Competencies and difficulties of undergraduate students facing a problem involving conjecture and proof
DOI:
https://doi.org/10.18172/con.6463Keywords:
conjecture, proof, Toulmin's model, ontosemiotic approach, undergraduate studentsAbstract
Despite the importance of proof for developing students' mathematical competence, understanding how its learning occurs remains a challenge for both researchers in Mathematics Education as well as teachers. In this study, we analyze the competence of first year university students in solving a problem involving the conjecture and proof of arithmetic properties. Adopting a predominantly qualitative methodological approach, we integrate the Toulmin's model with tools from the Onto-semiotic Approach to characterize and analyze the practices developed by students. Specifically, we identify the objects and processes involved in the argumentations, relating them to the elements of the Toulmin's model, and examine the degree of generalization achieved. This integration enables us to gain deeper insight into students' competencies with proof and the difficulties they encounter. The results of our study show that, while most students use correct deductive argumentations to validate or refute conjectures explicitly stated in the given problem, they face significant difficulties when formulating conjectures that are not explicitly provided and in developing their proofs. Furthermore, many students fail to achieve the expected level of formalization at the university level, and when they do, it does not necessarily translate to greater relevance in their solutions. These findings indicate that the instruction that the students have received thus far has been insufficient to develop a solid understanding of proof. We conclude by emphasizing the need to focus on how proof is addressed in current instructional processes to tackle the identified difficulties and to create opportunities for meaningful learning.
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