Classroom Changes

I think that as a future educator, I am expected to do many things. I think that the Process and Practice Standards require teachers to be well rounded in math and education as a whole. The goal is to also instill that same idea into the students.  One thing that I think I will have to put a lot of emphasis on is communication. Communication is a big part of several of the SMP as well as one of the Process Standards. I think that communication is very important and I can see why this is stressed in many standards. I also will be expected to make math much more applicable to the real world. Many of the content standards included applying skills to real-world situations. This is always important because it helps keep the students engaged and also helps them to gain a better understanding of the topic. I think that teachers are being expected to include more and more in their classes every year and it will be difficult to always include these changes.

Technology Reflection

My favorite piece of technology used in this class was the Smart Board. Although we only briefly presented simple tasks on the Smart Board, I found it to be very helpful. Prior to this class I had barely used a Smart Board. I know that I will likely be using one often in my career so it was really nice to be able to practice and experiment with it a little. I also really liked the idea of the blogging. I think that it saved a lot of class time. By blogging reflections, we did not have to spend a lot of time discussing how assignments/readings went which allowed us to move forward at a quicker pace. I also found the videos to be interesting. Although I did not agree with everything shown in the videos, I still think that it was entertaining and good to see. Watching other educators and critiquing them is one way to understand how to better improve ourselves as educators.

Manipulative Reflection

I found the manipulative activity in class quite informational. I had not seen some of the manipulatives before, so it was interesting in the way that I was learning about new things. It was also informational through hearing the ideas of my peers. We were able to come up with so many uses for something that at first glance may actually seem useless. It was great to see how some manipulatives can be quite versatile. This is something that teachers (especially new teachers) should really think about because we will not always have a lot of materials to work with. 

1.      How do you know students deepen their understanding while using manipulatives? 
 I know that students deepen their understanding while using manipulatives because they are able to make their own discoveries. Students who are able to explore and discover things on their own will always have a better understanding than students who were simple lectured on a process. Also, I think that when using manipulatives, the information is more likely to be moved to their long term memory. Students have an easier time remembering things that are hands on type activities.

2.     How do you know if the students can transfer their understanding from manipulatives to other situations?
I think that if the students can explain the topic/process in their own words, it shows a good understanding. If the students can do this, they are likely to be able to transfer their learning to different situations. It is important to assess the students' learning along the way always, especially when working with manipulatives. Posing different types of problems during manipulative activities is another way to ensure this. 

3.     How can you assess that understanding or growth?
 You can assess student understanding using manipulatives in many ways. Discussion is the most readily available and likely way to assess understanding often. Students may also be assessed by completing certain activities where they may have to draw what the manipulatives would look like, write written responses to questions about the manipulatives, etc.

4.     When students work in groups, how do you hold each youngster accountable for learning?
It is important to always make sure that each student in the class is learning. This can be achieved by listening to what each students has to say, having each student in a group answer a different question, have the students each write reflections, etc. This will depend largely on the age of the students. 

5.     When students work in groups, how do you assess each youngster’s depth of understanding?
The students' depth of understanding can be assessed in a similar way. Students in groups may each have to come up with one additional problem, or an extension onto the activity. The students may have to explain their thinking out loud or in writing, etc. 

6.     How are you improving students’ problem solving skills with the manipulatives?
Students' problem solving skills can be improved greatly through using manipulatives. When using the manipulatives without much guidance, the students will have to come up with ways to implement them. Students will also be able to complete the problems in a completely different way, which will add to their skills. 

Assessment Reflection

Throughout my math career, assessments have been essentially the same. After learning a large chunk of material, one class period would be devoted to one written, summative assessment. I was always the type of student who disliked projects and group activities, so this never really bothered me. However, knowing what I know now, this was probably not the best way for my teachers to get a good idea of what I was understanding. Through this math methods class we have been assessed in many ways, none of which were a traditional written assessment. We have been assessed based on our blogs, presentations, discussions, and peer evaluations. We were also required to assess ourselves on various projects. I think that this is an effective way to assess students. In these types of assessments, you can hear what the students are actually thinking. Students also have the opportunity to tell you what they are having difficulty with. The peer and self evaluations also give the students a sense of responsibility which I think is important in the classroom.

Error Analysis

Having the chance to look at several different errors that students make was a great opportunity. I found it to be interesting to see all of the different misconceptions that students possessed on different topics. I also enjoyed that we had to answer some questions as if we were the students with those misconceptions. It was very easy to see how the students could think the way that they did when looking at the problems. I think that this was valuable because this is something we will have to do a lot when we are in the classroom. It is important to notice patterns in the types of problems that the students are getting wrong. Some misconceptions may go unnoticed if the student is still getting the majority of the problems correct. However, not noticing the issues as they arise can cause them to turn into a bigger issue down the road in the student's math education.

Assessment Journals

Open-Ended Assessment
Questions are considered open ended when there are multiple ways to solve it as well as multiple correct answers. Creating these types of problems is time consuming and often difficult. These questions are much more difficult to come up with. You have to be sure when creating these problems that the students will understand what the question is and that they need to show how they came up with their solution. Implementing these questions on assessments in my classroom would allow for me to gain a better understanding of where the students are at in their understandings of the topics.

Assessment Options
There are many different types of assessments for teachers to choose from. Assessments are designed to help students learn as well as to help teachers to see what the students are understanding. Assessments should be chosen specifically to evaluate students on each topic. Using the same type of assessment over and over is not useful to the students or the teacher. By using different types all of the time, the students will be able to demonstrate their strengths in different areas and the teacher will be able to get a clearer picture of the students' current performance levels.

Understanding Student to Open-Ended
This article focused on several different student responses to the same question. Their responses showed that the students who were in sixth grade were able to thoroughly explain their processes in writing. The responses also give the teacher a great understanding of how the students are working through the problems. Asking for an explanation of how the students found their answer is a great idea. I think that this will even help the students, because they may notice that something does not quite make sense when they are writing their explanations, which gives them a chance to go back and make adjustments.

Conversations
This article starts out with a great example of why classroom discussions are valuable. The students are answering questions aloud which lets the teacher know that some of her students are confused about a topic before the one which they are currently working on. This allows for the teacher to go back and clarify that basic topic and then continue to move on. Conversations in the classroom can be helpful in directing instruction. The teacher can listen to how students explain various topics to see where they need the most help.

 Portfolio Assessment
When I think of portfolios, I think of something that shows growth and development. After reading this article, I realize that there are more things to look for in math portfolios than growth. Students can show their attitude, problem solving skills, growth, writing, and connections all in a portfolio. Portfolios can be assessed based on their organization as well as the math content within them. I think that portfolios are a great assessment tool which can be shown to future teachers to help understand the levels of their new students. This is also a great thing to show to parents so that they can see the growth as well.

NAEP Reflection

I really enjoyed this assignment. I found it so interesting to be able to see so many different student responses to the same problem. I think that it was valuable to be able to see how greatly the thought processes of students can vary, even within the same grade level. I enjoyed picking out specific samples and judging them to determine what score should be given. It was very eye opening to try to assess them based on the rubrics provided. Many of the rubrics were vague or had overlapping ideas for different scores. I think that the rubrics could have been more specific in order to allow for more objective assessment. I think that creating a plan of how to continue student learning after their work on the problems was also quite interesting. We chose three students who had very different abilities and showed different strategies in solving the problem. This caused it to be slightly challenging to create a plan of what to do next for the class as a whole. I think that this was an authentic activity because in our future classrooms our students will always have varying abilities and talents.

Video Reflection #2

This video starts out with the teacher asking the students what they know about multiplication and division. The teacher allows the students to talk to each other and then asks them what ideas they have about the two operations. I think that it is a good idea for the students to talk out the ideas that they have with their peers, however, the students may not have been ready to do this just yet. I think the teacher could have taken a little more control in guiding the conversations at this point in the lesson.
Next, the teacher asks the students to complete some mental math problems. She then has the students discuss with their peers again about how they solved the problems. During this time of sharing strategies, many misconceptions were shown. Students seemed to get confused between different operations and were unsure of which to use when. The teacher discusses the different ways which students solved the problems. The teacher shows show the students could use drawings to enhance their answers. When the teacher asks the students to write about what they will remember about Charlie's way they need a lot of guidance to come up with something to write down.
During the faculty debriefing the teacher talks about how the students all said that division is related to subtraction. I also found this to be very interesting because that is not what I would have thought to say. I agree that the students must have heard that somewhere or simply thought that since multiplication is similar to addition, division must be similar to subtraction.
Overall I found the video interesting. I think that there could have been more guidance and structure in the lesson, but I also like the large emphasis on their individual exploration.

Applets & App Review

Applet #1
Catch the Fly
 http://hotmath.com/hotmath_help/games/ctf/ctf_hotmath.swf
This applet is designed to practice working with x and y coordinates. This is just a quick activity that provides simple practice, this is not an applet that the students would need a lot of time to go through. The premise of the game is that you are a frog and a fly lands somewhere on the coordinate plane. You have to type in the coordinates of where the fly is in order to eat it. The applet itself is on a plain grayish blue colored coordinate plane with the frog a vibrant green color. The answers which are typed in by the students are placed on the plane in red for incorrect answers and green for correct answers. The plane is easy to see and has clear numbers which are easy to read.
I think that this applet can be used anytime when the students finish work early or are allowed to have time on the computers. This applet is perfect for middle school students and I think that the students would be entertained by it. The feedback is very clear, the point which the students selected is highlighted until the correct point is selected. The coordinates are also written next to that point to help them see how they are set up. The feedback is immediate and helpful.

Applet #2
Factor-Tris
http://hotmath.com/hotmath_help/games/factortris/factortris_hotmath_sound.swf 
This game adds a math spin to the traditional game of Tetris. In this game, the students select what size blocks will come down based on their factoring of numbers given. For example, the number 72 will show up on the screen, the students then must select 72 units in some way (a rectangle which is 6x12). That rectangle will then come down the other side of the screen in a Tetris type game. This game is really fun and great for middle school aged students.
The main focus of the game is factoring. This is a great way for the students to keep practicing their multiplication and division skills. There is also some spacial reasoning skills required for the Tetris side of the game. This can be used in the classroom again during free time as a skill builder for students.

App #1
iTooch 7th Grade Math
http://www.edupad.com/itooch/middle-school-app/
This app is awesome! It includes so many different topics all geared toward 7th graders. The app is appealing to children and has many bright, vibrant colors. The topics include; numbers and operations, algebraic structures, statistics, functions, and geometry. Within this app, the player starts out answering a few questions from the selected category. After getting these questions correct, it will allow you to take a test within that category. The student can then unlock new levels and tests to take. This game has great sounds and visual effects. It is fun and exciting to play. This is something that the students can play for a very short amount of time or even a longer time because of the way it is set up. The teacher can select what category the students should play in or the students can select whichever they would like. The app is available for Apple and Android devices.

Thinking Through a Lesson Article Reflection

This article talks about Think Through a Lesson Protocol or TTLP. This is something that I have never heard of before and at first sounded kind of silly to me. The protocol seemed to be sort of common sense to me. I think that teachers should be naturally following the three stages of the TTLP without having to really think about it. The goal of teaching is to ensure that students are learning, and I think that this protocol does ensure that, but I do not see why a teacher would need to be told to do this. For example, in figure 2 the section of questions dealing with stage one include, "How will you group students?", "What errors might students make?", and "How will students record their work?". These are questions that teachers should always be asking themselves when planning lessons for any subject.
I think that the TTLP has a lot of great ideas in it, but I think that any good teacher would already be implementing those ideas even without ever even knowing what the TTLP is. I think that asking these questions and providing students with higher-order thinking problems is always a good idea and will benefit the students greatly. If the TTLP can help teachers to do this, I think it is a good idea.

A Model for Understanding Article Reflection

I found this article quite interesting. I think that we, as teachers do in fact need to have a vision of what understanding looks like. This article gives us many ideas of how to check for understanding when teaching mathematics. It was very surprising to me that the article was written thirty years ago, because it is still very relevant today.
I greatly enjoyed all of the example questions that can be asked to check for understanding of concepts, processes, and facts. I think that it is hard to determine whether a student is truly understanding what the material is on a deep level, or if they are simply memorizing what they can and pretending to understand. One example I found to be a good idea was when talking about prime numbers, asking "I have a prime number written on the back of this card; what can you tell me about it?" This is an excellent way to determine if the students truly understand what makes a number prime. The ability to generalize what they have learned and talk about it in their own words will often show that they have a deep understanding.

Rich Task Reflection

Through completing this assignment, I have learned a lot. I realized that there needs to be a lot more thought into planning "rich" activities. It was difficult to find an activity that included parts which would work for all different types of learners. Our task was the "How Big is Barbie?" activity. I really enjoyed working with this activity and changing it slightly to meet the needs for the assignment. I think that this is an activity I could definitely see myself implementing into my classroom one day.

I also greatly enjoyed the activities which the other groups presented. I think that the biggest thing which I learned through all of the presentations was that it is very important to run through the full activity before trying it in a classroom setting. There were some minor hiccups during all three of the activities that I think could have been fully avoided if we had all ran through the activities beforehand.