I believe one of the main ideas of the article is that a teaching form which enforces purely instrumental understanding can permanently damage a students' attitude towards mathematics. The article suggests that because Benny wasn't given understanding of why the rules for fractions and decimal conversions worked, he developed his own meaning that made sense to him.
This is extremely applicable to us as math education majors because when we are teachers we'll need to make sure that our students know the reasons behind what we are teaching. If we don't, they may dislike math and believe that we are just making it up. They will work only towards "cracking the key" to our tests, and then fail to retain a deep understanding that will be lasting throughout the rest of their education.
Monday, January 25, 2010
Thursday, January 14, 2010
Blog Entry #2
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.
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.
Tuesday, January 5, 2010
Blog Entry #1
1. Mathematics is the study of problem solving.
2. I learn math best through repetition. It helps me to memorize the steps to solving the problems, and puts it into my long term memory.
3. They will learn math the best by applying it to real life situations. This allows the students to experience math firsthand; once they realize that math can be used in places other than school, it will seem more fun and more practical.
4. Some current practices in math classes that I think help students are those that use technology and fun computer programs. Especially in geometry. I think these make math more exciting as well as hands on.
5. When teachers give tests and then move on to the next subject without individually helping the students who do poorly or fail, the students become permanently behind and never learn the material as well as they should. Tests should be a way of helping the teacher see where the students needs help.
2. I learn math best through repetition. It helps me to memorize the steps to solving the problems, and puts it into my long term memory.
3. They will learn math the best by applying it to real life situations. This allows the students to experience math firsthand; once they realize that math can be used in places other than school, it will seem more fun and more practical.
4. Some current practices in math classes that I think help students are those that use technology and fun computer programs. Especially in geometry. I think these make math more exciting as well as hands on.
5. When teachers give tests and then move on to the next subject without individually helping the students who do poorly or fail, the students become permanently behind and never learn the material as well as they should. Tests should be a way of helping the teacher see where the students needs help.
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