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.

Summary and Synthesis

This semester we have learned numerous concepts and easy ways to use them in our own classrooms. We've been taught how to teach in a different way. We've learned many tools both hands-on and in groups. I've been really interested in learning about why we do certain things in math, such as flip the fraction when we are dividing. I think these types of concepts are importnat to learn because many students will ask why we do something. I think these concepts have taught me to appreciate math more because in the past no teacher has told me why we do things the way we do them. Most of the time we are just given math equations, a few examples, and then given a bunch of math problems to complete. I've never liked that way and it made me hate math. The way we've been learning how to do math makes me more interested in the material. This semseter has made me more confident in math and I hope my confidence continues to grow by the end of the semester.

Summary and Synthesis

Throughout this semester we have worked a great deal on taking a constructive approach to teaching. We have learned that students will have a deeper understanding of what they are learning in math if we take the time to teach students why or how a formula is used instead of simply telling them to memorize it and use it. Students should be given multiple strategies to solve a problem and realistic problems for which they can relate to. All throughout my primary, middle and high school educational career I had never been told why a formula is used. Teachers would simply tell me to memorize a specific number of formulas and use them to find the correct answer. Students should be given multiple routes or strategies that they can use to solve a problem. Additionally, teachers should include problems that also have multiple solutions instead of a simple right or wrong answer. It is important that teachers do not put down students when they make a mistake. Students need to learn that everyone makes mistakes and making mistakes is how students strengthen their mathematical thinking. From this class, I have taken away a number of instructional strategies for which I can use in my future classroom. For instance, when teaching fractions to students, I would use manipulatives, such as money, to help teach students about fractions, decimals, and percents, because money is something they can relate to. Students will also discover why knowing about fractions, percents, and decimals is important. I hope that by the end of this semester I will feel more confident about teaching math and have less anxiety with my math skills.

New Insights and Their Implications

In this class I have learned a lot of new information that has benefited myself. I have learned new concepts I did not know before. Learning how to use different manipulatives helps with learning and making a visual can help conceptualize the concepts. Doing the LPU it forced me to really think about how to do the problems. In the order to explain it to someone else I really had to think about the process I was completing. I also learned well from watching other classmates do other problems on the board. Doing the extra research for the LPU showed me what we are learning in class is being done in other books and that it is important to change the style of math teaching that is happening in the classroom. Starting with an already known concept and adding on to the concept is the best way to teach students a new math concept. Not only starting with an already known concept, but even starting with a simpler problem helps students understand a concept and find the algorithm them self. I know i have gained more of a conceptual knowledge about concepts about topics i was just taught the formula and just had to go do a worksheet. Starting with a concrete example to a more symbolic meaning is the best way to teach students math. Students need to have a visual and other ways to justify their answer then just the basic old formal or rule to complete a math problem. I know with the division of fractions, I had no idea why we inverted and multiplied. In high school I took the upper math class and I really did not know why we were learning all of the math because we had no real world connection. I could figure out an easier way to look at the problem, so I could solve the problem. I was also able to explain the process to my classmates, but I had no idea why we were doing what we were doing. Students need to get the conceptual understanding and be able to apply it to real world situations to under stand the importance of math concepts.

New Insights and New implications

First of all my new insight is that I am not good with google docs, it took me 40 minutes to figure how to post this. On a mathematical topic, i found out that just because i knew the rule to something did not mean that i knew the reason for the rule. Many kids do ask 'why' in school and they are shunned by their teacher with a simple answer that has no meaning which is 'because I said so' I am also finding out that as I go through the class I get more and more frustrated because it forces me to think aside from the rule that was rudely inserted in me as I learned math. One thing that I found interesting in class today was the reason as to 'why' when we are dividing fractions we 'flip' the second fraction and multiply across. I had NOOO idea the reason why, and it always bothered me, but know I understand and I feel as if I could explain to that 'why' child the reason as to why that rule is set up.

Summary and Synthesis

Throughout the semester we have been learning a lot of different concepts and how to apply them into the classroom. This class has provided different hands on activities both individually and as group work. We are being taught how to break questions down, how to answer our own questions, and we are just being told why to use a certain procedure, but we are being taught why it is important and how it actually works. When working with fractions in class, we were able to learn why you use certain formulas and how to use visuals to come to an answer. In the past, I have always just been given a formula, and than told to use it. Never was I ever told why or how this formula actually produced the correct answer. Sometimes I become irritated with myself when it comes to math, I wish I was able to produce an answer faster and with more confidence. I hope to have that confidence by the end of the semester.

New Insights and Implications

From this class the main thing I have learned is that just because you know how to use an equation, it doesn't mean you know how to teach it. I've learned that there are a lot of steps a person has to go through in order to teach one concept. Things are not as simple as just teaching an equation and expecting students to apply it right away. Sometimes I forget that my students can't pick up on things as fast as I can because they don't have as much background knowledge. I need to teach things as though students are starting from scratch.

Personal Concerns and Next Steps

Until taking this course, I genuinely thought that I understood math enough so that I could teach it to other people. I've always been a big fan of math and enjoyed doing it on most levels. I even tutored math for several years in high school and part of college. By taking this course, I feel like I am starting from scratch and don't know anything about math at all. It's really disappointing to find out that I'm not prepared to teach my favorite subject. I'm getting more into English because of this class. I'm trying my hardest to understand why we are learning what we're learning, but I don't quite get it. I hope that things will soon click for me.

Personal Concerns and Next steps

I am HORRIBLE when it comes to any math! specially Geometry! working on the polygons is terrifying to me. As a future teacher it scares me to even think about not being able to help the students with math. English is my second language and when I was force to change my mathematical knowledge to English was difficult because i had not learned the right vocabulary. I hope this class can help me catch up with the math vocabulary and understand ways that math can be solved and hopefully math will be easier for me to understand and to teach.

Summary and Synthesis

So far this semester's Tuesday and Thursday classes have opened my eyes to a different way of teaching. One thing I have always wondered about was how am I going to teach math to my students so that they are able to understand it in a better way than I was. This class has given me a new way of thinking and has produced me with some new ideas about teaching math. For instance working on one quality problem with extensions can help a student become more confident in their math skills. The trick is to make sure the task is meeting a standard and is at the appropriate level for the students. Another thing I have learned is that the teacher needs to stay out of the way of students learning. Students learn by making mistakes and as a future teacher I will try to balance the line between encouragement and support and taking over. I remember reading some article where the teacher taught his entire class by asking questions like; what shape is this? and how would you find the area? Asking questions helps students make meaningful connections to them that will develop into quality math skills. The recent experiences I have had in this class has encouraged me to become a more quality teacher and I am excited to see what else I will learn.

Summary and Synthesis

In class so far this semester, we have been exploring geometry and new ways to discover how to scale shapes to a larger size, find the area of a triangle with out the formula and making observations about triangles. I would have never thought about teaching math in the way we are learning to problem solve and determine and justify answers with out the answer given to the students. Students get to work through the problem with out a formula. I think that it is a good way to get students problem solving and remembering how to do math problems.

New Steps and Personal Concerns

To be blunt, I am terrible at Math, and I always have been. I think a big part of my problem has been, I have never had a "good" math teacher. Teachers have always lectured, lectured, lectured. And expected, me as the student, to understand what they are saying, without providing that extra help and explanation. My personal concerns for the future are that I will struggle when comes to Math. I don't want my students to struggle as I do, and I want them to feel comfortable enough were they can ask questions, without feeling like they will be looked down upon because of their confusion. This class however is helping me to conquer my fear of math, by giving me specific and help skills that can be used in a classroom. I hope to feel more comfortable with the word "MATH" and everything that comes with it by the end of this course.

summary and synthesis

Summary and Synthesis – This class is one of my favorites. I like how we actually get to participate in hands on activities as we are taught to teach. In all my other classes we are simply talked at and told to use these techniques in our classroom but we never get to see them implemented. One issue I have, which is probably just because I like to know things, is how we are given problems but not provided the answers, even after several weeks of working on it. I find it frustrating and a bit discouraging but I know that it is also because I am not used to this way of teaching. I learned a lot. One thing is the names of the strategies that are used and also different ways to ‘attack’ a problem. I also learned that as we are going out into the school the standards are all going to be changing over to the core standards rather than state standards. I have learned that I speak too quietly and am always unsure of my answer, therefore causing me to not get involved very often with the conversation in class. I am one of those people who is afraid of being wrong.

New Insights and Their Implications

I have learned from my peers what it would be like for my student working with their peers in a classroom. One person has the simple easy solution and then we begin to feed off of that and understand the more complex parts of the math problem. I feel like I am always looking for the simple easy solution and it takes more time for me to remember how I solved problems like this back in Elementary/Middle School. Through the methods class I have learned that there is more than one way to find a solution to every problem. A student can use manipulatives, models, drawings, and many other ways to see a problem visually and solve it. As a teacher we can find a way to extend every problem for further learning or gifted students in the classroom. I feel like I have learned many great new strategies for problem solving that I will one day implement in my classroom. I will give the students time to work and struggle on the problems instead of just giving the answers or working through the problems right away.

New Insights and their implications

I feel as though I have already learned new teaching methods in this class. I have learned from the teacher, my peers, the textbook, and our online resources that we have been using. A lot of these concepts and methods that we have been learning are new to me. Alot has changed since I first began learning math. It seems like when I learned math I was taught one way to do it and then we just practiced that method using a multiple of different numbers. We never had to explain why we did what we did or how we got our answer. It was simply just two plus two equal four. The only manipulatives we used were little counting blocks. We didn't have manipulatives for triangles, fractions, etc. Through the new methods that I have been learning in this class, I have learned how to reflect on the math problems and how to explain why were are doing the steps we are doing. I am learning how to explain myself in the mathematical problems that I conduct as well as figuring out alternative ways to do problems. One problem can be expanded up on so much and that can definitely help students understand math much better. By reflecting on problems and explaining problems, students, as well as myself, can better understand the problem and be able to apply that method to other stituations and/or concepts. With a stronger understanding, students will gain much more confidence in their learning. They will want to reflect on their problems and they will become excited when they figure out new problems. I already experienced that within this class. When we were using different shapes to make a hectagon for example, it was exciting to achieve the correct scale factor. I want to implicate a lot of excitement and hands-on learning in my classroom. I want my students to be confident and motivated to learn! I am excited to gain even more ideas from this class and learn more methods that I can implicate into my own classroom someday.

New Insights and Their Implications

I have learned that there are many different ways to teach subjects. Students are going to learn in many different ways, but if we can get students to actively engage in the activities, learning, and classroom discussion than there might be a greater chance that the information will be learned more effectively. I believe this is the case with what we are doing in Math this semester. It is a new way to attack math and one, at least for me, that is very new and uncomfortable. I loved my math teacher and I loved most of my math classes in high school. But seeing this approach to teaching/learning I wonder if I would have enjoyed Geometry rather then dreading it. Like I said above I really thought this new methods was uncomfortable and to be honest I thought it was wasting my time. But I would say the day we went over the manipulative assignment it all changed for me. I began to see the reason and the positive effects of this teaching method. So I guess now i am ready to learn how to effectively teach this way.

Questions and Answers

I wonder if my oldest daughter would feel better about the subject of math if she had an instructor that had engaged her in problem solving?
The process of learning through problem solving that we have experienced in class so far this semester has made it very clear to me why my oldest daughter hates math. Math for her was a struggle because she constantly felt that she was failing by making errors. If she had an experience where she could have applied the knowledge she had (background knowledge) and engaged in a problem solving process where mistakes were supported she may have learned to love math. This way of teaching and engaging students with math makes math a subject that is directly related to their lives and not just something that they either excel at or fail at. The use of math in our daily lives is overlooked in most math classes which makes the learning really not relevant to the students. It is my hope that by taking this class and striving to teach math in an engaging and problem solving way that students like my daughter will not leave my class hating math, but instead feel confident and successful.

New Insights and their Implications

I have learned many thing already in K-8 Math Methods. I learned from my peers, teacher and the book that there are many ideas and concepts dealing withmethods that were new to me. I learned to tackle a problem with many different ways to solve that same problem. We can use the multiple ways to check our work as well. I have learned that a good problem will have many entering points for different learning levels the students may have. I have learned that math can be more fun and interesting instead of cut and dry, question and answer. I have learned how to help my students when they begin trying to solve new problems. I have learned that students do not always need to know the answers, but we as teachers, should help foster the confidence in our students to want to try to solve a problem, with thinking and discussion. The students should be praised for any connections they try to make, instead of being afraid to fail. I have learned about the content standards and aligning my lessons with these standards. I hope to learn even more strategies to use in my own classroom to help my students to become more confident in their problem solving.

Personal concerns and next step

The person concerns i have always had with teaching math is that i will not be able to teach it effectively because I as a student have always struggled in that subject. I hope that i am able to learn many concepts and ideas to help math strugglers like myself and I become confident in my math teaching. The next step is make sure that i develop great ideas and concepts so that i can become an effective math teacher, this class will defiantly help me become more confident in teaching math. My next step will also to make sure i am engaged in the activities so i can effectively understand that so i can teach them someday.

Personal Concerns and Next Steps

I have never been very good in math. I have also had to struggle through my math courses. I have tried everything to be good at math. This class has offered me new insights into understanding math. I think this course would have been very helpful to take before Math Concepts because this course has helped me understand math better than those courses did. I worry as a teacher that I will not be able to teach my students math very well but this course is helping me learn very valuable techniques. I cannot say that this class has made all my fears go away but I'm hoping later on in the course I will learn to be more confident in my math skills.

New Insights and Their Implications

This course has already helped me to gain a deeper understanding of what a successful math curriculum should look like. I have learned new knowledge from my peers, the instructor, and the readings in the book. From my peers, I have learned that there are a number of ways to comprehend and solve a problem. I have learned new strategies for which I can use to answer a problem. Through the group work in I have learned that everyone begins the solving of a math problem in a different way. In other words, there is not a right or wrong way to understanding and solving a problem. Math has multiple routes for which students can take and also multiple solutions. From the instructor I have learned that there are four types of methods for solving a math problem which include: George Polya's four steps to problem solving, drawing a diagram, solving a simpler problem, managing your point of view, constructing a table or chart, and the process for mathematical inquiry. In order for our students to be successful problem solvers, we must show and discuss with them these multiple routes for which they can chose from when working to solve a problem. They need to develop a sense of confidence in their math skills. If we are unable to help them develop the confidence at a young age, math will only become more difficult for them. I have learned from the class that teachers in the United States need to try a different approach to their instruction of math. Teachers should not just give students the answer right away. They should give students time to reflect and work with others around them before simply giving them the correct answer. Students need to develop stronger critical thinking skills not just for math purposes but for all subject areas. I believe that the process of specialization is a key building block for helping students understand math. Specialization is simply starting with a random problems and moving to more organized problems or in other words systematically changing the problem to make it more complex. Students again need to gain confidence in their math skills. We should allow them with the opportunity to get up and front of their peers and work out a problem and congratulate them even if they do not get the right answer. We should first discuss what they were doing right before criticizing what they are doing wrong. Overall, I believe this class will help me to be a stronger and more confident teacher of math.

New Insights and their Implications

There are many things I have learned from my peers, the instructor, and the readings about elementary school students and the teaching and learning of mathematics. From my peers, I have learned some different ways students may think about problem solving other than the ways I thought they would. Through small group and whole group discussion, I have learned other ways, routes, and strategies to problem solving that elementary students are going to use. I have also learned that I will not always be able to know or guess what routes students are going to take to solve problems since there is no direct route to problem solving. Students are going to use their own experiences and past knowledge to find a strategy that works best for them. As a teacher, I need to be flexible and let them explore these strategies. Through discovery, students will be able to build on their own schema's and become better problem solvers over time. From the instructor I have learned that it is best to have students explore and justify their answers through using multiple strategies. Mathematics is not about teaching students the way to arrive at the answer through a set of processes but rather have the students discover for themselves how to arrive at an answer. Mathematics teaching has changed dramatically this way since I was in elementary school. Teaching mathematics today is about facilitating students on how to develop problem solving skills and have students use their metacognition to think about what they are thinking about. Problems should be open ended in that they should have more than one solution, they should draw on the student's mathematical knowledge, they should address Common Core Standards, and they should be presented in realistic/authentic context. From the reading, I have learned that teachers are lacking focus and coherence while students are lacking reasoning and sense making. As future teachers, we must instill that students are able to draw conclusions on the basis of evidence and have the ability to develop understanding by connecting it with prior knowledge. I have also learned the many tools that can be used such as George Polya's steps to problem solving, ten problem solving strategies, and processes of mathematical inquiry. By learning these new strategies, as a future teacher I plan to implement them into my classroom so my students can feel better about approaching and solving mathematical problems.

February 4th by 5pm.

Summary and Synthesis: So far throughout the semester, I have learned a lot about math and me. This subject is the one I am nervous about teaching in the future, because of my own personal struggles. This has been apparent in some of the activities we have done thus far such as the scale factor, area, and perimeter chart and questions. I understood everything sooner or later, however I do not want this to happen in my future classroom. As a teacher I should know what I am teaching and understand in order to teach them from the beginning. It just makes me nervous. I hope by the end of this course my thoughts about math and teaching it will have turned.

New Insights and Their Implications

From both the article and classroom discussions, I have learned that there is a completely different way of teaching math that is not seen in the typical classroom. Mathematics is more than just being told the formula, plugging in random numbers, and finding the answer. Mathematics is being able to discover and understand the problem and investigating different ways the problem can be solved. Through allowing the student to work through the problem on their own, they are able to see the components that make up the problem and understand the solution and how they found it. A way that teachers can develop lessons to ensure that their students are understanding mathematics is by teaching it in a constructivist way. Constructivism in the classroom can be seen by: being child centered, teachers are acting as guides for student learning, students are working with real world problems, communication is open between student and teacher, students are building previous knowledge, and students are actively working through the problems. When students are given a real life problem to solve and the teacher helps guide the student by using open communication, instead of telling, students are able to truly learn the answers on their own. Constructivism is a theory that dates back hundreds of years ago and is now starting to be recognized as a method for teaching mathematics. In fact, the NCTM Mathematical Standards are designed around this theory. After having the discussions, reading the article, and actually seeing what constructivism looks like in the classroom, I hope to be able to provide my future students with the opportunity of actually being able to explore a math problem instead of being trapped to using the textbook and falling into the same routines of so many teachers. I want my future students to be able to completely understand the problems that are presented and know how to solve them.

Personal Concerns and Next Steps

Math has always been my weakest subject. Every math class I have ever taken, has always been a struggle for me. It always took a little extra time and a little extra explaining for me to understand. Because of this, I am concerned about teaching math in my own classroom. I know how a lot of those children feel when the math concept just doesn't click. The only problem is, is that I'm not sure if I could be of much help. I am so used to and stuck in the traditional ways of teaching math that its going to be hard to change my ways. I am excited for this class to teach me new and more significant ways to teach math. So far, it has been difficult for me in grasping this different concept in teaching math. I know that the way I learned math originally cannot be the way I teach math in the future. It is up to me to make sure that my students are given significant mathematical tasks and that I provide a lot of opportunities for my students to succeed in math. Changing the ways we teach math is going to be difficult but necessary. I want my students to feel confident in math, unlike I ever did. This is probably one of the most important classes I will take here at USD and with every one's concerns on math scores, it is up to me to do my best in this class and get the most out of it. I hope I can take a lot out of this class and apply it to my own classroom.

Summary and Synthesis

Recent experiences that have impacted me through my math course on Tuesdays and Thursdays include, new ways of thinking when teaching and learning, new ideas to think about when planning for lessons and tests, and new ways to encourage students to learn. Over the past couple Tuesdays and Thursdays I have learned about what teachers and students are currently lacking such as, teachers are lacking focus and coherence when teaching students, and students are lacking the skills of reasoning and sense making. I have learned that it is vital for students to be able to make sense of what they are learning and be able to give reasoning on why they did a problem the way they did and how they come up with the solution. In the past and recent years, teachers have been failing in teaching students how to make sense of mathematics and how to provide reasoning for their thoughts. Teachers are also lacking when it comes to focusing on the number of math topics covered. Other countries teach fewer content areas and cover them in a deeper manner. I believe, after being in this class, that here in the United States, we do the opposite. Rather than teaching our students what they need to know now and for the future, we teach our students what is in a textbook and then move on. This does not allow students to develop a deeper level of thinking. We need to focus more on certain skills that will help our students succeed now and in the future. I have also learned through this course that mathematics needs to be learned through hands-on activities that use manipulatives and the problems need to be relateable for students. The students can learn math so much easier and can understand different processes better if they have manipulatives to look, touch, and use. This allows students to use their hands and their brains and discover many different routes to solving problems as well as, helping them be able to explain their reasoning for what they did. I have also learned about the 5 strands of math and the 4 processes of math and how to help my students be successful for each one. The impacts these experiences have had on my will affect the way I teach math to my future students. I will not teach right out of a textbook and I will allow my students to focus their attention on a realistic problem for as long as it takes for them to all understand it and be able to explain what they did and why. I will also make math more exciting for students by allowing them to be creative, work in groups, and us manipulatives. Through my experiences in this course, I truly feel that I will be an excellent math teacher for my future students.

-Danyel