The current debate about standards assumes there to be one form of knowledge that is unproblematically assessed in these tests. This assumption is contradicted by a large body of research that shows the existence of different forms of knowledge. Evidence also shows that students can be very successful on standard, closed tests with a knowledge that is highly inert and that they are unable to use in more unusual and demanding situations. In a three-year study I found that students who attended a school that used the "progressive" maths teaching methods that are being widely criticised developed more effective forms of knowledge than students who attended a school using traditional methods. This resulted in their attaining higher GCSE grades and, importantly, being able to use their school knowledge in real-life situations.
The traditional school in my study used exactly the sort of methods encouraged by international comparison researchers such as David Reynolds. The teachers taught maths using whole-class teaching and textbooks, and the students were tested frequently. There were high standards of discipline at the school, students worked hard and they were setted by ability. At the "progressive" school the students worked on open-ended projects in mixed-ability groups in every maths lesson, there was very little whole class teaching and discipline was extremely relaxed.
Over three years, I monitored a group of students at both schools (300 students in all), from the beginning of year 9 (when they were 13) to the end of year 11 (when they were 16). I watched 100 lessons at each school, interviewed students, gave out questionnaires, conducted assessments of their mathematical understanding and analysed their GCSE responses.
At the start of the three years there was no difference between the students in terms of mathematical attainment, sex, ethnicity or social class. At the end of the three years the students had developed in very different ways. One of the results of these differences was that students at the project-based school attained significantly higher GCSE grades. This was not because the students at this school knew more maths, but because they had developed a different, more effective form of knowledge.
At the textbook school many students could not adapt the formal procedures they had learnt to anything other than textbook questions. This was because the maths lessons had focused on set methods and rules - in the traditional style that is being advocated by many. The students believed that mathematical procedures were a specialised type of school code, useful only in classrooms. They did not regard maths as a thinking, flexible subject. These students were not unusual in this regard. Many studies have shown the limitations of formal maths teaching for active knowledge use.
At the project-based school the students were not introduced to any standard rules or procedures (until a few weeks before their GCSE examinations) and they did not work through textbooks of any kind. Instead, the students were given extended problems to solve. In solving these problems the students adapted and changed the mathematical methods they knew, they learnt about new areas of maths and they were encouraged to think critically at all times. As a result of this approach the students developed an ability to use maths in a range of different situations, including the GCSE examination. In interviews with approximately 40 students from each school, more than three-quarters of the project-based students reported that they used their school-learnt maths in situations outside school. This compared with none of the textbook students.
The difference between the students at the two schools related to the flexibility of their knowledge. The textbook students had developed a form of knowledge that they could only use in limited tests and exercises, whereas the project-based students had developed a more general knowledge. This study was only focused on two schools but the textbook school was not unusual in the way it taught maths and the detailed nature of the study meant that it was possible to consider the reasons why students responded to these two approaches in the way that they did. The differences in mathematical competence were not to do with good and bad teaching, they were to do with the effectiveness of problem-solving approaches in developing a broad mathematical competence.
The results of this study are worth considering because they suggest that formalising the way in which maths is taught is likely to decrease the capability of students in real-life situations. In a recent international test assessing problem-solving, British children came first. This capability is under threat. Earlier this year the maths department in the project- based school was forced to return to textbook teaching, partly because of fears about Ofsted inspections. It seems there is no longer a place for teachers who want to try new approaches or strive towards something more than test success. Formal teaching may result in enhanced performances on international tests, but surely the aim of schools must be to equip students with a capability and intellectual power that transcends the boundaries of the classroom?
The author is a lecturer and researcher in mathematics education at King's College London.Reuse content