New Insights and Their Implications

In class we have looked at math in a completely different way than I have previously looked at it. I have struggled with math for as long as I can remember. Math is not an easy subject for me. With the block activity that we did in class, I found it incredibly difficult to follow through with the activity. After having discussed the assignment in class, I am coming to a better understanding of it. Having the students work with their limited command of the material to find their own answers and solutions is an interesting concept. The activity helped me to understand what the students go through in the process of learning a new concept in a problem-based learning situation. The activity also helped me to have a better understanding of the concept. I am looking forward to using a problem-based learning environment in mathematics when I am a teacher. I think it will provide the students an opportunity to think through their problems without directing their approach.

New Insights and Their Implications

Today in class we were to discuss the state and national standards that we located in response to our scale, area, and perimeter activity. After breaking down the wording of the state standards I found it quite interesting that we really did not have a standard specifically for that activity. I found this to be quite disappointing because I learned greatly from that activity, but would not have used it in my classroom due to the fact that a standard did not dictate that I should teach it. It was interesting to see just how many different paths could be taken when deciphering the standards. The wording was so vague that it was difficult to pinpoint what the focus of the standard was supposed to be. Standards are the basics in teaching and if we cannot decipher their meanings now, then what does this mean for our future students?

New Insights and Their Implications

Teaching and learning math is way more than I thought it was. I have learned a lot about problem solving and how it should be taught in the classroom. Students need to be able to brainstorm the problem and create different ways of coming up with one solution. It is clear to me that not every student is going to learn in the same way so by having students communicate the different ways they found a solution to a problem, it might help other students understand the concept. Also, after talking about the South Dakota standards today, it made me think about what I am doing when I am picking standards. I feel that I will analyze the standard more after this class period as well as look at the National Standards and Curriculum Focal Points.

New Insights and Thier Implications

Doing the scale factor activity really thaught me alot about where I am in my mathematical education. I understood alot more concepts when I talked about it with my peers and with other groups. It is important to rach out to other peers when you are struggliung and sometimes this is not asy for me to do because I like to do things on my own. I also learned that I understand concepts but am unable to express them verbally. I can do it on paper but when I tried to help my classmates I was unsuccessful. This has taught me to remmeber that while being a teacher it is not always best to have studnets working independantly. It is important to let studnets converse and figure things out together because they might be better able to help one another.

New Insights and Their Implications

I couldn't believe that the State Standards and the National Standards are so different from each other. This was the first time that I have looked at the National Standards so I was very shocked at how the State really don't make sense when you read the National ones. I believe that now after seeing the National ones that every teacher should take a look at these before creating a lesson. You do however have to still make your lessons match the State standards. This was a very important class for me.

New Insights and Their Implications

Today in class we discussed the NCTM standards compared to the state standards. I was shocked to see how the state standards are so vague compared to the national ones. I had never looked at the national ones before. In every class we use the state standards for all subjects so I had never thought to look at the national ones. It is a big eye opener to see that it is much better for the students to look at the national ones and then at the state standards. The national standards are easier to look at and tell what the standard is. It tells you exactly what you need to do. There is no reading into it, guessing or thinking maybe this is what it means like with the state standards. I think that it would be so much better as a teacher to use the national standards. The national standards say exactly what you need to have the students learn. This would make it so much easier to find activities that fit the standard! I know that I will have to look at state standards and plan lessons around them but the national standards will help me also. I can look at them first and then look at the state standards to pick my standards that work best

New Insights and Their Implications

South Dakota state standards have been the focus of so many of my methods courses here at the U. We, as students becoming teachers, are told that we have to follow the state standards in each content area because that is what our state's main high-stakes tests cover. Choosing state standards has been almost a kind of guessing game in many of my classes. You know the activity that you want to do and you go through the state standards and cherry-pick the ones that "sound good" or the ones that you think the activity touches on. It takes quite a while! It is hard to find standards sometimes because, as we talked about in class today, they are written in such vague terms. I was not there when the standards were written, so how do I know what exactly they mean?

Today we discussed the benefits of using national standards and curriculum focal points to help in the process of choosing certain standards that apply to an activity, or to find an activity that applies to the standards. Honestly, I very rarely have looked at the national standards for most content areas when making lesson plans for other courses. It astounded me how much easier it was to pick one or two of the state standards after looking at the NCTM document and at the curriculum focal points. It will be extremely helpful in the future to be able to look at all of these documents when choosing standards for lessons.

New Insights and Their Implications

Today in class we talked about standards. I was shocked to look at South Dakota's state standards and compare them to the national standards. From the scaling activity we did in class, we found specific detailed national standards that supported our activity, but when we looked at the state standards, it was very difficult to find one that matched and if we did it had to be stretched out or further explained. By looking at these standards, it made me realize that there is more to be covered then just what the state of South Dakota is asking. This was a great activity and I believe it would be very beneficial to students in upper middle school, but when looking at the South Dakota state standards, I am not sure this activity would have fit in. It was a very good discussion in class, and interesting to hear what other students had to say. If I was a teacher in South Dakota, I would probably not have done this activity because the state standards are to be met, and that's what the students will be tested on. Also learning from the scale activity, it was a great way to work with our peers. We had the actual blocks and had to design the shapes. It was very interactive, even though it was frustrating at times, but we learn from each other. After the activity was done, the scale factors were then explained. At first I was a little confused, but after today I have a better idea on how to explain and figuring out how to enlarge the scale factors to find area and perimeter.

Personal Concerns and Next Steps

My personal concern is if I'm going to meet all the requirements for my students by providing them enough activities, meeting the standards and knowing the subject matter to its entirety. I have not done my intership so with this class it seems like becoming a teacher is a lot harder than it looks like. I know one way to solve this fear is to go through my intership and practicum and see if my fears change. Math is not a strong point so maybe I could go to a math camp or practice math more often or volunteer in a classroom to gain my knowledge. I had never seen the Smartboard technology and I had no idea there was National Standards that need to be met until taking this course. By going to my classes, I can converse with my peers about technology and ideas they have.

Constructivism

Constructivism is where a student builds on prior knowledge to create their own ways to solve any problems and establishes their own learning styles.

What is Constructivism?

Please copy-paste your response to this question in a comment to this post. Do not start a new post. Thank you.

New Insights and Their Implications

New Insights and Their Implications - What did you learn from your peers, from the instructor, and/or the readings, about elementary school students, and/or about yourself, and the teaching and learning of math and what are their implications to teaching and learning mathematics?

During the past month in class I was amazed on the many different tools that can be used to help students learn fractions. I really enjoyed learning how to incorporate models into the classroom as well as experimenting with them myself. I thought the interactive sites on the web were outstanding and look forward to using them in the near future. I have always enjoyed fractions, but I do realize they may be difficult for some to understand so with the help of what we have learned in class as well as the models I believe I can now more effectively teach fractions. I look forward to learning more about math during the last month of class and can't wait to adapt this information to my own classroom.

Question and Answer

When we started the section on fractions I knew I was going to have problems understanding this topic because it has always been something that i have struggled with. As we went through the weeks about discussing fractions and learning more about them a lot of my questions were answered. The first question that i had was, how am I suppose to teach about a subject that I am not comfortable with. The solution I have come to is I will use my resources. Dr. Reins has pointed out a couple websites that I can use as resources to help my future students understand fractions. I will also use my book as a reference because they have many good ideas that I can use in my class room. The second question that I had was, what other models are our there that I can use to teach fractions to my students. This question was answered when as a class we were introduced to fraction bars, pattern blocks, dot paper, cuisenaire rods, and the cake pan method. I quickly caught on how to use each of the models and I will be using them in my classroom as well to help further my students understanding for adding/subtracting fractions and multiplying/dividing fractions. After the lessons on teaching fractions the only question that I still have is with all of the methods that we have been learning about is there one we should start with first or is it OK to start with any and then cover them all eventually? I feel that I would want to start with the one that know the best so I can give the students a foundation and then branch of in showing them all the other methods.

Summary

I enjoyed learning how to use all the manipulatives we used in class and the different ways that they can be used to teach. I liked learning why we do the steps we do to solve math solutions instead of a rote method of trying to memorize. I enjoyed watching the CGI video in class, it was good to see an example of how it is supposed to work in the classroom environment. I think the CGI method could be very beneficial for students as long as the teacher can guide the lesson and engage the students in the activity.

Summary and Synthesis

I actually liked this last unit on fractions. I liked that we were able to work with different manipulatives like the pie pieces and fraction bars. Most of what I know (with Math) is due to rote style learning. I like that this last unit gave us different ways to teach fractions. I agree that teachers need to start showing their students how to develop an understanding of "why" to do things, rather than just doing things without a reason and just off of memorization. I believe that students will have a better chance of understanding and remembering different mathematical rules and operations if they understand why they are doing what they are doing. I liked learning about the CGI program as well and it was nice to watch the video to see the classroom with that learning approach. I think that it is important to give students many options when solving problems and I thought this unit did a great job of showing us future teachers how to achieve that goal in our own classrooms!

Summary and Synthesis

We have been learning about addition and subtraction problem types. I have never heard of join, separate, part-part whole or compare.I think this is an interesting way to look at problems. It took me awhile to understand the differences of and what each one meant. These four are different structures of a problem and they describe the way a number sentence will be written. This is good for teachers because they know where to start when giving their students addition and subtraction problems. The teacher will also know what types of problems to give to the students and also what types of problems the student struggles with.

Summary and Synthesis

During the last month we have discussed the different ways students can use manipulatives to help them solve addition, subtraction, multiplication, and division of fractions. This is a very important skill for students to have, because it requires the students to think about fractions without just looking at number and memorizing steps used to solve for the solutions. I thought it was very beneficial to have multiple manipulatives, especially the Cuisenaire Rods online. It is a known fact that students have a hard time learning fractions and I agree that building the skills using manipulatives in the beginning will help the students with the skills they need when solving fractions throughout their education.

Summary and Synthesis

This last unit over fractions introduced me to new ways to teach math. There are many different manipulatives available to help make math more hands on and gets students involved. I learned that manipulatives that are not labeled are better when teaching fractions because they can be used to represent different whole units. I really wouldnt have thought about that unless it was brought to my attention. We have also been discussing how a CGI classroom is set up and how it runs. I think this is really interesting because I know personally I like directions and clear and concise explanations, however according to this method of teaching it is all about student exploration and figuring things out on your own. In some ways I agree with this and in others I dont think that it is right. I think that to an extent students need to be taught basic skills and concpets about math and then five them problems to solve on their own. Personally I am not a birg fan of the whole exploration technique because I am HORRIBLE with math concepts and why they work and how you do things. I like steps and directions... but I guess that is something I will have to come to appreciate!

Questions and Answers

One question I had at the beginning of this unit was, how can I teach students about fractions so they have a relational understanding of the process? It was obvious to me that I only had a conceptual understanding of most processes and did not understand WHY we follow certain steps to solve a fraction problem. As a student, I do not remember ever using the manipulatives as extensively as we have in this class to understand the fraction concepts. As a future teacher, I now understand the importance of using those manipulatives and following certain steps to develop the concept of fractions in my future classroom. While I know the steps are not a foolproof manual that will give the students immediate and complete understanding, I realize that the steps can be used as a guide to get students thinking and building on their own ideas about fractions.

Questions and Answers

As our ELED 330 class went into the fractions unit, I had many questions about fractions because of my background with adding, multiplying, dividing, and subtracting fractions. When I learned fractions in grade school, I don’t remember using so many models. It’s refreshing to know that when I teach fractions, that I will be able to provide and use many different types of models in my classroom. Also, I have a clearer view of the different methods that can be used in helping solve fraction problems dealing with addition, subtraction, multiplication, and division. My question at the beginning of this unit was, what method should be taught to help children learn more about fractions? I now know that the students should have the opportunity to look at a problem first and find their own methods. Since I want to teach elementary, a good way to let children explore their own methods in fraction problems would be through literature.

Throughout my time in K-8 Math Methods, I have learned a ton of valuable information when it comes to the teaching of mathematics. Looking back, I can say that when i first entered this course, I was scared to death. I had very little confidence my my mathematic abilities. However, I have definitely seen a change in my confidence over the last three months. I am no longer doubting myself when it comes experimenting with different mathematic concepts. My understanding and confidence in the area of math has definitely increased. However, I still have one question. How well will I transfer this new understanding for teaching math into the classroom? When I think about this question, I think about incorporating these new learning experiences into the classroom while trying to meet state standards, preparing my students for state mandated testing, and more. However, as I mentioned above, when i first entered this class, I was scared. Yet over time, my confidence in mathematics grew. I believe the same process will happen when it comes to incorporating these teaching strategies in the classroom. Over time, my confidence in the teaching of mathematics education will increase as I gain experience in the incorporation of these strategies.

Questions and Answers

Many of the questions that I have had have been answered throughout the last couple of class periods. I had always wondered what types of tools are appropriate when instructing students and trying to help them understand a concept. After talking about addition, subtraction, multiplication, and division models and processes, I understand now that there are so many ways to teach these applications. My question still unanswered, is how do you choose which model or method do you teach first? I know no matter what that when I choose method, I'm going to have to teach a different student a different method because not every student will understand the same way, but how do you know which model will be the most successfull within your classroom. This is my only remaining question, otherwise I feel that I have a good understanding of the various ways in which to teach these processes.

New Insights and their Implications

Learning about Cognitively Guided Instruction has really helped me to understand how I can apply the math techniques we are learning to my own classroom. Reading about the CGI classroom was very helpful to realize the importance of the student's learning process and how a teacher needs to take that into consideration. It was very helpful to watch the video of the two different classrooms to see how those teachers integrated CGI into their classrooms. It is helpful learning about it in class and reading about it in our textbook, but it was nice to see a real classroom and be able to see how it all works. I think the CGI approach is one that I strongly believe in and I think it will benefit my students as long as I incorporate it the right way. I also think that reading the article written by Ashlock was very helpful. I never really thought about how important it is to notice not just that your students are getting answers wrong, but why they are getting answers wrong. Working on the error patterns from Ashlock's book really helped me get a hands on experience to start noticing student's error patterns and being able to recognize them so in turn you can provide activities or strategies for them that will help them to overcome these errors. I think Ashlock's book has a lot of valuable information in it and I look forward to reading it and learning more about strategies that I can incorporate into my future classroom.

Concerns and Next Steps

Wow! Each day of class I learned something new and different that gave me a whole new perspective on teaching mathmatics. Through class discussion, hands-on activities and trying things out, I have gained new knowledge and an understanding of how to solve problems and why we solve the problems this way. I feel like I have moved from a procedural understanding to a conceptual understanding of problem solving. This shift in knowledge has made me more aware of and given me a deeper understanding of why things are done the way they are in math. Rather than just doing the steps because thats what we were taught, I undestand why the steps are what they are and why we follow the steps to arrive at an answer. I also learned that there are multiple ways to do a problem. This class helped me try things out and work them out on my own and STRUGGLE, rather than having the teacher just give me the answer. Having a conceptual understanding of the "hows" and "whys" in mathematics, I feel that I will be able to better teach students in the near future. From the new techniques, strategies and how-abouts discussed and learned from this class, I hope to give students a deeper and greater undestanding of mathematics through this new approach to teaching. As a future teacher, my "next step" is to shift student's knowledge of mathematics from a procedural understanding to a conceptual undestanding. This of coruse will be accomplished by applying the techniques, strategies and way of teaching math that I have learned in this class to my own future math instruction. My one and biggest concern is that I won't be able to steer away from the traditional way of teaching matehmatics to teaching this new approach we have learned in class. I fear this because this is what teachers are trained to teach and only know how to teach because thats the way the textbooks outline it. I am concerned that I won't be able to shift students understanding of mathematics from prodcedural understanding to a conceptual understanding. However, after being introduced to CGI classrooms, I feel that this would be a great place for me to start to make this change happen. I hope that from the new knoweldge and insight I have learned from this class, I can change and make a difference in mathematics instruction and students understanding of math.

Concerns & Next Steps

When we first began this class I figured it was going to be like all other math classes where we would be taught something and be forced to repeat that process one million time over until it was just in our memory. I enjoy this class because it is the exact opposite, we are learning new ways and techniques to approach and solve problems, as well as rethinking a whole new way to teach our future students. For some things in the class i find difficult at first (such as Pick's Theorem) but after discussing it in class and completing some examples in class I find myself understanding it much better. I find it very helpful to take my own notes during class so that I am able to look back and see how we completed something. I like that we are able to discuss things in class, we may not get everything covered but I would rather understand something than move so quickly to cover everything and end up feeling left behind!

Summary and Synthesis

This month flew by and I have gained a lot more knowledge on certain areas than I had before. Most recently was the knowledge I have been gaining on fractions. To be honest I never analyzed how we teach fractions, I feel I never paid attention because fractions has always been an area that confuses me at times. I can't wait to find out why we flip a fraction for division because I really have no idea why we do that, or how even multiplication works. I know how to do all of that I just don't know why we do it. I think pointing out 3/4 to one whole is completely different 3/4 to a different whole. When I first learned about fractions this confused me until I got to Junior High, it is something I am going to make sure is clear when I become a teacher.

Summary and Synthesis

Summary and Synthesis - Summarize and synthesize the impact of recent experiences, ideas, and/or issues encountered on Tuesdays and Thursdays in class from your perspective since the previous blog.

During my first blog, I really had no idea where this class was going. I was unsure the point of the way you were teaching, yet I did know it had it's purpose. Now my understanding is that you are teaching us the same way we need to be teaching our students. Asking questions, and finding solutions to our own problems is kind of the way the course is going, and is the way we want our students to be thinking. Instead of being spoon fed information, you are providing us with problems where critical thinking skills are needed, and where we have to be motivated to figure out the solutions to the problems presented. Although it is a different way of thinking, the point is now becoming clear for this course, and our future in teaching math and other subject areas. This course is enabling us to find the reasons behind the spoon fed formulas provided to us in math- the way we were taught growing up. Now I understand a particular concept in math (for example: why area of a triangle is 1/2 bh) and can apply it to many other areas of math (for example, using triangles to find the shapes of polygons). This course has showed me there are many different routes a person can take in coming up with a solution, and our future children need to see this as well. This course is providing me with the necessary skills needed to be able to get my children to think critically, and problem solve in math. The way this course is being taught is challenging, but now I have deeper understanding and meaning behind the math that I originally learned. I now understand how being set in the traditional way of teaching math can offset students in not truly understanding concepts being taught. The whole saying, a mile wide and an inch deep makes more sense now. Because there are so many topics that need to be covered in math, teachers rarely get to touch base with students to know if they are truly understand the meaning of concepts being taught. This new way of learning helps students see the true meaning of mathematical concepts, which can be applied to multiple areas in math, as we are presented with a problem and have to come up with the solution.

New Insights and Their Implications

I really liked what Dr. Reins taught us about teaching new information, and getting students to construct new meaning based upon what they already know. For example as I was learning about polygons, area, and constructivism I was able to draw conclusions, and make connections to build my understanding of the topic based upon my own comprehension. The idea of parroting back information for a test or an assignment seems to be useless for long term memorization and overall comprehension. I believe that the movement from covering information in breathe to covering information in depth is tough for many teachers to try as they are pressured by schools and standards to become accountable for the students learning; yet it is most useful to get students to become accountable for their own learning. Yet, it is proven that students who can create meaning and make it valuable to their lives are more likely to retain in long term memory. I would like to make this part of my own teaching motto, to get students to understand using scaffolding as a motivational tool in the classroom.
I also liked what Dr. Reins has showed us about drawing conclusions based upon Pick's theorem, the chop method, and drawing rectangles to find the area. This helped me move my thinking about area from strictly formulas and numbers to more theoretical and higher leveled think about areas. I truly enjoy learning this new approach to mathematics.

Personal Concerns and Next Steps

Through out the last month I have had a great time continuing to build my knowledge on how to correctly teach math. I am beginning to understand how this method of mathematics instruction can be used to the classroom but do have one concern related to this technique. That one concern is how to teach math using this method and still find time to cover all the standards and continue to stay on pace with the curriculum. I understand that its important to make sure all students get a solid foundation set and then build upon that but it is difficult to teach this way and still cover what needs to be covered. If it was up to me I would spend as much time as possible teaching mathematics the way we have been learning in class but in order to keep my job I understand that I will have to meet all standards and cover all of the curriculum. Teaching mathematics in a constructivism manor may be better for the students but I'm still unsure how I can speed up the process in order to reach the goals that is set forth my the school district I'm teaching in. I look forward to learning more in the next two and a half months and improving my teaching skills.

Personal Concerns and Next Steps

Personal Concerns and Next Steps - You may blog about genuine personal concerns created as a result of the experiences in this course. (Please note: If you blog on this item be sure to provide sound rationale for your concerns and what steps you are planning to pursue to address your concerns. This is not an opportunity to vent, be constructive and professional about it. Don't complain about something unless you offer a better alternative or solution.)

Personally Math has never been a favorite of mine because I have struggled with it. This last test was long and covered a lot of material that I thought I knew, but now I'm not sure that I knew it well enough. I know it's important that I learn and understand this material and I think that I am going to take more notes and make sure I understand the questions that are being asked at the begining of each chapter. I do like that this class is an hour and fifteen minutes because it allows us more time to go over everything and discuss.