Assessment at the End of the Component

The test set by teacher consisted of the trigonometry questions from past P1 papers [See §7.9]. These questions required analysis to determine what was required, an ability to disseminate a trigonometrical function into its component parts and familiarity in dealing with radians and surds. This was an accumulation of the different aspects of the topic presented and the students were required to consider the different aspects within the whole. As they answered the questions the students were observed by the teacher and the researcher however, since the teacher wanted the students to attempt the questions under examination conditions i.e. no conferring and within a time constraint, it was not suitable for the researcher to ask the students about their thinking processes as they went. This was a shortcoming of the pilot study that was reconsidered for the main study [§5.8]

However the student’s papers viewed after the lesson showed that P2 had no understanding of the period of the sine wave, wrote out procedural answers in full taking 10 lines or more showing an inability to flexibly short cut procedures. She did not use spatial visual representations flexibly and was muddled by a question where the graph y=A +B Cos 2x was shown with coordinates given for a maximum and minimum value for which the values of A and B had to be determined. Surprisingly she factorised 3sin2_{x +sin}

x-2 =0 wrongly and did not give any values for sin x=-2/3.

P3 started most of the questions with a correct strategy but had problems with algebraic misconceptions, for example, she wrote 2sin (2x+k) = 2sin 2x + 2sin k. When she could

unit circle and the graphs to refocus her thinking.

P4 did not attempt most of the questions. When he did he frequently made errors in algebra manipulation. His solution strategies were random, for example, in the question: Given that f(x)≡ 3+2Sin (2x+k) and y=f(x) passes through (15,3+√3) show that k=30, he substituted 15 for x but failed to substitute for y. This suggests a stimulated Action approach that has not been properly thought through as to the purpose of the Action. P1 started parts of the questions except the one where the graph was shown (described above) which she did not attempt at all, but frequently abandoned her solution process even when it was correct. She used the unit circle to find solutions. P5 was absent this lesson.

The students did badly in this test. Their marks were:

P3: 20% P1: 12% P4: 10% P2: 8%

The results indicated that none of the students would pass the pending AS level examinations.

Despite the fact that in this test, rather than being the highest ranking student, P2 achieved the lowest mark and P3 the highest, P3 remained convinced that P2 had a better style of learning since it involved the memorization of formula. P3 therefore still rationalised that she should follow P2’s style:

20% is not going to get me very far is it! If I am going to pass this exam I need to learn the formulae like P2 does. I mean it’s no good liking maths if you fail the exam. From now on my priority must be to get serious about this exam!

(P3)

P2’s perceived style of learning appeared to match the general philosophy that pervaded the teachers admonishment to “learn” and that of the class, explicitly at least, to subscribe to it. This became the perceived route to success for P2 and P3.

Despite the fact that all of the students within this sample had been awarded an A* or A in mathematics within the GCSE or its equivalent, the class teacher gave the following explanation for the poor performance. He said that he could:

“… put this down partly to the fact that the class are not ‘mathematicians’ because they are doing a mix of A-level subjects and partly down to the hardness of the subject”.

The conclusion of these results is that the past Pure1 questions were beyond the ability of this group. They were unable to consider trigonometry as a whole, unable to connect the component parts of an algebraic representation of a compound function with the series of geometric transformations it represented. They were unable to switch between representations and poor at dealing with radians. The also had poor algebra manipulation skills.

However, the AS mathematics results for this group generally present a different picture, one that confirms P2’s approach to mathematics and confirms P3‘s approach to elements of the P1 course.

Table 5.2 Pilots Group’s AS Mathematics Results

The exceptionally high marks achieved by P2 and to a lesser extent P4 appear to show the benefit of practising past paper questions extensively as a preparation for the assessment. However the marks here give indications of ability that are in marked contrast to those indicated by the concept maps. It is worth noting here that P1 decided to continue the course through A2 but not into Higher Education. She said it would look good on her CV to have A-level mathematics but she did not like mathematics enough to study it thereafter.