Instrumental vs. Relational Understanding
One of the ideas that stood out to me while reading this article was the idea that "the most important thing about an activity is its goal". It seems to me that this is applicable not just in math but in life. Without a goal, an activity has no point. Goals are what make actions meaningful. For example, I am not a big fan of running. Running as a form of exercise seems like torture to me. It feels pointless. However, when I am running around playing with my dog I enjoy running and the activity has meaning. The goal, whether it be to get in shape or have fun with my dog, determines not only why the activity is performed but also how. It shapes the experience. Thus, I agree with Skemp that it is important to be conscious of whether your goal is instrumental or relational understanding when teaching mathematics. Not only will the goal effect why you teach and what you teach but also how you teach.
Another thing that resonated with me was the idea that while relational understanding is pretty clearly the preferable choice in most situations, getting there can be difficult in practice. Two of the challenges Skemp highlighted were the problems of an over-burdened syllabi and difficulty of assessment. Throughout the first few weeks of classes, I have noticed a theme that often times one of the biggest obstacles for teachers is the need for efficiency. Lack of time and resources pushes us to look for teaching methods that are quick and easy. For that reason, it seems to me that an exceptional teacher is one that is able to work around this system and find creative ways to balance quality and efficiency.
I was also struck by the importance of context. In general, relational understanding seems preferable but there also may be some contexts in which that level of understanding is not required. If we had to fully understand every concept we learned then it would take a very long time to learn anything. I think that there may be different levels of relational understanding as well. For example, in a high school physics class you would want your students to understand why they are using certain formulas in certain contexts. However, you do not need to teach them the calculus that is required to understand how these formulas were developed. Context is important here too. In a university physics class, calculus is required to achieve the desired level of understanding. This comes back to the importance of establishing your goals before engaging in the activities required to achieve them. The context determines the goal which determines how the activity is approached.
Another thing that resonated with me was the idea that while relational understanding is pretty clearly the preferable choice in most situations, getting there can be difficult in practice. Two of the challenges Skemp highlighted were the problems of an over-burdened syllabi and difficulty of assessment. Throughout the first few weeks of classes, I have noticed a theme that often times one of the biggest obstacles for teachers is the need for efficiency. Lack of time and resources pushes us to look for teaching methods that are quick and easy. For that reason, it seems to me that an exceptional teacher is one that is able to work around this system and find creative ways to balance quality and efficiency.
I was also struck by the importance of context. In general, relational understanding seems preferable but there also may be some contexts in which that level of understanding is not required. If we had to fully understand every concept we learned then it would take a very long time to learn anything. I think that there may be different levels of relational understanding as well. For example, in a high school physics class you would want your students to understand why they are using certain formulas in certain contexts. However, you do not need to teach them the calculus that is required to understand how these formulas were developed. Context is important here too. In a university physics class, calculus is required to achieve the desired level of understanding. This comes back to the importance of establishing your goals before engaging in the activities required to achieve them. The context determines the goal which determines how the activity is approached.
Lovely!
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