YMaW chapters 5-7

Before reading these chapters I had never really considered drawing models to represent my work when multiplying and dividing fractions. It was very interesting to see what some of the students came up with. For example when Dawn Selnes asked her students how much chocolate each of six children would get if she shared five bars among them evenly, many of the studetns drew models. When I had first read this problem and before I looked at some of the student's strategies I was trying to solve this problem in my head and then also on paper and because fractions are not my strong point in math I was having a very difficult time solving this problem. I was unable to picture this problem as a model. Once I looked over the different models I was surprised at how well I understood the problem. Seeing this problem in drawing helped me see how much of a candy bar each student would get. This problem reminded me of the submarine problem we did in class. When I had to solve the problem I couldn't because I didn't use models, however when we watched the video of the students solving this problem and they were drawing models, it became so much more clear to me.

One thing YMaW helped me realize was how much work I need to do with fractions. When I tried to solve the problems on the gray math activity sheet I had a lot of difficulty. It was easy for me to understand multiplying and dividing fractions when I saw students models and their explanation, however when it comes to me attempting I am still unsure of a lot of things. It is confusing to me as to why I have such a difficult time with fractions, decimals, and percentages, because if I were dealing with whole numbers I would have a much easier time making models, etc. I also have a much better understanding of how to multiply and divide whole numbers. One good thing about these chapters was that I understood them much better than the last chapters in YMaW, which shows me that I am making progress!

Young Mathematicians at Work Chapters 5-7

Prior to reading these chapters in the book, I would not have been able to draw diagrams for the problems in the Math Activity: Multiplying and Dividing Fractions worksheet. At first, I did solved the problems in my head by either multiplying or diving the fractions. However, it is very helpful to see what the problems look like to really understand the mathematical processes in the multiplication and division. I did however have trouble representing 1/2 x 6 in problem set A number 3. It is hard to figure out how to draw 6 x half of something using a bar or strip. However, it was relatively simple to represent these problems visually.

When I moved to problem set B it was much more difficult to represent the problems with drawings. I think this is the case because the numerator is 5 instead of 1. Also, there were times when you did not have whole numbers and you had to divide two fractions which is sometimes difficult or tedious to represent. I hope that we can work on these in class so we can practice effective ways to represent these problems with students.

Young Mathematics at Work: Chapters 5 and 6

Overall, on thing I think that has really struck me about this book is how the problems which are posed to a class of students are relevant. The children become emotionally attached to the problem and want to find the right answer. They are not just doing the problem because the teacher said to do it. The “Pizza Patterns” problem in this chapter really caused the children to become involved in mathematics in a way that was exciting and fun. Also, this type of problem allows for a lot of small group and large group discussion. For example, one group began noticing that the whole matters. One child snipped a little off of his paper to make the thirds equal but noticed the whole matters when comparing it to another students work. On page 77 of Young Mathematics at Work they state, “The whole matters – you can not change the whole by snipping off a little bit – it changes everything!” This is not something that comes naturally to students, but through problem solving and discussion this one particular child began to understand this concept more fully.

Another interesting point that I found in chapter 5 is that people use models as a tool to help them mathematize. People have to work through the problem to gain a greater understanding of the concepts within the problem and models are a way of doing this. In chapter 5 the authors also stated that people have to move from models of thinking to models for thinking. At times people solve the problem and then create a model for their thinking, but as teachers we have to move our students to using models as a way to help them through their thinking process.

Chapter 6 in Young Mathematics at Work was very different from Chapter 5, but it caught my interest because of the different algorithms which were used. I feel that we live in a world where the “traditional” algorithms are what is accepted and known. Before reading this chapter I had no clue that there were other algorithms that worked and people used hundreds of years ago. Even though the algorithm we use today may be the most efficient the beauty of all algorithms is “that they [are] generalized procedures that [can] be used as efficient computation strategies for all problems” (p. 99). It is important to remember that there is no correct way of solving a problem, the most important thing is that people understand that mathematics they are doing and why they are doing it.

At the end of chapter 6 the authors state, “ Teaching algorithms is in fact harmful to children’s mathematical development” and “ when algorithms are taught as procedures to use for any and all problems children necessarily give up their own meaning making in order to perform them” (p 105). I feel that both of these quotes are something to think about when working with children and choosing the different mathematical task to teach to your students.

chapters 3 and 4

I have read some information about early number systems and it has always amazed me. It is so fascinating to read and learn about the many oral and written systems that have emerged across cultures over time. The only thing that threw me off was when I came to the section that discussed unit fractions. Fractions have always scared me and I have never had a good understanding of fractions. Still to this day when I am cooking and working with measurements I have to ask which measurement is bigger.

This chapter helped me learn about the different strategies that students use when solving for equivalence and working with fractions. It was confusing for me to completely understand how the students solved the problems and it showed me that I have ways to go when it comes to fractions and not only teaching them, but understanding them myself. One thing that I really like that this particular teacher did in the beginning of the chapter was that she firsted asked Helaina and Lucy to tell the class what they did. After they provided the class with their answer the teacher went on to ask if anyone could explain the strategy used by Helaina and Lucy. I believe this is a great thing to do because as the students explain their thinking to each other, they are thinking more deeply about the mathematics and they are learning how to communicate mathematical ideas to each other. This very beneficial because and it will help the students think more deeply about math and result in better understanding.

While reading through the students explanations I would get really confused and almost lost during their explanation and I will have to go back and read these chapters a couple more times to get a full understanding of the students strategies. These two chapters emphasized even more to me how important it is to provide students with opportunities to explain their answers, discuss, and reflect. In order for students to become able to mathematize their world, they need to be allowed to do so in their own meaning-making ways as they are learning. It is important to allow students to practice with big ideas and progessively refine their strategies. I have a ways to go when it comes to fractions and I know these chapters will (once reread a couple of times) will help me better understand fractions, decimals, percentages, etc.

Young Mathematicians at Work Chapters 3 and 4

Something I found interesting in chapter 3 was how the Mayans and Egyptians dealt with a problem similar to the sub problem from earlier in the book. I thought it was interesting that their thinking allowed them to arrive at a similar solution as the students who had the sub problem. They both used the concept of unit fractions to divide the sub. I think this concept is interesting because it almost seems instinctive to divide a sub this way before you have knowledge about other ways to use fractions. I am still wondering, however, why 2/3 seemed to be the magic number. Why didn't the Myans/Egyptians divide bread into halves so there would be 5 halves to distribute with a remaining half that they could have divided into tenths? This seems like it would have been easier, especially when using unit fractions.

In Chapter 2 I was interested in the part where James was trying to divide a strip of paper into thirds. Made 3 pieces and threw the extra piece away. This idea ties into my math lesson that I completed yesterday. My lesson was on part-part-whole addition and subtraction. In the beginning of the year, my students could not do a part unknown subtraction problem. After doing the lesson, I can see that a lot more of the students understand the relationship of parts to a whole, but I am interested in observing how the think when they start to use fractions. I am hoping to observe some fraction practice before the end of the year so I can reflect on this later.

I also thought Tanya's case of comparing fractions was interesting. When she got to the point of comparing 2/3 and 5/8 she really struggled because she did not have a consistent whole. Her bar graph seems to be useful and it makes sense to her, however, the problem lies in the fact that she is comparing fractions of two different wholes. I think this is a common misconception that students have. She saw the 3 in the denominator of 2/3 and realized that 3 is smaller than 8 (5/8) so she made the 5/8 bar larger than the 2/3 bar. This seems to make sense, but to help scaffold, I think it would be useful to give Tanya two concrete objects such as strips of paper that are the same size (so the whole is equivalent) and have her divide them into 2/3 and 5/8 so she can see that the whole has to be the same. You could even compare her drawing to a real life situation. You could ask her how big her whole is and eventually scaffold to the idea that her representation is similar to comparing 5/8 of a large sub to 2/3 of a small sub.

Young Mathematics at Work: Chapters 3 and 4

After working on project three I really focused on the questions that the teachers asked in the lessons on the book. I really liked how they asked things like "Who can explain the strategy Helaina and Lucy used?" This is a question which is open ended, but does not make one specific child answer the question. I really feel that paying attention to these aspects of lessons will help me become a better teacher in the aspect that I will learn how to better faciliate children thinking.

I also noticed that throughout the book the children had to do a lot of problem solving. I feel that problem solving, the way they talk about it in the Young Mathematicians at Work, is not always a task that is empasized in school. In the book the children have to problem solve by figuring out a way to solve a problem, not using an algorithm to solve the problem. This is a very different way of solving problems than I have been used to in my school experience. I really like the begining of chapter 4 when they talked about the student trying to create three equal parts out of a strip of paper. I also liked how they talked about how the child did succeed in the task, but how the question became what the thirds were of. It is through reading the math books this semester that I learned that digging into this deeper thinking is what causes children to move to the next level of mathematics.

DMI chapter 6

While reading DMI chapter 6, I began thinking about what I understand and don't understand about teaching math. I am beginning to feel much more comfortable when it comes to addition and subtraction, however I am still very much confused when it comes to multiplication and division. I am not completely sure why these two are so difficult for me, maybe because I don't understand how they work myself. Math used to be a subject that I thought would be so easy to teach, however I am now beginning to realize that it is not and it has become the one subject that I am very interested in mastering. I would love to continue taking math classes and going to conferences, etc. One thing about math, depending on the teacher, is that it can either be one of the most boring classes, or it can be fun and engaging. I want my math classes to be fun and engaging, and not a tidious subject that my students want to just get over with.

Young Mathematicians At Work and My Autobiography

After reading the first chapter in this book, I cannot help reflecting on some of the things I wrote at the beginning of the semester in my math autobiography. I wrote that I always struggled with math because I could not see the creative aspects to it. I loved how this chapter compared math with art because there are so many ways to solve problems. When reading about how students solved the sub problem we worked on in class, I immediately thought about the relationship of math to art. The students in the book exhibited many interpretations using different materials, just as an artist would. I really like seeing this link, because it makes math meaningful. I also like how this chapter discussed the old way of teaching. I feel that some of my mathematics instruction was teacher centered where students were considered a blank slate. I believe this is one of the many reasons I could not find creative aspects of math. Exploring how the teacher used this problem in the classroom and how we used it in our classroom gave me great insight into teaching in a way that math becomes creative for students.

Young Mathematicians at Work Chapters 1 and 2

After reading this first two chapters of this book I began to think about what I have learned about teaching mathematics over the course of this semester. The first chapter focus on the field trip problem which we had worked on in class. It was interesting to see the different ways the children came up with doing the problem. The ways they decided to do the problem were much different that what I had done. I believe this may have been due to my knowledge of mathematics beyond fractions. Both doing this problem and reading about how a classroom of students chose to solve this problem caused me to see the problem in a different light. I gained an understanding of mathematics, how to make things equals, etc. This problem, as I noticed from working on it myself, was a high level thinking problem. When working on the problem I had to think about how to begin to solve the problem, solve it, and then work through my thinking. On the other hand, when reading chapter 2 about the 18/24ths money problem, I saw a huge difference in students thinking. I had not previously done this problem, but I could see from reading the text that the problem was more about the answer than how someone came to get the answer. I also quickly noticed that the problem was confusing. I believe this may have been due to the way it was written, but it also may have been because it is not likely for someone to find out the faction of money they spent.

With all this said, I feel that I have gained a greater ability to pick out high and low mathematical thinking word problems. Before this class I would have never thought that there were differences in the type of word problem which is given to students. Now, when look at the problem children are working on and notice there thinking process I find myself analyzing the type of thinking which is done in the word problem.

The one question is where does this higher-level thinking in word problem correlate with teacher instruction. Do teachers using this framework ever instruct their students other then when they are working on word problems?

I would love to learn more about how mathematics is taught over the course of a week in a classroom which uses higher level thinking word problems.