# Hard times

Underlying all the recent criticism of standards in mathematics teaching is the unchallenged assumption that GCSE is much easier than O-level used to be, since the "new maths" revolution of the late Sixties and early Seventies.

But is it true that things are easier? I decided to compare the 1968 London University syllabuses with the draft 1998 ULEAC syllabus. I chose 1968 because that was the year I did my O-level. There were two syllabuses in 1968, A and B. A was the "easy" one (that is, it contained no calculus). I also looked at the LEAG (SMILE) 1983 O-level, because it comes halfway through the 30-year gap between 1968 and 1998.

Here are some of the comparisons I was able to make:

* In 1968 syllabus A comprised Arithmetic and Trigonometry; Algebra; Geometry (practical and theoretical). Syllabus B comprised Numbers; Mensuration; Formulae and Equations; Graphs, Variation, Functionality; 2-D figures; 3-D figures; practical applications

* In 1998 the equivalent list of topics is: Number (which includes all the 1968 number topics), Algebra (includes the 1968 "algebra" topic and the "graphs, variation and functionality" topic); Shape, space and measure (includes the "trigonometry", "2-D" and "3-D" and "Mensuration" from 1968); Handling data (which was not in the 1968 syllabus at all, apart from a brief mention of averages)

* Assessment in 1968 for syllabus A was three papers, two at 2 hours and one at 2.5 hours; in 1998 there are three papers, all at two hours (if you do the board marked coursework option) Q a reduction in timed assessment of 30 minutes, or 1 minute a year on average

* There are, taken over all three periods, 96 different mathematical topics mentioned. The 1968 syllabus A covers 46 of them, 1968 syllabus B, 57; 1983 O level 55; and GCSE in 1998 64

* syllabus A 1968 was four pages, including a prohibition on slide rules. 1998 has 20 pages just for subject content at higher level; it includes the requirement that efficient use of calculating aids be taught

* It is argued that students were better prepared for A-level in the old days. The 1998 syllabus specifically requires the manipulation of surds and the conversion of recurring decimals to fractions and back again Q both items are specifically excluded from the 1968 syllabuses

* Just taking the number part, in 1998 there are 10 occurrences of the words "understand and use" or "understanding" Q they are not used at all in the 1968 syllabuses. Comparing the requirements just for number 1968 and 1998 we find that 88 words are needed in 1968, but a staggering 421 in 1998.

But it is not just number work that comes under attack: algebraic skills are particularly prone to criticism. In 1968 syllabus B, which was the harder one in terms of both length and content, we have two headings under which we find algebra: "Formulae and equations" and "Graphs, Variation and Functionality". This latter heading includes some elementary calculus, some of which has been lost in 1998, in the sense that Leibnizian notation is not specifically mentioned, but not all. Tangents and areas under graphs are still in the "Further material section".

Another example: in 1968 we have "Simple equations, quadratic equations and linear simultaneous equations in two variables"; the 1998 equivalent is: "solve a range of linear equations, simple linear simultaneous equations, inequalities, and quadratic and higher order polynomial equations selecting the most appropriate method for the problem concerned, including trial and improvement methods". What is important here is not just the longer specification, but the greater intellectual demands being made, particularly by the underlined items.

It is in the area of manipulative skills that we find professors of mathematics and engineering complaining most vigorously. But the specifications in 1968 and 1998 are hardly different: Construction of a formula through symbolic expression of a functional relation . . . or through generalisation of an arithmetical result. Interpretation, evaluation, and very easy manipulation of a formula (1968 syllabus B) Construct, interpret and evaluate formulae and expressions . . . manipulate algebraic expressions; form and manipulate equations or inequalities (1998) Later on in the 1968 syllabus B under the graphs heading, we find this: Simple cases of the function y = Ax3 + Bx2 + Cx + D + ExP1 + FxP2 where the constants (sic) are numerical and at least three of them are zero In 1998 it is hardly different: Plot graphs of the form

y = Ax3 + Bx2 + Cx + D + ExP1 where at least two of A, B, C, D and E are zero

(although it may be argued that the 1998 specification only allows for 10 variants, and the 1968 specification allows for 20)

What about depth? It may be argued that although the syllabus is deeper and more specific, the examinations are easier than they used to be. At a recent teachers' conference at which I spoke on this topic I showed three pairs of similar questions, one from 1968 and one from 1998 and asked the conference to say which was which. Everyone identified the geometry questions correctly, by virtue of the phrase "diagram not to scale" in the 1998 question. The content was identical. With the number questions, the length of the question was the giveaway; in fact the 1998 question required higher literacy and interpretation skills:

1(ii). If A = 5.7 x 105 and B = 3.8 x 104 find the values of AB and (A - B)1000 11. The distance from the Earth to the Moon is 250,000 miles (a) Express this number in standard form: The distance from the Earth to the Sun is 9.3 x 107.

Calculate the value of the expression: distance from the Earth to the Moon distance from the Earth to the Sun giving your answer in standard form The third pair of questions, controlled for clues like typeface, length and language, was harder to tell apart. Only about half the conference got it right.

2. (a) Solve the equation 7x + 8 = 35 - 2x (b) Solve the simultaneous equations 5p - 2q = 16 3p - q = 9 (1998 syllabus) 1. Solve the following equations: (a) (3x - 2) - 4(5 - x) = 27 (b) y - 9y2 = 0 (c) 5x - 3y = 3x + y = 7 (1968 syllabus) If we compare the assessment demands made we find in 1968 the examination for Syllabus A had three papers that required nine solutions each: 6 compulsory and three chosen from 5. A total of 27 questions in six and a half hours Q a rate of 4.2 per hour; in 1998 the two written paper require 34 questions to be answered in four hours Q a rate of 8.5 per hour, with no choice.

The main items missing from the 1998 syllabus are formal calculus most of the constructions and all the Euclidean proof. But over the years the following topics have been added Transformations Bearings Vectors Probability Iterative processes Matrices Set Theory Group Theory Relations The four topics in italic at the end of the list were in the 1983 syllabus, but have disappeared explicitly from the 1998 syllabus. They are still there, however, suffusing the entire syllabus, apart from the group theory. It seems to me that these new items are at least as intellectually demanding and probably more important to modern developments in mathematics than the rote learning of Euclidean proofs.

It is important to remember that I am comparing two syllabuses from the same board. Other boards, and indeed other syllabuses from the same board, had different content in 1968. It was possible to take an examination that included more statistics, or matrices, for example.

The claim that we have watered down the content of GCSE does not stand up to close scrutiny. We need to look elsewhere for the reason for the perceived (but nowhere proven) decline in mathematical standards in England.

What has happened is that GCSE has raised expectations; all my comparisons above are based on the higher level GCSE specification, because this is what should more closely match the old O-level. A candidate who passes GCSE with a grade between A* and C will have covered more mathematics to a greater depth than the O-level candidate of 1968. But these students are not appearing in the departments of mathematics and engineering (where they are going is another debate Q probably into business and accounting, where the money, power and status are to be found), so these are now accepting students with lower starting qualifications that they would have in 1968.

It is extremely unlikely that the average level of ability in the population will have changed much in the last 30 years; what has changed is the part of the normal distribution curve from which university entrants are drawn. In 1968, the top 20 per cent of the population would have taken O-level mathematics, representing students from about one standard deviation above the mean; now GCSE is tackled by 80 per cent of the population, so from about one standard deviation below the mean.

There are complaints from higher education that students are less well prepared. There is no concrete evidence for this, only anecdotal. What is true is that students are less prepared to accept ideas without question. In the past, students who did not understand what the lecturer was saying would not have made their ignorance public. Modern students are much more likely to admit that they do not understand. In the recent report on standards from the London Mathematical Society et al there is a bizarre little footnote in which an unnamed professor complains that a student did not believe his explanation of a mathematical identity; this is supposed to illustrate the weakness of the student, but in fact it demonstrates the weakness of the teaching.

I am not, however, claiming that we don't have a problem. What I am claiming is that we do not have a new problem, that the problem is not the one that the popular Right would have us believe it is, and that the solution is not a return to the practices of the past. To all those who claim that New Mathematics and the teaching methods of the Sixties and Seventies are the cause of all our present woes, I would address these questions: * Why were teaching methods reviewed and changed in the Sixties and Seventies? Was it because traditional methods were so hugely successful it was felt necessary to destroy them?

* Why were maths syllabuses changed and revised in the Seventies and Eighties? Was it a deliberate effort to tear down a functioning edifice and replace it with something new and useless?

To conclude, I should like to list some quotes from an earlier report: "Much failure and backwardness in arithmetic is due to pushing pupils along too fast, that is dealing with figures without having formed a number sense for them. "

"Practice without understanding pays poor dividends . . ."

"Research results show that children who are in the first place permitted to build up their tables from the concrete by using counters, and so 'discover' them . . . are in general quicker and more accurate than those who learn them blindly as tables provided by the teacher."

"Much of the backwardness in arithmetic . . . would disappear if the curriculum were based on the recognition of the fact that over 80 per cent of the problems with which adults have to deal in everyday life involve only the four fundamental processes with numbers under 100, fractions Q 12, 34, 14, 13 Q and percentages. Nearly 90 per cent of the arithmetic is in connection with shopping."

These quotes are taken from Diagnosis and Remedial teaching in Arithmetic by Schonell and Schonell, published in 1957. One of the most telling quotes is this: ". . . in no country would syllabus requirements acceptable in 1926 or even 1936 be acceptable in 1956. Of course, we have little exact evidence of how much of the extensive primary school syllabuses were really understood in 1926."

Add 40 years to the dates and the statement is still true. In 1956 there were complaints that standards were falling and pupils were not learning enough of the basics; the whining has not changed.

Paul Garcia is head of maths and computing at Harlow College, Essex