Otten, S., Herbel-Eisenmann, B.A., & Males, L.M. (2010). Proof in algebra: Reasoning behind examples. Mathematics Teacher, 103(7), 514-518.

Throughout this article, the main focus is on how proving an idea or rule is much clearer and provides for greater understanding than simply providing examples of the rule does. It is mentioned that proofs should not only be used for geometry class, but also for other areas of math such as basic algebra. They do not need to be complex, college-level proofs, but they need to be adequate to prove the rule is true for all cases. The majoritiy of the article is spent on a classroom example of the cross-multiply rule That is, if two fractions are equal, you can multiply the numerator of the first with denominator of the second, and that will be equal to the numerator of the second multiplied with denominator of the first. The teacher first goes about showing that this works through providing a few examples. The article points out that students are easily convinced so they would believe the rule to be true. They might eventually think that examples are a proof or that proofs are unnecessary. When a general case of the rule is presented after examination of several different examples, the students will know that the cross-multiply rule works for all cases and will be most benefited because they will know a certain rule is true.

Including more proofs in lower-level math classes would be beneficial to the students then and in the future. Most of my math experience has been taught through examples, like in the article, which showed that the rule was true for the numbers in the examples. Sometimes I would get confused and assume the rule is true for all numbers when that is really not the case. Doing a proof of a rule would teach students how to find out when a rule works and when it does not. Like the article said, proofs do not really come up until geometry. The idea of a proof is so new to students in geometry that they do not understand how to use them and get frustrated. This leads to a general aversion towards proofs, which I have seen many times. If proofs were introduced sooner, students would be more comfortable with them and be more inclined to use them rather than avoid them. Introducing them sooner would also show that proofs are necessary and can not be replaced by examples. Including proofs would also connect ideas together and make mathematics seem more logical. Often math can seem like a random bunch or rules and procedures, but using proofs can connect ideas and show why a rule works instead of just knowing that the rule works. This would be beneficial to students because they would more likely enjoy math instead of resenting it.