Friday, January 15, 2010
MthEd 117 Assignment 2
When we talk about "understanding" in mathematics, it takes us to a very grey area where there are different types of understanding. Richard R. Skemp, in his article "Relational Understanding and Instrumental Understanding", discusses this grey area in the context of two different types of understanding that happen in mathematics, especially in educating students. Skemp states that relational understanding is what can be considered a "true" understanding of the subject matter; it is knowing not only how to solve problems and what to do, but also knowing the why behind it all. Instrumental understanding, however, is knowing a list of rules, and then knowing when to use them to solve problems. At first, it may seem that relational and instrumental understanding are completely disjunct from each other. However, after a closer examination, one realizes that instrumental understanding is actually a smaller part that is completely contained within relational understanding. If someone has a relational understanding of a math concept, then they understand the instrumental side of it as well, knowing rules and shortcuts to use when solving problems; but, they also understand why those rules work and can even derive other rules from their relational understanding. In this way, Skemp shows that teaching mathematics for a relational understanding is a much more empowering way of helping students. There are advantages and disadvantages to both though. Instrumental understanding, in general, is much easier to "understand"; some topics, like multiplying two negative numbers together, are very difficult to understand relationally. Instrumental understanding also allows someone to get the right answers, and get them quickly; which, in turn, gives the students immediate rewards that are quite satisfying to them. However, with that type of understanding, students never understand the "why" in the math they are doing, and will never be able to formulate mathematic concepts on their own. They will always rely on someone else to give them another list of rules and procedures to follow in order to get the right answer. Relational understanding gives students quite an advantage in mathematics because it is much more adaptable to new tasks; when given a new, but related concept in mathematics, students with a relational understanding can reason and manipulate what they know to help them solve a problem that could be considered "outside of the box". Relational understanding is also much easier to remember, and, when gained, motivates students to explore new areas within mathematics on their own steam, greatly reducing the need for the teacher to try and motivate students when teaching. On the other hand, relational understanding is much more difficult for students to grasp at first, and it is also very hard to evaluate whether or not students have gained a relational understanding. Overall, Skemp's article brings up the question, "Which type of understanding should the term 'mathematics' fall under?" To Skemp, only a relational understanding should really be called teaching "mathematics", but the ideas within his article continue to be discussed, and need to be in order for teachers to reach a way to teach the students in the best way possible.