This work is part of a wide-ranging long-term proje ct aimed at fostering students' acquisiti... more This work is part of a wide-ranging long-term proje ct aimed at fostering students' acquisition of symb ol sense through teaching experiments on proof in elementary number theory (ENT). In this paper, in parti cular, we highlight the positive effects of our approach a nalysing the written reflections that the students involved have produced at the end of the project. These refl ections testify an increased level of awareness, de veloped by students, about the role played by algebraic lan guage as a tool for thinking and a positive evoluti on in their vision of algebra. A focus on meaning in the teaching of algebra: intr oducing our approach A central aspect of a teaching aimed at fostering a mature conception of mathematics is helping studen ts "conquering" the meanings which are connected to the learning of concepts and to the use of mathematic al instruments during school activities. The main prob lems related to a "warped" conception of mathematics have be...
Games of interpretation, anticipating thought and coordination between verbal and algebraic register: key-aspects in the analysis of students' proofs in elementary number theory
This work is part of a wide-ranging long-term proje ct aimed at fostering students' acquisit... more This work is part of a wide-ranging long-term proje ct aimed at fostering students' acquisition of symb ol sense through teaching experiments on proof in elementary number theory (ENT). Our aim is to analyze the use and the role of algebraic language in the development of such proof s. In this paper we present the analysis and classi fication of students' behaviour in facing the proof of a conjec ture while working in small groups. The analysis of students' protocols was made by reference to the following in terpretative-keys: the application of specific conc eptual frames, the games of interpretation between different frames, a nticipatory thoughts, the use of conversions and tr eatments and coordination between different registers of represe ntation. Our analysis highlights the incidence of a nticipatory thoughts and of the flexibility in the coordination between different frames and different registers of represe ntation in development of proof in ENT.
This work is part of a wide-ranging long-term project aimed at fostering students' acquisition of... more This work is part of a wide-ranging long-term project aimed at fostering students' acquisition of symbol sense through teaching experiments on proof in elementary number theory (ENT). In this paper I present some excerpts of students discussions while working in small groups on activities of proof construction. My analysis of these transcripts is aimed at highlighting the incidence of anticipating thoughts and of the flexibility in the coordination between different conceptual frames and different registers of representation in the development of proof in ENT. In particular, I singled out four main sources of interpretative blocks, highlighting the strict interrelation between anticipating thought and students' difficulties in the interpretation of the algebraic expressions they produce.
i) After a brief overview of the studies which brought to the rise of early algebra, (ii) the iss... more i) After a brief overview of the studies which brought to the rise of early algebra, (ii) the issues of socio-constructive teaching as well as of the more complex role of the teacher are discussed and the importance of reflecting upon teaching and learning processes for the professional development of teachers is underlined. (iii) Our view on early algebra and our studies about it are then introduced, highlighting the role played by metacognition in both pupils' conquest of the mathematical meanings that emerge from the classroom-based action and teachers' development of a capacity of having control of and predicting the effects of their own actions. (iv) We dwell on both instruments and modalities we used to lead teachers to reflect upon the impact of their own behaviours on pupils ' mathematical constructions, and (v), in this concern, we underline the high value of analytical comments made by teachers in written form on the transcripts of classroom-based processes.
Improving Awareness About the Meaning of the Principle of Mathematical Induction
This work is based on our conviction that it is possible to minimize diffi- culties students face... more This work is based on our conviction that it is possible to minimize diffi- culties students face in learning the principle of mathematical induction by means of clarifying its logical aspects. Based on previous research and theory, we designed a method of fostering students' understanding of the principle. We present results that support the effectiveness of our method with teachers in training who are not specializing in mathemat- ics.
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