There is a branch of philosophy that deals with logic, and since mathematics has evolved from this branch of philosophy logic is also taught in mathematics. The focus of logic, the desired result, is to determine when an argument is valid, and when it is invalid. Once some basic concepts are mastered one can determine if an argument makes sense when presented by another person, and when an argument might be convincing to another person.
To set the rules for arguing successfully it should be noted at the outset that emotion is not only unnecessary, but may hinder a well presented argument. Having a listener receptive is a first step, so avoid an unwanted reaction by remaining calm.