Curriculum Plan

              This project was very time consuming, but also very rewarding. There was a lot that went into this final project. I had an awesome group though that helped me out every step of the way. We worked together so fluently to get this plan accomplished! This certainly showed me just how much collaboration will help in the future. It will be important for all of the grade levels to collaborate and be on the same page so that we are sure the students are learning what they need to in order to be successful.
              A second thing I learned was that it is important to make sure that everything is covered standards wise when it comes to standards. This was challenging at first to make sure that two different sets of standards were covered. This is also true when it comes to the content areas. This is because each content area had to be covered at least one quarter for each grade. This meant we had to go back and check our work to make sure that we had everything covered. This also goes along with us having to make sure that the order we put the content orders in. This is because we wanted to make sure that we would cover certain content areas first so that the students had the base knowledge for some of the other content areas.
             A third thing I realized was how important making sure everything built off one another whether that be quarter to quarter or grade to grade. This is true for many of the mathematics that is covered in any grade. There are topics that need to be reviewed as well sometimes before you move onto a concept that is built off of that concept that was reviewed.
                Overall, this project was very rewarding. I learned so much that I can put to use when it comes to my own classroom in the future. This includes knowing how to even start going about a math curriculum for any grade. It also made me realize just how much work goes into planning and starting to implement a curriculum plan. I also realize that whenever you plan for a curriculum, you always need to be reflecting to make sure that your plan is best suited for your classroom. This may mean that as you go through a year you might adapt and change when you go over a certain content area for instance. You may also have to make modifications as you go too.This can depend on the students' prior knowledge, etc. All of these different components contribute to making sure that every single student you have come through your future classroom doors succeeds! This is especially true when looking at this from a K-8 perspective as we saw the other groups curriculum plans in class. it just goes to show how important that each year all the standards be covered, and that each teacher for each grade level makes sure and is double checking their students so to speak so that they are truly prepared to tackle the next academic year by the end of that school year. The presentations also showed a lot of overlap. This is because each school year is basically building off the year before. A lot of the concepts the students have had before. However, each concept is  built upon the next year. There are also usually a few new concepts that are introduced to the students each year. I also realized that teachers need to make sure that they utilize each content area and not just focus on certain ones each year. I feel that would not help the students in the long run at all.

Classroom Changes

                Teachers these days are expected to change a few things in regards to mathematics teaching. This is especially true with the implementation of the common core. There are certain areas that may not be expected to cover anymore. There are also topics that are still going to be covered but in a different way. For instance, growing up I was always taught to add and subtract up and down. Now, they are wanting us teachers to teach those mathematical concepts across. Things like that are going to take time to get used to doing that way. They are also putting a lot more emphasis on inquiry based math. This includes more real life situations, and more investigations that are student lead. Another aspect of that is modeling. This could include, but is not limited to, using manipulatives, etc. to model. Another big thing that is changing is the assessments. Teachers need to start reflecting a lot more to make sure that they are on the right track so that their students do get that deep understanding.  Error reflection also goes along with this. So, overall, everything we have covered from assessments on down in this class is showing us what will be expected of us. It is also expected of us to keep up to date. This means that we need to make sure that we are constantly looking at articles and websites and take classes to make sure that we are in touch with any new strategies etc.

Technology Reflection

                    This blog will be all about technology in the classroom. We have certainly used a lot of technology in the course of this month and a half. One of the technologies we used was the smartboard. We had to sign in on the smartboard at the beginning of each class. We also had to present a tip on the smartboard that we all could use. These assignments helped us get use to using the smartboard so that when we get into our own classroom we are able to utilize this technology. A second type of technology that we had to use was the jiing. This allowed us to record our voices and implant it into our prezi presentations. This will help us learn how to implement sounds or our voices into our presentations when we start teaching. A third type of technology that we have explored is math apps and applets. These are always a great way to implement technology into the classroom because this is a way to help students not only review, but also keep their skills sharp. We also are going to have to use the internet and our computers for our curriculum plans. We will have to create a video and upload it to the internet onto a website such as youtube, etc.
                       Overall, technology is very important to incorporate in the classroom these days. Students are more and more into learning with technology as the days go by. More and more different learning technology comes out all the time. Technology also gives us teachers another strategy to use when taking ELLs etc. into consideration. I will certainly be implementing a lot of these technologies into my classroom when I begin my teaching career. I have not a doubt that this will help my students succeed!!!

Error Analysis

                This blog will be about error analysis. Errors are made many times by your students throughout your teaching career. They could  be really simple errors that are easy to catch and fix, or they could be a little more complicated and need a little more analyzing to figure out. Analyzing errors is important not only for the students sake, but for the teachers sake too. From the teachers perspective, it is important because it shows what you need to go over again or help students with so that way they are not making that mistake again in the future. From the student's perspective, it's important because if the mistakes they are making are not caught, then they will continue to make those same mistakes throughout their school careers. That will in turn make it difficult to get through the other math classes that they will face.
                   It seemed to me like there are many regrouping and place value mistakes that are being make from the problems that we analyzed in class. Some of the students were borrowing from the wrong place (example borrowing from the tens when they should be borrowing from the hundreds), regrouping wrong by not crossing out the number which causes them to be off by one. As mentioned before, even though these mistakes are not very bad mistakes, they still effect the student to solve the problem correctly. I will certainly have to keep these in mind as I get my own classroom in the future. I will make sure that I analyze my students' work as I am grading them to make sure that they are not making any of those simple mistakes!

Assessments in Math Reflection

             This blog will be a reflection on the assessments in mathematics. We certainly have covered a lot about assessments in the last month and a half. I have learned so many things that I will take with me when I get my own classroom. This is even true regarding my experiences with mathematics growing up. Growing up, I was always assessed by the traditional tests and worksheets. Especially when it came to multiplication facts. We would have timed assessments for those. Throughout this semester though, I have found out just how important assessment really is in mathematics.
              The first item we discussed regarding assessment was the different kinds of assessment. This made me realize that my teachers could have used many more different types of assessment when it came to mathematics over the course of my school career. It also made me realize the many options I have as a teacher to pick from when I am assessing my students in mathematics. This class has also shown me that I need to be extra careful when it comes to choosing which assessment I will choose.  Not every assessment will be right for every type of project of problem.
                The second item discussed is interpretation and reflection when it comes to assessment. When we looked at several articles, I realized just how important reflection really is. Even if you created the rubric, you should always be reflecting on it and see if you need to make changes to make it better or more precise/clear. This will eliminate any confusions or misconceptions your students will have regarding the rubric.When it comes to interpretation, its all about making sure you know exactly what you will be assessing your students on. This will help you decide on just what will constitute what each number will represent.
               The third item discussed is what kinds of assessment she(Dr. Nugent) has used to assess us throughout the course of this class. She has used a lot of different assessments throughout the course of this class. She has had us blog, do evaluations, projects, and participation (in-class activities). All of these have been a way for her to assess not only our mathematical understanding in some instances, but our pedagogical understanding as well.
                Overall, the topic of assessment has been covered extensively throughout this class. It is extremely important. It is one of the the ways that you are able to gauge your student's progress throughout the year. It helps you decide what you need to change your strategies towards when it comes to teaching mathematics. It also helps you modify your instruction so that every student in your classroom can succeed!

Problem/Project

                    This blog is going to be about the problem/project. This was a very interesting and eye opening project.We did a project that was centered for fourth graders. We covered three content areas throughout the process of this project. The project was the students had to find the mean, median, and mode for their sports team, design a stadium for their team, and build a model for their team.  One of the reasons for that is because before this project I never realized how much work went into designing a project for mathematics. There are lots of areas that have to be covered when designing a problem or project like this. However, this project also shows how important collaboration can be when it comes to making sure that everything is covered in the project. This project is also very beneficial to show us what kind of standards need to be covered depending of what grade level you choose to teach and create the problem for. It is also important to make sure that the standards you chose line up with the objectives you have for the problem. You also have to make sure that you take into account any 504s, IEPs, or ELLs you might have in your classroom when designing these problems.
                     Overall it was a very rewarding project to be apart of. The two partners I had were perfect, and we worked together so well to make sure this project was accomplished. We certainly took the time and went over every requirement for this project. This also showed me, as mentioned previously, how much time will be needed to be put into planning these projects for the students in my classroom. This project also showed me how you can cover several content areas when planning and teaching this problem/project.

Content Standards

             This blog is all about the content standards project. I will be discussing a couple of points regarding this project. The first area I will be discussing if when looking at my area from a K-8 stand point, are the standards covered enough throughout the curriculum. The answer to this question is no. As I was looking through the content standards that my other group members turned in, I noticed from third grade on to eighth that there seemed to be a lot of standards either not addressed or not even covered in the textbooks. This will certainly need to change if we want our standards to be covered efficiently. This also shows that we as teachers need to be taking the initiative to make sure we know the standards ourselves so that we can make sure that we cover the standards that are nit being covered to satisfaction.
                 For my particular area, all of the standards were covered in the textbooks. This shows me that I need to make sure that these are utilized to the fullest because of the fact that they are covered so well in the textbooks. I was also surprised that because mine were covered so well that grades three through eight were not covered as well. This whole project has taught me that each textbook in mathematics that is used by teachers will need to be looked at to see what standards need to be covered more or addressed if they are not addressed.

Manipulatives

             This blog will discussing manipulatives in mathematics. I will be discussing several points about the use of manipulatives throughout this blog. The first question that will be discussed is the question of how do we know if students are deepening their understanding when using manipulatives. As we have gone through this class, we have used several manipulatives ourselves. By doing so it certainly helps us answer this question. Students' understanding is deepened, because they are able to use the manipulatives and be hands on. This allows the students to actually visually see the math being worked out as they are trying to solve a problem. It also allows the students to understand why a formula or equation works the way it does.
                The second question that will be addressed is about how we know if our students can transfer their information from the manipulatives to other situations. This is apparent not only in other math problems, but in math areas in the different core subjects such as social studies when you make a map, and have to scale it. This shows that they are transferring their knowledge by using what they learned by using the manipulatives in math to solve not only other math problems, but other subject problems as well. You can also see if the students are transferring their knowledge by having them working in partners, and when you walk around you hear the students explaining to each other how to do it because they remember from using the manipulatives.
                 The third question that will be discussed is how we can assess that understanding or growth. You can save the students work from the unit. Then you can keep constantly checking that portfolio that you have made for that student to see if that particular student is understanding and growing, or if you have to go back and either reteach or change strategies to better suit that student. You can also have them write a math journal. This can also be checked daily, weekly, or however often you want to check it. Like the portfolio, the journals will also let you know if you are headed in the right direction or not. Both of these will allow you to ensure your students success!
                   The fourth question that will be discussed is how can we hold each youngster accountable when the students are working in groups. One way is by each student coming to teacher to meet her when they are done, and they would have to explain their reasoning and work that they show their group did. You could also have an individual part to the assignment which would make them accountable because they will have had to learn it in order for them to accomplish the assignment. Again, you could do journals, and in the journals have them explain the process their group and themselves went through in order to accomplish the problem.
                  The fifth question that will be discussed is how can we assess the depth of understanding of a student when they work in groups. One way would again be a portfolio. Collect the work and keep constantly checking it throughout the project to see how deep their understanding is. You can also using pre and post assessments to check their depth of understanding. This will also tell you whether the students are on the right track. As previously stated, you can also do the journals. All of these  options will tell you if you need to revisit an idea if your students' understanding is not quite there.
                      The sixth and final question to be discussed is how am I improving my students problem solving skills by using the manipulatives. One way is by having them use the manipulatives, they are learning to break the problem down and analyze it to where they can solve it. They are also being helped because they are learning to take problems step-by-step if they need to in order to conquer the problem. I am also helping their problem solving skills by having them visually represent problems if they need to in order to solve problems. All of these points and questions that have been discussed are all ways to ensure that your students are successful when working with manipulatives in mathematics! This was even shown with the four different manipulatives Dr. Nugent had us use in class. It really had us thinking about how each of the specific manipulatives that she had out could be used with our students in any grade. It was certainly challenging on some of the manipulatives, but we talked and collaborated as a group of three to figure it out. Now I certainly know at least four manipulatives that I would love to have my students use for various reasons in my classroom!

"Problem Posing", "Assessing Problem solving thought", and "Assessment Design:Helping Preservice Teachers Focus on Student Thinking"

             For this blog, I will be discussing three articles. The three articles are call "problem posing", "assessing problem solving thought" , and "assessment design: helping preservice teachers focus on student thinking". The first article I will be discussing is "problem posing". This article is all about how problem posing has not gotten the attention it deserves in recent years. However, according to the article, it is now starting to receive more attention. The author talks about how if they are wanting this to be part of mathematics classrooms, then there needs to be criteria about how we are going to assess this both with problem posing and of problem posing. The author then goes on to say how there are three criteria that should go into the assessing of this. They are quantity, originality, and complexity. Each of these are explained in full detail. The article concludes by saying that assessment is a very important part of math, and that as it becomes more widely used in the classroom teachers need to make sure they are looking at the ways they are assessing it.
           The second article I will be discussing is "assessing problem solving thought". This article deals with the fact that this author thinks that teachers should go through the assessment process first before they have their students do the problem. According to the author this is to help teachers understand and go through the problem themselves and see how their students might go about doing the problem. It also goes into detail on how to create a good rubric to use to assess your students. It also discusses the difficulties that might be associated with the assessment process. The big thing from this article is that we as teachers need to make sure that we are not letting our emotional attachments to the students get in our way from assessing what is actually on the page that the students have demonstrated.
               The third article that I will be discussing is "assessment design: helping preservice teachers focus on student thinking". This article is all about an assessment project that a woman  designed for preservice teachers so they can understand that math "should make sense to students." Each person involved in this project had to go through a specific process that involved picking out one of the CCSSM content standards to focus on. Their job was to focus on this standard and evaluate how the students were thinking about the content they were teaching. They then had questions that they were to be asking themselves as  they were evaluating their projects. They even had one woman give her testimony as to how her experience was while going through the process of this project.
               I personally thought all three of these of these articles were eye opening. They really made me think of how important assessment really is when planning any sort of lesson. I will certainly have to implement some of these projects and ideas into my own classroom especially the one that pertained to preservice teachers as I will be student teaching this fall. These will certainly help me be a better mathematics teacher throughout my teaching career!

"Connecting the threads of area and perimeter"

                This blog will be about the article " connecting the threads of area and perimeter". It is from the March 2014 issue of Teaching Children Mathematics.This particular article was all about a quilt project that can be done pertaining to the subject of area and perimeter. The author starts out by saying she got the idea for this quilt project because her students would always ask when they would ever use what they learn about perimeter and area. So, that go her to thinking that she needed to find a way to give the students more real life connections when it came to perimeter and area. This is where the quilt project came into play. The students where to go through a process. Then once they went through that process they had to get it approved by the author before they could start the actual construction of their quilt square. During the process though, it allowed the author to go around and do formative assessments on her students to see if they completely understood the project and what was being asked of them. This worked out extremely well because not only was she able to help struggling students get back on the right path, but the also gave her the opportunity to let the students conversate and collaborate with their fellow classmates if they had questions. She was also able to address all of the common core standards as well as all of the CCSSM standards relating to this project. This project took a week to complete.
                     This article was very interesting to read. I was very impressed to find out how to incorporate more real life connections when it comes to perimeter and area. It also really opens your eyes when it comes to how creative you can  be when it comes to hitting those standards. I also really loved how you don't even need very many materials when it comes to planning this project. I will certainly keep activities like this in mind when it comes to certain subjects like perimeter and area to where my students will always be able to make those real life connections to things they might not think they will ever use.

"Decimal Fractions: An Important Point"

               For this blog I will be discussing the article " decimal fractions: an important point." This article is from the March 2013 issue of Mathematics Teaching in the Middle School. This article starts off talking about how decimal points are a point of frustration for students. This is because, according to the author, students  have a lot of misconceptions about the decimal point. The author then goes into some background knowledge to help us gain some insight into how the author came about in looking into this subject of decimal point fractions. The author even gives some research to back up why students might have misconceptions about this subject. The next part of this article is about different strategies that came about from investigating and researching this subject. It even has a nice table to look at that has four misconceptions on it and explains them in a couple  different ways on the table. Those four misconceptions are longer is larger thinking, zero makes small thinking, shorter is larger thinking, and money rule. This article also talks about different student samples of work, and some of them shown on different tables.
               This article was very educational and eye opening for me. I was certainly one of the students who struggled with decimals and fractions growing up, so I am glad that I have this article to take notes on and utilize if I am ever teaching this subject or helping a student with this subject. I also love how it give samples of student's work to understand what the author is talking about. This article also made me realize just how important decimals and fractions are to everyday thinking and life. I will certainly implement this into my classroom by making sure that I have those common misconceptions learned, so that way if I see my student struggling I will have a nice starting point to go from. I will also be watching what strategies I use when teaching this subject and try some of the other strategies mentioned in this article to make sure all of my students succeed!

Video Analysis 2

 This blog will be a video analysis. The video that I will be analyzing is a lesson for fourth grade that deals with multiplication and division, and it was called "number operations". The planning was the teachers and students going over just what constituted multiplication and division.The students had to talk with their partners before sharing as a class. During this time, the teacher was trying to get the students to understand the big idea that the groups were equal.She was also trying to get the students to understand how creating pictures to solve problems worked. Then the students did mental math before the teacher had the students complete the word problem. The specific activity was the students solving the problem of Maria saving $24 and Wayne saved three times less than her.
                  The teacher had the students look at the problem then gave each student a piece of paper that had the problem on it. After each student was given the piece of paper, they were instructed to take it back to their desks and work on it individually for a minute than they could look at their partners. After they students completed the problem, they came back together as a class and discussed ideas and answers.
                     The faculty debriefing was very interesting. It consisted of the teacher and three observers. The teacher discussed how the students had been almost conditioned to understand that multiplication is like addition, and division is like subtraction. Along with that idea, she said that especially the division is like subtraction idea was interesting because most people don't see division as like repeated subtraction. One of the observers brought up the point that peer pressure might have something to do with why some of the students had their answers the way they had them. The teacher than discussed how she had to drag the idea of the equal groups out, and she had to almost prompt them to expand of the addition and subtraction ideas. One of the implications that I noticed was the whole misconception about the 24 or 32 dilemma. It looked like the observers and the teachers automatically thought they would be on the right track. The student debriefing was also useful because it showed just what students knew what they were doing because I noticed that a lot of the same students were answering or sharing ideas during the discussions which is why the teacher had to prompt for other students to share.
                      Overall, I thought the video was an excellent resource and was utilized perfectly. It shows just how important collaborating can be by having the observers in the classroom, and involved in the planning and reflection processes. The video especially is an excellent tool for reflection in many areas. For example you can tell where certain things may not have gone as you had planned. A perfect example was the confusion about the 24 or 32, and the kids relying on their peers answers or peers mathematical thinking at times. This also proves that by debriefing with your colleagues, you can also find out just what ideas need to be revisited. I also liked how the video was split into segments. It allowed me to look at them a lot closer, and it made it easy for me to watch certain parts more than once when I needed to.So, all in all it helped me realize just what I need to do when I start teaching math, and it also taught me some things that could be adapted.

Math Applets

                For this blog, I will be talking about two applets and one app so three altogether. The first app that I am going to talk about is the smart exchange. This is specifically designed for smartboards. However, you can look at them on your computer before you try it on the smartboard. I chose an app on this site that is for the 3-5 grade category. The particular app has to do with fractions it goes through different ways for the students to review fractions. For example, on of the activities on this app is match the fractions. This would be perfect for students to use not only as a review for the test, but also as a way for them to get more practice with fractions if they are struggling.
                   The second app is all about angles and it is designed for grades 6-8. This particular app was found on shodor.org/interactivate. This can certainly be implemented at several times throughout a unit in geometry on angles. This could included using this as a pre-test, using this as a review for before a test, or using it as an intervention technique to give students extra practice. Throughout the app, it gives you different angles to look at and you have to answer how many acute angles you think there are, etc.
                    The third applet is on http://illuminations.nctm.org/. The actual applet is called grouping and grazing. It is designed for k-2. It has different activities that you can do and it involves animals that graze in groups. You can count by 5's, 10's or add/subtract. Again, as with the other two apps, you can certainly implement this in your classroom by having the students doing this several times not only when you are teaching addition and subtraction, but as review and intervention  technique if needed. All three of these apps that I have discussed I think will be great additions to my future classroom and will help so many students succeed to their fullest potential!

Student Work Reflection

        This blog will be about the student work project that my group and I did. It was certainly an eye opening project. I never knew just how much information you can really gather from looking at student work. This includes what their previous knowledge is, what they might still be struggling with, etc.It is important to see this in all of your students work, especially in math, so that way you can make sure that all of your students succeed to their fullest potential. It is also important to analyze your students' work, especially in math so that you are able to plan your strategies to where they work for all of your students to succeed.
         Overall, you need to make sure that you are constantly going over and analyzing your students' work. I know I will certainly implement this as soon as I get into my first classroom. I have also found out that you need to look at multiple works in order to even start getting a handle on the information that the works provide for you. I also realized that it might also be useful to collaborate with other teachers for the same grade because you can have second opinions to think about as you are going through them. You also might find out many different strategies that students might use to solve a problem. The biggest thing I noticed was that interpretation is key when looking at how you classify the samples.

"Thinking Through a lesson" and "A model for understanding"

                  For this blog, I am going to discuss two different articles. The first article is called "thinking through a lesson." I found this article very interesting. This is because the topic of this article is a process that teachers can use when planning a math lesson that involves higher or more complex thinking. It is a three part process. The three parts are selecting and setting up the mathematical task, supporting students' exploration of the task, and sharing and discussing the task. The article then goes into detail about each part, and explains what questions teachers should be asking their students at each part during the process. This article also gave an example of the type of problem that would be great to use this process with. Another part of this article was talking about how you should be looking at your students prior responses and responses to the other tasks you give them so you know how you can help your students succeed even more.
                 The second article was called "A model for understanding." The first item discussed was the definition of understanding. The article gave seven signs that you will exhibit if you truly understand something. They are you are able to state it in your own words, give examples, recognize it in various situations, make connections between that topic and other topics covered or discussed, use it in multiple ways, foresee some of its consequences, and be able to state its opposite.The author also discusses understanding as a process. This process involves organizing and integrating knowledge according to a set of criteria. Another idea discussed is the idea that understanding is also a continuum. This means that students only have partial understandings. The rest of the article goes into detail by explaining how to understand different concepts in math and gives examples to go off of when planning lessons or activities. You can implement both of these articles into my teaching career by taking these processes to heart when I am planning my activities for my students. For example, if I was going to plan a problem that dealt with colored candies in a bag, I would go through these processes. From reading these two articles, I know now that it is important to keep these processes in mind to make sure that I am pushing each of my students to challenge themselves so they can succeed at their fullest potential.

How the process standards are related to the CCSS M standards

                This blog will be talking about the CCSS M standards and the process standards and how they all intertwine. A lot of these standards tell us the same thing. We especially found that out when doing our poster activity today in class. It showed that a lot of them are intertwined to almost every single one of the other set and vice versa. It is important to know how important it is that these two sets of standards intertwine because when you start planning and teaching lessons you will want to make sure that you are keeping that in mind. One of the most interesting things that came up out of both sets of standards, to me, was communication. This tells me, and the rest of my classmates, that we really need to be paying attention to how we are wording things to our students, and how we need to model that skill so our students know what to do.
Another big concept that was brought up a lot was the reasoning portion of the sets of standards. This is important because if the students are not able to reason and understand exactly why a mathematical concept or idea works the way it does, than they will experience frustration as they get further along in their academic years.
         All in all, I noticed that for each one of the process standards, at least one of the CCSS M standards. I found that very interesting because it has really opened my eyes to how I need to be planning and going about the problems that I will be giving and preparing for my students. By seeing how these sets of standards connect, I also see how important modeling the right thing for my students is. I feel like, if we as teachers don't show the students what they need to be doing, than we are ultimately setting them up for difficulties. This goes for when the students use manipulatives etc. as well. The main beauty, as previously mentioned, is that when you hit one set of standards, you are more than like hitting another standard from the other set which makes the lesson twice as rewarding.

Video Analysis 1

                   This blog will be a video analysis. The video that I will be analyzing is a lesson that deals with cost analysis, and it was called "comparing linear functions". The planning was with the teacher and observers collaborating and talking about the two previous activities that had been done, which lead them to the conclusion that they needed to do this particular activity that they were going to do in the lesson. During the planning, they also discussed that they would have to cover any confusions or misconceptions that the students might have during the lesson. The specific activity is called DVD plans.
                  The actual lesson the teacher went through with the students step by step. He had the students look at how they constructed their table and started it. Then he had the students open their packets and look at different students tables. As he had them look at the work, he had the students talk to their partners and see if the data that they were looking at made mathematical sense, and ask themselves why the data did or did not make sense. The students found out that even though the data may not have matched the plan, it still could of made mathematical sense. One of the students brought up the concept of zero. The class had a discussion about whether there should even be zero listed in the data. Most students saw that there should be zero listed because even if you do not rent or buy any movies you are still paying the flat rate for the month. One of the big confusions that the students had was the plan where it was the $13 plus $1 per extra rental. I noticed though that the students eventually realized this concept and became not as confused about it. I also noticed when looking at student work from the two previous cost analysis activities before this lesson I noticed that on the second page there was lots of confusion. Some of the students did not even answer any of the questions.
                     The faculty debriefing was very interesting. It consisted of the teacher and three observers. The teacher discussed how the students had the instantaneous connection to the prompt. Along with that, he discussed how he wanted to delve deeper into that connection for next time. The three observers then discussed their groups that they were observing. The first observer said that she noticed that one of the students set his data up in three charts, which she found interesting. Along with that, she said that the idea of just what a t-chart is etc. needs to be discussed more thoroughly next time and compare whether they are the same, different, etc. The teacher than discussed how he would like to discuss the idea of zero more with his students, and that he liked the fact that his students brought that idea up. They, as a group, also discussed how maybe the teacher should just include all of the charts into the packet instead of having the big posters up on the board, because the ones one the board the students rarely referred back to compared to the packet in front of them. One of the implications that I noticed was the whole t-chart situation. It seemed like he automatically thought that all of his students knew exactly what a t-chart was etc.
                      Overall, I thought the video was an excellent resource and was utilized perfectly. It shows just how important collaborating can be by having the observers in the classroom, and involved in the planning and reflection processes. The video especially is an excellent tool for reflection in many areas. For example you can tell where certain things may not have gone as you had planned. A perfect example was the confusion about the t-charts, and the kids not using the big posters like he had thought. This also proves that by debriefing with the students, you can also find out just how much growth they actually made, and just what ideas need to be revisited. I also liked how the video was split into segments. It allowed me to look at them a lot closer, and it made it easy for me to watch certain parts more than once when I needed to.So, all in all it helped me realize just what I need to do when I start teaching math, and it also taught me some things that could be adapted.

Article 2 "How did the answer get bigger?"

             For this particular blog, I will be discussing an article from the journal Mathematics teaching in the Middle School. The name of the article is "How did the answer get bigger?" This article addresses the fact that when students begin to work with dividing fractions, they tend to have trouble comprehending the idea that the answer could be bigger than the dividend or divisor. So a big point of this article is that number sense includes your students understanding the significance of the size of the fraction. Another big point is that your students need to understand the connections between the whole and an individual fraction and between other fractions.This article also talks about how your students need to have a flexible mind to understand certain concepts involving fractions.
              This particular article is important for fifth to sixth grade because those are the academic years when students start getting into dividing fractions. This is important because up till these academic years, students are only use to doing whole number dividing, which is why it makes it difficult for the students to understand the concept of how an answer got bigger. You can certainly implement this in the classroom by using several of the strategies mentioned in the article. Most of them include the use of manipulatives. I have noticed the more I research different topics and articles in math that hands on and the use of manipulatives are very popular strategies. So, to implement this in your classroom, according to the article, please remember to look at multiplying and dividing fractions from the areas of modeling, equivalence, and symbolism. An example would be to use individual cubes to show the division in a visual sense so the students have the opportunity to physically see how it turns into a big answer.

Article #1 "From whole numbers to invert and multiply"

               For this post, I am going to discuss an article from the journal Teaching Children Mathematics. The article name is "From whole numbers to invert and multiply." This article talks about the different standards that get developed in grades 3-6 when it comes to dividing fractions when you need to use the inverting property. This article talks a lot about using various strategies to use when going about teaching these standards to your students. One of the strategies was using modeling to show whole number division. Another strategy that was discussed pertaining to modeling was the partitioning model. Other areas that were discussed throughout this article were multiplicative inverses as unit rates, visualizing fractions, and understanding the invert and multiply algorithm. As you go through this article, you can certainly see how these standards build upon one another in order for the others to be understood in the upper elementary level as mentioned previously.
                This article is important because this article helps teachers realize the importance of making sure that the students at their grade level understand the concepts before moving on so their students can be successful throughout the later school years. You can also certainly implement this in you classrooms, especially in grades three through six because that's what grade levels this particular article is targeting. This article also makes you realize just how important manipulatives can be  when it comes to those harder concepts for your students to grasp. This is also true for the hands on aspects of this manipulatives. So, in conclusion, you need to implement this by making sure you use manipulatives to help students grasp any concept that might be difficult for them. An example would be the invert and multiply concept from the article.

NCTM Process Standards

                  There are several NCTM process standards that are listed on the NCTM website. However, for the purpose of this blog I will only be discussing five. The first one is problem solving. According to the article, problem solving means that you are having your students participate in a task or problem where the solution is unknown in the beginning. It talks about how if students have good problem solving skills they are able to analyze the situation. This also helps them naturally become able to ask questions that are based on the situations that they encounter. You, as the teacher, play an instrumental role in helping the students develop problem solving skills according to this particular process standard. This means that the problems that you give students need to be engaging, create an environment that allows the students to experience and share their successes and failures, and discuss why the failures worked out that way. You can certainly put this into practice when teaching. An example the article gave was middle school students given different fruit juice mixtures to see which one is fruitier to introduce the topic of proportions.
                  The second process standard that I will talk to you about is reasoning and proof. This particular process standard is important because students need to know why a particular process or mathematical concept works. This standard is also important because this standard is used across the content areas. One of the biggest things with this standard is that it needs to be developed as the students progress throughout their school years. You can do that by having your students consistently use it in many contexts. You can implement this in your class by making sure that you are reviewing the concepts consistently all year so that they are confident and able to build on it the next school year.
                  The third process standard that I am going to discuss is communication. Communication is also a process standard that is important no matter what subject you are talking about but especially in math. This is because if you are not precise in your communication it will be all confusing to your students when you are explaining the concepts. This means that you need to make sure that when you are explaining a new concept to your students, or if students are explaining a justification for an answer to you, or if your students are labeling that you are correcting them if needed so they know the correct precise communication skills for math. You also need to make sure that you encourage your students to express themselves clearly and effectively especially as they get further along in school. There is also lots of benefits to your students working collaboratively together. This is because,according to the article, the students get the opportunity to see the perspectives and methods of their fellow classmates. They also,according to the article, should become more aware of their audience as they continue to get further along in their school careers. You can implement this standard into the classroom by correcting students when they are not using the correct communication. You can also implement this standard by making sure that you personally are modeling the correct communication so that your students know what to do.
                     The fourth process standard that I am going to discuss is connections.Connections, according to the article, is important because it helps students come to the realization that mathematics is not a set of isolated skills. It also helps kids understand that it relates in someways to the other core subjects. As the students go along in their school careers, they should, according to the article, come to expect and exploit the connections using what they have learned in one area for justification on how something does or doesn't work in another area. You also, as a teacher need to make sure that you have the connections that the students can make in their real lives as well. You can implement this standard in middle school, for example, when you have them collect graph data. This is also true in social studies. An example will be when going over measurements on a legend of a map of the united states, and you are trying to figure out which state has the biggest land area, etc.
                     The fifth and final process standard that I am going to discuss is representation. According to the article, this is important because it allows your students to communicate mathematical approaches, arguments, and understanding to themselves and others. It also is important because it allows students to recognize the connections that can be made between similar topics to the one they are studying, and apply those concepts to real life like situations or problems. With this particular standard, you need to make sure that you let your students represent their ideas in their own way even if it is not the standard way to do it. This is so that it makes sense to them. However, the article says that you should still teach the standard way to do it so that way that still get that opportunity to see it. You can implement this as a teacher by having the students represent a math problem such as 16-4 by drawing pictures. You can tell them a standard way to do them but don't take away points when grading if they do it differently but get the same answer.

Group Noteworthy Ideas about Math and Rich Activities

5-21-14

             There were pdf articles that we had to read. The first pdf was called rich activities. This pdf starts off and explains what to consider when looking at whether an activity is rich or not. The first aspect to consider is students age. The second aspect is the student's gender. The third aspect is what the student's expectation of the class is. The last two aspects that are worth considering is the knowledge and experiences that the student might have.This pdf goes on to explain that the student's cognitive demand must be at a higher level. There are four different areas. According to the pdf, they are memorization, procedures without connections to concepts or meaning, procedures with connections to concepts or meaning, and doing math. According to the pdf, we need to be making sure as mathematics teachers that we are not just having kids memorize mathematical facts. This is because, when we, as math teachers, do that are not allowing our students to expand their knowledge and see the whole picture of just what the mathematical concept is or how we got from one point to the next. Another aspect of this pdf is taking into consideration that we really need to look at the problems we give to our students. This is because "looks can be deceiving" so to speak meaning that it might look like a higher level problem, but when you actually break it down and look at it, the problem is really not a higher level problem. This pdf also gives great questions to think about when planning lessons. Some of the questions include in what ways does it build on prior knowledge, previous life experiences, and culture?, what are all the ways the problem can be solved?, what particular challenges might the particular activity present to struggling students and ELL students?, etc.I will certainly keep all of these in mind when I start planning math lessons for my classroom. This also really puts into perspective just how much thinking and reflecting goes into planning a lesson or making sure an activity is rich enough.

         The second pdf that I looked at was group noteworthy ideas. This gives different math habits to keep in mind. They are explore ideas, orient/organizing, think in reverse, generalizing, representing, justifying, math language, and check for reasonableness. Each one had a small explanation by them. Also, as you look at each of these ideas, you notice how important each one is to the whole big picture on how students understand mathematics. Without any one of these ideas going through their heads, especially if they are working on a collaborative assignment, they will have difficulty, whether it be great or small, in completing the assignment or task. This is also true, in my opinion, when it comes to planning as a teacher. You need to make sure that you keep all of these ideas in mind, especially when making groups, to help make sure that all of your students can have success in doing the assignment.  

Attend to precision and make sense & perservere

5-21-14

             There  are several CCSS standards when it comes to teaching math to elementary students. The two that I am going to discuss are attend to precision and make sense & persevere. These two, along with the other CCSS standards, will help your students be successful when it comes to math. Also, having the students proficient in these two standards will help their anxiety levels lower when it comes to math. 

             The first standard I will discuss is make sense and persevere. There are many things that you can say about this standard. The first thing that needs to be said is that not every problem that your students will in contact with will have a clear cut solution to figuring it out. This means, that the students will have to have several chances to attempt the problem. With that being said, you need to consider several different questions when you are examining the types of problems you are giving students. This is because, when you are giving students problems, your job as a teacher is to have your students understanding the many different mathematical concepts so well that you are confident that even if they don't see the solution right away, they will still be able to think through a problem based on what formulas, etc. they already know. 

          The second standard that I will be discussing is attend to precision. This is important because it pertains a lot to how the students communicate in the subject of mathematics. This includes labeling answers, being clear and concise when using definitions, using definitions properly, etc. This also goes for calculations and procedures. These are particularly important because students need to be specific especially when it comes to, for example, conversions in later elementary years. You would certainly apply this,for instance,in fourth grade when dealing with measurement, units of time, etc. You could take off a point for each time they don't use a label to accompany their answer. You can also relate this to your students everyday lives by getting them involved in a discussion on how confusing it would be to explain to someone increments or specific numbers with no labels, etc. 

              There are two articles that I read pertaining to these two specific standards. The first article is called "Reasoning and Sense." This article pertains to the standard make sense and persevere. The article gives a very good definition of what the word sense making means. The authors, Kasmer and Martin, say that reasoning and sense making are intertwined. Another good point of the article is that there no is  the need to shift from students just memorizing formulas ,etc. and actually understanding the big mathematical ideas that are the foundation in their younger elementary years for when they reach  the secondary years. That also hits home when it comes to persevering because if the students are able to fully grasp the concepts or big ideas in the elementary years, then they will be able to persevere and work through much more complicated problems in their late elementary to secondary years. Also, a way to apply this particular standard to your classroom is by creating problems that will challenge the  students and make them persevere by having to make sense of the what the problem is asking and think of different solutions to try like in story problems.

    The second article I read was "Multidimensional Teaching." This pertained to attend to precision. This article goes into detail on how you need to pay particular the social and cultural aspects of your classroom in order to implement multidimensional teaching into your classroom. This pertains to this particular standard because you are paying attention to fine detail and precision by looking at the cultural and social aspects, which is then modeled to your students through mathematics when it comes to the fine details like labels, etc. This article even gives a teaching example on how to implement this into the classroom.