At a time of transition, when the increasing availability and affordability of Computer Algebra Systems (CAS) presents mathematics educators with new challenges, this thesis explores two facets of students' abilities and understanding that impact on the use of CAS in teaching and learning mathematics. In this thesis, these are called 'Algebraic Insight' and 'Effective Use of CAS'. A framework is presented and described for each construct and then the frameworks are explored within the context of a course in introductory calculus, taught by the researcher to a class of 21 undergraduate tertiary students. Algebraic Insight is the subset of Symbol Sense required when using CAS for the mathematical solution phase of problem solving. The framework breaks Algebraic Insight into two aspects: ability to Link Representations (symbolic, numeric, graphical); and Algebraic Expectation, the cognitive skill required to monitor symbolic work (comparable to arithmetic estimation for monitoring numeric work). The framework of Effective Use of CAS is also divided into two aspects: Technical, using syntax and program features; and Personal, the willingness to use CAS in a judicious manner. Throughout the 15 week course a bank of tests, surveys, observation and interviews assessed students' levels of Algebraic Insight and Effective Use of CAS. The instruments successfully monitored changes and demonstrated class improvement, a finding clarified by 7 detailed case studies. Effective Use of CAS and Algebraic Insight are inter-dependent. First, sufficient Algebraic Insight is needed to begin to use CAS. Second, CAS can be employed successfully as a learning tool for exercises designed to improve Algebraic Insight provided the student demonstrates at least a moderate level of Effective Use of CAS. Third, Algebraic Insight helps students to use CAS in a strategic manner. The new outcomes of this study that will be of use to teachers and curriculum planners are the frameworks of Algebraic Insight and Effective Use of CAS, the quick Algebraic Insight Quiz and the CAS use survey. Of the terms coined for this study, perhaps the most useful will be Algebraic Expectation. The new terms and new frameworks provide a structure for the new focus that the teaching of algebra must adopt in a CAS environment.
This study investigated students' understandings of class inclusion concepts in Geometry. The purpose was to identify a developmental pathway leading to an understanding of the interrelationships among two- dimensional figures and their properties. The design involved a tightly focused investigation of the manner in which geometrical class inclusion concepts evolve, in particular, relationships among triangle and quadrilateral figures, and relationships among their properties. Empirical evidence is provided to explain the difficulties students face in understanding of class inclusion notions. This evidence has theoretical as well as practical implications. The theoretical base for this study is the van Hiele Theory, which comprises five levels of development in Geometry. Numerous studies have involved a focus upon the holistic aspects of the first four van Hiele levels and this has resulted in supportive empirical evidence of the existence and nature of the levels. Pertinent to this study, the level associated with a student who accepts and utilises notions of class inclusion is described as Level 3. This aspect of Level 3 is regarded as both a difficult concept to acquire and a prerequisite for formal deductive reasoning. This study extends research into the van Hiele Theory by narrowing the microscopic lens and providing a focused analysis on the understanding and development of class inclusion concepts in Geometry. In an attempt to refine the characteristics of the development of this concept, this study utilised the SOLO model to provide deeper insights into the van Hiele levels. The investigation comprised three studies. The first of these, Study 1, explored the context of triangles, and included two main components. These components were relationships among triangle figures, and relationships among triangle properties. Study 2 extended the baseline data of Study 1 via the investigation of students' understanding of relationships among quadrilateral figures and relationships among quadrilateral properties. Each of these studies involved in-depth interviews with 24 students of higher mathematical ability, purposely selected, within Years 8-12 (ages 13-18 years) in two secondary schools. Study 3 also consisted of two parts. The first of these, a quantitative synthesis, based upon the application of ACER's QUEST analysis program, utilised Rasch measurement theory. This part of Study 3 also considered developmental changes from a longitudinal perspective. The second part of Study 3 considered developmental changes in the form of four case studies. A central finding of this study was the identification of a broad generic framework which describes the developmental pathway leading to an understanding of class inclusion notions. This pathway characterises student growth in understanding of relationships among figures, and relationships among properties. The pathway was characterised by two cycles of responses of the concrete symbolic mode (SOLO), and two cycles of responses of the formal mode (SOLO). The existence of this pathway has challenged accepted characterisations of van Hiele's Level 3. Behaviours previously described as requiring Level 3 thinking have been found by this study to include Level 3, Transitional Level 3/4, and Level 4. This study identified student difficulties associated with attaining Transitional Level 3/4. Here, students need to focus upon relationships that are not supported by visual cues. This is identified as formal thinking. The characterisation of transitional groups, evident at Level 3/4, provides guidance concerning teaching activities and implications, to assist students' in their progression from Level 3 to Level 4. In general, the known property relationships assisted students in the formation of sub-class relationships. In addition, property relationships did not emerge as an identifiable sequence; instead they appeared dependent upon student familiarity with individual properties. However, developmental patterns were evident in terms of language-use where property descriptions appeared to hinder the formation of relationships at Level 2, and property descriptions were conducive to the utilisation of relationships at Level 3, Transitional Level 3/4, and Level 4. Of surprise were the similarity of results for two different contexts of quadrilaterals and triangles. This finding providing support for the notion that thinking at a particular level in one context assists the progression to the same level in other contexts. The quantitative synthesis across contexts validated the chosen instrument and the developmental trends highlighted by the application of the SOLO model. There was consistency across the triangle and quadrilateral contexts concerning relationships among figures and relationships among properties. The longitudinal student responses, over the two-year period, were interpreted along the previously identified developmental path. Evidence presented in the case studies indicated that individual student responses to similar tasks within different contexts were not always at a consistent SOLO level, dependent upon individual familiarity of triangles or quadrilaterals. It was also evident that some students responded at a higher SOLO level concerning either relationships among properties or relationships among figures. The research highlights the reasons students find class inclusion concepts in Geometry difficult to grasp. Secondary-school (ages 12-18 years) curriculum content concerning such notions have been identified as requiring thinking at van Hiele's Level 3, Level 3/4 and Level 4. Thus the hurdles encountered by many students are detailed through the characterisation of the development of relationships among figures and relationships among properties. In addition, this study highlights the use of the SOLO model as an interpretive tool for research in Mathematics education.
Dull classroom environments, poor students' attitudes and inhibited conceptual development led to the creation of an innovative mathematics program, the Class Banking System (CBS), which enables teachers to use constructivist ideas and approaches. To assess the effectiveness of the CBS, actual and preferred versions of the Individualised Classroom Environment Questionnaire (ICEQ), the actual version of the Constructivist Learning Environment Survey (CLES), the Test of Mathematics-Related Attitudes (TOMRA), and conceptual map tests were administered to two groups of fifth-grade students as pretests and posttests during the course of two academic years. To enrich the data collected from those questionnaires, three case studies were undertaken based on observations and interviews of selected students. A comparison of Class Banking System (CBS) students with non-CBS students suggested that CBS students experienced more favourable changes in terms of mathematics concept development, attitudes to mathematics, and perceived classroom environments on several dimensions of the Constructivist Learning Environment Survey (e.g. Personal Relevance, Shared Control) and the Individualized Classroom Environment Questionnaire (e.g. Participation and Differentiation). Furthermore, a direct comparison of the CBS and control classes on the posttest confirmed higher scores among the CBS group for classroom environment, attitudes, and concept development scales. The study also replicated some patterns from prior research. First, data analyses generally supported the validity and reliability of the classroom environment instruments in terms of indices such as internal consistency reliability, discriminant validity, factor structure, and ability to discriminate between the perceptions of students in different classrooms. Second, students preferred a more positive classroom environment than the one perceived to be actually present. Qualitative information based on classroom observations and student interviews reinforced and enriched the patterns of results obtained from the concept test and questionnaires in that it supported the effectiveness of the CBS in providing elementary mathematics students with a constructivist and individualized classroom learning environment that promotes both conceptual development and positive attitudes.
This study was designed to explore the computation choices made by 78 students in Years 5 to 7. The ability to choose and use a repertoire of computation methods is an important goal of mathematics education. While one might expect to find a great deal of research evidence outlining the computation choices students make and why they make them, this was not the case; and as such it was decided to explore what computation choices students make and why they make them. When examining the literature dealing with computation choice few studies were found that directly discussed the issue. There were many studies of computation and discussion of factors that might affect computation choice. The literature also outlined the need for the computation focus to change from purely the development of skills, particularly with paper-and-pencil, to enhance the ability of students to make considered computation choices. Several models of computation were reviewed along with literature dealing with metacomputation. This prompted the need for a fresh look at computation in terms of a non-linear computation model that better reflected the computation process students pass through when solving a computation problem. In particular the role of metacomputation as a means of choosing a computation method, then guiding and monitoring the computation was explored. Students in Years 5 to 7 were chosen to participate in the study as it was felt 10-12 year-old students would have had enough exposure to various forms of computation so as to be confident and competent in using all forms of computation. Students were asked to complete a series of computation items using their preferred computation approach. Clinical interviews were conducted to determine why students made particular computation choices. Observational data and field notes were used to collect data on what computation choices were made and how successful students were in executing their chosen method of computation. Data were analysed and it was found that students made appropriate computation choices in slightly over 50 percent of cases based on the success rate experienced when completing computation questions using their favoured method. In some cases computation choice was limited by a lack of competence in all forms of computation. In particular it was noted that many students were unable to make use of simple calculators. Interview data indicated that students make computation choices with little hesitation and based on a set of rudimentary criteria such as the magnitude of the numbers involved, or the operation required. There was little evidence to suggest that students looked beyond these simple criteria when making a decision about which form of computation to use. The implication of the research is that teachers may better understand how students make computation choices and what hampers the making of computation choices. As a result of understanding of the process students use for making such choices, teachers should be able to raise student awareness of the process of making a computation choice. The thesis concludes with a recommendation that in much the same way that teachers have been encouraged to focus on developing mental computation strategies, they should also encourage students to discuss their criteria for making particular computation. choices. In doing so students will be encouraged to broaden their thinking about the computation process. A suggestion is also made that time spent in the classroom developing each of the computation alternatives, mental, written and calculator, needs to better reflect the usage patterns of adults. Students who have a better understanding of how to use all types of computation will be in a better position to make appropriate computation choices.
It is clear from the literature that attention should .be paid to teachers' beliefs about how the mathematics classroom operates and about the nature of mathematical problem solving. The introduction of a mathematical problem-solving curriculum which ignores these aspects is likely to be frustrated. This research seeks to learn more about the use mathematical problem solving by Tongan teachers and what these teachers consider to be good teaching of mathematics and mathematical problem solving. Five mathematics teachers in the Free Wesleyan Church of Tonga education system were interviewed about their beliefs using the Kelly Repertory Grid technique through Enquire Within software. Surveys were also conducted of the material produced for schools by the Tongan Government's Ministry of Education. It was found that the Ministry of Education has implemented a minimal problem-solving curriculum throughout the Kingdom and that the beliefs of the majority of teachers interviewed were compatible with this minimalist model. Recommendations for the Free Wesleyan Church's education department are made on possible approaches to fulfilling the Ministry of Education's stated aim of preparing students to apply the principles of mathematics to unfamiliar situations.
The present study investigates students' gendered achievement-related choices and behaviours in mathematics and English, using the Expectancy- Value theory of achievement motivation developed by Eccles and colleagues, which is the most prominent current model predicting academic choices in the form of course enrolments. The central social issue addressed in this thesis is the differential participation in higher-level mathematics by girls and boys, in both their senior high school years and intended careers. Key questions investigated are first, to what extent do boys plan to participate in maths to a greater extent than girls, both in senior high and in their planned careers? Second, what are the predictive influences of gender, expectancies, values and task demands on maths participation and achievement-related behaviours? Third, what is the nature and development of boys' and girls' trajectories for expectancy, value and task demand variables from junior through to senior high school? Fourth, what causal sequencing among expectancy and value constructs can be discerned? These questions are addressed using longitudinal data from three cohorts in an overlapping cohort sequential design (N's= 428, 436, 459). Parallel analyses are conducted for English to assess domain specificity of findings. These four major questions comprise the extensive quantitative survey phase of the study. Further questions comprise the intensive qualitative interview phase, namely: what factors are facilitative of, and detrimental to high-ability girls pursuing high levels of mathematics? How are their self-perceptions of talent derived? And how do students explain greater male participation in maths at school and in the workplace? Self-perceptions and values were identified as the most important factors related to gendered achievement-related choices and behaviours in maths and English, where participation in maths and English HSC course levels as well as intended maths- and English-related careers comprised choice outcomes, and performance and self-reported effort exerted comprised behaviour outcomes. Findings support research within the Expectancy-Value framework, indicating that students' maths values are the strongest predictors of mathematics course enrolment in high school, while success expectations and competence beliefs most strongly predict mathematics achievement. In English, in contrast to maths, self- perceptions but not values predicted choice outcomes, while similarly to maths, self-perceptions predicted performance outcomes. Following identification of key predictors of maths and English achievement- related outcomes, gender differences favouring boys in maths and girls in English were identified across many of the self, task and value constructs assessed. Further, optimal grade levels for intervention were identified through inspection of when changes in boys' and girls' growth trajectories for key Expectancy-Value perceptions occurred through grades 7 to 11. This was achieved through an overlapping cohort sequential design, enabling an 'accelerated' longitudinal study spanning grades 7 to 11, as well as providing evidence of replicated effects across cohorts. Directions of relationships between Expectancy- Value self-perception and values constructs were also clarified through structural equation modelling techniques using longitudinal data. Directional paths for maths confirmed those proposed in the Expectancy- Value model for maths, with self-perceptions influencing values, whereas for English, reciprocal influences occurred between self- perceptions and values. The qualitative component of the study provided rich insights into the bases for high-achieving girls' disproportionately low talent perceptions, through comparing their responses with carefully chosen contrast groups. This group of girls is also arguably the group with whom we should be most interested, being the group for whom both 'waste of talent' and social justice arguments are most strong. The present study provides guidance as to how interventions may enhance self-perceptions related to maths, and suggests valuable insights as to how increasingly equal participation of males and females in maths may be achieved. Findings provide theoretical and measurement contributions for researchers within the Expectancy-Value framework, as well as implications to educators concerned with enhancing achievement-related choices and outcomes, particularly for girls in relation to maths.
This thesis examined individual differences in cognition and affect in secondary school students' multiplicative knowledge structures of basic mathematics problems. The study was set in a metropolitan State High School in Brisbane, Queensland, Australia which draws students from a mix of demographic and socio-economic areas. Four quantitative studies examined models of cognition, affect and knowledge structures of basic mathematics problems and their inter-relationships. The first study used Luria's 'whole brain' theory of Information Processing abilities or preferences. This theory describes two ways in which people input, store and analyse sensory data. According to Luria's model, information is processed successively (sequential and primarily temporal) and simultaneously (continuous and primarily spatial). The second study used Marsh's multi-dimensional model of Self-Esteem which describes academic and non-academic dimensions as measures of Self-Image. The academic dimensions of Maths and Verbal Self-Esteem can be linked directly to performance in mathematics problems. Basic mathematics items used in Queensland and other parts of Australia provided a measure of student performance on four arithmetical operations of addition, subtraction, multiplication and division of positive integers and non-integers. The fourth study showed that individual differences in Information Processing Preferences and the academic dimensions of Maths and Verbal Self-Esteem were related significantly to increasingly complex multiplicative knowledge structures in basic mathematics problems. A later Phase 2 qualitative study supplemented and complimented the significant findings of the four Phase 1 studies. This fifth study incorporated semi-structured interviews of 22% of the cohort. This was a selective sample drawn from Years 8 and 9 only and was informed by the findings of the Phase 1 quantitative results. The interviews focused on individual differences of Simultaneous and Successive Information Processing Preferences and affect in the inter- connectedness of multiplicative knowledge structures that students use in solving basic mathematics problems. Procedural and Conceptual knowledge aspects of multiplicative structures were analysed within a proceptual framework described by Gray and Tall. Simultaneous Information Processing Preferences were found to be essential in enabling connectedness of the more conceptual aspects of multiplicative knowledge. Successive Information Processing Preferences were found to be essential in enabling connectedness of the more procedural aspects of multiplicative knowledge inherent in algorithmic processes. An expanded model of Proceptual Multiplicative Knowledge proposed that flexibility, inherent in Simultaneous Information Processing Preferences, is the catalyst for enabling more complex multiplicative knowledge structures in basic mathematics problems. The relationship between these cognitive aspects and the academic dimension of Maths Self-Esteem, while generally related positively to differences in Simultaneous Information Processing Preferences, is not as clearly defined with idiosyncratic behaviours apparent in higher levels of Simultaneous Information Processing Preferences. This significant finding of individual differences in Information Processing Preferences and Maths Self-Esteem in proceptual aspects of multiplicative knowledge structures of basic mathematics problems provided a sound basis for further major research within the disciplines of psychology, mathematics and mathematics education.
This study is in response to the documented need to develop alternative assessment procedures which reflect current conceptions of the nature of school mathematical learning. From the literature a conceptual framework was developed that reflected key aspects of assessment of school mathematics. The nature of school mathematical learning was defined and the types of educational decisions arising from assessment were identified. The nature of conventional and alternative assessment paradigms was also defined. The initial theme of the study was to examine the support provided by conventional and alternative assessment for classroom educational decisions. The second theme of the study was to consider the theoretical bases for assessment, in particular Vygotskian theory, which explicitly integrates assessment with teaching and learning. The third focus of the study related to the operationalisation of Vygotskian theory as dynamic assessment. It sought to define conditions for the practical use of dynamic assessment related to a specific domain of school mathematics, understanding of place-value numeration. These research themes were explored through the development and implementation of a dynamic assessment procedure for use by school psychologists within a particular school and curriculum context. The support for classroom educational decisions from the dynamic assessment procedure was compared to that provided by conventional assessment. Outcomes show that differences in content validity account for discrepancies in information about student learning derived from the conventional and dynamic assessments. The dynamic assessment procedure was able to maintain content validity through reference to cognitive definitions of learning and the use of a set of cognitive research tasks. In contrast, the conventional assessment instruments had lower content validity due to the restricted nature of the item format. The study also demonstrates the contribution of Vygotskian theory to the development of clinical assessment procedures for school contexts. Vygotskian theory informed the development of the dynamic assessment procedure and underpins the integration of teaching, learning and assessment evident in the study. In response to documented resistance to dynamic assessment in schools, conditions under which dynamic assessment can be implemented in schools are identified. The attitude of the assessor to the inclusion of teaching in an assessment procedure, the importance of the assessor- student relationship and the assessor's knowledge of the curriculum domain of learning are all defined as essential aspects of dynamic assessment. Professional development to introduce practicing school psychologists to dynamic assessment is recommended by the study.
This thesis focuses on the investigation within a primary school of an emergent classroom community of practice that is in accord with a sociocultural approach to promoting learning and development through the processes of collective argumentation. One outcome of the thesis is the development of an emergent community model of understanding classroom learning. The design of this model involved a synthesis of methodological approaches to researching classroom learning that share a compatibility with Vygotskian assumptions that relate to the role of semiotically-mediated social interaction in promoting development. It is within this 'emergent community' framework for understanding learning that a qualitative study into a Year 7 classroom's ways of coming to know, do, and value mathematics was conducted over the course of one school year. The study was concerned with providing insights into (1) how communal practices emerge and are sustained within a classroom, (2) how students construct and display certain identity positions within a collaborative learning environment, (3) how the social processes of the classroom interact at the small group level to motivate and guide students to 'speak' and 'act' as members of a collaborative learning community, and (4) how students, at the whole- class level, employ communal practices and social interactional forms to challenge and extend their ways of knowing, doing, and valuing mathematics. The study was also concerned with exploring the relationships between the communal practices, processes, and products of an emerging classroom community of practice and those practices, processes, and products promoted with various high school classrooms. Through employing detailed analyses of video/audio-taped transcripts, teacher/student journal entries, students' responses to pre- and post- participation questionnaires, and records of students' self-selected seating arrangements within the classroom, the findings of the study provide support to Vygotsky's contention that the form of interpsychological functioning has a determining influence on the resulting form of intrapsychological functioning and index the centrality of communicative processes in the emergence of a classroom community of practice. In particular, the study found (a) that the emergence of a classroom community of practice is marked by an irreducible tension between the teacher, the tools that are employed to facilitate the emergence of community, and the context in which the emergence is situated, (b) that students participate differently in the social processes of a classroom community and construct identities that are marked by changing relationships between participants, (c) that the culture of the classroom motivates and guides students' moves toward or away from more mature ways of participating in the community, (d) that the whole-class discourse genre of an emerging classroom community is co-constructed by the teacher and students over time as they challenge each other to extend the quality of learning and teaching that takes place in the classroom, and (e) that participation in an emerging classroom community of practice provides worthwhile long-term learning benefits for students in terms both of their personal needs and in terms of the goals of national curriculum documents. These findings are presented and discussed in a manner that demonstrates how sociocultural theory may function to inform and describe the emergence of a classroom community of practice. The implications of these findings are presented in terms that have the potential to provide the educational community with a set of pedagogical and research tools that may facilitate the establishment and investigation of classroom learning environments that actively involve students in meeting personal needs and civic expectations.
The research presented in this study deals with the use of graphics calculators in Mathematics education at the pre-calculus level. It has two broad aims: to document student misconceptions associated with the use of graphics calculators; and to investigate whether informing teachers about likely student difficulties might favourably affect the teachers' use of graphics calculators in Mathematics teaching. Student misconceptions are discussed in two aspects: conceptual misconceptions, which concern how well students are able to relate the information shown on the graphics calculator screen to the other mathematical concepts they are learning; and technical misconceptions, which describe students' ability to remember which keys to press in order to achieve specific graphical results, and the students' general understanding of the fundamental operation of the graphics calculator. The research study was undertaken in two overlapping phases - one focusing on students and the other focusing primarily on teachers. In Phase One, three intensive clinical interviews were conducted with each of twenty-five students from Year 10 and Year 11 high-achieving Mathematics classes who had limited experience in operating a graphics calculator. The students considered a series of tasks designed to create situations of cognitive conflict by directly exposing some of the graphics calculator's limitations. The results of these student interviews indicated that many difficulties in using a graphics calculator may be due to inadequate understanding of some fundamental mathematical ideas including scale, accuracy and approximation, and the link between different representations of functions. These weaknesses point to shortcomings in the present curriculum which may have adverse affects whether or not graphics calculators are used. The results also suggest that students may need to acquire some technical understanding of how the graphics calculator works. It is particularly crucial that students learn how each pixel is assigned coordinates and how these coordinates are related to the coordinates displayed on the screen. In Phase Two of the study, a two-day workshop was designed for a group of twelve Mathematics teachers who had not previously used graphics calculators in their teaching. The workshop was based on the findings of the Phase One student interviews. The teachers were taught how to use a graphics calculator and became acquainted with students' conceptual and technical difficulties. Six of these teachers were subsequently observed using graphics calculators in their teaching, and a group of fifteen students from the Year 10 and 11 classes taught by these teachers was randomly chosen. These students were interviewed using the Phase One protocols. The Phase Two results attest that teachers also exhibit calculator misconceptions. With careful instruction, however, teachers can gain sufficient confidence and expertise to make effective use of graphics calculators in the classroom. The Phase Two students achieved some significant gains in their understanding of the calculator's basic operation which indicates that the intervention with this group of teachers was largely successful.
The primary purpose of this research was to examine individual differences in learning from worked examples. By integrating cognitive style theory and cognitive load theory, it was hypothesised that an interaction existed between individual cognitive style and the structure and presentation of worked examples in their effect upon subsequent student problem solving. In particular, it was hypothesised that Analytic-Verbalisers, Analytic-Imagers, and Wholist-Imagers would perform better on a post-test after learning from structured-pictorial worked examples than after learning from unstructured worked examples. For Analytic-Verbalisers it was reasoned that the cognitive effort required to impose structure on unstructured worked examples would hinder learning. Alternatively, it was expected that Wholist- Verbalisers would display superior performances after learning from unstructured worked examples than after learning from structured- pictorial worked examples. The images of the structured-pictorial format, incongruent with the Wholist-Verbaliser style, would be expected to split attention between the text and the diagrams. The information contained in the images would also be a source of redundancy and not easily ignored in the integrated structured- pictorial format. Despite a number of authors having emphasised the need to include individual differences as a fundamental component of problem solving within domain-specific subjects such as mathematics, few studies have attempted to investigate a relationship between mathematical or science instructional method, cognitive style, and problem solving. Cognitive style theory proposes that the structure and presentation of learning material is likely to affect each of the four cognitive styles differently. No study could be found which has used Riding's model of cognitive style as a framework for examining the interaction between the structural presentation of worked examples and an individual's cognitive style. Two hundred and sixty-nine Year 12 Mathematics B students from five urban and rural secondary schools in Queensland, Australia participated in the main study. A factorial (three treatments by four cognitive styles) between-subjects multivariate analysis of variance indicated a statistically significant interaction. As the difficulty of the post-test components increased, the empirical evidence supporting the research hypotheses became more pronounced. The rigour of the study's theoretical framework was further tested by the construction of a measure of instructional efficiency, based on an index of cognitive load, and the construction of a measure of problem-solving efficiency, based on problem-solving time. The consistent empirical evidence within this study that learning from worked examples is affected by an interaction of cognitive style and the structure and presentation of the worked examples emphasises the need to consider individual differences among senior secondary mathematics students to enhance educational opportunities. Implications for teaching and learning are discussed and recommendations for further research are outlined.
This thesis is an institutional study, attempting to account for the current situation of mathematics within the Australian vocational education and training (VET) sector. Contextualisation is provided in the first place through pertinent issues concerning professionalism, teaching, learning, and research - arising from the author’s own and other vocational mathematics teachers' reflections on practice. The construct of institution is employed to set a more theoretical foundation with regard to, respectively, the discipline of mathematics, the field of mathematics education, and the field of vocational education and training. Technology emerges as a unifying construct for the complex relationships between mathematics and industry, in both production and management discourses, and between mathematics and vocational education. From the meta-analytic stance of this thesis, vocational education- itself has become an industry and its political and social structures are explored to elucidate the apparently ambiguous position of the discipline of mathematics within the sector. A recurring theme is public image - firstly of mathematics, secondly of vocational education and training in an increasingly deregulated sector which relies on segments of public opinion for its continuing survival, and thirdly in relation to the discourses of lifelong learning. Literature reviews are undertaken with respect to mathematics in and for the workplace, adult learners of mathematics, and issues associated with curriculum and teaching - not least the trend towards flexible delivery in the face of a chronic lack of discipline-based professional development. In this sector, where technology is, inter alia, a tool, an object (rarely a subject), and a vector for transmission, there is a dearth of research related to mathematics education. The question arises as to whose interests are being served by the apparent under- theorisation of mathematics education in the Australian VET sector. The latter part of the thesis attempts to theorise these developments, drawing upon a range of intellectual work, but finding that Basil Bernstein's concepts of symbolic control, pedagogy, and identity provide the most coherent framing for the terrain covered. The thesis concludes with discussion of unresolved policy, research, and practical issues, briefly considering a selection of relevant vocational and mathematics education research - being undertaken in comparable European countries. Possibilities are explored for developments on a structural level and, drawing on the literature of pedagogies intended - to combat racism, on an individual level for teachers and students to address issues of identity formation. It is hoped that these might contribute to an enhanced public image of both mathematics and vocational education and training. However, it is argued that even public image has become technologised.
This study is a multiple-perspective analysis of teaching and learning in two Year 11 classes. Both classes were studying topics on vectors and all students owned graphics calculators. In one class, the author was a replacement teacher for one term and the research was a critical inquiry into her teaching in relation to three goals. The goals were for students' to participate actively in class, that the author would be inclusive in her actions and teach in real-life contexts. In addition, she was interested in students' use of graphics calculators for learning. Conversation analysis revealed a diverse range of modalities of student participation, and how, as the teacher, she mediated students' participation in ways she had not recognised previously. The text of the classroom conversation illustrated also both inclusive and non-inclusive aspects of the enacted curriculum. One issue was how real for students were the 'real-life contexts' in the vector problems implemented. Circumstances deemed that specialised use of graphics calculators was limited, but examples are reported of equation solving and calculation. The second class was that of a colleague. The author observed whole-class work, and acted as an assistant teacher at other times. The research purpose was broad: to understand more about students' mathematics learning, including how routine use of graphics calculators could mediate mathematical advancement. To achieve this goal the author tracked over one month two students' conceptual development in relation to the vector topic they were studying. Use of graphics calculators was an intrinsic part of their learning, and could be seen to enhance their progress. For one of the students, the author considered also the constitution of her mathematical identity, namely, the constitution of her mathematical competence and incompetence in the wholeclass domain.. This research inquiry had other facets besides the teaching and learning of vectors. It is an autobiographical account where the chapters are written in a variety of genres, which are indicative of how the author's writing evolved from relatively closed, objective reporting to more open, dynamic inquiry. A feature of the last chapters to be finished is their dialogic rather than monologic tone. A second evolutionary aspect of the dissertation is that it represents how the author's thinking on learning changed, so that she cast initial analyses in constructivist and social constructivist terms, while analyses produced later in the study are underpinned by the assumptions of phenomenology. The study concludes with an overview of the outcomes identified in each chapter and a process commentary of dilemmas associated with the research. This self- reflexivity and the other characteristics of the study place it as belonging to the postmodern era.
This study is a multiple-perspective analysis of teaching and learning in two Year 11 classes. Both classes were studying topics on vectors and all students owned graphics calculators. In one class, the author was a replacement teacher for one term and the research was a critical inquiry into her teaching in relation to three goals. The goals were for students' to participate actively in class, that the author would be inclusive in her actions and teach in real-life contexts. In addition, she was interested in students' use of graphics calculators for learning. Conversation analysis revealed a diverse range of modalities of student participation, and how, as the teacher, she mediated students' participation in ways she had not recognised previously. The text of the classroom conversation illustrated also both inclusive and non-inclusive aspects of the enacted curriculum. One issue was how real for students were the 'real-life contexts' in the vector problems implemented. Circumstances deemed that specialised use of graphics calculators was limited, but examples are reported of equation solving and calculation. The second class was that of a colleague. The author observed whole-class work, and acted as an assistant teacher at other times. The research purpose was broad: to understand more about students' mathematics learning, including how routine use of graphics calculators could mediate mathematical advancement. To achieve this goal the author tracked over one month two students' conceptual development in relation to the vector topic they were studying. Use of graphics calculators was an intrinsic part of their learning, and could be seen to enhance their progress. For one of the students, the author considered also the constitution of her mathematical identity, namely, the constitution of her mathematical competence and incompetence in the wholeclass domain.. This research inquiry had other facets besides the teaching and learning of vectors. It is an autobiographical account where the chapters are written in a variety of genres, which are indicative of how the author's writing evolved from relatively closed, objective reporting to more open, dynamic inquiry. A feature of the last chapters to be finished is their dialogic rather than monologic tone. A second evolutionary aspect of the dissertation is that it represents how the author's thinking on learning changed, so that she cast initial analyses in constructivist and social constructivist terms, while analyses produced later in the study are underpinned by the assumptions of phenomenology. The study concludes with an overview of the outcomes identified in each chapter and a process commentary of dilemmas associated with the research. This self- reflexivity and the other characteristics of the study place it as belonging to the postmodern era.
This study investigated the feedback practices of an experienced year eleven Mathematics B teacher. Key documents such as the Senior Syllabus in Mathematics B, A National Statement on Mathematics for Australian Schools and the Assessment Standards for school Mathematics called for the use of feedback to improve student learning. Examination of the literature demonstrated the need for research investigating the implementation of feedback procedures in the classroom. The limited existing literature examining teachers' actual use of feedback suggested the related factors of mathematics assessment, teaching practices and classroom culture need to be considered. Due to the absence of definitive literature describing teachers' use of feedback in the classroom, an exploratory, interpretive, qualitative approach was adopted. The descriptive case study of one teacher provided the depth of investigation necessary to describe the teacher's feedback practices and to illustrate the factors influencing these practices. There was found to be a close relationship between the nature of the classroom culture and the type of feedback that was used. In this classroom the focus of feedback on marks, the correct solution and examination recommendations demonstrated the value placed on achieving the correct solution during assessment. The classroom culture was not conducive to the provision of informative feedback or its use by the students. In its place was a definite culture of marks, tacitly agreed upon by both the teacher and the students. The depth of description of the case enables teachers to compare the context of the case to their personal situations and to make judgements about heir own feedback practices. Future research may investigate the conditions necessary for a culture where feedback is valued. Alternatively the theoretical feedback procedures may be investigated to examine how they may be implemented in a culture valuing marks and the correct solution.
Understanding base-ten numbers is one of the most important mathematics topics taught in the primary school, and yet also one of the most difficult to teach and to learn. Research shows that many children have inaccurate or faulty number conceptions, and use rote-learned procedures with little regard for quantities represented by mathematical symbols. Base-ten blocks are widely used to teach place- value concepts, but children often do not perceive the links between numbers, symbols, and models. Software has also been suggested as a means of improving children' s development of these links but there is little research on its efficacy. Sixteen Queensland Year 3 students worked cooperatively with the researcher for 10 daily sessions, in 4 groups of 4 students of either high or low mathematical achievement level, on tasks introducing the hundreds place. Two groups used physical base-ten blocks and two used place-value software incorporating electronic base-ten blocks. Individual interviews assessed participants' place-value understanding before and after teaching sessions. Data sources were videotapes of interviews and teaching sessions, field notes, workbooks, and software audit trails, analysed using a grounded theory method. There was little difference evident in learning by students using either physical or electronic blocks. Many errors related to the 'face-value' construct, counting and handling errors, and a lack of knowledge of base-ten rules were evident. Several students trusted the counting of blocks to reveal number relationships. The study failed to confirm several reported schemes describing children' s conceptual structures for multidigit numbers. Many participants demonstrated a preference for grouping or counting approaches, but not stable mental models characterising their thinking about numbers generally. The independent-place construct is proposed to explain evidence in both the study and the literature that shows students making single-dimensional associations between a place, a set of number words, and a digit, rather than taking account of groups of 10. Feedback received in the two conditions differed greatly. Electronic feedback was more positive and accurate than feedback from blocks, and reduced the need for human-based feedback. Primary teachers are urged to monitor students' use of base-ten blocks closely, and to challenge faulty number conceptions by asking appropriate questions.
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Children with learning disabilities often struggle to learn that the counting process will tell them 'how many'. The research suggests that this may be due to their limited informal mathematics understandings, which other children develop during their preschool years. This thesis seeks to understand the nature of children's informal mathematics knowledge and how this is linked to their understanding of counting and their understanding of number as a representation of quantity. The study was carried out in two parts, an investigation of the literature and an empirical study. The literature in relation to children within the normal range suggests that children's informal mathematics knowledge consists of protoquantitive schema, subitising and part whole understandings. A model is proposed by the author, of how these quantity understandings are interdependent and support children to develop an understanding of numbers and counting. An empirical study was conducted to investigate the informal mathematics understandings of a group of 25 primary aged children with learning disabilities. This data revealed an interrelationship between protoquantitive schema, subitising, part whole understanding and counting. It seems that each supports children's developing understanding of number as a representation of quantity. The proposed model was slightly modified to reflect what the data was showing. The data also showed that some children have less informal mathematics understandings than others, suggesting that they could benefit from a curriculum which includes activities designed to help them develop these understandings.
The teaching of probability in schools provides a good opportunity for examining how a new topic is integrated into a school curriculum. Furthermore, because probabilistic thinking is quite different from the deterministic thinking traditionally found in mathematics classrooms, such an examination is particularly able to highlight significant forces operating within educational practice. After six chapters which describe relevant aspects of the philosophical, cultural, and intellectual environment within which probability has been taught, a 'BroadSpectrum Ecological Model' is developed to examine the forces which operate on a school system. The Model sees school systems and their various participants as operating according to general ecological principles, where and interprets actions as responses to situations in ways which minimise energy expenditure and maximise chances of survival. The Model posits three principal forces - Physical, Social and Intellectual - as providing an adequate structure. The value of the Model as an interpretative framework is then assessed by examining three separate aspects of the teaching of probability. The first is a general survey of the history of the teaching of the topic from 1959 to 1994, paying particular attention to South Australia, but making some comparisons with other countries and other states of Australia. The second examines in detail attempts which have been made throughout the world to assess the understanding of probabilistic ideas. The third addresses the influence on classroom practice of research into the teaching and learning of probabilistic ideas. In all three situations the Model is shown to be a helpful way of interpreting the data, but to need some refinements. This involves the uniting of the Social and Physical forces, the division of the Intellectual force into Mathematics and Mathematics Education forces, and the addition of Pedagogical and Charismatic forces. A diagrammatic form of the Model is constructed which provides a way of indicating the relative strengths of these forces. The initial form is used throughout the thesis for interpreting the events described. The revised form is then defined and assessed, particularly against alternative explanations of the events described, and also used for drawing some comparisons with medical education. The Model appears to be effective in highlighting uneven forces and in predicting outcomes which are likely to arise from such asymmetries, and this potential predictive power is assessed for one small case study. All Models have limitations, but this one seems to explain far more than the other models used for mathematics curriculum development in Australia which have tended to see our practice as an imitation of that in other countries.
This thesis examines the knowledge of problem solving of two groups of students in Year 5 at primary school. The knowledge of problem solving possessed by a small group of teachers is also examined. The research is related to the Key Competencies description of Solving Problems and is set within a framework of a constructivist view of learning. The research design incorporates two groups of children, one was a class group of students in Year 5 and the other was a group of Year 5 students chosen from four schools on the grounds that they were judged to be academically gifted. Five teachers who taught the students in the Regular group completed a questionnaire about their views of problem solving and the differences between successful and unsuccessful problem solvers. A major concern in this thesis was the adequacy of the Mayer Report's representation of Solving Problems. A second major area of concern was to assess what teachers and students know about the Key Competency of Solving Problems. The Mayer Report states that the Key Competencies are not bodies of knowledge. The responses of students and teachers indicate that they have formed views of problem solving as a body of knowledge. The descriptions of problem solving given by the Regular students and Gifted students were compared with their behaviour on a problem solving task. The explicit views of the Regular students did not match their problem solving behaviour. The Gifted students were more able to describe their problem solving behaviour and their descriptions matched their behaviour on a problem solving task to a greater extent than was the case for the regular students. They showed an explicit awareness of their problem solving behaviour. The Gifted students indicated that they had a better command of problem solving terminology than the Regular students. The author argues that explicit discussion of the processes of problem solving could improve students' knowledge and use of problem solving terms, as well as assisting teachers to discuss problem solving with students. The findings have implications for teaching and learning. The assumptions made by teachers during teaching have implications for teaching actions and for student learning. There were discrepancies in the way teachers and Regular students viewed problem solving as a general or specific construct. The Regular and Gifted students regarded problem solving as specific to curriculum areas while the group of teachers indicated that they viewed problem solving as a general construct. The discrepancy in terminology used by the Regular students and their teachers suggested that there is a need for an explicit framework with common terminology to be used by teachers and students in discussing classroom work on problem solving. It is argued that the explicit awareness of problem solving knowledge of the Gifted students contributed to a positive motivation for problem solving and an effort orientation in their problem solving. The findings support a view of problem solving as a body of knowledge that can be taught in each curriculum area rather than being a set of simple cross curricular skills as proposed in the Key Competencies Report.
This thesis is about gender and the culture of selected computer based secondary mathematics classrooms in Victoria. The behaviour, engagement, and attitudes of boys and girls, and for high and low achievers, were explored. A year 10 class participated in a pilot study, and two classes, a year 8 and a year 9 mathematics class in which computers were regularly used, were selected for the main study. Mathematics lessons were video taped, documents collected, and two girls, two boys and the teacher from each class interviewed. Data were analysed qualitatively and case studies for each classroom presented. A questionnaire, distributed to all students in the main study, was analysed quantitatively and compared with case study findings. Whilst each classroom was culturally different, the pedagogy, learning relationships, power relationships, participation and role of computers advantaged, in general, the learning of boys and high achievers. Boys were more likely than girls to compete, take control of their learning, share their knowledge of computers, view computers as a male domain, and be perceived as computer experts. For boys, computers provided pleasure, relevance and/or success in mathematics. Girls' expertise was denied visibility, yet they were successful peer tutors. They were more likely to be concerned about whether computers aided learning and enabled success in mathematics. Within gender differences were observed. High achieving boys displayed dominant behaviour; low achieving boys strove to identify with the computer experts. High achieving girls persisted as 'outsiders within'; other girls 'backed off'; exceptional girls challenged gender stereotypes. Teachers' methods and beliefs, student ownership of laptop computers and gender differences in students' experience with computers, contributed to these classrooms being viewed less favourably by girls. Research into students' perception of computer based mathematics as a male domain, and computer based teaching methods that engage low achieving students, is needed.
The present project investigated tertiary mathematics students' perceptions of their learning environment, and whether such perceptions were linked with achievement. Perceptions were evaluated using the University Mathematics Learning Environment Questionnaire, developed as part of the research, and including forms directed to the actual learning environment and the preferred one. The questionnaire showed satisfactory reliability, and scales formed by factor analysis of the items discriminated between groups whose environment showed observable differences. Discrepancies between actual and preferred environment scores were significant, in a direction indicating agreement between students' preferences and perceptions hypothesised as favourable. Multiple regression showed strong correlation between learning environment questionnaire scores and students' achievement. An additional independent achievement comparison was made, between two enrolment groups that had observable differences in their learning environments, as well as significantly different learning environment scores. Results found significantly higher achievement in the group that showed more favourable perceptions of their learning environment.
This is the report of research into attitudes of students in transition from primary to secondary school. The focus of the report was to ascertain the attitudes of students towards learning and enjoyment of learning mathematics in upper primal)/ and lower secondary school. Surveys were conducted with the students in upper primary and the same surveys were administered to the same students in lower secondary school. The research was intended to compare the attitudes of these students towards mathematics and learning. The research was undertaken to evaluate students' perceptions of external and internal influences on their ability to learn and enjoy mathematics. The students were asked to answer questions in relation to their attitudes to mathematics and the perceived influences of parents, teachers and peers on their ability to learn and enjoy mathematics. The data collection procedure involved two surveys. The results were analysed by way of interpreting box plots, linear graphs and tables. The results of the surveys indicated that students in both upper primary and lower secondary had positive attitudes towards mathematics and a desire to succeed in the subject. The students at both levels also recognised the importance of mathematics in terms of obtaining employment and achieving entry into tertiary education. The comparison between the responses to student attitudes and perceived influences of parents, teachers and peers towards mathematics from upper primary and lower secondary school revealed that the students at lower secondary were more positive towards mathematics than in upper primary. This result was true for every category in the research. This was an unexpected result. These findings provided valuable insights into the importance of developing both primary and secondary mathematics curricula. The results also highlighted the need for teachers to be more aware of attitudes of students in relation to mathematics education.
Over the past two decades mathematics education research has drawn attention to the inadequacies of traditional transmission approaches to teaching and learning and has thereby presented an alternative theory. The essence of this view is that mathematical understanding in dependent on deep conceptual development. This type of intellectual growth is possible within environments where interaction is the underlying component and where the social functioning allows learners to actively construct mathematical knowledge. The most outstanding characteristic of this view, referred to as 'social constructivism', is the essential role of language throughout the learning process. The traditional school routine of memorising fixed procedures prescribed by teachers has resulted in superficial development within many areas of the mathematics curriculum. One mathematical domain, which has consistently caused intellectual dilemma for teachers and learners alike, is that of fraction ideas. It has been proposed that these ideas are particularly difficult for children to develop as their occurrence and utilisation is far less common by comparison to whole numbers. This investigation explored whether the difficulties experienced during the development of common fraction and percent ideas at the year six and seven levels could be both highlighted and overcome within a classroom adoptive of a social constructivist philosophy. The study was conducted with two classes, one at the year 6 and 7 levels and the other at the year 7 level only. All programs began with a pre-test based on prior common fraction and percent instruction in order to determine appropriate starting points. The social constructivist environments in which the programs were implemented were created through the utilisation of specifically designed mathematical instruction games. This activity resulted in highly intellectual mathematical interaction which in turn led to the formation of deep, flexible conceptions of the ideas considered.
In 1997 the graphics calculator, a digital hand-held device with the ability to draw graphs, and perform other important calculations, was permitted by the Victorian Board of Studies into external mathematics examinations in Victoria. Through case studies of two secondary school mathematics departments within Melbourne, the research aims to identify the areas of mathematics in which graphics calculators are being used two years later, and gain an understanding of the influences and obstacles teachers face in adapting to this technology. This research reveals that teachers in these schools use graphics calculators from years 8 to 12 for topics including graphical work, statistics, equation solving and calculus. Very little calculator programming takes place. Despite the fact that the teachers participated in professional development, obstacles on technical difficulties, management problems, departmental culture, and personal reluctance to use this device still persist.
The main advantages and disadvantages of the Telematics environment for talented mathematics students were investigated through a case study. The case study considered the interaction of, and opinions of 11 Year 9 students and the teacher/researcher. Participants were from nine schools in regional Western Australia, and were withdrawn from face-to- face classes to attend mathematics transmissions. Qualitative data were collected through student interviews, an anonymous questionnaire, tape recording of lessons, and teacher field notes. Students all agreed the main disadvantage occurred if timetabling for Telematics transmissions did not align with their local school class times for the same subject. The teacher perceived the main disadvantage was that during lessons, for various reasons, students chose not to contribute, making it difficult to gain responses from students and create productive class discussions. Many students felt intimidated to contribute during class discussions. Allowing time during lessons for social interaction and encouraging students to reply directly to each other's contributions led to many students feeling more at ease to talk during lessons. The researcher perceived the greatest advantage of learning through Telematics was that with the small classes she was able to work more on an individual basis with students and to check that students understood individual concepts. Towards the end of the data collection period, some students were offering comments on other students' work and their thoughts on the material being developed, without prompting. The research concluded that for a particular issue, student responses were often in opposition. Consequently although some students found one aspect of his/her learning environment a distinct advantage, others found this a disadvantage; and for others, the same issue was unimportant. As such, it is not possible to categorise most aspects of the Telematics learning environment as either an advantage or disadvantage. Therefore rather than the teacher concentrating on eliminating or utilising certain aspects of the Telematics environment, the teacher needs to consider what individual students perceive as advantages and disadvantages and cater for those individual needs.
Recent research has shown that integrated worked examples can reduce cognitive load and facilitate learning. The present research investigates these laboratory findings by replication and extension in a classroom setting. Cognitive load theory suggests that many conventional methods of instruction in mathematics are not effective because they deploy cognitive resources away from activities relevant to learning. Solving equations in algebra is a complex problem- solving task which imposes a heavy cognitive load. It is suggested that the conventional, multiple practice method in a classroom situation is one such method of ineffective instruction. The focus of the present research was based on the theoretical framework that students ability to solve algebra equations would be enhanced if cognitive load is reduced and a cognitive understanding approach is adopted as the dominant method of learning. Students exposed to this model of instruction were given worked - example models to study. Instructional emphasis was on understanding the process and the steps to solution, to facilitate cognitive understanding, rather than on the completion of large numbers of problems as multiple practice. All experiments were based on in class experience over a normal timetabled week. This involved the application of homework based on the instructional approach used in each class. In this way experiments were not simulated laboratory tests but were as close as possible to normal classroom environments. It is suggested, from these findings, that using a cognitive understanding approach can enhance the skills involved in solving some algebra problems. These findings reinforced previous laboratory based research and emphasise the importance of the theory of cognitive load, in classroom situations. Research results suggest that there is a need for changes in actual classroom practice in some aspects of the mathematics curriculum. Further research is needed to investigate issues raised as an outcome of the research encompassed in the present thesis.
The Third International Mathematics and Science Study (TIMSS), the largest comparative study of its kind, was administered to approximately 500,000 students worldwide. In Australia, the results of this study are being used to compare our students and schools to other students and schools around the world. The results may also influence decisions about curriculum reform and allocation of educational funding within Australia. This thesis sets out to investigate the TIMSS test items for Population 2, with the objective of determining the degree of validity of these test items to Australian mathematics teachers and their students. By eliciting feedback from a sample of Australian mathematics teachers, their thoughts on the validity of the TIMSS test items were documented. This was achieved through a mail out questionnaire that included a representative sample of 32 TIMSS test items from population 2. The results from the questionnaire were used to establish the overall validity of the TIMSS test items to Australian Mathematics teachers and the students they teach. In total, 154 teachers, representing Government, Catholic and Independent schools, from around Australia replied to the questionnaire. The study found widespread variability in the type and amount of content taught by teachers to their Australian students. Consequently, differences in content validity of the TIMSS study were found to exist across Australia. These differences appeared to be more apparent between states and territories than between school sectors. Respondents also expressed concern about the general appearance and layout of the TIMSS test items. In particular, some of the language used in test items relating to Proportionality, appeared not to be used in Australian classrooms. In addition to this, teachers reported that the TIMSS test items were not particularly useful for ascertaining student competence. This casts doubt over the value of any inferences made from the results of TIMSS. Furthermore, this research found significant variability in student familiarity with the item formats used in the TIMSS study. Overall, students were found to be most familiar with the short answer format and least familiar with extended response and performance assessment formats. This is a particularly important result as the TIMSS designers placed great emphasis on the use of extended response and performance assessment formats.
This study investigated lower secondary mathematics classroom learning environments in Brunei Darussalam and their associations with students' satisfaction with learning mathematics among a large sample of 1565 students from 81 classes in 15 government secondary schools. This sample size was large enough to permit meaningful analyses at the class level. Students' perceptions of the classroom learning environments were assessed with a version of the My Class Inventory (MCI) that had been modified for the Brunei context. A measure of student satisfaction also was included in the study to permit investigation of satisfaction- environment associations. Three types of scale validity analyses were conducted with student responses to the MCI. First, principal components factor analysis with varimax rotation was used to check whether the a priori factor structure held up with mathematics classes in Brunei Darussalam. Second, the internal consistency reliability of each scale was estimated using Cronbach's alpha coefficient for two separate units of analysis: the individual and class mean. Third, in order to test the ability of each scale to differentiate between the perceptions of students in different classes, a one-way ANOVA was conducted with class membership as the main effect. The exploratory nature of the present study lent itself to describing the nature of a typical lower secondary mathematics classroom in Brunei Darussalam. Mean scores were generated from the data to provide a profile of what a typical classroom is like. Sex differences in classroom environment were explored using a one-way MANOVA with the set of MCI scales as the dependent variables. The unit of analysis was the within-class gender mean obtained by calculating the male mean and the female mean for each class. Associations between student satisfaction and classroom environment were investigated using simple and multiple correlation analyses. Regression coefficients were used to identify which MCI scales were related to student satisfaction when the other MCI scales were mutually controlled. Both the individual student and class mean were used as units of analysis. The study revealed a satisfactory factor analysis for a refined three-scale version of the MCI assessing cohesiveness, difficulty and competition. This finding is noteworthy because the factorial validity of the MCI has not previously been established in past research in other countries. Each scale displayed satisfactory internal consistency reliability and discriminant validity and was able to differentiate well between the perceptions of students in different classes. Overall, the study suggested that students' perceived positive learning environment in mathematics classes. Associations between satisfaction and the learning environment were statistically significant both at student and class levels for most MCI scales. Also the study supported earlier studies suggesting that boys and girls hold different perceptions of the same classroom learning environments.
This research study used action research to investigate changes in a teacher’s practice which occurred with the implementation of teaching strategies aimed at increasing students’ autonomy. The study involved students in the Year 12 mathematics classes at a Catholic coeducational college in a Queensland provincial city. An initial study over one term was carried out in 1997. This informed the major study of 1998. The changed teaching approach emphasised collaborative learning, the use of self-regulatory learning strategies, making sense of rather than memorising mathematical concepts, and reflecting on learning behaviours and on understanding of concepts. The students were encouraged to be actively engaged in their learning and to accept responsibility for it. Action research allowed the students to contribute to the teacher’s planning and evaluation of the teaching and learning in each cycle. Data were collected from students’ journals, interviews, classroom observations, questionnaires, and the teacher’s fieldnotes. Narrative analysis centred on three themes: issues affecting the students, issues affecting the teacher, and issues associated with carrying out research into practice.
The students perceived themselves as being more autonomous than they had been previously. Nearly all accepted the responsibility for their own learning, and were able to work together to improve the learning of all class members. They also believed that the changed approach had contributed to improved results. The teacher was able to act as a facilitator of the students’ learning, rather than feeling responsible for ensuring that all learnt. Action research proved a suitable methodology for a full-time teacher seeking to improve her practice.
Students’ responses to the study support the need to explain the purposes of changes to the ways in which they are taught and expected to learn. It proved important to allow the students time to adapt to the changes, and to allow them to have input into decisions about their learning. They needed time to reflect on their learning for them to become aware of what they were doing and whether it was proving effective. The teacher realised the need to be flexible in her teaching approach, and to provide sufficient scaffolding for the students as not all students wished to become more autonomous learners.
The purpose of this study was twofold: to investigate factors which influence the pedagogical practices and beliefs of beginning teachers of mathematics in primary classrooms, and secondly to evaluate a professional development support model for beginning primary mathematics teachers. A model for professional development and support of beginning teachers of primary mathematics was designed from critical characteristics of effective teacher support obtained from the literature. The model was designed via the use of a 'fellow worker' to help beginning teachers implement constructivist ways of teaching mathematics in their classrooms. It was implemented over a period of the first year of teaching for the five participants. The research took the form of an interpretative, qualitative study. The main methods of data collection were interviews, observation, journals, case methods meetings, repertory grids, RADIATE categories, questionnaires and characterisation scales. Data were analysed using techniques of qualitative analysis and incorporated the use of the NUD*IST computer program. Findings suggest that the professional development, support model was successful in helping beginning teachers implement and sustain a more constructivist philosophy in mathematics teaching. It appeared to provide an effective framework to meet the individual needs of teachers within specific contexts. It was an effective alternative to the isolation and 'sink-or-swim' attitude of the first year of teaching felt by the participants. Beginning teachers used reflection in their teaching and generally began to implement less teacher- directed, traditional methods of teaching after emerging from a foreshortened 'survival' period. The major influences acting on pedagogical practices were the children in the classrooms of the beginning teachers. Other factors such as limited pedagogical knowledge, traditional ways of behaving as a teacher, beliefs about mathematics, mathematics teaching and learning, and time also influenced classroom practice in primary mathematics. The major finding of the research is that, with personal and context-specific support, beginning teachers can start to implement pedagogical practices in primary mathematics consistent with recent recommendations. This thesis recommends that the support must come from both the general system and the school levels and must address the needs of the individual teachers rather than mass induction methods. Distinction and separation must be made between beginning teacher support and the assessment of the beginning teacher's competence.
This thesis is based on research conducted to investigate the effects of computer programming on cognitive and affective outcomes in two upper primary classes. The aims of the research were to establish whether a particular type of methodological intervention, which reinforced strategies developed in a programming context, could improve the likelihood that problem- solving strategies acquired through programming would be transferred to mathematical problem-solving. In addition, the research set out to investigate whether programming affected individuals with differing personality traits in different ways. Students worked over a twelve-month period with the programming software MicroWorlds. In the first term they learned basic semantics and syntax of the programming language and thereafter completed five tasks that were research assignments from a range of key learning areas. These tasks were called Lap-T tasks and as part of the overall curriculum were completed and presented for evaluation. Pre and post- tests in mathematical problem solving which sought evidence of ability to obtain correct answers, identify appropriate strategies and articulate strategies used, were administered at the beginning and end of the year. In addition, students completed a questionnaire at the beginning, middle and end of the year to establish attitude change to aspects of learning with computers and learning through programming. Other data was obtained through the Rosenberg Self Esteem Test and the Eysenck Personality Inventory. Students kept journals in which they reflected on their programming experience. A series of class lessons for reinforcing strategies developed in programming, called strategy training was directed towards one group only, the Strategy Training Group (STG) for a total of 18 hours. The other group, the Independent Learning Group, was not assisted to make connections beyond those automatically acquired while programming. The findings of the research were an affirmation for the value of the strategy training for improving the likelihood that strategies acquired during programming would transfer to other problem-solving contexts. In addition, there was confirmation that programming is a valuable addition to an upper primary curriculum, contributing to students' perceptions of control of their own learning and providing challenge and satisfaction while developing transferable problem-solving skills. The research indicates that strategic use of programming in the primary school curriculum can be utilised in the development of mathematical problem-solving skills.
The thesis reports upon an investigation into the specific action theories of upper primary teachers of mathematics towards common classroom teaching behaviours. The initial phase of the study described and defined six common upper primary mathematics classroom teaching behaviours through the use of descriptive scenarios. These classroom teaching behaviours were: the use of a prepared stencil; the use of group work; the use of a textbook for homework; the student use of calculators; the use of the standard multiplication algorithm; and, the use of a pen and paper short answer test. Using a qualitative approach (phenomenography) the research identified the range of conceptions and normative beliefs held by the teachers towards each of six specific classroom behaviours. The results were able to confirm and expand the data uncovered by the review of literature. The Theory of Planned Behaviour (TPB) was then used to analyse the conceptions that arose from three of the specific classroom behaviours. The theory provided the framework for the teacher action theories for each behaviour. Teacher action theories were described and examined for the classroom behaviours: use of stencils; use of group work; and student use of calculators. As teacher action theories were postulated to be the basis for the performance of common specific classroom behaviours, teachers were categorised according to their intentions to perform as well as their performance for each specific behaviour. The design of the study emerged from the review of literature associated with intention, the predictors of intention (beliefs, attitudes, subjective norm, perceived behavioural control) and the classroom behaviours. The research uses approaches from scientific and postmodern traditions. Together with the methodological considerations, a total of 9 research hypotheses and 10 research questions were postulated, examined, discussed and reported. The thesis concludes with a consideration of the implications of the findings as they relate to values research and to preservice and inservice teacher education, and the development of more effective reflective practice in schooling.
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