There's a hole in the bucket
Recent publication of the Third International Mathematics and Science Study, and the ensuing criticism, can have only added to the demoralisation of maths teachers in England. Primary teachers may feel especially let down because numeracy is seen as their responsibility, and the cry of "innumerate" seems a rallying point for critics.
Responses to the report seem to suggest (again) a belief that mathematics teachers are ineffective. The report acknowledges we "did relatively well" in data representation, analysis and probability and geometry, but as items in these categories together make up only a quarter of the total for population 1 (nine-year-olds in Years 4 and 5) this does little to boost our rating.
There seems a desire to compare expectations of children's mathematical performance with those of children in other countries. Whether this also means we accept what other countries regard as curriculum priorities may be open to question (although reactions in the media suggest otherwise).
The School Curriculum and Assessment Authority (now the Qualifications and Curriculum Authority) has been quick to commission a study to evaluate the TIMSS items for population 1 and population 2 (13-year-olds) against key stage 2 and key stage 3 national tests in mathematics for 1996.
A stated aim of the project is to inform any changes in the national curriculum for mathematics (and science) in 2000. If one aim is to identify discrepancies between our curriculum and those of other countries, we need do little more than examine the categorisation of items within the TIMSS tests and the proportion of items in each of the categories, and compare these with national curriculum requirements.
The table below shows the number of items in each category.
Category Number of items
Whole numbers 25
Fractions and proportionality 21
Measurement, estimation and number sense 20
Data representation, analysis and probability 12
Patterns, relations and functions 10
The second category, fractions and proportionality, provides a significant clue to our apparent under-achievement as it constitutes just over a fifth of all items. This suggests an emphasis on two aspects of number and related concepts that are not reflected in key stage 2 of the national curriculum.
The National Foundation for Educational Research report on the TIMSS study acknowledges this. It points out that while all countries took the same tests, no account was taken of the differences in curricula. Accordingly, a test-curriculum matching analysis was carried out by the NFER to determine the relevance of each sub-set of items for English pupils. While the official report acknowledges this, it has not been more widely publicised.
It is highly relevant when this sub-set of items is examined more closely for, in the opinion of the test-curriculum matching panel, none of the released items (21 in all) was considered representative of the national curriculum for this age group. The mean score for English pupils was two percentage points below the international mean on these items. Three items not released but which formed part of the total sample were, though, deemed representative of the national curriculum for Year 5 pupils and are included in the report.
The scores of English nine-year-olds in Year 4 were above the international mean in two out of three of these items, but in only one of the items for Year 5 pupils. The results on items overall in this category are anomalous and need to be examined in relation to the expectations implicit in the national curriculum for pupils in the two year groups.
Examination of items in the sub-set of fractions and proportionality shows the extent to which demands of the national curriculum for nine-year-olds differ from what seems expected in other countries.
A closer look at what is meant by "proportionality" turns out to be the mathematics that so often appears in what used to be known as "word problems". The first example of the released items that falls within this category is shown below.
Mario uses five tomatoes to make half a litre of tomato sauce.
How much sauce can he make from 15 tomatoes?
A. a litre and a half
B. two litres
C. two litres and a half
D. three litres
This is a challenge for English nine-year-olds. The first step is to determine the increase in proportion (three times as many) of tomatoes needed. The second is to multiply one half by the answer.
Assuming pupils get the first step right, they could then work out 3 x 12 or make some sort of pictorial representation such as: Or they could simply guess.
Compare the demands of this question with the level descriptors for key stage 2.
Level 2 - identify and use halves and quarters, such as half a rectangle or a quarter of eight objects.
Level 4 - recognise approximate proportions of a whole and use simple fractions and percentages to describe these.
Explore and describe number patterns, and relationships including multiple, factor and square.
Level 5 - calculate fractional or percentage parts of quantities and measurements using a calculator where appropriate.
The leap in demand comes at : Level 6 - (be) aware of which number to consider as l00 per cent, or a whole, in problems involving comparisons, and use this to evaluate one number as a fraction or percentage of another.
Understand and use equivalences between fractions, decimals and percentages, and calculate using ratios in appropriate situations.
Teachers are guided more by programmes of study than levels in selecting what they teach. And the programme of study for mathematics says pupils should "understand and use, in context, fractions and percentages to estimate, describe and compare proportions of a whole".
But compared with the demands at the various levels there is a mismatch between the expectations of the programme of study and the level descriptors for key stage 2. Most pupils are also expected to achieve level 4 at the end of key stage 2. (The figures for 1996 national tests were 40 per cent achieving level 4 and around 12 per cent achieving level 5.) A further indication of expectations in this area is given in the non-statutory mathematics test for Year 4, which SCAA produced for the first time this year, and which adds to the confusion. It falls into two parts) and teachers are told part B has "more challenging questions" and teachers should "use their judgment" to decide which children should complete these.
Below are two questions from the paper, the first from part A and the second from part B.
Part A: The jug holds 12 litre The bucket holds 5 litres.
How many full jugs of water are needed to fill the bucket?
Part B: Tick each of the cards that shows more than a half The question from part A lies untidily between levels 2 and 4 and "levelling" would probably be done according to whether a pupil solved the problem pictorially or simply by multiplying 2 x 5.
The part B question assumes some knowledge of equivalence between fractions, decimals and percentages. This is explicitly mentioned for the first time at level 6, and teachers would probably think hard before inviting many eight and nine-year-olds to attempt this part of the test. The suggestion implicit in the question is that some pupils will have been taught this (some undoubtedly will have) but many will not.
Demands placed on English nine-year-olds by TIMSS questions of the kind at the start of this article are, to say the least, unrealistic compared with the requirements of the national curriculum. This begs the question of how appropriate it is to enter pupils for such a study. But it also (perhaps more importantly) highlights some confusion in the requirements of the national curriculum with respect to a crucial area of mathematics.
Any implication that the poor performance of English nine-year-olds on this major category might be the result of ineffective delivery of the maths curriculum is unjust.
The mathematics represented by almost all of the TIMSS items under fractions and proportionality forms no part of the requirements of the national curriculum for this age group and teachers are not required to teach it. In fairness to teachers, it would have been appropriate to draw the attention of the public to this fact when reporting results of the study.
Marilyn Nickson is head of the primary assessment unit, research and evaluation division at the University of Cambridge Local Examinations Syndicate