If found this to be quite an interesting read. I've been somewhat aware of the difference between a student completely comprehending subject material vs. just memorizing the formulas and not being able to extend the information beyond a few rigid examples in class, but I haven't consciously thought about how the ambiguity of the word "understanding" can make discussion about this topic challenging. I'll have to keep this in mind when talking with students about their "understanding" of course material. I also really liked the analogies Skemp used, such as the comparison to trying to learning music without listening to any of the notes. These helped put into perspective the relative absurdity of forgoing providing students with a strong "relational" foundation in mathematics which they could use to apply to other areas of math inside and outside of the classroom.
I agree with Skemp about the overall importance of relational understanding for students. I think this type of understanding sets them up for success in their ongoing math education as it encourages to seek out patterns and make connections between the different areas of their learning. Without the bigger picture, I feel that students will find math to be a tedious and frustrating subject in which one simply has to memorize rules and apply them in a few specific circumstances. I do also agree with Skemp, though, that having an instrumental learning of a subject is sometimes all that can be reasonably expected or provided. In most cases, it is good to have an instrumental learning in addition to a relational one so that you can solve problems more confidently and efficiently while knowing that you could provide proper explanation of each individual step if it was asked of you. However, there are certain formulas that come up in high school mathematics for which their origins go beyond the scope of the classroom. For example, I think a proof of the Pythagorean Theorem would only serve to confuse students at an 8th or 9th grade level with no background whatsoever in writing proofs. Even still, I think it is useful to be transparent with students about moments like this when their relational understanding of a topic might be postponed, and that there are still ways to motivate them and to provide them with intuition about the material.
You’ve written a thoughtful response here — I especially like your point about the ambiguity of the word “understanding” and how that shapes classroom talk. Bringing in the music analogy and your example with the Pythagorean Theorem shows you’re thinking about the practical side of teaching. To push this further, you might ask: when should we insist on relational understanding, and when is instrumental “good enough”? That extra layer of critique would take your post into the next level. Nicely done.
ReplyDelete