As the article pointed out, many results can be obviously seen and experimentally tested thoroughly enough to eliminate the need for a rigorous proof to establish their truth. However, hyperbolic geometry is a largely unfamiliar context where intuition may be deceptive. Therefore, there is a genuine need for rigorous proof to verify and to convince students of the validity of the results of discarding Euclid's Fifth Postulate. Some examples are " All triangles have angle sum <180" and "For every line l and every point P not on l, there are infinitely many parallels to l through P"

A derivation proof can provide a helpful answer for why a formula works. While the familiar quadratic formula can easily be tested empirically, its accuracy does not explain why it works. In fact, without background , the quadratic formula can be seen as a magic, plug-and-chug trick. Showing students how to derive the formula by completing the square on the general equation ax^2 + bx^2 + c = 0 to force the simpler form (x-d)^2 = f can put the formula in the appropriate context and give students a tool for deriving it if they forget it.

The first example that comes to my mind is an abstract algebra proof that my teacher described as a "constructive" proof. We set out to prove that the algebraic structure F[a] was a field, in particular that each element had some multiplicative inverse. I was expecting a slick "existence" proof that only showed abstractly that such inverses existed. However, by the time we finished, we had discovered an explicit arithmetic algorithm for finding the inverse of a general element. As a student, I found this types of proof accessible and satisfying because it provided extra useful information.

The axiomatic system of Euclidean geometry is the best example of where this function of proof plays a role. One proves familiar facts like "Two intersecting lines have exactly one point in common", "Vertical angles are congruent", and the SSS triangle congruence theorem to analyze what axioms and results are absolutely basic and to understand how other results follow from and depend on others. The result is fitting a collection of relationships, facts, and theorems into an organized, logical basis for the geometry that we are familiar with.

A test or assignment might include a problem to solve and ask for a proof that the solution found is valid. For example, "Find all the real roots of x_ - 2x_ +3x –1 = 0 and prove that you have found them all." By requiring a proof to accompany an answer, the teacher is asking the student to communicate his understanding of the pertinent mathematical concepts, to practice explaining and organizing his mathematical thinking, and to become comfortable with providing and seeking justification for answers.