# Every unsuccessful problem solver is unsuccessful in his or her own way: affective and cognitive factors in proving

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## Abstract

It is widely recognized that purely cognitive behavior is extremely rare in performing mathematical activity: other factors, such as the affective ones, play a crucial role. In light of this observation, we present a reflection on the presence of affective and cognitive factors in the process of proving. Proof is considered as a special case of problem solving and the proving process is studied adopting a perspective according to which both affective and cognitive factors influence it. To carry out our study, we set up a framework where theoretical tools coming from research on problem solving, proof and affect are present. The study is performed within a university course in mathematics education, where students were given a statement in elementary number theory to be proved and were asked to write down their proving process and the thoughts that accompanied this process. We scrutinize the written protocols of two unsuccessful students, with the aim of disentangling the intertwining between affect and cognition. In particular, we seize the moments in which beliefs about self and beliefs about mathematical activity shape the performance of our students.

## Keywords

Affective factors Cognitive factors Beliefs Problem solving Proof University## Notes

### Acknowledgements

Research program supported by MIUR (PRIN 2005019721_002 ‘Meanings, conjectures, proofs: from basic research in mathematics education to curricular implications’).

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