<p>University of Southampton, England</p><p>Reports on teacher training, classroom instruction, and dissemination activities</p><p>A) The teacher training </p><p>Designing the training to meet the needs of UK teachers</p><p>During the earlier phases of the project, it became clear that various forms of graph-plotting software, and/or graphing calculators, are quite widely used in the UK in the teaching of Calculus. As a result, the Southampton team explored various forms of software (such as Sketchpad and Autograph, in addition to EucliDraw) with which to realise the aims of the training course within the specification of the CalGeo project. Some examples of the use of these forms of software were discussed at the third and fourth project meetings. </p><p>Given the established use of software by UK teachers of Calculus, pre-trialling of elements of the training material was undertaken to ensure that the project training materials were relevant, both to teaching approaches in England and to models of effective in-service training for teachers. Pre-trialling took place in October- November 2006 with two teachers. One was not only an experienced teacher of Calculus to UK students aged 16-18 years old, but also a keen user of software; the second was also not only an experienced teacher of Calculus to UK 16-18 year olds, but also had experience with the EucliDraw software.</p><p>The outcome of the pre-trialling was clear: - some of the proposed course material was liked (for example, ideas to do with introducing students to the idea of a limiting processes and of ‘local straightness’) but the use of such approaches is already fairly widespread in UK schools - some of the material was considered to be what might constitute useful background material for teachers (for example, proposed material on differentiability and continuity) but what UK teachers are looking for as part of a training course is practical ideas for use in the classroom; and given that the topics of differentiability and continuity, and issues such as the epsilon/delta definition of a limit, are not taught until University-level courses, training material on such topics would not be considered very relevant by UK teachers of pre-University Calculus</p><p>Hence the task of the Southampton team was to design a training course that took the objectives of the CalGeo project, and the relevant elements of the training material produced during the initial phases of the project, and adopt and adapt these to meet the needs of UK teachers of pre-University Calculus.</p><p>Timing and location of the training</p><p>1 The training course took place during 18 and 20 January and 14 and 17 February 2007 at the University of Southampton, UK. The total number of hours of training was 20.</p><p>Number of teachers trained</p><p>15 teachers were trained.</p><p>The content-program of the training</p><p>The training consisted of three elements: - Teaching Calculus: ideas and concepts - Activity-based approaches to teaching Calculus - Using ICT in teaching Calculus</p><p>The first two of these elements provided the introduction to the main element, focusing on the use of ICT, all within a framework that sought a particular emphasis on learners’ intuition and visualization within the Calculus</p><p>The parts of the training devoted to the ideas and concepts of teaching Calculus provided the means for the participating teachers to discuss the following: - The teaching and learning of introductory differential Calculus, including the issue of the definition of limits - The teaching and learning of introductory integral Calculus, including the issue of proofs - The place of ICT in the teaching and learning of Calculus, including what might be suitable forms of technology and where, and how, use of such technology might aid, or hinder, learning</p><p>The parts of the training devoted to activity-based approaches to teaching Calculus provided the teachers with the opportunity to explore the use of Calculus in mathematically-modelling various situations (for example, acceleration, exponential growth) and to think about, and decide, whether or not the use of ICT might be advantageous (or not).</p><p>The major part of the training was devoted to the use of ICT in teaching aspect of the Calculus. Several themes were explored in the training course, including: - approaches to introducing differential Calculus, especially the relative merits of using graph-plotting software (such as Autograph) as compared </p><p>2 with dynamic geometry software (such as EucliDraw or Sketchpad), perhaps with such decisions being based on the aims of the teacher (and probably their expertise with the specific software) - the place of using ideas of ‘local straightness’ and the tangent as the limit of a chord in differential Calculus and ideas of area under curves in introductory integral Calculus</p><p>Equipment</p><p>The training session took place in a well-equipped teaching room with high- powered computer and data projector (beamer). The participating teachers were provided with laptops for use during the training sessions. A total of 25 laptops were available, plus a PC and data projector/beamer for the presenters..</p><p>Evaluation of the training program by the trained teachers</p><p>The teachers rated the training quite highly; the mean evaluation score was 2.1 on a 1-5 scale with 1 being “excellent”. Particularly highly rated were the elements of the training that provided the teachers with the opportunities to discuss approaches to the teaching and learning of Calculus and those that showed effective uses for ICT for teaching. Less successful were the elements of the training that focussed on the place of mathematical modelling, even though such an approach is encouraged within the curriculum specification in the UK (possibly because, at the current time, the modelling approach is probably not carried through sufficiently well to the type of examination questions for which students are prepared).</p><p>B) Classroom Instructions 1 The training was followed up with all the participants of the training. In two schools, the implementation of the test application was followed in some detail. </p><p>Both these test applications were concerned with students in Grade 11 (the students were aged 16-17). In both schools, the teachers worked with full classes of about 20 students each. </p><p>The two teachers tried, as a result of the training, different approaches to introductory differential Calculus. One also tried, as a result of the training, a different approach to introductory integral Calculus. </p><p>The approach of one of the test applications teachers was to have students in pairs investigating the gradient of y = x2 by calculating the limit of the gradient y/x chords as x tends to zero, using a ready-made table to aid the calculation. This was repeated for y = 2x3 prior to the result being demonstrated algebraically. Then the teachers and students used Cartesian graph-plotting software. In the case of this teacher, the preference was to use the software Autograph. The teacher used a data projector/beamer in the class to demonstrate the calculation the students had carried out at a point, leading to showing the gradient of the moving tangent as the point changes.</p><p>3 In the other test application, the teacher decided to try using dynamic geometry software, In the case of this teacher, the choice was Sketchpad software. As a result of the training, the teacher found that dynamic geometry software gave distinct benefits in comparison to Cartesian graph-plotting software such as Autograph. The benefits, the teacher found, were associated with more ‘dynamic control’ over the form of diagram projected to the class and which students in the class could adjust various elements of in order to see the effect – such as exploring the differentials of polynomials of the form a*f(x) and f(x)+a.</p><p>In each case, the teaching took place in a regular classroom set up with computers and a data projector/beamer.</p><p>The students said that they gained a lot of understanding from the lessons by being able to experiment with the software. The teachers agreed that that their students seemed to be able to gain a better understanding of Calculus concepts through use of geometry software. </p><p>C) General comments on the implementation and follow-up to the training in the UK </p><p>The teaching context in the UK (such as the widespread use of ICT in teaching, including the use of graph-plotting software, and, to a lesser extent, dynamic geometry software in the teaching and learning of mathematics, including Calculus) necessitated careful pre-trialling of the training material to adapt these to meet the needs of UK teachers of Calculus in the pre-University education sector. </p><p>Attracting teachers of post-16 students to courses, and for these teachers to get permission (from their Principals) to attend, are both notoriously difficult in the UK. It took a good deal of work to plan a suitable course and ensure suitable attendance.</p><p>The over-riding preference of many (if not most) UK teachers is for training that is perceived as having a direct impact on their classroom teaching and which provides ideas that they can quickly adapt to their own classroom. This is, in the main, very understandable. A drawback, however, is that training elements that attempt to get teachers to ‘rethink’ some aspect of teaching appear to be most beneficial when these training elements form part of a long-term professional development process, especially if the ideas in the training are going to be taken up by teachers to any deep extent. As such, it would be helpful if the training material developed for the CalGeo project could be incorporated, or could form a component, of a long-term professional development process encompasses other aspects of teaching mathematics to students in the 16-19 age range and/or preparing from University-entrance level examinations in mathematics.</p><p>4</p>