Constructivism

Constructivism is where a student builds on prior knowledge to create their own ways to solve any problems and establishes their own learning styles.

What is Constructivism?

Please copy-paste your response to this question in a comment to this post. Do not start a new post. Thank you.

New Insights and Their Implications

New Insights and Their Implications - What did you learn from your peers, from the instructor, and/or the readings, about elementary school students, and/or about yourself, and the teaching and learning of math and what are their implications to teaching and learning mathematics?

During the past month in class I was amazed on the many different tools that can be used to help students learn fractions. I really enjoyed learning how to incorporate models into the classroom as well as experimenting with them myself. I thought the interactive sites on the web were outstanding and look forward to using them in the near future. I have always enjoyed fractions, but I do realize they may be difficult for some to understand so with the help of what we have learned in class as well as the models I believe I can now more effectively teach fractions. I look forward to learning more about math during the last month of class and can't wait to adapt this information to my own classroom.

Question and Answer

When we started the section on fractions I knew I was going to have problems understanding this topic because it has always been something that i have struggled with. As we went through the weeks about discussing fractions and learning more about them a lot of my questions were answered. The first question that i had was, how am I suppose to teach about a subject that I am not comfortable with. The solution I have come to is I will use my resources. Dr. Reins has pointed out a couple websites that I can use as resources to help my future students understand fractions. I will also use my book as a reference because they have many good ideas that I can use in my class room. The second question that I had was, what other models are our there that I can use to teach fractions to my students. This question was answered when as a class we were introduced to fraction bars, pattern blocks, dot paper, cuisenaire rods, and the cake pan method. I quickly caught on how to use each of the models and I will be using them in my classroom as well to help further my students understanding for adding/subtracting fractions and multiplying/dividing fractions. After the lessons on teaching fractions the only question that I still have is with all of the methods that we have been learning about is there one we should start with first or is it OK to start with any and then cover them all eventually? I feel that I would want to start with the one that know the best so I can give the students a foundation and then branch of in showing them all the other methods.

Summary

I enjoyed learning how to use all the manipulatives we used in class and the different ways that they can be used to teach. I liked learning why we do the steps we do to solve math solutions instead of a rote method of trying to memorize. I enjoyed watching the CGI video in class, it was good to see an example of how it is supposed to work in the classroom environment. I think the CGI method could be very beneficial for students as long as the teacher can guide the lesson and engage the students in the activity.

Summary and Synthesis

I actually liked this last unit on fractions. I liked that we were able to work with different manipulatives like the pie pieces and fraction bars. Most of what I know (with Math) is due to rote style learning. I like that this last unit gave us different ways to teach fractions. I agree that teachers need to start showing their students how to develop an understanding of "why" to do things, rather than just doing things without a reason and just off of memorization. I believe that students will have a better chance of understanding and remembering different mathematical rules and operations if they understand why they are doing what they are doing. I liked learning about the CGI program as well and it was nice to watch the video to see the classroom with that learning approach. I think that it is important to give students many options when solving problems and I thought this unit did a great job of showing us future teachers how to achieve that goal in our own classrooms!

Summary and Synthesis

We have been learning about addition and subtraction problem types. I have never heard of join, separate, part-part whole or compare.I think this is an interesting way to look at problems. It took me awhile to understand the differences of and what each one meant. These four are different structures of a problem and they describe the way a number sentence will be written. This is good for teachers because they know where to start when giving their students addition and subtraction problems. The teacher will also know what types of problems to give to the students and also what types of problems the student struggles with.

Summary and Synthesis

During the last month we have discussed the different ways students can use manipulatives to help them solve addition, subtraction, multiplication, and division of fractions. This is a very important skill for students to have, because it requires the students to think about fractions without just looking at number and memorizing steps used to solve for the solutions. I thought it was very beneficial to have multiple manipulatives, especially the Cuisenaire Rods online. It is a known fact that students have a hard time learning fractions and I agree that building the skills using manipulatives in the beginning will help the students with the skills they need when solving fractions throughout their education.

Summary and Synthesis

This last unit over fractions introduced me to new ways to teach math. There are many different manipulatives available to help make math more hands on and gets students involved. I learned that manipulatives that are not labeled are better when teaching fractions because they can be used to represent different whole units. I really wouldnt have thought about that unless it was brought to my attention. We have also been discussing how a CGI classroom is set up and how it runs. I think this is really interesting because I know personally I like directions and clear and concise explanations, however according to this method of teaching it is all about student exploration and figuring things out on your own. In some ways I agree with this and in others I dont think that it is right. I think that to an extent students need to be taught basic skills and concpets about math and then five them problems to solve on their own. Personally I am not a birg fan of the whole exploration technique because I am HORRIBLE with math concepts and why they work and how you do things. I like steps and directions... but I guess that is something I will have to come to appreciate!

Questions and Answers

One question I had at the beginning of this unit was, how can I teach students about fractions so they have a relational understanding of the process? It was obvious to me that I only had a conceptual understanding of most processes and did not understand WHY we follow certain steps to solve a fraction problem. As a student, I do not remember ever using the manipulatives as extensively as we have in this class to understand the fraction concepts. As a future teacher, I now understand the importance of using those manipulatives and following certain steps to develop the concept of fractions in my future classroom. While I know the steps are not a foolproof manual that will give the students immediate and complete understanding, I realize that the steps can be used as a guide to get students thinking and building on their own ideas about fractions.

Questions and Answers

As our ELED 330 class went into the fractions unit, I had many questions about fractions because of my background with adding, multiplying, dividing, and subtracting fractions. When I learned fractions in grade school, I don’t remember using so many models. It’s refreshing to know that when I teach fractions, that I will be able to provide and use many different types of models in my classroom. Also, I have a clearer view of the different methods that can be used in helping solve fraction problems dealing with addition, subtraction, multiplication, and division. My question at the beginning of this unit was, what method should be taught to help children learn more about fractions? I now know that the students should have the opportunity to look at a problem first and find their own methods. Since I want to teach elementary, a good way to let children explore their own methods in fraction problems would be through literature.

Throughout my time in K-8 Math Methods, I have learned a ton of valuable information when it comes to the teaching of mathematics. Looking back, I can say that when i first entered this course, I was scared to death. I had very little confidence my my mathematic abilities. However, I have definitely seen a change in my confidence over the last three months. I am no longer doubting myself when it comes experimenting with different mathematic concepts. My understanding and confidence in the area of math has definitely increased. However, I still have one question. How well will I transfer this new understanding for teaching math into the classroom? When I think about this question, I think about incorporating these new learning experiences into the classroom while trying to meet state standards, preparing my students for state mandated testing, and more. However, as I mentioned above, when i first entered this class, I was scared. Yet over time, my confidence in mathematics grew. I believe the same process will happen when it comes to incorporating these teaching strategies in the classroom. Over time, my confidence in the teaching of mathematics education will increase as I gain experience in the incorporation of these strategies.

Questions and Answers

Many of the questions that I have had have been answered throughout the last couple of class periods. I had always wondered what types of tools are appropriate when instructing students and trying to help them understand a concept. After talking about addition, subtraction, multiplication, and division models and processes, I understand now that there are so many ways to teach these applications. My question still unanswered, is how do you choose which model or method do you teach first? I know no matter what that when I choose method, I'm going to have to teach a different student a different method because not every student will understand the same way, but how do you know which model will be the most successfull within your classroom. This is my only remaining question, otherwise I feel that I have a good understanding of the various ways in which to teach these processes.

New Insights and their Implications

Learning about Cognitively Guided Instruction has really helped me to understand how I can apply the math techniques we are learning to my own classroom. Reading about the CGI classroom was very helpful to realize the importance of the student's learning process and how a teacher needs to take that into consideration. It was very helpful to watch the video of the two different classrooms to see how those teachers integrated CGI into their classrooms. It is helpful learning about it in class and reading about it in our textbook, but it was nice to see a real classroom and be able to see how it all works. I think the CGI approach is one that I strongly believe in and I think it will benefit my students as long as I incorporate it the right way. I also think that reading the article written by Ashlock was very helpful. I never really thought about how important it is to notice not just that your students are getting answers wrong, but why they are getting answers wrong. Working on the error patterns from Ashlock's book really helped me get a hands on experience to start noticing student's error patterns and being able to recognize them so in turn you can provide activities or strategies for them that will help them to overcome these errors. I think Ashlock's book has a lot of valuable information in it and I look forward to reading it and learning more about strategies that I can incorporate into my future classroom.

Concerns and Next Steps

Wow! Each day of class I learned something new and different that gave me a whole new perspective on teaching mathmatics. Through class discussion, hands-on activities and trying things out, I have gained new knowledge and an understanding of how to solve problems and why we solve the problems this way. I feel like I have moved from a procedural understanding to a conceptual understanding of problem solving. This shift in knowledge has made me more aware of and given me a deeper understanding of why things are done the way they are in math. Rather than just doing the steps because thats what we were taught, I undestand why the steps are what they are and why we follow the steps to arrive at an answer. I also learned that there are multiple ways to do a problem. This class helped me try things out and work them out on my own and STRUGGLE, rather than having the teacher just give me the answer. Having a conceptual understanding of the "hows" and "whys" in mathematics, I feel that I will be able to better teach students in the near future. From the new techniques, strategies and how-abouts discussed and learned from this class, I hope to give students a deeper and greater undestanding of mathematics through this new approach to teaching. As a future teacher, my "next step" is to shift student's knowledge of mathematics from a procedural understanding to a conceptual undestanding. This of coruse will be accomplished by applying the techniques, strategies and way of teaching math that I have learned in this class to my own future math instruction. My one and biggest concern is that I won't be able to steer away from the traditional way of teaching matehmatics to teaching this new approach we have learned in class. I fear this because this is what teachers are trained to teach and only know how to teach because thats the way the textbooks outline it. I am concerned that I won't be able to shift students understanding of mathematics from prodcedural understanding to a conceptual understanding. However, after being introduced to CGI classrooms, I feel that this would be a great place for me to start to make this change happen. I hope that from the new knoweldge and insight I have learned from this class, I can change and make a difference in mathematics instruction and students understanding of math.

Concerns & Next Steps

When we first began this class I figured it was going to be like all other math classes where we would be taught something and be forced to repeat that process one million time over until it was just in our memory. I enjoy this class because it is the exact opposite, we are learning new ways and techniques to approach and solve problems, as well as rethinking a whole new way to teach our future students. For some things in the class i find difficult at first (such as Pick's Theorem) but after discussing it in class and completing some examples in class I find myself understanding it much better. I find it very helpful to take my own notes during class so that I am able to look back and see how we completed something. I like that we are able to discuss things in class, we may not get everything covered but I would rather understand something than move so quickly to cover everything and end up feeling left behind!

Summary and Synthesis

This month flew by and I have gained a lot more knowledge on certain areas than I had before. Most recently was the knowledge I have been gaining on fractions. To be honest I never analyzed how we teach fractions, I feel I never paid attention because fractions has always been an area that confuses me at times. I can't wait to find out why we flip a fraction for division because I really have no idea why we do that, or how even multiplication works. I know how to do all of that I just don't know why we do it. I think pointing out 3/4 to one whole is completely different 3/4 to a different whole. When I first learned about fractions this confused me until I got to Junior High, it is something I am going to make sure is clear when I become a teacher.

Summary and Synthesis

Summary and Synthesis - Summarize and synthesize the impact of recent experiences, ideas, and/or issues encountered on Tuesdays and Thursdays in class from your perspective since the previous blog.

During my first blog, I really had no idea where this class was going. I was unsure the point of the way you were teaching, yet I did know it had it's purpose. Now my understanding is that you are teaching us the same way we need to be teaching our students. Asking questions, and finding solutions to our own problems is kind of the way the course is going, and is the way we want our students to be thinking. Instead of being spoon fed information, you are providing us with problems where critical thinking skills are needed, and where we have to be motivated to figure out the solutions to the problems presented. Although it is a different way of thinking, the point is now becoming clear for this course, and our future in teaching math and other subject areas. This course is enabling us to find the reasons behind the spoon fed formulas provided to us in math- the way we were taught growing up. Now I understand a particular concept in math (for example: why area of a triangle is 1/2 bh) and can apply it to many other areas of math (for example, using triangles to find the shapes of polygons). This course has showed me there are many different routes a person can take in coming up with a solution, and our future children need to see this as well. This course is providing me with the necessary skills needed to be able to get my children to think critically, and problem solve in math. The way this course is being taught is challenging, but now I have deeper understanding and meaning behind the math that I originally learned. I now understand how being set in the traditional way of teaching math can offset students in not truly understanding concepts being taught. The whole saying, a mile wide and an inch deep makes more sense now. Because there are so many topics that need to be covered in math, teachers rarely get to touch base with students to know if they are truly understand the meaning of concepts being taught. This new way of learning helps students see the true meaning of mathematical concepts, which can be applied to multiple areas in math, as we are presented with a problem and have to come up with the solution.

New Insights and Their Implications

I really liked what Dr. Reins taught us about teaching new information, and getting students to construct new meaning based upon what they already know. For example as I was learning about polygons, area, and constructivism I was able to draw conclusions, and make connections to build my understanding of the topic based upon my own comprehension. The idea of parroting back information for a test or an assignment seems to be useless for long term memorization and overall comprehension. I believe that the movement from covering information in breathe to covering information in depth is tough for many teachers to try as they are pressured by schools and standards to become accountable for the students learning; yet it is most useful to get students to become accountable for their own learning. Yet, it is proven that students who can create meaning and make it valuable to their lives are more likely to retain in long term memory. I would like to make this part of my own teaching motto, to get students to understand using scaffolding as a motivational tool in the classroom.
I also liked what Dr. Reins has showed us about drawing conclusions based upon Pick's theorem, the chop method, and drawing rectangles to find the area. This helped me move my thinking about area from strictly formulas and numbers to more theoretical and higher leveled think about areas. I truly enjoy learning this new approach to mathematics.

Personal Concerns and Next Steps

Through out the last month I have had a great time continuing to build my knowledge on how to correctly teach math. I am beginning to understand how this method of mathematics instruction can be used to the classroom but do have one concern related to this technique. That one concern is how to teach math using this method and still find time to cover all the standards and continue to stay on pace with the curriculum. I understand that its important to make sure all students get a solid foundation set and then build upon that but it is difficult to teach this way and still cover what needs to be covered. If it was up to me I would spend as much time as possible teaching mathematics the way we have been learning in class but in order to keep my job I understand that I will have to meet all standards and cover all of the curriculum. Teaching mathematics in a constructivism manor may be better for the students but I'm still unsure how I can speed up the process in order to reach the goals that is set forth my the school district I'm teaching in. I look forward to learning more in the next two and a half months and improving my teaching skills.

Personal Concerns and Next Steps

Personal Concerns and Next Steps - You may blog about genuine personal concerns created as a result of the experiences in this course. (Please note: If you blog on this item be sure to provide sound rationale for your concerns and what steps you are planning to pursue to address your concerns. This is not an opportunity to vent, be constructive and professional about it. Don't complain about something unless you offer a better alternative or solution.)

Personally Math has never been a favorite of mine because I have struggled with it. This last test was long and covered a lot of material that I thought I knew, but now I'm not sure that I knew it well enough. I know it's important that I learn and understand this material and I think that I am going to take more notes and make sure I understand the questions that are being asked at the begining of each chapter. I do like that this class is an hour and fifteen minutes because it allows us more time to go over everything and discuss.

Personal Concerns and Next Steps

When we first began this course, I was unsure of how it would help me understand and apply math methods in classrooms. Rather than focusing on definitions and formulas, this course aims to explain how definitions and formulas are formed through understanding the process that it takes to create them. For example, in the Pick's Theorem activity, we did a number of examples to demonstrate how the formula is formed, and how it applies to different shapes. Another example would be the "What's my Rule" Activity where we described the properties of shapes. Although I found it frustrating at first, having us explore what we know about concepts through worksheets, and then discussing them in class has proved to be helpful. As we near the second half of this midterm, I plan to take more in-class notes and to ask for help more if I don't understand a concept that is being covered in class.

Personal Concerns and Next Steps

There are many concerns that I have with this new method of math. I understand that exploring the concepts of math is important but it makes me wonder how the students are supposed to learn the basic skills of math. When I think of math I think of a set of rules and formulas and that help us find an ending number. Personally I am not a big fan of math to begin with so I dont exactly enjoy exploring math. How do you motivate children to want to explore math? I guess to me it is something very different and it is something that I will need to get used to. However, I do see a lot of classes besides math getting more into the explore and discover method of teaching rather than telling children what to know. It will be interesting to see how teaching changes in the next few years.

Summary and Synthesis

I was really confused on what we were doing and how we were doing everything since the last time I blogged. I felt that we did not cover the material we were going over in much depth and I was not getting what I was suppose to be getting out of class. The mid-term exam then kind of snuck up on me and I did not know for sure what to expect from it. I remembered talking about a lot of the things that was mentioned on the review guide, I just did not know for sure what Dr. Reins was expecting. Since the last time I blogged and now that the mid-term is over, I really have a better understanding of how this kind of teaching has an impact on somebody and how we need to relate this kind of teaching when we teach. I know how to have students problem solve, think critically, and remember the information that I teach them. If they do not remember the information that I teach them ten years down the road, such as equations, they will know how to come up with the equation on their own. I am gaining a better perspective of this class and now I do believe I am getting what I am suppose to get out of this class.

New Insights and Their Implications

I thoroughly enjoyed the instructors way of teaching how to find the area of polygons. At first I didn't understand what was going on but looking back I understand what he was trying to do. It was a great way to show us how to teach a topic so students actually understand what they are actually doing. I had to look up the the Picks -Therum. I needed to see it to understand. I am not one of thoes people that can pick up a math concept without being shown how to do it. I understand it better now but I needed to see what I was to be doing. Implications include a better understanding of how to teach concepts about math so students understand what and why things work. I can teach in this style with some small changes for the different ways students learn.

Summary and Synthesis

I have learned a lot from math methods, since we started this course. The newest topic I learned about was fractions. I never knew how I missed out on when I was kid. I thought all kids learned fractions the same way, but by learning them in a way of halves and that's it, it doesn't help them in the end. Teachers need to help students construct the idea of fractional parts of the whole and equal-sized portions or fair shares. By teaching these two concepts first students will gain a better understanding of fractions. To teach these two concepts teachers should begin with sharing tasks when developing lessons about fractions. These are just a couple of the things that I have learned so far. In class we were also talking about 6 important steps that we need to know. In class we did not get through all six, but I am looking forward to learning more about fractions. Another things that I have learned from the lesson this week was how many magazines there are for teacher to get resources to help teach concepts such as fractions. In the magazines there are so many manipulatives that teachers can buy to help students better understand the concept of fractions. I am glad we were able to keep the magazines we received in class because it will be a good resource for me when I become a teacher. Fractions is a tough topic to understand for some students, but with the right help from teachers and from the the use of manipulatives learning fractions can be fun.

Summary and Synthesis

Sitting in the first day of K-8 Math Methods, I felt somewhat apprehensive. Math makes me nervous. I have never felt confident in math. When we received our first math assignment, I was nothing but nervous. To be honest, I was not accustomed to this type of problem solving. I was expecting the formula and process method of learning. However, I soon began to open up to this new form of mathematical teaching. The experience of different activities within our class periods have helped me to gain an understanding for the hands on approach to solving mathematics we have been learning about. One specific learning activity that has stood out in my mind deals with finding the areas of a polygon. Experimenting with different methods of finding an area of a polygon helped to open my mind to a more constructivist approach to learning mathematics. I realized the importance of exploration in mathematics, and the impact is has on students learning. Another learning experience that has impacted my understanding for exploration in mathematics would be the combing of formulas to find the areas of a trapezoid or a parallelogram. These experiences have also helped me to understand the concepts behind the processes of mathematical inquiry. My understanding for these mathematical problems stand out for one reason, because I now understand the relational concept behind the process. Overall, I have realized the importance behind helping to students to develop conceptual understanding in regards to problem solving. Many of the discussions that have taken place within our class periods have helped me to become more confident in providing a more hands on learning experience for my future students.

Summary & Synthesis

Math Methods has been an eye opening experience for me. Learning math was a hard concept and subject to learn. I have always had a somewhat negative attitude towards math, because it has been a trying subject. Since taking Math Methods I have learned a different way to approach a problem and learned the strategies to solving a problem. I have also learned that the way I was taught was more less repitition and learning the information for a test and less learning on how to discover strategies and patterns to problems. In Math Methods I have learned that students need to be able to have a solving strategy that they can justify and have multiple routes to explore for an answer. I like the alignment process. This alignment component provides a continious loop of modifications and improvement of learning and teaching.

New Insights and Their Implications

It was interesting learning about the different steps you have to teach in order for students to be able to learn fractions. There are some things that we talked about in class that I never really thought about when working with fractions. Fractions was one concept that I thought was very difficult to learn when I was in the younger elementary grades, so that is why I found this to be so insightful. Talking about this in class gave me a new perspective on how to teach fractions and also made me realize the different new terms that you need to introduce to students when learning about fractions. I also thought that looking through the magazines was very helpful to get some ideas on some hands on activities that you could use in your classroom. Learning about this was very helpful for my future teaching of fractions and ratios.

Summary and Synthesis

Before thursdays' discussion on fractional concepts I thought of a fraction as a number over a number, the top being the numerator and the bottom number the denominator. A fraction simply tells us what part of a whole. After discussing fractional concepts in class, I see now why I, and many students sturggle with fractions; its because we lack the foundational understanding of fractions. There is so much more to fractions that I was never taught. I just learned the basics. Thrusdays discussion on fractional concepts gave me a whole new perspective and knolwedge of fractions, for my own persoanl understanding, but also what I need to do as a teacher to help students understand concepts of fractions. I am now aware of the 6 steps that need to be followed/taught for the development of fraction concepts. In class we discussed the first 3 steps. Step 1: Introduce Sharing and The concept of Fractional Parts and Understanding of the Unit, we discussed students notion of fair sharing and the teacher's job to break students away from using a halving strategy. I learned the best way to do this is to change the number of things being shared and the number of sharers (odd division) to force students to confront their halving strategies. Students must also understand a "unit" of a fraction before they can compare parts of a fraction to a whole. Step 2: Introduce Models for Teaching Fraction. During class we looked through two different magazines that had a wide variety of different models/manipulatives for students to use to help them understand fractional concepts. Choosing what manipulatives to use is very important for teachers to consider. It was brought to my attention, that what we see isnt always what the student sees. Therefore, we must choose manipulatives properly, introduce them to the students and ensure they will enhance and assist students learning. I also learned from class discussion that the lables on the models can sometimes take away from students true understanding by dening them the oppprtuinity to reflect on the size of the peice relative to the whol and decide for themselves what the fraction is. Lables on pieces also remove flexibility in the materials. Teachers need to consider these factors when using manipulatives. Step 3: Intrdocue Fractional Symbols. Here we discussed the importance of building ideas and notions first and then introducing symbols. In class we also determined the difference between fractions and ratios; which sadly none of us could explain the difference when asked during class. After comparing and discussing the two, we arrived at the notion that a fraction is a realtionsip between a part and a whole and a ratio is a comparision of two distinct sets. After thursdays class discussion, I have a better understanding of fractional concepts as far as the first three steps go. The topics discussed in class have helped me understand the way students learn and what specific concepts and ideas I need to teach them about fractional concepts prior to teaching them fractional computation.