title: Alternative assessment in mathematics education: a case study in St. Vincent and the Grenadines

authors: Trotman, Sandra M.

abstract: Following changes in mathematics in recent years and the way it is advocated that it be taught in schools, research in mathematics education has begun considering changes in the way student mathematical learning is assessed. Over the years, many teachers and educators have equated assessment with written tests, often using short answer questions and multiple choice items. This has led to ongoing debates about the place of tests in assessment. Some researchers have argued that written tests only assess a sample of the mathematics felt to be important. They have indicated that there is an increasing mismatch between the status of school mathematics and current assessment practices. This has resulted in the move towards the use of alternative assessments for formative purposes, such as self-assessments, interviews and journals. While these alternatives are positive developments, there is need for more research in the area of assessment of mathematics. For example, little is yet known about teachers’ views of assessments, whether they would be receptive to alternative forms, whether they could implement them, and what effects such implementation might have on teachers’ teaching and students’ learning.

The present research was designed to investigate the above issues in a Caribbean secondary school context. Three local teachers were investigated to determine whether they could use suggested alternative assessment strategies, as intended, with (13 to 15-year-old) students of three third form classes in the context of Number and Number Theory. Eighty-two students from one secondary school in St. Vincent and the Grenadines in the West Indies took part in this research. The main data were collected over a 12 week period in 1996 during which the researcher was a participant-observer-she was one of the three teachers. The study was qualitative and involved interviewing and observing (i) the students before, during and after instruction (instruction designed to include the students as active thinkers in the instructional and assessment processes), and (ii) the two fellow-teachers. Data were collected by means of interviews, field notes of classroom observations, teachers’ and students’ journal entries, and from an analysis of the St. Vincent and the Grenadines third form mathematics curriculum. Additional data regarding teachers’ views about assessment were collected, by means of a questionnaire, from 20 teachers in 15 secondary schools on St. Vincent. 

In this research, conducted within a constructivist perspective, four alternative assessment approaches were tried by the teachers following introductory workshop sessions. One teacher trialed self-assessment, while the remaining two teachers both trialed journals and interviews as alternatives for formative assessment in the mathematics classroom. Although observation was not formally selected to be researched, it was a complementary aspect of the three teachers’ assessment processes. 

The questionnaire data suggest that, rather than being resistant to change in traditional manner of assessment, many teachers who responded already seemed to be exploring alternative forms. The findings also suggest that teachers will need strong support systems to assist them in implementing alternative forms of assessment.

The findings from the main data indicate that mostly positive changes were experienced by the students and teachers who took part in the study. Specifically, the data showed that: 

(a) Student self-assessment required students to determine their strengths and their weaknesses on a given mathematical task, set up their own criteria for good work, grade their weekly tests and compare self-grades with the teacher’s grade, all with the aim at improving learning. The data revealed that students struggled with this at first, but with feedback from the teacher, researcher and peers, they were able to reflect on and monitor their own learning. The findings indicate that students need support and opportunities to evaluate and reflect on their own mathematical understanding and performance. Student-teacher discourse helped facilitate improvement of student performances.

(b) Although observation of students is part of teachers’ daily practice, it was surprising to note that six of the twenty secondary school mathematics teachers surveyed indicated that this approach was not applicable for assessment purposes. Nevertheless, the main data revealed that the observation of students can help teachers to (i) understand students, (ii) learn from what they see, and (iii) help students move to another level of thinking or performance. Further, the data indicated that observation operates as an informal, spontaneous assessment activity that can reveal patterns of errors. It was also noted that immediate feedback should be given to students. Two concerns that were voiced were that caution was needed when interpreting observations and that informal observation needed to be structured. 

(c) Analysis of the data collected on the use of journals revealed that when asked initially about writing in mathematics, students did not recognise word problems and story telling as aspects of writing to solve mathematics; hence they could not at that stage link writing, particularly journals, to mathematics. Initially, students were hesitant about the role of journals in mathematics. They objected at first to having their journals assessed. These concerns diminished as the trial progressed. ‘Postie notes’ in conjunction with student journals evolved as a means of vital communication between teacher and students to discuss student thinking. The two teachers involved in trying out student journals used the data collected to inform their teaching. Their experience indicated that journal entries need to be read frequently if they are to have value. 

(d) With respect to student interviews, at the end of the trial only two of the 54 students who experienced this means of assessment retained some concerns about it. The rest were positive. For their part, the teachers learned to listen to their students thinking aloud as they explained how they solved problems. They also learned to acknowledge their students’ ideas and to hold back rather than immediately tell students the answers. The third form mathematics syllabus, lack of time, external examinations and traditional teaching and learning practices were viewed by both teachers as deterrents to successful implementation of the interview approach. 

(e) As students became active partners in the learning process, most believed that they gained a fuller picture of their own achievements and progress and a better sense of themselves as writers and thinkers in mathematics. The findings give further support to the constructivist view of learning, as the use of the four alternative assessment approaches influenced student learning by getting them to be more active constructors in the learning and assessment processes. 

The teaching practices of teachers were affected in different ways, and this seemed to be related to their beliefs about mathematics learning and teaching, and hence assessment. While one teacher was able to implement self-assessment successfully, another was concerned about the pressure of time and other commitments. Students also were concerned that this intervention would detract from the time needed to cover the mathematics syllabus.

This research, highlights the importance of secondary teachers’ perceptions to successful implementation of alternative methods. The data revealed that while it was not possible to have the alternatives implemented entirely as intended, teachers and students nevertheless benefited from their use.

text: Alternative assessment in mathematics education: a case study in St. Vincent and the Grenadines Following changes in mathematics in recent years and the way it is advocated that it be taught in schools, research in mathematics education has begun considering changes in the way student mathematical learning is assessed. Over the years, many teachers and educators have equated assessment with written tests, often using short answer questions and multiple choice items. This has led to ongoing debates about the place of tests in assessment. Some researchers have argued that written tests only assess a sample of the mathematics felt to be important. They have indicated that there is an increasing mismatch between the status of school mathematics and current assessment practices. This has resulted in the move towards the use of alternative assessments for formative purposes, such as self-assessments, interviews and journals. While these alternatives are positive developments, there is need for more research in the area of assessment of mathematics. For example, little is yet known about teachers’ views of assessments, whether they would be receptive to alternative forms, whether they could implement them, and what effects such implementation might have on teachers’ teaching and students’ learning.

The present research was designed to investigate the above issues in a Caribbean secondary school context. Three local teachers were investigated to determine whether they could use suggested alternative assessment strategies, as intended, with (13 to 15-year-old) students of three third form classes in the context of Number and Number Theory. Eighty-two students from one secondary school in St. Vincent and the Grenadines in the West Indies took part in this research. The main data were collected over a 12 week period in 1996 during which the researcher was a participant-observer-she was one of the three teachers. The study was qualitative and involved interviewing and observing (i) the students before, during and after instruction (instruction designed to include the students as active thinkers in the instructional and assessment processes), and (ii) the two fellow-teachers. Data were collected by means of interviews, field notes of classroom observations, teachers’ and students’ journal entries, and from an analysis of the St. Vincent and the Grenadines third form mathematics curriculum. Additional data regarding teachers’ views about assessment were collected, by means of a questionnaire, from 20 teachers in 15 secondary schools on St. Vincent. 

In this research, conducted within a constructivist perspective, four alternative assessment approaches were tried by the teachers following introductory workshop sessions. One teacher trialed self-assessment, while the remaining two teachers both trialed journals and interviews as alternatives for formative assessment in the mathematics classroom. Although observation was not formally selected to be researched, it was a complementary aspect of the three teachers’ assessment processes. 

The questionnaire data suggest that, rather than being resistant to change in traditional manner of assessment, many teachers who responded already seemed to be exploring alternative forms. The findings also suggest that teachers will need strong support systems to assist them in implementing alternative forms of assessment.

The findings from the main data indicate that mostly positive changes were experienced by the students and teachers who took part in the study. Specifically, the data showed that: 

(a) Student self-assessment required students to determine their strengths and their weaknesses on a given mathematical task, set up their own criteria for good work, grade their weekly tests and compare self-grades with the teacher’s grade, all with the aim at improving learning. The data revealed that students struggled with this at first, but with feedback from the teacher, researcher and peers, they were able to reflect on and monitor their own learning. The findings indicate that students need support and opportunities to evaluate and reflect on their own mathematical understanding and performance. Student-teacher discourse helped facilitate improvement of student performances.

(b) Although observation of students is part of teachers’ daily practice, it was surprising to note that six of the twenty secondary school mathematics teachers surveyed indicated that this approach was not applicable for assessment purposes. Nevertheless, the main data revealed that the observation of students can help teachers to (i) understand students, (ii) learn from what they see, and (iii) help students move to another level of thinking or performance. Further, the data indicated that observation operates as an informal, spontaneous assessment activity that can reveal patterns of errors. It was also noted that immediate feedback should be given to students. Two concerns that were voiced were that caution was needed when interpreting observations and that informal observation needed to be structured. 

(c) Analysis of the data collected on the use of journals revealed that when asked initially about writing in mathematics, students did not recognise word problems and story telling as aspects of writing to solve mathematics; hence they could not at that stage link writing, particularly journals, to mathematics. Initially, students were hesitant about the role of journals in mathematics. They objected at first to having their journals assessed. These concerns diminished as the trial progressed. ‘Postie notes’ in conjunction with student journals evolved as a means of vital communication between teacher and students to discuss student thinking. The two teachers involved in trying out student journals used the data collected to inform their teaching. Their experience indicated that journal entries need to be read frequently if they are to have value. 

(d) With respect to student interviews, at the end of the trial only two of the 54 students who experienced this means of assessment retained some concerns about it. The rest were positive. For their part, the teachers learned to listen to their students thinking aloud as they explained how they solved problems. They also learned to acknowledge their students’ ideas and to hold back rather than immediately tell students the answers. The third form mathematics syllabus, lack of time, external examinations and traditional teaching and learning practices were viewed by both teachers as deterrents to successful implementation of the interview approach. 

(e) As students became active partners in the learning process, most believed that they gained a fuller picture of their own achievements and progress and a better sense of themselves as writers and thinkers in mathematics. The findings give further support to the constructivist view of learning, as the use of the four alternative assessment approaches influenced student learning by getting them to be more active constructors in the learning and assessment processes. 

The teaching practices of teachers were affected in different ways, and this seemed to be related to their beliefs about mathematics learning and teaching, and hence assessment. While one teacher was able to implement self-assessment successfully, another was concerned about the pressure of time and other commitments. Students also were concerned that this intervention would detract from the time needed to cover the mathematics syllabus.

This research, highlights the importance of secondary teachers’ perceptions to successful implementation of alternative methods. The data revealed that while it was not possible to have the alternatives implemented entirely as intended, teachers and students nevertheless benefited from their use.

year: 1999

title: Evaluation of composite laminates interleaved with nanofibre and microfibre veils.

authors: Collins-Gargan, Rosalie

abstract: The research covered in this thesis aimed to investigate the use of nanofibre and microfibre veils in carbon fibre reinforced composites and assessed the potential of the veils to improve damage resistance during impact and fatigue loading. It was hypothesised that the interleavings would increase the amount of energy required for crack propagation because of toughening due to fibre reinforcement mechanisms such as crack deflection, fibre pull out and fibre breakage. The work was undertaken as a combined project between the University of Waikato (Hamilton, New Zealand) and Revolution Fibres Ltd (Auckland, New Zealand). 
During this investigation, six thermoplastic polymers were chosen (acrylonitrile butadiene styrene (ABS), acrylonitrile styrene acrylate (ASA), polystyrene (PS), chlorinated polyvinyl chloride (CPVC), polymethyl methacrylate (PMMA) and polycarbonate (PC)) that could potentially be used for the electrospinning of polymer nanofibre veils. Nanofibre veils were successfully produced from PMMA, and a polymer blend of polyamide 6,6 (PA6,6) and PMMA, (referred to as 'nanoNyplex'). These veils, along with three other nanofibre veils (nanoPA6,6, poly vinyl butyral (nanoPVB), and poly ether sulfone (nanoPES)), three microfibre veils (polyphenylene sulfide (microPPS), polyetherimide (microPEI), and woven polyamide 6 (microtricot)) procured from other manufacturers, and three veils combining one of the nanofibre veils with each of the microfibre veils (microPPSnanoPA6,6, microPEInanoPA6,6, and microtricotnanoPA6,6) were then used as interleaves in the manufacture of carbon fibre reinforced epoxy composite panels. Interleaves were placed between every ply of prepreg. After curing the panels, test specimens were created to assess fatigue, vibration damping and compression after impact performance.
From the vibration damping study, it was found that the nanoNyplex interleaving improved damping the most. It was thought that energy dissipation was due friction brought about by the movement of the interleaving fibres in the matrix, resulting in friction due to weak adhesion between the nanoNyplex fibres and the epoxy matrix.
From the compression after impact (CAI) section of this study, it was found that specimens interleaved with nanoPA6,6, microPPS and microPPSnanoPA6,6 had the highest CAI strengths. From optical inspection, it appeared (in general) that as the CAI strength of the specimen increased, the length of the damage region also increased. However, those identified with the highest CAI strengths had shorter damage regions.
From the fatigue section of this study, it was found that the use of most interleavings, (apart from microtricot) increased the number of cycles to failure. Post fatigue test scanning electron microscopy confirmed that crack deflection was present for most interleaved specimens. Some evidence of pull out and breakage of the interleaving fibres was seen on the fracture surfaces of the nanoPA6,6, microPPS, microPEI, microPEInanoPA6,6 and microPPSnanoPA6,6 interleaved specimens.
For both CAI and fatigue, it was found that improvement was generally greater with veils that had a large number of fibres per unit area and high adhesion strength with the matrix. However, for CAI it seems that high fracture toughness was also desirable.

text: Evaluation of composite laminates interleaved with nanofibre and microfibre veils. The research covered in this thesis aimed to investigate the use of nanofibre and microfibre veils in carbon fibre reinforced composites and assessed the potential of the veils to improve damage resistance during impact and fatigue loading. It was hypothesised that the interleavings would increase the amount of energy required for crack propagation because of toughening due to fibre reinforcement mechanisms such as crack deflection, fibre pull out and fibre breakage. The work was undertaken as a combined project between the University of Waikato (Hamilton, New Zealand) and Revolution Fibres Ltd (Auckland, New Zealand). 
During this investigation, six thermoplastic polymers were chosen (acrylonitrile butadiene styrene (ABS), acrylonitrile styrene acrylate (ASA), polystyrene (PS), chlorinated polyvinyl chloride (CPVC), polymethyl methacrylate (PMMA) and polycarbonate (PC)) that could potentially be used for the electrospinning of polymer nanofibre veils. Nanofibre veils were successfully produced from PMMA, and a polymer blend of polyamide 6,6 (PA6,6) and PMMA, (referred to as 'nanoNyplex'). These veils, along with three other nanofibre veils (nanoPA6,6, poly vinyl butyral (nanoPVB), and poly ether sulfone (nanoPES)), three microfibre veils (polyphenylene sulfide (microPPS), polyetherimide (microPEI), and woven polyamide 6 (microtricot)) procured from other manufacturers, and three veils combining one of the nanofibre veils with each of the microfibre veils (microPPSnanoPA6,6, microPEInanoPA6,6, and microtricotnanoPA6,6) were then used as interleaves in the manufacture of carbon fibre reinforced epoxy composite panels. Interleaves were placed between every ply of prepreg. After curing the panels, test specimens were created to assess fatigue, vibration damping and compression after impact performance.
From the vibration damping study, it was found that the nanoNyplex interleaving improved damping the most. It was thought that energy dissipation was due friction brought about by the movement of the interleaving fibres in the matrix, resulting in friction due to weak adhesion between the nanoNyplex fibres and the epoxy matrix.
From the compression after impact (CAI) section of this study, it was found that specimens interleaved with nanoPA6,6, microPPS and microPPSnanoPA6,6 had the highest CAI strengths. From optical inspection, it appeared (in general) that as the CAI strength of the specimen increased, the length of the damage region also increased. However, those identified with the highest CAI strengths had shorter damage regions.
From the fatigue section of this study, it was found that the use of most interleavings, (apart from microtricot) increased the number of cycles to failure. Post fatigue test scanning electron microscopy confirmed that crack deflection was present for most interleaved specimens. Some evidence of pull out and breakage of the interleaving fibres was seen on the fracture surfaces of the nanoPA6,6, microPPS, microPEI, microPEInanoPA6,6 and microPPSnanoPA6,6 interleaved specimens.
For both CAI and fatigue, it was found that improvement was generally greater with veils that had a large number of fibres per unit area and high adhesion strength with the matrix. However, for CAI it seems that high fracture toughness was also desirable.

year: 2015

title: Transformative pedagogy and language learning in Maori and Irish contexts

authors: Glynn, Ted

subjects: transformative pedagogy

abstract: Establishing positive learning communities in classrooms where pedagogies are socially and culturally responsive and centred on shared life experiences is critical for the revitalisation of minoritised languages such as Irish and Māori, which lack the dominance and power of the wider languages of government and communication. Transformative pedagogy and pedagogical re-positioning of teachers are essential to legitimising and affirming minoritised languages in the classroom. Three small-scale New Zealand studies of emerging literacy in Māori are introduced, exemplifying a responsive and transformative pedagogy that enables teachers to position themselves as socially and culturally responsive participants within classroom contexts. These studies offer strategies that might facilitate the development of similar transformative pedagogy in Irish contexts also.

text: Transformative pedagogy and language learning in Maori and Irish contexts Establishing positive learning communities in classrooms where pedagogies are socially and culturally responsive and centred on shared life experiences is critical for the revitalisation of minoritised languages such as Irish and Māori, which lack the dominance and power of the wider languages of government and communication. Transformative pedagogy and pedagogical re-positioning of teachers are essential to legitimising and affirming minoritised languages in the classroom. Three small-scale New Zealand studies of emerging literacy in Māori are introduced, exemplifying a responsive and transformative pedagogy that enables teachers to position themselves as socially and culturally responsive participants within classroom contexts. These studies offer strategies that might facilitate the development of similar transformative pedagogy in Irish contexts also.

year: 2009