Summary and Synthesis

When learning about fractions, i thought that there was only one or two ways to learn about them but i was definately wrong. After this class it really has shown me especially in the area of fractions that there are numberous ways to teach it using pattern blocks, fraction bars, caesianaire rods, etc.... I think that being a visual learner I would have understood fractions better given these objects unlike what i did have to work with when i was in elementary school. This made me realize just some of the options that i have to help me teach my students about fractions.

New Insights and Their Implications

I never knew about all the different steps when learning about measurement. It was interesting for me to learn about the six different steps and how the students really should be learning about measurement, not just be given the tools and use them right away. It made me reimence about my education, and all i remember is using the ruler and calculators and not really understanding why. I think that sometimes this would be more work for the teachers to make sure that the lessons include these steps but i think that its work the students education to let them fully explore these math concepts. It makes me also think that maybe if i would have learned somethings this way my view towards math would be better, and i would feel more confident in this area.

Summary and Synthesis

I was really impressed with our book we have been looking at and using in class lately. We have been talking about (CGI) or the teaching techniques of Cognitively Guided Instruction. I really enjoyed watching the videos because they gave you a first hand look at how this teaching technique is used in a classroom. I also believe that asking a question to the whole class and allowing them to work to come up with an answer is great. It is important in my eyes that they come together and discuss what they did to figure out the question as group. This was also demonstrated in the video. By allowing students to use a variety of manipulatives to aid in their solving students will come up with unique ways of discovering the answer. This technique helps students learn the material more in depth rather than just memorization. Watching these videos helps me see how I can use this teaching technique in my future classroom.

New Insights and Their Implications

I never new until this class all the different methods of teaching fractions. I was very impressed with all the different techniques that the other students in the class came up with. Dr. Reins was very helpful in opening up my mind to new ideas for teaching fractions. It is difficult to think about new ways when you were only taught one. To me it just seemed easier using the ways I knew until I tried some of the techniques everyone else came up with. It was nice being able to use the fraction bars, pattern blocks, and Cuisenaire rods because they are all helpful methods to help students see the fractions and to do the math problems. I was able to use these things to discover new ways for solving problems. It is important to know that children learn different than eachother and using several methods to solve a problem will help all the children succeed. I will make sure that I use these different techniques when teaching fractions in my future classroom. I will also keep an open mind to new ideas of solving problems. I will also allow my students a chance to discover new ways to solve problems.

Summary and Synthesis

These last weeks we have been learning about fractions. We have been using the (CGI) Cognitvely Guided Instruction book. I like this book because it shows very good insites to teaching math. I also liked watching the videos that came in the book. I agree with how the teachers had many different manipulatives that kids could use to solve the math problems. I like how they let the children solve the math problems their way instead of just giving them one way. I also liked how she had them teach the class how they solved it. I am still amazed by how many ways there are to teach fractions. I really like this about them because it could be easier for a student in a way...but it could be harder for a student to do it that way....so there are numerous ways you could teach the students in order for them to understand the problems.

Summary and Synthesis

I did not realize until the last couple weeks of class how many different ways we as teachers can teach fractions. I can honestly say that I still struggle with adding, subtracting, multiplying, and dividing fractions. It seems as though every time I am presented with these I am having to be retaught each and every time. It is good that there are so many different ways that we can teach this and make each student understand. Using the fraction bars, pattern blocks, or Cuisenaire rods are all helpful methods to help students see the fractions and to do the math problems with. I am not sure if we had these when I was learning this in school, but I can definately see how helpful they are now. You know, Math is my worst subject. The reason is becuase no one can ever tell me why we do the things we do in math. I was always just told, "That's the way it is! I have a really hard time when things are not explained to me or if I can't see it or physically do it. I think that these manipulatives are great learning tools becuase students can actually see why and how. Working with partners has also been a learning experience. Watching how other groups members solved the probems compared to my style was interesting. It showed that not every students will go about the problem the same way.
There are many methods that can help students learn about fractions. Teachers and students will actually have to work together to come up with methods and teaching strategies for the best learning to occur.

New Insights and Their Implications

These past few weeks in class we have been introduced to methods of how to teach students about addition and subtraction and multiplication and division. Dr. Reins has recently introduced the Cognitively Guided Instruction (CGI) to us. Although this method is fairly new to us, the basis of the approach is to use concrete materials, such as fraction bars, pattern blocks, or other manipulatives, to help students see or "act" out the problems. I think this method is a wonderful way for students to get a visual of what is happening during an addition, subtraction, multiplication, or division process. The CGI approach is a great way for students to figure things out on their own also. They have manipulatives to use to demonstrate 4 + 2 or 7 x 4. All students have different ways of thinking, and by giving them blocks, a time line, or fraction bars, they are able to show their process of thinking. This can be beneficial to the teacher so that he/she can see how their students learn best. Another point I would like to make about CGI is that the teacher should have all the students model how they arrived at a solution and their way of thinking. The teacher should be there to guide and instruct the students, and also encourage them.

New Insights and Their Implications

I have a great experience because I work at a daycare in Sioux Falls and while taking these methods courses this is helping me to apply and understand what is being taught to us in class. During study time, I get to work with students on their math skills so as we talk in class I am able to think about the students that I am working with and help them apply the strategies. I see them trying to apply what they have learned from teachers and are unable to correctly use what has been taught to them as well. I feel that for experienced teachers, changing their way of teaching must be difficult, however as a new teacher I will be able to help students construct their own understandings as this will only benefit them.

My biggest concern is that students will not be able to correctly construct their understanding. Will my teaching help them or hinder them?

New Insights and Thier Implication

I have a great experience becuase I am doing my SPED intern this semester while taking these methods courses and this helps me apply and understand what is being taught to us in class. In the resource room that I am in I work with a group of students on their math skills so as we talk in class I am able to think about the students that I work with and the strategies they apply to thier math. I see that many times they are taught how to do it (a process) and not given the chance to brainstorm on their own. I do wonder with this group of remedial math students if they would be able to constuct thier own understanding with out any guidance. But I do see them try to apply what they have learned from teachers and are unable to correctly use what has been taught to them. It is hard for experienced teachers to change the way they teach but as a new teacher I will help students construct thier own understanding so that they can benefit on thier own. My concern is, as I stated before that these students will not be able to construct the correct understanding. So am I hurting the students more than I am helping them?

Summary and Synthesis

During the past few days we have been discussing the teahing techniques of Cognitively Guided Instruction (CGI). I enjoyed viewing the videos shown in class demonstrating this model of teaching. I agree with asking a question, and allowing students to discover a solution on their own, coming back as a whole group and having the students explain their rationale for their answer. Allowing students to use a variety of manipulatives for their discoveries is helpful. Peer teaching can be helpful for those students in need of additional practice. Many students are taught knowledge and never fully comprehend the why of the problem, manybe the teachers don't fully understand the why, Who knows. Understanding the CGI model and seeing it demonstrated will allow me to let my students to explore and demonstrate on their own.

Summary and Synthesis

In the past few class we have been working on CGI. It is a teaching strategy to helps students learn in a different way. I feel that this is a wonderful way to guide students' learning. I think that by teaching this way students will be more motivated in math class and also have a better attitude when it comes to math. However, I do think that when teaching math, a teacher must also used a instruction strategy at some point. There is such a variety of students in the classroom, and some learn best with CGI where others need more direction. Another reason why you can not just teach in a CGI approach is because the standardized test that are required do not allow students to use anything other than their fingers, so students can not rely on manipulatives. It think that it is important for students to have some CGI in the classroom though because it helps students to understand the WHY of math. When they have discovered this they can move in to the instructional understanding of math so that they can do well on the standardized tests.

New Insights and Their Implications + Question

We are now reading the CGI book, which has given me new insight to students' learning of math as well as the teacher's role. Cognitively guided instruction, to me, means that the teacher allows the students to explore ways of solving problems, choose a method that works for them, and explain how they used that method to reach a solution. The book focuses on addition, subtraction, multiplication, and division, giving a variety of examples of word problems that can be used to help students understand each. The CGI book states, “It is when children decide upon a strategy to represent a mathematical situation and implement that strategy that problem solving takes place.” I have now realized that students can develop their own understanding through the use of good problem-solving questions, manipulatives, and other tools. The teacher guides the process by giving appropriate problems and asking the students questions about their thinking and how the problems were solved. Reflecting on their strategies helps students improve their understanding as well as value the thinking of their peers.
As also stated in the book, "Gradually, children come to recognize that their thinking is important, and they come to value the process of doing mathematics." We have been learning from the beginning of this class that students should "do mathematics," and learning about CGI has helped me to verify this in my own thinking. And just as Dr. Reins told us, that we won’t get the class at first but will eventually…I finally have that understanding. My philosophy of teaching mathematics has therefore changed. I will now create a classroom environment that supports doing mathematics, encourages communication, and allows students to actively explore strategies on their own and develop their own understanding so that they can form connections and value their learning. I do wonder, however, why we did not begin the class with the CGI book?? I think the class would have made more sense to me from the beginning if we would have started out learning about cognitively guided instruction.

Summary and Synthesis

The most recent work we have been doing in class with the fraction bars, pattern blocks and cuisenaire rods has really helped to shape my new understanding of mathematics using fractions. I have always been fairly decent when dealing with fractions, but until this class have never thought about common demoninators in this way. Using the manipulatives we have been using really helps focus my attention on why to use a certain denominator. As mentioned in class, I always just used factors of the two denominators to find the least common multiple. That way is probably alot faster, but does not show how or why you got the LCM. Using these manipulatives makes it clear to see how important it will be to use them with my students in the future. By having them explore fractions using them, they are able to get a visual picture of what a specific fraction means. I think this will help them to be successful at mathematics with fractions.

New Insights and Their Implications

When working on fractions during the past week in math methods course I enjoyed working with the different manipulatives to help me understand the material (fraction bars, pattern blocks, and Cuisenaire rods). When I was in school our learning of fractions came from worksheets and taking apart a pizza to see what fraction was left. It never involved the different activities we are doing in college.

The worksheet we did have to do in class involved working with Cuisenaire rods to find the answer, it was a little bit of a struggle at first to figure out the rods since they are not labeled, but than it helped me think more about the different fractions and how they are related. So I think by the instructor us rods that were not labeled we had to find out for ourselves how much each rod was worth.

This will be a great way for students to experiment and “do math”. They will not be given the answer by having labeled parts, they will have to take the time to figure each measurement on their own. I use to think that providing labeled parts to students would help them because that is what is normally seen in classrooms.

Our unpacking the standard was also related to fractions so by reading the text, doing class activities, and researching fractions I think I have a better idea of how to teach fractions to the students in my future classroom.

Summary and Synthesis

During the last few classes, we have been using manipulatives to help with the understanding of fractions. I can fully understand why we would use these in teaching fractions to our students because they still help me when I am comparing fractions. When we were doing that worksheet, it was much easier to figure the answer when we used the manipulative to work it out. I found that it was easier for me to mentally figure out the pattern blocks than it was to figure out the cuisenaire rods. That was probably so because there are less pieces in the pattern blocks and they are different shapes. I will definitely use these types of manipulatives in my class in the future.

Summary and Synthesis

I have been really impacted recently about how many different ways fractions can be taught. I think learning to add, subtract, multiply, and divide fractions can be a difficult subject as well as trying to teach it to all students. Using the fraction bars, pattern blocks, or Cuisenaire rods are all helpful methods to help students see the fractions and to do the math problems with. Also, I think it is important to not teach students about the rules we learned when dealing with fractions and allow students to form these on their own. I have been surprised with the different ways my classmates are able to do these problems because my mind thinks so narrowly about these problems. I was never taught to think outside the box about these problems, but I think it’s important I do not do this to my students. There are many methods that can help students learn about fractions and I think the models can help all students learns as they can work hands-on and see the actual product.

Personal Concerns and Next Steps

Before this class started I assumed it was going to be writing a bunch of math lessons and going over what is important in each grade. While the semester has been going I have realized that is not what this methods course is at all. I think that what we are learning is important but I am also afraid of starting my student teaching next fall and realizing that I have no idea how to teach specific math skills. I am personally not very good at math making my confidence level go down and my anxiety level rise when I even walk in the classroom. If I could avoid teaching one subject in my classroom it would be math. One thing that I have realized throughout the semester is that I do not want to pass this fear or frustration of math on to my students so I hope to be able to teach them math skills on a deeper level so they are not relying on their memorization of equations to solve problems. So all in all I am concerned I will not be able to reach this goal I have set for myself.

Personal Concerns and Next Steps

My personal concern created by this class is that I would like to learn more about how to teach all areas of math and multiple ways of doing this. I think as a class we all understand the importance of constructivist teaching and about having students understand math concepts and the reasons behind the formulas. Now that we understand this, I would like to learn more about all of the different areas in math and how to teach the underlying methods behind these formulas. I understand this may not be possible and we are learning new methods everyday about different areas, but I think we still have a long way to go. Knowing that for some of us this may be the last semester we have to learn better ways to teach then we were taught, I think we need to spend all the time we can on learning about all the topics in math. I have been so shocked already about how many areas in math I am not as knowledgeable about as I thought and how math methods can be taught in so many different ways. I think it would be beneficial to everyone in class if we spend as much time we can during each math class covering different areas in math and if Dr. Reins would share his great ideas about how to teach these to students so that they really understand each of the methods they are being taught.

Summary and Synthesis

This week we started learning about fractions. I thought I knew alot about fractions, but man oh man am I wrong. There are so many ways to learn about fractions. I would teach fractions in numerous ways. I like that you can use many different things: pattern blocks, fraction bars, and ect. to help student's learn about fractions. Many student's learn better by seeing things done in many different ways. I think it would help student's learn how to do them rather than "just because". In all honesty I learned fractions in the "just because way". This is making me realize that there are alot more to fractions.

Summary and Synthesis

This last week in class we have been discussing fractions. I have learned that teachers should never give the rules for solving problems with fractions right away. Students need to use visual manipulative for understanding what a fraction is, how to compare fractions, and how to add them. Manipulatives such as dot paper, fraction bars, and shapes should not be labeled so students can figure out the fraction themselves. Students also must understand that the denominator is the number being counted and the numerator is the number of parts under consideration. Also, using benchmarks such as 0, 1/2, and 1 is a great way for students to know where a fraction is on a number line, how big a fraction is, and compare the fraction to another. I have learned that students need to bring their prior knowledge of fair sharing and build their own knowledge of fractions using benchmarks, manipulatives, and other strategies.

New Insights and Implications

In our class this past week we have discussed the topic of using manipulatives to help students understand fractions. I know that manipulatives are useful and that there are different kinds available to address almost every subject found in school; however, I didn't know what to look for to pick out one manipulative over another. For example I, at first, did not realize fractional manipulatives should not be labeled. After our class discussion I now understand that it is important that students create/construct this understanding of fractions through working with the manipulatives instead of being told what it is. This is just one more example of how allowing students to construct their own understanding will better their learning of the material and help them to retain the information better than they would if it was just told to them by the teacher. ~Dustin Mees

Summary and Synthesis

In class we have been discussing fractions and how to divide them. One approach that we took was dividing candy bars equally for a certain amount of kids. Thinking back to how I was taught, I do remember my teachers taking this approach. I have never thought about using this approach to teach halving in my classroom some day. This is one of the easiest things to do, so I feel that my kids will benefit from this. Looking through that book was very interesting, to see the different types of manipulatives that you can use for fractions is just astonishing. I am very excited to be able to use manipulatives to teach my students fractions. I never thought about not having the fractions labeled would be detrimental to the kids. I now feel that having maniuplatives with no labels on them is the most beneficial.

Summary and Synthesis

Fractions have been our math topic for a few class periods now. When I was learning about fractions as a young child I always did what the teacher showed us how to do. We were given a problem and had to follow the “right” steps that the teacher had taught us to find the solution. Never were we asking to explore or to find our own way of steps or processes of coming to the answer. I never knew that fractions were so complex. There are multiple ways of thinking about how to solve fractional problems. I can now see how the fractional education that I was taught, was very much limited in its approaches. For my classroom I can now see that there isn’t only one path to take to get an answer but multiple paths to take. By allowing my students to become independent and explore fractions on their own (to an extent) I will only be helping them get a greater understanding of why certain processes are done and how they are done. These last few classes have really opened my eyes to knowledge that is new to me. The limited approaches that my math teachers taught to me so long ago is not the way that I want to teacher my students.

New Insights & Implications

After discussing fractions the last couple of class sessions, the way I would go about teaching the students about fractions has changed quite a bit. I was taught how to work with fractions by using rules, such as multiplying by 2 to find an equivalent fraction, but there are different ways that let the students explore other possiblities without using these rules. I don't recall using manipulatives a whole lot when I was in elementary school, although I'm sure they were used at some point, so working with them in class was kind of a eye opening experience. Getting to work with them and actually seeing how you can use them to find common denominators really helps the student understand what they are doing. For instance, a red one (1/2) equals three green ones (3/6). It shows them visually what they are doing when they have to add or subtract fractions, which hopefully will lead to them being able to eventually solve the problems with little or no effort and have it come naturally.

Doing Math

Doing Math: I believe doing math means students are understanding the thinking behind the process they are taught. The act of doing math means students are not just plugging in numbers to an equation they were taught but actually knowing why they put the numbers where they do in the equation and where those numbers came from. Sometimes doing math means students will have to struggle through a problem to arrive at the final answer. They need to investigate different strategies to arrive at the final answer. By trying a number of strategies the student will make a genuine connection with the problem and finding a way to solve it. By trying different ways to solve the problem students make sense of the math they were taught and this is what I believe it means to be “doing math”.

Personal Concerns and Next Steps

Personal Concerns and Next Steps: After being in this course I am very concerned about how teachers are teaching mathematics to their students. I discovered that in many cases teachers show how to do a problem and give the formula and then students just mimic what the teacher did and plug numbers into the formula they were given. Unfortunately this means many times the students do not actually know the reasoning behind the way they figured out the problem. They have the formula and know how to plug the numbers in but they have no concept of why they are solving the problem that way. This really concerns me because I am one of those students. I know the formula but not the reasoning behind doing the formula. I can find the right numbers to plug in to get the correct answer but don't really know where the reasoning is coming from. The next step for me is to try to not teach mathematics to my students in this way. I will make a conscious effort to give students the extra push to try and work through and solve the problem on their own. I will encourage them to struggle a little and try different methods to arrive at the answer. By doing this they will truly understand the why behind the problem.

Questions and Answers

As a results of taking this class lots of questions have been raised. The main question I have is how do we go about finding the time to teach using different techniques then we were taught. One of the reasons we were taught math the way we were had a lot to do with time. Teachers are given a whole list of standards they have to meet and not nearly enough time to teach everything they need to. When you throw in new teaching techniques it does not seem like there would be enough time to teach your students that way even though it is a better way. You want to give your students the best possible education they can get and that means creating a deeper understanding of math for your students. Instead of just showing students formulas and how to plug in numbers you want students to have a deep understanding of why math processes work the way they do. It will be interesting to see how teachers teach this way for deeper understanding and still have to to fit in all the standards they are supposed to teach. My question is how do we as upcoming teachers do this in our classrooms?

New Insights and Implications

After talking about manipulatives in class, I can not believe how many manipulatives are available to teachers to use in the classroom. I think that they more a student is able to work with the manipulatives and use them the more they will understand what is being taught to them. I wish that when I was in grade school that my teachers had the manipulatives or used the manipulatives that were available to them, I think that if I was able to use them learning math that I would understand some concepts a lot better today. So this makes me understand that when I am in a teaching position that I should use the manipulatives available and let my students use them and learn new concepts with using the manipulatives.

New Insights and Implications

After the past week of class I have realized an important aspect of teaching, especially in mathematics. In is essential, for teachers, to know and understand all the terminology located in the standards. At first glance, the standards seem simple; although after this week of class I have realized that the standards are full of loaded terminology and are complex. As a teacher, I need to be able to understand how to appropriately dissect each standard in order to effectively teach students. I think standards can be extremely confusing and complex, so as teachers, we need to make sure we understand what is expected of us in order for our students to learn what is expected of them and perform well on state and national standardized tests.

New Insight and Their Implications

In class we were discussing fractions and how to divide the candy bars equally for a certain number of kids. I have not thought about breaking away from the halving strategy. It is the easiest thing to do, so I have always just done it. I also thought that looking through that book was interesting because I would have never guessed that there were so many different types of manipulates that we could use to teach fractions, and some of them are not very expensive. I never would have realized that buying the ones without the fractions labeled will be more worth while.

New Insights and Their Implications

On Tuesday, we discussed using manipulatives in the classroom for a variety of reasons, one of which is fractional understanding. I've always known manipulatives are useful, but have been wary of them because some times they seem to be overused. But many are very helpful to students. When we were talking about them, it was mentioned that the manipulatives with the fraction written on them aren't very good to use. At first I didn't understand it, but now it makes sense. Though the size of the manipulative piece doesn't change, the fractional value does, depending on what is considered a 'whole'. Using it the other way would confuse students in the future. I think it is also good to not label them so the students can experience creating the fractions on their own. They can 'construct' in their mind how fractions work, and relate that to which ever manipulative piece they are working with. Students understand better and longer when they discover the solution on their own, through their own strategies. I'll definately remember this when teaching.
Mary Fink

New Insights and Their Implications

While I was reading chapter 20 I noticed a section about measuring time and found myself really interested. During my internship this semester for SPED I have been watching first hand how teachers teach students to read a clock. There are many different ways to do it and it made me think about how I was taught to tell time or read a clock. It is a difficult skill especially for students who do not think it is necessary with digital clocks all over. Our book made a good point about some common confusions students have. Students are taught one hand at a time and then are expected to put them together which is more confusing to them than just learning both together in the first place. If students learn about exact hours or hour and half they will not be able to tell any other time. I noticed in the classroom last week that one student was supposed to look at multiple clocks and write down what time it was and she did fine when it was exact hours or half past an hour but when it came down to counting minutes in between she was completely confused. I think that the book gives a good alternative to teaching clock reading which begins with a one-handed clock and then moves onto discussion about what happens to the big hand has the little hand goes from one hour to the next. Students need to predict where the minute hand should be if the hour hand is in a specific location. The book also suggests teaching time in 5-minute intervals. This is a skill that I would not have expected in my math book but I think that I it gave a lot of good information and I know now from experience that this is a tough skill for students and different methods of instruction are necessary.

New Insights and Their Implications

Class this past week as been very insightful. It has been really interesting to see a new perspective of teaching. Dr. Reins has been showing us the differences in reading "critically" and "literally." I thought that he stated something that is fairly obvious to the majority of us by saying "Some people are book smart, while others are life smart." When I heard this comment I immediately thought of how I look at problems and my approaches to solving them. I know that everyone learns differently, but it never really occurred to me that I am going to have to get all of my students on the same page by approaching problems from many different perspective. I won't be able to show all of them my approach to a problem and expect everyone to understand as I do. Therefore, in order to be a good teacher who benefits their students' learning, a teacher needs to approach education from many perspectives. This can give students the opportunity to figure out how they learn best.
Going back to reading "critically" and "literally," this was fairly frustrating to me. I understand the purpose of reading something and then finding the answer. I also understand the purpose of reading something and looking for the meaning or interpretation. I believe that this will be one of the most major obstacles a teacher will have to face. Students should not be discouraged from reading critically or reading literally. I am not sure how to approach this, except to respect each student's take on what they read and learn.

Doing Math

I think doing math is problem solving using patterns and numbers. Math is always looked at as formulas and the outcome that is produced. The process is the most important in math. When teaching math I think you have to build on the concepts and prior knowlege the students have. Math is going more in depth rather than covering all areas. By going in depth other areas will develop themselves from the childrens' interst.

Personal Concerns and Next Steps

I too have many concerns when it comes to my future of teaching. I see all of the teachers in their classrooms and wonder how they became so knowledgeable because I have no idea still what I'm doing. I still question whether or not I would be a good teacher and more so, I am too worried that I will let people down. There is just so much too learn and so little time to learn it. But I have been very thankful and have learned a lot so far, I am just excited to see what the future holds and how smart *someday* I will be. I just cannot wait to have my own classroom and students and be able to decorate my classroom, but I know all of that will be here before I know it. :)

Personal Concerns and next steps

As do many of my classmates I have many concerns about going into to next semester of student teaching. Many of my concerns stem from not fully understanding the use of standards and how to know for sure if your students are learning what they need to be. As a first year teacher do I teach what I think are the main ideas of the standards and wait to see if my class passes or fails or do we have a education class that teaches correct application of the standards. I have written many lesson plans with standards listed on them but I am not truely sure if I am covering in depth what needs to be taught per each standard. When will I know for sure if what I am teaching is what my students need to know. The only solution I see to this with less than six months before I am in a classroom is creating more lesson plans and trying to understand the true depth behind all the standards in all the content areas.
Another concern I have is in specialization in content areas. Even in second grade now the students are switching teachers for certain classes. I do not feel as if I am strong enough in any one area to teach in depth about it. I hope these are the fears that all near student teachers face and my solution is to work hard and do my best to understand what my students need to learn to be successful.
With concerns,
Hannah

Summary and Synthesis

I have found this math class to be very interesting. In our last class we learned how to find the area of some geometric shapes. I had already been taught the formulas to find area but never looked at it the way we did in class. I found it interesting the way we really used problem solving to find area rather than just plugging numbers into formulas. I really like to work through problems, so the way we did the problem solving suited me well. It is cool how you work through a problem first and then see how it fits into the formula. I believe that doing problems and teaching this way will help students learn and think about problems as more than just finding answers.

Summary and Synthesis

In the first classes I thought we were mainly focusing on the standards and teaching to them. I soon discovered that we are just looking at them so we know and understand their meanings more in depth. I did not really know how they were listed. I also think the main focus of the class is for us to think outside the box. Learning how to use problem solving in our teaching and creating lessons that show the process and why it is the process. I am a little concerned with what is going to be on the test and our further assessments. The Ponca Park Project was a nice idea, but I think it did not get the point acrossed that the teachers were hoping for. It was fun to get out of the classroom though.

Summary and Synthesis

During class we have been talking about finding the area of any polygon on a geoboard and got to do some hands on activities for this concept. One activity we did was writing an algorithm that would work for any polygon that was made on a geoboard. In class some students talked about their different algorithms and showed the different ways this concept could be learned. After I got to see the different solutions I decided my algorithm needed some changes that would make it simpler. (I could take off a few of my steps. Ex: one of my steps was having a trapezoid, but the trapezoid would be divided into a triangle and rectangle). I think by allowing the students time to discuss their solutions in class, we learn a different view of the concepts and also gives a time to justify our answers.
The class also did a more hands on activity by cutting the triangle out of the rectangle and used string to make different shapes to see what would give more area. I believed the rectangle would have more area because of the length of it. I was wrong though, it was the circle.
I realize by having the students do different activities with one concept; they are getting a more in-depth learning. They are starting to realize why the different approaches work and see a variety of strategies to complete a task. We are not focused on memorizing one formula for a problem, we are allowed to explore and talk about our learning with others. I know in my elementary/high school that is how we learned math, memorized a formula and practiced it for the test. We never did look back at the formula once we got past the test. Now i find myself struggling with math classes in college, the educator show how but next explained why and had us make the connections.

New Insight and Their Implications

I have learned a lot already in this course about math issues as well as teaching in general. I have always thought of myself as being able to learn math concepts and understand them fairly easy, but I am struggling in this math course with the new way of learning these concepts. I didn’t ever think deeply about the way mathematics is taught in the school systems and how I was taught the different formulas. Now knowing how repetitious our math program is and how we never really understand the concepts behind what we are learning, I know I won’t be able to teach that way. I have become more insightful and already learned and retained information on many math formulas that I would have never been able to remember previously.
Taking this course has allowed me to really think about teaching to every child and how using different methods to teach the same concept can really benefit every student in the classroom. Having students memorize formulas without understanding why they are like that and how the formula is made hasn’t helped students retain the information learned. As a teacher, it is going to take a lot of time for me to look deep into the formulas and make lesson plans that teach this way. In the long run though, I will be teaching student’s information they will remember and hopefully reaching every student by teaching to all the different learning styles.

Personal Concerns and Next Steps

I am truly concerned with my ability to connect standards to lessons correctly. This is one of the most important things we as teachers have to do. I feel as though we have not been taught enough about this subject in previous classes. In Dr. Reins class we are talking about it, but I still feel like I am not able to go out into a classroom and start writting lessons with standards without wondering if I am doing it correctly. I hope that in this class we will talk about standards more and in more depth. This way we can feel comfortable using the standards in our lessons. The question that I am wondering about is why if these standards are so important how come we haven't talked about them in all subject areas. We have been required to look them up, but nobody has ever helped us learn the true meaning of them. I am going to work hard in Dr. Reins class by reading the material and taking notes in order to learn as much as I can about the standards and how to use them correctly. Hopefully this will help me feel more comfortable using these standards in the future.

New Insights & Implications

The most prominent new insight for me would have to be the fact that many mathematics problems can be solved using many different routes. In my previous math classes, we were taught formulas and specific ways of solving problems. However, this is not the best way to teach since every student thinks in a different way. Teaching this way merely forces students to use numbers in a certain way every time, go through the motions, and never really understand what they are doing or why they are doing it. When teaching, I now believe that the students should be allowed to come up with their own method of solving a problem by using the knowledge and understanding of certain concepts that they will hopefully gain through the teacher's guidance and activities. It will be a difficult task for me, since I was never forced to think deeply about mathematics before. I feel like I am starting over, but the more I am forced to think in this way in this math methods course, the more practice I will have, and eventually I will constantly be thinking in this way. When I start teaching, I will be learning right along with the students. But if I am learning mathematics in a different way with deeper understanding, I will know that my students should be as well.

Personal Concerns & Next Steps

My biggest personal concern for Math Methods is being able to completely grasp what is going on in class. I feel that since this is a different approach mathematics that what I have normally been taught, that I may not fully understand and grasp this new concept.

My solution to this problem, is to continue reading the book, doing my assignments, talking and visiting with Dr. Reins with concerns and problems I'm not understanding, and talking with classmates to better understand this concept.

So far I've liked this class and the approach it's taking and hope that I can turn my thoughts around so I can fully understand what's going on in class.

New Insights and Their Implications

Based on the mathematical instruction that I have had in my past schooling, I thought that the only way to succeed at math was to memorize as many formulas as possible and hope that I didn't forget them. However, it was very easy to forget these formulas because no one ever took the time to explain where the formula came from and why it worked to solve for the correct answer. Even in elementary school I remember memorizing charts and multiplication tables. In class I like that we work on the theory behind these formulas and processes so that we can better understand how to apply them and why they work. Math is definitely not my strongest subject and I think it would have been easier to understand if I was taught in a way that I could make connections and put reasoning behind what I was trying to do.

As a future teacher I look to teach math to my students in a different way than I was taught growing up. I think that by teaching students how and why something works they will be able to better understand the application and be able to retain the information. I can understand that it may be more difficult to teach in this way, especially since math is not my strongest subject, but I can still properly prepare for instruction and I believe that I can still accomplish teaching my students in this way. Students will be more motivated in mathematics if they are taught in a way that enables them to understand the "why" and the "how" and be able make connections from one concept to the next.

Summary and Synthesis

In class, we discussed perimeter and area. I thought I had a good idea of how to do them, but I realized that was all I knew...HOW, not WHY. I thought it was interesting what results we got when we were given string and used it to find the perimeter/circumference of different shapes. I always thought the square would be larger than the circle, but instead it was the other way around. I thought this was a great way to demonstrate this fact to students rather than just stating the fact that it is. I think learning this will help my students to understand the concept more than just performing the problem "because I said so"

I also thought it was interesting learning about the algorithms and using the geoboards. By forcing myself to come up with an algorithm, I am more likely to understand it than to just perform it. I liked how we could compare algorithms, so we could see how everyone else solved the problems and how different they all were.

Personal Concerns and Next Steps

I too do not know if I should put this under this heading or under questions and answers, but i feel that this is more of a concern than a question. When we were at Ponca Park we talked as a class about standards. We talked about the differences in state and national and then we also pulled apart the stands so that we could understand what they actually meant. Now in math class we are learning how to understand the language. My concern is that we are pretty far in our careers of becoming teachers, and we all know that we have written many many lesson plans for other classes. Why is it now that we are learning how to read and understand the standards? I am now asking myself if I have connected all the lesson plans to the right standards. I feel that this is something that should be done in a earlier class. Maybe education foundations? what do you think??
Megan

New Insights and Their Implications

I learned by finding the area of a triangle and polygons that there is more than one way to come up with the same answer. Instead of plunging numbers into an equation. I can exactly see that you can make difficult concepts easier by using hands-on activities. If students can visually see how a concept works it makes the concept easier to understand and most students will remember it. This activity reinforced my belief that hands-on activity is a great way to teach math. I knew that hands-on activities were great with lower elementary students but this activity showed me that hands-on acrtivities are also a great way to teach hard concepts to older students.

Insights and Implications

Insights and Implications- Learning math in school has always been receiving the formula and finding the correct answer. I never really knew why we learned formulas or how math was used in the real world. This course has shown me that students need to understand the "how" and "why" of math. Students need to be allowed to get into groups and find different ways of getting answers. I have also learned that students can not be scared to give wrong answers because they can lead students in a different direction towards a solution. As a teacher I need to teach deeper into the concept instead of teaching a lot of concepts. I am finding out that math is about finding ways to solve problems and understanding how math concepts are used in the real world.

While learning about area, we got the opportunity to use cut-out shapes, find the areas of polygons on a geo-board, and use string to have a deeper understanding of the concept. I want my students to learn math by finding the answers through exploration. Students can use past experiences, peers, and models to explore different roads towards solutions of problems. Being able to find solutions on their own, students will have an increased motivation and positive attitude towards math.

New Insights and Their Implications

Math has never been my favorite subject, it is fun when you understand it, however as I got to higher grades, and more difficult topics it seemed that my understanding dwindled (as did my interest level). I think that the reason why I have difficulty with math is that I never had a good basis to grow as a student. This class has taught me the importance of teaching math in a way different from how we were taught. Students need to be taught WHY we do the formulas WHY we do in math. On Thursday (Sept, 27), this really became evident to me because students need to know why they are using the formula for the area of a triangle. In my education, my teachers stopped at just telling us the formula. I think these insights mean that I need to teach math differently from how I learned. I want my math class to investigate math, and make sense of mathematics rather than telling them how to do it. Math has become too much of an independent study that is only taught through completing assignments solely from a textbook. Learning math, I think, should be more broad and a group effort (at times). At first I think that this takes more effort and is time consuming, but it builds a core understanding of math which makes the more complex/difficult problems easier to complete.

Summary and Synthesis

This semester, thus far, has been very informative and eye opening. We have explored many different topics in the short period of time we have had. We began with the topic of constructivism and what we thought it meant. Different meanings were investigated along with the history and the true meaning of constructivism. After the thorough study of constructivism, we were introduced to problem solving and the different methods involved in teaching a student. I found it interesting that as a pre-service teacher, I myself have never questioned as to why certain mathematical problems are approached the way they are. I have always been taught what to do but never knew why (regarding mathematical processes). Following problem solving we took a field trip as a class to Ponca Park. Integration, concept mapping, and standards were the focus of this trip. Unfortunately, the weather did not cooperate with this experience and we were inside for most of the day. I think if the weather had cooperated, the experience would have been more fulfilling than it was. After the Ponca Park experience, and currently, we have been working on geometry and all that it has to offer. Dr. Reins has given us different theories, articles, and methods to all look at to help benefit our learning in becoming a teacher. Overall, this class is helping me to explore different ways of thinking and essentially benefiting me by opening my mind up to several methods and approaches in teaching mathematics at an elementary grade level.

Summary and Synthesis

Last week we focused on finding the areas of triangles and polygons. In the past, I have always encountered short lessons on the area of triangles. The teacher simply gives the formula, shows how to "plug" numbers into it, and then gives an assignment over it. I have never spent so much time discussing and digging deep into the formula itself. I feel like I now fully undertand not just how to use the formula A=1/2Bh, but what it actually means. I like how we took time and did several different activities that helped us uncover the formula's actual purpose.

I also think it has been helpful to discuss problems with our peers. I think everyone has different views on things and it is very helpful to hear the way others solve problems. If I can look at a problem and be able to identify several different paths to the same solution, it will be easier for me to make sense of the problem. Talking with others helps me to become aware of other possible strategies I would not have thought of on my own.

Summary & Synthesis

On Tuesday 9-25, in class we talked about constructivism and how it is a way of how kids learn. To take this further we did an activity to help us understand the area of triangles and constructivism by doing the surrounded activity. In this activity we took a piece of construction paper and we cut a rectangle out of it, then we inscribed a triangle with in the rectangle and then cut out the triangle. We then discussed the different observation that we noticed when we took the two scraps of paper and put them on top of the inscribed triangle. I think that this would be a good activity to have a class do to understand the area of triangles, and help them understand the area of triangles. After this activity I now understand how constructivism is entangled in everything that we learn and the we will teach in the future. It is just a matter of how students learn and how the subject is taught.

Summary and Synthesis

On Tuesday (9/25) we discussed finding the area of a triangle on a geoboard using different methods. We were then given an assignment where we were to use what we learned in class to find the area of several different shapes on a geoboard and come up with an algorithm for it. For me, the easiest way to solve the problem was to find the area of a rectangle that encompassed the shape and subtract away the areas that were not shaded. This involved finding different triangles and/or rectangles that could be made to find the area of the unshaded region. When finding the area of the triangles I used the formula A=1/2BH. After I found the area of each unshaded region, I would then total up those areas and then subtract it from the area of the rectangle and come up with my answer.

After class discussion on Thursday (9/27) about the formula for a triangle, I found out that some students (any age) don't really understand the formula and what the base and height is referring too. This makes it extremely difficult for them to understand what to look for when trying to find the area of a triangle. I think that all the different examples showing that triangles with the same base and same height will have the same area is a good way for students to gain an understanding of the formula. They will then have a better idea on what to look for and it will hopefully be easier for them to solve.

Doing Mathematics

Chapter two of Van de Walle talks about doing mathematics. What does it mean to "do mathematics"?

Questions and Answers

One of the assignments for today's math class was to find a algorithm that can be used to solve the area of any polygon. We talked about two different methods that have been found to work. They are Pick's Theorem and the Chop Strategy. I think that I found another one but I'm not completely sure if it will work in all cases.
I start out by listing all the coordinates in order and listing the first one again at the bottom so it completely closes off the polygon. Then I multiplied all the diagonal numbers slanting down and to the right and all the diagonals slanting down and to the left. I added both of these columns and subtracted the smaller one from the larger one and divided it by two.
The first example looked like this.
0 4
8 2 3 0
12 4 4 8
16 4 0 0
0 2 2 8
0 0 0 0
0 0 4 0
= =
36 16

36-16=20
20/2= 10
The area of the first polygon on the sheet is 10...

Summary and Synthesis

Today in class (Thursday 27th) we discussed area of polygons and triangles. First we learned how to find the area of a triangle by figuring out the area of the rectangle and dividing that by 2. Then we were given the task of finding the area of the polygons. At first I tried finding the area of the rectangle for each polygon and then dividing just like we did for triangles. That just gave me numbers that didn’t seem to fit. So then I tried to divide each polygon into triangles then find the area of those triangles rectangles and dividing by 2. That didn’t work either. But I never thought about taking the unused portion of the geoboard and finding the area of each of those then adding that up and subtracting that from the total geoboard to find the area. It all seems so easy now but that was one way that I would never of thought of on how to do it. The right way to solve for area is actually way easier then I was trying to make it when I was trying to come up with my algorithm.
I think it was good for the students to have to work on and try to figure out algorithms because it really got our minds thinking of all the different aspects to think about. To use this same strategy with younger students I think will benefit them just as it did us as college students. They will get a chance to look at and think about possible solutions and then will be taught the “right” solution and they will have a better understanding of why that is the “right” solution and why theirs didn’t work.

Personal Concerns and Next Steps

I wasn't sure if I should put this blog under this title, or under questions and answers. While reading the text, I completly agree that students need to be able to explore questions on their own, without the teacher giving them the answer. But where do we, as teachers draw the line? If the students aren't getting anywhere, their frustration level grows. Even mine, while working on formulas for irregular polygons, was eventually shot. And I'm a college student. Obviously they wouldn't be working on something quite so high of a level, but for them, multiplication will be as difficult as advanced algebra is to me. Hints and helpful, and very useful up to a point. But what if I have a student who just can't figure it out? Would it be better to try and have a peer explain it? Or would they feel worse because their peer understood it, but they didn't? There's a difference between giving them the answer, and explaining the process. Explaining the process is okay to do, isn't it? They still find the answer and their own. And just looking at the incredible time constraint teachers are under to help students understand math, I'm not sure that students could realistically explore too many processes on their own. I know I'm rambling a bit, but I'm trying to work it out in my own mind. Is giving an explaination as to why a formula works enough, or are the students supposed to figure out the formula themselves? I want to be a great math instructor for my students, and give them a good, solid foundation in math. Having them actually enjoy it is one of my main goals, since I never got that as an elementary student. I'm just not sure the best way to go about it. I need to balance the needs of my students against the needs that the administrators and state force me to adhere to. Thanks for any help you can give me!
Mary Fink

What is constructivism?

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