Summary and Synthesis

On Wednesday, October 14, 2009, our class went to Beresford Elementary for Operation Coyote. While at Beresford, I taught not only one math lesson but two math lessons. The first math lesson was in first grade and we introduced the number line and worked on addition and subtraction. The second math lesson was in third grade and we worked on the number 5 times tables. At first, I struggled with the thought of teaching a math lesson without adequate preparation time, but once I started teaching the first lesson (in first grade), I become confident in how I was presenting the information. Later in the afternoon during my second lesson (in third grade), we taught a lesson about the number 5 times table. We gave the students a problem of the day and had them work on. Then, we had the students talk about the answers they got. Most of the students had the correct answer but some had different ways of getting the answers. We had the students talk about the different ways in which they found the answers and told them that just because you had a different way of getting the answer, does not mean your way is the only way. I found experience to be a great connection to what I am learning in this class, math methods. When I was doing math in elementary, there was one way to do a problem and that was it. When this type of instruction is introduced to students, more students will "get" math problems.

New Insights and Their Implications-Blog #2

Throughout class this month, I have learned a lot of new things about teaching students math. However, the one thing that sticks in my mind and that I found extremely interesting was breaking a shape into other shapes. For instance, it was really interesting when we found the formula for the trapezoid by breaking the trapezoid into two triangles and a square. I had never looked at shapes in that way. I had no idea that I could develop a formula on my own and didn’t believe I could until that assignment. I also found it helpful to use tangrams to see how shapes can be composed and decomposed. Overall, I learned a lot about teaching math to students and am looking forward to learning how to teach the more difficult concepts that students struggle with in the upcoming classes.

blog #2: Personal Concerns

This class is definitely unlike any other math class that I have had. Although I have gotten frustrated with this class a few times, I have learned a lot too. For example, when we did the taking apart area formulas, I struggled with this concept a little at first (proofs). Now that I can see where the formulas are coming from it will definitely help me to teach the formulas to my studnets. Going through my previous education years, I definitely just memorized formulas, but actually being able to produce the formula even if I can't remember it is a huge benefit. I just hope that as the class goes on I keep reconstructing my previous notions.

Summary and Synthesis

Throughout this course, I have altered my thinking about what constitutes best practice for teaching mathematics. Like many of my classmates, I have been challenged by this course to think in new ways and make sense of ideas without taking shortcuts. Each concept we have experienced in this course has been linked and built upon other concepts, which are all part of a "bigger picture." Teaching math is not about drilling isolated concepts and formulas to students, which are impossible to remember or understand, but is about allowing students to construct their own ideas and create a conceptual understanding about why they are carrying out procedures. Like Bridgette mentioned, I also learned the benefits of cooperative learning. In the past, working with others has usually been considered a form of cheating. From this class, I have been able to see how everyone benefits from sharing their thinking and justifying their own ideas, which strengthens each student's individually constructed understanding.

New insights and their implications

Through this math class so far I have learned a lot about how kids should learn math and how I learn math. I really like this class because I am learning math in a way that is like I have never learned before. I have always been someone who struggled to learn and memorize formulas and then know how to use them in a situation or problem becuase I did not understand them. This class is teaching me math the way I should have learned it, and is showing me how to teach kids math so they actually understand what they are learning. I have learned that students are not actually learning anything when they memorize formulas, but learn much more when they understand why and how to solve a problem rather than just memorizing solutions. I have also seen through this class how tough it may be to change the way math is taught, but in my classroom I will be able to help students learn math in a different and most likely a more effective way.

Summary and Synthesis- Blog #2

This Math Methods course has definitely caused me to think about teaching math in a new light. I have been struggling with quite a few of the concepts that we are learning, and this may be because I am used to math teachers in the past walking me and my class through all the steps that need to be taken to complete a problem. In this class we are learning through accessing our prior knowledge and using critical thinking skills to solve problems without the instructor as the facilitator. I had a hard time with finding the area's of the shapes on the geoboard, and after I asked for help I am much more confident with the concept. I have learned that I need to think outside the box and focus on what I need to be doing to solve the problem in front of me and not get overwhelmed, because that is when I run into trouble. This new way of teaching and learning has been a stuggle for me but I am able to see the positive effects that will result. If students are finding their own methods for problems and going through the entire thought processes, like we are in this class, they will be more likely to retain that information and make their learning more meaningful. This class is a struggle for me but each day I am learning in new ways and I am hoping that it will have positive effects on my future teaching of mathematics.

personal concerns...

Throughout this semester so far there have been different ways to think and understand this subject. I get frustrated pretty easy when I don't fully understand something and I have been on a rollercoaster of frustration and confusion. I try to cope with all the new learning strategies and fit them into everyday situations. This class is definitely different than any other math class previously taken. It is good to accept a challenge and work at it, but when I continue to try and don't grasp a concept, I get frustrated with myself and find myself stopping at what i'm doing and coming back to it to make sense or just leave it all together. I need to be a little more patient and look outside the box to find the answers, sometimes this is difficult for me.

New Insights and Their Implications - Blog #2

Throughout the last month of class, I have learned many valuable lessons. I believe the module and lesson on geoboards and area of a polygon were the most beneficial activities. I found it challenging to try to formulate different methods for solving the area of a polygon. During this activity, I learned how to cooperate with peers in order to solve more of the difficult problem. When we brought the worksheets back to the classroom, I realized how students could use several different methods to solve an area of various polygons. Even though I was able to view the different methods/strategies my colleagues and the professor used to solve the same problems, I understood my method best.
After completing this specific activity, readings, and modules, I realized that students can learn how to solve problems in various ways. I will use this information and carry it into my future teaching. I believe it is important to allow students the opportunity to show their method/strategy for solving a problem. In addition, I realized that I learned best by using my own method to solve these problems. I have concluded that students will learn best by using their own methods to solve problems. In my future teaching, I will give students problems and ask each student to come up with a method of solving. Even though I struggled through this activity, I believe I learned much more. In conclusion, I believe students will learn a concept more thoroughly if they apply their own ideas to finding the solution.

Personal Concerns and Next Steps- Blog 2

Math has always been extremely difficult for me and this class is a challenge too. It is difficult to look at math in a different way when I have been doing math one way for so long especially when I struggle with the way I was taught. I do believe that teachers who struggle with a concept and then finally understand it are going to be able to teach the students well. I think that teachers who struggle will teach the concepts to the students well and be able to help them with their problems. I know that it will take more than just this class to understand this type of math especially when I still have problems understanding the old way to do math.

Personal Concerns and Next Steps

This semester I have gained a lot of new insights and ways to understand math in a completely new way. I have been challenged, confused and frusturated a lot this semester because I am being taught to think about math in a totally different way than I have ever been taught. It is diffucult to try and learn something in a completely new way. I know what it feels like to be a student learning a brand new concept. The concepts that have been introduced are important but are still hard to look at in a new way.

Personal Concerns and Next Steps -Blog2

I have always enjoyed doing math and being in math class. This class has definitely been a different view on math and challenging me in different ways. I really like the activities we have been doing in class. Like finding the formulas of a trapezoid and parallelogram by breaking down the shape into simpler shapes. I have never thought of doing it this way before. I have always just memorized formulas and never knew where they came from. Looking at in a different way was very challenging for me, but it made sense. I also liked using the chop strategy for finding polygons. I have never thought of surrounding it with a rectangle and decomposing shapes from it. I really like all the new ideas that we have been learning in class.

Blog # 2--Summary and Synthesis

We have been talking about finding the area of polygons. I find this topic interesting and hard to understand. I did not learn this in grade school and never used a geoboard until now. With listening to other student methods and knowing there is more then one way to solve the problem, makes it more understandable. I also find it interesting that there is only one true written down formula known as Pick's theroem and by reading the modules and class discussion, no one knows his thinking that made him come up with the formula. Pick's theroem works wonders and anyone can find the area of a polygon on a geoboard really fast. I am learning that there is not just one way to solve a problem and students can help each other. The teacher does not need to provide the answer, but look at the student's method of solving the problem.

New Insights and Implications

Since this class started I've learned several things in regards to concepts about math education. I've always viewed math as a very dry subject with only one right answer; when in fact there are several ways to approach one problem. It's important to encourage students to come up with a solution in their own way rather than paving the path for them. When doing this we teach students problem solving skills. We also talked about the importance of using the right math standards for both state and national. We also learned about what standards mean and how to analyze each standard and put an appropriate assignment with each standard.

New Insights and Their Implications

Over the past few weeks, we have learned many aspects about math. The block activity stands out the most for me and I feel I learned a lot during those two days and trying to solve the problem. The activity helped me develop problem solving skills and really think about how I was thinking. I was shocked when we tied this activity to the standards. It fits multiple standards but only pieces of each standard. When we broke down the standards, I found it very interesting how you need to really look into the wording of the standards to get the real meaning out of them. I think it is crazy how much money is spent to revise the standards when we already know in two years they are going to change.

New Insights and Their Implications

This class has been quite an eye-opener so far. I thought this class was going to be fairly simple because math has always been one of my better subjects but I have come to find that this class is more than math concepts. Instead of simply finding an answer and moving on, I've learned to take a step back and think about the why's and how's. For instance, the activity we did with the shapes seemed very simple, until we discussed it in class. I never noticed a pattern with our data at all until the class discussion. Even then, it wasn't about finding the pattern, it was why and how this pattern existed that was important. It's not always about the right answers but the methods and strategies used to get that answer. Also, I've learned that the content standards used for mathematics in South Dakota are not very well written. The discussion we had the other day about state and national standards was pretty interesting. I never really thought about standards in that much detail - always took them for what they were. I have really learned to consider both standards in order to give my students a more thorough education.

The past few weeks in the class has been something I have not experienced before. I usually do very well in classes, I normally understand concepts and complete them to the best of my ability. I have never have a major struggle with any school work, until this class. I am used to just looking for the answer and then once I get the answer I could truly care less about how I got the answer, as long as I got the credit for the right work and right answer. This class has turned that all upside down for me, which I feel is something that all students should experience. The way you teach math is the way that math should be taught, I am actually building on my knowledge instead of just "floating" through the class with the correct finished product. In the past few weeks I have really been testing my knowledge with a different way of thinking, even though it is extremely frustrating that it does not just come to me, I feel that I will benefit from this class. While sitting in class I try to think of ways to teach like you, as in to really get my students thinking, not just saying "here is the answer."

New Insights and Their Implications

Through this class I have a better understanding of the standards, state and national. I had thought I did understand them and what they ment, but after this week I am shocked as to how much I do not know. When we looked at the activity and tried to find standards that would fit with the activity, I thought I had done alright. As a class, we looked at other standards and talked about them. This was wonderful, as I got to hear what other studetns were thinking when they chose the ones they did. With the implications standards being what they are, I am supprised that they are so vague in their attempt to let the teacher know what to teach. I now know that I need to look at the national standards with the state standards to produce quality lessons that my studetns can learn from. What I need to do now is to take this new found information and apply it to all my classes.

Blog One: Insights & Implications

I have already learned a lot in this class about how to teach math and how students learn. First off I have learned that giving students time to solve problems themselves is the best way for them to learn. Like my peers in class I too was taught math in the manner of here's the problem, here's how to do it, now practice by doing a worksheet filled with problems. Even though I may have been able to learn math this way, not every student thinks the same way as their neighbor. I have also learned a lot already this semester about the standards. It is now quite clear to me that the state standards are much different from the national standards. It is interesting that the state standards are so vague. I had never realized that to be so until we spent time picking them apart in class last Thursday. What I hope to continue learning in this class is how to connect the standards to my teaching successfully. Successfully to me means that a lesson is one that the students were able to relate to, connect prior knowledge to, and understand.

New Insights and Their Implications (Blog #1)

This month I have learned a lot about how to teach mathematics. Going into this class I was unsure how to explain concepts to other students. I really enjoy how we are discussing with each other how to explain different concepts. Every student learns differently and by showing us all of the different ways we can solve problems we can keep this information in our notes so when we teach it to students we can show them that there are more than one way to solve a problem. I also learned that it is important to have a variety of questions to show examples of when teaching mathematics. As a teacher, I need to have students perform a multitude of easy, moderate, and difficult questions and also show them how to do some of each for examples. I also need to find different ways to get students to have hands-on learning when it comes to mathematics so they can perform trial and error for themselves without the aid of the teacher. The final thing I learned this month was that teachers need to give little help to students when they begin their work. We need to let them try things for themselves and not just give them the answer.

New Insight and their Implications

I am starting to gain new insight into the standards. I was not aware that one needs to take into consideration both the National and State standards when planning a lesson. So far with my education I have looked at the standards and written down every one that might pertain to the lesson plan. I see how this does an injustice with our students. It is better to pick a couple of standards, both from the state and national standards, to focus your lesson on. This way one can make sure that you have taught all standards required. Listing a bunch of standards, on a certain lesson, may lead you to believe that your already taught a standard but you have not.
I have new insight into realizing that I have not had math for a long time, and the old addage 'if you don't use it you will lose it' may be applicable. This class will help me learn and remember what I have already learned. I have noticed in the schools that they are teaching differently from when I was taught in school. My children are in 3rd, 5th, and 7th grades and their homework is a bit different than I remember. For example, my 5th grader is not allowed to line up math problems on top of each other for multiplication. There are at least two other ways they are learning to do multiplication. They use a table to solve the problem. I am anxious to learn these new ways both for teaching and helping my own children with their homework.

New Insights & Their Implications

Since this class has started, I have already gained many new insights in regards to teaching. I now understand how important it is for students to be able to think for themselves and engage in a learning process in which they are able to independently create solutions from the information and material given. As we have talked about numerous times in class, it is beneficial to students' learning if the teacher lets them struggle a little to figure out solutions instead of giving them the answers. This way, the students are more active learners and will take responsibility for their learning. We also discussed in class how it is extremely important to not only refer to state standards when planning lessons, but also consider national standards. It was made very clear in class how the state standards are very vague and may be missing vital components. The national standards are more detailed and help provide a better description of the learning that is expected to take place at each grade level. In other classes, we have not talked much about national standards, so it was very helpful that we covered this in class. These are some of the new insights that I have gained so far in class.

New Insights and Their Implications

Since the beginning of this class, the main idea that I have learned is how to teach math. Growing up I was taught math, but ALL my teachers taught me the same way. They would present the problem, do an example on the board, and make us do like 15 for practice. However, since being in the class I have come to understand there is another way to teach math. If more students were able to understand math, by struggling through math problems and using their own solutions then math would not be a scary subject like it is now.
Another insight I have really learned a lot about through this class is standards. Before coming to this class, I though of standards as state standards and that is the only information I NEED to worry about teaching. However, once again I was wrong. When I teach, I need to look at National as well as State standards. If I look at both, as well as APPLY both it will allow my students to score better on tests, and dig deeper into certain material, rather than of the idea of "a mile wide and an inch deep."
The last insight I have really learned from this class already is the wording and choosing standards. With the exanple we did in class, it allowed me to realize to choose one or two standards and teach those, rather than trying to teach seven different standards as one time. In addition, I need to research the meaning of the words in the standards and not just use my defination of them.
I have learned about myself that I am closed minded when it comes to math. I was always taught a certain way, that I assumed if I ever taught math that would be the right way to do it; however, I need to be more open to math and allow my students the opportunity to struggle, learn from others, as well as from the teacher.

New Insights and Their Implications

I have lerned many things from my peers and the instructor so far in this course. I have learned that there are a variety of ways to teach math, and no one way is the correct way. I have learned that you have to allow students (and yourself) to struggle in order to come to a solution to a problem, and to truly understand the concept. I have also learned that there are many ways to approach problems, and various ways (or ramps) to take when solving a problem to get to the correct solution. Some implications for teaching and learning mathematics are to allow the students (and yourself) time to figure out the problem independantly. The students and individual must struggle a bit if they are to understand the concept presented before them. Students must be allowed to test different ways of reaching a solution to a problem, as well as understand how they came to the solution. The same can be said for learning mathematics; you must concentrate, and struggle with problems before coming to a solution to further your understanding. Through this course, I have also learned a lot about myself and my own struggles with mathematics. I am not the best math student, but I find that talking with peers and asking questions about reaching solutions is a great way to learn because it allows me to understand the concept for myself, not just simply coming up with a correct answer. As the semester progresses, I hope to obtain more insight into the teaching and learning of mathematics.

New Insights and Implications

I have always enjoyed math and can't wait to teach my future students the subject. Before taking this math methods class, I have never thought about the little details that go along with teaching math. I have come to realize, more now than ever before, that teaching math may be more difficult than it seems. The scale factor activity made me learn a lot about how to teach students in a way so that they are learning for themselves. In addition, linking the standards to the scale factor activity was a great way for us students to analyze each standard to see how they fit or didn't fit into the activity. I feel like this class will truly teach us how to help our students learn math instead of just doing math. So far in the few weeks we have had class, I have learned what not to do and what to do in terms of teaching math to students.

Summary and Synthesis

Throughout the beginning of this course, I have learned to view mathematics through a different eye. By participating in classroom discussions, group work, and small group assignments, I understand that teachers need to instruct students on problem solving, not the actual problem itself. I now believe that teachers should work harder on developing students who are unique and individualistic thinkers. In addition, teachers should allow their students to make conjectures about mathematical problems in order to better understand their mathematical thought process.
When we completed the scale factor activity, I realized how difficult it must be for some students when they do not understand a certain math concept. During the activity, I became frustrated with certain drawings and equations, but I was able to work through the problem with the help of my peers. I believe this was one of the best ways for me to understand this difficult concept. As a future teacher, I believe we should all students to collaborate with other students to better understand each other's thinking as well as their own.

New Insights

Over the past few weeks we have spent some time going over our area and perimeter with scale factors worksheet and I have seen a completely different style of teaching that I personally haven't seen or experienced first hand. Instead of running to our rescue right as we have a question or perhaps start to veer off the path to the correct answer, Dr. Reins lets us stray from the correct path and allows us to struggle to work our way back to the right path. This is something that for some students may be discouraging, however it has helped me tremendously. I have never been scared to ask questions however through this new way of teaching, it has allowed me to first think about the question, where I need to go and then from there I begin to work backwards almost to try to fit the two pieces together. For the students who get frustrated or discouraged easily, a small hint every once in a while may help build their confidence and slowly from there give them less and less help. Overall, I have found that although this method of teaching is quite difficult to get used to at the beginning, the rewards at the end are completely worth it.

New insights and implications

Last week we did an activity which involved area, perimeter, as well as scale factor. All of these terms were familiar to me, but as the activity progressed, it became more difficult. I found that it was beneficial to work together and talk in a group to come up with a solution to the problem. Talking with peers helps because, for me, it is sometimes easier to understand their point of view rather than the instructors; however, when you the instructor explained in more throughly, it was easier to understand and your graphic organizers helped as well. I learned that sometimes when things don't come as easy, I should try to talk to peers inside and outside the classroom. This week, we also discussed standards and the several kinds (state, national, and focal points). It was interesting because the focal points seamed to be very specific in which I think standards should be measured. I learned from my peers and instructor how to look at standards as well as determine if how certain standard (s) fits a certain activity.

New Insights & Their Implications

Throughout the first few weeks of class, I have altered my thinking of what makes good mathematical teaching. As typical of many of my peers, I learned math in the basic way of being told the correct way to approach a problem and then completing a number of similar problems. We learned in class that there are multiple entry points for every math problem, and one strategy should not be considered superior over others. By showing students that there is only one correct or best way to approach a problem, we as teachers may be boosting the confidence of one set of students, but we will also be discouraging other students who puts lots of effort into their work. I love Math because it challenges me to think critically and look at problems from multiple perspectives, but I know all my students will not have the same opinion. It is my duty to learn and understand how to use all different strategies of approaching problems. This knowledge will enable me to reach all my students at their current level of mathematical understanding and encourage them to apply strategies that work best for them. In addition, I also find it interesting that Jolley requires their teachers to be accountable for reaching each of the state standards at least three times. Even though we learned state standards are not as reliable as national standards, I think this is a great idea. When teachers address a standard one time, they may not reach every student. By being accountable for all standards and reaching each standard several times, teachers increase their chances of having all students meet the standards.

New Insights and Their Implications

After having a few weeks of this class, I have realized that teaching math is harder than it looks! I have always enjoyed math, so I assumed that teaching it would come somewhat easy for me. However, I've learned that applying mathematical concepts is more important that always getting the right answer. It's more about the process than the answer. This class is forcing me to think about math from a different perspective; my students'. In order to teach students, I must understand where they are at, how they are thinking, and what strategies they struggle with. Understanding how my students interpret math will make it easier for me to teach them skills like problem solving. Students will benefit more from struggling to learn the process more than they will if I just gave them the answers. I'm excited to learn more about teaching mathematical strategies to students.

New Insights

There have been several new insights this semester that I find useful and interesting. The last couple class sessions we have been talking about the state, national and focal point standards that guide math curriculum. Before this class I never realized that they were more standards to follow or use as a guide beside the state standards. I find it very helpful and useful knowing that there are more standards out there to use besides the one for your state.
This class has also opened my eyes into a new way of teaching math that I never considered before. Growing up in all of my math classes we did a lot of worksheets with examples on the board to teach the process. As a student, I remember that if I struggled I could look to my teacher for help. I can't say that they would just give me the answer, so I do believe that they used some of the similar teaching techniques in which we are being taught in class.
Solving problems can have a lot of different steps to take and I have also learned how to approach teaching story problems. I believe it all begins with the approach in which a teacher delievers to his/her students.