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?