## Developing first-year mathematics student teachers' understanding of the concepts of the definite and the indefinite integrals and their link through the fundamental theorem of calculus : an action research project in Rwanda.

##### Abstract

This thesis describes an Action Research project within the researcher's practice as a teacher educator in Rwanda. A teaching style informed by the Theory of Didactical Situations in Mathematics (Artigue, 1994; Brousseau, 1997; 2004; Douady, 1991) and by the Zone of Proximal Development (Gallimore & Tharp, 1990; Meira & Lerman, 2001; Rowlands, 2003; Vygotsky, 1978) was conducted with first-year mathematics student teachers in Rwanda. The aim of the teaching model was to develop the student teachers' understanding of the concepts of the definite and the indefinite integrals and their link through the fundamental theorem of calculus.
The findings of the analysis answer the research questions, on the one hand, of what concept images (Tall & Vinner, 1981; Vinner & Dreyfus, 1989) of the underlying concepts of integrals student teachers exhibit, and how the student teachers‟ concept images evolved during the teaching. On the other hand, the findings answer the research questions of what didactical situations are likely to further student teachers' understanding of the definite and the indefinite integrals and their link through the fundamental theorem of calculus; and finally they answer the question of what learning activities student teachers engage in when dealing with integrals and under what circumstances understanding is furthered.
An analysis of student teachers' responses expressed during semi-structured interviews organised at three different points of time - before, during, and after the teaching - shows that the student teachers' evoked concept images evolved significantly from pseudo-objects of the definite and the indefinite integrals to include almost all the underlying concept layers of the definite integral, namely, the partition, the product, the sum, and the limit of a sum, especially in the symbolical representation. However, only a limited evolution of the student teachers' understanding of the fundamental theorem of calculus was demonstrated after completion of the teaching.
With regard to the teaching methods, after analysis of the video recordings of the lessons, I identified nine main didactical episodes which occurred during the teaching. Interactions during these episodes contributed to the development of the student teachers' understanding of the concepts of the definite and the indefinite integrals and their link through the fundamental theorem of calculus. During these interactions, the student teachers were engaged in various cognitive processes which were purposefully framed by functions of communication, mainly the referential function, the expressive function, and the cognative function. In these forms of communication, the cognative function in which I asked questions and instructed the students to participate in interaction was predominant. The student teachers also reacted by using mainly the expressive and the referential functions to indicate what knowledge they were producing. In these exchanges between the teacher and the student teachers and among the student teachers themselves, two didactical episodes in which two student teachers overtly expressed their understanding have been observed. The analysis of these didactical episodes shows that the first student teacher's understanding has been triggered by a question that I addressed to the student after a long trial and error of searching for a mistake, whereas the second student's understanding was activated by an indicative answer given by another student to the question of the student who expressed the understanding. In the former case, the student exhibited what he had understood while in the latter case the student did not. This suggests that during interactions between a teacher and a student, asking questions further the student's understanding more than providing him or her with the information to be learnt.
Finally, during this study, I gained the awareness that the teacher in a mathematics classroom has to have various decisional, organisational and managerial skills and adapt them to the circumstances that emerge during classroom activities and according to the evolution of the knowledge being learned. Also, the study showed me that in most of the time the student teachers were at the center of the activities which I organised in the classroom. Therefore, the teaching methods that I used during my teaching can assist in the process of changing from a teacher-centred style of teaching towards a student-centred style.
This study contributed to the field of mathematics education by providing a mathematical framework which can be used by other researchers to analyse students' understanding of integrals. This study also contributed in providing a model of teaching integrals and of researching a mathematics (integrals) classroom which indicates episodes in which understanding may occur. This study finally contributed to my professional development as a teacher educator and a researcher. I practiced the theory of didactical situation in mathematics. I experienced the implementation of some of its concepts such as the devolution, the a didactical situation, the institutionalization, and the didactical contract and how this can be broken by students (the case of Edmond). In this case of Edmond, I realised that my listening to students needs to be improved. As a researcher, I learnt a lot about theoretical frameworks, paradigms of study and analysis and interpretation of data. The theory of didactical situations in mathematics, the action research cyclical spiral, and the revised Bloom‟s Taxonomy will remain at my hand reach during my mathematics teacher educator career. However, there is still a need to improve in the analysis of data especially from the students' standpoint; that is, the analysis of the learning aspect needs to be more practiced and improved.