I think the most important point that he is trying to make is that the IPI system is bad because it is not built off of understanding. He has quite a few arguments to support this. Benny uses patterns rather than understanding to find his answers. This means that he uses a lot of memorization instead of actual understanding. Benny has not forgotten anything, but rather he has developed consistent methods for different operations. This means that it is not that he is confused, but that he has entirely learned the wrong things. The IPI system does not require or even encourage the help of teachers in the classroom. Because of this, the students are not even watched to make sure that they are learning the material correctly. The only way students get this help is if they specifically ask the teacher for it. Instead, they learn what the exact answer to each problem is from a test. This does not require knowledge again it is just memorization. This means that they are learning specific rules for every single problem which does not allow for understanding, but rather makes it impossibly difficult to learn all of the answers. As well as not having help from teachers the students don't work with one another. This causes many of the same problems that they have due to not talking with the teacher.
The argument that I would like to focus on is the role of the teacher in the classroom. This is valid today because it helps teachers to better see the impact that they have on students as opposed to tets impact on students. Teachers are a crucial part of the classroom. In the IPI system teachers were intentionally left out of the learning process this was supposed to let the students move at the pace that was best suited for them. However, as can be seen in the case of Benny, this will lead the students to learn the information incorrectly and it will not allow them to understand the underlying concepts.
Monday, January 25, 2010
Thursday, January 14, 2010
Blog Entry #2
While instrumental understanding is necessary the focus, for both students and teachers, should be relational understanding. Instrumental understanding is the way in which students work through problems. It is very similar to the term “plug-and-chug”. If you are given point A and they want you to find point B, it is through instrumental understanding that you will get there. It is quick and easy to teach this; however, if relational understanding is taught (and learned) then the basic understanding of how to get to from A to B is apparent and may even be reached through different methods. Even more importantly the operation of moving from A to B has meaning and value. Relational understanding is allowing the students to fully grasp the concepts and the meaning in the math. Skemp uses the analogy of a city, where instrumental understanding is specific directions from of place to another. This is the fastest route and can be memorized quickly. In this analogy relational understanding is becoming familiar with the city this allows the individual to find the fastest way from point A to point B, and it may take him longer to find that route; however, once he is familiar with the city he can move from place to place with a full understanding of where he is and he can not only move from A to B but also to F and G without getting lost. Instrumental understanding has value but only in the short term. If we want progress in mathematics for students then we need to focus on teaching and learning relational understanding.
Tuesday, January 5, 2010
Blog Entry #1
First off, I’m sorry if I have a lot of excess in my writing. Writing is not my forte.
I’ve never really thought about what math is, and after thinking about it a while I’m still not sure. I would say that it is the study of numbers, but numbers may very well be just something that has been made up and simply a tool created in the field of mathematics. While most sciences would be the study of one specific thing, math is primarily a basses for other forms of study. Many different sciences use math in them. So it is hard to say that math is the study of change or the study of space (area). I guess to me math is a tool itself created or discovered in order to form a foundation for the study of many different things (physics, astronomy, etc).
I think that I personally learn mathematics best when I first learn the concept and the ideas behind it, then I learn the jargon terms and then I actually learn the arithmetic. If I learn it in this order I will more readily remember it and in many cases I will be able to re-teach myself the arithmetic if I first have a solid understanding of the concept. After learning all three parts of any given subject I also need to practice doing the arithmetic over and over again. This is how I feel I have best learned mathematics.
I suppose this is a fairly good structure for most students learning mathematics; however, I understand that everyone is different and that as a teacher I may need to find ways that will better suit my classes. I will also try to help my students figure it out for themselves. I have found that if I have to struggle a little and think about things on my own I will learn the material on a deeper level (if that makes any sense).
I haven’t seen how it is taught in schools since I graduated, but I remember a few things that they did that I thought were helpful. One of my teachers was good at making what seemed to be contradictions in math. This would catch the attention of the students and they would be interested to learn the new material that would explain the “contradiction” he gave at the beginning of class. Another way to catch the student’s interest was to tie it to real life uses. This gives meaning to what you are doing. Rather than “plug-and-chug” you are actually learning how to do something meaningful. Other than that all I can remember is like I said before, repetition and pushing students to struggle through learning the material themselves rather than just explaining everything to them. One thing that I didn’t see much of when I was in school, but there seems to be more of is the use of technology in the classroom (“smart” boards, computer programs, etc). If used correctly technology can help keep students interested and it can make it easier to do hard arithmetic in a way that allows students to learn the concepts which can give them a solid foundation for learning the arithmetic later.
I think that one of the worst things that was done in classes for me was some teachers did not try to build on what you had previously learned. If I learn something well in math and then have another class that builds on that previous knowledge it made it a lot easier for me to learn based on my solid foundation from the previous class. I had a lot of classes that felt like they were not at all correlated to any other math class I had ever had.
I’ve never really thought about what math is, and after thinking about it a while I’m still not sure. I would say that it is the study of numbers, but numbers may very well be just something that has been made up and simply a tool created in the field of mathematics. While most sciences would be the study of one specific thing, math is primarily a basses for other forms of study. Many different sciences use math in them. So it is hard to say that math is the study of change or the study of space (area). I guess to me math is a tool itself created or discovered in order to form a foundation for the study of many different things (physics, astronomy, etc).
I think that I personally learn mathematics best when I first learn the concept and the ideas behind it, then I learn the jargon terms and then I actually learn the arithmetic. If I learn it in this order I will more readily remember it and in many cases I will be able to re-teach myself the arithmetic if I first have a solid understanding of the concept. After learning all three parts of any given subject I also need to practice doing the arithmetic over and over again. This is how I feel I have best learned mathematics.
I suppose this is a fairly good structure for most students learning mathematics; however, I understand that everyone is different and that as a teacher I may need to find ways that will better suit my classes. I will also try to help my students figure it out for themselves. I have found that if I have to struggle a little and think about things on my own I will learn the material on a deeper level (if that makes any sense).
I haven’t seen how it is taught in schools since I graduated, but I remember a few things that they did that I thought were helpful. One of my teachers was good at making what seemed to be contradictions in math. This would catch the attention of the students and they would be interested to learn the new material that would explain the “contradiction” he gave at the beginning of class. Another way to catch the student’s interest was to tie it to real life uses. This gives meaning to what you are doing. Rather than “plug-and-chug” you are actually learning how to do something meaningful. Other than that all I can remember is like I said before, repetition and pushing students to struggle through learning the material themselves rather than just explaining everything to them. One thing that I didn’t see much of when I was in school, but there seems to be more of is the use of technology in the classroom (“smart” boards, computer programs, etc). If used correctly technology can help keep students interested and it can make it easier to do hard arithmetic in a way that allows students to learn the concepts which can give them a solid foundation for learning the arithmetic later.
I think that one of the worst things that was done in classes for me was some teachers did not try to build on what you had previously learned. If I learn something well in math and then have another class that builds on that previous knowledge it made it a lot easier for me to learn based on my solid foundation from the previous class. I had a lot of classes that felt like they were not at all correlated to any other math class I had ever had.
Subscribe to:
Posts (Atom)