Goodman, T., (2010). Shooting free throws, probability, and the golden ratio. Mathematical Teacher, 103 (7), 482-487.
I think the author is trying to get across the point that contextual problems can give students the opportunity to use the math that they already know to model that problem as well as others like it. He spends the article explaining one problem in particular. He gives a probability problem with shooting free throws. He showed the different ways that the students solved the problem. This is a very flexible problem so it allows for many different variations on the contextual problem. So he then showed some different variations and each of the ways that students could solve them. He doesn't spend much time tying the problem to his main idea other than to simply say that the ways students chose to solve the problems depended on what their previous knowledge was.
I agree with the authors main point, but I am confused as to what the purpose of the article was. I thought the task he presented to his students was good for learning probablity; however, it seemed like the students already understood probablity, and thus they were not learning so much as testing already aquired knowledge. Students will use previously learned techniques for contextual problems, but understanding this should only be the first step in how to teach students (I would have liked him to explain how this connected to the way teachers should teach).
Thursday, March 25, 2010
Thursday, March 18, 2010
Journal Article 1
D'Ambrosio, B.S., Kastberg, S.E., Viola dos Santos, J.R., (2010). Learning from student approaches to algebraic proofs. Mathematics Teacher, 103 (7), 489-495.
I think one of the main points that authors are trying to make is that analyzing students work is not only a way, but an important way, to learn the depth of students understanding. They use the example of 12 grade math students and their understanding of algebraic proofs. They go into detail with one question in particular. Prove that if a number ends with a 5 that the number squared will result in an answer that ends with 25. They then give a hint that should start students on an algebraic proof of this topic. They show the work of many different students and they analyze the work of each of the students, and they show what you can learn about each students understanding of how to do an algebraic proof.
I agree that students work should be analyzed in order to ensure that what the teachers are teaching is getting through to the students. It is obvious from the paper that students are having a difficult time understanding specific parts of algebraic proofs. They are having a hard time with understanding the purpose of a variable in an algebraic proof, and they are giving explanations in some places where they could be showing arithmetic. Analyzing the mistakes they make will make a large difference in the teachers ability to help correct the mistakes that the students are making.
I think one of the main points that authors are trying to make is that analyzing students work is not only a way, but an important way, to learn the depth of students understanding. They use the example of 12 grade math students and their understanding of algebraic proofs. They go into detail with one question in particular. Prove that if a number ends with a 5 that the number squared will result in an answer that ends with 25. They then give a hint that should start students on an algebraic proof of this topic. They show the work of many different students and they analyze the work of each of the students, and they show what you can learn about each students understanding of how to do an algebraic proof.
I agree that students work should be analyzed in order to ensure that what the teachers are teaching is getting through to the students. It is obvious from the paper that students are having a difficult time understanding specific parts of algebraic proofs. They are having a hard time with understanding the purpose of a variable in an algebraic proof, and they are giving explanations in some places where they could be showing arithmetic. Analyzing the mistakes they make will make a large difference in the teachers ability to help correct the mistakes that the students are making.
Wednesday, February 17, 2010
Blog Entry 5
There are many advantages to the style of teaching used by Warrington in this paper. Students were given the opportunity to explore the material with fellow students. This led to a lot of student interaction and the exchange of ideas in order to help them come up with the best solution. This helps the students to learn how to converse in mathematical discussions. Secondly, since they were not taught any new type of arithmetic they had to build off of the knowledge that they already had. By this time the students had a firm understanding of division and created understanding built on that solid foundation is good. Students also were able to adapted their ideas during the class discussion. Another advantage that I think is really good is that for me personally when I have struggled through the learning process and figured something out on my own then I have been able to remember and recall that knowledge much more easily than when I have simply had things explained to me. Lastly the students were able to strengthen their ability to solve problems. They started by estimating, and then they used this knowledge along with their knowledge of division of whole numbers to solve problems.
Along with these advantages comes many disadvantages that need to be considered. The first of which and probably the greatest disadvantage is that it took more time. If the students spend a week on this subject alone then it will cut into the amount of time that can be spent on other topics. The invert and multiply idea is very easy to teach very quickly. If every topic were taught with as much depth as the division of fractions was in Warringtons paper then you would not be able to cover nearly so much material in one year. Another possible disadvantage is that if this was not done correctly then students may never come to an understanding of how to come up with the proper solution. I know that there have been a few times in Math Ed 117 where I thought I had the right thought process and later found out that I didn't even have the right answer. So teachers teaching using the method discribed in Warringtons paper would need to know how to guide the students to coming up with the right answers and/or have a way of ensuring that the students were reaching the correct conclusion.
Along with these advantages comes many disadvantages that need to be considered. The first of which and probably the greatest disadvantage is that it took more time. If the students spend a week on this subject alone then it will cut into the amount of time that can be spent on other topics. The invert and multiply idea is very easy to teach very quickly. If every topic were taught with as much depth as the division of fractions was in Warringtons paper then you would not be able to cover nearly so much material in one year. Another possible disadvantage is that if this was not done correctly then students may never come to an understanding of how to come up with the proper solution. I know that there have been a few times in Math Ed 117 where I thought I had the right thought process and later found out that I didn't even have the right answer. So teachers teaching using the method discribed in Warringtons paper would need to know how to guide the students to coming up with the right answers and/or have a way of ensuring that the students were reaching the correct conclusion.
Tuesday, February 9, 2010
Blog Entry #4
Contructing knowledge is an action. Von Glasersfeld explains that you do not "gain" or "acquire" knowledge because these are terms that are too passive for explaining the act of increasing ones knowledge. He explains that the way that people increase their knowledge is through constructing it, and not through receiving it from someone else. Meaning that they experiment and find contradictions and through this process they find meaning or construct knowledge. Also, since knowledge is constructed it must be built on what we already know. So it is important to recognize that knowledge is built on or seen through our previous understanding.
As a teacher who believes in constructivism I would try to be aware of the way students learn. This would include giving the students plenty of opportunities to experiment in different mathematical contexts. I would try to help them find their way to various contradictions allowing them to revise and construct around these problems. Another part of being aware of the way students learn would be to understand the importance of knowing what the students have already learned. Since they will only build on what they have previously learned it is important that the way the teacher teaches is compatible with their current knowledge.
As a teacher who believes in constructivism I would try to be aware of the way students learn. This would include giving the students plenty of opportunities to experiment in different mathematical contexts. I would try to help them find their way to various contradictions allowing them to revise and construct around these problems. Another part of being aware of the way students learn would be to understand the importance of knowing what the students have already learned. Since they will only build on what they have previously learned it is important that the way the teacher teaches is compatible with their current knowledge.
Monday, January 25, 2010
Blog Entry #3
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.
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.
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.
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