Formative and summative assessment problematic in mathematics education

This axis of research is linked to a thesis which has just begun (Maud Chanudet). In Geneva secondary schools, in 10e classes (grade 8 – students aged 13/14 years) students with ‘scientific profile’ are taught an annual one hour about development in mathematics, ‘aimed at teaching that contributes to the reinforcing and development of students’ capacities and skills in the strategy of problem resolution and the activities of mathematical situations,’ according to the guidelines of the liaison document of the Cycle d’Orientation (secondary school). This hour comes on top of the weekly 5 hours of mathematics can, depending on cases, lead to pooling students from different classes together, sometimes with a different teacher than the one who usually teaches the mathematics class; even so, the hour is always restricted to 10 to 15 students. Moreover, according to official guidelines, this hour has to lead to a separate summative assessement, which requires at least two marks for each semester, meaning one mark for every 4 hours of class! It is also noted that ‘the research and its writing up will account for 2/3rds of the final mark – and therefore, 1/3 will be content.’ We are therefore confronted with a contradiction. Indeed, on the one hand this type of teaching requires a great of autonomy on the student’s part, who has to develop general competences linked to the resolution of open problems, implying a research design which requires time and maturity. On the other hand, the teachers have to answer to the institutional requirement of having a highly segmented summative assessment, covering very short periods.

 Given this difficulty, the group of teachers concerned, along with the DiMaGe team has chosen to concentrate the assessement on the research narration device. A new contract is signed with the teacher: the student commits to recollecting the steps of her research as well as possible, to point out her mistakes, how the ideas came to her mind; in exchange, the teacher commits to focusing the assessment on these specific points rather than on the solution.’ (chosen definition in the cantonal liaison document). This practice of narration of research is not recentand has given rise to a number of works particularly in the IREM and APMEP networks since the 1970s (Bonafé et al. 2002, Chevalier 1992 and Sauter 1998), but also recently in some textbooks, or the Sésamath website. It was not worthwhile to attempt to revisit it within the current trend of research design, especially given our implication in the PRIMAS project (Dorier 2012 and Dorier and Garcia 2013).

 First of all, the practice of research narration, especially with student of this age, requires that they develop certain linguistic skills and that they learn how to describe what they are doing; that they watch themselves doing in a way and that they be able to write it down. We can therefore ask ourselves if this isn’t a little demanding, especially given the multicultural context that Geneva usually offers. Moreover, certain students are sometimes capable of finding the solution right away and therefore don’t have much to say. In contrast, the risk would be of developing capacities in students of ‘explaining for the sake of explaining,’ all the while forgetting the mathematical relevance of the research or its objective (the goal of the problem). In order to circumvent these issues and to be able to establish a contract with students that also takes into account the problem of research narration assessment with a formative goal, we have worked on creating an assessment matrix given to the students at the start of the year which becomes de facto a common tool for teachers and students, in order to learn through the research narration to better solve the mathematics problems. This matrix is therefore an assessment tool which is both formative and summative. On the basis of this first step, Maud Chanudet’s thesis work will concentrate on a partnership with a group of teachers in order to see how this matrix (which will be subject to changes) can become effective. Secondly, we are contemplating the idea of observing certain teachers in the day to day in class usage of this matrix, in order to see how the students change their practice of the research narration.

 Beyond this precise thesis work our team work in partnership, via Sylvie Coppé, with the European project ASSIST ME (Assess Inquiry in Science, Technology and Mathematics Education) and Professor Lucie Mottier-Lopez’s team EReD Evaluation, Régulation et Différentiation des apprentissages.