special scottish issue || school assessment in scotland

School Assessment in ScotlandAuthor(s): Bill RichardsonSource: Mathematics in School, Vol. 28, No. 1, Special Scottish Issue (Jan., 1999), pp. 27-29Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211956 .

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School Assessment in

Scotland

by Bill Richardson

The purpose of this item is to explain and illustrate the main structure of examinations that pupils in Scotland take towards the end of their time in secondary school. It does not set out to be a comprehensive history of school examinations in Scotland. Readers who want detailed information might benefit from reading either of the books by H. Philip. Scotland has, in the view of many, a distinct advantage over England, in that it has just one statutory examining body which, until recently, was called the Scottish Certificate of Examination Examining Body (SCEEB). Thus there is no question of'shopping around' to find an Examination Board which has a syllabus or examination structure which suits better than another. A few schools do take English examinations but these schools are from the independent sector, perhaps with a tradition of sending pupils on to Oxbridge.

In common with the rest of the UK, in Scotland pupils are required to stay at school until they reach their 'leaving date' which is related to their sixteenth birthday and, since this was raised in 1972, the demand for school assessment has in- creased dramatically. But, before explaining how our assessment works, I want to describe the main structures in the Scottish school system. Pupils start primary school at the age of five, as elsewhere, but the change to secondary school is after seven years in the primary sector. (To the best of my knowledge, there are no middle schools in Scotland.) Thus pupils in first year in Scottish secondary schools (designated as S 1) are Year 8 as other parts of the UK would label them. The first level of SCEEB is towards the end of S4 (Year 11) and in step with GCSE. After that there is a divergence as in S5 (Year 12) the target for most pupils is to prepare for sitting subjects at Higher Grade. A good pupil will sit five Highers and entry to tertiary education is largely based on results in Highers. A significant number of pupils leave at the end of S5. Some go into employment, some to local colleges and some to universities (of this group, many will live at home and go to their local university-a prominent feature in Scotland). Those who stay on into S6 (Year 13) take a variety

Mathematics in School, January 1999

of courses, some will retake Highers to improve grades or take up fresh Highers to increase breadth, whilst many take one-year courses designed to assist in the transition to university. These courses are called the Certificate of Sixth Year Studies (CSYS)-a name which does not trip off the tongue -and are, in broad terms, comparable to A level examinations. It may also be worth noting that there is no overt selection on entry to the secondary sector: state schools are comprehensive but, by virtue of their catchment area, some are more academically inclined than others.

So, moving to the assessment, it is not appropriate to go back to the origins of the Higher Grade (over a hundred years ago) but to 1962 when the system of O Grade courses for pupils in S3 and S4 was started (prior to that, pupils took Lower Grade examinations, but in S5, alongside Highers) and the SCEEB came into being (previous assessment had, to a greater or lesser extent, been the province of the Inspec- torate). The courses leading to O Grade examinations were intended to be suitable for the upper 30% of the ability range. In the area of mathematics, there were two main subjects. O Grade Arithmetic was, eventually, sat by about 60% of a year group and was sufficiently well thought of to be part of entry qualifications to jobs and colleges. O Grade Mathemat- ics was attempted by some 40% of the age range and provided entry to employment and to more advanced courses. It was assessed by two end-of-course examinations, the first paper was multiple choice and the second required extended answers. It was a fairly typical syllabus and assessment and I include examples later. With the pressure caused by the rais- ing of the school leaving age to 16 in 1972, it became clear that more pupils wanted to have courses which provided certificates. With O Grade Arithmetic available, mathematics departments did have something which could be tackled by more pupils than was the case in other departments but, even so, something was lacking and many schools made use of the CSE courses available through the Northern Regional Examining Board. However, things were moving in Scotland to address the issue of across-the-board assessment and, in the second half of the 1980s, O Grade Mathematics was replaced by Standard Grade Mathematics. (The Standard Grade courses were phased in over several years but, as ever, mathematics was one of the first.)

Standard Grade courses, which are the current courses in Scotland, differ significantly from O Grade and also from GCSE. There are three levels of assessment: Foundation, General and Credit (in ascending order of difficulty). Presen- tation is for adjacent pairs (Foundation/General or General/ Credit) and the vast majority of candidates sit both papers for which they are entered. It needs stressing that the courses and examinations are designed to cover the whole ability range. In 1996 Standard Grade Mathematics was taken by 96% of the S4 cohort (Scottish Examination Board, 1996, p. 4). The results are given on a seven point scale. An award of a 1 or 2 is available only from the Credit paper, General papers provide a 3 or a 4 and Foundation paper a 5 or 6. An award of a 7 is described as 'course completed'. In 1996, the number of awards in Mathematics were:

27



Award 1 2 3 4 5 6 7 Tota I

Number 9 412 8 578 13 936 10 930 13 938 4 126 596 63 246 Percentage 13.8 12.6 20.5 16.1 20.5 6.1 0.9 96

[Scottish Examination Board, 1996, Table 1, p. 7]. As well as recording the overall award, the certificate contains awards for separate elements. In mathematics, these are Knowledge and Understanding; Reasoning and Application; and Investi- gating. The first two each count for 40% of the overall award and are assessed in the end-of-course examination whilst the 20% for Investigating is assessed within the school, usually by means of 'extended problems' provided by the SCEEB. However, there are modifications being debated at present which will result in some changes a few years hence.

Moving now to Higher Grade Mathematics which, as mentioned, is a one-year post-16 course and, as such, cannot be compared directly with A level. The syllabus includes rou- tine items of coordinate geometry (straight line and circle); introduces vectors and goes as far as the scalar product; cov- ers work on sequences (via recurrence relations); extends work on simultaneous equations to include one quadratic; introduces ideas and notation of functions as far as inverses; looks briefly at quadratic, exponential and logarithmic functions; looks at the remainder theorem; radians are introduced; the standard trigonometric addition formulae are covered, as is work on expressing a cos x + b sin x as k cos (x - a). Finally, the Calculus is introduced and standard methods and results are covered, the Chain Rule being included but not the Product Rule. Integration is applied to a variety of contexts including area under a curve (but not to solids of revolution). Currently, assessment of Higher Grade Mathematics includes an in-class investigation (worth 10%) and a pair of end-of-course examinations and examples of these are also given later.

Examples of Assessment

O Grade Mathematics (from 1973 Paper Two) Q10. ABC is a horizontal triangle with AB = 24 cm, AC =

15 cm and angle A = 50. Calculate the length of BC correct to one decimal place. (5 marks)

1518

%cm

24 D

M is the mid-point of AB and MP is a vertical line of length 18 cm. Given that angle MCP = 570, calculate, correct to one decimal place, (i) the length of PC, (4 marks) (ii) the size ofangleACB. (3 marks)

Q11. (a) The electrical resistance R ohms of a given length of copper wire of circular cross-section varies in- versely as the square of the diameter d cm. If the resistance of a wire of diameter 0.06 cm is 0.05 ohms, find the formula connecting R and d and calculate the resistance of a wire of the same length and diameter 0.3 cm. When dis halved, what is the effect on R? (8 marks)

(b) Express p in terms of q, given that

2 q=1 +

p (4 marks)

28

Standard Grade Mathematics (1998 Credit Level) Q3. A skip is orism shaved as shown in Fiy. 1.

3m

<

1.4 m

Fig. 1

The cross-section of the skip, with measurements in metres, is shown in Figure 2.

3

x x

1.4

1.5

Fig. 2

(a) Find the value ofx. (1 mark) (b) Find the volume of the skip in cubic metres.

(3 marks) Q4. A sequence of terms, starting with 1, is

1, 5, 9, 13, 17, Consecutive terms in this sequence are formed by add- ing 4 to the previous term. The total of consecutive terms of this sequence can be found using the following pattern. Total of the first 2 terms: 1 + 5 = 2 x 3 Total ofthe first 3 terms: 1 + 5 + 9 = 3 x 5 Total of the first 4 terms: 1 + 5 + 9 + 13 = 4 x 7 Total of the first 5 terms: 1 + 5 + 9 + 13 + 17 = 5 x 9

(a) Express the first 9 terms of this sequence in the same way. (2 marks)

(b) The first n terms of this sequence are added. Write down an expression, in n, for the total.

(3 marks)

Standard Grade Mathematics (1998 Foundation Level) Q.11 (a) Raymond earns L6500 per year.

He is given a 3% increase. Calculate the increase he is given for the year.

(b) Raymond had wanted an increase ofLL4 per week. Does he get as much as he wanted? GIVE A REASON FOR YOUR ANSWER.

Q 13. These necklaces are made with black and white beads.

(a) Complete this table.

Number of black 1 2 3 4 5 6 12 beads Number of white 6 9 beads

(b) Write down a rule for finding the number of white beads if you know the number of black beads.

Higher Grade Mathematics (from 1998 Paper Two) Q7. The functionfis defined byf(x) = 2 cos x - 3 sin x.

(a) Show that f(x) can be expressed in the form f (x) = k cos (x + a)' where k > 0 and 0 < a < 360, and determine the values ofk and a. (4 marks)




(b) Hence find the maximum and minimum values of f (x) and the values of x at which they occur, where x lies in the interval 0 < x < 360. (4 marks)

(c) Write down the minimum value of (f(x))2. (1 mark) Q8. A gardener feeds her trees weekly with 'Bioforce, the

wonder plant food'. It is known that in a week the amount of plant food in the tree falls by about 25%. (a) The trees contain no Bioforce initially and the gar-

dener applies 1 g of Bioforce to each tree every Sat- urday. Bioforce is only effective when there is continuously more than 2 g of it in the tree. Calcu- late how many weekly feeds will be necessary before the Bioforce becomes effective. (3 marks)

(b) (i) Write down a recurrence relation for the amount of plant food in the tree immediately after feeding. (1 mark) (ii) If the level of Bioforce in the tree exceeds 5 g, it will cause leaf burn. Is it safe to continue feeding the trees at this rate indefinitely? (4 marks)

Q 11. (a) The variables x and y are connected by a relationship of the form y = aebx where a and b are con- stants. Show that there is a linear relationship between logey and x. (3 marks)

(b) From an experiment some data was obtained. The table shows the data which lies on the line of best fit.

x 3.1 3.5 4.1 5.2

Y 21 876 72 631 439 392 11 913 076

The variables x and y in the above table are connected by a relationship of the form y = aebx. Deter- mine the values of a and b. (6 marks)

References Philip, H. A Short History of National School Certificates in Scotland. Philip, H. The Higher Tradition, ISBN 0901256846. Scottish Examination Board, 1996 Examination Statistics.

Author Bill Richardson, Elgin Academy, Elgin IV30 4ND.

TRIPLETS

by A.P. Mason and B. Murphy

Two seemingly unconnected problems:

1. Find all the cuboids with integer length edges whose volumes and surface areas are numerically equal.

2. Find all combinations of three regular polygons which fit together exactly at a point.

1 A Volume = abc Surface area = 2ab + 2bc + 2ca

So we need to find integer values a, b and c satisfying abe = 2ab + 2bc + 2ca * Y2= Y + Y + + (Equation A)

Without loss of generality assume a < b < c:

a = 1,2 No solution.

a= 3 then Y + c = Y,

a = 4 then Y + V = Y,

b = 3,4,5,6 -= No solution b = 7 c = 42 (3, 7, 42) b = 8 - c = 24 (3, 8, 24) b = 9 -c = 18 (3, 9, 18) b = 10 -= c = 15 (3, 10, 15) b = 11 -= No solution b = 12 -= c = 12 (3, 12, 12) b 2 13 -=c < b

b = 4 ~ No solution b = 5 - c = 20 (4, 5, 20) b= 6 c = 12 (4, 6, 12) b = 7 - No solution


b= 8=> c= 8 (4,8, 8) b > 9 - c < b

a = 5 then X + ) = , 0 b = 5 c = 10 (5, 5, 10) b = 6 -~ No solution b27c < b

a = 6 then b + Y = X b 6 c = 6 (6, 6, 6) b 7 c < b

2. A regular n-sided polygon has interior angles of 180' - 3600/n So we need to find integer values a, b and c satisfying the

equation:

(1800 - 3600/a) + (1800 - 3600/b) + (1800 - 3600/c) = 3600 = * Y + Yb + Yc~ = Y22

This is identical to Equation A. Hence the same 10 triplets satisfy both problems. Algebraically this is clear, but geometrically.. .?

We leave the interested reader to look at the 2- and 4-dimensional analogues of problem 1 and to see if these compare with other numbers of regular polygons and polyhedra meeting at a point.

Authors A.P. Mason and B. Murphy, Fullbrook School, Selsdon Rd, New Haw, Weybridge, Surrey KT15 3HW.

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