I think the main idea of Erlwanger's article is that the system of teaching students is broken. The evidence I have for this opinion if that the author spends a lot of time discussing how Benny understood math and how it was the wrong way to do problems but to him it made sense. Benny is a really smart kid, bases on his knowlege he was doing things correctly. However the system said he was understanding the math correctly in fact he was one of the best students in the class. This shows that the system is broken because it passes kids through who don't know what they are doing.
I think the main point in paragragh one is that people who the schools and teachers say are passing math classes and understanding math problems really arn't. This is valid today because a lot of children in today's schools just remember enough to regergitate information on the test but the forget it and do not understand what they are doing and why they are doing it. this system is broken and must be improved by better teaching methods and better more conceptual understanding based tests.
Monday, January 25, 2010
Thursday, January 14, 2010
Scemp's Article: Relational and Instrumental Understanding
This blog entry is to summarize Skemp's ideas about relational and instrumental understanding. What each one is, how their related (if at all) and the pros and cons of each one. According to the article instrumental is to have an equation or concept and how to plug things into it or use it but to not know where the equation or idea came from. Relational understanding is to know not only how to use the equation but to know where the equation or idea came from and how it related to other ideas in Mathematics.
There are overlaps between these two types of understanding. Meaning that these two understandings are not mutually exclusive. Relational understanding of mathematics is a deeper form or Instrumental understanding. With instrumental understanding you can know how to use the equation. Relational understanding takes instrumental understanding a step further. It lets you know where the concept and equation comes from and why plugging in numbers into the equation works.
There are pros and cons to each type of understanding. Some pros for instrumental understanding is that it is quick to learn at first. If the problem does not deviate from the set pattern, answers can be obtained fairly simply. Some of the cons for instrumental understanding is that not knowing why an equation works or the general ideas behind it and where it comes from can create problems when presented with problems out side of the set equation. Another con with instrumental understanding is that you will have to remember every equation because you might not be able to see where each equation comes from and the a general idea can be formed to give you various individual equations.
Some pros for relational understanding are that it allows you to see where the equation comes from and why it works thus allowing a student to do problems that are outside any given equation and students can derive the equations from understanding and not have to memorize each equation for every situation like they would have to do if their understanding was only instrumental. The main con for relational understanding is that it takes more time to learn than instrumental and that some times relational understanding might not be as obvious to understand as the instrumental idea of "here is the equation, it works, use it."
There are overlaps between these two types of understanding. Meaning that these two understandings are not mutually exclusive. Relational understanding of mathematics is a deeper form or Instrumental understanding. With instrumental understanding you can know how to use the equation. Relational understanding takes instrumental understanding a step further. It lets you know where the concept and equation comes from and why plugging in numbers into the equation works.
There are pros and cons to each type of understanding. Some pros for instrumental understanding is that it is quick to learn at first. If the problem does not deviate from the set pattern, answers can be obtained fairly simply. Some of the cons for instrumental understanding is that not knowing why an equation works or the general ideas behind it and where it comes from can create problems when presented with problems out side of the set equation. Another con with instrumental understanding is that you will have to remember every equation because you might not be able to see where each equation comes from and the a general idea can be formed to give you various individual equations.
Some pros for relational understanding are that it allows you to see where the equation comes from and why it works thus allowing a student to do problems that are outside any given equation and students can derive the equations from understanding and not have to memorize each equation for every situation like they would have to do if their understanding was only instrumental. The main con for relational understanding is that it takes more time to learn than instrumental and that some times relational understanding might not be as obvious to understand as the instrumental idea of "here is the equation, it works, use it."
Tuesday, January 5, 2010
Mathematics is the way in which numbers fit together. It is a way of counting things and solving problems. Mathematics allows you to figure out any given problem relating to numbers and how they relate to each other.
I think that I learn math the best by first having a concept explained to me. Then seeing an example problem that relates to the concept being learned. Then being given a similar problem and being asked to solve it to see if I have learned the concept being taught. If I don't quite have the answer to the problem right, I then ask questions of someone who knows (an instructor for instance) who can point out to me the part of the concept (equation, whatever) I forgot to apply while doing a sample problem of the initial concept explained. Then trying another similar problem and repeating the process.
I think this is true because hearing a concept is not enough, you have to practice what you think you are learning so you can know if you missed anything and learn again what you may have missed the first time. You must do problems to test you otherwise you can not be sure you are learning the information.
Current practices in school mathematics classes that are being used to promote student learning are having students do problems from a book or an other source to see if they have understood what the instructor has been saying. Also mathematics is promoted in classes by having students do problems related to concepts learned which really make them think and challenges them so they are no just mindlessly plugging in numbers into given equations. By doing problems and not being spoon fed answers students can learn mathematics because it challenges them to think for themselves and not be given an answer from the instructor. Although the instructor is there in case the students get really stuck.
Some of the current practices used in mathematics classrooms that are detrimental to students learning is giving them equations for solving things and having them mindlessly plug in numbers into that equation. This is detrimental to students learning mathematics because this does not allow them to see where the equations come from just how to use them or plug numbers into a calculator. Students should have a basic understanding of where equations come from AND how to use them, not just how to use them. If you know how to use the formula a^2+b^2=c^c and know it is referring to the sides of a right triangle that is good, but it is detrimental to just plug in numbers into this equation not knowing where it comes from or what the symbols represent. So just teaching equations and not where they come from is detrimental because any one can plug in numbers into a calculator, but to know where the concept behind the calculator, but not knowing general ideas behind it is detrimental to a students learning.
I think that I learn math the best by first having a concept explained to me. Then seeing an example problem that relates to the concept being learned. Then being given a similar problem and being asked to solve it to see if I have learned the concept being taught. If I don't quite have the answer to the problem right, I then ask questions of someone who knows (an instructor for instance) who can point out to me the part of the concept (equation, whatever) I forgot to apply while doing a sample problem of the initial concept explained. Then trying another similar problem and repeating the process.
I think this is true because hearing a concept is not enough, you have to practice what you think you are learning so you can know if you missed anything and learn again what you may have missed the first time. You must do problems to test you otherwise you can not be sure you are learning the information.
Current practices in school mathematics classes that are being used to promote student learning are having students do problems from a book or an other source to see if they have understood what the instructor has been saying. Also mathematics is promoted in classes by having students do problems related to concepts learned which really make them think and challenges them so they are no just mindlessly plugging in numbers into given equations. By doing problems and not being spoon fed answers students can learn mathematics because it challenges them to think for themselves and not be given an answer from the instructor. Although the instructor is there in case the students get really stuck.
Some of the current practices used in mathematics classrooms that are detrimental to students learning is giving them equations for solving things and having them mindlessly plug in numbers into that equation. This is detrimental to students learning mathematics because this does not allow them to see where the equations come from just how to use them or plug numbers into a calculator. Students should have a basic understanding of where equations come from AND how to use them, not just how to use them. If you know how to use the formula a^2+b^2=c^c and know it is referring to the sides of a right triangle that is good, but it is detrimental to just plug in numbers into this equation not knowing where it comes from or what the symbols represent. So just teaching equations and not where they come from is detrimental because any one can plug in numbers into a calculator, but to know where the concept behind the calculator, but not knowing general ideas behind it is detrimental to a students learning.
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