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.

DMI chapter 5

This chapter was very interesting and it opened my eyes to how important it is to understand how your students think about and understand multiplcation. One thing that I have learned so far throughout this course is providing your students with opportunities to show you what they know and or understand. In the first case Eleanor gives her student a homework page and has them go home and solve the problem in two different ways. It was interesting that she did not give them any ideas on how to solve them, but rather left it up to them in order to see what they know, understand, or dont know. While reading this I started reflecting on my own strategies and the one thing that I realized I need to work on is allowing my students to solve the problem how they think they should or how they understand. I feel as though when I am teaching math I try and teach the students ways in which I learned or understand math. I understand by reading this chapter that in order for me as a teacher to really learn about how my students think about and understand math I need to let go of the leash and allow them to explore different methods and understandings. One thing that concerned me while reading this chapter was that I did not always understand how students thought or solved the problem. For instance, I did not understand how the students used arrays to solve 23 x 4. This method seemed confusing and like I said, I did not understand it. Also I was a bit confused with what was behind Michael's thinking when he tried to solve 25 x 9. While reading this chapter, my biggest concern was the fact that I did not always understand the logic behind students thinking and it made me worried that when I have my students solve a math problem the best way they know how or understand, I won't be able to understand their thinking and be able to guide them in the right direction. I know that I shouldn't have all of these answers right now, but they are thoughts that were going through my mind while reading this chapter. All in all I really enjoyed this chapter and thought it was really interesting.

DMI chapter 5

DMI Chapter 5

One thing I got from Eleanor's case in chapter 5 of the DMI book is how important it is to have an understanding of where the children are in their learning. Taking notes and work samples are one way to keep track of how their thinking has progressed (or not progressed). It is important to analyze your own teaching, what the children are learning, and how you can better support them in their development. The question I have is how you support all the children in their learning so the information being taught is not to hard or to easy for one particular child.

Also, with the DMI Chapter I really thought about how the type of problems that are given to students need to be well thought out. In Lauren's case when she gave the problems:

2 x 4
3 x 4
2 x 40
20 x 4
23 x 4

She allowed for her students to think through the problems and analyze how a problem they already know can help them with a "harder" problem. The other thing I liked about this case is how the children were excited about learning and instead of the learning process being a struggle it became a game as to how big of a multiplication problem they children could solve.

Overall, I really liked this chapter because I feel that many of the ideas that were in the text could be pulled out and be useful in other circumstances or grade levels. Also, being able to read the case studies allows for a context which a certain idea was used. This gives me more of an understanding of how it could be used in my future classroom.

DMI Chapter 5: Interesting Ways to Multiply

Chapter 5 shows some very diverse thinking when it comes to multiplication. When reading Eleanor’s case I found it interesting than none of the children used the standard algorithm to multiply 27x4. A lot of the strategies are very conducive to mental math. I thought it was interesting how Mark changed the problem into an addition problem by adding all of the 20’s and then four 7’s. I also thought Mika’s interpretation of 4x25=100; 2x4=8; 100+8=108. She is basically using the distributive property by solving this way. These strategies as well as using the arrays in Lauren’s case study were much easier to understand for student than using the standard algorithm as seen in Susannah’s case. Some students tried to follow a procedure rather than really understanding the math behind multiplication. An example of this is:

1

49

x 2

108

In this case the student “carried” then 1 and added 4+1 to get 5 and multiplied 5 by 2. This use of a familiar procedure proved detrimental because she made a small mistake that lead to an incorrect answer.

When reading Lauren’s case an issue came up for me. She was troubled by the language students were using when they multiplied by a multiple of 10 (4x20=60). The students would have explained that 4x2=6 and you “add a zero.” I find this language to be a bit troubling as well. I think the concept is very important, especially to doing mental math, but I wonder if there is a better way to describe the mathematical processes that are actually occurring.

Robert’s At Risk Article

The article I read was “Problem Solving and At Risk Students” by Roberts. The teacher explains how difficult it was moving to an urban school where the students were angry and underprivileged. Her teaching methods just didn’t seem to work. One thing this teacher did was to activities that integrated team building. This article discusses that children “think as a byproduct of the activities, assignments, and so on we ask them to do.” She decided that she needed to create activities that allowed the students to feel successful so they do not become angry with failure and give up. She also started by using problems that had only one answer so students didn’t feel like they were taking a risk to make a claim. Roberts learned to model many ways of solving problems to reach the wide variety of student thinking. With her guidance, students became more accepting of the different solutions for one problem. She also build up student confidence because she gave rubrics in class and had students do reflections of their work to show progress, even though their report cards may not have been the reflection of the progress they were indeed making. Her strategy here was student centered assessments such as dialogue journals between her and the students. All of these strategies helped to foster positive thinking from students about mathematics and they serve as great practical suggestions for classrooms with at risk students.

A concern that I have after reading this article is that if I have at risk students I won’t have the answers about what to do to help them. Children are so diverse in their needs that I worry I may not find a way to connect with and help at risk students. I think what I can take from this article is to never give up trying and to always use my resources to seek help with at risk students.

Meeting the Needs of All Children

One of the most interesting long term experiences I am able have in my field placement is the ability to see the progression of the GLCEs and also note how the skills are introduced, worked on, and mastered. This may be due to my knowledge of the kindergarten math GCLEs, but it may also be because the curriculum at the elementary school where I do my field placement is based on the GLCEs. With that said, I also see teacher support for children who are struggling in particular curriculum areas. Assessment is very important to this school and they work hard to make sure the students have accomplished certain tasks at particular times throughout the school year. This focus on assessment helps the teachers see where a child is struggling and needs more support. It is also helpful when thinking about the needs of children throughout their school career. Since I am in a kindergarten classroom, many of the students have not been diagnosed with special needs. Some of these students are being labeled and others are being worked with to make a decision about what their needs are to succeed to the fullest extent.
For students who have special needs we typically work on-on-one with them when there is enough support in the classroom. Each child has different needs and being able to work with them one-on-one allows for greater support in the areas which they need to further develop. In the article, “Learning-Disabled Students Make Sense of Mathematics by Jean L. Behrend she states, “ Encouraging Evan, who generally guessed the answer, to think about the problem changed his perception of mathematics….He realized that he could solve the problems.” Every students has their own needs as an individual and working with them through these problems whether they deal with self-confidence, academics, or anything else, is important for ALL children’s growth and development .

Corduroy

DMI chapter 4

After reading this chapter it amazed me that these 2nd graders thought the traditional algorithm was challenging. I remember when I was in elementary school this was the only way to solve a problem. I remember finding other ways to be more challenging, which proves that each student learns best in different ways. I have spent that last couple of weeks wondering how students think and after I read chapter 4, I took time to think back to how I used to think about math. When it came to the traditional algorithm it was very simple 49
+ 58 _______
107
For me it was easy to regognize that you start with the ones and add 9+8 which equals 17. I had learned that when the ones number equaled greater than 10 you "carried" the tens to the tens. So I would carry the 1 over to the tens and add 4, 5, and 1 together to get 107. For me, these types of algorithms were easy, where as solving it like these second graders liked best (40+50=90) (9+8+17) (90+17+107) was more challenging. For me, solving a problem like this took more thinking and more time. I have noticed in my placement that the students solve problems in the traditional way, but problems that don't require regrouping, which makes me wonder if they have even been exposed to such problems. After reading this chapter I started thinking about which way I think is the best and I came up with all of these ways shown on page 64 are important for students to be exposed to, practice and understand. Like I said previously I was great at traditional algorithms, but I never understood regrouping and still sometimes have a hard time wrapping my mind around it. I think it is important for teachers to give their students opportunities to experiment with different addition algorithms like Lynn did in this chapter, because all students do think and understand things differently.

DMI and Field

I always knew that there were different ways and methods to do traditional algorithims, but due to my education I always thought the traditional algorithm was the best. I felt that this was the algorithm which needed to be taught in school. When reading chapter 4 in the DMI book one teacher, Lynn, states, “I think a very important difference this year was that by the time these ten children were exposed to the traditional algorithm, they had successfully constructed their own understandings of addition with regrouping.” I found this very interesting when thinking about my own experience with algorithms. Even though I have not seen algorithms being used in my field placement due to the fact that they are kindergarteners, I feel that this statement makes a lot of sense when thinking about mathematics. Without having an understanding of what you are doing or why you are doing it the children will just believe it is nonsense. Therefore, it is important that they create their own understandings about mathematics before being pushed to use the traditional algorithm.

Throughout the time I have spent in my field placement I have begun to see them use word problems with the kindergarten classroom. These word problem really help them think about the mathematics, but they traditionally are done as activities right before the end of the day. I feel that it might be helpful for some students to have manipulative to use and not just their fingers. Lynn states, “A significant difference this year was that I included lots of word problems along with rods-trading through the whole year.” The children need to be given different modes of solving problems and be allowed to use the modes throughout the whole year. Scaffolding can be used to help these children gain a better understanding and be able to do harder math problems as the year progresses.

Overall, I find the DMI book very interesting. I enjoy going into my classroom and talking to my CT about what we are learning in class and what it states about kindergarteners in the books we are reading. She usually does an activity with the students to allow me to see that the students really can do word problems, etc. This has really allowed me to see how the information we are learning in class can be applied to the classroom and that the children really can perform certain tasks when they are put into context.

DMI Chapter 4

I was amazed at how this reading really mirrors the second grade classroom I am in for this semester. The students I am working with are subtracting two digit numbers and learning the standard algorithm. In the reading, the first group of students really struggled because the algorithm was pushed onto them before they really understood the idea of tens and ones and regrouping. Although the second group didn't seem to enjoy the algorithm, they were driven to make sense of it because it came later in the year when they had a better understanding of adding tens and ones. In the case of Fiona (37-19) she lost track of the operation she was doing because she took 10 away from 30 and then she took 9 away from the 20 and forgot what to do with the 7. She confused her operations likely because she had several steps to solving this problem. Lynn's asking whether the 7 birds stayed or left was a helpful question because Fiona really had to think about what was happening with the numbers. Lynn said that Fiona usually got other problems like this after asking her this question once. I think Lynn pushed Fiona to ask herself what she was doing with the numbers so this knowledge transfered to later problems.

At the end of chapter 4, I found it interesting that Lynn noted that there was a lot less subtracting up which I think is because her students were working with the math behind the algorithm instead of just memorizing steps to complete a problem. As I said before, I am seeing some of the exact same things in my field placement as Lynn, especially subtracting up. Some students have a firm grasp on adding tens and ones and regrouping so the algorithm makes sense to them. However, many of the students struggle when I ask them to show me a problem in a different way that writing it as an algorithm. Students really struggle when subtracting two digit numbers that require regrouping. If the "bottom" number is larger than the number on top, they quickly resort to subtracting up. I strongly feel that this is because they do not see how the standard algorithm relates to tens and ones. The problem, I feel lies in the rushed manner in which teachers have to teach the curriculum. My CT is an amazing math teacher and she really pushes the students to use manipulatives or other strategies for addition and subtraction (making 10's, compensation, and so on) but the curriculum almost demands that students know the standard algorithm. Their math work books are almost entirely composed of problems written in the standard algorithm. This is a topic that is very interesting to me because I see it every week, and I am very curious to see how the students view the standard algorithm at the end of the year as they head toward 3rd grade!

I must admit, it isn't often that I truly look forward to reading for classes, however each week I truly look forward to our math readings because they have completely opened my eyes up to math thinking and understanding. On the other hand, I am now very scared to teach math, because I am starting to think that it is the most challenging subject to teach. Throughout my math education I have had some wonderful teachers and some not so good teachers. My wonderful teachers came in high school and college and therefore all throughout elementary and middle school math was a challenge. I was pretty good at math and scored well on homework and tests, but when it came to understanding what I was doing or why my answers were correct, I could not tell you. I made many mistakes, but did not learn from those. I loved the Eggleton and Moldavan article about the value of mistakes and I thought it was so interesting about the father that said he had two sons in two different learning environments. The one son who had a traditional atmosphere (like myself) would say "I don't get it!" where as the other son who had learned math through discovery and exploration would say "I don't get it yet!" I know that I will definitely have to explore how to allow my students to learn from there mistakes because I do not want my future students to give the correct answers without understanding and if they give an incorrect answer I want to work on understanding their logic, for instance in chapter 3 of DMI when the class was going through the days chart and one student responded that after 59 came fifty-ten. Even though this is not the correct answer I see his logic behind his answer. Most kids know how to count from 1-10 and therefore why when they get to 59 should it not be fifty-ten. I see this child's logic, but as an aspiring teacher how would you fix this child's reasoning? Would you just let him explore his answer and discover that it is really 60? There is still a lot I need to learn about teaching math, but these readings are definitely providing me with more insight.

Making Mistakes Equates to Learning

I am always amazed at how our readings seem to fit my field experience each week! I noticed a lot of mistakes in my student interviews today, but I could understand their logic. In the Eggleton and Moldavan article, the authors express how students need to think through their errors to understand why they are wrong and that it is important for students to try to show counter examples. I have noticed in the field, especially after my math interviews that my students always ask, “Is this right?” without really thinking through math problems. It was so tough to explain that I was not looking for right answers, but logical thinking! In the DMI chapter for this week, all of the student answers to Dawn’s number chart made sense. When the student replied “fifty ten” to follow 59 it makes sense. The student knew that they were working with the fifties and that 10 follow nine so this answer would make sense to a student. I also noticed in Marie’s section a student represented the number 127 with one flat, two units and 7 rods. I think this idea goes back to a topic we talked about recently in class. Students have trouble seeing the representation of the number 127 with manipulatives. There is logic behind Mary’s reasoning because she showed one object in the hundreds place, two objects in the tens place, and 7 objects in the ones place. The problem is that she simply sees the base 10 cubes as objects and not a representation of numbers.

Logical Thinking

As adults, I believe, we are more close minded about academic subjects such as math. We have an understanding of how to do specific types of math and it comes naturally to us. This causes the mistakes children make to sometimes seem odd to an adult or a teacher. Even though this is may be true, it is important to realize that these mistakes can be little lessons (Eggleton and Moldavan, 2001). These errors are typically logical and can be used as a way to explore mathematical concepts. For example, until reading DMI: Building a System of Tens I had never seen a child write 10095 for 195. To an adult this answer may seem completely wrong, but when thinking about it in a child’s point of view the answer begins to seem more logical. The child knows the number 100 and the number 95 and therefore puts the two numbers together to create 195. This answer tells a teacher a lot about the student. For example, this particular student does not have an understanding for place value.

From the case studies I have also noticed that sometimes it is hard to completely understand the logic behind a child’s thinking. Children may not be able to explain their understanding completely and therefore the teacher is also likely not to completely understand their thinking. I believe that children come up with their own insights into mathematical thinking through experience and exploration. It is therefore the teachers job to nourish this for all the students in his or her classroom.

DMI chapter 2

After reading chapter 2 in DMI, I must say how interesting it was to read the different scenarios to get a better insight on how children think. I do have to be honest though and admit that it is difficult for me to try and understand why they solve a problem a certain way. I have worked with some student in my placement on math and when I see them solve a problem that seems more confusing than beneficial I have to stop myself from offering a different solution. I have to keep in mind that it isn't about what I want them to be able to do, but rather what they understand. One thing that I noticed in this chapter was that most of the students got the answer right, but all of their strategies were different. One thing that I found particularly interesting was how Kim (page 28-29) solved (18 + 34 =). When I was in elementary school I was taught to go from right to left. I would have solved this problem by adding 8 and 4 and getting 12 I would have carried the one and added 1, 1, and 3 to get 52. Kim instead, added the 1 ten and 3 tens to get 40 and then 8 and 4 to get 12, so 52. I just thought this was the most interesting way to solve this problem because I am so used to adding from right to left and carrying. The biggest thing that I got out of this chapter was that I do not believe there is a right way to do math. I know that the one thing I have to get adjusted to is not trying to switch a child's thinking. I need to sit back and observe their strategy and question why they did something the way they did.

How Children Think About Math

After reading Chapter 2 in DMI, I learned a lot about how children think. I saw few examples of children's thinking didn't make sense. For the most part, every case study has logic to it. I have to remind myself about what I learned in the Thompson article: teaching is not about what you want students to be able to do, it is what you want them to understand. Although in the DMI chapter a student insisted on counting tally marks (sometimes more than 100), it is important to understand that this makes sense. There is nothing "incorrect" about how this girl was adding numbers. From there you can aid in showing students a variety of ways to solve problems (especially in more practical ways).

Something I also found useful was how many students added two digit numbers from left to right. From my schooling I have the common addition algorithm drilled in my head where you add the ones first and "carry" if necessary and then add the tens. This is very tedious and confusing and part of the reason I struggle with mental math. As I read the case studies, I practiced some of the addition problems and I added the tens first an then the ones. If I had to regroup it wasn't as confusing as if I had to "carry". This enlightening experience has really motivated me to help my students to understand place value and see how numbers really work. I hope to observe the ways they add and subtract and so on and teach these ways of thinking to other students such as those who count on their fingers, or use tally marks for difficult problems.

Complex Mathematical Thinking

As teachers we have to begin to think like children. Many times since as adults we have mastered the idea of numbers and counting it can be hard to understand the struggles children are having in math. Richardson does a good job explaining this by stating, “When they count, they must ignore the physical properties of what they are counting. Number can’t be seen. Number is an idea. We are often unaware of just how complex an idea is until we see very young children work to figure out number concepts.” It is therefore important to remember that we have to understand the struggles children may face in learning mathematics. Four of the things Richardson talked about in his article are inclusion, one-to-one correspondence, conservation of number, and number sense and relationships. From my experiences in teacher education classes at MSU I have a good understanding of one-to-one correspondence and conservation of number. The two ideas that I was unaware of were inclusion and number sense and relationships. I will discuss these two ideas separately.

Inclusion: Before reading this article I had never thought of the idea the children would only bring you the 3rd object they counted rather than bringing all three objects. When someone says “Bring me three oranges” they mean to include all of the oranges that were counted to get up to the number three. I have never seen a child do this but would like to see a child before they have a grasp on inclusion.

Number sense and relationships: Richardson stated “Children need lots of practice before counting becomes a genuinely meaningful and useful tool.” These experiences and practice need to be meaningful experiences and not for any old reason. “Given meaningful counting experiences, children will develop a strong sense of number and number relationships as they simultaneously develop facility with counting.” I feel that he gave good examples of how to create meaningful experiences by counting children in the bus line, the number of people in each group, etc.

Overall I really enjoyed this article and learned a little more about children’s mathematical thinking.

I read the Woleck (2001) piece for the jigsaw reading. In the article he stated, “Drawing emerges as a powerful medium for discovering and expressing meaning; for the young child, drawing brings ideas to the surface.” I really think this quote allows teachers to see how drawing can be useful in a mathematical context. The drawings can be used for children to work through their thinking process but can also be used to help explain their ideas to the whole class. Drawings can become a way for mathematics to become more static and less complex.

I must say I, I have found everything that I have read in CGI very interesting. I think it is fascinating to learn about the different strategies in which children learn. However, these strategies also confuse me and I know that it is going to take some practice for me to learn and understand the different strategies. One thing that I read that really stuck out was in chapter 4 when it talks about how there are good reasons for introducing multiplication and division word problems early in the mathematics curriculum. I would never have thought before reading this chapter that you could introduce multiplication and division to kindergarten children. I remember when I was in elementary school I did not start learn about multiplication until 2nd or 3rd grade. It was very structured. I learned addition first, then substration, multiplication, division, etc. I also think it is fascinating how children use different strategies to solve the same kind of problem. It is so cool to see how children think through problems. Before reading these chapters I thought that there was only one type of way to solve a math problem, however this misconception has greatly changed and I am really excited to further my education in mathematics methods.

CGI Chapters 4-6

Something I found particularly interesting was how partitive division is harder for students than measurement division. I thought that finding the number of groups is a tough concept to grasp, so clearly I thought students would find measurement division to be harder than partitive. I now understand that it is harder to figure out how many are in a set of groups because this allows students to use trial and error which can make solving the problem much harder. This is definitely something I learned.

I also thought the section on remainders in division was very interesting. I remember that when doing division we would write our solution with R1 (for remainder 1). We never did much with the remainder until we started to be more versed with fractions. However, I learned about how the remainder can be the solution to the problem such as in the following problem from page 44:

Mrs. Baker has 17 cupcakes. She wants to share them equally among her 3 daughters so that no one gets more than anyone else. She also wants to use all the cupcakes up. If she gives each daughter as many cupcakes as possible, how many cupcakes will be left for Mrs. Baker to eat?

I really like this approach to using the remainder within division problems, because it is a different way of thinking of what is left over than what I am used to from my schooling. I like it because you don't feel as though the remainder is wasted.

CGI: Multiplication and Division

The idea that problems can be “related but differ in what is known and what is unknown” is a new idea to me (Carpenter, 34). When reading three different problem types I first saw them as the same “problem”. In all the problems, no matter what the unknown was, I was able to differentiate the numbers to create the same problem.

(On page 34 of CGI)
Ex. Megan has 5 bags of cookies. There are 3 cookies in each bag. How many cookies does Megan have all together.
5 x 3 = ?
Ex. Megan has 15 cookies. She puts 3 books in each bag. How many bags can she fill?
3 x ? = 15
Ex. Megan has 15 cookies. She put the cookies into 5 bags with the same number of cookies in each bag. How many cookies are in each bag?
5 x ? = 15

These problems are all similar to me, but to a child first learning mathematics they have trouble differentiating the different types of problems. Therefore, it is through this book that I learned that teachers need to pay particular attention to the type of word problem they are giving to their students.

Another aspect of CGI that I found interesting is that “there are god reasons for introducing multiplication and division word problems early in the mathematics curriculum.” I had always thought of multiplication and division as being something that was taught at the upper elementary grades, but in Children’s Mathematics: Cognitively Guided Instruction they state “With experience, many kindergarten children can solve simple multiplication and division problems by using counters to model the groups described in the problems. By first and second grade, many children use a variety of strategies to solve multiplication and division word problems” (44). This came as a shock to me, but through thinking about younger children being able to do multiplication and division I have realized that the different strategies available to children (and that children known) have an impact on their ability to solve the problems.

Even with the knowledge that young children can do multiplication and division problems, I question how much this is done in elementary schools. In my field I have seen the kindergarten children doing mental math with addition, but wonder if structurally this is what the children should be doing. The other issue is that in kindergarten the children cannot read quite yet. This causes issues with the children being able to do word problems without the help of an adult. I am wondering what you see during your time in the field and if you feel that the children are as advanced as what is shown in the CGI book?

CGI Blog

Before I read the first three chapers of CGI I had never really stopped to think about how children learn math. I have always just thought of math as something you get or you don't. It was very informative to read about the different ways in which students learn math such as; join, separate, part-part whole, and compare (graph page 12). I thought it was neat how the book stated how strategies such as counting come naturally to young children. Children do not have to be taught that a particular strategy goes with a particular type of problem. I thought this little fact was so cool and it got me thinking about the students I work with and how each of them use different strategies for the same problem. One student isn't more right than the other, it just shows how children learn things in different ways. Another thing that I found particularly interesting was the effort that goes into creating story problems for students. I had never stopped to think about how the wording in word problems can make a problem harder or easier for a student. Problems are easier if their wording corresponds to the action sequence. For example:
Janice had 9 cookies. She ate 3 of them. How many cookies does Janice have left?
VS
Janice just ate 3 cookies. She started with 9 cookies. How many cookies does Janice have not?

Clearly the first one is easier because the starting quanity is given first. All in all I really enjoyed these three chapters and learned a lot about math, however I do have one question and that is at what age do you try to get students to stop counting up or down and just doing the math? How do you teach students to SOLVE a math problem, rather than just counting up or down?

More on Cognitively Guided Instruction

When reading the first three chapters of Cognitively Guided Instruction it was refreshing to review the different ways to construct addition problems (as well as subtraction). I found it particularly helpful to have the chart (page 12) that explains the different types of problems (Join, Separate, Part-Part-Whole, and Compare). I was also helpful to include the result unknown, change unknown, and the start unknown sections. These ideas are new to me. I think that understanding how to construct math problems using a variety of these methods is beneficial to students because it allows them to use critical thinking skills even if they are solving the same problem. In my field placement, my students struggle with word problems so I think it is important to introduce all of the problem types and to give them practice in completing such problems.
Another thing I noticed that in my students' math workbooks, the students always skip the story problems. I think they really struggle with the language which may be detrimental to their learning. I found the Chapin and Snock article to have many interesting points about how the language of word problems can negatively influence student understanding. Having these two readings in mind, I hope to make sure my future students are well versed in the language of math as well as well practiced in the many problem types.

Cognitively Guided Instruction

In many college level courses students find the reading to be useless, unneeded, or boring. For me, this was not the case when reading about Cognitively Guided Instruction. When reading the first three chapters of the book I became excited know that I could use some of this information in my future classrooms. The author of the book states, “Over time, direct modeling strategies give way to more efficient counting strategies, which are generally more abstract ways of modeling a problem” (Carpenter, 3). I was struck by this quote, not because I did not realize that children moved from direct modeling to more abstract thinking of mathematical concepts, but because I questioned if all students do this on their own time or if some need more direct instruction on how to begin to think abstractly (for example: doing mental math). This quote also reminded me of the subtle things teachers need to be looking for during mathematical thinking. An observer may think a child who gets the answer right by using direct modeling has a better understanding of mathematics than a child who uses mental math and gets the answer incorrect. In reality the teacher needs to think about the student’s mathematical thinking and where they need to grow from there.

In the book the authors state, “Variations in the wording of the problems and the situations they depict can make the problem more or less difficult for children to solve” (Carpenter, 10). I found this extremely interesting. I had never thought about how a word problem can be manipulated to change its difficulty. In the book the example the authors gave (in case you do not remember) was:

Janice had 9 cookies. She at 3 of them. How many cookies did Janice have left?

Janice just ate 3 cookies. She started with 9 cookies. How many cookies does Janice have now?

The problems are the exact same, just worded a little differently. This information visually showed me how a teacher needs to be careful when writing his or her own word problems, or when picking word problems from a book. This is because the ways the word problems are written affect the way a person perceives the problem. Another way a teacher can use these two word problems are by using one or the other for an extension or simplification. Depending on the needs of the child the same word problem can be written in a harder or easier way.

I also really enjoyed learning about the different types of word problems. I had never thought of word problems as having so many different types. I am not sure if I have completely memorized what each of the word problem types are called, but I know have an idea that when teaching a classroom a teacher can not just pick a word problem arbitrarily. A good teacher has to plan ahead!

Overall, I really thought this book had a lot of helpful information. I am interested to see what others thought about the book and the information they found important. The only thing I am worried about is being able to spend enough time to pick the correct word problems for my students. I am also questioning myself on how I will be able to remember all the different types of word problems and recognize them by sight. Hopefully this will come from experience and time, but as of now I feel that this task may take a lot of time and effort for beginners.

There are many learning goals that I would like to accomplish throughout this semester. I have always been pretty average when it comes to math and I have always enjoyed it, however I did not always feel safe in the enviroment in which I was learning math. I was always scared to raise my hand and give an answer because I was too scared of being wrong and being made fun of. I want to create an environment much like the one described in the Lester article. I want my classroom environment to allow the students to be curious, question things, explore and most importantly feel respected. I believe that when you create an environment such as this, that all students have a better opportunity to succeed. Another goal that I have is I want to walk away from this class and be able to explain how something is done. For example, I know that 4+4=8, but when a student asks me "why?" I want to be able to provide them with a confident explanation. I am very good at solving math problems, but I have not yet mastered WHY math problems work out. I would also like to walk away from this class knowing different methods and strategies to teach math. I can remember from my own math experience that I didn't always learn math the same way as my classmates and therefore I think that it is important to provide students with different ways to learn math. The most important goal that I would like to learn from this class is how to help my students appreciate math. I do not think that many students appreciate math or think that it is important, and therefore I want to make math fun, and engaging for my students.

Learning Goals

After reading the syllabus and the readings for this week, I developed some learning goals for this semester. I was really interested in the article that discussed how to gradually create a learning community for math. One of my goals is to learn how to create a safe environment such as the one in the article. I want my students to feel that they can take a "risk" in the classroom to express new ideas. I also want to learn more about how to get parents involved in their child's learning, if possible, pertaining to math. Another goal that I think is important is learning about different strategies to teach the many methods of solving a math problem. In the article where the teacher used the unifix cubes, I learned a strategy for teaching addition and subtraction that is hands on and engaging. Although it is impossible to learn every strategy for teaching every type of problem, I feel that individual examples in case studies add up and can help me to develop my own strategies for in the classroom.

A last learning goal, is that I want to be able to see math problems from many different perspectives. Before taking Math 201, I only solved math problems in one way that was comfortable to me. I want to be able to look at a math problem and imagine the many ways students can solve it. I want to be skilled in looking for patterns in solutions and seeing if that method will work every time.

Welcome!

Hello!
I look forward to discussing with all of you over the course of the semester about issues relating to education. I feel that this blog is a way for all of us to get to know one another and also talk to each other in a professional manner. I hope we can learn from each others ideas and work together as a team to push our thinking beyond the surface. With that said, I would like to begin by discussing my ideal classroom and goals for this semester in TE 402.

Throughout my experience as an undergraduate in at Michigan State University I have had some good experiences and some that could have been better. This is also true when thinking about my experiences in Elementary School, Middle School, and High School. I believe that most, if not all, students come to school with expectations for themselves and their teachers. Students want their time in the classroom to be worthwhile and memorable. The content that is taught may change from year to year but these expectations for the teacher do not. I believe that the things that I am stating here deal with my ideal classroom for TE 402 and in the field. As a student and a teacher I want my classroom to be welcoming and hands-on. I also believe that the academic content which is taught during class should be molded by the interests of the students. This makes the learning process more authentic.

Students come into the classroom with prior experiences that have molded who they are as a person. Some students may be quiet while others may choose to talk a lot in class. Other students may struggle with certain academic content and become quickly frustrated with themselves or their peers. As a teacher you have to take all these different learning styles and create a learning community where all students can excel. This is not always easy, but through TE 402 and our field placement this semester we can hopefully learn how to create lessons and activities for children with a variety of learning styles.

Overall, I am confident that TE 402 will meet my expectations and that we, as a class, will benefit from open discussions that connect class content with what is going on in our field placement.

Have a wonderful week!

Kathryn