What knowledge have you gained about how children learn mathematics from this course?

1. I have gained a lot of knowledge about learning math in this course. Children all learn differently, we all already knew that, but what was unknown was different ways to teach how to solve a problem so that many different children can find the correct answer based on what they already know. Children can come to the same conclusion, same answer, even if they start at a different spot of the problem because it depends on where they are at and how they are attacking the problem. It is important to let students know that they can make mistakes and that's ok, it is also important to let children explore a problem before you give them too many limitations, the "correct" way to solve the problem or the answer. It is also valuable to let children discuss and verbalize what it is they are thinking, how they are getting to their solution.

What knowledge have you gained about how children learn mathematics from this course?

What knowledge have you gained about how children learn mathematics from this course? I have learned that children truly learn in different ways and those ways have to be met the best way possible. There is more to mathematics than showing students "this is how to do it" and then hand them a worksheet to work out similar problems. This procedure or way of teaching makes future math teachers to underestimate their students abilities because they only know how to think of a problem one way. Teachers need to become aware of what their kids know and then go further indepth on the topics that need to be. Children are really curious at different ages and I think teachers need to take that into consideration as well. When using manipulatives, question the students about them and about how they are thinking. Who knows how many ways will turn up to solve one problem.

How has this course changed or not changed your perceptions of teaching math?

I originally thought that math was as basic as introducing a basic formula to students and building on that. I never thought much about using manipulatives past the younger ages of students. I liked how the class forced me to look at problems the way my students would and break them down in order to be able to teach them to my students. This class frustrated me several times because it took me so long to understand the importance of breaking problems down the way Dr. Rein's did. Students need to be challenged and given opportunities to approach problems in different styles. Each student learns a different way and at a different rate, which means that my instruction should cater to the differences of my students. In order to teach a problem, it is important to start with the basics and work my way up.

Last two Questions--Kendra Lundeby

1. What knowledge have you gained about how children learn mathematics from this course?

• While being in this class I have learned many important aspects on how to teach math. I learned that is important to teach students the strategies and steps to reach the understanding and solution rather than just stating a formula. Explain why the formulas are set up the away they are and plan many activities to learn the stages of the problem. Teach students the foundation of the math so they create a strong foundation of the material before they need to use it in their worksheets and activities. I have gained a better ideas on how to properly teach math so that more students are understanding the material. In this course I have gained an understanding on how the teach math better while being in this course.
  

(2)

How has this course changed or not changed your perceptions of teaching math?

This course has changed everything I have thought about math and doing math. I now know that helping students to reach inside to help themselves build on prior knowledge is a great way to learn math. Once given a problem, students are able to figure out a problem with out equations that they would have memorized in the old way of teaching math. Often students, if given time to think and struggle, will find a way to solve a problem.In the same classroom, with the same problem, there will often be many different ways that will be shared for solving the problem. Students will then learn that with the different ways to solve a problem, they can use other strategies to check their work. One of the things that really stuck out to me was that students should be praised, even if their answer is wrong. I think this is a great concept, because if a student is afraid of making a mistake, they will not try very hard. Overall, I think this class was great because it helped me see a whole different outlook on teaching math.

Knowledge gained

What knowledge have you gained about how children learn mathematics from this course? I have learned that some educators underestimate the abilities of children to solve mathematical problems. Teachers do this by sticking to the material in the book and only assigning worksheets instead of creating a classroom where students are able to really show what they know and the processes they take. To truly learn mathematics, children need to be presented with real life problems and allowed to discover the answers on their own or with classmates or at least attempt them. Too often we are just taught the rule but are not informed of why the rule is the way it is when knowing the reason would have students understand mathematics much more. Every child has the ability to learn, but it is up to the teacher to create material that is pulls students in and has them interacting with the material.

From this class I take so much knowledge and I feel more confident not only on how to nurture every students learning styles but I also how I learn as an individual. I have gained confidence regarding teaching math. Math is one of my weakest points in school and it has always been. I was terrified to get into a classroom and teach math, I felt unprepared and not capable to introduce any sort of math. I may not be able to nurture every students learning ability but I feel confident and prepared enough to reach more than 90 percent. I have gained a tremendous amount of knowledge in this class and I cannot wait to put it all in practice.

Perceptions of teaching math

How has this course changed or not changed your perceptions of teaching math? I feel more confident about teaching math. When I was in high school all my teachers taught math pretty much the same way. They would do the examples out of the textbook and show them on the chalk board, and then give us an assignment from the textbook or worksheet. There was never an explaination of 'how' or 'why' we solved the problem that way just here it is learn it method. I now know that there are many different ways to teach math to students. It is good for the students to struggle with the material and work on it by themselves to find multiple ways to solve a certain problem. The students may also use (electronic) manipulatives to help them work out the problems intead of just paper and pencil. I will always find exciting, hands-on ways to teach math to my students so they become interested in it!

How has this course changed or not changed your perceptions of teaching math?

This course has definitely changed my perceptions of teaching math. Before taking this class, I always thought that maybe it wouldn't be so hard to teach math to young children. I have now realized that it can be quite the process. There is a huge difference between children understanding math symbolically and children understanding math conceptually. I have learned how important it is to gain a conceptual understanding of math. We need to understand why we do what we do. When I learned math, I learned it the symbolic way. It was easy for me to understand, but I know that will not be the case for all children. Some students will need further instruction when it comes to math and will need to be taught in different ways to fully understand math. I feel as though this class has changed my perception of teaching math in a good way. Sometimes this class was frustrating because I had a difficult time understanding some concepts; but I definitely understand now why the strategies that we did were important. Gaining a conceptual understanding of math and using manipulatives is extremely important and I plan on teaching students math in this way.

How has this course changed or not changed your perception of teaching math?

This course has changed my perception of teaching math in many ways. Before taking this course, I thought the way I was taught math in school was going to be the way I would also teach it to my future students. Until this course, I thought math was about rote memorization, worksheets, and formulas. This is probably why for many people who are not good at memorization do not enjoy math and therefore dread having to teach it. After taking this course, my perception of teaching math has really opened up to the possibilities of making math fun and enjoyable for students. Engaging students in hands-on activities, using manipulatives, working in groups, and having students think about what they are doing and justifying their work are all important concepts I have learned to do when teaching math. Before this course, I was also unaware of the power in using concrete, semi-concrete, and symbolic models were. If using these models helped my grasp a deeper understanding of math concepts then I know teaching it to my students will help them as well. I feel better today about teaching math then I did previous to this class.

How has this course changed/not changed your perceptions of teaching math?

This course has completely changed the way I thought about teaching math, and the way I thought about math in general. I now realize the importance of allowing opportunities for students to understand the reason why you do something in math (area formula) instead of simply telling them what to do and giving them a worksheet. I also now realize how important it is to have a few projects that are real world related. When I was in school I would often wonder why I had to learn certain math things; with opportunities like these students will know why they are learning the things they are. I have also learned that children want to be challenged. Sometimes educators think that a math concept will be to hard for their grade level of students. I believe that with the right instruction students will have the confidence to tackle some of the hard problems and even learn form their mistakes.

How has this course changed or not changed your perceptions of teaching math?

This course has changed my perception about teaching math in many ways. The prospect of teaching interests me in many ways because of my desire to engage students creatively with the material and information that they need to learn. Math has been a subject that I didn't think that I would be able to approach in that manner. Based on my history of learning math it has all been rote memorization and replication of work. The worst part of math for me is when I could find the answer but didn't follow the steps that the instructor required and was counted as incorrect for the work. This class has taught me that if I am willing to put the work into my lesson designs math can be taught in a way that it engages students and supports their actual learning, not memorization. This belief that math can be taught in a way that allows the students to construct their knowledge and possibly find enjoyment and challenge in the subject of math motivates me. This motivation will lead to creating lessons that challenge students through creating significant math tasks, creating an environment that supports mistakes, and supports students in doing math. This class has definitely changed my perspective on teaching math, but more than that has supplied me with tools and knowledge to continue to grow professionally in this area of teaching.

How has this course changed or not changed your perceptions of teaching math?

This course has changed my perceptions of math greatly. Before I took this course I really hated math. It was not a subject I enjoyed and I would never take math classes unless I had to. I now understand that the reason I hated math is because I was being taught the wrong way. Math is actually quite easy when it is broken down. The different methods that we were taught in this course have greatly helped me at math and has made me a better math teacher. I know can say that I enjoy some math. I am not as scared to teach math and I feel more confident to explain math concepts to my future students.

What knowledge have you gained about how children learn mathematics from this course?

I have gained a lot of valuable knowledge about how children learn mathematics through this course. Before this class, I knew that how I learned math growing up was not how we were supposed to teach it now. I now know some great ways to teach math and I have learned in what ways children learn math best. I have learned that children do not learn best through rote memorization. We should not teach formulas and other mathematic problems to just memorize but we should teach the reason to why they use those formulas, etc. Children learn best through the use of manipulatives with concrete examples and then move into semi-concrete examples and finally symbolic examples. We should not use worksheets and lecture, but we should use hands-on and interactive activities. The children should feel comfortable in taking risks while learning and sharing their answers. This class has taught me some valuable lessons and techniques to use that will help students become successful learners in math.

How has this course changed or not changed your perceptions of teaching math?

This course has changed my perceptions of math instruction. Before this course, I had never thought about the purpose behind solving a problem or any key math concepts. I simply plugged numbers into an equation until I found the correct answer. Teachers did not let us or tell us of multiple routes to solving one problem. My teachers would explain on strategy and ask us to follow the strategy when working on our homework. Additionally, manipulatives were hardly ever introduced in my elementary math classes. My hope is that I can help strengthen the math skills of my future students by introducing the use of manipulatives to better support their learning. It is important that we allow students time to think through a set of problems or mathematical questions before we give them the answer. In other words, we should guide students through the mathematical thinking process without giving away the answers. Lastly, students should be placed into groups where they are asked to discuss their individual ideas and solving strategies with their peers. All students come to the classroom with different levels of knowledge which they should share with others. Students should not be afraid to make mistakes or ask for help.

Changed Perceptions of Teaching Math

How has this course changed or not changed your perceptions of teaching math?
This class made me realize ways students learn mathematics the best way. Before this class I did not know that manipulatives were the best way to teach students how to do many math concepts. Working with concrete models help students understand the basic process of a concept and then moving to semi-concrete helps the students see the whole procedure. Students should have the opportunity to discover math concepts on their own and work with each other to explain concepts. It is important to have students explain and know why they are doing every process. Not all students come in with the same knowledge and students start a concept at all levels. If students are given the opportunity to explore the concept, they can enter the problem at different levels and can discover different ways to solve the problem.

How has this course changed or not changed your perceptions of teaching math?

This class has changed my perceptions of teaching math. Up until this class I never really second guessed what and how I was taught growing up in school. I always liked math, it seemed to come easier for me than some of my other classes. I guess that might be why I never gave it a second thought. I have seen that there are new and different ways to teach math to students. I agree now that there should be a progression on how to get to the main goal. The only concern of mine is becoming comfortable enough with it to effectively teach this way in a classroom. Hopefully this will just come with practice and time.

March Blog-Personal Concerns

Math is a confusing topic for me and my biggest concern is that I will not be able to teach it effectively to my students. Through this class I have learned more about how to teach math but only to a certain extent. I obviously will not learn how to teach every math topic through this class but that is definitely a concern for me. I feel more comfortable with teaching the topics we have touched on in class, but I'm still concerned with how I will teach the rest of the topics in math. I am also concerned about how to use the games and activities we have done in class in my future classroom. What if the school does not agree with the way I would like to teach math? It concerns me that I will have the correct tools to use from this class but how will I go about using these tools in a school that may not agree with them?

How Children Learn Mathematics

Through this course I have learned many new strategies to use when teaching math, and information about how children learn math best. I have learned that math learning takes place when a child is as early as 6 months old and it is best to start teaching math through basic strategies as soon as possible. Math may be confusing to many students and it is a teachers duty to teach math through the best approaches. I believe I have learned a variety of approaches that will help my as a future teacher. I know there are certain steps to teaching every math topic and it is best to start out with learning what the students already know about the content and build from there. I also know that it is best for a teacher to start out with a hands-on activity that does not deal with symbols or vocabulary words. This way students can develop their own understanding about the content and the teacher can build their knowledge based on the strategies the students are using. I have also learned from this class that it is vital to ask the students questions about their learning process so the teacher can have a full understanding of what the student knows and what may need more focus. Overall, this call has taught me a variety of games, tools and strategies to use in my future classroom.

What knowledge have you gained about how children learn mathematics from this course

This course has better helped me understand the level of student thinking when it comes to mathmatical concepts. Math can be a very confusing subject for many differnt students. Be able to understand how students think, apply, and comprehend material, will better help me to present new material. Math can be taught at an very early age. It is important to help children learn basic counting skills in order to build upon in later years. Children have greater critical thinking ablitities than sometimes credited. Its important to allow students to make mistakes and help them learn from the mistakes. As a teacher its important to ask questions like, "how did you solve this, why did you do this this way?" Asking questions helps us, as teachers, understand why a child solved or applied a concept. This class has provided many new ideas and games to present to all levels of students. I have learned alot from the LPU's and plan on applying them in the future in my classroom.

New Insights and Their Implications

Throughout the semester I have learned a lot of new things that will help me be a better math teacher. There are many different semi concrete models that will help me teach my students complicated subjects like dividing fractions. I have also learned that it is important for students to understand WHY they have to do something (like flip the second fractions when dividing). A better understanding of concepts like these will help students gain confidence in their math abilities.

New Insights and their Implications

I have really began to see how this type of teaching can be very beneficial to students. I really wish that I was taught this way when I was in elementary and high school. I believe that if this were the case then, all of the "rules" that I learned just because might have been learned for a reason and a truer understanding. I have also seen that this might be difficult to teach this way. We have see a couple of groups teach in class now and it seems a bit more difficult. But it may seem that simply because we have taught that way. I am hoping that is the case.

New Insights and Thier Implications

From class, I have learned how much different it is to truly learn math. I feel that I have missed out on a lot of much needed information and as a result I don’t have a concrete understanding of reasoning’s in mathematics. As a result of only knowing how a procedure is done and not ‘why’, it is making this class a little more difficult to grasp these understanding. While it is stressful trying to understand the ‘why’ of the question, when I do get to that point in the understanding I feel as though I am in fact learning it for the first time and the right way. I have learned from my peers, that everyone does learn in a different way and it is critical as a teacher to be aware of that. Also, I have learned how useful the right manipulative is while teaching a lesson. Manipulative are able to help further the knowledge by allowing students to visually see and touch the objects. Through these interactions, students are able to learn math in a much greater, better way.

New Insights and Their Implications

In class I have learned that is best to give students problems and let them struggle and manipulate them to solve them. The students may work with partners or on their own to figure out the problems, and give them an adequate amount of time to work on the problems. I am doing my internship in Newcastle, NE at the junior high level. Wednesday I taught my first lesson to 8th grade algebra, and I taught them how to graph linear inequalities. First I explained what they were and how they were similar to graphing linear equalities, which they did the day before. I then showed a few examples on the board while explaining the material and getting their input from previous knowledge. They did great as a class on the board, but when I gave them problems to manipulate at their desk with graph paper I got a lot of blank looks even working with partners more than half of the class was lose. I had one of the groups of girls come up to the board to share their answer (which was correct) and explain to the class how they got it. The boys were still confused. I went over the steps and process again and they seemed to understand or were just pretending. I applied all the knowledge I have learned about the learning process and still feel that my lesson was unsuccessful to fifty percent of the class.

Summary and Synthesis

Since my last posting it has really hit home for me that it takes a commitment from the student to be willing to work through the process in order to truly construct their knowledge. This became very apparent to me during the work with fractions that the first group presented to the class. I knew that I could find the correct answer if I resorted to my existing knowledge of the steps needed to resolve the problem. I also knew that if I wanted to experience the learning the way that a student who wasn't taught the procredures but instead was provided with a conceptual way of learning that I would have to struggle with the math. Trying to stay away from the procedural steps and work with the concrete and semi concrete models in order to find the reason for the procedural steps I went back through the models and tried to evaluate what I was learning about the problem. Due to spending the time struggling I did come to an understanding of moving from the semi concrete model into the symbolic problem. Experiencing this process instead of reverting to the methods that I knew would give me the right answer was definitely beneficial to me. The willingness to be uncomfortable during the process is something that as a teacher I will be asking my students to do; I think that I need to be willing to experience it also. This really brought the message of constructing our knowledge about math to the forefront for me.

Summary and Synthesis

When we came to class the first day, I was honestly scared about this class. I was to figure out how to find the answers without being told how to find them, was a scary concept after being told what to do to find the answer for so many years. I have actually enjoyed class a lot, I like being challenged, and being able to find those answers on my own or within the group work we do. I like that we are learning many different approaches to the same problem or question given to us. I am excited to be a part of this new way of teaching, because I think it is the right direction in helping students understand what they are doing. I have already shared some of the concepts with my oldest daughter, and cannot wait to teach my future students in a way that is easier to understand and realize why they are doing what they are doing.

Personal Concerns and Next Steps

I have always enjoyed doing math and taking math classes. I think the reason I liked math was because the problems had a set of steps and as long as you followed those steps in the order you were told you would get the correct answer. I have always been good at following steps and therefore received good grades in math. Until this class, I never thought about the reasons behind doing the steps and having to follow them in certain order, I always just did them. I memorized the rules and followed them. After taking this class, I am now concerned about all the things I really didn’t know about math all along, the “why.” Why do the steps result in the correct answer or why does that rule work? For so many years, I have done math the traditional or instruction way without really knowing why. I know habits are hard to break and I want to make sure my future students know the reasons behind why math works and be able to construct their own rules. I want math to be relational for them not instructional like it was for me. In class, it has been hard trying to reconstruct what I already know about math and not resort to just using the formulas to find the answers. So far I have been able to follow along pretty well and after taking a look at the processes of why the steps and rules work and multiple entry points of problems I now have a deeper understanding of math. I am just afraid I won’t be able find a way to make all math relational to my students. I now know how to make the things we learned in class relational but what about math topics we didn’t cover. Will I be able to teach in a relational way or will I resort back to what I already know and teach the traditional way. How will I know what to do or how to teach? This class has really opened my eyes into a math world I didn’t know existed and now it is up to me to find the confidence to re-teach myself and find out more of the “why” behind the rules.

Summary and Synthesis

Well, I have learned things the hard way and sometimes that is the best way to learn them but sometimes it shakes you right down to the bone. I really have learned a lot about Math in a more relational way which is how Math should be taught. Kids will learn better especially when they relate and make connections to the material. It is hard to go from a way of learning you learned throughout school and just all of a sudden switch. That is why it is SO important for children to have this type of relational learning from the beginning. This class has given me a new insight into Math (one that I honestly didn't even look at because of the way Math was taught to me). Even though the traditional route is the most common form of teaching; it doesn't make it the most influential! I am still scared that my fears about Math will cause problems in my future class but it is a challenge I am trying to face and conquer (somtimes it is not as nearly successful as I want it to be). Finally, I think it is important for children to know the HOW and the WHY when learning Math. I honestly don't remeber wondering why we invert the second fraction when dividing; I think that is because of how long I was taught through an instrumental way. Tons of children ask this question though and what do teachers say; honestly? After this class, I do have a better understanding of why to do certain steps.

New Insights and Their Implications

So far, this course has given me some great insights as to the new way of teaching math. I have learned that we need to work towards giving our students more of a relational understanding instead of an instrumental understanding. It is imperative to make connections between what they are doing and why they are doing it. From the discussions from my peers and from Dr. Reins, as well as the research I have done through the LPU, I have learned that students learn math best when concrete experiences are worked into symbolic experiences. This can first be done by first working on concrete examples through manipulatives and visuals. Then once the students obtain and understanding of the concept, they can move into semi-concrete examples by still working with manipulatives but by adding number sentences, etc. Then finally, once the students have grasped the concept, they can learn the symbolic meanings of what they have already learned. By doing this step by step, the students can gather their own conclusions and meanings on that topic and develop their own understanding--rather than it just being given to them. All of this is going to be extremely helpful to when I teach in the future. Since I am used to more of a instrumental way of understanding, it is important for me to switch my ways and develop my own teaching into a relational way of understanding. By doing this, my students will become successful learners in mathematics.

Personal Concerns and Next Steps

Upon entering this class, I really enjoyed Math. It has always been one of my favorite subjects and I felt as though I understood the basic concepts. For my internship, I helped in a 7th grade math classroom and I loved it. After that experience I had positive intentions of teaching Math in my future. Before taking this class, I was excited to learn how to break down math and learn more about teaching it to younger kids. After participating in this class, it has really made me look at math a whole different way. Some good, some bad. It makes so much more sense to teach kids on a relational basis rather than an instrumental basis. I came into this class strictly knowing only an instrumental basis of math. To me, the instrumental understanding makes sense because that is how I have always learned it. It also makes much more sense to teach kids a relational understanding of math. I am definitely more concerned now about teaching Math in my future because even I struggle with understanding a lot of what we are currently doing in class. If I have a hard time understanding it then I am worried about teaching it to my students. My concerns have been lessened a bit by understanding the process of teaching through concrete, semi-concret, and symbolic ways. I now understand better how to teach in those ways and how beneficial they are to students. I still struggle with it, but it is beginning to make sense. My next steps include trying to work harder in our math class to understand your styles of teaching and understanding the "why" part of mathematics. I think it is important for us to understand why we do stuff so I am going to continue working hard to understand that. Since this class I have developed a lot more concerns with teaching math in general as well as my own knowledge in math, but my next steps will include working hard to develop a more thorough understanding.