Conceptions of basic education teachers about math proof: influence of professional experience

A prova é uma atividade que desempenha um papel fundamental na construção do conhecimento matemático, razão pela qual adquire relevância nos programas escolares de Matemática. Admitindo que as conceções dos professores sobre a prova afetam a forma como ela é tratada em sala de aula, procuramos averiguar as conceções de professores portugueses de Matemática do 3.º ciclo do Ensino Básico (do 7.º ao 9.º ano) sobre diferentes aspetos da prova matemática e a influência que a experiência profissional tem nessas conceções. Adotando uma abordagem metodológica mista, recolhemos os dados através de um questionário, respondido por 72 professores, e de uma entrevista a duas professoras com experiências profissionais diferentes. Os resultados revelam que os professores, sobretudo os que têm menos tempo de docência, consideram que a prova matemática tem uma natureza distinta da de outras disciplinas, é uma atividade essencial para a construção do conhecimento matemático, tem como função verificar e explicar a veracidade de uma afirmação e permite desenvolver o raciocínio e a comunicação matemática. No que respeita à participação dos alunos na atividade de provar, são os professores com mais tempo de docência que mais o destacam, o que permite aos alunos perceberem a natureza desta atividade. Em termos curriculares, são os professores com menos experiência docente que mais concordam com a presença da prova logo nos primeiros anos, embora salientem que esta atividade só faz sentido em alguns tópicos programáticos.Proof is an activity that plays a key role in the construction of mathematical knowledge, which is why it acquires relevance in mathematics programs. Admitting that teachers' conceptions about proof influence how it is handled in the classroom, we tried to investigate the conceptions of Portuguese mathematics teachers from the 3rd cycle of basic education (grade 7 to 9) on different aspects of mathematical proof and the relationship that the teachers’ experience has in these conceptions. Adopting a mixed methodological approach, we collected data through a questionnaire answered by 72 teachers, and an interview with two teachers with different professional experience. The results show that teachers, especially those with less experience, consider that mathematical proof has a distinct nature from other disciplines, it is an essential activity for the construction of mathematical knowledge and its function is to verify and explain the truth of a statement by developing reasoning and mathematical communication. With respect to student participation in the activity of proving, teachers with more teaching experience are the ones that most highlight it, which allows students to realize the nature of this activity. In curriculum terms, teachers with less experience are the ones that most agree with the presence of proof in early grades, while emphasizing that this activity only makes sense in some program topics.Este trabalho contou com o apoio de Fundos Nacionais através da FCT – Fundação para a Ciência e a Tecnologia no âmbito do projeto PEst-OE/CED/UI1661/2014, do CIEdUM e do projeto UID/Multi/04016/2016info:eu-repo/semantics/publishedVersio

How to think about informal proofs

This document is the Accepted Manuscript version of the following article: Brendan Larvor, ‘How to think about informal proofs’, Synthese, Vol. 187(2): 715-730, first published online 9 September 2011. The final publication is available at Springer via doi:10.1007/s11229-011-0007-5It is argued in this study that (i) progress in the philosophy of mathematical practice requires a general positive account of informal proof; (ii) the best candidate is to think of informal proofs as arguments that depend on their matter as well as their logical form; (iii) articulating the dependency of informal inferences on their content requires a redefinition of logic as the general study of inferential actions; (iv) it is a decisive advantage of this conception of logic that it accommodates the many mathematical proofs that include actions on objects other than propositions; (v) this conception of logic permits the articulation of project-sized tasks for the philosophy of mathematical practice, thereby supplying a partial characterisation of normal research in the fieldPeer reviewedFinal Accepted Versio