New Insights and Their Implications

Throughout the semester I have learned a lot of new things that will help me be a better math teacher. There are many different semi concrete models that will help me teach my students complicated subjects like dividing fractions. I have also learned that it is important for students to understand WHY they have to do something (like flip the second fractions when dividing). A better understanding of concepts like these will help students gain confidence in their math abilities.

New Insights and their Implications

I have really began to see how this type of teaching can be very beneficial to students. I really wish that I was taught this way when I was in elementary and high school. I believe that if this were the case then, all of the "rules" that I learned just because might have been learned for a reason and a truer understanding. I have also seen that this might be difficult to teach this way. We have see a couple of groups teach in class now and it seems a bit more difficult. But it may seem that simply because we have taught that way. I am hoping that is the case.

New Insights and Thier Implications

From class, I have learned how much different it is to truly learn math. I feel that I have missed out on a lot of much needed information and as a result I don’t have a concrete understanding of reasoning’s in mathematics. As a result of only knowing how a procedure is done and not ‘why’, it is making this class a little more difficult to grasp these understanding. While it is stressful trying to understand the ‘why’ of the question, when I do get to that point in the understanding I feel as though I am in fact learning it for the first time and the right way. I have learned from my peers, that everyone does learn in a different way and it is critical as a teacher to be aware of that. Also, I have learned how useful the right manipulative is while teaching a lesson. Manipulative are able to help further the knowledge by allowing students to visually see and touch the objects. Through these interactions, students are able to learn math in a much greater, better way.

New Insights and Their Implications

In class I have learned that is best to give students problems and let them struggle and manipulate them to solve them. The students may work with partners or on their own to figure out the problems, and give them an adequate amount of time to work on the problems. I am doing my internship in Newcastle, NE at the junior high level. Wednesday I taught my first lesson to 8th grade algebra, and I taught them how to graph linear inequalities. First I explained what they were and how they were similar to graphing linear equalities, which they did the day before. I then showed a few examples on the board while explaining the material and getting their input from previous knowledge. They did great as a class on the board, but when I gave them problems to manipulate at their desk with graph paper I got a lot of blank looks even working with partners more than half of the class was lose. I had one of the groups of girls come up to the board to share their answer (which was correct) and explain to the class how they got it. The boys were still confused. I went over the steps and process again and they seemed to understand or were just pretending. I applied all the knowledge I have learned about the learning process and still feel that my lesson was unsuccessful to fifty percent of the class.

Summary and Synthesis

Since my last posting it has really hit home for me that it takes a commitment from the student to be willing to work through the process in order to truly construct their knowledge. This became very apparent to me during the work with fractions that the first group presented to the class. I knew that I could find the correct answer if I resorted to my existing knowledge of the steps needed to resolve the problem. I also knew that if I wanted to experience the learning the way that a student who wasn't taught the procredures but instead was provided with a conceptual way of learning that I would have to struggle with the math. Trying to stay away from the procedural steps and work with the concrete and semi concrete models in order to find the reason for the procedural steps I went back through the models and tried to evaluate what I was learning about the problem. Due to spending the time struggling I did come to an understanding of moving from the semi concrete model into the symbolic problem. Experiencing this process instead of reverting to the methods that I knew would give me the right answer was definitely beneficial to me. The willingness to be uncomfortable during the process is something that as a teacher I will be asking my students to do; I think that I need to be willing to experience it also. This really brought the message of constructing our knowledge about math to the forefront for me.

Summary and Synthesis

When we came to class the first day, I was honestly scared about this class. I was to figure out how to find the answers without being told how to find them, was a scary concept after being told what to do to find the answer for so many years. I have actually enjoyed class a lot, I like being challenged, and being able to find those answers on my own or within the group work we do. I like that we are learning many different approaches to the same problem or question given to us. I am excited to be a part of this new way of teaching, because I think it is the right direction in helping students understand what they are doing. I have already shared some of the concepts with my oldest daughter, and cannot wait to teach my future students in a way that is easier to understand and realize why they are doing what they are doing.

Personal Concerns and Next Steps

I have always enjoyed doing math and taking math classes. I think the reason I liked math was because the problems had a set of steps and as long as you followed those steps in the order you were told you would get the correct answer. I have always been good at following steps and therefore received good grades in math. Until this class, I never thought about the reasons behind doing the steps and having to follow them in certain order, I always just did them. I memorized the rules and followed them. After taking this class, I am now concerned about all the things I really didn’t know about math all along, the “why.” Why do the steps result in the correct answer or why does that rule work? For so many years, I have done math the traditional or instruction way without really knowing why. I know habits are hard to break and I want to make sure my future students know the reasons behind why math works and be able to construct their own rules. I want math to be relational for them not instructional like it was for me. In class, it has been hard trying to reconstruct what I already know about math and not resort to just using the formulas to find the answers. So far I have been able to follow along pretty well and after taking a look at the processes of why the steps and rules work and multiple entry points of problems I now have a deeper understanding of math. I am just afraid I won’t be able find a way to make all math relational to my students. I now know how to make the things we learned in class relational but what about math topics we didn’t cover. Will I be able to teach in a relational way or will I resort back to what I already know and teach the traditional way. How will I know what to do or how to teach? This class has really opened my eyes into a math world I didn’t know existed and now it is up to me to find the confidence to re-teach myself and find out more of the “why” behind the rules.

Summary and Synthesis

Well, I have learned things the hard way and sometimes that is the best way to learn them but sometimes it shakes you right down to the bone. I really have learned a lot about Math in a more relational way which is how Math should be taught. Kids will learn better especially when they relate and make connections to the material. It is hard to go from a way of learning you learned throughout school and just all of a sudden switch. That is why it is SO important for children to have this type of relational learning from the beginning. This class has given me a new insight into Math (one that I honestly didn't even look at because of the way Math was taught to me). Even though the traditional route is the most common form of teaching; it doesn't make it the most influential! I am still scared that my fears about Math will cause problems in my future class but it is a challenge I am trying to face and conquer (somtimes it is not as nearly successful as I want it to be). Finally, I think it is important for children to know the HOW and the WHY when learning Math. I honestly don't remeber wondering why we invert the second fraction when dividing; I think that is because of how long I was taught through an instrumental way. Tons of children ask this question though and what do teachers say; honestly? After this class, I do have a better understanding of why to do certain steps.

New Insights and Their Implications

So far, this course has given me some great insights as to the new way of teaching math. I have learned that we need to work towards giving our students more of a relational understanding instead of an instrumental understanding. It is imperative to make connections between what they are doing and why they are doing it. From the discussions from my peers and from Dr. Reins, as well as the research I have done through the LPU, I have learned that students learn math best when concrete experiences are worked into symbolic experiences. This can first be done by first working on concrete examples through manipulatives and visuals. Then once the students obtain and understanding of the concept, they can move into semi-concrete examples by still working with manipulatives but by adding number sentences, etc. Then finally, once the students have grasped the concept, they can learn the symbolic meanings of what they have already learned. By doing this step by step, the students can gather their own conclusions and meanings on that topic and develop their own understanding--rather than it just being given to them. All of this is going to be extremely helpful to when I teach in the future. Since I am used to more of a instrumental way of understanding, it is important for me to switch my ways and develop my own teaching into a relational way of understanding. By doing this, my students will become successful learners in mathematics.

Personal Concerns and Next Steps

Upon entering this class, I really enjoyed Math. It has always been one of my favorite subjects and I felt as though I understood the basic concepts. For my internship, I helped in a 7th grade math classroom and I loved it. After that experience I had positive intentions of teaching Math in my future. Before taking this class, I was excited to learn how to break down math and learn more about teaching it to younger kids. After participating in this class, it has really made me look at math a whole different way. Some good, some bad. It makes so much more sense to teach kids on a relational basis rather than an instrumental basis. I came into this class strictly knowing only an instrumental basis of math. To me, the instrumental understanding makes sense because that is how I have always learned it. It also makes much more sense to teach kids a relational understanding of math. I am definitely more concerned now about teaching Math in my future because even I struggle with understanding a lot of what we are currently doing in class. If I have a hard time understanding it then I am worried about teaching it to my students. My concerns have been lessened a bit by understanding the process of teaching through concrete, semi-concret, and symbolic ways. I now understand better how to teach in those ways and how beneficial they are to students. I still struggle with it, but it is beginning to make sense. My next steps include trying to work harder in our math class to understand your styles of teaching and understanding the "why" part of mathematics. I think it is important for us to understand why we do stuff so I am going to continue working hard to understand that. Since this class I have developed a lot more concerns with teaching math in general as well as my own knowledge in math, but my next steps will include working hard to develop a more thorough understanding.

Summary and Synthesis

This semester we have learned numerous concepts and easy ways to use them in our own classrooms. We've been taught how to teach in a different way. We've learned many tools both hands-on and in groups. I've been really interested in learning about why we do certain things in math, such as flip the fraction when we are dividing. I think these types of concepts are importnat to learn because many students will ask why we do something. I think these concepts have taught me to appreciate math more because in the past no teacher has told me why we do things the way we do them. Most of the time we are just given math equations, a few examples, and then given a bunch of math problems to complete. I've never liked that way and it made me hate math. The way we've been learning how to do math makes me more interested in the material. This semseter has made me more confident in math and I hope my confidence continues to grow by the end of the semester.

Summary and Synthesis

Throughout this semester we have worked a great deal on taking a constructive approach to teaching. We have learned that students will have a deeper understanding of what they are learning in math if we take the time to teach students why or how a formula is used instead of simply telling them to memorize it and use it. Students should be given multiple strategies to solve a problem and realistic problems for which they can relate to. All throughout my primary, middle and high school educational career I had never been told why a formula is used. Teachers would simply tell me to memorize a specific number of formulas and use them to find the correct answer. Students should be given multiple routes or strategies that they can use to solve a problem. Additionally, teachers should include problems that also have multiple solutions instead of a simple right or wrong answer. It is important that teachers do not put down students when they make a mistake. Students need to learn that everyone makes mistakes and making mistakes is how students strengthen their mathematical thinking. From this class, I have taken away a number of instructional strategies for which I can use in my future classroom. For instance, when teaching fractions to students, I would use manipulatives, such as money, to help teach students about fractions, decimals, and percents, because money is something they can relate to. Students will also discover why knowing about fractions, percents, and decimals is important. I hope that by the end of this semester I will feel more confident about teaching math and have less anxiety with my math skills.

New Insights and Their Implications

In this class I have learned a lot of new information that has benefited myself. I have learned new concepts I did not know before. Learning how to use different manipulatives helps with learning and making a visual can help conceptualize the concepts. Doing the LPU it forced me to really think about how to do the problems. In the order to explain it to someone else I really had to think about the process I was completing. I also learned well from watching other classmates do other problems on the board. Doing the extra research for the LPU showed me what we are learning in class is being done in other books and that it is important to change the style of math teaching that is happening in the classroom. Starting with an already known concept and adding on to the concept is the best way to teach students a new math concept. Not only starting with an already known concept, but even starting with a simpler problem helps students understand a concept and find the algorithm them self. I know i have gained more of a conceptual knowledge about concepts about topics i was just taught the formula and just had to go do a worksheet. Starting with a concrete example to a more symbolic meaning is the best way to teach students math. Students need to have a visual and other ways to justify their answer then just the basic old formal or rule to complete a math problem. I know with the division of fractions, I had no idea why we inverted and multiplied. In high school I took the upper math class and I really did not know why we were learning all of the math because we had no real world connection. I could figure out an easier way to look at the problem, so I could solve the problem. I was also able to explain the process to my classmates, but I had no idea why we were doing what we were doing. Students need to get the conceptual understanding and be able to apply it to real world situations to under stand the importance of math concepts.

New Insights and New implications

First of all my new insight is that I am not good with google docs, it took me 40 minutes to figure how to post this. On a mathematical topic, i found out that just because i knew the rule to something did not mean that i knew the reason for the rule. Many kids do ask 'why' in school and they are shunned by their teacher with a simple answer that has no meaning which is 'because I said so' I am also finding out that as I go through the class I get more and more frustrated because it forces me to think aside from the rule that was rudely inserted in me as I learned math. One thing that I found interesting in class today was the reason as to 'why' when we are dividing fractions we 'flip' the second fraction and multiply across. I had NOOO idea the reason why, and it always bothered me, but know I understand and I feel as if I could explain to that 'why' child the reason as to why that rule is set up.

Summary and Synthesis

Throughout the semester we have been learning a lot of different concepts and how to apply them into the classroom. This class has provided different hands on activities both individually and as group work. We are being taught how to break questions down, how to answer our own questions, and we are just being told why to use a certain procedure, but we are being taught why it is important and how it actually works. When working with fractions in class, we were able to learn why you use certain formulas and how to use visuals to come to an answer. In the past, I have always just been given a formula, and than told to use it. Never was I ever told why or how this formula actually produced the correct answer. Sometimes I become irritated with myself when it comes to math, I wish I was able to produce an answer faster and with more confidence. I hope to have that confidence by the end of the semester.

New Insights and Implications

From this class the main thing I have learned is that just because you know how to use an equation, it doesn't mean you know how to teach it. I've learned that there are a lot of steps a person has to go through in order to teach one concept. Things are not as simple as just teaching an equation and expecting students to apply it right away. Sometimes I forget that my students can't pick up on things as fast as I can because they don't have as much background knowledge. I need to teach things as though students are starting from scratch.

Personal Concerns and Next Steps

Until taking this course, I genuinely thought that I understood math enough so that I could teach it to other people. I've always been a big fan of math and enjoyed doing it on most levels. I even tutored math for several years in high school and part of college. By taking this course, I feel like I am starting from scratch and don't know anything about math at all. It's really disappointing to find out that I'm not prepared to teach my favorite subject. I'm getting more into English because of this class. I'm trying my hardest to understand why we are learning what we're learning, but I don't quite get it. I hope that things will soon click for me.