The number of countries now involved in the project (Appendix 1) has continued to increase, with possible future participation from Russia and the Ukraine.
The methodology of this research is based on regular testing of pupils from age 13+ for a period of 23 years, monitoring the progress made by groups of pupils of similar ability, and then investigating further those schools, classes (i.e. teachers) and pupils which show progress much above or below the norm.
At the start of the project, all pupils take a mathematics Potential Test. This is a 40 minute test with 26 questions of increasing difficulty. It is taken only once and aims to assess the pupils' aptitude for mathematics with questions based on mathematical logic and spatial awareness. The test is designed for pupils aged 1214 years and was trialled extensively before being used in this project.
The pupils then take tests in
These 40 minute tests, each with 50 marks available, are designed to show progress in attainment and are based on the intersection of the English, Scottish and German mathematics curricula for pupils aged 1316 years. In National Curriculum terms, they cover the basic topics from Levels 4 to 9 in the relevant attainment target. Most questions are relatively short and precise with few followthrough marks and, unlike Key Stage 3 assessment, relatively contentfree. Similar tests, with questions either repeated or replaced with equivalent questions, are taken at the beginning of each of the following years of the project.
Testing began in 1993 with pupils aged 13+ in England, Scotland and Germany and these English and Scottish pupils are now taking exams (GCSE in England and SEB Standard Grade in Scotland) this summer. Other countries joined one year later, either with pupils aged 13+ (equivalent to Y1 of the project, enabling comparison to be made retrospectively) or 14+ (Y2 of the project).
As well as test data (each response entered on computer) we collect further information using School, Teacher, Class and Pupil Questionnaires.
To complete the picture we visit schools to:
(i)  observe mathematics teaching, recording the important characteristics of the lesson. 
(ii)  interview teachers and pupils individually (usually concentrating on pupils who have made either very good or very poor progress over the year). 
We are guided on where these indepth visits should be made by Performance Indicators, which are calculated for each pupil on progress made over Y1Y2 and Y2Y3. This progress is compared with the average progress made by groups of pupils with similar potential and similar topic test scores at the start of the year. The formula used is
Here T1 and T2 are the total scores on the three tests (Number, Algebra and Shape and Space) in Year 1 and Year 2, and d is the average increase for the group of pupils with similar potential and first year attainment.
In England, our representative sample consists of 12 schools taken from a total of 40 project schools. It is chosen to be representative according to geographical location and type of school (independent, grant maintained, l.e.a. comprehensive). As a final check, a weighted average according to cohort size of the 1995 Mathematics GCSE results of all the schools is taken and compared with national 1995 GCSE Mathematics figures using a chisquared test. It should also be noted that the results are relatively insensitive to minor changes in the schools in the sample, and we have confidence that our sample is a good representation of the whole country.
Our Scottish sample is produced in a similar way to that of England  but using the 1995 SEB Standard Grade Mathematics results as a final check. Our sample in Germany is based on three of the Landers (= counties or regions), with 50% of the sample in the lander of Hessen. This is regarded by many as being the weakest region educationally in Germany, so our figures may be a little low for a good representation of the whole of Germany. It also includes pupils who have repeated one or two years, since they have been held back due to insufficient progress made in German, Mathematics or both subjects. This could influence the results for the lowest 10% of the sample and we are investigating this aspect further. However, this factor is not present in the results of the other countries presented here such as Poland, Holland, Hungary and Singapore.Some overall results for representative samples of pupils for Year 1 (age 13+) and Year 2 (age 14+) are summarised below.
Note that the final representative samples for England and Scotland will be based on the 1996 GCSE/SEB results.Country  Number  Algebra  Shape and Space  TOTAL 
Finland  21.1  8.0  8.8  37.9 
Greece  20.6  11.8  8.3  40.7 
Scotland  18.2  8.8  14.0  41.0 
England  17.6  11.3  15.4  44.3 
Germany  23.5  12.5  11.3  47.3 
Poland  24.0  16.6  13.6  54.2 
Holland  26.3  13.3  19.2  58.7 
Singapore  33.4  23.9  18.1  75.4 
Country  Number  Algebra  Shape and Space  TOTAL 
Norway  20.9  12.4  12.1  45.4 
Finland  22.6  12.0  12.5  47.1 
Scotland  22.1  12.7  18.6  53.4 
England  20.2  14.4  19.9  54.5 
Germany  26.9  17.6  17.3  61.8 
Poland  29.2  24.9  22.4  76.5 
Hungary  29.2  25.7  22.5  77.4 
Singapore  34.6  30.7  26.9  92.2 






Not only were England and Scotland well behind some of the other European countries in total attainment for these core topics on the first testing but the progress made during the year was less than in other countries. It should also be noted that Singapore is doing so well that it begins to become more difficult for many of the pupils to show any real progress  so their increase of 16.8 over the year is an excellent result.
The results do indicate underperformance by English and Scottish pupils, particularly in comparison with Poland and Singapore. We can gain further insight into this possible underperformance by looking at particular ability groups. For England, Germany, and Singapore, we can divide our representative sample into three equal parts, low (L), middle (M) and high (H), and monitor the progress of each group.
Our observations of teaching in these countries (and others such as Hungary which also showed very high attainment in Y1) have led us to make draft recommendations for mathematics teaching and learning in England in order to raise expectations and Appendix 2 gives details of these. We are not saying that we have 'proved' that these recommendations will work  only that our data indicates
underachievement and our observations led us to suggest that these factors might enhance performance.
These recommendations are being implemented through the Mathematics Enhancement Programme(MEP) demonstration project.
We are in the position to publish confirmed Year 3 results for England. These are given below
Country  Number  Algebra  Shape and Space  TOTAL 
England  22.4  18.0  24.2  64.6 
Scotland  24.0  19.0  24.3  67.3 
These show a remarkably similar trend to our previous results. The total average increase for England is again 10 marks, and for Scotland it is 13.9 marks. The progress inside the three ability groups is very consistent with Year 1  Year 2. We had anticipated that focusing on teaching for GCSE exams in Years 10 and 11 in England might improve the rate of progress, but over Year 10 that has not proved to be the case, although the significant increase in the progress made by Scottish pupils should be noted.
Finally it should be noted that during this year we have also tested some of our pupils on Applying Mathematics. These tests pose contextual questions, the answers to which need more thought and careful analysis than those in our basic tests. We have not yet had a chance to analyse fully these results, or to compare with pupils in other countries, but our preliminary analysis indicates, as might have been expected, a strong correlation between results in basic and applied tests.The results given here are not published in order to demoralise further the mathematics teaching force but simply to provide sound evidence on which to base recent claims of underperformance, and more importantly to indicate some ways to improve mathematics teaching and learning in this country. It is time to support and invest in mathematics and teaching by providing the means to obtain enhanced performance.
Mathematics is a key school subject and its neglect could cause problems for generations to come. In an increasingly global society, in which we compete for trade throughout the world, it is crucial that our future workforce, at all levels, should be proficient in basic mathematical skills.
Although we must improve attainment, we must be careful not to undermine the gains that have been made over the last decade or so. Mathematics classrooms in the UK are happier places now and there is less dislike of the subject than in the past, but it does appear that this has been acheived at the expense of attainment. What is needed is a balance.
England  Czech Republic 
Germany  Brazil 
Scotland  USA 
Finland  Australia 
Norway  Japan 
Greece  Singapore 
Holland  
Hungary  Thailand 
Poland  Russia and Ukraine 
Mathematics Teaching
Mathematics Assessment