Making the A grade
During last summer's Channel 4 reality TV programme That'll Teach 'Em, I played the role of maths chief examiner and it gave me the opportunity to compare 1950s O-level papers with today's GCSE. I was surprised to discover that the O-level syllabus was significantly smaller than the current key stage 4 higher programme of study.
The extra topics pupils study today include more difficult calculations involving standard form, transformation geometry, transformations of graphs and, most significantly, all aspects of data-handling, which occupies 25 per cent of the current syllabus. However, maybe because the syllabus was smaller, the questions were more demanding than current GCSE questions with respect to arithmetic and algebraic fluency, and also required candidates to solve problems, use mathematical reasoning and understand the concept of proof, particularly in geometry.
In the short time available for maths in the programme, the teacher was able to improve significantly the pupils' ability to work without a calculator and solve problems such as the first O-level question shown in the box.
Young people are sometimes criticised for their poor numerical skills. This experiment seems to show that, if we choose to spend more time teaching numerical skills, young people will soon acquire them. As exemplified by the first question shown for each syllabus, current GCSE questions tend to test conceptual understanding of number rather than technical fluency, which seems sensible when electronic devices can do the calculations for us.
The other criticism levelled at pupils today, particularly those who go on to study for science and engineering degrees, is that they lack fluency in algebra and the ability to reason geometrically. My experience of marking the pupils' O-level papers confirmed a weakness in algebraic manipulation (see question 2). Question 3, which required a geometric proof, defeated all but two pupils. However, it is interesting to note this comment in the O-level examiners' report at the time: "The most noticeable trends are an increasing weakness in basic manipulation in elementary algebra and a decline in the quantity and quality of the work in formal geometry. Many attempts at factorisation and change of subject are pitiful and a number of schools produce geometry of little value." So, "falling standards" in algebraic manipulation and geometric reasoning are not a new phenomenon.
The current GCSE specifications, first examined last year, include more emphasis on these skills. Are the exam papers reflecting this? GCSE question 3 asks candidates to prove two triangles are congruent, but there is no clear motivation within the question to do this, because no use is made of the result to deduce other geometric facts. The O-level question 3, on the other hand, expects candidates to be able to work out for themselves that they have to use congruent triangles to show that the angles are equal and the question concludes with an interesting geometric result.
So the O-level geometry question is more demanding and more interesting than the GCSE question. How do the algebra questions compare? Not only is the GCSE question 2 on applying algebra to solve a problem more straightforward than the O-level question 2, but the algebra skills needed to solve it are also simpler. Again, the O-level question is more demanding.
I was also struck, when marking the papers, by the pupils' inability to solve a non-routine multi-step problem for which they had to decide what strategy to adopt. I do not know how many pupils in the 1950s would have been able to solve such problems, but my experience of teaching this age-group since the early 1960s leads me to believe that there has been a steady decline in young people's ability to tackle with confidence an unusual or challenging mathematical problem.
Perhaps teachers these days tend to play safe, because they have to meet targets for their classes. If they put their energy into helping as many pupils as possible achieve a grade C or better, it may be at the expense of helping pupils to develop their ability to think mathematically by giving them non-routine and challenging problems.
One question explored in That'll Teach 'Em was whether standards had fallen since the 1950s. Of the 28 pupils involved in the programme, 11 "passed" O-level maths and 27 passed GCSE. But in the 1950s, what data I could find indicated that only 12 per cent of the exam cohort passed O-level maths. In 2001, 12 per cent gained A or A* so it is more meaningful to compare the number of O-level passes with the number of As and A*s achieved. Of the 11 who "passed" O-level, seven achieved A* and four an A in GCSE. A further five gained an A, but "failed" O-level. So it would seem on the basis of this very small sample that the O-level pass was harder to achieve.
The GCSE syllabus has made maths more accessible, so many people have opportunities that would have been denied to them in the 1950s. But is this at the expense of stretching our most mathematically able young people? Many believe so, and Making Mathematics Count, the report of Professor Adrian Smith's inquiry into post-14 maths education, acknowledges this (www.mathsinquiry.ork.ukreport). One of its recommendations is that the QCA develops an extension curriculum for more able pupils. If itdoes this, and follows another of the report's recommendations that handling data is taught elsewhere in the curriculum, we shall return to a maths curriculum that resembles that of more than half a century ago.
Barbara Ball is the professional officer for the Association of Teachers of Mathematics
* A second series of That'll Teach 'Em will be broadcast this summer.
O-level, Cambridge (Syllabus B), 1959
1. Mathematical tables must not be used in this question.
(i) Express 2 ft 5 in as a decimal of 1 yd, correct to three decimal places.
(ii) Simplify 3 V (2 - 1 ).
2. A woman buys x pounds of apples each week for 4s. When the price of apples rises by 1d per pound she gets (x - 1) pounds for 4s 1d. Obtain, and simplify, an equation for x.
Do not solve your equation.
3. ABCD is a rectangle and equilateral triangles AXB, BYC are drawn, X and Y being outside the rectangle.
Prove that (i) the angle AXD = the angle BXY; (ii) the triangle XDY is equilateral.
GCSE, OCR (QCA pilot), 2003
1. Which of these fractions can be represented by a terminating decimal?
Explain how you can tell without working them out.
145 150 155 160 2. David collected 1p and 2p coins in a jar.
There are 374 coins in the jar.
The coins have a total value of pound;5.02.
Use algebra to find how many 1p and 2p coins there are in the jar.
3. A, B, C and D lie on the circumference of a circle. AB is parallel to CD.
BC is a diameter.
Prove that the triangles ABC and DCB are congruent.