This is about the tutoring I did today (mainly for a report I will have to write soon)
This afternoon I helped two kids, one boy and one girl, cycle 3.
Starting with the girl (manisha)
Manisha had to work on a sheet with word problems and two pages from her math textbook. I realized that I have learned a lot about teaching math from the Van de Walle textbook, even compared to last semester. While tutoring last semester I would want to encourage children when they did something right, saying things like good job and you're right. This semester instead of praising them for doing it "right", I asked why they were doing it that way and I found that often they actually had no idea. They just thought it was the right thing to do. In fact when working with Manisha, I found that she was focused on which mathematical operation she had to use rather than really trying to understand the problem. She would read the question out loud and then ask me if she was supposed to divide, or multiply, or add. When I would ask her what she thought she should do and why, she got a little frustrated. Granted this tutoring is taking place at the end of a long school day, so it's normally for kids to be tired and less enthusiatic, but I think it was more that she was perhaps not used to doing that kind of thinking. One problem was something about a tennis player who needed 100 balls. Balls come in cases, 12 boxes with 3 balls in each. How many cases would he need and how many balls would he have extra. The biggest difficulty here was trying to get her to see everything in a case as a group. She was focused on doing 4 X 3 = 12 but she did not know where to use it or how. So what I tried with her was to draw a picture, how many balls were in each box and how many boxes were in a case. I found that this helped although the student found it kind of long. The same thing with another problem (a window is 30 meters off the ground, a tree is planted and grows at 1 1/2 meter (or inch..) per year, how many years before the tree reaches the boys window? Here was did not know what to do and she also was not sure how to add fractions 1/2 + 1/2. I used drawing again and she kind of understood how to add the fractions but it was hard for her to match fractions with whole numbers. So she knew 1/2 and 1/2 made 2/2 which reduces down to 1/1 but she did really know how to add 2 + 2/2 or 1/1. What we did was we added it year by year how the plant would grow. We made a chart together. In both problems we drew pictures and used charts. She found this a little tideous, which it is. But I figured that it was better for her to understand what she was actually doing , rather than to just tell her she could add, divide, multiply, or whatever. I let her try out her own ideas and test out different theories. The thing I found the most difficult was that I did not have as much time as I would have liked to go over things with her and discuss. ALSO i'm sure she didn't really like that we were taking a long time. (at the community center, if you do not get all of your homework done in the alloted time, you have to go to overflow to finish it. And while I do not think this is bad, I can understand that kids would rather get their homework done in time and have fun). While helping her I could see that she was really looking to me to see if she was doing things right or to see what she should do. This is where I can really see the benefits of problem-solving as discussed in the Van de Walle text. She should not always be looking to me to see if she is right or to see what to do next. I found it a really great experience and it was super relevant to everything that I am reading about in math class. I was really glad for the opportunity to practice the things we are learning about. It's really tempting just to tell the student how to do it, but i realize that students have to construct their own knowledge and if you always just tell them, they will become dependant on you and will not figure things out for themselves. ALSO students can come up with creative ways to do things that we as teachers may not have thought about before. FOr some work in her mathbook, she used a calculator and I can see where the whole use /dont use a calculator debate stems from. The book told her to estimate some things and she told me that her teacher tells them not to estimate but to use a calculator. As someone who has quite a challenge with mental math, i can see how I would rather use a calculator, but I would never get better at my mental math skills.
With the boy, Christian, we looked at fractions and did a dictee. What he had to do with fractions was compare two and say which one was bigger or smaller. I liked this exercise as an intro to fractions because it came with little rectangles divided into parts that they could shade to get a visual look. Christian could shade no problem, but he was trying to compare to fractions by finding the common denominator. He, however, did not really know why or how to do this. Sometimes he would X by 2 to one fraction and they would then both me over the same denominator, but other times he would X by 2 the side that was already bigger. So he knew it was something to do with division but he did not understand that he was trying to get them over the same denominator in order to be able to compare them. I think that through discussion, he started to understand what he was doing and why a little bit more. To see, i had him do 3/10 and 5/6 and he actually put them both over 30. That was kool. With Christian too though, he just wanted to get the work overwith without actually understanding what he was doing. And I think that rather than loading kids up with work, we really need to focus on getting them to understand concepts and the rationale for what they are doing. In the end, that is what will help them succeed in math and in problem solving in general.
This afternoon I helped two kids, one boy and one girl, cycle 3.
Starting with the girl (manisha)
Manisha had to work on a sheet with word problems and two pages from her math textbook. I realized that I have learned a lot about teaching math from the Van de Walle textbook, even compared to last semester. While tutoring last semester I would want to encourage children when they did something right, saying things like good job and you're right. This semester instead of praising them for doing it "right", I asked why they were doing it that way and I found that often they actually had no idea. They just thought it was the right thing to do. In fact when working with Manisha, I found that she was focused on which mathematical operation she had to use rather than really trying to understand the problem. She would read the question out loud and then ask me if she was supposed to divide, or multiply, or add. When I would ask her what she thought she should do and why, she got a little frustrated. Granted this tutoring is taking place at the end of a long school day, so it's normally for kids to be tired and less enthusiatic, but I think it was more that she was perhaps not used to doing that kind of thinking. One problem was something about a tennis player who needed 100 balls. Balls come in cases, 12 boxes with 3 balls in each. How many cases would he need and how many balls would he have extra. The biggest difficulty here was trying to get her to see everything in a case as a group. She was focused on doing 4 X 3 = 12 but she did not know where to use it or how. So what I tried with her was to draw a picture, how many balls were in each box and how many boxes were in a case. I found that this helped although the student found it kind of long. The same thing with another problem (a window is 30 meters off the ground, a tree is planted and grows at 1 1/2 meter (or inch..) per year, how many years before the tree reaches the boys window? Here was did not know what to do and she also was not sure how to add fractions 1/2 + 1/2. I used drawing again and she kind of understood how to add the fractions but it was hard for her to match fractions with whole numbers. So she knew 1/2 and 1/2 made 2/2 which reduces down to 1/1 but she did really know how to add 2 + 2/2 or 1/1. What we did was we added it year by year how the plant would grow. We made a chart together. In both problems we drew pictures and used charts. She found this a little tideous, which it is. But I figured that it was better for her to understand what she was actually doing , rather than to just tell her she could add, divide, multiply, or whatever. I let her try out her own ideas and test out different theories. The thing I found the most difficult was that I did not have as much time as I would have liked to go over things with her and discuss. ALSO i'm sure she didn't really like that we were taking a long time. (at the community center, if you do not get all of your homework done in the alloted time, you have to go to overflow to finish it. And while I do not think this is bad, I can understand that kids would rather get their homework done in time and have fun). While helping her I could see that she was really looking to me to see if she was doing things right or to see what she should do. This is where I can really see the benefits of problem-solving as discussed in the Van de Walle text. She should not always be looking to me to see if she is right or to see what to do next. I found it a really great experience and it was super relevant to everything that I am reading about in math class. I was really glad for the opportunity to practice the things we are learning about. It's really tempting just to tell the student how to do it, but i realize that students have to construct their own knowledge and if you always just tell them, they will become dependant on you and will not figure things out for themselves. ALSO students can come up with creative ways to do things that we as teachers may not have thought about before. FOr some work in her mathbook, she used a calculator and I can see where the whole use /dont use a calculator debate stems from. The book told her to estimate some things and she told me that her teacher tells them not to estimate but to use a calculator. As someone who has quite a challenge with mental math, i can see how I would rather use a calculator, but I would never get better at my mental math skills.
With the boy, Christian, we looked at fractions and did a dictee. What he had to do with fractions was compare two and say which one was bigger or smaller. I liked this exercise as an intro to fractions because it came with little rectangles divided into parts that they could shade to get a visual look. Christian could shade no problem, but he was trying to compare to fractions by finding the common denominator. He, however, did not really know why or how to do this. Sometimes he would X by 2 to one fraction and they would then both me over the same denominator, but other times he would X by 2 the side that was already bigger. So he knew it was something to do with division but he did not understand that he was trying to get them over the same denominator in order to be able to compare them. I think that through discussion, he started to understand what he was doing and why a little bit more. To see, i had him do 3/10 and 5/6 and he actually put them both over 30. That was kool. With Christian too though, he just wanted to get the work overwith without actually understanding what he was doing. And I think that rather than loading kids up with work, we really need to focus on getting them to understand concepts and the rationale for what they are doing. In the end, that is what will help them succeed in math and in problem solving in general.
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