Questions and Answers

One of the experiences during class that I had questions on was the assignment regarding Pick's Theorem. I really had difficulty with this assignment regarding how to go about figuring it out. Honestly, the first thing that came to my mind was calling my friend who is a Math major. She ended up being the one to help me out, and I was glad she did. She showed me and a few others how to figure out the areas of the different shapes.

My next question was what I was going to need this information for. I went to class the next day and shared my answer. Dr. Reins told the class that if they had payed attention, then their following homework assignment would be completed. Then it hit me, I needed the previous information in order to know why Pick's Theorem worked. This was a good feeling.

My last question regarding this experience, was when was I going to use this information again. I was answered during the Midterm. I got to show how I understood Mathematics and how this theorem worked.

I was happy with the many opportunities I was given to use the material I learned and explored multiple times.

Summary and Synthesis

Thus far, I have learned a lot about how to better teach mathematics. I agree with the fact that the traditional way of teaching math is outdated and too procedural to accomplish major goals in mathematics, but I must say that I think it will be difficult to transfer my thoughts into a new way of thinking and teaching. It will definitely take time to accomplish major goals for how to better teach mathematics because it is first crucial for myself to become familiar with procedures. This way of teaching math does make more sense to me. I think that it is important for young students to be able to ask questions and be active participants in their learning. I think they should have the opportunity to find answers on their own and make sense of the information that is being given to them. I feel that procedures and always following rules makes students think that there is only one way to accomplish tasks and they should know that for full understanding, they need to learn all about the problems instead of just the formulas. I know that I have a lot more to learn, but I am willing to learn for the sake of my future students. I want my students to be taught the best way that I know how to teach.

Class on 10.21.08

Today in class we went over our quiz questions for Chapter five and six in our book.  The quiz is due tomorrow (10.22.08).  It was very nice of the teacher to go over the questions with us before we took the quiz because in the past not everyone has done well on the online quizzes.  The benefit of going over the questions the first part of class is that now we all know what exactly we need for the quiz, that way we can do well this time around. 

The second half of class we did an experiment.  The name of the experiment was “The leaping Egg (or Ping Pong Ball)”.  You take two glasses and put a funnel in one of the cups to simulate a wine glass, (We decided that wine glasses would be inappropriate for elementary students).  You then blew on the ping pong ball and it jumps into the next cup.  You can measure how far apart the cups can be before the ball gets out of control.  It is a very good experiment for student to understand air flow.  

Summary and Synthesis

Throughout the semester I have learned a lot of new methods of teaching mathematics. The midterm exam was an excellent way for me to put together the information we have learned so far this semester and apply it to teaching mathematics. I feel as though occassionally a method of summarizing what is happening in class, that way you can put a grasp on what you have learned and apply to teaching.

summary and synthesis

What I think so far about this class has changed in a way. I first thought that I was just in no way going to be able to get it or grasp any concept, but I have and that is good. The teaching method that you have adapted and have taught this class in is making a ton of sense now that I see how it is supposed to be done and what is expected. I do feel better know that I have the first test done with and now I know what to expect. I am sure that I still have so much more to learn and am still excited to do so. I think I need to understand more about the whole philosophy but think I will get there.

Summary and Synthesis

I have found as the course goes on, that I am remind myself time and time again that the way I originally thought teaching math or the way math was taught to me is not very beneficial. The course has taught me that there needs to be more analysis and time taken in math by the educator as well as the student. I was used to being given the right way to do a problem and then I was supposed to mimic that way within each related problem. The course has taught me that it is necessary for the student to know why they are using the formulas or why the formulas are the way they are. This is why I also have learned some activities that practice this idea, such as the more recent one where we were to develop a formula for a given shape.

Personal Concerns and Next Steps

This class made me struggle a lot at first. I feel I am still trying to grasp some ideas in this class though but not as bad as at first. This new teaching style is new to us and it is hard to take something in one semester when we have been taught pratically the exact opposite our entire lives. I think that this is a good teaching method from the expierience that I have had but yet I can not wrap my head around it completely. I might know and understand more than I give myself credit for because I thought I had gotten a C on the test but really I got an A. My personal concern is that after this semester this new teaching method will be difficult to take on or escape my mind on how to do it. My step is to absorb as much information from class and find other ways after this semester to keep the teaching method fresh in my mind and also able to use it.

Summary and Synthesis

One of the things that I have learned from this class is that they way I was taught math when I was growing up was not necessarily the right way. I was taught to use the wrong techniques, and that is why math is so difficult for me today. In class now, we are learning how to teach in a contructivist way, which is totally new to me, and sometimes frustrating- simply because it is a brand new method. In the past, I remember having to do a lot of memorization of procedures and formulas; with the constructivist method, there is much less emphasis on memorization. The students will have to do a lot of figuring out on their own, sometimes without seeing a formula first. This gives each child a chance to work out the problem for themselves first. I think this method is better because it gives the students a chance to really dig deep into the concepts of mathematics and figure things out for themselves.

New Insights and Their Implications

I think the best thing that I learned from this class is how to teach differently.  We learned from a study called TIMSS, which showed the United States that we are behind in our way of teaching that we need to do something different.  From this study we found that we need to teach like some of the other countries are teaching which is in a constructivist manner.  We also talked about why constructivism is important in the classroom.  We learned that students and everyone comes to know, with their knowledge.  Everyone needs to build off of there existing knowledge that we have processed.  If I can learn to teach in a constructivist manner then I will be helping future students. 

Personal Concerns and Next Steps

Some personal concerns for me is being able to teach math this way and do a good job at it. I do not always understand what we are talking about or how we figured out to do the problem. I think that this is because I never had to think outside the box when doing math and math has always been hard for me. I will continue to try and understand math, keep up with the readings and work through the problems and hopefully by the end of the class I will have a better grasp of things. I think that this is the best way at teaching math and hope that I will be able to successfully teach my students this way.

Summary and Synthesis

So far, this math class has taught me that teaching mathematics to students is no longer about memorization and following the steps exactly as outlined in the book. Because of studies done between countries, primarily focusing on Asian cultures and the U.S., math curriculums are being evaluated and revamped. Students are now learning in a more constructivist manner, including more hands on assignments and class discussions about problems. The math textbooks have changed in order to facilitate and encourage student learning. Unfortunately, I remember having textbooks with paragraphs of explanations about the problems which would show up later in the chapter; however, once I got to the problems, the time I spent reading was a waste because I could not figure out how to apply the book knowledge to the problems.

I have learned that much of what students gain from their school experience does not stick with them; students may also have their own preconceived notions which interfere with the correct information. This is why teaching math properly at a young age, and continuing with it all throughout their school years, is incredibly important.

Summary and Synthesis

All of the work and exploration we did with geometry and the area of shapes was really interesting. The way the learning was left to us was a good experience to have. We talk about constructivsm and student-centered classrooms in so many classes, but I have never been in one until this class. When studying for the midterm test, it was not like other classes where you go to the class notes and find definitions typed out. This class requires exploration which is honestly a new experience for me. It is a great experience though. As a teacher, I want to run my classroom similar to this because student retain information better when they have to work to find the answer and it's not just simply given to them. One more thing I want to sum up is that the midterm test was rewarding. I was expecting it to be difficult and to much, but because I studied and had learned in the class, I felt great after taking it. I hope my students can have a feeling like that. By working hard and really understanding a concept, assessments are intrinsically motivating.

Summary and Synthesis

I have learned interesting ideas from this math methods class. The most important part for me, is learning about the United States educational system. In class we have learned that the US is behind in mathematics when compared to other countries; yet, the US spends more money than these countries. I found it fascinating to learn why the US trails so many countries. Countries that are ahead of the US have a smaller focus (meaning they have fewer topics to cover) but the ideas they do teach have a higher degree of coherence (meaning the topics are linked together). The US tries to cover everything and the ideas have little coherence. Since the last blog, I have learned more ways to teach in a constructivism manner. I learned more about constructivism; it is a way that students construct their own knowledge about something. It is an inquiry and discovery manner of teaching and math learning. It was nice to have constructivism modeled for us in class. I liked the lessons and how they gave me examples of how to teach using problem solving. I have never been exposed to this much hands on math learning and I believe that this method of teaching is a ton better than the old method of teaching. I believe that if teachers adopt this method in the US, we will see improvement in our country's ranking in math comprehension. Since the last blog, I have become more of a believer in this new method math learning.

Summary and Synthesis

My perspective of this class has changed a lot over the semester. At the beginning I was really worried it was going to be extremely difficult, but now since I am somewhat comfortable with your method of teaching my perspective has become a lot more positive. I still do not know if I completely understand the philosophy of the teacher, but over the year I am coming to understand it more and more. After having the first test done makes me a lot more comfortable as well because I always stress out about the first test the most with a new teacher because I do not know what to expect.

Summary and Synthesis

Overall, I feel completely different about this class now than I felt after the first week of class. After the first week of class I felt as though this class was going to be extremely challenging and overwhelming. I no longer feel that way, however there are times where I struggle to complete assignments fully. I think that is the point...that is the best way to learn. At the beginning of the year I also felt as though the class website was too cluttered with information to provide much help to students, however I have adjusted to that as well and it proves to be an excellent resource.

Summary and Synthesis

My experiences with class so far this semester, have been meaningful but at times frustrating. I feel the material we cover is somewhat difficult, but the way we participate and investigate various activities helps me understand the content. To find what is expected of me for this class, I realize looking at the session notes is the most beneficial. Taking the test on Tuesday also helped me understand that the session notes are the only way of obtaining what is expected of me to know and retain. I now know how and what to do before each class period to be fully prepared each day for discussion.

Summary and Synthesis

My experiences with the class have been all right. I find that the worksheets and the hand outs really do help me study and understand the content better. I have conquered my fear of the chop strategy, and I have found that scale factor is a weakness. Some of the experiences have been frustrating, because some of the topics covered are still difficult to handle. The directions are sometimes hard to follow, but a question is never turned down in class. I feel as thought the questions asked and answered really have helped in my understanding at times. The most important thing for me to remember is that thinking outside the box is OK, and to take risks is important when teaching and learning math.

Summary and Synthesis

To be honest at the start of this class I was very skeptical about the methods that Dr. R was teaching us. I wondered to myself how would I ever be able to teach in a manner that I had never learned myself? Throughout the course of these past weeks though, I have started to come around to the idea of it. I have been learning things that I already knew how to do, but did not know why I was doing them or why they worked the way they did! Essentially, I have always had an instrumental understanding of math and now I am beginning to (slowly) have a more conceptual understanding of math. Already I can see the benefits to this type of teaching. I hope with this new attitude I will have a more positive outlook and learn even more from this class.

Summary and Synthesis

I have learned a lot of information so far this semester in math methods. I have learned a variety of new ways to present and teach material to students. I have learned what constructivism is and how to teach in a constructivist manner. It is important to have children learn to solve things on their own without the equation or formula given to them. Students can explore the problem first to try to develop the formula on their own. When students work this way it gives them a better understanding of how to solve the problem, how the formula works, and how they acheived the answer that they got. By working this way, students are making meaning out of what they are doing. Teachers need to make sure to use the students prior knowledge and to connect that to what the students are learning. We have learned different ways in which to teach scale factor to students. We have also learned different ways to approach problem solving and ways to help students solve problems in multiple ways. Right now we are learning about different polygons and finding the area on them. I have learned a couple of different ways to find the area of a polygon; such as, Pick's Thereom, chuncking the interior, and chunking the exterior area of the polygon. We are also learning the different formulas for the polygons. This class has required to think on my own and to use my prior knowledge to connect to new ideas.

Summary and Synthesis

I have learned so many new and interesting concepts this semester through math methods, such as constructivism. I have learned that it is important to be a constructivist teacher when teaching math because students need to learn how to do mathematics on their own. They need support and guidance also, but helping them to actually do mathematics is the best thing for them. When students learn new concepts on their own they make meaning out of it and learn much better. Using prior knowledge to help construct new knowledge is extremely important and can be used to help students develop new ways of thinking about math. Constructivism is just one of the many different concepts that I have learned this semester. Another concept that I have dealt with is finding areas of different shapes and polygons. I have used my own prior knowledge to help me with finding area formulas and coming up with algorithms for areas of polygons. Pick's Theorum is just one of a few methods that I learned for finding the area of a polygon. I found it to be very useful and easy to use. Throughout this class, I have been required to think on my own and find answers to questions that I did not know before. This entire course is based in constructivism and it truly does help in the learning process.

Summary and Synthesis

Since my last blog, I have remebered and discovered new methods in finding area of shapes and what the definition of shapes are. The definition of a polygon is one of the first things that I learned by doing looking at different shapes and figuring out if they were polygons are not. I also learned different methods in figuring out the area of any polygon. One formula I learned was Pick's theorem which as a pattern toit when drawing shapes on a geoboard. There were also other qualifications for quadileterals and rectangles that I had to remember since grade school. Many of my textbooks had defined a rectangle but they defined it in a way that it was so broad and it could be another shapes definition. I think it is important to define shapes well enough so students understand exactly the characteristics of that shape and what makes that shape unique compared to the other shapes.

Summary and Synthesis

I have learned from my instructor that I am not bad at math but was taught math with all of the wrong techniques. I now realize that I should have been taught math through a constructivist manner. I was always taught math content in a broad and general way. It seemed as though I had to hurry and memorize formulas, definitions, procedures, etc. the fastest way possible just so I could keep up with material my teachers would be teaching. I now know that children need to build upon their prior knowledge in meaningful ways in order for them to gain a deeper understanding of the information that is presented to them. I also now realize that I need to implement constructivism into my own teaching style if I want my students to really and retain most of the material that is presented to them. Implications of learning and teaching mathematics is that it can have some very difficult. This is why it is important for teachers to understand how children learn best.

New Insights and Their Implications

I think the biggest thing I've learned in this class is that the way I was taught math growing up is completely wrong! I always had a very traditional way of learning math; my teachers talked about the new concept for awhile, gave us a worksheet to do, and after enough lessons, tested us on it. There weren't many hands-on experiences provided to us students. When it came time to take a test, we just memorized the equations and answered the questions. After the test was done, pretty much everything we memorized was forgotten. That would probably explain why I'm not very good at math now.

I guess I never really knew that there was a different way to teach the information. But thanks to this class, my conceptions on teaching math are changing drastically! Although there are times I get frustrated in this class, I can honestly say I'm learning so much and actually retaining the information because we're doing hands-on activities and working on constructing our own knowledge. I really see the benefits of teaching math in this manor. I hope to be able to adopt this method in my own classroom. I think my students would really benefit from learning this way!

Personal Concerns and Next Steps

My convern in this class is a personal concern which I have struggled with a lot but even more so in this class. I understand the concepts and the materials you are presenting us but I have a hard time explaining it to other people sometimes and think it could be benefital to me how to explain the whys some more rather then just presenting the right kind of problems. To help students understand the right connections.

Personal Concerns and Next Steps

Some personal concerns that i have are that I am unable to think out side the box. I have always learned a certain way to mathematical formulas and how to problem solve. I was never really given opportunities to be challenged and find different solutions. Another concern of mine is that it is so hard for me to take tests and comprehend and make connections while studying that it could eventually hurt me in the long run. Some steps that i can take into account when i am studying is to try and make those connections to real life situtations, and maybe start studying a couple weeks before the text, rather than a couple days before. It is just very hard to study so far in ahead, when we are required to do some much homework in all of my other classes. I am not complaining, but sometimes school gets a little overwhelming and i seem to fall behind in all of my studies. Some other steps i can take in order to improve would be to give myself challenges and make my self solve problems using different strategies.

Summary & Synthesis

The impact of the recent experiences throughout the Tuesday and Thursday blogs have really opened up my eyes to thinking "outside" the box. Many math classes encountered today in elementary school have made way into the old thinking of how math is run. Through this math class I am in, I have had a tremendous impact on the fact that as teachers we need to guide our students into their own thinking and help them to model and demonstrate their prior knowledge in order to build upon their new experiences in math. The thinking in itself in these classes makes known that there are MANY different strategies and skills used to be able to solve ONE problem. This is effective to the classroom because it allows students to pick and choose and experiement with all in order to find out which problem-solving tactic works for them. This is a wonderful idea, because it focuses on the independent child, and their individual needs! The issues that are brought up in the classroom, with the research encountered through Japan and other countries of the region, look at math a completely different way. The U.S. lacks in math and needs certain progress. This class helps explore dicussions through models, student to student interaction, student to teacher interaction, geoboards and many manipulations that will help us to better understand the methods of teaching math.

New Insights and their Implications

I think that the most important thing that I learned during this first half of a semester is how to teach mathematics in a more reformed way by taking a constructivist approach. By looking that the TIMSS data, I learned that we do need to take a different approach in mathematical instruction in the United States. But knowing a change is necessary and knowing how to make that change are two different things. That is where the constructivist approach comes in. From what I have learned about constructivism during class and from the readings, I believe that this approach is a good step toward reformed mathematical instruction. By allowing students to construct their own knowledge we will be allowing them to again a deeper understanding of the concepts. Also, a classroom following the constructivist approach will have better focus and coherence. I feel that if I am able to implement a constructivist environment for my students to learn in, I will be doing my part in the reformation of mathematical instruction. I feel that the learning that takes place during a lesson built with a constructivist approach mirrors that type of learning we want our students to be having based on the results of the TIMSS study.

What does it mean to "Do Mathematics?"

Doing mathematics means that you are involved in the science of pattern and order (pg 13).  You are always exploring different ways to solve math problems.  You have to look at multiple angles of the problem to see if it is the correct way to solve it.  The old way of doing mathematics is “listen, copy, memorize, and drill” (pg14).  This is one way to do mathematics however the new way is more efficient and helps you learn why you get an answer not just, this is the formula, solve the problem.  The new way to do math is to have the students think of the problems and formulas and start from the beginning.  When you are doing this the students actually remember the process more and understands why something is the way it is.  This is doing mathematics because you are looking at the science of pattern and order.    

What does it mean to "do mathematics"?

When students are doing math they are engaging in the science of pattern and order. The students are exploring and explaining different ways that they used to come up with the answer to the problem. They are making sense of and figuring out the problem. The environment also has to be one where math is not threatening and every student is respected for their ideas.

What does it mean to "do mathematics?"

According to Van De Walle, "doing mathematics" means that students are engaging in the science of pattern and order, and doing mathematics is effortful and often takes alot of time (13). I think that in order to be able to"do mathematics" children should be actively involve in the process of understanding how the problem works, how they can connect it to their lives, and how they can solve the problem in not just way but in maybe several ways. We are taught in school that there is only one way to do a problem, but in all reality there could be several ways to solve the problem and get a probable solution. "Doing mathematics" also means that it shouldn't always be easy, we should be able to dig into it and struggle a little bit. According to Van De Walle there are many verbs we can correlate to "doing mathematics." For example, we can correlate explore, investigate, conjecture, solve, justify, etc (13). When students are engaged in these verbs it is very impossible for them to be passive learners or observers (13).

What does it mean to "do" mathematics?

Please add your comments to this question raised by chapter 2 of Van de Walle. Comment directly to this post about what it means to be involved in "doing mathematics."

What does it mean to "do mathematics?"

Doing mathematics means that one is actively engaged in solving problems. While doing mathematics, one is looking for patterns and order to come up with the answer. When doing mathematics, one is not just following rote procedures that were previously taught by a teacher. By doing mathematics, one is able to understand why they got the answer or why a formula found works in the given situation.