The student videos were very interesting to watch. I thought it was pretty cool how the different students used different strategies to solve the multi digit problems. I could see how the U.S. standard algorithm would be somewhat confusing to these students because the algorithm is not necessarily easy to understand, and in some cases, the strategies the students used are a lot quicker than the algorithm.
Another lesson I learned from watching the videos and the class discussion, is to reread a portion or all of the question if the student asks for the number again. I like reiterating this point and keeping the numbers tied to the context of the question because I think this keeps kids thinking about the context rather than just trying to pull numbers out of everything. I agree that the more the kids get used to thinking in terms of context and not just numbers, the more easily they will be able to understand math problems with multiple steps that they will encounter in upper grades.
I think I deserve the full 2 points for the math blog!
Tuesday, June 29, 2010
Saturday, June 26, 2010
6/23/10
From reading my previous 6/21 post, I saw that I did not give my student the correct way to solve his math problem. I am still glad that the student was able to figure out the math problem, even if I explained how to get there incorrectly. Hopefully it will at least help if the student can think in terms of "sharing". This shows me that I still need a lot of practice with division problems and thinking about which word problems go with their correct solutions, or the solutions students often come up with.
Moving on to multi digit word problems is easier for me than division because I do not necessarily need to figure out if I should be dealing or chunking in the problem. With that being said, I found it extremely interesting to learn that so much of figuring out multi digit problems has to with understanding how to work with 10's. Since I was pretty much just taught to use the U.S. standard algorithm to figure out multi digit problems, I did not really think about 10's very much, or if I did I didn't know I was. Now that I know how important understanding how to operate with 10's is, this will be an area in which I will really have my students work with.
Another extremely important part of my math curriculum will be to show number sentences in different forms. It was crazy to think that so many students could see that 3+1=4 was always true, but they couldn't see 4=3+1 as being true because they've only been shown the first type of number sentence. I am curious now to see how many problems in the current elementary math books are written in the simple a+b=c form and how many are written in other forms. I am very happy that I have this information and will hopefully be able to teach my students the importance of the information so they are more easily ready when they get to algebra.
Moving on to multi digit word problems is easier for me than division because I do not necessarily need to figure out if I should be dealing or chunking in the problem. With that being said, I found it extremely interesting to learn that so much of figuring out multi digit problems has to with understanding how to work with 10's. Since I was pretty much just taught to use the U.S. standard algorithm to figure out multi digit problems, I did not really think about 10's very much, or if I did I didn't know I was. Now that I know how important understanding how to operate with 10's is, this will be an area in which I will really have my students work with.
Another extremely important part of my math curriculum will be to show number sentences in different forms. It was crazy to think that so many students could see that 3+1=4 was always true, but they couldn't see 4=3+1 as being true because they've only been shown the first type of number sentence. I am curious now to see how many problems in the current elementary math books are written in the simple a+b=c form and how many are written in other forms. I am very happy that I have this information and will hopefully be able to teach my students the importance of the information so they are more easily ready when they get to algebra.
Thursday, June 24, 2010
6/21/10
My brain was a bit fried in math class so it took me quite some time to come up with a word problem for a PPW PU. I finally figured it out and was able to write a word problem for my partner. I do think that I will remember most of the information taught in this course, but I know I need to keep up with it and always try to implement the information learned. This does not necessarily mean that I will be able to teach using the CGI method all of the time, but I will always try to teach to what the students already know. I will also always try to keep in mind that a lot of students are going to think differently and I should be aware and ask them questions to see if I can modify my instruction to help as many students as possible truly be able to learn.
I was excited because earlier that day in my practicum class, the students were working on a measurement division problem in which they needed to figure out how many 2cm legs of a table could be made with a piece of wood that was 8cm. The boy I have in class that really struggles with everything started to write this problem as a subtraction problem like 8-2, but then he just sat there lost. I asked him what he was going to do next and he didn't know, nor did he know the answer. I verbally read the question to him and he still couldn't get it. Then I asked him verbally, "If you have 8 pieces of gum and you want to share the gum with your friend so the 2 of you both get the same amount, then how many pieces of gum would each of you get?" He thought for just a second and said, "4". I was excited that he got the answer. I then explained to him that the 8cm stick has to share itself equally to make legs that are each 2cm. I am not quite sure if I explained the stick/leg thing to him correctly, but at least he got further than he would have and maybe now he can think of division in terms of sharing. I think if I had more time to work with him then I could get him to understand math. Unfortunately my practicum will end next week. The good takeaway point from all of this for me is that I now have some tools to explain things in different ways to kids that might help them. I am also more cognisant of how the students are thinking!
I was excited because earlier that day in my practicum class, the students were working on a measurement division problem in which they needed to figure out how many 2cm legs of a table could be made with a piece of wood that was 8cm. The boy I have in class that really struggles with everything started to write this problem as a subtraction problem like 8-2, but then he just sat there lost. I asked him what he was going to do next and he didn't know, nor did he know the answer. I verbally read the question to him and he still couldn't get it. Then I asked him verbally, "If you have 8 pieces of gum and you want to share the gum with your friend so the 2 of you both get the same amount, then how many pieces of gum would each of you get?" He thought for just a second and said, "4". I was excited that he got the answer. I then explained to him that the 8cm stick has to share itself equally to make legs that are each 2cm. I am not quite sure if I explained the stick/leg thing to him correctly, but at least he got further than he would have and maybe now he can think of division in terms of sharing. I think if I had more time to work with him then I could get him to understand math. Unfortunately my practicum will end next week. The good takeaway point from all of this for me is that I now have some tools to explain things in different ways to kids that might help them. I am also more cognisant of how the students are thinking!
Monday, June 21, 2010
6/16/10
I really enjoy learning fractions with the use of sharing. It does make fractions more understandable. I am still concerned with the notion of adding, subtracting, etc. fractions with each other. I thought the examples of the children solving the "slices of cake" problems were extremely interesting. It is interesting that children with more math knowledge will try to use a math problem to figure out which table to sit at in order to get more cake, and the children without math knowledge will just think about the problem and figure it out. I would think that both groups of the students would think about the problem before they try to put it into a math problem. But this just goes to show that most of the math students are taught just to memorize ways to solve problems and not necessarily taught to think about what is being asked first. I have noticed that after some teachers read word problems to their students they first ask what math strategy should be used. I think the teachers should instead be asking for the students to think about the word problem and what it is asking. Then if the children need a math strategy to figure out the problem, the teacher can go into that. But first, the children should be taught to THINK about problems.
Tuesday, June 15, 2010
6/14/10
I am starting to feel more comfortable with the multiplication and division word problems. I am nervous about dealing with fractions because I have always seemed to have issues with fractions. I am glad that Dr. Shih said a fraction is simply a partitive division problem with the numbers changed, but I am still feeling uneasy about learning fractions.
I like the way Dr. Shih presented the partitive division problem by stating, "Show how you would share 16 pieces of pizza with 4 kids." I could have actually used the information earlier that day. I have a student in my class and there was a partitive division problem that asked how many pages a student would read per day if they had 6 days to read a total of 12 pages, and they read the same number of pages each day. My poor student was completely lost. I tried to help him as much as possible. I had him draw out the number of pages and write out 1-6. With a lot of leading questions, my student finally got the answer, but I still do not really think he had it figured out. I think that if I would have presented the problem to my student in "sharing" terms, then the student would have more easily figured out the problem. I was going to try the problem with my student again today but he was not in school. I'm hoping to try the problem with him again tomorrow. I know it will be a small piece of the huge gap of information he is missing, but I guess every little bit helps!
I like the way Dr. Shih presented the partitive division problem by stating, "Show how you would share 16 pieces of pizza with 4 kids." I could have actually used the information earlier that day. I have a student in my class and there was a partitive division problem that asked how many pages a student would read per day if they had 6 days to read a total of 12 pages, and they read the same number of pages each day. My poor student was completely lost. I tried to help him as much as possible. I had him draw out the number of pages and write out 1-6. With a lot of leading questions, my student finally got the answer, but I still do not really think he had it figured out. I think that if I would have presented the problem to my student in "sharing" terms, then the student would have more easily figured out the problem. I was going to try the problem with my student again today but he was not in school. I'm hoping to try the problem with him again tomorrow. I know it will be a small piece of the huge gap of information he is missing, but I guess every little bit helps!
Sunday, June 13, 2010
6/9/10
After reviewing measurement division and partitive division more, I am now feeling pretty comfortable with figuring out which type of problem it is. It does take me longer than the addition and subtraction, but I think I've got the hang of it! Grouping is the easiest and for me, seems to give the most explicit directions so I don't think I'll have any issues with that.
I enjoyed working with some of the math learning stations to see how kids might learn from the stations. My group was able to figure out most of the problems pretty quickly. A few of the problems stumped us, one specifically is the tootsie roll and block counting for a certain number of tootsie rolls. We didn't get it at first, but then we figured it out. I liked that learning station the best because it made the students use part of one of their answers to figure out the other answers. I think stations like this help students become critical thinkers because they have to first figure out an answer to a problem and then figure out they need to use that information to figure out additional problems.
It was also very cool to watch the kid explain why a+b-b=a. Something like this leads students to become critical thinkers as well because they start out with a few number problems that make the statement true. As the student thinks about the problem more and sees the statement holds true for all numbers they use, they start to think about the problem differently. Then, finally, they break the problem up into two conjectures to prove, without numbers, that the problem will always be true. I liked that Dr. Shih mentioned that the greatest prediction of kids knowing conjectures is kids having practice with more conjectures. I hope when I become a teacher I will be able to give my students practice with conjectures so they can become critical thinkers and understand why problems like the one above will always be true.
I enjoyed working with some of the math learning stations to see how kids might learn from the stations. My group was able to figure out most of the problems pretty quickly. A few of the problems stumped us, one specifically is the tootsie roll and block counting for a certain number of tootsie rolls. We didn't get it at first, but then we figured it out. I liked that learning station the best because it made the students use part of one of their answers to figure out the other answers. I think stations like this help students become critical thinkers because they have to first figure out an answer to a problem and then figure out they need to use that information to figure out additional problems.
It was also very cool to watch the kid explain why a+b-b=a. Something like this leads students to become critical thinkers as well because they start out with a few number problems that make the statement true. As the student thinks about the problem more and sees the statement holds true for all numbers they use, they start to think about the problem differently. Then, finally, they break the problem up into two conjectures to prove, without numbers, that the problem will always be true. I liked that Dr. Shih mentioned that the greatest prediction of kids knowing conjectures is kids having practice with more conjectures. I hope when I become a teacher I will be able to give my students practice with conjectures so they can become critical thinkers and understand why problems like the one above will always be true.
Tuesday, June 8, 2010
6/7/10
I am really understanding the different types of addition and subtraction word problem types so the quiz was fairly easy for me. It is nice to see something you know and be able to identify the word problems and know what makes them easier and more difficult to solve.
I am extremely happy that Dr. Shih mentioned that he has not seen any type of research that shows if you take away student's tools then they will get to abstraction quicker. This is wonderful news because I think in today's teaching society so many teachers think the opposite. I have seen teachers try to make the students do the problems in their heads instead of using their fingers. And I have seen the student's reactions; it seems as though the students feel bad about themselves because they feel that they should not be using their fingers to count. I am happy there isn't any evidence supporting the taking away of student's tools. All students learn at different paces and they all learn in different ways. When I sit back and think about it, I still sometimes use my fingers to count with. I've always excelled at math, and the fact that I still sometimes use my fingers to count just goes to prove that students don't need their tools taken away in order to abstract other math calculations.
I'm ready to get more practice with the multiplication and division. I can't seem to tie partitive to dealing and measurement to chunking. I understand each and can tell you if you have to deal or chunk the problem, but I can't seem to remember which is partitive and which is measurement. But, I am sure with more brain stems I will be able to remember which goes to which!
I am extremely happy that Dr. Shih mentioned that he has not seen any type of research that shows if you take away student's tools then they will get to abstraction quicker. This is wonderful news because I think in today's teaching society so many teachers think the opposite. I have seen teachers try to make the students do the problems in their heads instead of using their fingers. And I have seen the student's reactions; it seems as though the students feel bad about themselves because they feel that they should not be using their fingers to count. I am happy there isn't any evidence supporting the taking away of student's tools. All students learn at different paces and they all learn in different ways. When I sit back and think about it, I still sometimes use my fingers to count with. I've always excelled at math, and the fact that I still sometimes use my fingers to count just goes to prove that students don't need their tools taken away in order to abstract other math calculations.
I'm ready to get more practice with the multiplication and division. I can't seem to tie partitive to dealing and measurement to chunking. I understand each and can tell you if you have to deal or chunk the problem, but I can't seem to remember which is partitive and which is measurement. But, I am sure with more brain stems I will be able to remember which goes to which!
Saturday, June 5, 2010
6/2/10 Math Day 4
I am extremely excited that I am now easily able to identify the different types of word problems. It still takes me a little bit of time to identify the problem, but I understand what problem type it is. I am also excited because I know that later we will be learning how to help the students understand math. I have always been pretty good at math, but I am just now realizing that I never really understood it. I got almost all A's in math, but remember very little. Like Dr. Shih says, I do not really remember math because I do not have many stems to tie it to.
I taught a math lesson on line plots on Thursday and it was pretty much a disaster. I thought after creating and reviewing the lesson that I had a pretty good idea on how to teach line plots to my practicum students. I was quite wrong. I stumbled on how to explain the line plots to the students and I stumbled when explaining the types of information that could be shown with line plots. I think the main reason I struggled is because I didn't really understand line plots myself. I get the concept and could easily figure out how to do one, but I don't really, really understand them. Because of this I don't think I was able to logically explain line plots to the students. I am going to observe my mentor teacher teach a lesson on graphs, so hopefully that will help me in some way.
To leave the blog on a positive note, I really enjoyed learning how Dr. Shih differentiated his instruction by giving his students the same math problems, but different numbers. I think the traditional mindset on differentiation has been that students at different levels need to have different assignments in which they are learning different things. Dr. Shih showed that the students do not necessarily need to be learning different concepts, they just need to be easier for certain students to understand. This is wonderful because all of the students are learning the same concepts and the lower level students are not falling farther behind!
I taught a math lesson on line plots on Thursday and it was pretty much a disaster. I thought after creating and reviewing the lesson that I had a pretty good idea on how to teach line plots to my practicum students. I was quite wrong. I stumbled on how to explain the line plots to the students and I stumbled when explaining the types of information that could be shown with line plots. I think the main reason I struggled is because I didn't really understand line plots myself. I get the concept and could easily figure out how to do one, but I don't really, really understand them. Because of this I don't think I was able to logically explain line plots to the students. I am going to observe my mentor teacher teach a lesson on graphs, so hopefully that will help me in some way.
To leave the blog on a positive note, I really enjoyed learning how Dr. Shih differentiated his instruction by giving his students the same math problems, but different numbers. I think the traditional mindset on differentiation has been that students at different levels need to have different assignments in which they are learning different things. Dr. Shih showed that the students do not necessarily need to be learning different concepts, they just need to be easier for certain students to understand. This is wonderful because all of the students are learning the same concepts and the lower level students are not falling farther behind!
Thursday, May 27, 2010
5/26/10 Math Day 3
I really appreciate that we are learning how kids think. It is difficult trying to "forget" what I have been taught throughout almost my entire education, but it is slowly becoming easier. When we go over the different types of word problems and see the videos of the kids, it makes it clear the reason kids think the way they do. This makes it much easier to understand how kids are solving, or trying to solve, word problems. I really like that I have learned to ask the kid how they came up with a specific answer. This will really help me understand what the kid is thinking and how they are going about solving different problems.
I actually put this into action in my practicum class today. The students were learning about dimensions and they were supposed to draw the top and spine of their books. The teacher told them to hold up their books and draw what they saw. One of the students drew a rectangle in the center of their page and four smaller rectangles around it. The drawing did not look like the other students' drawings so I asked the kid to explain it to me. He told me that the big rectangle was the front of the book, and the other four smaller rectangles were the top, back side, and spine. Since the teacher didn't say to just draw two sides, this students drew all four while most of the others only drew two sides. If I just assumed he was just drawing a bunch of squares then I would have been wrong and the student would not have understood if I told them their drawing was incorrect. I am thrilled that I now have the mindset to ask the student to explain before I just assume the student does not understand the question or what the answer is supposed to be.
I actually put this into action in my practicum class today. The students were learning about dimensions and they were supposed to draw the top and spine of their books. The teacher told them to hold up their books and draw what they saw. One of the students drew a rectangle in the center of their page and four smaller rectangles around it. The drawing did not look like the other students' drawings so I asked the kid to explain it to me. He told me that the big rectangle was the front of the book, and the other four smaller rectangles were the top, back side, and spine. Since the teacher didn't say to just draw two sides, this students drew all four while most of the others only drew two sides. If I just assumed he was just drawing a bunch of squares then I would have been wrong and the student would not have understood if I told them their drawing was incorrect. I am thrilled that I now have the mindset to ask the student to explain before I just assume the student does not understand the question or what the answer is supposed to be.
Tuesday, May 25, 2010
5/24/10 Math Day 2
Class was very interesting. Figuring out the word problem grid was challenging and I am not sure my group would have gotten as far as we did without the "action" clue from Dr. Shih. Even after we received the clue it was still difficult for us to figure out what problems went into each section of the grid. It seemed like we just stared at the problems for a long time. Then something happened! Some of the problems started to look similar, i.e. they had similar actions taking place inside of them. Once we figured out the first action of, as we called it, getting then it was somewhat easy to figure out the first 3 sections. It took us a little longer to figure out the next 3 sections but we figured out that they were basically just opposites of the first 3 sections and instead of getting, there was giving taking place in the problems. The last 5 sections were a bit tricky. We think we have the rows correct, but we are not exactly sure of the column placement in the last row.
It is crazy to think that all word problems fit into one of these categories. It is even more crazy to think that not a lot of teachers know about this. Knowing about the different types of word problems will definitely make it easier to help students with word problems. We now know which word problems to start out with that will be easiest for kids to understand and that is really amazing!
It is crazy to think that all word problems fit into one of these categories. It is even more crazy to think that not a lot of teachers know about this. Knowing about the different types of word problems will definitely make it easier to help students with word problems. We now know which word problems to start out with that will be easiest for kids to understand and that is really amazing!
Sunday, May 23, 2010
Math Day 1
I enjoyed Dr. Shih's first class. Before the class I was wondering how interactive 2 1/2 hours of math was going to be each day. I was pleasantly surprised that the class was extremely interactive and the time seemed to go by very quickly. Everything Dr. Shih taught really kept my attention and I was able to really understand what he was teaching. This was a relief as I do not really remember much about school math. I enjoyed learning that teaching math is not so much about memorizing math all over again, but instead learning how to teach kids.
The most exciting aspect of the class is that we are actually going to learn how kids think and learn. I think this is the most important part of being a teacher because if you cannot figure out how to help the students learn, then you are probably just going to be spinning your wheels. It was very interesting to see how Dr. Shih first had us try to solve a word problem and then how we worked backward from there to see how kids think. Being introduced to the four solution strategies kids utilize was eye-opening. I have always wanted to understand how to help kids learn but I haven't really had many professors who were actually able to teach that. I think the information we learn from Dr. Shih will be invaluable and I am thrilled to be able to learn a small portion of what he knows.
The most exciting aspect of the class is that we are actually going to learn how kids think and learn. I think this is the most important part of being a teacher because if you cannot figure out how to help the students learn, then you are probably just going to be spinning your wheels. It was very interesting to see how Dr. Shih first had us try to solve a word problem and then how we worked backward from there to see how kids think. Being introduced to the four solution strategies kids utilize was eye-opening. I have always wanted to understand how to help kids learn but I haven't really had many professors who were actually able to teach that. I think the information we learn from Dr. Shih will be invaluable and I am thrilled to be able to learn a small portion of what he knows.
Monday, May 17, 2010
Laura Hamilton's Math Autobiography
My name is Laura Hamilton. I am in the Initial Licensure Program for elementary education. Because of this, my recent experience with school math is extremely limited. I graduated from the University of Wyoming in 2000 with a bachelor's degree in communications, so it has been 10 years since I have seen textbook mathematics.
From elementary to high school I do not remember much of anything math related. I do remember taking statistics in college, but I do not remember any specifics. I also remember having to buy an expensive scientific calculator and using that to figure out certain math problems. I still have the calculator, but now I just use it for basic math calculations.
Throughout my professional career I have used a lot of math, but I have often set up Microsoft Excel to do all of my calculations. I have just come up with the formulas to plug into Excel to make it do what I want it to do. I usually utilized Excel to figure out budget tracking calculations. I also used Excel to come up with other calculations depending on what I was working on at the time.
Now that I am attending school full-time again, I usually use Excel to keep track of my household budget. I do like math and I really love Excel. I unfortunately just do not remember a lot about math as it relates to learning in school settings. I am looking forward to learning best practices to utilize in teaching elementary mathematics!
From elementary to high school I do not remember much of anything math related. I do remember taking statistics in college, but I do not remember any specifics. I also remember having to buy an expensive scientific calculator and using that to figure out certain math problems. I still have the calculator, but now I just use it for basic math calculations.
Throughout my professional career I have used a lot of math, but I have often set up Microsoft Excel to do all of my calculations. I have just come up with the formulas to plug into Excel to make it do what I want it to do. I usually utilized Excel to figure out budget tracking calculations. I also used Excel to come up with other calculations depending on what I was working on at the time.
Now that I am attending school full-time again, I usually use Excel to keep track of my household budget. I do like math and I really love Excel. I unfortunately just do not remember a lot about math as it relates to learning in school settings. I am looking forward to learning best practices to utilize in teaching elementary mathematics!
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