Longitudinal Proof Project

A Longitudinal Study of Mathematical Reasoning:
Student Development and School Influences
1999 - 2003


This research project followed on from our project Justifying and Proving in School Mathematics that ran from 1995 to 1999, which investigated high attaining Year 10 students' understanding of proof in the areas of algebra and geometry.

The project followed the timetable as detailed in the original proposal. Dr Dietmar Küchemann worked as the Research Officer for the project. All the instruments used in this research were designed and tested in collaboration with a group of teachers in five 'design schools' who were identified at the beginning of the project and who met regularly with the project team. The first Year 8 proof test was designed in late 1999 and piloted in the spring of 2000. The test was administered in June 2000 to high-attaining Year 8 students in top sets or bands in randomly selected secondary schools from within nine geographically diverse English regions (3083 students from 114 classes in 63 schools). All those students still in top sets or bands were tested again using a specially designed Year 9 test in the summer term of 2001, and, for the final time, using a specially designed Year 10 test in June/July 2002. The Year 10 test instrument included some questions that were the same or similar to those in the Year 8 and 9 surveys in order to assist in assessing progress, as well as some questions used in a previous ESRC project (Justifying and Proving in School Mathematics, R000236178) so useful comparisons could be made. Each of the three tests aimed to assess both geometrical and algebraic reasoning in questions that were presented in both open and multiple-choice formats.

In addition to the collection of student data, a school and teacher questionnaires were also administered annually. The former provided information on the location, intake, curriculum, examination and procedures of the school, and the latter, completed by all the teachers of the classes involved in the survey, provided relevant personal data on qualifications and experience. All the data collected each year were analysed using descriptive statistics. Multilevel statistical techniques were also used to identify any school, curriculum and student factors associated with competence or progress in mathematical reasoning. In Year 8 responses on a mathematics test (administered in May 2000) comprising selected items from TIMSS (the Third International Mathematics and Science Survey) were used as a baseline in the statistical models. In Years 9 and 10, students' scores on Key Stage 3 tests were also included as baseline data.

All the data from the Years 8, 9 and 10 proof tests have been coded and analysed using descriptive statistics and multilevel modelling. All the Year 8, 9 and 10 analyses are now available in Project Technical Reports and various papers on different aspects of analysis have been presented at conferences or are in the process of publication.

The Year 8 multilevel modelling analysis was used not only to highlight significant predictors of students' responses but also to identify schools that were statistically above average in their promotion of their students' mathematical reasoning (in comparison to baseline performance). Case studies of some of these schools, consisting of descriptions of policies and practices of the school and the mathematics departments, were undertaken. A similar exercise took place in the summer of 2002 and Spring of 2003 focussing on 'outlier' schools identified in the Year 9 and Year 10 analyses respectively. Throughout the project, ongoing feedback was sent to participating schools and the research team had also been in contact with the mathematics advisors of the areas included in the sample for advice and feedback.