Blog #7 Response: NCTM Article - Fractions without Understanding

Philipp, R. A., and Vincent, C. (2003). Reflecting on learning fractions without understanding. ON-Math, Vol 2, No. 2, 1-6.

The main idea of this article, “Reflecting on Learning Fractions without Understanding,” is that students learn less if they are taught procedural followed by conceptual, compared to students who learn more from only being taught conceptual instruction. Routine procedures actually inhibit the learning of concepts. This idea was apparent in a study done in a classroom of only conceptual understanding methods. The experiment involved the teacher giving a purely procedure-based lesson about converting between improper fractions and mixed numbers. The experiment was video-taped, students given an assessment, and then interviewed. Students realized that this was a different teaching method as the class went along. A few weeks later, the conceptual side of converting fractions was taught and similar procedures followed. After calculating the results of the assessments, it was apparent that students retained a far greater deal of information after a long period of time when taught conceptually rather than procedurally. When some students were taught procedurally followed by conceptually, results showed that students’ understanding dropped incredibly. One student, Rachel, said she does not really figure out the problems herself when she tries to do problems procedurally; she is just following the way the teacher tells you to figure it out. On the other hand, when she solves problems conceptually, she understands. This is what she had to say: "So when I figure that out, it's easier, and, um, once I figure it out, it's, it stays there 'cause I was the one who brought it there. So, and it is just easier to do when you figure it out yourself, instead of having teachers telling you." When Rachel was asked to solve a problem in the interview, she first attempted it with the procedure taught by her teacher. She arrived at the wrong answer. When told so, she tried the same problem her own way, arriving at the correct answer. When asked why she solved the problem procedurally first, she said it was because that was the way she learned it first. Thus, the conclusion was made that procedure followed by conceptual teaching is not as effective as purely conceptual teaching.

I thought this was an incredible finding. I do not, however, agree that teaching conceptually is the ultimate method of teaching out there. Obviously, as the article outlines, strictly conceptual teaching is better than strictly procedural teaching. I think that the best way for students to learn is by teaching conceptual and procedural hand-in-hand. This is different from what was discussed in the article because the two types of teaching were done weeks apart from each other. We have already discussed why purely conceptual teaching may not work – it takes more time, not all the material gets covered, not all the students get involved, etc. If some procedural is taught along with the conceptual, then students will really understand the material (from the conceptual part of the lesson) and they will know how to quickly arrive at answers (the procedural part of the lesson), and thus they will do well in school and get throughout the required amount of material. For me, I like it when teachers have taught the conceptual side of concepts so that I can truly know what is going on, followed by the procedural side of things so that I might be able to compute answers quickly and efficiently. I think the key part of this idea is for teachers to teach how the conceptual and procedural connect to each other. This will result in optimum understanding, in my opinion at least.

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.