Regrouping and Calculators

I read Lynn's case study in the DMI book this week with a lot of interest because I am also in a second grade classroom where they are working with regrouping in much the same way that she talked about doing in her classroom. I understood some of her frustration when she said there was a disconnect between the games they would play with trading ones and tens to when they actually tried to do addition with regrouping on their paper. Line 41-45 discusses this scenario where students that had a good grasp of ones and tens still wanted to count by ones when faced with two digit numbers.

Today our class worked on regrouping with their rubix cubes for awhile until they went back to their desks to work on a worksheet and it took a little convincing for them to not all discard their cubes right away. My students probably would not know what to do with a traditional algorithm right away though because they are in the stage where they are trying to really solidify what it means to "regroup" and to have a really good understanding of the tens and the ones place. They are really able to make sense of it like Lynn's students in her class. I was very impressed when I did my math interviews with my students today. I had selected some word problems that asked for multiplication and division. I knew they hadn't discussed that in class yet, at least not while I was there or that my CT had mentioned. Even so, every student was able to correctly solve the problems and was able to explain it to me. What made me even more excited was that I interviewed 3 students and there were 3 different methods used. These students were really able to make sense of the numbers to solve problems they might not have experienced before.

In the Groves article about calculator use I was pleasantly surprised to see how young and in what manner they were using calculators. I liked the quote, "There is no evidence that children became reliant on calculators at the expense of their ability to use other methods of computation" (128). The students in K-2 were able to explore numbers far beyond their reach if they did not have a computer that allowed them to experience many of the patterns that are very important in math. Looking back on my own math experience with calculators, I don't remember having access to them until about fifth grade. Even so, I remember being strongly encouraged not to use them and we definately did not spend time exploring different math ideas on the calculator. Since calculators are far from a new invention it has puzzled me as to why few teachers have really embraced them as a resource. I was impressed that when the teacher asked students to divide 64 by 7 the students were quickly able to look at their response in decimals and think of it as being a bit more than nine. The teacher made the point that this equation would have been over their heads if they weren't allowed to use a calculator and it also provided an important look at decimals. I know this article definately gave me a new outlook on the use of calculators, especially in the younger grades.

DMI Ch. 4- Erin

“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 understanding of addition with regrouping” (Schifter, 67). They constructed their OWN understanding. It is this that I feel is one of the most important factors in math. I know in my own personal experience, I have not learned by what the majority would necessarily consider the “norm”. I always tended to find my own ways of understanding in academics, and especially in mathematics. I thought it was very interesting the different ideas that come up throughout chapter four of “Building a System of Tens”. Especially interesting was the idea of adding in order to subtract that Paul brings up in lines 254 through 267. I reread his way of thinking over and over and I, a senior student at Michigan State University, am easily confused with the way he solved his problem. It made me wonder if this was an example of this child’s actual mature level of thinking.
Something else that came up that I have strong opinions on is the expectations put on us by the district, parents and coworkers. I often find myself wondering if I can successfully pull off some of the techniques I’ve learned about in the College of Education. I understand the use of expectations, but I do wish that some of the politics of school weren’t so strict. I feel that we need to be more open minded about our students learning, yet it’s not going to happen when there are so many standards and expectations to meet.

DMI Ch.4

After completing this week’s readings, I found Ch. 4 of our DMI textbook to be very insightful. Throughout the two separate case studies, Lynn has her students experiment with a variety of problem solving strategies when completing diverse addition problems. Each student appears to have his or her own way of completing these problems and becomes slightly confused when the standard borrow/carry or regrouping algorithm comes into play. Many of the children solved the addition problems beginning with the tens column. On page 64, we can see a chart that Lynn comprised for her class along with responses from the children saying, “I did it that that way! Or “I solved it the green way”. For example, one of these strategies utilized when adding 38 and 25 was 30+20=50, 8+5=13, 50+13=63. In this strategy, the student begins by adding the tens column then adds the ones column. The final step is adding the two answers together to receive the total. Another strategy used was 38+2=40, 40+20=60, 60+3=63. In this particular problem the student rounded 38 up to 40 by adding two. They then added the tens columns together 40+20 to get 60. Rather than adding the remaining five in the ones column, the student remembered they had to first subtract the 2 that was initially added and thus, add only 3 to retrieve 63. Though many of the students were familiar with adding the tens column first and disliked the standard way of doing things (adding the ones column first) Lynn states, “What amazed me was that they could all make some sense of the pink way” (66). Given all of this information, we can identify the various ways and multiple strategies utilized by these students in order to obtain the correct answer to the given addition problem. In Case 2, students also used a variety of methods to obtain the correct answer to two-digit subtraction problems. For example, when Paul was asked to solve 39-17 he decided to take the 17 apart in 3 steps. He first computed 39-10 to get 29. He then took 29-4 to receive 25 and then 25-3 to get 22. Essentially, he broke 17 up in 10, 4, and 3, and subtracted each portion from 39 individually. Nathan on the other hand, approached this problem quite differently and used addition to solve this subtraction problem. He first added 17 and 3 to get 20 and then added that 20 to ten to get 30. Nathan then added 30 to 9 to get 39. In essence, Nathan begins with the lower number and continues to add a variety of numbers to it until he gets to the higher number. He then takes each number he added to the originally number (17) and adds those together to retrieve the answer. These are two very different and unique strategies utilized by two distinct students.

In the second case, Fiona is trying to complete the subtraction problem 37-19 and in the process, loses track of the numbers she is supposed to be subtracted. She begins by subtracting 10 from 30 to get 20. (She rounds 30 down from 37 in order to make the number more manageable.) She then takes the 20 and subtracts 9, the remaining portion of 19 left after the 10 had already been subtracted. Fiona then gets confused because she is unsure if she should add or subtract the initial 7 that she dropped from 30 in order to make the number more manageable. With the teachers prompt, “Did those 7 pigeons leave or stay” she realizes the 7 needs to be added in order to obtain 18. This is a strategy that children implement and often results in the inability to correctly track numbers throughout their thought process.

Figuring out how each of these children solved the given equations and implemented reason and logic in a way that makes sense to them truly allowed me to view math through the eyes of a child. Many of these strategies are ones that I would not think of on my own, yet they are astonishingly accurate and sensible. By walking through the process each child took, I was able to understand the wide array of strategies present and will be able to keep these in mind when teaching in the future.

Making Mistakes

I found this week's reading to be very interesting. It was nice to read through "The Value of Mistakes" because something I see teachers doing sometime in rushing in to correct students right away when they are doing school work instead of allowing students time to think through their own mistakes. In case 11 when Dawn was recording student responses to what number they thought would come next on the calendar she recorded all the responses on the board without comment. Then she allowed time for the students to try to reason out their answers without telling them if it was right or wrong. When students have to really think about their answers this forces them to analyze why it is right or wrong and helps them think about whether their answer makes sense or not. The Eggleton article says, "[students] may make similiar mistakes in the future, but they have now had the experience of analyzing the mistake to know why it is incorrect." This really is the goal of the teacher because they will not always be over the students shoulder to correct any errors they might have later in school and life.

I liked the example the article gave of the students with various numbers who were supposed to line up in order. I think this was more useful than on a worksheet where students have a series of numbers they are supposed to order. When used in this interactive way students have to talk to each other and justify their thinking. This means students have to decide if their math makes sense and they are also inadvertently engaging in group work because they work with each other to complete their line.

Case 14 in the DMI text where the students were discussing the importance of zero reminded me of something I watched in field today. Students were working on taking money written in words and turning it into decimal. One of the problems was something like four dollars and two cents. This was the only problem where the amount of cents was under 10 so many students struggled with it, writing $4.2 instead. This prompted a nice discussion in the class about the difference between .2 and .02 which would not have come up if the students had not made the mistake or if the teacher had just quickly corrected the error. Muriel writes in the case, "I wonder how maing sense of zero affects children's understanding of place value. Actually, it's probably more to the point to wonder how not making sense of zero affects children's understanding of place value," (56). While many of the students in my class seem to grasp how zero works, many others still do not fully understand the role of zero as a "placeholder".

I found the idea of the difference between "I don't know it" and "I don't know it yet" to be rather powerful. If you can get your students to the point where they know they can solve any problem if they try long or hard enough then they will probably have a lot more enjoyment and success with math.

The Value of Mistakes/DMI CH 3

After reading through this week's casebook studies present in DMI's chapter 3, there are several errors and identifiable misconceptions present within many of the children's thinking. Coinciding with these misconceptions, there is still absolute logic and reasoning that exists. For example, in Case 11 (Dawn), the kindergarten teacher is using a hundred's chart to track the number of days the children have been in school. The chart is completed up to the number 59 and the teacher poses the question to the students, "What number should I write on our chart today?" One student in particular named Andrew replies "fifty-ten". The teacher records this as 510. Though this answer is incorrect, we can identify the logical reasoning that exists within Andrew's mindset. Page 46 states, "Once again, physical involvement seemed a necessity as Andrew moved up to the chart and dragged his finger across the row with the numbers 51, 52, 53, 54, 55, 56, 57, 58, and 59. Andrew seemed to be making use of the number sequence; thus his response of fifty-ten made perfect sense" (46). In Andrew's mind, the sequence of 5's or 50's as the first numeral should remain constant and carry over into the next space, while the second numeral should increase by one interval in order to continue the pattern in place. Thus, fifty-ten, is a perfectly logical, though incorrect, solution to this problem. In addition, in Case 13(Marie), many students make similar errors and struggle when regrouping numbers and/or accurately representing place value amounts. For example, it states, "Lou showed 99 by laying out 9 rods and 9 units. His written number sentence to represent this amount was 99=9+9" (50). In this example, though Lou is able to use the manipulatives in the correct manner, he is failing to understand that 99 is, in actuality, 90+9. While this is a common misconception found in children, it is one that is logical and makes sense in the minds of young children. The number 99 is quite literally comprised of two 9's thus alluding to the fact that 99 would simply be equivalent to one 9 plus another 9. Other errors found within this particular case study are that of ones and tens placement reversal. For example, "Mary showed 127 as one flat, 2 units, and 7 rods, which she arranged in the same order. Her number sentence was 127=100+27" (51). Similarly, "Darryl's number was 237, which he showed by lining up 2 flats, 7 rods, and 3 units in this order" (51). Both of these students are accurately representing each number with the correct amount of manipulative (for example, in 127, representing the 2 as 2 units) however, they are using the incorrect manipulatives and failing to recognize the place value that the number in the ones and tens place holds. All of these mistakes, though erroneous, are actually quite logical and make sense in the mind's of young children.

The article, The Value of Mistakes, states, "When we teach children that mistakes can be the beginning of the learning process, that they are to be used as stepping stones leading to success, we are much more likely to engender children who say 'I don't get it yet'. As one student shared in a journal entry, 'Mistakes are like little lessons all in themselves. If you face them as learning opportunities rather than reminder of your inabilities, then you're bound to be a better mathematician" (47). All of the children in the case studies can thus use their mistakes as segues for success and take the initiative to deconstruct their thought process in order to determine where the error occurred and how they can improve next time. If all of these steps are taken, students can surely attain academic success.

Eggleton and Case Studies

I have very mixed reactions to this weeks readings. On one end, I'm very understanding and get what they are saying about children making mistakes. On the other end,I feel that as teachers it is our job to fix student's mistakes. At the same time I have questions about everything. I am mostly confused after reading "The Value of Mistakes" by Eggleton and Moldavon. I do agree with some of what they are saying about children needing to make their own mistakes and fixing them. In fact, one of my favorite quotes come from this article. A father says about his two separately taught children, "One son's reaction to a problem might be, 'I don't get it,' whereas the other son's reaction would more likely be, 'I don't get it, yet!'" (Eggleton, 47). It is this type of "go-get-them" type attitude that I want to instill in my students. I would like them to learn from their mistakes because I do agree with a lot of what is said in this article. I believe that children who learn from mistakes made in math will certainly understand and comprehend the math better than a child who is just told what to do. I also agree that student's very rarely think back on these mistakes as negative experiences. In fact, I feel the complete opposite. From my own personal memory, I remember feeling extremely accomplished when I figured out the answer after so many mistakes and so much work. I do wonder though, as a teacher, when is the right time to interfere? How long do you let a child work and wonder before you come in and show them how to do it?
I also really liked the readings from DMI. We have a school day chart like the one talked about in Dawn's case in my placement classroom. I am placed in a kindergarten class and I always wondered why the chart was not ordered by week. I now understand the point and thought behind the ten to each row. I found it interesting that Andrew physically moved to show the rows of ten. With this idea AND the seashells, it just goes to show how many different ways one idea in math can be shown. I think for young children especially, it is important to expose them to concrete, abstract and physical ideas. I also really liked the idea of the division chart in Donna's case. I am always curious about new ways to teach about ten and I feel this would be a new and great way to teach student's about patterns and tens. As I was reading though, I found myself to be frustrated with Marie's case with Lou, Jose, etc. I would have loved to seen how they would have reacted with different manipulatives. I would like to know if it was the place value they were confused about or the actual blocks. If they were more aware of which piece represented which amount, they may have been able to understand better.

Groups of Tens

I was excited to get into some of the case studies in the reading this week. In school, it was always hard for me to understand math out of a book and I have experienced some of that even doing just readings about math. When I read the CGI readings they became much more real to me. The case studies in DMI book gave me a lot of concrete ways to think about some of the ideas that we are learning about. I really liked case 6 where Lucy was working on how to translate the algorithms that she knew on paper where she had to "carry" to making the same problem with unifix cubes. This reminded me of the discussion that we had in class last week where we were debating the merits of the "borrow and carry" method. Once Lucy had gone through the entire problem with the blocks it really seemed to solidify the idea in her mind and that just reinforces the idea that borrow and carry does not have any real math connotations other than what we assign to it.

When I was reading the post before mine I really enjoyed the real world example of the toddler that was "naming" the crackers that she was eating. Before I read the "What You Need to Know About Beginning Number Concepts" article I had never really thought about that before. I don't really remember my thoughts about math when I was that young and I suppose I just haven't spent much time working with very young children about math. It does make sense though that children would try to use numbers to name things around them just like they use words to name other objects that they see. It makes things more concrete instead of the abstractness of numbers.

When this article also talked about the conservation of number I was thinking about ways that I had seen this demonstrated in some of the younger elementary classrooms I had been in before. I had seen students before when the objects they were looking at were bigger they assumed there were more of them than if they were really small.

I'm looking forward to continuing to work my way through the casebook to find some examples that I can try to integrate into field or possibly my student interview in the near future.

Beginning Number Concepts

After completing this weeks readings I found the article, "What You Need to Know About Beginning Number Concepts" to be especially interesting and applicable to the students within my particular classroom. After carefully observing strategies that students within my classroom use (as well as children within the daycare that I currently work at), I often found myself asking, I wonder why they chose to do that? or How did they come up with that answer? This article offered several answers to these questions that I have had and has made counting from a child's perspective much more feesible. For example, I noticed a 2 and half year old child at my daycare take an interest in numbers. At snack time, I asked her to count out five goldfish crackers. She completed this task with accuracy. I then, jokingly said, "Can I eat those 5 goldfish?" She responded by simply picking up the very last goldfish she had counted and offered it towards me. According to the article, this child was failing to "realize that the number they say when they count the last object in a group includes all the objects previously counted" (2). It was so interesting to be able to reflect upon this minor incidence that occurred and truly be able to capture this child's thought process within the moment. She, in essence, had "named" the last goldfish cracker 5, and thus, decided that to hand me five goldfish crackers meant to hand me the one titled 5. According to the article, this child is also failing to "recognize that the order in which we count the objects does not matter" (2). It will be extremely interesting to work with this child in the future and monitor how her thought process changes and progresses over time.

In addition, I have also noticed the concept of conservation of number come into play in a variety of situations both within my classroom and within the daycare setting. An excellent example of this occurred during arts and crafts time last week. The children were each given four cut-out hearts to make valentines for their family members. Some children's hearts were larger than others and many of the children began to complain saying "But he/she has more than I do!" The children were unable to separate what they were visually seeing with their eyes and logical reasoning. At the time I was a bit confused and was unsure why the children, all fully capable of counting, thought they were distributed unequal amounts. I now realize that children are extremely reliant upon their visual sensory field and tend to believe what their eyes tell them. Overall, I found this article to be of great importance, especially when working with children in lower elementary grades or those functioning cognitively at the k-2 level. Being able to enter into the minds of children and truly deconstruct their thought processes is essential and leads to more purposeful and tailored instruction.

Week 5 Readings-- Erin

This weeks readings definitely gave me better insight into how children think as they learn more about mathematics. I never would have thought about inclusion until reading the article, "What You Need to Know About Beginning Number Concepts". I always assumed that children would naturally know that all objects are included when counting. I did not realize it was a concept that developed, and we as teachers need to be sure our children understand before moving forward. I suppose that most of the understandings talked about in this article were things I thought were common sense. It's hard for me to think like a child but this article certainly helped me realize it will be necessary in order to help my students. We assume that our students will come to us with the natural talent and ability to so what we need them to do. This article raises the point that we might need to help our students if they have not yet reached this level. In the article, the author writes, "Given meaningful counting experiences, children will develop a strong sense of number and number relationships as they simultaneously develop facility with counting." I find myself wondering though if this is enough, and if not, what are some techniques to help our student's understanding?
When reading "Building a Systems of Tens" book, I realized that this book was more about specific examples rather than “in your face” type of information. I didn’t quite know how to process this type of approach to learning. I found myself questioning more, but not necessarily finding the answers. I did not like how the teacher writing even asked questions. I suppose this shows us how we will question what we do and must think about our teaching as it is happening. Despite my critiques, I found each case to be uniquely interesting. One in particular caught my attention because the word “carrying” came up. After our conversation in class a couple weeks ago, I have been paying more attention to math terms used. I do agree that rebundling and bundling. In the case described, the students Carmen and Melissa do not seem to be bothered by the term but I do wonder whether it would have made a difference if it had been explained otherwise.

CGI Ch. 4-6 Reflection-- Erin

Despite my initial negative feelings of reading about division and multiplication, I actually found these three chapters to be both interesting and helpful. I have found that this book explains math in a way that you have never thought about before. I personally find it confusing as I am reading it, but when I sit back and think about what I read, it actually ends up simplifying the subject for me. Like the previous chapters, the chapter on multiplication and division taught me more about the structure and types of problems than I ever thought possible. One of the more interesting points I found in the chapter was something I don't believe was made to be a big idea. On page 439 the book is discussing skip-counting as a counting strategy. I found it so interesting that children generally skip-count using three. I can see five or two but I was surprised with the number three. I tend to believe that sets of three tend to be more difficult than these children obviously see them. As this chapter went on, we see on page 42 that children sometimes have to figure out what to count by. They explain that trial and error is used in a situation like this. As they explain on the same page, "... it is significantly more difficult to use a counting strategy for a Partitive Division problem than for a Multiplication or Measurement Division problem". I found myself wondering when children tend to grow out of it and have this realization. Does it take showing other techniques and strategies or do they naturally realize that there are better ways to figure it out? I also found myself wondering whether there is a correct order to teach area and array problems. If one should be taught before the other one, how do you build into and relate to the other one?
As the book moved into multidigit number concepts, I found myself being even more amazed by all the different problems and strategies. I like the idea of base 10 because I know that it can simplify problems to the point where children can truly understand. I remember feeling like I could do math so fast when I learned to count by tens. In reality, I'm sure it was no faster than I should have been doing it, but the confidence allowed me to enjoy math more in elementary school. What did surprise me a lot in this chapter, was all the different students abilities. For some students, they understood right away with no manipulatives, others could get it with some direct modeling and others were not able to use the idea of base ten at all. I was also surprised to see that kindergartners were working with base ten. In the kindergarten class I am currently placed in, some students can not even count to a hundred. I wonder though if I were to ask them to count by groups of ten to one hundred, if they would be able to do it. Throughout this chapter they look at so many different modeling strategies for all of addition, subtraction, division and multiplication. I do not understand though if there is a correct order to teach all of this in. For instance, there is a large section on algorithms. The invented algorithms are obviously something that come to the children naturally since it is their way of looking at a number. They go through so many positives that come from using these algorithms but where and when do they start using them? Is there something that must be taught before they can begin to implement them? How early do these kind of rationalizations come to children? Although these chapters brought me many questions, it also once again calmed me. I feel like I am slowly grasping a better idea of what math is and how children see it.

CGI Ch. 4-6 Reflection

Like chapters 1-3, I found chapters 4-6 to be extremely interesting because of the presentation of diverse types of problems that can be used within a classroom setting. Though I have heard of the terms Multiplication, Measurement Division, and Partitive Division, it has always been difficult for me to adequately distinguish the difference between them. I know realize that when the number of groups are unknown it is a Measurement Division problem, while when the number of items in each group are unknown (or parts in each group) it is a Partitive Division problem. This knowledge is extremely helpful because it allows one to establish which type of problem may be more advanced and which would be more suitable to a child performing at or below grade level content expectations. For example, it is much easier for a child to be given a Partitive Division problem because the child can then establish how many groups there are are and equally distribute the given quantity of items one by one into each group until there are none left. This strategy was used by Ellen on page 37 of our CGI textbook. I also became aware of the fact that some students use separate objects or counters to represent each group. For example if the problem reads, Mr. Franke baked 20 cookies. He gave all the cookies to 4 friends, being careful to give the same number of cookies to each friend. How many cookies did each friend get? A child may first count out 20 objects to represent the cookies but then also count out 4 additional separate objects to represent the 4 friends while diving the 20 cookies. I found this to be extremely interesting but also wondered if this strategy may confuse children who choose to use it. Although the book suggesting it does not, I am still a bit skeptical.

Another concept I found extremely interesting, is that children can compute two and three digit numbers even if they have a limited notion of grouping by tens. I always thought quite the opposite in that children must have a keen understanding that the number 21 is really 2 sets of ten joined with the single digit 1. However, the book states that this assumption is invalid and "as long as children can count, they can solve problems involving two digit numbers even when they have limited notions of grouping by ten." This knowledge is extremely helpful because had I not known this I think that I would have prolonged teaching students how to compute two and three digit numbers in the future if I did not feel that they fully grasped the concept of grouping and regrouping tens. Now that I know it is acceptable to advance regardless of their knowledge of this concept, I will be able to do so in the future.