Theses abstracts
In this study the relationship between numeracy and language-based errors when solving mathematical word problems was investigated at the Year 7 level in one school. Participants were students from a large, single campus, coeducational secondary college in the south-eastern suburbs of Melbourne, Victoria. The school runs a combined numeracy and literacy intervention program at year 7 and uses a multi-step selection process to determine students who require intervention, including the use of separate numeracy and literacy test instruments. The aims of the study were twofold: to determine if the two testing instruments used by the school, PATMaths Revised for numeracy and TORCH for literacy, were selecting the appropriate students for their combined numeracy and literacy intervention program and to compare the proportion of language- based errors that occurred when students attempted mathematical word problems in this study with those reported by Newman's 1977 study. A sample of 84 students from a cohort of 258 year 7 students consented to participate in the study and data from the school's numeracy and literacy testing for all 258 year 7 students in the year of the study, 2005, were also used. The 84 consenting students were tested using the Newman Language of Mathematics Kit to determine which students were 'at risk' and then these 'at risk' students were interviewed using the diagnostic interview process as outlined in the Kit to determine the proportion of language-based errors they made. The findings show that the test instruments used by the school to select students for their intervention program are useful as a first stage in the selection procedures used by the school and that the test used in the Newman Language of Mathematics Kit is not an accurate substitution for the school's testing instruments. The findings also indicate that the types of questions on tests may have an impact on student's results. When comparing the test in the Kit with the PATMaths Revised test it was found that the PATMaths Revised test had a wider variety of categories (number, space, measurement, chance and data) than the Newman test (number, spatial, logic). In relation to language-based errors it was found that students are still making a significant proportion of language-based errors when attempting mathematical word problems when compared with those reported by Newman. [Author abstract]
The nature and function of Task Complexity, in the context of senior secondary mathematics, has been identified through: a search of the research literature; interviews with experts that focused on the nature of task complexity; expert use of the Williams/ Clarke Framework of Complexity (1997) as a tool to categorise the complexity of a task, and observation and analysis of the responses of senior secondary mathematics students as they worked in collaborative groups to solve an unfamiliar challenging problem. Although frequently used in the literature to describe tasks, ‘complexity’ has often lacked definition. Expert opinion about the nature of mathematical complexity was ascertained by seeking the opinions of experts in the areas of mathematics, mathematics education, and gifted education. Expert opinion about task complexity was stimulated by questions about the relative complexity of two tasks. The experts then categorised the complexities within each of these tasks using the Williams / Clarke Framework of Complexity. This framework identifies the dimensions of task complexity and was found by experts to be both useful and adequate for this purpose. A theoretical framework was developed to assess student ability to solve challenging problems. This theoretical framework was used to design a test to assess student ability to solve challenging problems. The information this test provided about the problem solving ability of the students in this study informed my analysis of student response to complexity. Case studies of two collaborative groups of final year secondary mathematics students were undertaken and these studies indicated the construct of a Discovered Complexity was a useful tool to analyse student response to complexity. This construct was formulated after preliminary observation of the video data. The task explored by the students was found to contain many potential complexities to discover but the two collaborative groups differed in the number and nature of complexities discovered. The discovery of complexities was found to add a dynamic element to the task as each new complexity altered the students’ perception of the task. The discovery of complexities was found to be associated with increased student engagement with the task and increased conceptual development. The interrelationships between Task Complexity, student engagement and conceptual development suggested by the findings in this study have been explained using a schematic representation I have named ‘Engaged to Learn’. This representation relates the concept of Flow and the concept of the Zone of Proximal Development to the concept of a Discovered Complexity thus relating the cognitive and affective aspects of learning.
Despite the vast research on the effects of acceleration programs on student achievement there is little quantitative confirmation of the benefits of these programs and there is no research that investigates the effects of acceleration on students' VCE Mathematics study scores. This research attempts to fill this gap by considering four years of data provided by the Victorian Curriculum and Assessment Authority (VCAA) relating to achievement in mathematics. Acceleration in this study means the completion of the Year 12 Mathematical Methods study during Year 11. The data constitutes experimental data for content acceleration and the results of students from schools without such acceleration programs provide the corresponding control data. However, the acceleration decision is not taken randomly by schools, so this data is only quasi-experimental in nature. The measures of mathematical achievement (Mathematical Methods and Specialist Mathematics study scores) are carefully audited, and are accepted as reliable and valid by the Victorian education system. Controlling for individual characteristics such as gender and prior knowledge, and allowing for moderation effects due to school sector (Government, Catholic and Independent) and school class setting (single-sex or coeducational), the effects of content acceleration are measured using multi-level modelling. This study examines the effects of acceleration on the VCE Mathematics study scores of students who completed both Mathematical Methods (Units 3&4) and Specialist Mathematics (Units 3&4) in Victoria, over a four-year period (2001-2004). On average this involved 5341 students from 341 schools in each year with 829 students included in a content accelerated program. The results suggest that content acceleration is beneficial, especially for students with higher prior knowledge scores. The quasi-experimental nature of the data means that a causal relationship between acceleration and students' mathematical performance can be claimed. In particular, this study showed that the effect of acceleration on students' Mathematical Methods (the Year 12 study taken in Year 11 by accelerated students) study score was not significant. However, the effect of acceleration on students' Specialist Mathematics study scores was significant. Accelerated students performed, on average, 2.7 points higher (on a 50 point scale) than equal ability age-peers who were not accelerated. Interestingly, for accelerated students who scored in the top 2 per cent for their General Achievement Test, in the mathematics, science and technology component, their Specialist Mathematics study scores were on average, almost 5 points higher (on a 50 point scale) than their equal ability age-peers. The statistical control of other factors means that these results can also be generalised to other states, other countries and, probably, to other subjects.
This study examines teacher and students' perceptions of teacher-student interpersonal behaviour in regional secondary classrooms. Teachers were then presented with this information so that a comparison between teacher ideal, teacher actual and student perceptions of their classroom could be examined. The results were then able to be used by teachers to reflect on and seek to improve their teaching practice. This study utilised the Questionnaire on Teacher Interaction (QTI) to collect data about the classroom learning environment of Australian regional mathematics and science classrooms. Qualitative information in the form of classroom observations and informal interviews has also been collected from a small subset of the student sample. This qualitative information was collected by the researcher in the dual roles of teacher and researcher. Triangulation of the methods of data collection sought to better validate the data collected, and assess multiple perspectives in the classroom. The study has involved a large sample of students from one country high school in Western Australia. All the mathematics classes from Years 8, 9 and 10 and all science classes from Years 8 and 10 have been included in this study. A particular focus for this study was the inclusion of both streamed and non-streamed classes from the mathematics and science areas. The value of this research has been enhanced in that the results have been used as a teaching feedback tool for participants involved in the study to examine, reflect and improve on their teaching practice. The research is a real world, authentic example of one instance where results from the study were used immediately on a local scale by participants. A unique feature of the outcomes from this project is that the teacher appears to play a greater role in determining the classroom climate than does the homogeneous or heterogenous grouping of students within a subject.
This thesis explores factors associated with the development of early learning difficulties in mathematics from two perspectives: issues relating to mathematics development and early mathematics education, and models of memory and information processing. A principal aim of the research was to better understand the puzzling question of why a significant number of students with otherwise good reasoning capacity are unable to develop efficient, effective strategies for solving basic number combinations. The very poor mathematical understanding and skills of students who have been described as mathematically disabled (MD) or as having an arithmetic learning difficulty (ALD) or mathematical learning difficulty (MD) have been described in the research literature over a period of thirty years. The profile of the characteristics of MD students is well established, having been described with a consistent prevalence of 6-8 percent students in several countries. In spite of having average conceptual understanding in some areas of mathematics, these students face an outstanding difficulty in mastering basic arithmetic facts, and continue to use very slow and ineffective strategies for solving even basic arithmetic problems, usually reliant on counting-by-ones finger strategies. More recent accounts of MD students also emphasize their poorly developed number sense and estimation skills. The development of effective interventions for mathematical learning difficulties is complicated by the fact that there are no distinctive diagnostic measures available for identifying MD. Furthermore, there are now strong indications that there are at least two distinctive forms of MD, with different instructional needs. Accordingly, one goal of this thesis was to identify significant predictors of early mathematical learning difficulties, as a guide to identification and intervention. Based on previous research exploring factors associated with MD, a comprehensive range of assessment measures was administered to 68 children in three Year 2 classes in metropolitan Brisbane, Queensland. The measures included detailed assessments of mathematical skills and understanding, and a range of memory and processing tests, to capture underlying factors constraining early mathematical learning. Children at risk of early mathematical difficulties were identified by a statewide process of diagnosis in Queensland administered in the second year of formal schooling, known as the Year 2 Diagnostic Net (Numeracy). In view of the high rate of comorbidity of mathematical and literacy learning difficulties, data were also collected about which students failed the Year 2 Diagnostic Net (Literacy). For the same reason, the children's speed of naming symbols, letters and words was measured at the same time as their fluency in identifying numbers. To understand better the nature of their learning difficulties, diagnostic teaching sessions were undertaken with ten of the students caught in the Year 2 Net (Numeracy), whom their teachers perceived as most in need of intensive learning support. In addition, an assessment and intervention were conducted with a Year 4 student at one of the study schools, who was facing significant mathematical and literacy learning difficulties. The study showed that forwards counting fluency and stage of strategy development (SEAL) on the Learning Framework in Number in the beginning of Year 2 were significant predictors of being caught in the Year 2 Net (Numeracy) at mid-year. Net status was also predicted by a student's performance on the Make 10 test, a novel task designed by the researcher to assess fluency in retrieval of Ten Fact combinations. As well as the level of strategy use (SEAL), a student's level of identification of written numerals was a further significant predictor of performance at the end of Year 2 on I Can Do Maths, a standardized test of early knowledge of Number, Measurement and Space concepts. The importance of identification and intervention for students at risk of MD was indicated by the increasing gap in performance between Net and normally achieving students on I Can Do Maths by the end of Year 2. The speed of processing of particular numerical skills were shown to be important indicators of early mathematical learning. Net students were significantly slower than their normally achieving peers in identifying 2-digit numbers on the CAAS 2-Digit task, and were slower on all the PAL rapid automatised naming RAN tasks - RAN Digits, Letters, Words, Words & Digits. Fluency in rapidly naming and switching between 2-digit numbers and familiar words, as measured on the RAN Words & Digits task was also shown to be a significant predictor of performance on the One Minute Basic Number Facts Test of Addition. The level of Highest Forward Digit Span and poor performance on a working memory Counting Span task significantly discriminated between Net and normally achieving students. Information processing models and models of working memory provide useful explanatory frameworks to account for particular error patterns, and for the characteristic failure of MD students to learn basic arithmetic facts. Insights from the mathematics development and mathematics education literature add to our ability to understand the early impasse in learning of students at risk of MD. Work samples from diagnostic teaching sessions with five of the Intensive Net students further illustrate the influence of a low stage of counting fluency and strategy use on a student's self-perception and ability to access the Year 2 mathematics curriculum. Processing strengths and weaknesses are discussed in relation to possible effects on the students' mathematical learning. The detailed case study of the Year 4 student demonstrates how a research-based intervention, which took into account cognitive factors, was successful in enabling learning of arithmetic facts and markedly changing the motivation and confidence of a student who was formerly very resistant to mathematics instruction. In particular, increased counting fluency and automatisation of addition facts appeared to free working memory resources for monitoring and self-correction by the student, and to facilitate a positive interest in exploring mathematical relationships. Based on the significant predictors of poor achievement in early mathematics, assessment measures are proposed which are critical for the identification of early mathematical learning difficulties. The thesis concludes with a number of recommendations for intervention, with accompanying requirements for pre-service and in-service training to assist effective implementation. Future directions for research in the identification and intervention of MD are proposed.
In recent times, technology is being used more and more in a variety of educational endeavours. This thesis considers the use of technology in the improvement of teaching and learning mathematics in higher education. In particular, it addresses, through three case studies, specific technological solutions to two well documented educational problems: the knowledge gap between secondary school and university level mathematics and how to provide higher level, honours or postgraduate, mathematics offering despite low student numbers. A literature review provides the context of the problem and an overview of previous attempts at solving this difficulty. The first case study addresses a common problem concerning students entering university from high school. It has been reported that students who enter university have insufficient mathematics knowledge and skills to take first year mathematics subjects and hence provide a sound mathematical foundation for other subjects. In this study at the University of Wollongong, video learning resources, predominantly worked solutions, have been developed to assist Engineering students in a first year mathematics subject. The four-stage evaluation model of Alexander and Hedberg (1994) was used to determine the effectiveness of the resources involved at each of the stages: Design, Development, Implementation and Institutionalisation. Evaluation addresses both the production methods and the learning outcomes. A mixed methodology combining surveys, interviews, and document analysis, was used to triangulate evaluation results at each of the four stages. In 2006, two different technologies were used to produce video resources for a limited number of mathematics questions. In the first instance this was to determine the ease of production and the students' preferences for the resources. Following this trial, a set of video resources which covered all the topics taught during the first four weeks of the teaching session was developed. These resources were used to examine whether or not they could be used to bridge the gap from high school to university. Analysis indicated that the resources were used by students whose mean baseline performance was lower than non-users (t 74 = 2.18, p = 0.033). In week four the results of Basic Skills Test 2 revealed that students who used the resources improved more than those who did not (t 72 = 2.43, p = 0.018), however, catching-up on fundamental mathematical skills was insufficient for there to be an impact on the final results in the subject. Surveys in the middle and at the end of the teaching session showed that students found the video resources were useful in helping them to learn and understand mathematics. Consequently, a set of video resources that covered all mathematics topics in the subject was developed for incoming students in 2007. To ascertain the impact of these resources, two cohorts of students were examined: students from 2004 with no resources and students in 2007 with a complete set of resources. Having established that baseline performance was the same, the students with resources were found to have improved their performance in all assessment tasks compared to students without resources: Basic Skills Test 2 (t 390 = 3.14, p = 0.002), assignments (t 456 = 2.80, p = 0.005), quizzes (t 456 = 3.49, p = 0.001), examination marks (t 456 = 3.03, p = 0.003), and final marks (t 446 = 2.38, p = 0.018), except in the Mid-Session Test (t 467 = 0.65, p = 0.519). The failure rate fell significantly in 2007 compared to the years between 2000 and 2006 (Z = 2.10, p < 0.05). Students' surveys suggest that while the primary gains have been in algorithmic learning, students consider they have better understanding of concepts. The final stage of institutionalisation reveals that the university has adopted and further extended the approach for the development of mathematics learning resources across disciplines. The second case study addressed the use of two-way communication technologies for teaching and learning for geographically dispersed students at the tertiary level and in particular, for mathematics. To do this, the author of this thesis worked in an apprenticeship model with the guidance of the Manager of Learning Facilities and Technologies at the University of Wollongong to compare a selection of Real Time Communication (RTC) technologies using several criteria. These were based on the needs of teachers and students, the institutional infrastructure and the literature. As there are a variety of Real Time Communication (RTC) technologies, a two-stage evaluation was adopted. In the first stage, a list of RTC technologies was composed based on criteria found in advertising and promotional materials. In the second stage, each of the short-listed RTC technologies was trialled to determine their effectiveness and efficiency in teaching and learning. At the end of this case study, one of the RTC technologies, which provides multiple video and audio tracks of all participants as well as application sharing such as shared desktop and whiteboard, namely Access Grid, was installed and trialled. With two other universities, also funded by the International Centre of Excellence for Education in Mathematics (ICE-EM), the opportunity arose for the third case study focussing on the question, 'How do universities provide a wide range of subjects to honours and upper level students when numbers of mathematics students and staff are small?'. The aim of installing a room-based node on the Access Grid was to teach and share the mathematics and statistics subjects with other Australian universities which had installed a room-based node. Three lecturers and eight students were interviewed and surveyed to evaluate the use of the Access Grid in teaching and learning these subjects. The lecturers and the students were tolerant of many technical failures, expecting them as part of the process of introducing new technology and recognising the opportunity provided by sharing classes. Two issues were identified: the need to train staff in the use of new pedagogical approaches and the fact that lecturers did not necessarily perceive the communication difficulties experienced by their students. The thesis concludes with a look to the future of technology in mathematics education and makes recommendations for embedding video resources within subjects in the other disciplines. Recommendations are also made for the use of synchronous technology such as Access Grid in teaching and learning.
As the need for educational reform is increasingly recognized, so too is the need for effective professional development. Historically the evaluation of professional development experiences has been limited to exit surveys, noticeably failing to examine the long-term impact of the effort. This study assessed the impact on the classroom learning environment of a yearlong, job-embedded professional development opportunity for middle-school mathematics teachers. The application of learning environment instruments to the evaluation of professional development is a unique feature of this study. The research employed the Questionnaire on Teacher Interactions (QTI) and a modified version of the What Is Happening In this Class? (WIHIC) survey with over 1000 middle-school mathematics students in 57 classrooms in the state of Washington. Both instruments were administered at the beginning and end of the school year. Teacher interviews were conducted with a sample of participants in order to further illuminate the impact of the professional development. Data from the study were examined for changes in the learning environment and to cross-validate the QTI and WIHIC with this specific population. Results indicate that the QTI and WIHIC are valid and reliable with the middle-school population is this study. Statistical analyses of learning environment data indicate that any pretest-posttest changes that were observed are mostly likely too small to be of educational significance. This study contributes to a better general understanding of the impact of this professional development, and its findings could be utilized in the preparation of future professional development opportunities.
Mathematics curriculum and policy documents in Australia and the US now place increased emphasis on problem solving, mathematical reasoning, and communication, and endorse small group work and discussion as a means of helping students to develop their understanding of mathematics. However, many fundamental theoretical issues surrounding the relationship between students’ mathematical thinking and these new forms of classroom interaction remain to be resolved. One issue which deserves continued research attention concerns the role of metacognitive processes in mathematical thinking. While the importance of metacognition is now widely acknowledged, previous research in this area has tended to focus on students working individually, in experimental settings, or on tasks prescribed by the researcher. In addition, few studies have attempted to examine the interactive characteristics of metacognitive activity occurring when students work together in natural classroom settings.
The research reported in this thesis investigated patterns of social interaction associated with metacognitive activity in senior secondary school mathematics classrooms. The study also sought to examine the teacher’s role in creating a classroom culture that supported students’ mathematical thinking and reflected the epistemological values of the discipline. The theoretical framework for the study drew on sociocultural theories of learning, and elaborated on four related versions of the zone of proximal development (ZPD), involving expert scaffolding, peer collaboration, interweaving of everyday and scientific knowledge, and participation in a community of practice.
Five teachers and their Year 11 or 12 mathematics classes contributed to the two year main study and an earlier pilot study which trialed data collection and analysis techniques. Multiple methods were used to gather data on features of classroom interaction and students’ individual thinking. Questionnaires and associated written tasks sought information of students’ beliefs about mathematics, perceptions of classroom practices, and metacognitive knowledge. Regular observation and videotaping of mathematics lessons recorded teacher-student and student-student interactions. Stimulated recall interviews were conducted with teachers and students to seek their interpretations of selected videotaped excerpts, and students’ views about learning mathematics were elicited in individual and whole class interviews, and in reflective writing.
Initial lesson observations and analysis of all students’ questionnaire responses helped to identify one classroom, more so than others, in which students appeared to be developing productive beliefs and metacognitive dispositions. Subsequent findings of the thesis were based on an intensive analysis of learning and social processes in this classroom. Analysis of social interaction patterns and the teacher’s and students’ espoused beliefs revealed assumptions about mathematics learning which were crucial to the teacher’s role in establishing and maintaining a classroom community of mathematical practice. These assumptions concerned the nature of mathematical inquiry as sense-making, and the function of teacher scaffolding and peer discussion in developing mathematical thinking and language.
Students’ metacognitive strategies were investigated through transcripts of interviews and videotaped lesson segments in which they worked on problems and read explanations and examples in mathematical text. Both the metacognitive function and collaborative structure of students’ dialogue were examined, in order to identify social processes which mediated joint monitoring and regulation. In problem solving contexts, students’ metacognitive activity was organised around routine monitoring of progress and recognition of, and response to, various types of impasse. These metacognitive “red flags” were raised by lack of progress, detection of an error, or an anomalous result. Further analysis of both successful and unsuccessful problem solving episodes revealed qualitative and quantitative differences in students’ social interactions that were associated with success and failure. Thus unsuccessful outcomes were characterised by poor metacognitive decisions exacerbated by lack of critical engagement with each other’s thinking. In practice, this was observed when students passively accepted unhelpful or misleading ideas and ignored potentially useful strategies suggested by peers. By contrast, successful problem solving was favoured if students challenged and subsequently discarded unhelpful ideas and actively endorsed fruitful strategies.
The study also investigated processes through which students came to understand mathematical text, and examined connections between different social contexts in which the class interrogated, critiqued, and validated texts created by the teacher and individual students. Here, collaborative metacognitive activity was structured by cycles of comprehension monitoring and jointly constructed explanations.
The findings of this study have both theoretical and practical significance. While there exists already an extensive body of knowledge about metacognitive strategy use in mathematical problem solving, the thesis sheds new light on the social processes of peer collaboration through which students monitor and extend each other’s thinking, conditions under which such interaction leads to successful or unsuccessful problem solving outcomes, and the metacognitive demands of mathematical reading. In reconceptualising metacognition as a social practice, this research also contributes to the growing number of studies applying sociocultural theories of learning to mathematics classrooms. From a practical perspective, the thesis argues that the teacher plays a crucial role in establishing classroom interaction patterns that develop mathematical reasoning, and identifies implications for preservice education of secondary mathematics teachers.
Falle, Judith
In this study, linguistic features were identified that could be aligned with the conceptual growth of students in the context of introductory algebra. The aim was to devise a model that provided explicit, objective evidence to support the subtle, interpretive judgements made by teachers.
Mousley, Judith
In this project, case study methods were used to capture four particular primary teachers’ ideas and actions in relation to the development of mathematical understanding. The teachers were from one public primary school.
Data collection involved videotaping of lessons, pre-and post-lesson interviews with the teachers, and one longer interview with each teacher. The video and audio data were first analysed in the light of literature on the notion of teaching for mathematical understanding. It was found that the teachers use the term “mathematical understanding” without thinking about what the term means for them or their classroom practice. However, they did use some of the approaches that are recommended in relevant research and teacher education literature. The development and uses of mathematical concepts were mediated by the social situations. In the latter part of this report, two pairs of lessons are used to demonstrate how activity, language, and tools affected the lesson outcomes as well as the teachers’ own understandings of mathematics concepts and how to teach them. It is argued that understandings of the term “mathematical understanding”, the notion of “teaching for understanding”, and understandings that shape practices in mathematics classrooms are all situated in time and place, as products of human activity.
Hughes, L.
The aim of the present research was to design and pilot test an instrument that would identify students' perceptions of the situational factors that contribute to mathematics anxiety. To achieve this, three separate yet inter-related research phases were conducted. In Phase I the Modified Revised-Mathematics Anxiety Survey (MR-MANX) was administered to 144 middle school students, whose ages ranged from 12 years to 16 years. From this the mathematics anxiety levels of the participants were ascertained and results revealed that females presented with significantly higher levels of mathematics anxiety than males. The 12 most highly mathematically anxious students were subsequently identified and in Phase II these individuals were interviewed regarding their mathematics experiences within the classroom, with particular reference to what factors they believed contributed to their mathematics anxiety. Findings revealed that the majority of the 12 participants attributed their mathematics anxiety to teacher factors, peer pressure, and prior knowledge of the mathematics concepts being taught. Test situations and mathematical topics such as algebra and problem solving, were also identified as contributing to mathematics anxiety. The results from these interviews together with the findings from a review of the literature on mathematics anxiety were used to develop a new instrument, the Situational Mathematics Anxiety Scale (SMAnS), and the new scale was pilot tested to examine its psychometrics in Phase III. Item affectivity indices were found to range from .27 to .68 and item discrimination indices from 0.25 to .80. The internal consistency of the SMAnS (Cronbach's alpha coefficient) was found to be .90. Overall, the results of the pilot test suggested the SMAnS to be a reliable instrument with the discriminating power to predict mathematics anxiety due to the situational factors.
The research reported in this thesis investigated the effectiveness of utilizing information technology (IT) in the teaching of mathematics in Hong Kong secondary schools from the perspective of serving teachers. The aim was to develop an understanding of the use, impact and possible future directions regarding the IT initiatives of the Hong Kong government. The collective case study method was used to achieve the aim of the research. Data were collected through semi-structured interviews with the teachers and documents from the Education Bureau (EDB) in Hong Kong and the respective schools. The steps of data reduction, data display, and drawing and verifying conclusions were followed for data analysis. The key findings of the study include (i) the use of IT enhanced the teaching process of specific mathematic topics, (ii) current use of IT could not help teachers to better achieve their roles as facilitators, (iii) the effective use of IT tools required a smooth and strategic integration into the traditional mode of teaching, (iv) subject culture and peer's sharing facilitated teachers' decisions on using IT tools, (v) the quality and proximity of IT resources had a collective effect on teachers' decisions on using IT tools, (vi) relevant training on both technical and application skills need to be provided to teachers on a continuous basis, and (vii) a more flexible work schedule was needed for teachers to devote more time and effort on the use of IT. The significance of this research is that it has provided implications for teachers to rethink what they need to effectively execute information technology in education (ITEd) and for policy makers to efficiently allocate resources and devise future policies to fit into practical teaching situations, from which to further promote more effective use of IT in Hong Kong education.
In recent years, teachers' pedagogical content knowledge (PCK) has become the subject of an increasing volume of research. Another body of literature has grown around the subject of reflection, and its effect on practice. Although a link between the two has often been assumed, few have attempted to map this link explicitly. This study investigates the possibility of such a link, and explores possible descriptions of the link. This case study compared two primary (elementary) teachers, investigating both their PCK and their reflection. The teachers were asked to complete a written questionnaire about mathematics teaching, including some questions requiring a response to hypothetical classroom situations. They were then interviewed about the questionnaire. The teachers were observed as they taught, and interviewed about those lessons. The questionnaire and first interview were used to investigate the teachers' PCK, and all interviews were used to investigate the teachers' reflection. One teacher was found to have both rich and well- connected PCK, and a strong tendency to reflect. The second teacher was found to have much weaker PCK than the first, and also demonstrated less reflection. Some examples of the first teacher's reflection were examined, to investigate the possibility that the teacher's reflection had an influence on the development of PCK. PCK was observed to develop during reflection, suggesting that reflection influences the development of PCK.
This paper describes a research study on the use of Computer Algebra Systems at middle secondary level. Four algebraic related topics were taught to Year 9 students at a Melbourne Secondary Catholic College using a symbolic calculator, the TI-89 as a mathematical tool for home and class use. Assessment at the completion of each topic included both a CAS-free and a CAS-permitted test. Tests, interviews and the Mathematics and Technology Attitude Scale questionnaire enabled comparison of student performance and understanding of mathematical skills in basic algebra. The data collected identified positive attitudes towards mathematics and the use of technology in the mathematics classroom through all levels of ability. This research study found that students improved their level of performance in mathematics when Computer Algebra Systems were permitted and did not lose their algebraic skills. It is the author's belief that if Computer Algebra Systems are implemented with solid pedagogical foundations and strategies, students' skills, attitudes, and opinions about mathematics will improve.
This paper examines the challenges of technology 'integration' into teaching by an attempt to incorporate dynamic geometry software into a portion of the Singapore mathematics curriculum. The focus of the study is a 2-lesson instructional plan on 'triangles'' taught to Year 7 students. Some of the problems faced with the design of the actual lessons are discussed.
This paper examines the challenges of technology 'integration' into teaching by an attempt to incorporate dynamic geometry software into a portion of the Singapore mathematics curriculum. The focus of the study is a 2-lesson instructional plan on 'triangles'' taught to Year 7 students. Some of the problems faced with the design of the actual lessons are discussed.
School climate is now considered such an important influence on student outcomes that large scale international comparative studies such as the Trends in International Mathematics and Science Study (TIMSS) have included measures of school climate in their contextual questionnaires. However, while there appears to be agreement that school climate is important, there are problems with much of the research into school climate which call into question the validity of many of the findings. As most of the problems encountered are to do with conceptualisation and the use of perceptual data, this has implications for the use of school climate measures in cross-national studies like TIMSS. In particular, these problems increase the likelihood that measures of school climate will not display measurement invariance. Measurement invariance requires that groups have a similar conceptualisation of a construct. Lack of measurement invariance is likely to affect the interpretation of group differences on measures of that construct. The purpose of this study was to investigate the measurement invariance of the Principals' Perceptions of School Climate (PPSC) scale which was introduced into TIMSS in 2003. Both confirmatory factor analysis (CFA) and item response theory (IRT) methods were used to assess the measurement equivalence of the PPSC across eight countries. These tests of measurement invariance found that not only was measurement invariance lacking but that the unidimensional model did not fit within the individual countries. This finding is discussed in terms of the implications for users of the PPSC and for the construct of school climate in general.
The research reported in this thesis aimed to comprehensively examine 3- to- 6-year-old children's rational number knowledge and to determine whether dynamic measures of performance can provide additional information about their learning efficiency and learning capacity in this domain. Study 1 assessed 96 preschoolers' informal rational number knowledge in a sharing task and a proportional reasoning task. In addition, the preschoolers' ability to represent formal fractional terms (i.e., half and quarter) was assessed. In Study 2, 140 children aged 3 — 6 years individually completed the static (unassisted) informal and formal rational number tasks, followed by a dynamic rational number task in which children received graded assistance until they successfully solved each problem. Static post-tests were completed one-day and one-month later. Study 3 examined whether the patterns of findings in Studies 1 and 2 were still evident when the proportional reasoning task was presented in a judgement task format, as opposed to a production task format. In study 3, 90 children aged 3-6 years completed the judgment task presented in static (unassisted) and dynamic (assisted) assessment formats. Static post-tests were completed one day and one month later. The results from the three studies indicated that young children's rational number reasoning can be significantly influenced by the quantity form, quantity size and fraction amount of the problem. On the proportional reasoning problems, children were more successful with discrete than continuous quantities, with problems comprised of the same quantity size than those comprised of different quantity sizes, and with those comprised of the fraction amount one-half than one- quarter. In addition, within a given quantity form, children's informal knowledge was related to their formal knowledge of fractional terms. The results from the dynamic assessment measures revealed that children aged 3-6 years had different learning profiles with respect to their levels of entering competence, the amount of assistance required to reach successful levels, and the amount of gain achieved on the post-tests. While the 6-year-olds did not appear to have fully acquired the concepts under investigation at the static pre-test, the dynamic assessments procedure revealed that almost all of the children had the capacity to acquire these concepts. The educational implications of the findings are discussed.
This study aimed to investigate secondary (grades 2, 3 and 4) students' learning difficulties and errors made in solving linear algebraic equations in Singapore, especially in regard to using the balance model. There were four parts to the data collections and analysis, specially designed for this study. Part one involved 452 students answering 14 relatively easy linear equations in a written test. Part two involved 452 students answering 16 harder linear equations in a written test administered separately from that in part one. Part three involved designing a new questionnaire measuring Attitude and Behaviour to Mathematics using the latest techniques applicable to Rasch measurement (N=452). Eight stem-items on goal setting, tasks and effort were answered in two perspectives (I aim to do this and My actual behaviour is) making an effective item sample of 16 for the questionnaire. Part four involved identifying the ten best students and the ten worst students measured on the Attitude and Behaviour to Mathematics linear scale and then interviewing them to investigate why they answered the 14 easy plus 16 harder questions on linear equations the way that they did. The data were analysed in four parts corresponding to the four data collections. In part one, all 452 written papers were marked and analysed. The errors made by the students were classified into 19 different categories and the students' errors made in the 14 easy equations were listed under these categories. In part two, all 452 written papers were marked and analysed. The errors made in the 16 harder equations were listed under these same 19 categories used in part one. In part three, the data were analysed with a Rasch computer program (world's best practice in measurement in the human sciences) to create a linear scale which was shown to be reliable. The Person Separation Index (akin to a Cronbach Alpha) was 0.91 and high. The item-trait chi-square (111, df=96, p=0.14) showed that a unidimensional trait was measured and 15 out of the 16 items fitted the measurement model with a probability better than p=0.08. Thus valid inferences could be made from the scale. In part four, the best ten students and the worst ten students on this linear scale were interviewed to investigate student thinking about how and why they answered the written questions the way that they did, and to gain their views on what they remembered about how they were taught to solve linear equations. The main results are listed below. (1) Many students didn't understand the common balance model for solving linear equations and, instead of adding or subtracting items from both sides of an equation, they just 'moved' the terms from the left hand side to the right hand side of the equation (or vice versa), thus leading to errors. (2) Some students didn't understand negative numbers very well and thus they made errors in transposing (or not transposing) the negative sign when dividing or multiplying two sides of an equation. This occurred in simple equations and in more complex equations involving brackets. (3) Some students were weak in arithmetic and made errors of an arithmetical nature causing them to make algebraic errors too. (4) Many students have difficulties in understanding equations involving terms in brackets and they cannot correctly apply the distributive law to these equations, and so they make errors. The main errors occurred when multiplying the constant to the second term in the bracket as compared to the first term. This occurred more often when there was a negative sign between the brackets. (5) Many students do not use the simplest method to solve linear equations and they often apply a harder method and this leads them to make errors. (6) It seems that some teachers tend not to emphasize the importance of the balance model (according to the students) and the teachers do not tend to use suitable teaching aids for teaching equation solving (according to the students). As a result of this finding, the researcher designed a computer program to teach the balance model for solving linear equations. She used this program to teach some of the weaker students during the interviews and found that this helped the weaker students considerably. (7) There seems to be a lack of interaction between the students and teachers partly because class sizes are too large, partly because there is too much syllabus content to be taught in the school time available, and partly because there is insufficient one-on-one teaching when students have difficulties. (8) Students with low attitudes and behaviour tended to make errors mainly because of a poor understanding of negative numbers, a poor understanding of what an equation means, weaknesses in arithmetic, and a poor understanding of the distributive law. (9) Some students made errors through carelessness, a poor motivation for studying Mathematics (and equation solving) or through laziness. (10) Attitudes were linked to and easier than behaviours for all eight stem-items relating to goal setting, tasks and effort in the linear measure of Attitude and Behaviour to Mathematics, thus supporting the theoretical view that attitudes influence behaviour. (11) Some students had low measures of Attitude and Behaviour to Mathematics but did well on the written tests because (they said) they knew that they had to do well in Mathematics and were highly motivated to succeed. (12) Some students had high measures of Attitude and Behaviour towards Mathematics but did poorly on the written tests because (they said) they were not much interested in Mathematics. In some cases it appeared that these students were also lazy.
Independent studies of teaching for number sense and problem solving have revealed that teaching for either of them separately poses a great challenge for the teacher. Yet research focussing on the relationship between number sense and problem solving was virtually non-existent, although the relationship between students' number sense and problem solving ability was becoming more and more evident through various modes and endeavours. Since both number sense and problem solving were being promoted as two of the major areas of emphasis in mathematics education, there was an urgent need to answer questions such as, 'How do they relate to each other in terms of how they are taught, learnt and utilised in solving mathematical problems?' Moreover, teachers were being challenged to ensure delivery of a balanced curriculum, while simultaneously having to develop the number sense and problem solving ability of the students. Hence, it was important to learn how they went about satisfying the latter. This implied that there was also a need to discover whether successful teachers of number sense and problem solving necessarily employed or had a specific teaching style. This study sought to explore what sort of relationships exist between students' number sense and their problem solving ability, and the contribution of the teacher's teaching style and the students' learning style towards students' performance in these two respective areas. The problem solving ability and number sense proficiency of three classes of Year 7 students, from three metropolitan primary schools, were compared to their learning style, and their mathematics teacher's teaching style. Sixty- eight students, comprised of twenty-six males and forty-two females, and their three Year 7 teachers were involved in this study. Of the three schools, two were private - a boys only and a girls only - and one was a mixed gender state school. A mixed methods design was employed through which a combination of the ethnographic approach, the case study, framework approach and grounded theory was applied, to investigate the relationships between students' number sense and problem solving abilities, and the teaching and learning style compatibility which promote such abilities. Hence, the method followed was both quantitative — by scoring of test results and quantification of qualitative data — and qualitative — through observations and tape-recorded interviews. Each teacher and respective students were observed in multiple teaching-learning situations as well as outside the teaching-learning context, and these observations and field notes were documented. Performance data were collected through pre- and post-tests of number sense and problem solving, and also through activities and exercises done in class; the latter were used solely for data validating purposes. A combination of the Think-Aloud and Stimulated Recall Interview (TASRI) protocol was used to gain an insight into how students solved the problems presented and to elicit responses about their thinking at the time the behaviour occurred. A ready-made Index of Learning Style (ILS) inventory was used as a means of ascertaining the learning preference modality of both the students and their teachers. The three teachers were also interviewed both formally and informally and a Teaching Style Inventory (TSI) was used to gather information as to their preferred teaching style and as a means of corroborating the data collected through classroom observations, field notes and interviews. The three principals, two deputy heads and two curriculum coordinators were also interviewed. On- site perusal of various written documents was also carried out. When triangulated, data obtained from the qualitative observations and interviews, and the quantitative teacher ILS and TSI, suggest that although these three teachers tended to use different teaching approaches, their focus was more on getting students to understand the rationale behind any concept and process under discussion. These teachers taught to the ability of the students first, and in so doing they considered individual learning preferences, although the former was given a lot more prominence than the latter. Classroom observations, student interviews and data gathered through the ILS tended to indicate that although all three teachers expressed a strong preference for receiving information through the verbal learning modality, they taught largely through the visual mode and employed the verbal mode mainly for discussions, with very little teacher exposition. This could be one reason why a large majority of students showed a preference for receiving information through the visual learning modality. This interpretation was supported by the results obtained from the Number Sense (NS) and Problem Solving (PS) tests.
The Intervention comprised three distinct components: Structured individual and small group work on pattern-eliciting tasks, 'patternising' the regular preschool program, and observing children's patterning in free play. Using a Framework of Assessment and Learning, children were placed on an individual 'learning trajectory' and progressed through an increasingly complex series of tasks. Analysis of children's progress focused on levels of structure and abstraction. Further, the Intervention provided on-going professional development of the importance of pattern and structure in early mathematical learning, which assisted teachers in modifying the emergent curriculum to incorporate patterning skills. Intervention children could successfully identify, construct and abstract the element within Repeating Patterns and calculate the number of repetitions. This was the dominant strategy used by Intervention children at Assessment 2 and sustained at Assessment 3 (12 months later). Many children used their knowledge of unit of repeat to extend and represent patterns in other forms. They were also able to Craw complex repetition from memory. The development of structural thinking about simple repetition, not just the modelling of simple repetition, advantaged the Intervention children. When dealing with Spatial Structures such as arrays of dots, Intervention children could perceive the structure of the patterns. In comparison, Non-intervention children's responses lacked any structural features. Another critical learning process observed during the Intervention was the children's development of transformation skills; they successfully used rotation to construct Hopscotch patterns and visualised simple and complex repetitions from different orientations. The assessment of counting and arithmetic development provided by the Schedule for Early Number Assessment (SENA 1), administered at the third assessment, showed that Intervention children's numerical strategies were more advanced than Non-intervention children. Some were quite advanced in their arithmetic strategies, using known facts and other non-count-by- one strategies. Further analysis of SENA interview data indicated that Intervention children recognised the structure of the patterns and partitioned the patterns into parts rather than counting individual items. Intervention children successfully symbolised, abstracted and transferred complex Repeating Patterns, and with no apparent exposure to Growing Patterns, many of these children could construct, extend, represent and justify these patterns 12 months after the Intervention. In contrast, Non-intervention children were unable to identify or extend Growing Patterns. They saw these exclusively as 'items' in simple repetitions in the same way as the simple repetitions that they were familiar with. These findings support those found by Warren, where 9-year-olds had greater difficulty with Growing Patterns than with Repeating Patterns. It was inferred that the difficulty with Growing Patterns was not necessarily the absence, or predominance of Repeating Patterns in early mathematics curricula. Rather, the inadequate or inappropriate development of repeating patterns without a sound understanding of the unit of repeat, limited, and possibly impeded the development of Growing Patterns. Children may be able to copy and extend patterns, but they may not necessarily identify a unit of repeat. The findings support Blanton and Kaput's conclusion that early algebraic learning is not developmentally constrained; young children have natural powers of generalisation and an ability to express generality. This study recommends that experiences in the first year of schooling focus on identifying, justifying and transferring various patterns, and using a variety of materials. Further, the study suggests repeating patterns should include not just 'recognising, copying, continuing and creating' simple linear patterns but rather, identifying the element within repeating patterns, the number of repetitions, drawing from memory, viewing patterns from different orientations, extending a pattern in multiple directions, and transferring a pattern to a different medium. Professionals must be aware of the natural patterning experiences in children's play and ask appropriate questions that will promote mathematical thinking. This can only be achieved through programs that integrate effective professional development that build teachers' knowledge and expertise and provide them with the necessary conceptual structures to take ownership of their planning and teaching.
This study is about a group of South African Grade 9 mathematics teachers and their professional knowledge. It looks at the problems that these teachers face when mediating a new, externally mandated assessment tool and how they choose to manage these problems. The study employed qualitative methods of data collection and analysis, and a theoretical framework based on a situative view of teacher learning. The setting for the study was the Common Tasks for Assessment (CTA) – a new externally mandated assessment tool initiated by the national education department as part of a curriculum reform process. Four Grade 9 teachers were observed in their classrooms as they mediated the new assessment tool. A framework of teachers' professional knowledge was used to analyse the data. This study, by examining the determining role played by context in the teachers' practices, showed that teachers' decisions are based on a consideration of their problems and that their identities, which have been forged in their experiences, determine the choices they make. This study also raises concerns about the use of real life contexts in the national assessment CTA tasks and recommends that more research is urgently needed around the use of real life contexts in national mathematics assessment tasks.
This thesis reports on the implementation of an Experimental Mathematics Program (EMP) in which finger gesture computation with its verbalisation in Portuguese and the main local languages was explored for early numeracy learning. Gestures correspond to local traditions, and allow for computations with ones, fives and tens, stimulating the use of computation strategies instead of elementary counting strategies. The EMP was carried out by four teachers in two primary schools in Beira, Mozambique's second largest city. The background for the experimental study comes from questionnaire data from 40 teachers working at the two schools, referring to their uses, at home and in school, of local languages and Portuguese, and to their uses of manipulatives in early arithmetic teaching. The most frequently used manipulatives in grades one and two are sticks, pebbles and strokes on paper or the blackboard, as recommended in teaching materials since colonial times and practised in teacher training colleges. However, a large minority of teachers resort to fingers and gestures in early arithmetic. In the EMP the verbalisation of pupils' computations was emphasised, as number meaning for children is linked to the spoken number names. They verbalised their computations not only in Portuguese, Mozambique's official language, but also in the main local languages. All local indigenous Mozambican languages have a numeration system with regular number names based on ten, where the numbers 11, 12, are pronounced as ten-and-one, ten-and-two. Regular and explicit number names are considered to facilitate children's arithmetic learning. Several Mozambican languages also use the intermediate unit of five, leading to the expressions five-and-one, five-and- two, etc for the numbers 6 – 9. The number names based on fives and tens were thought to give additional support to children's computation strategies. The thesis reports four case studies, focusing on how the four teachers implemented the EMP. The teachers succeeded having their pupils calculating, at the end of Grade Two, sums and differences within the limit of 100, using gestures and explaining all steps of the computations. The gestures allowed for powerful visualisations for recomposition methods around five and ten. The four teachers were unanimous in their conclusion that the results of the EMP are much better than the results they used to have in the past, when they and their pupils were working with unstructured manipulatives like sticks, pebbles and strokes, and using counting strategies. The main focus of the EMP was on the use of gestures and the corresponding verbalisation of computations. However, attention was also paid to aspects of classroom interaction like having pupils trying to solve problems on their own without previous explanation by the teacher, and ways of helping pupils with difficulties. In general the four teachers showed interest in their pupils' computations and difficulties, instead of emphasising their own explanations. This is a major shift as compared to earlier classroom observations, reported in 1990 for mathematics classrooms in the cities of Maputo and Nampula.
This study was conducted with eight grade 5 classroom teachers and their students to examine the classroom learning environment, geometry content knowledge of elementary classroom teachers and geometry content knowledge of their students and student and teacher attitudes toward mathematics. Associated with this study was a 5-day professional development program designed to enhance teacher content knowledge and pedagogical knowledge in geometry. Eight teachers in the study were interviewed as to their attitudes toward learning and teaching geometry and administered the Amended Set of Mayberry Test Items to determine their van Hiele level of geometric thinking before and after implementation of the professional development program. The pre and post assessments administered to the 205 students in the study were a Geometry Content Knowledge Test, two scales from an adapted instrument Test of Science-Related Attitudes (TOSRA) to measure students' attitudes to mathematics, and five dimensions of The What is Happening in This Class? (WIHIC) questionnaire, assessing student perceptions of seven the classroom learning environment. Teacher outcomes resulted in an increase of the level of geometric thinking between the pre and posttest on the Amended Set of Mayberry Test Items and a more positive attitude towards teaching geometry. In terms of student outcomes, changes in students' perceptions of their learning environment occurred in terms of more student involvement, greater task orientation, and more cooperation. However, there were no significant changes in student attitudes toward mathematics between pre and posttests though interviews with students contradicted these quantitative findings. Larger differences in student achievement in geometry were reported. The positive findings of this study support the need for a professional development program that will increase teacher content knowledge and pedagogical knowledge to help their students meet rigorous academic standards.
There are many approaches to teaching mathematics. This paper examines one student-centred approach called the Booklet System used by a number of secondary schools in Brisbane from the mid-1990's for about a decade. Claims made by those advocating this system include improved achievement in mathematics and a better attitude to mathematics in general. Year 9 students in three schools were given a Pre Test at the beginning of the school year and a Post Test in the second last week of the school year. Two of these schools operated a traditional teacher- centred textbook system and one school used the student-centred Booklet System. All students participating in the study in the three schools also completed Attitude Questionnaires at the same time as the Pre Test and Post Test. Statistical analysis revealed that the Booklet System is successful in improving a student's attitude about their own progress in Mathematics, but does not improve their attitude to homework or Mathematics in general. It is not surprising then that this study shows that the Booklet students do not achieve any better academically than the NonBooklet students.
Statistical covariation refers to the correspondence of variation of two statistical measures that vary along numerical scales. Reasoning about covariation commonly involves translation processes among three representations: (1) numerical data, (2) graphical representations, and (3) verbal statements such as 'taller people tend to be heavier'. Two well-known translations are graph production and graph interpretation. Less well known is the process of speculative data generation, involving translating a verbal statement into a possible graph or other data representation. This study explored school students' reasoning involving these three translation skills through various tasks in surveys and interviews. Evidence is presented concerning methods to assess these skills, and concerning how students as young as third- grade can engage covariation tasks involving familiar contexts. Interviews involved prompting for cognitive conflict using responses from other students, and provided evidence of limited engagement of ideas that were slightly more sophisticated than their own responses. Responses for each of the three translation skills were described within assessment frameworks involving four levels - Nonstatistical, Single Statistical Aspect, Inadequate Covariation, and Appropriate Covariation - distinguished by the structure of combining correspondence and variation. Distinguishing features of the levels suggested stages of development that may inform instruction. For development from prior beliefs to data-based judgements, tasks involving counterintuitive covariation were designed to prompt students to engage data. For development from single variables to bivariate data, time was observed as a natural covariate, implicit in statements such as 'it's getting hotter', with a connotation of order that supported pattern recognition of passing time being associated with corresponding change in a measured variable. For development from single cases to global trends, many students represented correspondence in a single pair of values, at the expense of representing variation. Tasks involving discrete data with few cases, and the use of case labels in responses, were observed to support the view of two data values each linked to the same corresponding case label. This consolidated view of correspondence supported consideration of additional bivariate cases involving variation. Students tended to articulate covariation using the language of comparison and change. Findings were related to issues in the historical development of coordinate graphing, to findings from educational research in statistics, algebra, science and psychology, and to recommendations within curriculum documents. Student representations of statistical covariation were observed to provide a window into statistical reasoning, and are advocated as a valuable basis for classroom discussions to help develop statistical literacy.
his study, which was conducted among middle-school students in California, focused on the effectiveness of using innovative strategies for enhancing the classroom environment, students' attitudes, and conceptual development. Six hundred and sixty-one (661) students from 22 classrooms in four inner city schools completed the modified actual forms of the Constructivist Learning Environment Survey (CLES), the What Is Happening In this Class? (WIHIC) questionnaire, and the Test Of Mathematics Related Attitudes (TOMRA). The data were analysed for the CLES, WIHIC, and TOMRA to check their factor structure, reliability, discriminant validity, and the ability to distinguish between different classes and groups. In terms of the validity of the CLES, WIHIC, and TOMRA when used with middle-school students in California, the factor analysis results attest to the sound factor structure of each questionnaire. The results for each CLES, WIHIC, and TOMRA scale for the alpha reliability and discriminant validity for two units of analysis (individual and class mean) compare favourably with the results for other well-established classroom environment instruments. A one-way analysis of variance (ANOVA) was also calculated for each scale of the CLES and WIHIC to investigate its ability to differentiate between the perceptions of students in different classrooms. The ANOVA results suggest that students perceived the learning environments of different mathematics classrooms differently on CLES and WIHIC scales. In general, the results provided evidence of the validity of these instruments in describing psychosocial factors in the learning environments of middle-school mathematics classrooms in California. The effectiveness of the innovative strategy was evaluated in terms of classroom environment and attitudes, as well as achievement, among a subgroup of 101 students. Effect sizes and t-tests for paired sample were used to determine changes in classroom environment perceptions, attitudes, and achievement for experimental and control groups. Pretest- posttest differences were statistically significant (p<0.05) for: the CLES scale of Shared Control for the experimental group, the TOMRA scale of Normality of Mathematicians for both the control and the experimental groups, the TOMRA scale of Enjoyment of Mathematics for the experimental group, and the achievement measure for both groups. Also ANCOVA was calculated to determine if differential pretest- posttest changes were experienced by the experimental and control groups in classroom environment perceptions, attitudes, and achievement. The results suggest that there were a statistically significant differential changes for Task Orientation, Normality of Mathematicians, Enjoyment of Mathematics, and achievement between the experimental and control groups. In each case, the experimental group experienced larger pretest-posttest changes than the control group. Overall, a comparison of the pretest-posttest changes for an experimental group, which experienced the innovative strategy, with those for a control group, supported the efficacy of the innovative teaching methods in terms of learning environment perceptions, attitudes to mathematics, and mathematics concept development. The results of simple correlation and multiple correlation analyses of outcome-environment associations for two units of analysis clearly indicated that there is an association between the learning environment and students' attitudes and mathematics achievement for this group of middle-school mathematics students. In particular, there is a positive and statistically significant correlation between: Normality of Mathematicians and Student Negotiation, Involvement, and Task Orientation with the individual as the unit of analysis; Enjoyment of Mathematics and all three CLES and three WIHIC scales with the student as a unit of analysis, and for the four scales of Personal Relevance, Shared Control, Involvement, and Task Orientation with the class mean as the unit of analysis. The multiple correlations between the group of three CLES and three WIHIC scales and each of the two TOMRA scales are statistically significant for the individual as a unit of analysis. Overall, the study revealed positive and statistically significant associations between the classroom learning environment and students' attitudes to mathematics. A two-way MANOVA with repeated measures on one factor was utilized to investigate gender differences in terms of students' perceptions of classroom environment and attitudes to mathematics, as well as mathematics achievement. A statistically significant but small difference was found between the genders for Student Negotiation and Task Orientation. Female students perceived their mathematics classrooms somewhat more positively than did the male students. There was no statistically significant difference between the genders on achievement and students' attitudes to mathematics. Qualitative information, gathered through audiotaped interviews, students' journal, and analysis of students' work, was used to clarify students' opinions about the new approach, classroom environment perceptions, attitudes, and conceptual development. These qualitative information-gathering tools were utilized to obtain a more in-depth understanding of the learning environments and the results of this study, as well as insights into students' perceptions. The responses from the students' interviews and students' reflective journals from the group that experienced the innovative methods generally suggested that introducing Cramer's rule as a method for solving systems of linear equations in the middle school can be beneficial and therefore might be considered for inclusion in the middle-school Algebra 1 curriculum more widely in California. Using only quantitative data would not have provided the richness that was derived from using mixed methods. Therefore, qualitative data obtained from students who experienced the innovative method generally supported the quantitative findings concerning the effectiveness of this method for teaching and learning systems of linear equations.
In a series of five experiments, the effectiveness of using worked examples to teach grade 8 and 9 students the process of translating a written sentence into an algebraic equation, was examined from a cognitive load perspective. The first experiment compared the use of worked examples with a problem-solving strategy. Both for higher and lower levels of prior knowledge in mathematics, the worked examples format group performed significantly better than the problem-solving group. In experiment 2 the worked examples format was compared with an 'algorithm' method for teaching students to write equations. No significant differences were found in performance on similar questions at either the higher or the lower levels of prior knowledge. However, for transfer questions and questions testing understanding, the performance of the worked examples format was significantly better than that of the algorithm format for the higher level of prior knowledge, though differences were not significant for the lower level. In experiment 3 worked examples using two different methods of checking the translation, the 'comparison' method and the 'substitution' method, were compared. No significant differences were found between the two methods for either knowledge group. In experiments 4 and 5 it was shown that grade 8 and 9 students were initially disadvantaged by the inclusion of a checking method. However, after a more substantial period of acquisition, for the students with a lower level of prior knowledge, those who received checking instructions performed significantly better than did those who did not receive such instructions. In contrast, higher knowledge students were continually disadvantaged by the inclusion of a checking method. Higher knowledge students receiving checking instructions experienced a significantly higher cognitive load than did those not receiving them, as shown by a measure of mental effort. The positive effect of checking for lower knowledge students and the negative effect for higher knowledge students in this domain is a further example of the, expertise reversal effect. Evidence was found that the inclusion of checking instructions led to a redundancy effect for higher knowledge learners and caused retroactive inhibition for all learners.