Young Mathmaticians 5-7

I think I have mentioned this before, but the fourth and fifth grade classrooms I tutor in are currently working through fractions right now too so it has been interesting for me to think about what I am seeing them struggle with compared to some of the examples raised in the texts. One thing that I think creates struggles for students with fractions is that it seems like the classrooms I observe usually go over one way to add or multiply fractions and it remains a very abstract idea for students. I have seen things like the double number line like they talked about in chapter 5, but this text just demonstrates how many ways you can approach looking at fractions. I looked at the array for quite awhile- I remember being first introduced to it in math 201- and while I can understand the answer I'm not sure I can really make sense of it for myself. I could easily plug the numbers in and make an array but I just can't relate it to anything else. Fractions are difficult for me just because I feel like I always think about them in abstract ways. It is hard for me to really make sense of dealing with complex fraction problems- I can do it- I just feel like I don't always understand exactly Why I'm doing what I'm doing. For some reason with the array in chapter 6 when they put a short story problem with the array it seemed to make more sense to me. I could kind of picture what was happening instead of just seeing tiles shaded in for no real apparant reason. Based on my own experience, making fractions as realistic to students as possible is an important part of teaching multiplying fractions. In chapter 7 I also thought it was advantageous for the teacher to have students work with ideas of multiplying (in this case groups of dots) before she introduced the idea of looking at it as fractions of the whole. Students already explored some math around the problem and found the whole before they started thinking about fractions. This also helps connect what students already know about multiplying and dividing whole numbers. I have heard of strategies like using the clock and money before because it is something very concrete to students. I am looking forward to talking more about this topic just because I feel like it is something I am weak in.

YMaW Ch. 5-7

First and foremost I just want to say that this week's readings were very tough for me to comprehend and understand. Not only is the material extremely heavy in content and intricate processes but it also founded upon one's basic knowledge and understanding of fractions. Because I have not taken a math class in 2 years it has been a struggle for me to understand the strategies students within the text are implementing and the rationale behind them. That being said, I did take away some very knowledgeable ideas and resources that could be used when teaching in the future. Primarily, I learned how important it is to develop a plethora of ways to model mathematical problems and situations so that students can mathematize effectively. Page 73 states, "Mathematizing-the human activity of organizing and itnerpreting reality mathematically-rather than a closed system of content to be transmitted or even discovered, mathematical models become very important." The book then goes on to provide a variety of direct modeling strategies that can be utilized to directly represent equations concerning fractions. I found the Ratio Table to be an extremely efficient way both organize fractions and rations and determine how to adjust those ratios by either making them bigger or smaller. The number line and double number line models, however, proved to be rather confusing to me and I am not sure I thoroughly grasp the concept. Another way to represent the multiplication or division of fractions is through use of an array. Arrays assist students in visually representing the direct action taking place in the equation and allow them to obtain appropriate answers through these means. On page 125 it states, "When children are given the chance to compute in their own ways, to play with relationships and operations, they see themselves as mathematicians and their understanding deepens." I found this quote to exceptionally summarizing of the preceding chapters in that its focus and emphasis remains on allowing children to explore mathematics in order to find what models and/or methods are most appropriate and conducive to their own, specific learning. As future teachers, this is one of the primary goals we should strive for.

April 16 Readings

I realized that this week's readings seemed to focus more on discussion and mathematical thinking than anything else. I found the article in particular to be extremely interesting. After the CGI book, I figured that discussion could be helpful to children's mathematical development, but I never realized the extent. I realized that the discussion process is just as important as the discussion itself. This article taught me about how important it is to have children make mathematical arguments. In order to do this, we as teachers, must set standards and expectations beforehand so that our students have a safe place to share ideas. I found it interesting that from revisiting their own thinking, "The student not only develops a through understanding of the ideas that she is grappling with, but she may also construct new understandings" (Whitenack, 525). I do worry personally about teaching these processes though. I do realize that children naturally question and form their big ideas and strategies, but I always find myself asking "what if?". The teacher in the article, "...focused on different aspects of the students' reasoning and continued conversations with different students in different ways" (Whitenack, 526). From this, I ask myself, what if I miss opportunities with children or are not able to recognize gateways to these conversations. I suppose this is something I will hopefully learn in experience.

Math content and processes

I read the Whitenack and Yackel article first where they were highlighting a classroom where the teacher had set up an environment where children discussed and defended mathematical ideas with each other. The teacher was able to facilitate the discussion to allow this to happen and you could see the process of sense-making in math really unfold for these children. In all of these articles it always seems so effortless in the classroom. As I prepare to teach my own math lesson, I wish I had more resources available to me about how some of these teachers started working to build this sort of environment. I wonder how much of it comes naturally and how much teachers have to help foster to make children comfortable with and eager to share.

While reading about fractions in Young Mathematicians the students had so many different ways to approach the problem. When I read about the "which cat food is cheaper" I immediately solved it the way that Helaina and Lucy did and didn't really think about it much further. The process skills in this classroom are a lot deeper though, these students were really pushing to find different ways to explore the problem. The two boys constructed a ratio table and figured out their answer that way, then two other girls came through and did it the same way with different numbers to find out the answer as well. I wouldn't have thought that this conversation could have led to so many different ways to discuss equivalent fractions but the teacher was able to steer the conversation from method to method to discuss different ways to represent the same idea. This teacher seemed to demonstrate the approach of getting as many students involved in talking about the math as possible through the use of many different solution ideas and the "pair talk" which I'm assuming was where students talked in pairs about a certain math idea so everyone had a turn to express their thoughts.

The big idea discussed in chapter 4- the whole matters- seems so obvious that it feels odd to me that this was never brought up when I have taken prior math classes ever. At some point of course I realized it, but it was never consciously in the front of my mind. Yet in Carol's class the students are free to explore this idea which really will help ground fractions with different denominators in their minds. Fractions are always in relation to one another. Again, in this case I thought it was so critical that the teacher had the students present so many different examples of the problems and were allowed to explore the idea in depth without her just giving an explanation that might not have made sense to them. This seems to be an integral part of having content make sense to students and to allow them to develop the process skills necessary to really "do" math.

Content & Processes in YMAT Ch. 3 & 4

Upon completing Chapters 2 and 3 of Young Mathematicians at Work, I can identify several content and process goals in place. First and foremost, like Carols classroom, Susan was focusing specifically on the content of developing fractions and reducing them to their lowest form. While the word problem she presented her students with did not necessarily constitute a context problem and provide opportunities for the students to engage in a plethora of strategies, they were still able to solve the problem using a few different tools. (After reading the chapter, I now know that the use of a variety of tools does not necessarily suggest multiple processes implemented. In the case of this classroom, tools vs. strategies was what ensued as a result of the provided problem) While some students chose to implement paper and pencil techniques, other were able to perform the necessary computations in their head, while still other implemented unifix cubes. However, this primarily represents a use of diverse tools rather than processes because all students approached the problem from the same angle knowing that it indeed required a fraction of 18/24 to be reduced. Perhaps had Susan framed the problem more similarly to Joel, who provided a context problem regarding the best buy of cat food, the students would have had a better opportunity to approach it from various angles. Looking at the problem that Joel presented to his class, we can certainly see that not only did his students use different tools, but used different processes and strategies in order to obtain the solution. Because he presented them with a context problem, one that had meaning to the students without an implied agenda, they were able to truly explore all of the options for solving. While some students began dividing 15 by 12 and 20 by 23, others began reducing the number of cans and price in half until they reached the number 1. Some students solved for 20 cans, figuring out how costly these would be at each store, while still others solved for one can in order to determine the difference and decide which store was the better buy. In addition, this problem lent itself to rich inquiry and truly engaged the students in the task at hand. Those these two teachers had the same general content goals in mind, the processes that each problem provide were drastically different. While Susan's word problem was primarily teacher-driven with more of a linear model of problem solving, Joel's was student driven and allow the students to explore a variety of feasible processes.

Young Mathematicians at Work

Upon reading the assigned chapters of Young Mathematicians at Work I have made a few discoveries. First, I realized that the ways that students' problem solve may be drastically different from the approach we may personally choose to implement. For instance, when presented with the "submarine sandwich" question in class, my immediate instinct was to whip out my calculator and begin computing division problems in order to obtain percentages. The student's within Carol's 4th/5th grade class, however, used a plethora of strategies such as manipulatives (unifix cubes) and visual representations (drawings, charts, etc.). Because these strategies are not ones that I, myself, was naturally inclined to implement, it was extremely difficult for me to make sense of their thought processes. I struggled while reading, and re-reading, the passages describing the various steps the children took in their solution strategies. As teachers, it is so important that we wrestle with mathematical problems ourselves and find multiple ways of solving it before introducing it to our students. In doing this, we are able to discover a variety approaches that can be taken to problem solve and anticipate responses and methods that students may offer. In addition, I also learned that though this story problem is directly related to fractions, decimals, and percentages, it is not necessarily appropriate or warranted to force these equations and ideas upon students. As Fosnot states, "Asking children to adopt multiplication and division shortcuts too soon may actually impede genuine learning. When introduced at the wrong place or time, good logic may be the worst enemy of good teaching" (5). This serves to say that though developing these concrete ideals is the focus and goal of the lesson, it is best to let children discover these processes on their own through the formation of connections and constant exploration. If this knowledge does not arise naturally, or out of genuine curiosity, then it may not be thoroughly understood or retained. Finally, the text stressed the importance of developing mathematical equations that are both relevant and meaningful to the student's lives. By introducing the "submarine sandwich" problem, the teacher was able to present a real-life dilemma that she encountered with a previous class and sincerely enlisted the help of her students in solving it. Fosnot states, "When children are given trivial word problems, they often just ask themselves what operation is called for; the context becomes irrelevant as they manipulate numbers, applying what they know" (2). In essence, through the development of problems that students can relate to, they will be more willing to invest the time and energy that it takes to truly explore the possible solutions rather than robotically computing meaningless information.

Division

Since we talked in-depth about the sub problem last week I wanted to start off with that. I was amazed at how frustrated I was while working on that problem. It's no wonder so many students do not understand fractions or spend a lot of time trying to understand how they work. I'm still at the point like many children where it is much easier for me to think about lots of fractions while drawing pictures. In the fifth grade classroom that I tutor in last Friday they were working with fractions by having fraction bars with each fraction laid out. I thought these were helpful because I was working with some of the students to discover what equivalent fractions were. We worked on taking things like the 1/3 bar and seeing what other bars would line up exactly equal with it, like 2 1/6 bars, or 1/2 and two 1/4 and four 1/8. Students could also explore how as the denominator got larger the size of the bar got smaller.

While I was reading the DMI chapter on division I was trying to pay attention to the different strategies that were being used. The problem like case 24 when the teacher was trying to get students to divide 134 jelly beans among 6 children was just like the division problems I was working on with a student in the fourth grade classroom that I tutor in. I was really amazed at all the different strategies that April came up with to try to solve the problem. Since it took her so many steps I was left thinking the same thing the teacher was at the end, when should we teach students the standard division algorithm? Or should we at all? Is it better for students to struggle and reason through it and then put it into an algorithm or is the algorithm even necessary? I wish I had read through this before working with the 4th grade student in my class on this subject. The teacher had only provided students with these flat counter chips that were rather difficult to work with because it was hard to create piles of them to keep numbers straight. I first tried to have the student model the division out but as the numbers quickly got a lot larger I realized there must be an easier way. The student seemed to sort of grasp dividing by separating into different piles but I'm not sure he completely grasped what he was doing at this most basic step. I think I was leading him too much and not really allowing him to explore the problem on his own. In retrospect I can see this, at the time I was just celebrating that he managed to sit in his desk and work on something for 30 minutes, which was a very large step for this student. I tried to work with him on trying to estimate for some of the larger numbers like April was doing but without a basic knowledge of his multiplication tables this quickly proved to be an impossible task. I worry about solidifying his division knowledge without a solid understanding of multiplication. In Betty's case she was using base 10 blocks with her students. Maybe that might have made this task somewhat less confused because a major part of each problem was sorting out the counters and then recounting and he was often one counter short which was very confusing when trying to arrive at a correct answer. I understood Betty when she was talking about how it was difficult to see the division when she counted out 10 7 sticks and 2 cubes to illustrate 72 divided by 3. I think just looking at that without seeing each one as a single I would not immediately think of my first step for division. I hope after our discussion in class about this issue I will have some more concrete ideas to take back to my 4th grade student next time I tutor him.

Young Mathematicians-- Chapter 1 and 2

I found the first chapter of “Young Mathematicians at Work” to be very interesting. After attempting the sub sandwich problem myself, I was eager to read about how some children would solve it. I found it comforting that almost all of the children drew pictures in order to help aid in their thinking since this was the way I went about trying to solve the same problem. I feel that looking at mathematics as a creative process can bring in different learners as help math appeal to more children. After doing this same problem though, I was able to understand how solving real life problems such as this can help. Like the authors point out, the children know that having all of the subs cut up into 1/8 pieces is unrealistic because the pieces would be too small. Being able to relate problems such as this to real life knowledge can help children think through a problem. I also liked the way that the authors explained how children think and learn. I liked the idea of children learning through the three different landmarks, strategies as schemes, big ideas as structures and models as tools for thought. I realize that the authors believe that each child must move through and understand each of these ideas. I find myself wondering though what will happen if a child struggles with one of the frameworks. Would you as a teacher just simply focus more on that particular framework or could you just focus more on what the child understands and the rest will eventually follow?
I did also like that the second chapter in the same book brought up the idea of community. I have always been an advocate for strong communities in a classroom because I have personally found that I have learned best in a classroom where I am comfortable. I do like that the authors point out that mathematics must be connected and involved in the idea of community. Community is not necessarily just about feelings and rules but it also creates a safe space for discussions. When discussions are had about mathematics, children can learn more from not only each other but from themselves as they explain their thinking out loud.

Groupwork

While we often read about the advantages of groupwork, it is always a helpful reminded that just places students in close proximity to each other does not always promote healthy groupwork. There is a difference between groupwork and cooperative learning. "The Dilemma of Groupwork" brought up some interesting ideas that I had not spent much time reading about before. The first time Mrs. Todd put her groups together without any prior planning I wasn't entirely surprised that it didn't work out, but I was surprised at all the different dynamics in a group that group members probably are not even consciously aware of. After reading the article it left me wondering how to solve the issue though. Cohen wrote that even when teachers tried to place students of similar abilities together they were still able to fine tune who they thought were the higher level students among themselves. If this is the case, what can teachers do to get all students involved? One thing that comes to my mind is giving each student a specific task within the group. This ensures that they have to get involved, especially if the tasks are integral to the groupwork. But about if students are just gathering to talk about an idea? Can you always structure groupwork so each student has a specific task? I wondered about the application of groupwork in math. In my field my CT always has the students in pairs. They are arranged so they have the same partners for a month or two at a time and students are paired with uneven levels of ability. Is this a good method of bypassing the internal ranking of groupwork? I wondered how it would apply to the work being done in the students in the DMI text this week. When Eleanor gave her lesson to the students it seems like they worked by themselves to come up with different methods of solving the problem. Would this many different methods have emerged if students had worked in groups? Would some of the students voices been ignored, thus losing valuable contributions? But perhaps even more refined ideas would have emerged. This is something that has really left me wondering. In Lauren's case she had a handful of students that were very confused about keeping place value in their head. Lauren was working with them but if they were all in a group together would they have stayed confused because they all had similar abilities? Maybe in a mixed ability group they could have solved the problem. Much of math has to do with making sense of how numbers work. It seems like groupwork would be ideal for this. I think group size is very important. For small problems like this I would think more than 3 would be too many. This limits the group dynamic issues that were brought up in Cohen. It is hard for one persons ideas to be ignored if there is only one other person in the group. Many students might think of math as a subject they don't enjoy because they see it as a solitary subject. If teachers really think of ways to integrate cooperative learning in a way that does not disadvantage some group members, all students could benefit from the activity.

Groupwork

After reading the article, The Dilemma of Groupwork, I could not help but find myself relating the information conveyed to my own, personal experiences within academic settings. The article construed the various problems coinciding with the act of group work within a classroom setting. These problems correlate with an array of statuses that are assigned to children inclusive of academic statuses, peer statuses, and societal statuses. When describing the concept of an academic status, the book states, "In the classroom it is impossible to compose groups where all members have equal status. Students generally have an idea of the relative competence of each of their classmates in important subjects like reading and math acquired from listening to their classmates perform, from hearing the teacher's evaluation of that performance, and from finding out each other's marks and grades" (28). In essence, when working in groups the children whom are highly respected academically and perform well on a plethora of given tasks are usually in a hierarchical position in which their opinions are more respected and thoughts and ideas taken to be truth. On the contrary, students who are known by their peers as being incompetent and unable to complete tasks with accuracy have few opportunities to offer their ideas and, when those thoughts are conveyed, are rarely taken seriously. Peer status is one that arises due to social hierarchies based on level of attractiveness, popularity, or athletic ability. Students seen as having a high peer status generally tend to dominate the conversations existing within group work while students possessing a low peer status have little opportunity to speak. Lastly, societal status's arise as a direct result of what culture is more valued within the community, school district, etc. For example, in the hypothetical situation depicted of Ms. Todd's class, there were only three African American students within her classroom and two of the three were non-participatory during group discussion and failed to convey their thoughts and ideas. This suggests that because there are so few individuals deriving from this culture (within this particular classroom) they maintain a low societal status. Overall, it is extremely important to consider and recognize these existing status's within a classroom. As Cohen states, "Those who do not participate because they are of low status will learn less than they might have if they had interacted more. In addition, those who are of high status will have more access to the interaction and will therefore learn more. It is a case of the rich getting richer" (36). He goes on to say, "If status characteristics are allowed to operate unchecked, the interaction of the children will only reinforce the prejudices they entered school with" (37). Because these outcomes are both extremely problematic and not only hinder children's academic learning but also socio-political mindset, it is increasingly important to make yourself away of these existing hierarchies and intervene as much as possible. In reflecting on my own experiences within the school system, I have routinely been one of the students who sits quietly during discussion and listens to the opinions of the rest of the group. This is not a result of my own incompetence nor of my inability to think of something constructive to say, rather just my own feelings of inferiority and shyness within a group. I now understand that in doing this I am essentially encouraging other individuals to view me as uneducated and assume I have nothing worthy to offer them within a group situation. This is certainly not how I wish to be interpreted by my peers and it is disheartening to think that, through my own actions, I have essentially perpetuated and encouraged this thinking. Because of the negative feelings that I have developed as a result of other's opinions of me, I know how important it is to intervene when these situations are occurring. In my own classroom, while I do intend to promote group work, I now have the insight to do so with a critical eye.

Dilemmas of Groupwork

I found the article “The Dilemma of Groupwork” by Cohen extremely interesting since I am a huge fan of using group work in my problems. I was well aware that group work could backfire considering students working with their friends were likely to fool around. I believed that if you assign groups made up of students with high ability levels and low ability levels, the high ability levels would be able to teach and help out those with low ability levels. After reading this article, I know that this is not necessarily all true and good. Like I had originally thought, there are students who are classified into different levels. I was not aware, though, that there was more to this classification. I found it interesting that they too had classified children with an academic status, but also an expert status. I liked that Cohen also went deeper to peer and societal status. I’ve seen multiple times in our classroom these classifications in play. During times of free time, the children automatically form their own groups. In these groups there is always a student who takes charge in the task at hand and one or two that sit back and usually observe. For example, there was a group of four students playing with blocks one day. One student who would not necessarily be classified as a high achieving student was taking the initiative and taking charge of the group. Although this child is not the one that the rest would normally see as being an expert or “high status member” in terms of academic level, this particular child is very popular with his peers because of his humorous personality. This shows that leaders are found all throughout school time and not only in assigned group work.
After reading this article, I found myself discouraged from wanting to use group work. I found myself analyzing and thinking about more examples from the classroom and my own experiences. The more I thought about it, the more I realized that Cohen had come to true conclusions. I have been involved in group work where these exact characters come to life. I have also watched group work where I have seen leaders, children being silly, children sitting back and not getting involved and students to shy or embarrassed to chime in. I have also been involved in group work where everyone truly worked together though. The more I thought about it the more I found similarities in all these instances. One was that there were no more than two or three involved. When you work in partners, it is necessary for both people to be involved. The other instances were when the activities were fun and all students were required to be involved. For example, last semester in my TE class, we did a lot of dramatic play in groups. In these instances everyone had to play a part so it was imperative that everyone became involved. The activities were also very fun and silly at times so everyone, even those who were naturally shy, enjoyed themselves and were more willing to put in effort.

A little late... but special needs students

I read the article called "Why Students with Special Needs Have Difficulty Learning Mathematics and What Teachers Can Do to Help" by David Allsopp. This article considered special need students to have one or more of four struggles. The first one was attention problems. These students have trouble focusing on one thing because they are constantly focusing on EVERYTHING making it difficult for them to catch the important details that the teacher may be saying.The second problems children sometimes face are cognitive processing problems. These children have trouble processing from what they see visually to what they are writing on paper. Metacognitive problems can also hinder children's mathematics learning. This means that the students me be unaware of other possible ways of learning. The fourth struggle is one that is extremely hindering in mathematics. Memory problems put students at a disadvantage because they have difficulty retrieving the information that the brain has successfully stored. In mathematics, children have a disadvantage because they require memory. For instance, in long division, it is necessary for the child to know addition, multiplication, subtraction and it takes memory to remember all the correct steps in the correct order.
In order to help these children, there are a number of ways we can plan and teach lessons. Allsopp explains some strategies to help with instruction with special need students. First, teach in authentic and meaningful contexts. It is helpful to directly model both general problem-solving strategies and specific learning strategies using multisensory techniques. This allows the teacher to cater to the different learners and show the children other ways to learn. Another strategy is to ensure that the sequence of instruction moves form the concrete, to the representational and then to the abstract. Teachers will find it helpful to allow the students opportunities to use their language to describe their mathematical understandings. It is also important to provide multiple practice opportunities to help students use their developing mathematical knowledge and build proficiency. Studies have also shown that using nemonic devices because they help retrieve problem solving steps from memory both independently and efficiently. For example, for the order of operations, we continually hear "please excuse my dear aunt sally". These "name games" are helpful for students. Most importantly, and I believe this is true for all students, is that we must continually monitor students performances and offer meaningful feedback.

Gifted Students ...

I was interested in spending some time reading about working with gifted students because in my experience, it seems to be an area that slips through the cracks more than ever. Chval and Davis raised the point that with an increased emphasis on passing standardized tests and making AYP teachers feel more pressure to raise up their lower students while not putting as much effort into challenging their gifted students. While I feel my CT works hard to create differentiated assignments, I still see quite a few behavioral problems coming from one student in my class that is probably the most gifted in math and quite a few other subjects that is stemming from boredom. In the same article, I felt very bad for Craig because I know I have seen many of those scenes repeated in many different classrooms.

The Wilkin's piece talks extensively about the Mathematics Investigation Center. I thought a lot of the guidelines they put down for using this piece would make it very useful in the classroom. The MIC is a set of about nine activities modeled around different areas of math that are kept in the classroom for students that need more challenging work to do. Some of the problems are easier so most of the class can work on them but some are designed to be less accessible to lower students in the class. I liked the idea that students were not asked to work at the MIC outside of math time so they did not feel like they were being forced to do extra work. The activities in the MIC are supposed to be designed to be integrated into the math unit the class is working on at the time so gifted students aren't just working on drop-in lessons. This seemed like a great solution to some of the problems that students complained about in the Chval and Davis piece where they did not like that they finished far before other students in the class and had nothing to do, or that the teacher did not want them working ahead of other students in the class. This instruction method is a great way to keep everyone on the same topic, but different students can work on the problems at different levels.

Chval and Davis bring up a similar idea when they discuss differentiated tasks. These tasks usually have several different "levels" that students can work on according to ability. Students that considered high level can work on more advanced parts of the problem but all students should be able to access the problem at some level. This is another good way to challenge gifted students without making them feel like they have been singled out to do more work.

I found some relevance within the "Behavioral" piece as well. Some teachers might make the assumption that since a student misbehaves and does not do his work that means he is not smart. When Carter was given the chance to do "real world problems," something that the gifted students interviewed in Chval requested, he responded well and really began to show interest in math class. While home issues that were out of his control disrupted his math learning, it shows how much effect trying to really get students to problem solve with different problems can have on students in the classroom. The cranium crackers got Carter more interested in staying engaged in class and gifted students need the same effort from teachers to keep them engaged when they already know the content their classmates are working on.

At-Risk Students

After doing this week's readings and, more specifically, completing my chosen article titled "Problem Solving and At-Risk Students: Making Mathematics for All a Classroom Reality", I found myself left with more questions than answers. This article was a personal narrative written by a fifth grade mathematics instructor who found herself teaching in an impoverished rural elementary school that performed significantly below both grade level content expectations and those expectations formerly met by students in her previous teaching position within a prestigious suburban school. On the first day of class, she implemented an activity that had been loved and admired by her former students. However, within the context of her new classroom, this task was deemed as much too difficult and the students became easily frustrated, angry, and humiliated for being proven incompetent. While I understand that this teacher was simply trying to confront her students with an exciting, challenging, and competitive new problem solving task, I cannot help but question her logic and reasons for assuming this activity would be appropriate. Did she not review the student's prior academic records to gain a feel for what the students are capable of doing? Does she not have access to school wide standardized test results? Had she not conversed with the children's prior mathematics instructor to obtain an understanding of the curriculum that they had been exposed to? While all of these sources of information inevitably must be taken with a grain of salt, they can still provide guidance and significant understanding regarding the children's current capabilities. I feel that had the teacher invested the time and energy to perform this research, this entire situation could have been prevented and the students' pride kept in tact. I do, however, think that this teacher took appropriate and helpful measures in order to remedy the situation and establish a classroom of "problem solvers". The teacher quoted a text titled Teaching for Thinking as saying, "If we are to think, we must dare to think. Daring implies confidence in ourselves and in our abilities. When we have confidence, we often succeed in doing tasks far beyond our expectations. When confidence is missing, we fail at tasks that seem well within our grasp. Confidence grows largely as a result of experience" (292). Through rereading this passage, the teacher was able to realize that in order for the students to perform academically, they first needed to feel confident in their ability to do so. The way that she decided to build this confidence was to reevaluate her original lesson plans and break them down into more simplistic tasks. In essence, she decided that it would be most effective to take baby steps in order to gradually build the children's self confidence vs. bombarding them with one large task. I think that these strategies are quite profound and would be especially useful within special education classrooms. So often, students labeled "special ed." begin to associate negative connotations to this term and perceive themselves as incapable of learning or "stupid". When working with these students in the future, it will be so important that I counteract these common stereotypes and build their self esteem so that they are both motivated to complete tasks and confident in their ability to do so. Overall, the most important piece of information that I took away from this article was that " If the activity does not work, if the papers get crumpled, if arguments begin or materials get abused, I may give up on the activity, but I never give up on my students" (295).

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.

CGI Chapters 4-6

I have really been enjoying the CGI text just because of all of the examples it gives of the many ways that students approach various math problems. Chapter 4 is a continuation of the principles established in the previous chapters but it applies to multiplication and division instead. I remember when I first came across partitive and measurement division I was an adult and I had difficulty figuring out what the difference between the two of them were and why it was even important because with my background the math in them seemed to be the same. Since this is my second experience learning about the differences I can now grasp that partitive and measurement division are uniquely different for children learning about math and that needs to be kept in mind. If you try to solve the problem by using counters or drawing pictures the difference becomes immediately obvious. I also liked the idea that the text raised about introducing simple multiplication and division problems in as low as K-1 so that students get used to the idea of using math in this way.

I also also surprised when the book addressed the common misconception that students should have a good grasp of base-ten in order to be able to work with two and three digit numbers. They raised the point that as long as students can count they can solve problems with two digit numbers. Of course, this is sped up immensely once students realize they can count in groups of 5 or 10, etc. Many of the students in my second grade class are in the process of refining their knowledge of the base ten system. They spend a lot of time with a chart they have that lists ones, tens, and hundreds and they have cubes they use to model problems that are two digit addition and subtraction. Most of the students can make blocks of 10 cubes and use them that way while other students still prefer to count out each block.

I also was interested in reading about the various strategies that students use when working with multi-digit numbers such as incrementing, combining tens and ones, and compensating. These are all still strategies that I use in mental math but I've grown so used to it that I don't spend much time thinking about what exactly goes into solving a math problem anymore. When working with students to have an awareness of all the different ways students can do something is helpful so you can help a student find the strategy that works the best for them.

CGI Ch. 1-3

I think of the initial ideas that I took away from these chapters was the statement that children learn in their own way that is uniquely different from adults. One way this was demonstrated was by giving examples of three different ways to look at the same math problem that was set up differently. Since I have had a lot of experience with math I see all of the problems as essentially the same whether I am adding 3 plus 5 or 5 plus 3. Every child can approach the math problem differently and teachers must be sensitive to that. In chapter 3 when the book detailed some of the various strategies that students use to solve problems I was a little bit surprised that the book said that children pick up on most of these strategies naturally and without being taught them. In my placement class my students are working on regrouping and a lot of them use the direct modeling strategy. They have a ones/tens/hundreds chart that they use with cubes and they will count each cube to come up with the answer. I remember in MTH 201 learning about partitive and measurement division (I think those were the correct terms) so some of that was coming back to me while I was reading about the difference that students perceive when they are splitting things into groups or when they have groups and they are trying to find the whole. There is so much more that goes into selecting appropriate math problems for the students in your class that it appears at first. In the video that we watched in class last week the teacher purposely picked problems where it was times 5 each time to see if the students noticed the pattern. As teachers it is important that we think about the different ways that students can approach a math problem when we create it so that it is easier to anticipate how they might go about solving it. The idea of having discussions in math seems so reasonable I'm amazed that I was never exposed to it really at all K-12. I hope to spend more time learning about how to implement discussions into my math classroom to further enrich my students learning.

CGI Chapters 1-3 Ideas

After reading the first three chapters of "Children's Mathematics" I already know that I will like this book. It's so interesting to watch a child count on their fingers for a particular problem and then go back and understand why this was helpful to them. Throughout the three chapters, I found multiple ideas and strategies that stood out but I found two of them most helpful and interesting. The idea of direct multiple modeling strategies seemed to be most helpful for younger and less advanced students. Since I am placed in a kindergarten class at this time, I was able to relate to these strategies better. Although the student's rarely do addition or subtraction, they are currently working on counting. I can see using physical objects being the best way to instruct them with join problems. The other theory I found really interesting was the whole idea of the join, separate, part-part-whole and compare problems. I never really realized there were so many ways to write simple math and addition problems. It is helpful to know these ways though so we can have variation in our work. It also helps us as teachers know that our students truly understand the multiple ways of performing addition and subtraction problems. Through the three chapters I only really came across one question. On page 22, we see two students performing the "counting down to" strategy. Both of the children were sure to either not count the first one or not count the last one to reach the correct answer. It had me wondering if this was a practice taught to the students or whether it was something the realized through trial and error. If it was taught, how would you teach something like that?

CGI Ch.1-3 Reflection

After completing this weeks selected readings from Cognitively Guided Instruction I found many concepts to be both helpful and extremely interesting. One of my greatest fears as a future educator has been my own confidence and knowledge of strategies to teach to children to utilize when solving mathematical equations. According to Ch. 2 of our text, however, it states, "All of the strategies we have described come naturally to young children. Children do not have to be taught that a particular strategy goes with a particular type of problem. With opportunity and encouragement, children construct for themselves strategies that model the action or relationship in a problem." While I questioned whether or not this could really be true, I reflected back upon experiences I have had within my own classroom. While I have never explicitly seen my CT teach the children any one strategy to use when solving mathematical equations, there still seems to be a wide array of techniques put to use by the children. For example, on one particular day there was a story problem written on the overhead projector that read " Jill has eight library books. She returns 5 of them to the library. Now how many library books does Jill have?" As I walked around the room to observe how each child chose to solve the problem, I was amazed at the wide array strategies that were being used. While some children obtained physical objects, or manipulatives, from the shelves, others were drawing pictures, writing equations, counting on their fingers, etc. As our textbook suggests, this is because children are intuitive and hard wired to develop a plethora of problem solving strategies independently. I find this to be both interesting and exciting because it enables children to become invaluable resources for each other.

I was also very interested in reading about the variety of ways to word a story problem in order to truly capture a child's cognitive ability. While we discussed in class how important the wording of a story problem is and the various benefits that differentiated wording ensues, I couldn't help but wonder HOW to do this. The reading, as well as the charts and graphs provided within the text, discussed in detail each specific type of problem and what that problem was testing. On page 12, figure 2.2. we are given a chart inclusive of eleven diverse ways to present the same exact story problem! Personally, I find this to be an excellent point of reference for me to continuously refer to in the future and fully intend on taking advantage of this information.

Lastly, though I do not have any questions regarding the reading, I do have a few questions regarding our class discussion about standards presented in Thursdays class. One of my classmates brought up the issue of private schools and wondered what standards they were required to adhere to. Having gone to private catholic school from k-12th grade myself, this was a topic I have also been curious about. Do national and state standards only apply to those schools which are governmentally funded? And if that is the case, then who creates and enforces standards within private schools? Also, there was never a special education program present within my schooling but it seems that nearly all public schools are inclusive of this. Is this also something that is mandated by the state for public schools? These are just a few questions I have had for some time now and our discussion on Thursday seemed like a perfect segue for me to present these ideas and gain some insight!

My Learning Goals

My learning goals for this class are fairly simple but perfect for me. I have never been very good at math which causes me to be nervous about teaching it. In this class, I would like to be given the knowledge and tools to do well and teach math to my students. I want to learn how to be a GOOD math teacher. I know it may seem as though this is a given, but in my experience I never really had math teachers that helped me understand math. I want to know how to help my students really understand math rather than just memorizing how to do it. I want to be able to know when they need help and how to help them. I also would like to be knowledgeable across all grade levels. In the past, our TE classes seem to be very specific to younger or older grades. Since I am not that good at math, I would like to be prepared no matter what grade I end up teaching. Along with math, I would like to continue my education on teaching in general. I always enjoy being exposed to fun activities or new ways to explain something to children. I believe a lot of this can happen in the field placement, but I would like to enhance that exposure in TE 402. I look forward to working with the other students in order to become a well educated math teacher.

Learning Goals

I'm coming into this course with high hopes for myself and the advances that I hope to make in my own teaching of math. While I felt that I was always proficient in math in elementary school, it was never a subject that really got me excited. I hope that is something that I can change for my students in the future. In this course I want to explore math in ways that I have not done before. In my placement my CT integrates partner and group work into math on a regular basis. As a student of math, I always viewed it as a solitary activity, which perhaps contributed to less enjoyment on my part. Integrating math into all subjects is important to me because I never want my students to have to ask, "How are we ever going to use this?" I definately want to spend some time focusing on how to reach students with varying levels of interest/ability in math. Even though I was a good math student in elementary school my lack of interest in math correlated to poorer math grades by high school. I have to admit I do occasionally worry about teaching math, not because I'm intimidated by the material itself, but that I will inadvertently resort to some of the same techniques used by teachers that did not peak my interest in the past. We briefly discussed the ineffectiveness of worksheets that asked students to repeat the same problems over and over in class last week and that just reminded me of so many experiences that I have had in math classes in the past. Above all, a goal that I really have for myself for this semester is to more deeply explore ways to captivate student's interest in math. I want to do this by reflecting on the types of math assessments that I can use to really determine a student's learning and how to tailor my teaching to reach the desired effect.

Learning Goals

My primary learning goal is to develop a sense of excitement and enthusiasm with regards to the subject of mathematics and try to lessen and/or eliminate the negative feelings that I currently have associated with this topic. For this evolution to occur, I think I need to begin by findings ways to motivate and encourage myself to become a logical thinker and learner. Though this is not my area of expertise, it will be important for me to constantly establish positives to this learning approach and consider the multitude of ways that this learning style will be helpful to me in the future. In order to change my thinking, I need to change my approach. I hope to discover some of these new approaches and strategies that I can utilize when performing mathematics and incorporate those strategies into the lesson plan that I will be teaching to my first grade students within my field placement. For example, when we were asked in class to comprise a list of the ways in which the number 16 can be reached, I was astounded at the variety of approaches and strategies that could be used in completing this problem. Not only were numbers incorporated, but graphs, charts, and drawings as well. These are the types of strategies that would have made my childhood experiences with mathematics more positive. I hope to continue to learn new approaches to mathematical equations and, as a result, be able to transform my attitude towards math in the process.