Curriculum Plan Reflection

Even though this project was quite intense, I really enjoyed it. This is something I have to get used to doing because I will some day be teaching middle school math. It was amazing to have the opportunity to work with two other math concentrations to work on yearly plans for 6-8th grade because we all have had more math training and we were able to share and hear about different ideas. For me, the most difficult part of this project was getting the video to work. I was having technology troubles with my video program so that added unnecessary stress to the project. I thought all three of us worked really well together throughout the entire project of planning, creating projects, and filming the video. I liked the opportunity we had to watch all three groups' videos. I thought it was interesting on how different each of the groups interpreted the assignment, as all three were very different. Again, as a math concentration, I was interested in seeing the other groups' papers so that I could gain ideas of math activities so that if I end up teaching at a lower level in elementary I will still have a variety of activities I could use that are age appropriate in my classroom. Out of all the projects, this was the most beneficial because once we begin teaching, we have to know how to create a yearly, quarterly, and then more detailed plan for our curriculum.

Classroom

There have been many changes to a math classroom in Illinois lately. With the recent introduction of the Common Core Standards, to also having Common Core SMP standards, and NCTM standards it can be intimidating how many standards teachers need to be aware of when teaching math. The Common Core State Standard - Standard Math Practice or SMPs are eight different standards that teachers should be aware of and have included into their math activities. These 8 standards are: Make sense of problems and persevere through solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model in mathematics, use tools strategically, attend to precision, look for and make use of structure, and lastly, look for and express regularity in reasoning.
There are 4 NCTM standards for math. When I was in middle school, it didn't matter how we came to the answer, as long as we got to the correct answer. The NCTM standards as well as the SMPs want the students to explain their thinking and reasoning and justify as to why their solution is correct. These standards make students who are more aware of their math abilities and know how to problem solve. Even if students get an incorrect solution, they can look back at their work and reasoning to see where they ran into a problem. The NCTM standards are: Reasoning and proof, communication, connections, and representation.
Even though I can see people having difficulty in changing the structure of their class to meet all of these standards, I am glad that when I began my education at Bradley that the common core standards were already created. This way my whole teacher education training has been done using the same sets of standards. Now I have been given the time to really become familiar with these standards before I have to be responsible in teaching them to my students. This will prepare me best when going into my own classroom once I graduate with my teaching license.

Technology

Technology in the classroom is always changing. I remember having chalkboards in the classroom when I was in early elementary. Then we switched to whiteboards. I also remember having the giant carts with overhead projectors and the clear transparencies teachers used to write on. Now classrooms are equipped with electronic white boards, regular whiteboards, ELMO projectors (digital version of the overhead projectors), tablets, smart response remotes, and computers. There are so many ways teachers can use technology to engage their students and to make learning fun. Even if schools don't have the funds to have all of the technology styles out there for their classrooms, I think the most useful are computers, smart boards, and the ELMO projector. The ELMO projector is a digital camera that displays whatever is placed in the view of the camera onto the smart board. This is very useful when demonstrating how to complete a task on a recording sheet, or when reading a word problem passage together. There are many different ways to utilize the smart board. Students love coming up to the front of the room to get a chance to use the board. This is definitely a great tool to use in any classroom. I would definitely use as many technology tools as I can for my math class. There are amazing apps and applets teachers can use in the classroom on the smart board, computers, or tablets. These provide new and fun ways for students to learn the math curriculum.

Manipulative Reflection

How do you know students deepen their understanding while using manipulatives?

• Students can deepen their understanding using manipulatives because the students are hands on and involved in the process of learning. When students are actually able to create and represent objects using manipulatives, they are able to visually see the process and are better able to understand the math concept.

How do you know if the students can transfer their understanding from manipulatives to other situations?

• I will know if a student is able to transfer their understanding by having open class discussions. Students will discuss and compare and contrast different ideas that came up during the manipulative session. Listening to the student led discussion will allow me to see if the students were able to take away information from the activity. I could then have the students split up into groups and see if they could solve a real world problem utilizing their knowledge from the manipulative activity.

 

How can you assess that understanding or growth?

• I could lead a class discussion about the topic, then let the students work with the manipulatives, and then have a closing discussion. This way I can see what thoughts and ideas existed before and after the manipulative activity to see what growth was acquired from the manipulatives.

 

When students work in groups, how do you hold each youngster accountable for learning?

• I would assign roles to each student in a group (recorder, directions, and two people using the manipulatives). Then every few minutes I would rotate the roles. This way each student has the opportunity to be hands on with the manipulatives, give ideas and directions as to how to use the manipulatives, as well as listen and record the group member's ideas. By setting these roles, each student has a job to do within the group. This was each student is accountable to learning and participating within the activity.

 

When students work in groups, how do you assess each youngster’s depth of understanding?

• I could assess their depth of understanding informally by listening to each group as they work together. I could also assess more formally by having the students submit a reflection assessment as well as answer a few questions regarding the topic covered by the manipulative activity.

Assessments in Math Reflection

Assessments in math can come in all different varieties, just like assessment in any other subject. I remember in middle school we would have daily assessment grades on our homework. In high school we would have homework checks for completion, however homework was a learning tool so we weren't counted off if we had the problem wrong. We were supposed to learn from our mistakes so we could be successful on the tests and quizzes.

In math methods we have learned many more assessment styles and tools other than homework grades, quizzes, and tests. There is a big push for switching classrooms over to inquiry based learning. This learning style is best assessed by a rubric. In methods we have learned about the importance of a strong rubric that isn't biased and the level on the rubric should be clear. I think performance based assessments and portfolio assessments are a great way to assess students. Performance based allows students to take the knowledge of the topic and apply it to a real world situation. Portfolio assessments allow students to see the physical growth of their learning by collecting various assignments throughout the semester or year. These assignments are much more meaningful to students because they are applying what they have learned. The traditional tests and quizzes are also of value, however these mainly assess memorization and don't represent the depth of knowledge a student may have when compared to all the levels of Bloom's Taxonomy.

In my classroom I want to have a variety of assessments for mathematics. Many students get discouraged in a math class, and with allowing students to see the connections of math to the real world, they will have a much better appreciation for the subject.

Error Reflection

In class we explored different math problems done by students. Each student modeled a different error within solving the problem. Since I am a math concentration, I find this really interesting. I enjoyed looking at each student's work and analyzing it for where they made a mistake. This is something I will have to be comfortable with when I begin teaching math whether it is at the middle school level or at the elementary level.
I thought this assignment was very similar to the NAEP reteach assignment. The only difference was that the NAEP assignment all students were solving the same problem, whereas this assignment each student was solving a different problem.

Assessment Readings

Open-Ended: This article talks about using open-ended problems to assess math students' work. I like this idea, because with the many possible solutions to the problem, students may think outside the box to find a strategy to solve the problem. Open-ended problems allow teachers to see the students thinking process and allows them to help the students where they struggle.

Options: This article talks about the different ways math can be assessed. This is often referred to as the menu choice. This article mentions how even though multiple choice may be an okay assessment, performance assessments with accurate rubrics are a much better tool to assess whether or not a student is able to apply the information in a real life situation.

Reasoning: This article is about students' ability to problem solve in mathematics. The teacher in the article had students explain what area was in their own words. This allowed the teacher to see what each student was thinking and how they were able to think or reason throughout the problem based on what the student wrote.

 Conversation: This article discusses the importance of assessing student conversations. In the example used in the article, a teacher was able to understand that the students didn't know the properties of rectangles. Then the teacher was able to do an impromptu lesson on what is considered a rectangle before moving on with the area lesson. I think assessing conversations is a great way to assess vocabulary terms. This way teachers will know if students know the correct terms and the teachers are able to clarify to make the topic more clear.

Portfolio: This article is about a teacher implementing the use of portfolios in her classroom. Portfolios are an important tool that allow teachers as well as students and parents to see the mathematical growth of the child. I love this idea of having a folder where the student can reflect back on their work. I also liked how this teacher used a certain criteria to assess the portfolio so that this was a project that encouraged the students in mathematics, instead of discouraging them.

2 Articles from our Groups

My first article was: Strategies to Support Productive Struggle. This article explains how teachers should approach students when students don't know how to start a problem, don't know how to continue on with a problem, or if they know the answer but can't explain why. This article showed many examples of dialogue between a teacher and a confused student. In each of the situations, the teacher used guiding questions to help the students break down the more difficult problem into a more simple problem until the student was able to comprehend the problem and move on.

I think this was a nice article on guiding questions. This provides examples for me that I can use as a reference for when I begin teaching. The article also included a chart that explained teacher questions vs. student questions.

My second article was: Counting on Using a Number Game. This article explains the difference between counting all and counting on. Counting all is when a student is given two numbers such as 4 and 2. The student would count, "one two three four, (then move to the next card) five six" to get the total number when these numbers are added together. Counting on is when a student is given the same two cards the student would say, " five six". The student would understand that four was already given to them and would know the number sequencing for the next two numbers.

I liked this article because it was geared towards younger students. With math I usually only work with upper elementary or middle school criteria because I am a math concentration and I will have a middle school endorsement. Having articles will give me resources to use if I do end up teaching in the younger grade levels.

Video 2 Reflection

This video started right in with the lesson. I was in the same position as the students because I didn't know what direction the video would take. The first video grouping had a meeting when teachers went over what the lesson would be about and how she would go about teaching the students. I liked how she was trying to get the student to use the term grouping when explaining the work regarding multiplication and division. I also liked how the teacher gave the students to share their ideas with their neighbor. I think these students need a lesson on mathematics terms and vocabulary. When listening to the students' ideas, you could tell some students really didn't know what some terms meant and in other times they would use the wrong word in replace of another term.

I liked how the teacher tied in "A picture is worth a thousand words" into the lesson for drawing a picture to explain their math, however she never explained what the quote really meant to those who didn't understand it. I liked seeing the engagement of the students during the lesson. They were able to drawn their own pictures and then had to explain how the picture related to the story problem. I also liked Charlie's way. I think the diagram could have been written in a clearer way so that the students could better understand what each of the boxes represented. Maybe next time the boxes could have labels such as Maria's money with her boxes, Wayne's money with his boxes, and then the words total money and a box to represent all of their money. The brackets in the initial diagram would have really confused me if I were a 4th grader. I like how in the debrief other teachers commented on the lack of a vocabulary foundation, the engagement during the lesson, and how some of the diagrams got the right answer, but the diagram did not explain what the math story was. If I were to teach this lesson, I would really emphasize the importance of pictures when doing math problems. I would also emphasize the importance of their connection to the math problem. The picture does no good if it doesn't show what is happening in the problem.

NAEP Reflection

I enjoyed this project. This project is really realistic to what I will be doing as a teacher in the future because I am a math concentration. I enjoyed looking at all of the student work samples  and giving them individual feedback on how they could improve their work. Through the process I learned about the importance of a good rubric. The rubrics we were supposed to use were confusing and each person could have given the same student a completely different grade. Rubrics should be so straight forward that it shouldn't matter who grades the work, that the student will always get the same grade because the rubric allows for no guessing.

Math Apps and Applets Geometry 6-8th Grade

1. The first applet I found is for teaching students about reflections.
http://nlvm.usu.edu/en/nav/frames_asid_206_g_1_t_3.html?open=activities&from=topic_t_3.html

Students can insert different shapes and drag the shapes around to see what the reflection looks like. Axes can be added to have the four quadrants with reflection. The reflection line can also be altered to be at any angle.

I think this is a great applet. It is really user friendly and allows the students to see what the reflection of many shapes would be.

2. The second applet I found is a geoboard for coordinate planes.
http://nlvm.usu.edu/en/nav/frames_asid_303_g_3_t_3.html?open=activities&from=topic_t_3.html

Each page in the geoboard instructs the students on what they are supposed to create. Once the students finish it they can move on to the next problem.

I think this is a fun applet. It is easy to use and is a great way to use geoboards and coordinate planes if these are not available in the classroom.

3. The app I found is a video geometry tutor app.
https://itunes.apple.com/us/app/video-geometry-tutor/id411902469?mt=8

This app has 80 videos teaching the many different aspects of geometry from points and lines to the Pythagorean theorem. Students would be able to access these videos anywhere and don't need access to the internet to do so.

This app is $1.99 to purchase. I think this would be worth the money to have downloaded to tablets for each student to use. This could be a very helpful resource for many teachers, students, and parents.

article 2

Thinking through a Lesson: Successfully Implementing High-Level Tasks.

This article starts off with a math problem about different bags with different amounts of red and blue marbles. The students are to determine the fractions of each, the percent of blue marbles, to scale up the ratios so all bags contain the same amount of blue marbles and so on. These types of problems are ones that don't have a required way to solve them. Students could be creative and solve these questions in any different ways from their neighbors. The rest of the article discusses TTLP, or Thinking Through a Lesson Protocol. This involves the teachers thinking about what questions they want to ask their students prior to the lesson plan as well as the teachers thinking and listing out all the possible ways students may: solve the problem, run into trouble and not know how to continue, or possible common errors students may make.

I thought this article had a lot of great information for current math teachers as well as students training to become teachers. Its important for teachers to realize that math problems can be solved in a variety of ways. So by the teachers solving the problem in these different ways will allow class time and questions to run more smoothly.

Article 1

A Model for Understanding: Understanding in Mathematics.

This article is all about how students are able to understand mathematics. The author, Edward J. Davis, created a list of items that people must be able to do in order to understand a topic. These were a list of seven things like, state in own words, state the opposite of the item, and give examples of it. Davis went on to saying that these items do not complete the list of items needed in order to thoroughly understand a topic. I like the format of the article how he would explain one of the items needed in order to understand, and then he would have a chart of questions and examples that would go along with the point needed for understanding the topic.

This is a beneficial article for math teachers because students all understand topics in various ways. With having these tools to help see whether or not a student has grasped the concept, students will be more successful in mathematics in the long run. If a teacher notices that the students do not understand the topic, they will change their teaching style so that everyone is able to get the math concept.

Rich Task Reflection

I enjoyed all three of the rich task lessons. The first activity with the coins is definitely something I would use if I teach math to younger students. These sticks with coins could be used in many different ways by altering the activity to have it coordinate with a slightly different math task. The second activity with the surface area measurement was a very cute idea. I could definitely see the students getting engaged and enjoying the math activity. I thought the Barbie activity was a great way to show students how to scale up an item by using ratios and rates. Overall, all three had many positives to only one or two negatives. These would be great activities to incorporate into math classrooms for students to be engaged in math.

CCSSM Standards Project Reflection

I found this project to be overwhelming in the beginning. Once we started working on it and breaking the project into smaller chunks it was less stressful. I liked how we were able to look up articles with teaching activities that coordinated with our standard. This helped me understand what the standard was really getting at. I also liked how we presented the standards in a jigsaw style. This way everyone in the class was able to gain from the research each group did.

Classroom Based Formative Assessments Article Reflection

The authors of this article discussed five base assessments that should be mixed in to math classrooms. The five formative assessments are: Observation, interview, show me, hinge question, and exit tasks. The first three, observation, interview and show me are assessments that are done during the day or lesson, whereas the last two, hinge question and exit tasks are done at the end of the lesson to assess the daily knowledge. I really like this idea of mixing up the assessments in math with using these five assessment tools. I think the show me is a great assessment. This is basically an assessment where students have to demonstrate how they are able to arrive at the answer to the problem. With math, teachers sometimes think that the only forms of assessment comes though quizzes and tests. This short article is a nice reminder that there are many ways to assess mathematics.

Video 1 Reflection

Planning:

I liked how the teachers, principal, and the math coach all sat down together to discuss how they were going to take notes from the lesson Ms. Lewis was about to teach. This meeting allowed everyone to be on the same page so they knew the expectations on how they are supposed to interact with the students and what the key ideas they want to take note of. Ms. Lewis and the math coach had planned out a lesson that re-teaches problem solving with the use of addition and subtraction. I liked their plan of going over poster examples as a whole class and then letting the students reflect on their work they had previously done and make any corrections or additions to their work after discussing the example problems.

 
Lesson:

Ms. Lewis started with the example posters with the students sitting up front on the rug. I thought this was good information and instruction, however it did not follow their plan of action and not many of the posters were covered since too much time was spent on the first one. Some of the same students kept participating, which is good. However, it would have been nice to see more overall participation from the whole class during the whole group poster activity. Ms. Lewis strayed from the plan when students got their work back. They had planned to only have the students work on 3 and 4 individually or to verbally say what they were doing. Teachers and observers were not supposed to interact in a way to aid or give answers. The small clip shown of the math coach showed her interacting with the student to understand their thought process, but she did not help the student or say whether their process was right or wrong. Ms. Lewis was helping a student through one of the problems, which wasn't part of the lesson plan.


Debrief:

I thought Ms. Lewis's reflection was spot on. She recognized that she took too much time in the beginning and was only able to get to a few of the posters. She also recognized that the lesson as a whole was not as she had planned. I like how they explained the rest of the posters that were not shown during the lesson so that the observers were then aware of what additional resources the intended lesson was supposed to be.
 

Overall:

I thought this video provided a nice example of how teachers collaborate and how they continually try to improve their teaching methods to help their students. I have already done my novice teaching experience and I can definitely relate to how the plan and actual lesson were different! No matter how much you plan and prepare for a lesson, you never know what may throw off the pace of the lesson. Students were struggling with the first poster, so Ms. Lewis spent more time on this than she had intended. I thought she did a great job adapting the lesson on the go. When teachers realize their students are not getting a topic, they need to step back and adjust the lesson plan from what they had initially planned. Same with if students are easily getting the topic, teachers can adjust the speed so that students move through the material at an appropriate rate.

Preserving Pelicans Article Reflection

The teacher in this article had discovered something many math teachers find out. She realized her students lacked the connection of mathematics to the 'real' world. She discovered there was a U.S. Fish and Wildlife organization that let students help them receive data. The teacher used this opportunity to teach an MEA or model-eliciting activity. These activities align with the common core for mathematics when it comes to problem solving. By connecting this math common core activity with a pelican colony research students were interested and engaged in this new math project. Students had to calculate the amount of pelicans in an area of nesting located on a map. Students were allowed time to brainstorm as a large class group and then as individuals. Students decided to use a square grid to calculate the area. When squares were only partially filled, they would cut out that area and move it to fill up a whole square. By using the modeling activity students were able to successfully show their understanding for the common core math standard that requires students to solve math problems related to realistic contexts.

I thought this was a great way to engage the students and get them excited about mathematics. Connecting the animal aspect would make the students want to solve the math problem even more because they can see how math can apply to real world ideas.

Articles that Relate to CCSSM Standards of Mathematical Practice 1 & 2

The first article I found corresponds with the first standard: Make sense of problems and persevere in solving them. The article is Quick Reads: Problem Solving with Laser Precision by Scott A. Goldthrop, published in Mathematics Teaching in the Middle School.

In this article, Scott explains how his students are given a problem and must find out a way to solve it. Students are divided up into small groups and are given a few tools to help them solve the problem. Two of the activities referenced in the article were to solve how high flags were in the gymnasium with the use of a mirror, ruler, and a laser pointer, and to determine the width of a soccer goal with only a paper towel roll, ruler, and trundle. These two activities definitely relate back to standard 1 because the students are having to plan out what they want to do and have to do some problem solving to get to their answer.

I thought these were great activities for a middle school classroom. In the beginning of the article the author discusses how with modern technology, students' attention spans are shrinking and students can readily gain information such as equations or formulas. He wanted his students to really grasp the concepts of mathematics, so he made his lessons more inquiry based and discussion based so students would be more likely to remember what they had learned.

I definitely agree with this. It will do students no good to just memorize formulas and to plug numbers into formulas to solve problem after problem on worksheets. With the use of technology, students would easily be able to look up a formula within seconds. I believe it is more valuable to teach students problem solving skills because these skills will be used every day for the rest of their lives. Using inquiry for students to learn concepts will allow students to become creative and discover on their own how to come to the solution. When classrooms are student centered, students are discovering about the topic on their own or in small groups by collaborating with others. When classrooms are more student centered, students will retain the information longer because they had discovered information on their own instead of just being given a formula. As a teacher, I want my students to retain as much information as possible as well as leave my classroom with various problem solving skills that they can use for the rest of their lives.

Goldthorp, S. A. (2013). Quick reads: Problem solving with laser precision.
     Mathematics Teaching in the Middle School, 19(2). Retrieved from
     http://www.nctm.org/Publications/mathematics-teaching-in-middle-school/2013/
     Vol19/Issue2/Quick-Reads_-Problem-Solving-with-Laser-Precision/

The second article I found corresponds with the second standard: Reason abstractly and quantitatively. The article is Develop Reasoning Through Pictorial Representation by Wendy P. Ruchti and Cory A. Bennett, published in Mathematics Teaching in the Middle School.

In this article, the authors discuss the importance of using pictures to represent student work. They explain how it is much easier to follow a student's train of thought by looking at pictures of how they went about solving a problem. The pictures allow the teacher to see their students' problem solving techniques. This way teachers will be able to find where a student is struggling as well as see the different ways students approached the problem. Not only is this technique helpful for teachers, but it is also helpful for students. Drawing out pictures allows students to visually see the problem. If the problem is more of an abstract idea, the picture will allow the students to create a visual to help them understand what is going on in the problem.

I am a visual learner, so I find drawing pictures to be extremely helpful. I liked the problems presented within the article because they showed the student work examples of different ways to go about solving the problem. This is also a strategy that can be used at a wide range of grades. Teaching students as young as kindergarten students to draw when problem solving will only increase their ability and creativity in problem solving down the road.

Ruchti, W. P., & Bennett, C. A. (2013). Develop reasoning through pictorial
     representation. Mathematics teaching in the middle school, 19(1), 31-36.
     Retrieved from http://www.nctm.org/Publications/
     mathematics-teaching-in-middle-school/2013/Vol19/Issue1/
     Develop-Reasoning-through-Pictorial-Representations/

Important Points about CCSSM Standards for Mathematical Practice

The first standard I researched was: To make sense of problems and persevere in solving them.
Here are some main points:

• It is important for students to gain and implement problem solving skills when it comes to mathematics. Instead of getting a math problem and jumping right into trying to solve it, this standard works towards students learning to understand the problem and to choose an approach that logically makes sense.
• Once the student solves the problem, students are to choose a different approach to check their answer. This shows students that there are many different ways to approach a problem and to still come to the same conclusions.
• This kind of mathematic problem solving will really show whether the student knows the topics well because the students must understand the different topics and be able to implement them in different situations.
• In this checking process, students are able to take a step back and look at their work. Students are then able to see whether or not their work answers the question or whether or not their answer makes sense.

The second standard I looked into was: Research abstractly and quantitatively.
Here are some main points:

• Students often struggle when it comes to thinking about abstract mathematical problems. The problem often lies in their ability to visualize what they are trying to solve.
• To overcome this, the standard wants students to use manipulatives and pictures to help symbolize these abstract ideas so the students are able to understand the problem and what they need to do in order to solve.
• Students should also be able to think quantitatively, or with numbers. For this portion, students will learn and will be able to understand the various units used to label numbers, convert from one unit to another, and understand what the number value is telling the students about the problem.