Richard R. Skemp presented an article in the Mathematics Teaching journal about two types of understanding that student can develop when learning mathematics. In the article, he describes one type of understanding as instrumental. From reading the article, my idea of what is meant by instrumental is an understanding of the rules and the steps to solving a mathematical problem, but having very little understanding of why those rules work. The other type of understanding presented is that of relational understanding. I believe that this is when a student can not only achieve the correct answer using the given rules and formulas, but also know where the rules and formulas came from and why they work.
These two types of understanding are not completely exclusive in that the students who have the relational understand also have the instrumental understanding. Additionally, I believe that students who have instrumental understanding may have an understanding of why things work, just not as extensively.
So if instrumental is included in relational, then why not just always teach relationally? For me personally, I learn better instrumentally because I am very good at memorizing, and storing what I memorize for a long time, rather than just forgetting it after a test. When a teacher spends a long time on how to get to the formula, equation, or rule that we will be using, the only thing I end up remembering is the end product. So for me, instrumental is preferable.
Since not everyone learns the way I do, both types of understanding have advantages and drawbacks. When learning relationally, the amount of material can be overwhelming and distracting. Students may learn how or why something works, but fail to know how to apply it to an actual problem. When learning instrumentally, students temporarily know the formulas and steps for say a quiz, test, or assignment, but that knowledge may not last, while if they had learned how to get to that formula relationally, they might otherwise recall.
Everyone learns differently, so it is hard to pick which way is ideal. Skemp clearly prefers relational, most likely because instrumental is included within it. I also agree because even though that's not how I learn, using relational can allow the student to decide for themselves what information they will work to recall and what is meaningless.
Subscribe to:
Post Comments (Atom)
This post was very interesting to read. You had a lot of experience and information to present which was nice. Your definitions of the two types of understanding were very clear and concise. I felt like there was a little bit missing when talking about the advantages and disadvantages, the main issue I found was just ease of memory but Skemp talked pretty extensively about many other issues that I didn't find. Thank you for a great post!!
ReplyDeleteI agree with Ali that this was certainly interesting to read. I enjoyed your own opinions that you shared but for this assignment. For this assignment though, I might have tried to focus more on what Skemp said than how I felt about what he was saying in his article. I felt the discussion of advantages and disadvantages that Skemp provided wasn't complete in your post and that the understanding of how the two types of understanding overlap was a little messy. I say this because I read at the end one statement on how instrumental is found within relational but hadn't read previously in your post just how one was found in the other. Great ideas though and your honesty was appreciated.
ReplyDeleteI agree with your point that instrumental is the more favored method students wish to be taught - it is a quick, easy way to get the right answer and just requires a little bit of memorization. However I am not so sure that it is the ideal method of understanding. For me, in certain instances, relational has been more favored - if I have forgotten a formula but understand the relational aspect of it, I can contrive the formula yet again. So in a way relational is more beneficial.
ReplyDeleteYou did a really nice defining the two types of understanding. It was very clear.
ReplyDeleteWhen students understand relationally, they also know how to apply it. Another benefit of relational understanding is students can continue to learn on their own, and discover new mathematics and formulas, instead of waiting for the teacher to give them another procedure to memorize.