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"?