### Modelling In-service Teachers’ Mathematical Reasoning in Constructing and Validating Proofs

#### Abstract

This study determines the mathematical practices of proof validation and proof construction of the in-service teachers. Currently, little attention in the literature about how one’s ability to validate correlates to the ability to construct proofs. Thus, we intend to fill this gap by providing a model about the mathematical proof reasoning of the 17 in-service teachers and to look into the connection between validation and construction of proofs using Toulmin’s paradigm of reasoning. Data were gathered from interviews and written responses through worksheets. The data analysis was grounded in the common categories that emerged which highlights uncertainty of their possessed skill when encountered difficulties which require many resources to be proficient in both validation and proof construction. It shows that proof construction is more challenging than proof validation. Thus, the data that we obtained suggest that, if an individual can successfully validate purported proofs then it is not an assurance that he can also construct proofs.

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