The timing for this article could not be better for me because it is related to my practicum experience. While I was on my short practicum, I was working with a boy in grade 8 who insisted on figuring out a new way to multiply or divide two 2-digit numbers, which was very different from the traditional approach. The fact that he was coming up with a new way to do multiplication/division was great except that some of his supporting methods were not leading him to the right answers and were extremely time consuming. In addition, this activity or this part of math was not the focus of the lesson rather that was the supporting part of the lesson. I was teaching the Pythagorean Theorem and discussion around multiplication was diverting attention from the main focus.
My SA was very frustrated with him as well. In fact, one day after school, he talked to him about perhaps taking Math essentials next. I had a completely different opinion. I thought he should just learn the algorithm and speed up a bit and, if he has time to invent new methods, he can as long as he is up to date with the course material.
So, it is fascinating for me to read this article now as I am also puzzled about this issue of how much freedom to promote and at what point. From the article, we can see that prior knowledge plays a role in the development of flexibility. In the Case of Computational Estimation, the students with greater knowledge demonstrated greater flexibility. In the example of my student above, I should show him a couple of different approaches and provide more information on background knowledge so that he can have more freedom in his strategy selection. Then he will be able to choose a strategy from a selection given or, based on provided supporting methods, he can either invent a new strategy or modify the given ones. The resolution of the matter regarding this student is similar to the article's discussion in the sense of strategy selection.
From the article, strategy uses can be about creativity, flexibility or adaptivity. In addition, there are three different teaching approaches mentioned. One of them is traditional in nature, the routine approach, and the other two are investigative and problem-solving approaches. The last two are more conducive to adaptive strategy use. Their difference lies in the accuracy of the final answers. The problem-solving approach is about inventing new while rejecting the given strategies. As a result, there are more errors in the process which lead to wrong answers 31% of the time.
How do you know that the problem-solving approach results in incorrect answers exactly 31% of the time? (Where does that precision come from?)
ReplyDeleteI would also be rather concerned that the teacher was suggesting the boy switch to Math Essentials the following year! If it were me, I would do as you suggested -- encourage the student to look for multiple approaches to a problem or procedure, while keeping him on track with approaches that actually work. There might also have been an issue of this student trying to distract from the lesson, and that would have to be dealt with as a separate thing.