Blog #6 Response: NCTM Article - Proof by Examples

D’Ambrosio, B. S., Kastberg, S. E., and Santos, J. R. (2010). Learning from student approaches to algebraic proofs. Mathematics Teacher, Volume 103, No. 7, 489-495.

In “Learning from Student Approaches to Algebraic Proofs,” a study showed how algebraic concepts are among the most difficult concepts for students to prove. Instead of proving statements algebraically, most students chose to “prove” statements by giving examples. The types of answers to the proof goes as follows: (1) “It’s fact. What is there to prove?” Students said the proof was not necessary, and the statement was self-evident. (2) Some students gave one or more examples that showed the proof held true there, saying this was sufficient for a proof. (3) Others showed the proof held for large numerical examples, therefore implying the proof always held. (4) Some students completely lost sight of the fact that they were actually trying to prove something, not solve something. When they saw the equals (“=”) sign, they assumed they just needed to solve an equation. (5) Very few students proved with algebra, followed by examples, and then the general case. It was apparent that the students who actually looked for the patterns in the examples were on their way to finding the generalized rule for the proof. The overall analysis of the results was this: “The data suggest that students need a much more robust vision of the role of proofs in supporting mathematical activity, making connections between mathematical ideas, and communicating findings. Students seem to have a limited repertoire of meth¬ods to use for proving those statements.” In order to fix this problem, teachers need to analyze their students’ written work carefully. In turn teachers will better be able to draw conclusions on the level of their students’ understanding and foundation in mathematical concepts.

I think this will allow for teachers to be able to lesson plan much more accurately. When teachers analyze their students’ work and find out where their weak points are, e.g. proving statements algebraically, then they will know which concepts need to be taught more fully in the classroom. Once students get a handle on proofs, they will have a much easier time with connecting and communicating their mathematical ideas. More specifically, once students gain a better understanding on how to prove mathematical concepts, they will have a greater “development of stu¬dent understanding of one another’s work and move toward the use of proof as a tool in exploring connec¬tions, relationships, and the veracity of mathematical statements and in communicating their findings.” I do not agree that giving examples as a solution to a proof is in reality a proof, but I can see why students would assume it to be true. They have not been educated on the matter. Teachers need to teach why giving only examples as answers to a proof is not sufficient for all cases. When teachers teach concepts only through examples, the processes may stick easier in the minds of the students. However, if teachers do give proofs behind the mathematical thinking, then students will have an easier time convincing others and themselves of the connections they make between their next mathematical ideas and the previous ones. It all comes down to the basic ideas discussed earlier this semester: the importance of relational teaching/understanding.