The test had questions on subtraction, long multiplication, long division, addition and multiplication of fractions, division of decimals, inverse proportion, finding a percentage of, and finding the percentage increase.
It thus required a mixture of skills necessary as a foundation for good mathematical understanding, but could also be seen as a test of basic numeracy.
The test results were poor. In many respects they confirmed worst fears. The only questions where more than 50 per cent of the school got the correct answer were the easiest ones of subtraction and long multiplication, and perhaps surprisingly, adding fractions.
Fewer than 20 per cent could divide decimals, do inverse proportion, or find the percentage increase. This is from a school in which, typically, 99 per cent pass Scottish Certificate of Education standard grade mathematics at age 16, and 80 per cent pass higher grade at 17.
The results by individual class showed that our setting structure is good: class averages decreased from set A to set F in each year, with few exceptions. But they also showed that, in general, averages for a particular set, say set C, decreased as children got older from Forms 1 to 4. It was only in Form 5, where pupils have chosen to take an intensive mathematics course leading to the Scottish Higher examination at the end of the year, that averages went back up to Form 1 levels, which were themselves not that impressive.
Clearly, a one-off test such as this cannot compare ability in arithmetic with that of pupils from 15 years ago. What it does show is that pupils in the school now are generally weak at basic arithmetic, when they do not have the use of a calculator. This point is worth stressing: I am quite confident that results would have been much better, had calculators been allowed. Every pupil would have got at least five of the questions correct.
But would that mean that they had a better understanding of arithmetic when they used a calculator? Obviously not, and it is the lack of understanding that is important.
The fact that this lack of understanding has not led to poorer external exam results than before says more about the nature of what is being examined, than about pupils' abilities.
The mathematics curriculum has changed significantly in the past 15 years, and pupils are simply not tested on the basic arithmetical skills, which are so important to other subjects as well as to mathematics.
As a consequence of the arithmetic test, and subsequent discussion throughout the school, we will be changing our mathematics curriculum next year.
We will continue to teach pupils to have a good knowledge of how to use a calculator, but will aim to make them less dependent on it. In particular, we will be introducing an arithmetic course into Form 1, to be done without calculators, with the aim of consolidating basic understanding at that stage.
In the rest of the school we will have more frequent revision of basic techniques, again without calculators. We will place more emphasis in our teaching on estimation, and judging the reasonableness of an answer, and we will demand a larger number of repetitions from our pupils by setting a larger number of examples on a particular topic.
The modern emphasis of "always provide a context, do a few examples, move on to something new in case they get bored" does not lend itself to the learning of what are fairly mechanical skills - and musicians accept the need to practise scales.
In short, we are going to devote a significant amount of time to teaching material that will not be directly examined. I am certain that those pupils who do not consider themselves to be "mathematicians" - and they are the majority - will be leaving school with a better education.
Many heads of mathematics in schools will, like myself, have been used to hearing concerns about maths standards within their own schools similar to those expressed nationally, perhaps over a longer period.
The concerns do not just come from within mathematics departments. Other departments, whose own teaching often relies on pupils having certain basic mathematical skills, are worried. University mathematics departments, and any other department whose courses have some mathematics content, are complaining of lower standards among each first-year intake.
The debate is a complex one. If standards of mathematical ability have really dropped, is this due to the ill-conceived content of new mathematics courses, or to the failings of teachers and new teaching methods?
Part of the problem lies in the fact that, national surveys apart, much of the "evidence" used in the debate is anecdotal. Also, many people talk about "mathematics" when what they really mean is arithmetic.
It is easy to demonstrate how lack of understanding of basic arithmetical skills can lead to difficulty with understanding later mathematics. Although perceived lack of arithmetical skills is only one aspect of the mathematics debate, it is a fundamental one. If pupils today are less skilled in arithmetic than, say, 15 years ago, then it is important to know this. The blanket use of calculators in the present day is probably concealing just such a conclusion.
The writer is head of mathematics at the Dollar Academy, Dollar, Scotland.
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