Connecting mathematics teaching with vocational learning

For many vocational students in England, mathematics is now a compulsory part of their programme, yet the inclusion of an academic subject within a vocational course presents challenges. In this paper, an analysis of a series of case studies of vocational student groups in Further Education colleges in England shows how contrasting practices in ‘functional mathematics’ and vocational classes reinforce perceptions that mathematics is an isolated and irrelevant subject. Some mathematics teachers made contextual connections by embedding mathematical problems in vocationally-related scenarios but distinctive socio-cultural features of vocational learning situations were often absent from mathematics classes. Addressing this disconnection requires a pedagogical approach and classroom culture that links mathematics learning with vocational values. The findings suggest that adopting mathematics classroom practices that reflect the surrounding vocational culture creates greater coherence for students and has positive effects on their engagement with mathematics learning

Connecting mathematics teaching with vocational learning

For many vocational students in England, mathematics is now a compulsory part of their programme, yet the inclusion of an academic subject within a vocational course presents challenges. In this paper, an analysis of a series of case studies of vocational student groups in Further Education colleges in England shows how contrasting practices in ‘functional mathematics’ and vocational classes reinforce perceptions that mathematics is an isolated and irrelevant subject. Some mathematics teachers made contextual connections by embedding mathematical problems in vocationally-related scenarios but distinctive socio-cultural features of vocational learning situations were often absent from mathematics classes. Addressing this disconnection requires a pedagogical approach and classroom culture that links mathematics learning with vocational values. The findings suggest that adopting mathematics classroom practices that reflect the surrounding vocational culture creates greater coherence for students and has positive effects on their engagement with mathematics learning

A study of the experiences of vocational students learning functional mathematics in further education colleges

The education system in England has long been characterised by a distinct separation between academic or vocational pathways in the post-16 phase. Until recent policy changes this division coincided with the point at which compulsory education ended and mathematics became an optional subject for many students. Those who failed to attain the widely accepted minimum standard of a grade C in GCSE mathematics, however, were often strongly encouraged to undertake a course to improve their mathematical knowledge and skills. This study focusses aged 16-19 and examines the learning experience of those who take a functional mathematics course alongside their vocational programme, as either a recommended option or a requirement of internal college policy. Research regarding the learning of mathematics for these students within the context of Further Education is limited. The study adopts a holistic view of the situation to explore the main factors that influence the student experience, with an emphasis on gaining insight and understanding of students’ perceptions of their learning situation. Using a grounded theory approach with multiple methods, the research includes a series of case studies of seventeen student groups across three Further Education colleges, from which within-case and cross-cases themes are identified. The research findings show how the student experience of functional mathematics is affected by a complex network of inter-linking factors associated with both the organisation and the individual. Although organisational factors such as policies and systems sometimes place constraints on opportunities, social and cultural influences shape student values and perceptions of functional mathematics. There is strong influence from individual teachers through differing interpretations of the curriculum and pedagogical approaches but social structures and relationships within the classroom are also important to students. In addition, the legacy of students’ prior experiences of learning mathematics has an effect on attitudes and emotions, despite the separation of space and time, indicating the significance of both cognitive and affective factors in this interaction of multiple influences. Many students approach functional mathematics in college with prior experiences of disaffection and low attainment but the study shows how attitudes and understanding are transformed for some students within the college environment. Fundamental to these changes is the functional curriculum which, based on the application of mathematics rather than knowledge-acquisition, facilitates teaching approaches that present a new image of mathematics as a useful ‘tool for life’. Using materials such as contextualised tasks to make meaningful links to student lives increases awareness of the relevance of mathematics, leading to greater engagement and understanding. In the transition from school to college there is a marked discontinuity of curriculum and environment, accompanied by value-changes indicating a stronger orientation towards adult life and vocational employment. Students respond positively to functional mathematics lessons where these values are embraced. Academic-vocational divisions, such as differences in values, culture, curriculum and approaches to learning, are evident at multiple levels in colleges and these produce tensions in the student experience of functional mathematics. Some effective bridging is achieved through appropriate classroom practices but coherence requires a multi-level embedded approach involving college structures and departmental policies rather than simply the actions of individual teachers. The research findings suggest that reversing trends of disaffection and failure with mathematics amongst students can be achieved in post-16 education but this is dependent on changes within the curriculum and learning situation. In the light of recent policy changes in England that will increase the numbers of post-16 students taking a mathematics course and the prioritisation of GCSE mathematics over alternative curricula, this study has much to contribute to understanding students’ perceptions of mathematics and the factors that influence their learning experience

The professional identity of mathematics teachers in Further Education

Professional identities may be viewed as narrative constructions in social situations but personal experiences and beliefs are fundamental influences in their development. Within Further Education colleges in England, mathematics teachers are typically expected to fulfil multiple roles, teaching a wide range of curricula and age groups, and this brings additional complexity to their professional identities. In this study, questionnaires and interviews with mathematics teachers in three Further Education colleges are used to examine their roles and professional identities. The findings show how teachers’ personal experiences of mathematics, in formal education and the workplace, influence their beliefs and are linked to their narrative and working identities. These teachers enact complex and varied roles but develop a ‘leading professional identity’ that can be linked to significant critical events in the past

A study of the experiences of vocational students learning functional mathematics in further education colleges

The education system in England has long been characterised by a distinct separation between academic or vocational pathways in the post-16 phase. Until recent policy changes this division coincided with the point at which compulsory education ended and mathematics became an optional subject for many students. Those who failed to attain the widely accepted minimum standard of a grade C in GCSE mathematics, however, were often strongly encouraged to undertake a course to improve their mathematical knowledge and skills. This study focusses aged 16-19 and examines the learning experience of those who take a functional mathematics course alongside their vocational programme, as either a recommended option or a requirement of internal college policy. Research regarding the learning of mathematics for these students within the context of Further Education is limited. The study adopts a holistic view of the situation to explore the main factors that influence the student experience, with an emphasis on gaining insight and understanding of students’ perceptions of their learning situation. Using a grounded theory approach with multiple methods, the research includes a series of case studies of seventeen student groups across three Further Education colleges, from which within-case and cross-cases themes are identified. The research findings show how the student experience of functional mathematics is affected by a complex network of inter-linking factors associated with both the organisation and the individual. Although organisational factors such as policies and systems sometimes place constraints on opportunities, social and cultural influences shape student values and perceptions of functional mathematics. There is strong influence from individual teachers through differing interpretations of the curriculum and pedagogical approaches but social structures and relationships within the classroom are also important to students. In addition, the legacy of students’ prior experiences of learning mathematics has an effect on attitudes and emotions, despite the separation of space and time, indicating the significance of both cognitive and affective factors in this interaction of multiple influences. Many students approach functional mathematics in college with prior experiences of disaffection and low attainment but the study shows how attitudes and understanding are transformed for some students within the college environment. Fundamental to these changes is the functional curriculum which, based on the application of mathematics rather than knowledge-acquisition, facilitates teaching approaches that present a new image of mathematics as a useful ‘tool for life’. Using materials such as contextualised tasks to make meaningful links to student lives increases awareness of the relevance of mathematics, leading to greater engagement and understanding. In the transition from school to college there is a marked discontinuity of curriculum and environment, accompanied by value-changes indicating a stronger orientation towards adult life and vocational employment. Students respond positively to functional mathematics lessons where these values are embraced. Academic-vocational divisions, such as differences in values, culture, curriculum and approaches to learning, are evident at multiple levels in colleges and these produce tensions in the student experience of functional mathematics. Some effective bridging is achieved through appropriate classroom practices but coherence requires a multi-level embedded approach involving college structures and departmental policies rather than simply the actions of individual teachers. The research findings suggest that reversing trends of disaffection and failure with mathematics amongst students can be achieved in post-16 education but this is dependent on changes within the curriculum and learning situation. In the light of recent policy changes in England that will increase the numbers of post-16 students taking a mathematics course and the prioritisation of GCSE mathematics over alternative curricula, this study has much to contribute to understanding students’ perceptions of mathematics and the factors that influence their learning experience

The professional identity of mathematics teachers in Further Education

Professional identities may be viewed as narrative constructions in social situations but personal experiences and beliefs are fundamental influences in their development. Within Further Education colleges in England, mathematics teachers are typically expected to fulfil multiple roles, teaching a wide range of curricula and age groups, and this brings additional complexity to their professional identities. In this study, questionnaires and interviews with mathematics teachers in three Further Education colleges are used to examine their roles and professional identities. The findings show how teachers’ personal experiences of mathematics, in formal education and the workplace, influence their beliefs and are linked to their narrative and working identities. These teachers enact complex and varied roles but develop a ‘leading professional identity’ that can be linked to significant critical events in the past

Using digital technology to enhance formative assessment in mathematics classrooms

In this paper, which is based on research from the EU-funded project Improving Progress through Formative Assessment in Science and Mathematics Education (FASMEd), we explore how iPads are used within formative assessment processes by six mathematics teachers and their classes in two secondary comprehensive schools in the Midlands of England. A design research approach is first used to develop and trial six lessons in which iPads are used in different ways within formative assessment processes. Through lesson observations, video analysis and teacher interviews we then examine how iPad technology contributes to these processes, the functions it performs and the distribution of activity between the main actors (teacher, technology and student). An analytic approach is developed which captures the interactions in visual representations, showing how the technology is often used as a form of communication but also performs more active functions which affect the role taken by the teacher in formative processes. The study offers insight into ways in which iPad technology contributes to effective student learning through formative assessment and introduces an analytic approach that may be useful for further studies

Effect of SiO2, total FeO, Fe3+/Fe2+ and alkali elements in basaltic glasses on mid-infrared spectra

A suite of basaltic glasses were examined to determine how subtle compositional changes affect mid-infrared spectra (650 to 5400 cm-1). Glasses with different SiO2, FeOtotal, Fe 3+/Fe2+, and alkali contents were synthesized in a gas-mixing furnace and analyzed using electron probe microanalysis, Mössbauer spectroscopy, and micro-reflectance Fourier transform infrared spectroscopy. The major mid-infrared spectral feature in silicate glasses is a broad peak located at ∼900 to 1100 cm-1 arising from Si-(Al-)O asymmetric stretching vibrational modes. To accurately compare spectra of different glass compositions, we have applied the Kramers-Kronig (KK) transform to our spectra and examined the resulting absorption peak (KK abs. peak). The location of the KK abs. peak shifts to higher wavenumbers as SiO2 content increases (1031-1054 cm-1 with SiO2 from 47.18 to 55.57 wt%). For basaltic glasses with near-constant Al/(A1+Si), the full-width half maximum of the KK abs. peak decreases as alkali content increases (235-188 cnr-1 with Na2O+K2O contents from 0.07 to 3.74 wt%). In contrast, the location and shape of the KK abs. peak are not affected by variations in total FeO (6.06-16.30 wt%) and Fe3+/Fe2+ (0.05-1.17). Our results show that KK transformed mid-infrared spectra of basaltic glasses may be used to determine the SiO2 contents in basaltic glasses, irrespective of FeOtotal and Fe3+/ Fe2+, and the alkali contents if Al/(A1+Si) is known. These observations will aid in the interpretation of laboratory and remotely sensed IR spectra