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.