Friday, January 15, 2010
Instant vs Delayed Gratification
According to Skemp, when we speak of understanding a Mathematical concept, there are two different types of understanding that we could possibly be referring to. The first, termed Instrumental understanding, is knowing what to do in order to get a correct answer. The second, Relational understanding, expands on this: it is knowing what to do, and the reasoning behind why this produces a correct answer. Both provide methods for solving Mathematical problems and both produce correct answers. The benefit of learning Math instrumentally is all to do with ease and speed. Given the right set of rules, a child can quickly, and correctly, produce a set of correct answers. It is easier for the child to understand what she or he has to do. However, there are drawbacks, many of which can be avoided by using relational teaching and learning. For every new set of problems, an instrumental learner must learn a new set of rules. On the other hand, a relational learner can adapt his knowledge of previous concepts and derive a new method. He is also more likely to remember something that he understands. Consider the 2 words "Happy" and "T8grP". Which are you more likely to remember tomorrow? The one that makes sense to you, that you can imagine using correctly in situations, that you feel confident adapting for a variety of uses. The learner will enjoy mathematics all the more for being able to solve problems without being dictated to at every step.
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I agree with a lot of what you wrote, but my favorite line in this whole thing is this: "For every new set of problems, an instrumental learner must learn a new set of rules". The reason I liked this so much is that it really simply states the problem with instrumental understanding. Somethings can be learned with instrumental understanding without creating too much of a problem, but I don't know how much you can memorize before you get confused on what is going on. I thought the two words analogy was good, but I thought one thing that might be lost in the analogy is that here it is actually easier for us to ready "happy". Where as relational understanding should be more difficult to teach and learn, though I do agree that once learned it is much easier to remember later, so in that way the analogy is good.
ReplyDeleteI have to say that I really liked how clearly and simply you pointed out the major flaw of instrumental learning. Personally, while I was writing my blog it took me awhile to finally express what you said in just a couple sentences. That "For every new set of problems, an instrumental learner must learn a new set of rules." I think this really portrays how instrumental learning can quickly become a daunting task.
ReplyDeleteMy only criticism, because I have to make one, is that the idea that relational understanding consists of instrumental understanding as well as expands on it could have been emphasized.
Besides that I thought your ideas were clear and concise which made it very easy to read. Also, I really liked the analogy of the two words at the end to bring all the ideas together.
When reading this, the part I really like was your definition relational understanding. The reason I liked it so much was that I felt like you were able to answer two of the questions at once. You not only defined relational understanding but you included how it was related to instrumental understanding by showing that it was first, just knowing how to do something, and then also the why. Good job.
ReplyDeleteThe only thing I would have maybe done different is I would have probably emphasized more of what Skemp said, I know that you did but relate what he had written, but since it was a review of his paper, I would have mentioned him more, that way it would feel more like it was what he wrote about in his paper, rather than you coming up with the ideas all by yourself. However, I really enjoyed reading your summary!
I think how you how you organized your paragraph was very interesting and very well done. I liked how you explained the definition of instrumental understanding and then built off of it to explain the definition of relational understanding. I also thought it was good how you explained the benefits of instrumental understanding and then built off of it again to explain the benefits of relational understanding.
ReplyDeleteI would have included more of the challenges of relational understanding that were mentioned by Skemp in his article. Such as that it takes longer to teach that way and may not be as successful with all students.
I think you did a fantastic job of explaining in a short space the difference between instrumental and relational understanding. You also were able to illustrate and explain the advantages of each type of understanding and even some of the disadvantages. Your blog was very easy to read and understand. One thing that would have helped me better follow the article might be to state that what you are explaining is what Skemp said, in order to give the blog credibility.
ReplyDeleteI really like your summary. It flowed really well and connected ideas with the article in a very easy way. I wonder if your example of the word happy and T8grP would be the best because happy is a word that can be remembered really easy so I would think that it applied to instrumental because instrumental is easier to learn and remember in a lot of ways.
ReplyDeleteI really appreciated your analogy to "happy" and a mumbled symbols. Understanding does make a big difference.
ReplyDelete