Thirty years of School Certificate maths

Press, 3 July 1978, Page 13

Thirty years of School Certificate maths

By

B. W. WERRY,

senior lecturer in mathematics educa*

tion, secondary division, Christchurch Teachers’ College.

The problem of comparing today’s School . Certificate mathematics paper with one from 30 years ago is similar to the problem of comparing motoring in its early days with motoring today. Not only are cars different in design today, but they are also available to more people, and are put to new oses.

The name “School Certificate” remains the same, but all else has undergone great change. The prescribed content for the examination, and the range

of ability of the people for whom the examination is intended, have changed radically. In 1947, School Certificate mathematics was for the mathematical elite only. About 63 per cent of fifth formers attempted the examination in 1947, and of these approximately 48 per cent sat in mathematics. Many of these candidates would have been in their fourth, fifth or later years at secondary school because School Certificate was intended originally to be a four year course for most pupils. Only

the very able were to attempt it in three years. Nowadays almost all fifth formers attempt School Certificate in their third year at secondary school, although, of course, many unsuccessful candidates repeat it in subsequent years. About 80 per cent of the total number of candidates now take mathematics for School Certificate. But there were other factors at work in 1947 to make the effective selection of those taking School Certificate mathematics even more narrow. About 15 per

cent of the primary school population did not enter secondary school, and only about'half of those who did ever reached Form 5.

By contrast, these days almost all children receive a secondary school education. They also stay longer — more than 80 per cent of the population remains for three or more years at secondarj' school. In. broad terms, then, the percentage of a given age group likely to sit School Certificate mathematics has risen from about 15 per cent in 1947 to about 70 per cent in 1977. We must also remember that roughly half of those sitting the examination pass; the other half fail.

Assuming a normal distribution of mathematical ability, a person who scored around 50 per cent in 1947 might be expected to score well over 75 per cent in 1977 because of the increased numbers of less able pupils attempting the examination. Whereas 30 years ago only the very able pupil reached School Certificate, today the examination must also cater for the average pupil. And because the range of ability is extended downwards to a greater extent, it is to be expected that the examination should reflect this too.

For these reasons it is not really possible to make a valid comparison between the standards of two papers 30 years apart. In addition, there have been substantial content changes in this time, together with a gradual reappraisal of the aims of teaching mathematics. The examinations of 1947 and 1977 are not only testing a different range of students, they are also to some extent testing different mathematics. We can, however, describe the contents of the respective papers and point out similarities and differences. Superficially, the papers appear quite different. In 1947, candidates sat two two-hour papers, one in arithmetic and algebra, the other in geometry and trigonometry. Today, there is one three-hour paper in mathematics in which the branches of the subject are integrated. On closer inspection, however, the weightings given to the separate branches appear roughly similar to those in the earlier paper.

The format of th# 1947 paper was uninviting. There were very few diagrams, and the small type gave a cramped appearance. The 1977 paper, by contrast, is more pleasantly spaced, and graphs and diagrams are plentiful. To help the candidates, questions are carefully

structured by being subdivided into smaller parts. All answers are to be written in spaces provided on the paper. These features not only help the candidate show what he has learned, but also tend to make marking easier, more consistent, and more reliable.

In the arithmetic questions there is much in common between the two papers. For example, from the 1947 paper we read: “On leaving school A and B obtain posi-

tions with the same business firm . . . B enters the Cash Sales Department, and the

manager gives him the following problems to work out: A machine is marked up for sale at £52 10s. If a paper: “A litre is the volume reduction of 5 per cent is to be made for cash payment, what does a customer, who pays cash, give for the article? . . The 1977 paper on the other hand contains this: “A shopkeeper buys calculators at the standard Trade Price. The Trade Price for one model if $2O. His profit is 40 per cent of the Trade Price. What is the shopkeeper’s selling price? . Computation itself has never been tested explicitly in School Certificate exam-

inations, but only indirectly in association with other skills. For example, we find this problem in the 1947 paper: “A solid metal sphere placed on the top of a monument contains 1437 1/3 cubic inches. What is the radius of the sphere?" Not only were candidates expected to remember the appropriate formula and to be able to rearrange it to fit the circumstances of the problem, but they also had to manipulate relatively complex fractions. In the 1977 paper there is less emphasis generally on complex multi-step problems such as this. The modern trend is reflecting the belief that a knowledge of concepts and principles can be tested more effectively in simple situations, not com- [ pl i cated by unnecessarily

complex manipulation or computation. For example, a typical problem from the 1977 paper: “A litre is the volume of a cube with sides each 10cm long. How many litres would a container hold which is 25cm by 20cm by 10cm?”

It is in the geometry section that the greatest contrast appears. In 1947 it was still the practice to ask for proofs of theorems, For instance: “If there are three parallel straight lines, and the intercepts made by them on any one straight line that cuts them are equal, prove that the corresponding intercepts on any other straight line that cuts them are equal.” The applications which followed such questions required candidates to construct their own diagrams. Candidates were also given a choice of four from six questions. In 1977 the choice was between “traditional” and “transformational” approaches to geometry. Exercises in both options tended to be more numerical and less theoretical than any question in 1947, and although no formal proofs were required, candidates were still expected to provide reasons for answers in many sections. The traditional alternatives required candidates to apply relationships to diagrams provided by the examiner. Much new termi-

nology and notation is to be found in the transformational section, for example, references to the symmetry properties of geometrical figures, and to the operations of translation, reflection, and enlargement. None of this was to be found in the Euclid-based approach of 30 years ago. To judge whether mathematical standards have changed since 1947 it would be necessary to take the top : 15. per cent of today’s pupils, teach them the con 1? tent of the mathematical syllabuses of 30 years ago before requiring them to sit the 1947 paper. Such a procedure is just not practicable. An abstract comparison of the two papers says nothing about the pupils sitting them. Nor do we know anything about the extent to which actual marks had to be adjusted to make them comparable to marks in other subjects in 1947. It is true, however, that until a few years ago raw marks in mathematics were scaled upwards by a considerable amount. A more appropriate comparison may well be with today’s University Entrance mathematics examination, but this would be complicated by accrediting procedures used in that examination.

But perhaps the last word should come from the Minister of Education’s Report to Parliament in 1947: .

changes in the composition of the upper classes of the

primary school and the lower forms of the postprimary school make it difficult to compare in any statistical way the average standards of work attained now with those achieved in the same classes 10, 20, or 30 years ago.

“The classes are the same only in the sense that they have the same names. It is a fact that employers must appreciate when electing staff. Thirty years ago some employers would select junior office staff from young people with only a Form II education, and might well have secured boys and girls of good aven age intelligence. Now, with rare exceptions, only the very dullest cease formal education at Form 11.

“A boy who had completed Form IV in 1917 could, in general, be relied on to be well above the average in native ability, since he belonged to a selected group; in 1947, as we have seen, he might be barely average, since most of the school population reach that level.

“I am convinced that many of the complaints from employers as to the poor standards of entrants to offices come from a failure to realise the change that has taken place in the constitution of post-primary school.” Many believe, nevertheless, that the best pupils today are at least as good as the best of 30 years ago.