Faculty of Education - Theses
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ItemAn investigation of the implementation of a problem-solving intervention in two primary classrooms( 2020)Problem-solving in mathematics is an important component of curricula around the world and it has been identified as essential that students develop this capacity in order to achieve success in mathematics. Studies have found that more teachers need to teach their students strategies to problem-solve in mathematics. The aim of this case study was to investigate the implementation of a problem-solving intervention by two primary school teachers over two lessons each. It focussed on their perceptions of the effectiveness of the intervention and how it might improve their teaching of problem-solving in mathematics in the future. It also focussed on how they implemented the intervention and how their students responded to the intervention. The problem-solving intervention was designed based on features identified in problem-solving literature and in discussion with the two teachers. Particular features that were incorporated into the intervention included: enabling and extending prompts; the provision of periods of time in which students were left to ‘struggle’ with trying to solve the problems themselves; and the provision of periods in which students shared problem-solving strategies with peers. The teachers were interviewed separately before and after teaching the lessons. The researcher observed all four lessons and collected student work samples from each lesson. Data was analysed using a content analysis strategy. The results suggest that the two teachers perceived that the intervention had both positive and negative impacts on their students’ problem-solving abilities. They found that the enabling prompts supported and extended their students’ thinking in the lessons and commented that their students enjoyed being challenged in the lessons. The two teachers perceived that it was often not beneficial for some of their students to struggle with problems in the lessons due to perceived resilience and confidence issues. Both teachers deviated from the intervention in the lessons in order to reduce the amount of struggle their students experienced. However, where students were given time to struggle in the lessons, they were able to formulate and record a greater range of problem-solving strategies. There appeared to be a tension for the teachers between providing time for their students to struggle with problems and preserving some of their students’ confidence. One of the teachers facilitated student share time in the middle of one of her lessons which allowed students to experience both struggle and success. This approach could serve as a compromise between these two tensions. The two teachers perceived that the intervention had a positive impact on their teaching practice. One teacher commented that she intended to implement problem-solving lessons based on the intervention in the future and the other suggested that she would incorporate more manipulatives in her problem-solving lessons.
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ItemPromoting change in teacher practice through supported differentiation of instruction in mathematics( 2019)Differentiated instruction has been shown to be effective in improving student learning outcomes; however, the resulting work load can be difficult for teachers to manage. A teaching package known as the NRP (Number Resource Package) was created to support teachers to differentiate their instruction, and used effectively in two classrooms. The package allows teachers to identify their students’ current understanding using a diagnostic test and a Guttman Chart, and then provides appropriate material for the area in which students need further consolidation. It assists teachers to identify, and provide instruction for, several different knowledge levels within the one classroom. Use of the NRP in the two experimental classes was compared with five classes that did not use the NRP and continued to follow their school’s mathematics curriculum. This study involved a quasi-experimental approach, using qualitative and quantitative data. Involved were an experimental group (two teachers) and a control group (five teachers) and a total of 147 year 7 students. The research took place in a large school in western Melbourne, Australia. The qualitative data consisted of three surveys and provided information on the effectiveness of the components in the NRP. The quantitative data consisted of a pre- and a post-test completed by students in both the experimental and control groups. These tests were completed at the beginning and the end of a nine-week teaching cycle and the learning gains were determined for each student (i.e. the difference between the pre- and post-test). There was a statistically significant difference between the experimental group and the control group when these learning gains were analysed. The results demonstrated that students in the experimental group who were taught using the NRP showed greater improvement on the post-test when compared to students in the control group. It was noted that those students who performed ‘below’ the expected level and those students who performed ‘above’ the expected level showed the most improvement in the experimental group, when compared with the control group.
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ItemTeacher questioning practices across a sequence of consecutive mathematics lessons: a multiple-case study of junior secondary teachers in Australia and mainland China( 2017)Question asking is one of the most common strategies used by teachers in their everyday classroom instructional practice. Over recent decades, many attempts have been made to categorise teacher questions asked during classroom instruction and to report on teachers’ skilful questioning strategies. These categorisations consider the context where the questions are asked, the appropriate use of different types of questions, the learning opportunities created in the sequences of teacher-student interactions and so on. This study was designed to extend our understanding of teachers' questioning practices in classrooms through a fine-grained analysis of mathematics lessons taught by four competent junior secondary teachers from mainland China and Australia. The study demonstrates the importance of examining teaching strategies over a sequence of lessons, the power of the IRF (Initiation-Response-Follow-up) framework as a basic structure for investigating classroom interactions, and the complexity of teaching practices, made evident through the focused investigation of the ubiquitous practice of teacher questioning. Based on the IRF framework, a comprehensive coding system was developed to analyse what kinds of verbal questions were initiated by the teachers to elicit mathematical information and in what ways the teachers made use of students’ verbal contributions in order to facilitate student construction and acquisition of mathematical knowledge. In particular, a distinction was made between Q&A question pairs, IRF (single) sequences, and IRF (multiple) sequences. Classification systems were developed for question types within each interactive category. Within IRF (multiple) sequences, the categories: initiating and follow-up represented a fundamental distinction, each category having its own suite of sub-categories. For each participating teacher, a whole unit of consecutive lessons was examined (from 6 to 10 lessons per unit). Analysis of the data suggested that: (1) Across the professional practice of the four teachers, two each in mainland China and in Australia, similarities and differences in the ways in which teachers employ questioning strategies were observed. The differences regarding questioning strategies across the consecutive lessons include: (i) number/frequency of questions asked in each lesson; (ii) the proportions of questions in IRF (multiple) sequences and the proportions of the questions in Q&A question pairs and IRF (single) sequences; and, (iii) the use of subcategories for initiation questions in each lesson. And the similarities are as follows: (i) the proportion of initiation questions in IRF (multiple) sequences out of all questions in each lesson; and, (ii) the use of subcategories for follow-up questions in each lesson. The essential point suggested by the comparison of similarities and differences regarding teacher questioning practices in this study is that the Chinese teachers and Australian teachers employed questioning strategies with similar forms but with distinctly different functions. (2) Regardless of the geographical location of the classroom, teachers’ questioning strategy choice is made rationally based on such contexts as the nature of instructional tasks and the constraints facing the teachers at the time. Those constraints might involve time limit and overemphasis on procedural fluency caused by the need to prepare students for high-stakes examinations, the demands of catering to students’ individual differences, the need for coherent delivery and explanation of sophisticated mathematics and the need to elicit information about student existing understanding. Unlike the two Chinese teachers who valued the achievement of lesson goals above any other factors, both Australian teachers placed greatest emphasis equally on students’ demands and lesson content. (3) In the case of the use of the three kinds of IRF (multiple) sequences (leading, facilitating/probing, orchestrating), the nature of teacher lesson planning – collaborative and institutionalised in the case of mainland China, and individually done in the case of Australia – affects how teachers make use of questions in class. These local educational contexts pose culturally-situated challenges, even though the teacher questioning strategies that are chosen and performed may reflect rational professional decisions by all four teachers, predicated on similar pedagogical goals. Teachers’ adjustment of their questioning routines in response to competing tensions in their classroom practices provided some of the most interesting features of the research. In addition, this study also suggests that teacher professional development program designers should ensure that novice teachers are given an opportunity to observe the teaching of a sequence of lessons and to observe closely how one expert teacher’s questioning strategies are strategically employed according to the demands of the particular lesson and its place in the topic sequence. Such strategic variation of questioning practice cannot be fully or correctly understood without the examination of the teaching of consecutive lessons.
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ItemDesigning and implementing an intervention program to improve under-achieving Year 8 students’ understanding of multiplicative thinking and assess associated gains in motivation and engagement( 2016)A fifteen-week intervention was carried out in order to examine the efficacy of the Scaffolding Numeracy in the Middle Years (SNMY) (Education Victoria, 2013b) instructional resources in improving the multiplicative thinking of eight under-achieving Year 8 students. The eight students participating in the intervention were contrasted against a similarly profiled control group of ten students. Participants were drawn from a co-educational high school on the lands of the People of the Kulin Nation in Northern Metropolitan Melbourne. Changes in students’ multiplicative thinking, as well as associated changes in motivation and engagement, were measured in pre- and post-intervention assessments. Multiplicative thinking was measured with the SNMY Assessment Booklets 1 and 2, with motivation and engagement examined by use of the High School Motivation and Engagement Scale (MES-HS) (Martin, 2003), also prior-to and following the intervention. These quantitative data sources were complemented with qualitative sources in the form of researcher’s notes and student work samples. Whilst analysis of quantitative data did not indicate significant changes in multiplicative thinking for either the intervention or the control groups, qualitative data sources indicated that students within the intervention group demonstrated modest gains in multiplicative thinking. No statistically significant changes to motivation and engagement were recorded for students in the control group, whereas those in the intervention group showed a significant decrease in both failure avoidance and anxiety, and a significant increase in disengagement between pre- and post-testing. Findings point to several barriers to achievement for mathematically under- achieving students in the middle years, as well as directions for the improvement of similar interventions in future.
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ItemUsing teacher capacity to measure improvement in key elements of teachers’ mathematical pedagogical content knowledge( 2016)School systems world-wide are investing increasing resources in assessment of students. The challenge is to gain value for teachers from this process. This study examined how we can use a construct of teacher capacity to identify improvements in teachers’ knowledge of Mathematics, their knowledge of the curriculum, their understanding of student’s mathematical thinking, and their ability to design and implement effective mathematics instruction as a result of using online diagnostic assessments (SMART tests- Specific Mathematical Assessments that Reveal Thinking.) Two principal challenges were addressed in this study: the first concerns how to translate a theoretical construct of teacher capacity in ways that truly reflect the professionally informed judgement and disposition to act. The second challenge was to design and use measures that would show improvement of teacher capacity over time as a result of using SMART tests. This study used innovative approaches involving teacher self-reports that were supported by evidence derived from a content specific questionnaire, related to the four elements of teacher capacity identified previously. The research study was carried out in the researcher’s school. 14 teachers used SMART tests over the course of one semester. All teachers showed improvement in teacher capacity as a result of implementing SMART tests, however improved teacher capacity was most evident amongst accomplished and expert teachers. The use of SMART tests also had a direct impact on teacher planning and informed classroom instruction. The study concludes with recommendations for future research in schools and in pre-service teacher education, utilising online diagnostic assessments of students. This study provides insight into what teacher capacity means in an educational setting, and how leaders in schools can effectively measure and improve teacher capacity in a school setting.
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ItemThe smart test system: teachers’ views about this formative assessment for mathematics( 2014)Given the continuous challenge of improving the quality of students’ learning of mathematics at the school level, and the consensus on the impact that teaching practices have on student learning, several countries have incorporated the development of assessments to directly influence teaching practices, known as formative assessments. However, this response has faced challenges, such as the need for more evidence about which particular formative assessment strategies may be beneficial for the learning of both students and teachers. Accordingly, this research is focused on providing evidence about teachers’ opinions from their experience as users of a particular formative assessment, the ‘Specific Mathematics Assessments that Reveal Thinking’ also known as the Smart Test System, developed at The University of Melbourne. The research aims to answer three fundamental questions. First, how do teachers perceive the quality of the Smart Tests items and the diagnosis provided from the Smart Test System? Second, to what extent, and how, do teachers utilize the information provided by the Smart Test System to change their teaching? Thirdly, to what extent, and how, do teachers report on their learning from their use of the Smart Test System? A mixed methodology approach was utilized according to the research questions, and an on-line self-administered survey was used as the method of data collection. The findings suggest that a majority of teachers who participated in this study have had a positive experience as users of the Smart Test System. Along with reporting that the Smart Tests items and the diagnosis provided are of high quality, many teachers mention formative uses of the Smart Test System. These uses mainly correspond to adjusting their planning and differentiation of their practices according to students’ needs. Almost all teachers in this study stated that they have learnt something from the Smart Test System. The findings of this study provide some understanding about how the Smart Test System fulfils its formative purposes. The results contribute to determining teachers’ opinions about the scope and the ways that they use the Smart Test System to adjust their practices. The results also identify some challenges that could be attended to in order to maximize the potential benefit from the use of the Smart Test System, and which can be considered to develop further formative assessment initiatives.
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ItemDiagnostic testing and changes to teaching practice in Year 9 mathematics classes( 2014)Teachers can use various means, including diagnostic tests, to determine their students’ knowledge. It is of interest to know the ways in which teachers interpret and act upon such diagnostic information. The aim of this study was to examine the use of a particular diagnostic testing system by six teachers in Year 9 mathematics classrooms. The focus diagnostic system was the SMART system (Specific Mathematics Assessment that Reveal Thinking), which provides teachers with an online diagnostic test, diagnostic analysis and teaching advice. This study focused on the use of the SMART system in two topics, linear equations and linear graphs. The participant teachers were interviewed before each topic to ascertain the ways in which they gathered knowledge about student understanding, current and intended teaching practices, and how they met individual student’s learning needs. On conclusion of each topic, participants completed a questionnaire and an interview to determine if any changes had been made. The teachers found the SMART system gave them some useful data on their students. The diagnostic analysis revealed gaps or misunderstandings in some students’ knowledge, the teachers realised that they could no longer assume that all students had the requisite prior knowledge. Through this discovery, teaching practice changed in a number of ways. First, the teachers were able to decide on a better starting point for the particular topic. For example, if many students did not have the expected prior knowledge the teachers began the topic with earlier concepts. Second, teachers could identify groups of students with similar learning needs and these students could be provided with activities that supported their learning. Furthermore, for some teachers it changed their view of students mathematical ability from, ‘some students do not have the ability to learn maths’ to, ‘these students have gaps in their knowledge and if these gaps or misconceptions are addressed they could progress to more complex concepts’. Most significantly, teachers reported becoming more prepared with appropriate materials for either individual students or groups of students. Hence the SMART system supported teachers to cater for individual student needs by highlighting the learning needs of students.
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ItemThe factors influencing secondary school girls' mathematics subject selections( 2014)In Australia, the number of enrolments of secondary school students in advanced Mathematics courses has been declining. This is at a time when mathematical abilities are considered to be a key component in the progressive and contemporary economy with the future prosperity of the nation depending on a significant proportion of the workforce to be educated disciplines including advanced mathematics. High technological skills and productivity are essential in a world that is becoming increasingly dependent on knowledge and innovation. Of particular concern is, while the gap between genders is closing (with the exception of the high performing students); the gender discrepancies in enrolments in some programs are substantial, particularly in the areas of mathematics and engineering. There continues to be an underrepresentation of females in mathematics courses and careers. With consideration of the previous literature, this research study explores the factors that influence girls’ mathematics subject selection in secondary school and discusses the findings and implications. The research was conducted at Hilltop Girls’ College, an all-female secondary school in Melbourne and employed qualitative methods to investigate the experiences and self-perceptions of the participants’ mathematics story together with their ability in mathematics. The data were collected via semi-structured interviews with 22 students from Year 9 to Year 12 and 3 teachers from Hilltop. The analysis of the language used by the girls and teachers in the study revealed the major influence on the students’ subject selection was the perception that mathematics is required as it is a prerequisite for many desirable university courses and it will contribute to increasing the students’ Australian Tertiary Admittance Rank. The students uncovered that studying mathematics is considered valuable primarily because of the understanding that it is important for university entrance, not because is it interesting or required for a career. One of the most interesting findings from this research was that the students had very little idea of how the mathematics they study at school is used in everyday life or in the workforce. The investigation of the culture of mathematics at Hilltop revealed the high status of mathematics at the College which can cause some students to experience anxiety about mathematics and the assessment of mathematics. The other factors that proved to be lesser influences on girls’ choice of mathematics subjects include, gender stereotype, pedagogy, parental influence, the value the girls’ placed on mathematics and the girls’ attitudes towards mathematics.
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ItemAdolescent perceptions of the concept of randomness( 1994)An investigation into adolescents perceptions of concepts of randomness, with a questionnaire trialled on 75 adolescent boys between Year 7 and Year 11 in Catholic schools in Melbourne, Australia.
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ItemAn insight into student understanding of functions in a graphing calculator environment( 2003)The introduction of graphing calculators into senior secondary schools and mandating of their use in high stakes assessment makes student expertise in finding a complete graph of a function essential. This thesis investigated the cognitive, metacognitive, mathematical, and technological processes senior secondary students used in seeking a complete graph of a difficult cubic function. A pretest of function knowledge was administered to two mixed ability classes in their final two years of secondary school. Five pairs of experienced users of TI-83 or 82 graphing calculators from these classes were audio and videotaped solving a problem task. Protocols were constructed and subjected to intensive qualitative macroanalysis and microanalysis using tools developed by the researcher from Schoenfeld’s work. The findings were: (1)all students demonstrated understanding of the local and global nature of functions and the synthesis of these in determining a complete graph; (2) a range of mathematical and graphing calculator knowledge was applied in seeking a global view of the function with their combined application being more efficient and effective; (3) an understanding of automatic range scaling features facilitated efficient finding of a global view; (4) all pairs demonstrated having a clear mental image of the function sought and the possible positions of the calculator output relative to this; (5) students were able to resolve situations involving unexpected views of the graph to determine a global view; (6) students displayed understanding of local linearity of a function; (7) when working in the graphical representation, students used the algebraic but not the numerical representation to facilitate and support their solution; (8) scale marks were used to produce more elegant solutions and facilitate identification of key function features to produce a sketch but some students misunderstood the effect of altering these; (9) pairs differed in the proportion of cognitive and metacognitive behaviours demonstrated with question asking during evaluation supporting decision making; (10) correct selection of xxi an extensive range of graphing calculator features and use of dedicated features facilitated efficient and accurate identification of coordinates of key function features.