Otten S., Herbel-Eisenmann B.A. & Males L.M.(2010). Proof in Algebra: Reasoning Beyond Examples. Mathematics Teacher Vol 103(7),514.
The main point of the article is to "provide an image of what proof could look like in beginning algebra, a course that nearly every secondary student encounters." There are divisions of ways that students pick up on proofs which was studied by Stylianides. First there is the externally based schemes, which Stylianides defines as convincing the student from an outside source. Second, there is empirical proof schemes which are justifications based solely on observations. And third, there is analytic proof schemes which consist of valid mathematical arguments such as a treatment of the general case or formal deduction within a system. However, from another study by De Villiers (1995), he made the point that proof should be more explaining and understanding than convincing. The argument that the authors iare making is that proofs in secondary schools don't go beyond geometry which many of the secondary students don't get to. So the authors try to take proofs beyond geometry to see what it is like in beginning algebra which almost every secondary student takes. There are two cases presented. In the first case the teacher showed the students the pattern slowly by using examples. In this article the teacher showed that cross-multiplication of a number equal to itself gives the same product. for example 1/12=2/24. If we cross multiply we get 1 because 1*24=24 and 2*12=24, so we get 24/24=1. So after seeing several examples the students notice a pattern, that is, taking the cross-multiple of an equivalent fraction produces the same number. In the second case, the teacher used the examples to show the general case of using this concept of cross-multiplication which the students constructed. They represented the fraction as a/b and the number equivalent as n/n, so it was na/nb and they justified it by saying that it is the same as multiplying by 1, the n's cancel out. So from these two cases the authors point out that mathematics is a state of mind that is "characterized by inquiry and a thirst for justification." They conclude that it is a goal that can be reached by sharing, discussing and analyzing experiences in proofs in the classroom.
So this is an attempt to have students think for themselves and not be so dependent on authority. I believe that it is a critical study because of the many students who are very dependent on authority. I was one of those students. I was very dependent on the teacher, and became very instrumental in my learning. However, I am learning that it is a state of mind that requires a thirst for justification as the authors put it. From my own experience in secondary school math, just like in the article, the only proofs that I was exposed to were those in geometry and none after that. I am coming to see more that the thirst for justification that is talked about in the article is the questioning why? Students need to be able to question why? and not just accept the information given by the authority. They need to read through and construct ideas about the concepts and procedures for their own. This way they will be open minded when being taught by the teacher, they will have a wider vision of the concepts and procedures because they will be able to see more and question why? when being taught by an authority.
