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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