The students received a homework packet on Thursday to work on over their fall break next week. The packet has questions on everything that they were studying about fractions, from reducing to lowest form, to adding & subtracting and multiplying & dividing. I was walking around the classroom while the students were working on their packets before they went home for the day and I stopped at one student's desk to watch him do his work. I noticed that he was counting on his fingers so I asked him what he was doing. He explained that the fraction was 74/3 so he was using long division. I asked what he was using his fingers for and he said he was counting by 3s to get to 74. I immediately saw what he was thinking and asked him if he thought the 3 had to go into the 74, not just the 7. It took no time at all for him to see what he was doing was unnecessarily difficult and he immediately wrote down 2 and multiplied that by 3 to bring below the 7 to subtract. He then continued through the rest of the problem to find the answer. After he got his correct answer, he asked if that would have worked, what he was doing before. I explained to him that it would work, but it is just a lot more work and very time consuming to have to count up to big numbers by a small number. I noticed he was counting on his fingers again and asked what he was doing. He said he just wanted to make sure and check his answer.
This was an example of a mathematical error, but not necessarily a misunderstanding. I think that the student knew the general way to do this operation because as soon as I pointed it out, he jumped right into doing it the correct way without any hesitation. I think that this shows that the student had an understanding of how to long divide, but definitely has not mastered it yet. What I would offer him as some extra help would be showing him how and why long division actually works and giving him practice problems to work on. I think with problems like long division, one of the best ways to become good and quick at it, is practice.
It is valuable that you are probing into students methods as they attempt to work on the problems and tasks that you present them. You might not need to point out to the student a different approach, even if it is more efficient. That is, it is important that the student continue to work on the problem in the way that is most comfortable for him, but then to give the student to compare his strategy with other strategies. My point here is to not necessarily be so hasty as to correct students when they use what may seem to be illogical or bizarre methods. At the same time though, it is essential to provide students with the opportunities to compare strategies and thought-processes.
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