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he new Scottish Curriculum for Excellence encourages teachers to create learning situations in which students can be open to new thinking and ideas, think creatively and independently, learn independently and as part of a group, link and apply new kinds of learning to new situations, communicate in different ways and in different settings and use technology in their learning. Students are to be encouraged not only to link aspects of their learning within any one subject but across other areas of the curriculum. These ideas are being promoted simultaneously with the release of TINspire technology which is specifically designed to allow dynamically linked multiple representations which are designed to enable students to see and make connections between images and concepts in mathematics and science. This research asks whether the use of this technology enhances students’ understanding of the mathematics now being experienced in multiple representations and goes on to investigate whether teachers are changing their classroom practice in accordance with the Curriculum for Excellence. In May 2008 the project got underway when training was provided for 12 teachers from 6 Scottish secondary schools. The schools were chosen to represent a range of types from a range of geographical locations and to be representative of most Scottish secondary schools, each being fully comprehensive, ranging from rural to city and covering a large range of socioeconomic backgrounds. The teachers have a range of background experience with one in her second year in teaching and others with many years of classroom experience. There is also a considerable range of experience with ICT in general and with mathematics software, especially in its use in classrooms. The use of TINspire with students was mainly focused from August 2008 to June 2009. All of the teachers used TINspire Teacher Edition software whereas all the students used the handheld version. The teachers completed 66 detailed lesson evaluations which were analysed for illustrations of the impact of the technology and changes in classroom practice and teaching methods. Introduction and Previous Related Research A section detailing the reasons which provided the researcher with the motivation for this particular study is followed by a review of literature relating to the use of multiple representations generally and specifically with respect to ICT. A smaller section considers the background literature relating to the wider aspects of this study and associated teachers’ classroom practice. Important Issues The literature review is followed by a description of the Curriculum for Excellence indicators and also a brief description of what is known in Scotland as Assessment is for Learning, which is an important consideration for all teachers. This research looks specifically at ‘relational understanding’ which is described and compared with the less thorough ‘instrumental understanding’. These terms are defined before the actual research questions are documented. A description of the TINspire technology follows these issues. 4 Research Methods and CPD for teachers The descriptive research design involving a ‘mixed methods’ approach, whose central premise is that the use of quantitative and qualitative approaches in combination provides a better understanding of issues under investigation than either approach alone, is justified in terms of valuing the thoughtful analytical lesson evaluation comments received from a highly trained body of professional teachers as well as the numerical data obtained in the study. A triangulation procedure is also used and involves teacher lesson evaluations, student questionnaires and lesson observations carried out by the researcher. The teachers were provided with a total of 6 days of training throughout the period of the research. These consisted of 2 days before beginning the teaching phase, 2 days in midproject (November 2008) and 2 days in March 2009. The nature and purpose of these training days is described and justified in this section. This training was extremely important and worthwhile and helped instil confidence in the teachers as well as give them an opportunity to share experiences. Research Questions Do teachers find that the use of dynamically linked multiple representations enhances their students’ relational understanding of the mathematics involved in their lessons or not, and what evidence do they provide to support their findings? The research also considers the following related issues: • In what ways is the learning and teaching of mathematics changing as a result of using the software plus handhelds? • When using the technology, are teachers conscious of changing the way they teach particular topics? • When using the technology, are teachers conscious of changing the way they teach in general? If so, what are these changes and how are they justified? • When using the technology, what is the impact on students’ motivation and engagement? • Is gaining mastery of the software/handhelds seen as a burden, barrier, or as valuable and motivating? • In what ways can the use of the handhelds assist teachers in the formative assessment of their students? The questions related to Curriculum for Excellence are: • Do teachers think the use of the handhelds helps students to become Successful Learners, Confident Individuals and Effective Contributors? • Which Curriculum for Excellence indicators provide appropriate criteria against which to measure students’ abilities in the above capacities? The Findings An initial survey was conducted to consider the teachers’ prior use of various forms of technology, including graphing calculator, graph drawing software, spreadsheets and 5 dynamic geometry. It was found that there was a wide spread of experience among the teachers. Most had made some use of a graphing calculator for teaching purposes, most used spreadsheets for their own purposes but not for teaching, about half made regular use of an interactive whiteboard. Ten of the twelve teachers had never used dynamic geometry. By linking dynamic geometry with lists and spreadsheets with the TINspire, the teachers were using technology that they had not previously used for teaching purposes. The next section describes a short survey conducted six months into the project. It confirmed that all lessons being evaluated did involve the use of linked multiple representations using TINspire, and that teachers evaluated these lessons for that very reason. They expressed a willingness to evaluate such lessons whether they were successful or not. The survey also elicited what teachers consider to be reliable evidence of relational understanding, pupil motivation and successful learning. The teachers in the study were representative of the highly qualified profession in Scotland and they demonstrated typical commitment to the project. The response rate for the return of completed lesson evaluation proformas was over 90% and the detail they provided is extensive. In 80% of the 66 lesson evaluations received, the teachers concluded that the use of multiple representation with TINspire enhances students’ relational understanding of the mathematics involved and they were willing to provide extensive evidence to support their argument. Only 3% contained a negative response. Among the evidence which teachers considered when making their decisions were; specific reference to the advantages of the use of multiple representations for the particular lesson topic, verbal and written responses from students, improved discussion, ‘seeing’ students’ understanding and improved retention. An equally large majority of the lessons involved a change of practice for the teachers. It appears that by being asked to use multiple representations with TINspire, the teachers think of new and different ways to teach the particular topic, put more emphasis on links within and across topics, expect more involvement from students and in some instances teach topics earlier than what would be considered normal. When considering changes to their practice in general, teachers highlighted a change in classroom dynamics giving students more freedom to investigate, allowing more discussion, making an effort to link topics and thinking of ways to use the technology to help deepen students’ understanding. In more than half of the evaluations the teachers emphasised the positive impact that using TINspire had on students’ motivation and engagement. A smaller number noted the positive contribution to pace and amount of learning. A variety of other positive comments were made and less than 10% of comments were negative. A convincing majority of almost two thirds argued that gaining mastery of the handhelds was not a problem for students and indeed that it was even perceived as valuable and motivating. Less than 20% of comments related to the difficulties experienced and some indicated the temporary nature of these. In contrast to the students, it appears that gaining mastery of the software and handhelds is more of a 6 problem for the teachers but still a majority of 70% argued that it was not a problem but valuable and motivating and worth the effort. With regard to the impact of TINspire handhelds on formative assessment, teachers were almost unanimously positive in their comments. Other than direct observation of handheld screens, teachers stressed issues such as increased discussion, more questioning, more open questioning, more student self assessment and more instant feedback both to students and teachers. Over 90% of the comments related to positive, beneficial observations regarding the use of handhelds for ongoing formative assessment purposes. Teachers were asked to indicate which Curriculum for Excellence indicators featured in each of their lessons. In more than 90% of the lessons, ‘Enthusiasm and motivation for learning’ was chosen. More than three quarters highlighted ‘Openness to new thinking and ideas’, ‘Learn independently and as part of a group’, ‘Make reasoned evaluations’ and ‘Solve problems’. Close runners up were ‘Relate to others and manage themselves’, ‘Think creatively and independently’ and ‘Apply critical thinking in new contexts’. Lesson Observations and Commentary This section of the report provides a description of each of the lessons observed by the researcher. Attention is focused on the use of dynamically linked multiple representations and the associated issues of classroom practice, teaching methodology and student engagement. The main points which arose were; • emphasis on linking representations and the impact of change in one of these • use of investigative approaches • work being tackled significantly earlier than would normally be expected • high level of questioning and teacherstudent and studentstudent discussion • high level of enthusiasm, enjoyment and interest displayed by students • variety of approaches to help students gain mastery of the handhelds such as ‘lockstep’ teaching, use of worksheet, use of animated PowerPoint slides • issues relating to management of the technology • new approaches to the teaching of statistics Conclusions and Recommendations from teachers’ lesson evaluations The teachers involved in this study, no matter what their background, length of experience as a teacher or extent of experience with ICT were convinced that the use of multiple representations of mathematical concepts generally enhances their students’ relational understanding of these concepts and were willing to provide extensive evidence to support their arguments. Also by virtue of using TINspire in their classrooms and by using files involving dynamically linked multiple representations which include geometry and spreadsheets, there was a significant increase in the use of both dynamic geometry and spreadsheets by the teachers in the sample schools during the project, in comparison to previous practice. A majority of the TINspire lessons involved a change from normal practice. It appears that by being encouraged to think about possible multiple representations of the 7 mathematics involved and by using the technology to assist with this teachers were more inclined to produce a different way of teaching each mathematics topic. Not only did the teachers change the way they taught particular mathematics topics but also the way they taught in general, moving largely from teacher exposition followed by textbook exercises to more practical and investigative approaches involving more active learning for the students together with more classroom discussion. The teachers provided evidence of improved motivation and engagement among their students. This may be a direct result of using the technology but may also be a consequence of the changing classroom practice or even a byproduct of improved understanding. Further study would be required to attempt to apportion causal relationships. A convincing majority of almost two thirds of the teachers’ comments regarding gaining mastery of the handhelds argued that this was not a problem for students and was even perceived as valuable and motivating. It appears that the teachers themselves experienced some difficulty in gaining mastery of the technology but argued that it was well worth the time and effort for the benefits which accrued. Some professional development time is therefore necessary for teachers before they can make effective use of the technology and feel comfortable and confident in its use in their classrooms. Regarding the use of handhelds for ongoing formative assessment purposes, over 90% of the comments relate to positive, beneficial observations. Teachers can as per normal observe students’ written work as well as what appears on their handhelds but it seems to be the changing classroom practice and in particular the increased level of discussion which is allowing teachers to ‘hear the children’s thinking’ and hence gain access to their levels of understanding. Such practice should clearly be encouraged. The teachers also indicated that using TINspire technology met several of the aims of the Scottish Curriculum for Excellence, especially in helping students to become ‘successful learners’ and ‘effective contributors’. In particular the following CfE indicators featured greatly in the teachers’ lesson evaluations; enthusiasm and motivation for learning, openness to new thinking and ideas, ability to solve problems, learn independently and as part of a group and make reasoned evaluations. Conclusions and recommendations from observed lessons • Time needs to be found initially to introduce students to the technology. It is recommended that the minimum necessary keystrokes are introduced for each lesson thus building up an improved facility over time. Some lessons may require little more than the ability to drag an object in a geometry page. • Teachers should experiment with and evaluate a variety of ways of introducing keystrokes to students. Some teachers used worksheets while others used animated PowerPoint presentations. Having these available allows students who fall behind to catch up again without interrupting others. • Teachers should understand the meaning of the phrase ‘dynamically linked multiple representations’ using the technology but should also be aware that other representations may exist. These may include physical representations such as ‘walking a graph’. 8 • When doing data logging using a motion detector it is recommended that students learn from the experience of creating a graph before being asked to interpret distancetime graphs. • Teachers should try to emphasise the impact on one representation caused by a change in another. • Some lessons involved the teaching of topics earlier than would have been expected in the Scottish mathematics syllabus. The technology made this possible and did not appear to cause any problems for the students. Teachers should bear this in mind when deciding how and when to teach each topic. • Students appear to be able to accept new terminology or notation which is used by the handhelds as long as they are reassured by the teacher that they will get a fuller explanation at a later date. • Students appear to benefit from “private investigation” with their own individual handheld. This is not so accessible when working only from the teacher’s edition software displayed on a whiteboard. Hence it is recommended that both approaches be used as each has its own advantages. • Teachers should consider the use of ‘sliders’ to assist students in situations where they are expected to investigate “what happens if …..? or when…..?” • Teachers should also consider carefully the interplay between handheld and the recording of results or findings. Some teachers asked students to record their results in a text page on the handheld while others preferred to have the results written in the students’ jotters (notebooks). • Teachers should be aware of file management (documentation) issues. If handhelds are issued randomly to students each time the class meet then students cannot be expected to store their work, their files. If the handhelds are numbered they can be issued to particular individuals to overcome this problem. However, if the handhelds are shared across a number of classes then this management issue becomes more complex. This can be used as an argument to encourage students to purchase their own handheld. Recommendations for Further Research Further research is needed to explore issues relating to teachers’ use and/or lack of use of technology in mathematics classrooms. Despite the rich body of national (UK) and international research literature providing evidence of the benefits of using technology to enhance students’ learning in mathematics, it appears that many teachers do not take advantage of this technology. We need to go beyond the obvious financial restrictions to enquire what aspects of a teacher’s working environment and conditions act as constraints to the introduction of specific technology and how might teachers be supported in order to reduce the impact of these constraints. We should also look at how teachers can be encouraged and supported to become action researchers. The teachers in this study were involved in action research in that they reflected on their practices and recorded their thoughts, experiences and findings in their lesson evaluations. It would be useful to find out how individual teachers might be assisted to produce further documentation of their experiences for the benefit of fellow teachers and educational research in general. Given the conclusion of the teachers in this study that the use of multiple representations with TINspire enhances students’ relational understanding, it would 9 be useful to investigate whether this is true for all such representations or whether there are specific areas of the curriculum where this approach is most productive and valued. This proved to be beyond the scope of this study. The teachers also indicated that using TINspire handhelds along with a more investigative and enquiring teaching methodology allowed the use of formative assessment strategies. It would seem sensible to conjecture that the introduction of classrooms of networked handhelds could provide even better access to students’ thinking and hence to improved formative assessment. Research is needed to enquire whether or not this hypothesis is correct. Lastly, it would be useful to investigate whether the findings of this Scottish study can be replicated internationally
