The direct proof method begins with a hypothesis, or "educated guess." The mathematician desires to prove that her guess is correct by way of mathematical axioms. Axioms are the basic rules of mathematics, such as "a number plus one equals the next number." By using the axioms in her field (there are approximately ten for "basic math," and more for other fields of mathematics), the mathematician can go through a step-by-step procedure to arrive at the statement of her axiom, thereby proving the hypothesis.
A mathematician may take a more off-course route to reasoning by supposing that the statement of interest is not true. The mathematician then takes steps using this assumption to show how other laws of mathematics would be violated. The fact that this supposition leads to the breaking of well-established mathematical laws shows the assumption cannot be valid. The implication is that the statement has been proven false through contradicting its existence.
Mathematical induction is a form of reasoning in which mathematicians only need to prove two statements to show that a series of statements is true. For example, if a mathematician sees a pattern and wishes to write it in mathematical language, he cannot do so without proving that the pattern makes sense for every instance of the language (in brief, because mathematical functions use integer variables, and integer variables are infinite in number, the mathematician must find a way around proving the function for every single value). Thus, the mathematician must first prove the function holds for the first value, and then that the function holds for the "next" value. That is to say, if the first value works and the "next value" works for any possible value, there is a cascade of proofs, showing that all values work.
The counterexample is a form of reasoning of disproof, as opposed to proof. The idea is simple: if one claims that a certain statement is true, a mathematician need only find an instance of the statement that does not hold true. The existence of such an instance destroys the entire statement, thereby disproving it.