Four specific research questions will be addressed to attain the aims specified in the section justification of the method.

1. Are there differences in the development of specific types of mathematical understanding and, if so, how do these influence pupils' progression across the Key Stage 1-3 period?

Learning to think quantitatively and learning arithmetic are two different studies. One previous study with relatively few children has shown that their mathematical reasoning in Year 1 predicts KS1-Maths performance independently of the children's knowledge of numbers at school entry (Nunes et al., 2007) and that improving children's reasoning in Year 1 significantly enhances their KS1 results one year later. An analysis of the ALSPAC data could provide a demonstration of a predictive relation between mathematical reasoning measured in Years 4 and 6 and KS2 and KS3 achievement, independently of knowledge of arithmetic. This could provide much needed evidence regarding later mathematics learning.

The ALSPAC dataset contains four types of predictors to be used in this investigation: (1) cognitive and (2) affective factors; (3) predictors related to teachers' and schools' practices; (4) a set related to the family and the socio-economic background.

The predictors relevant to this first research question are the cognitive measures, which include assessments of the children's:

a) mathematical reasoning (Years 4, 6 and 8 assessments);

b) knowledge of arithmetic (a sub-test of the WISC);

c) working memory (backward digit span), short-term memory (non-word repetition and forward digit span) and attention (a specific measure of attention and a scale within the Strengths and Difficulties Questionnaire);

d) general intelligence (Weschler Intelligence Scale for Children: WISC).

Regression analyses will be used to investigate whether each of the first three of these measures makes a distinct contribution to predicting students' mathematics achievement at each KS assessment, after controls for differences in general intelligence and in the other two variables. We hypothesise that children's mathematical reasoning will make a specific contribution to the prediction of KS results beyond what is predicted by the other factors. A positive result would provide the basis for a programme to promote children's mathematical reasoning with the aim to improve their mathematics achievement.

Separate scores for items within the mathematics assessments will be used to distinguish (a) additive and multiplicative reasoning; (b) reasoning about quantities and reasoning about relations; and (c) problems about extensive and intensive quantities. These will be used in analyses to investigate the extent of their relative power in the longitudinal prediction of maths and science attainment.

2. Is the development of some skills more important for progression than others?

There is a debate in education regarding the relative emphasis that should be given to computation and reasoning skills for later mathematics learning. It has also been suggested that spatial and visualisation skills are important for mathematics learning. The WISC contains sub-tests that allow for considering how arithmetic and spatial skills contribute to mathematics attainment.

There is also increasing interest in the role of attention in mathematics learning. The WISC subtest "Coding" is considered a measure of attention. Previous analysis of the ALSPAC data showed that the correlation between this sub-test and KS1 achievement was significant, though low. There are more specific measures of attention which will be used in this analysis.

3. Are there differences in the development of skills by gender, ethnicity, or socio-economic factors, such as parents' education or SES?

The dataset will be used for the analysis of these demographic variables considering both the impact of these variables at school entry and the possible augmentation of these effects if the home environment contains little academic incentive (e.g. books) and if the parents have low educational aspirations for the children.

Data from the teachers' questionnaires will be used to assess the effect of some school practices (streaming) and environment (number of pupils per class; number of students receiving free school meals; a specialist maths coordinator available) on students' mathematics progression.

4. How is development of understanding affected by children's self perception of their own ability in mathematics?

The ALSPAC measures of children's affective reactions to mathematics include questions on pupils' self-confidence, their liking for mathematics and its perceived usefulness, and their perception of the teacher's abilities and attitudes towards maths and teaching the subject. Different items were included at the different data collection points but it is possible to assess whether, according to different reports (i.e. by parents or by children), the students showed consistent liking, or not, for mathematics.

Some research has shown that pupils are reasonably accurate in identifying their position in their class as mathematics learners but their teachers' perceptions of them is just as important as their measured ability in influencing their self-perception as learners (Pretzlik, Olsson, Nabuco, & Cruz, 2003). Teachers' perceptions of their pupils' abilities in mathematics are influenced by the pupils' reading ability; the latter tend to be more accurate than their perceptions of the pupils' mathematics abilities. We need, therefore, to explore how pupils' abilities, their self-perceptions and their teachers' perceptions of them interact in the prediction of KS2-3 mathematics outcomes.

Final notes

In all of these analyses, we propose to test whether pupils' general ability and general self-confidence are better predictors of their attainment and later liking of maths than their specific mathematical reasoning capabilities and self-confidence. For this reason, measures of attainment in reading and in English, and self-confidence in and liking of English and Maths are required measures. It will also be necessary to analyse the data both including and excluding children with special needs (sensory, reading and language). The attainment of these groups of possible outliers could affect the results of the analyses.

The list of relevant variables is appended.