Pilot Study: Sample Details

The purpose and methods of the research were explained to the students and their consent for involvement within the study was obtained just before they started the trigonometry component of their AS course. As a preliminary each student was asked to draw a concept map for trigonometry and how the different aspects of it linked up in their minds. Only one of the 6 students was familiar with concept or mind maps so the researcher gave a demonstration of a possible map for circles that included formulae, spatial images and possible applications to cylinders etc including links as an indication of how a map may be constructed.

During the following four weeks, three lessons a week were observed as the class covered the chapter in the module. After each of the first three lessons two of the students were asked to wait behind and participate in interviews through which some insight into what each student had learned and understood about the aspect of the module that had just been taught was gained. For example the students were asked ‘Tell me about what you learned today?’ then depending on the response further follow up questions were

does that relate to what you know about sines, cosines and tangents?’ etc. The intention was to gain an indication of whether or not what had been learned contributed towards a modification of their original schema. Towards the end of the trigonometry component of the course (lesson 9) the students were given the set of task questions [§5.7.1] and invited to draw a second concept map. The following lesson (lesson 10) the teacher gave the class an ‘End of Module’ test which he had constructed from past P1 questions. During lesson 11 the test of the previous day was considered and after this a further interview, to investigate issues that arose from the second map, was then held with each student.

5.3.2 The School.

The school is a mixed, State grammar school within Southern England. It has approximately 1 000 pupils, of which 250 are in the 6th _{form. Pupils are awarded a place in}

the school on the basis of the outcome of an examination in their final year of primary school. This exam, nationally known as the 11+, is intended to consider students achievement and potential on the basis of mathematics, verbal and non verbal reasoning and written English.

The examination is no longer supported nationally but the local authority has opted to maintain it against most of the prevailing trends over the past thirty years. Approximately 30% of the children within the area administered by the Local Authority currently attend the grammar schools but this includes a large proportion within the 6th _{forms (median age}

16.5 -18.5). The Grammar schools have large 6th _{forms in comparison to with other}

secondary schools. College numbers do not contribute to the education authorities figures. Thus the number of children selected for the grammar schools from the state primary schools is far less than 30% and represents children at approximately the 75% percentile or above. All may be identified therefore as the more able.

At the end of key stage 4 students (age 16 years) take 10 or 11 subjects at GCSE level. Within the school reported within this study the pass rate for students achieving A* to C grades is currently 88% compared to the Local Authority average of 59.1% and a national average of 45.8%. Pupils who have performed well, that is gained at least 30 points at GSCE (A*-5 points, A-4, B-3, C– 2 and D-1) across the range of their subjects with a B or above required in the subjects that would be studied at AS and A2 level, may apply for entrance into the 6th _{form. In line with most schools a pre-requisite to studying}

mathematics at A-level is that the Higher paper [§2.2.2.1] was taken at GCSE (available grades: U, C, B, A and A*) and a grade of at least a B obtained.

5.3.3 The Students.

The students who participated within the Pilot Study were awarded grades of A or A* at GCSE or their equivalent. The consequence is that the sample within this study represents a small subsection of the population of English students who were selected by ability at 11 and then again at 16. In this particular instance they were all confident about their own ability in mathematics particularly at the start of the course but this was to change. The pilot study sample was selected by the Head of Mathematics at the school on the grounds that he thought they would be more interesting since they were not “full time mathematicians” but were taking mathematics AS/A2 level as part of a mix of subjects that included Psychology, Art, PE, English, French and History. The sample consisted of four students who had been at the school from the outset of their secondary education and two Chinese students who had entered the school at the start of the year. The Chinese students were able to understand English but had more difficulty explaining clearly their thinking and the reasons for taking a particular action. They paid little attention to expositions but were alert to any resultant formulae or axiom. The English students, in contrast, engaged in and contributed to the expositions and took notes of the background rationale as well as copying down notes from the board.

5.3.4 The Teacher.

The teacher was just starting his second term teaching after retraining via a Graduate Training Scheme, his degree was in Business Studies. He had previously had a career in market research which culminated in him running a small company. He was very hard working and had an informal, relaxed attitude in the classroom that was appreciated by the group, but his focus was to teach the students what they needed to know to get the best grades possible in the examinations. He frequently made comments such as “I’ll start you off but I won’t be able to sit in the exam with you”; “make sure you learn this question as there is a very good chance it will come up in the exam”; “Unfortunately you will have to learn this for the exam” [§5.6.4]. To achieve his objective he taught specific procedures that were detailed and served to solve a particular type of problem [§5.6]. Each lesson considered a new type of problem and an associated procedure to achieve an answer. He was not, he said, a visualiser and so preferred to use algebraic means to solve problems whenever he could. He explained that he didn’t like the topic associated with graph transformations and had never found much point to them. Where it was clear that images were a preferable alternative in part of a solution process, such as the use of the unit circle to find obtuse or reflex angle solutions, the image was presented complete [§5.6.4]. Privately the teacher thought the Chinese students were the strongest in the group and would perform best in the examinations. Interestingly he also was diffident about his own

a real mathematician!”

5.3.5 Lesson Format

In the classrooms where the group was observed, there was a board and a projection screen. The lessons were timetabled as doubles and were 100 minutes long. The pilot study teacher followed a lesson format that started with an exposition, either at the board or using power point projected on to the screen, followed by questions chosen by the teacher from the set text. The questions in the main were technique exercises. The class either worked on the questions by themselves or conferred quietly whilst the teacher moved around the room and answered queries put by individual students. Sometimes the teacher felt the need to call the attention of the class to consider an issue that had arisen that he felt was relevant to the whole class but mostly the groups were left to explore the problems, and attempt the solutions themselves. Homework was set at the end of each lesson and usually took the form of finishing the set questions which were to be handed in to the teacher at the start of the next lesson.

5.4 Outcomes of Pilot Study