Conclusions and Refinements

The emphasis of this pilot study was to see the way in which trigonometrical concepts develop. To do this a knowledge base was established in terms of the initial concept maps and then another in terms of the subsequent maps. In between the content and the

Pure 1 Statistics 1 Decision 1 Total mark Final Grade

P1 64 62 82 69 C P2 100 76 90 89 A P3 68 47 48 54 D P4 79 77 62 73 B P5 20 30 44 31 F P6 46 55 80 60 C

understanding, P1, P2 and P4 provide evidence in their concept maps and interviews that they are making little effort to understand trigonometry relationally [§3.1.2] and for them an instrumental understanding is all that is required. Their knowledge of core properties has not been derived by consideration of the process but learned by rote. It is a point of interest whether their teacher’s style of delivering operational knowledge that is already condensed [§3.1.4] and his emphasis on operational problem solving procedures suits their learning method or dictates it. There is little evidence of a Process conception as defined by Dubinsky [§3.1.5] being developed and little evidence that the spatial-visual imagery introduced during this component is being employed to give a deeper awareness of the core concepts [§3.2.3].

The evidence in P3’s first concept map pointed to an interlinked schema and during the lessons she clearly attempted to find a meaning to what she was learning in the light of what she already knew. This points to an attempt at relational understanding. She constructed the special angles triangles each time using Pythagoras theorem to find the lengths rather than memorising the details but consistently made the error of giving the perpendicular a value of 2 instead of the hypotenuse. This suggests an attempt to understand the spatial–visual representations as procepts that simultaneously represented both operational and conceptual elements as described by Gray & Tall [§3.1.6] but the teacher emphasis and P2’s high score in the task questions led her to abandon these attempts at relational understanding in favour of memorising facts and formulae [see Delice & Monaghan §3.3.5]. Despite the fact that P2’s test scores were very erratic, P3 insisted this would give her a better chance of success in the assessment. The conclusions drawn from the evidence in P5’s maps show that no schema modification has been made to include the new content. [See Harel & Tall §3.4.2].

With regard to issues of the assessment objectives, the conclusion that may be drawn from the second concept maps and the second interviews is that the reality of what is being learned in this class room may be falling short of the assessment objectives stated in the AS/A level modular syllabus (2004) to ‘Develop an understanding of coherence and progression in mathematics and of how different areas of mathematics can be connected’ and again under synoptic assessment which states:

‘Synoptic assessment in mathematics addresses candidates’ understanding of the connections between different elements of the subject. ...Making and understanding connections in this way is intrinsic to learning mathematics’

the new AS/A Level Mathematics, the post 16 mathematics advisory group stated that:

We wondered whether it would be appropriate for the word “function” to be properly defined at AS since many students will have used it loosely at GCSE. [Porkess, R. Commentary on QCA’s draft proposal for AS/A Level Mathematics (2002) p 12]

The evidence shown here is that for these students the lack of a definition for function has led them to believe that it relates to the format in which it is presented i.e. f(x) =sin x is recognised as a function but y=sin x is not. There is no connection between functions and properties in any way.

With respect to the methodology, the concept maps had given some good insights into the student’s schemas [§3.1.3] and these were confirmed by the interviews. However the interviews required a greater structure in order to compare and contrast answers so that evidence of any patterns in the students developing schemas might be identified. It was therefore decided that a questionnaire would be devised that would provide the framework for investigation of students understanding of core issues. In addition a failing of the pilot study was that no analysis had been made of student’s initial competence in trigonometry. An assumption had been made that the student’s ability in the subject were roughly comparable because they had all achieved high grades at GCSE (or equivalent). In the main study this needed to be verified and not assumed. Finally although looking over the students attempts at the past P1 questions was interesting it would have been more useful to have asked the students to talk through their thinking as they attempted the questions. This could have provided a deeper insight into any tentative links, algebraic or spatial- visual, that the students were aware of but did not pursue and, if this was indeed the case, why they did not pursue them. It was therefore decided that the same questions would be put to the students in the main study but under supervision where they would be encouraged to talk through their thinking.