Error Blog

This was somewhat of a difficult activity as we were to assess student errors with working with mathematical problems, but were not able to actually have access to students that we could question for their reasonings that were not particularly displayed in the work. This activity did make me realize once again, however, how difficult it is to understand students' work and errors. We have practiced with work like this in the past while working with the NAEP problems and rubrics, but it is still noted that this is something to be practiced a great deal before going into the classroom. This activity was also a reminder that a teacher must understand the students' thought processes to give quality feedback that can be followed up with all of the student in the classroom to deepen understanding.

Manipulative Reflection

How do you know if students deepen their understanding while using manipulatives?
Students deepen their understanding while using manipulatives because they have something physical and real to relate their ideas to while working with a mathematical concept. If you were to simply give students an example on paper and ask them to apply the concept they have just been taught moments ago without any real-life relation or physical manipulative work, it would be extremely hard for the student to understand what they were writing and working with. Using manipulatives helps give students real and tangible proof that the concept they're working with has reason and meaning.

How do you know if the students can transfer their understanding from manipulatives to other situations?
Teachers can begin to check on their students' abilities to transfer their understanding from manipulatives to other situations by slowly working away from the manipulatives towards real-life situations. For example, a teacher could start with 3D shape manipulatives while working with area, then move to having the students draw the shapes themselves or use the Smart Board, and then finally having students find objects of these 3D shape in or outside of the classroom to work with for a real-life connection.

How can you assess that understanding or growth?
Teachers can first assess the understanding and growth by asking students to give reasoning and proof of why and how they are using their manipulatives. We can also assess growth beyond this by having discussions about what their manipulatives can relate to in the real-world, and the importance of understanding that connection and the connection to the mathematical concept they are working with. If students are able to relate to these ideas, they will have shown their understanding and growth beyond working with simple manipulatives. 

When students work in groups, how do you hold each youngster accountable for learning?
An activity that I like to do with students working in groups is having students assign themselves a "job" for that group discussion/activity. Whether it be simply through discussion or writing down what is happening in their group, each student will have the opportunity and responsibility to participate. For example, if I had students in a group of 3 working on problems with manipulatives, one student may have the job of discussion leader, another might be a connector, and the other student may be asked to represent the work, etc.
When students work in groups, how do you assess each youngster's depth of understanding?
As I had stated before, if students were to have specified jobs in their groups, it would be beneficial for the students to come up with some written work of what they accomplished during the group work as somewhat of a reflection and checklist for understanding. Once the students had done their own write up or reflection, there could be a discussion or group write up of how their parts all came together and what they understand of the overall concept that would be beneficial.

How are you improving students' problem solving skills with the manipulatives?
Manipulatives greatly help students with their problem solving skills as it gives them physical objects to work with while learning and practicing various mathematical concepts. It also helps students begin to make connections with the manipulatives to real-world situations, and thus will lead them to work with this practice throughout higher levels of mathematics.

Curriculum Plan Reflection

Creating this curriculum plan was by far the most challenging project of the semester. However, it was also the most beneficial to my learning as a future teacher and brought to light a lot of personal strengths and weaknesses I had not noticed before. When our group started the curriculum plan we were extremely overwhelmed. Once we really began looking through different activities and projects that we wanted to include for each grade level, however, things started to progress a lot more. We were really able to get into the standards for our 3 grade levels and look at what things were repeating and what things were building off of each other from the previous years, and this led our group to some wonderful ideas about review activities for the start of each school year. Overall, working with this project was a good experience for our group and was especially helpful in getting us fully involved with the standards for grades 3-5.

Watching the other groups' curriculum plans was really great. It was tremendously obvious that our group was a little technologically challenged after watching how great everyone's videos were. One thing I did notice, though, was how it was somewhat difficult to really understand everyone's curriculum plans form just watching what they did in the video. We had to go back to Sakai and really look through the written plans to get the whole picture and see how it was connecting to the other grade levels. Once we did this, we could really connect a lot of the ideas from grades prior and after ours together and see how a lot of standards we building off of each other and how some were very redundant. Once again, this was a great experience to see how the standards were fully worked in to the other grade levels that we did not focus on.

Standards and Classroom Changes to Deepen Math Learning Reflection

Throughout this course, we have been consistently working with the Standards of Mathematical Practice and the NCTM Standards. The utilization of these standards is in regards to changes made in the K-8 curriculum by introducing the Common Core Standards for mathematics and language arts into the schools. This change in the curriculum is extremely beneficial for both students and teachers as it allows both parties to utilize many different ways of learning, as well as tie in the process standards into our everyday learning.

NCTM Process Standards:

• Problem Solving
• Reasoning & Proof
• Communication
• Connections
• Representation

The new Common Core Standards easily connect to these 5 process standards and have the instructor focusing on fulfilling each of these to deepen mathematical understanding in the classroom. For example, a 4th grade teacher may be working with her students on problem solving with fractions. Instead of incorporating just one way of looking at this concept with just one process standard like we might have seen in the past, the Common Core Standards ask the teacher and the students to show reasoning and proof behind their thought process while problem solving, communicate with peers about ideas, connect these ideas to real-life situations and past mathematical concepts, and finally represent their ideas and mathematical skills in many different ways besides simply filling out a worksheet.

Assessment Reflection

There are many different types of assessments that teachers can use in the classroom. What is most striking to me is how much the idea of what kinds of assessments to use has changed over the years. When I was in elementary school, and even somewhat into high school, a lot of my teachers were solely focused on formal assessment. However, as I am now going through classes, I am getting the opportunity to research, read about, and practice the many different types of assessments that can be much more beneficial than old-school formal assessments. For example, there are many different authentic assessments that can be utilized in the classroom. Authentic assessments "ask students to read real texts, to write for authentic purposes about meaningful topics, and to participate in authentic literacy tasks such as discussing books, keeping journals, writing letters, and revising a piece of writing until it works for the reader." These types of assessment can help teachers better realize how much their students are actually understanding on a real-world level and work to grasp their students' though process behind the work they have completed, just as much as the finished product. Authentic assessments such as these also open up the floor to much more of an opportunity for discussion amongst peers and teacher-student discussion. As we have seen throughout this semester, whether it be through videos, articles, or simply through discussing our own experiences, children tend understand and grasp a concept more when there is communication happening in the classroom. Likewise, this communication can be a form of assessment on its own. 

Additionally, I still believe that creating a form of structure to assess the students it necessary, as it helps not only the teacher stay aware of what is to be looked for, but it guides the students in the right direction as well. Even with authentic assessments, some form of rubric should be provided for the students, especially in mathematics. Using rubrics in assessments will also help us as teachers to be able to go back and reflect on what the students have done, give feedback, and follow through with that feedback to create a richer learning experience and more beneficial assessments that both measures and strengthens understanding in the classroom.

Technology Reflection

Throughout this course, we have been shown and had to research our own ways to use many different forms of technology in the classroom. Using technology in today's classrooms is extremely important and it is important to stay up to date with how to use the different technologies. One thing that I really loved about this course is that we worked with the Smart Board a lot and in many different ways. This was especially important for me because I was not used to the Smart Board and had the opportunity to learn a lot of useful techniques that I can take to my future classroom. Having us as the students research Smart Board activities was also very helpful because we could learn from our colleagues as well as teach ourselves. Another activity that involved technology that I know was extremely beneficial was allowing us to research math apps and applets. So many of us brought wonderful ideas to the table and we were all able to explain how to use these in the classroom. I know that I would be able to use every one of these technologies in my future classroom. Overall, this class was extremely helpful in teaching many different technologies to take to the classroom and I believe this is so important for us as future teachers to continue working with.

Moving Beyond Brownies and Pizza

"A lack of fractional understanding is a well-documented obstacle to student achievement in upper elementary and middle school math" says the National Center for Educational Statistics. This article begins by pointing out their main focus as being about helping 4th graders to use number lines to develop a rich understanding of comparing fractions. The article then goes on to discuss that there are 3 main goals they are focusing on:

1. Build students' understanding of fractions as numbers with a definite magnitude
2. Increase students' understanding of measuring with fractions
3. Develop fraction number sense by avoiding early introduction of traditional fraction algorithms

Then, as they move onto trying to reach these 3 goals, they tell us that they will be focusing on 6 different types of lessons that identify ways that 2 fractions might be related to one another:

1. Both fractions are one unit fraction more than a half
2. Both fractions are one unit fraction less than a half
3. Both fractions are one unit fraction less than one
4. Both fractions can be used in the context of money
5. More of a bigger unit fraction or less of a smaller unit fraction
6. One fraction can be expressed in terms of another

There was then 2 different examples from students and their thought process about different perspectives of the same number line. Even though both students thought differently about the problem, it was clear that the number line was central to their thinking. The same process was shown for different students a few more times, as well. After the lessons, the teacher was able to see that all of the students in the class had made progress in relation to the original 3 goals. 

I had some extremely good thoughts about this article. I really liked the overall idea of getting the students to build an understanding of fractions as numbers with a definitive magnitude. Also, I think this teacher's example of her plan for the class was very well thought out and put into motion the best way it could have been. It also really got me thinking about the troubles students commonly have when being introduced to fractions. It's hard to realize that once some misconceptions or misunderstandings are made right off the bat, it is difficult to help the students get back on track and fully understand the concept they're working with. And this is common with any mathematical concept. This is the main reason I liked this article and it really brought that idea to life. 

Girls Build Excitement for Math from Scratch

This article begins by focusing on the idea of increasing "digital literacy and mathematical fluency" through a workshop dealing with computer coding. Computer coding helps people to advance mathematical understanding and spatial reasoning. Coding is significantly important with numbers showing that five of the top ten jobs in the United States are in information technology. However, they state that girl's attitudes towards math related careers diminished diminish at a far more substantial rate than boys, highlighting the need to focus on girls and STEM disciplines, and this led to the creating of the computer coding unit designed to girls in grades 6-8. The program that the girls use is called Scratch: "a graphical two-dimensional, drag-and-drop programming approach, with easy to understand graphical icons representing basic programming elements." They began using this program in the classroom by implementing lessons that may address how to create a new project, add and modify, move projects around the screen, etc. Scratch was also designed to help emphasize the role of mathematics and science in computing and technology. Three mathematical topics are focused on during the lessons: mathematical descriptions of position and movement, scaling and percentages, and mathematical abstraction of physics concepts. The article continues to talk about how Scratch could easily be implemented in your own classroom one to two hours a week. It talks about how math teachers could design a multidisciplinary coding unit with science, technology, and mathematics.

I was very surprised when I began reading this article that this is what it was about. Overall, though, I think this is a very interesting concept. It reminded me that in our high school we had a class that was similar to this, although they only considered it to be a "computers class." When I think back on it now, however, I remember how much math really went into the coding. You had to have a solid understanding of each mathematical concept before you could even begin to understand the program. This is somewhere that I could see trying to implement the Scratch program hitting a bump in the road. Even though this program is fairly interactive and usable, students would need to have a complete and thorough understanding of the concepts you would want them to use. I was extremely surprised that they wanted to implement this project in middle schools, because I just think they would have a very rough time being able to actually use this to its full potential. I like the idea and the goal they are working to accomplish and it is a great program, however I think something like this would be much more useful at the high school age once they have mastered these concepts and are closer to choosing their career path in college.  

Problem Project Reflection

This project was really something interesting. For starters, beginning this project took A LOT of time. It is extremely difficult to come up with an idea that students will be able to relate to and at the same time connect to the standards as it should. But we finally were able to come up with our great idea for a game day. We had found a similar idea to this on Pinterest and decided that it would be something really fun to play around with and extend it into a problem project for 4th grade students. Another thing that I thought was super fun about this project was working to tie in interdisciplinary connections. I love when projects or lessons can tie in multiple subjects, and we were excited about being able to include English and Visual Arts into ours.

Additionally, it was very interesting to listen to the other groups presentations and what they game up with for their grade bands. One thing that stuck with me after hearing the other groups' ideas was how as you get up into the higher grade levels, the more ill-structured you can make the whole project and you can give a little more creative freedom with the project still running smoothly. I feel as though groups dealing with younger ages felt the need to include a lot more information and instruction than the groups with older grades. Overall, I think our group as well as both of the other groups came up with some amazing ideas for such a difficult project.

NAEP Reflection

This project started out to be very challenging and really had us working hard and thinking about helping the students we were working with. It was interesting to see that even though we have worked with creating and reflecting different assessments throughout the education program, it is something different to really give the students feedback and think about how to create a follow-up lesson that will positively effect the entire class. I think that one thing we did very well while doing this project was choosing 3 students who had difficulties that were somewhat similar in their work. They weren't all exactly the same, but they were related enough on different difficulty levels that we were able to create a follow-up lesson that would help each one of these students' difficulties. 

It was also very interesting to listen to how other groups decided to go about this project and how they interpreted rubrics differently/what kinds of feedback they would give to their students. Although a lot of ideas others had were different than my own, I was able to take away a lot of good ideas and examples from them. All in all, I think a project like this is very beneficial for future educators and should be done more than once with different topics.

Number Operations: Multiplication and Division

The first thing I noticed about the beginning of this video is that the teacher did a very good job of introducing the lesson. She used a variety of techniques to get the students thinking about the idea of multiplication and division when she had them think about a well known quote, share ideas about what the quote meant, and then having them talk to a neighbor about what they already know about multiplication and division. She also went on to have students share their ideas about what they already know about multiplication and division with the entire class. This way students were able to see how others were thinking about problems that were actually the same, and could give students another way to go about solving problems in the future. Additionally, she went on to build on her beginning topic of "a picture is worth a thousand words" by showing the students how they can show their grouping with pictures.

As the teacher was going through the problems with students, I liked that she would take everyone's ideas and write them down on the board whether they were correct or incorrect so that she could work through the errors with the whole class. She continued with this concept through the division part and here is where you could really see how helpful talking to each other was for the students. It was interesting how much the students were sharing their ideas with each other and listening to their peers. This would be extremely helpful with some students who are struggling to put the idea into their own words because they can talk it out with a partner. As the teacher went back to her concept of drawing pictures to explain division, it definitely reminded me of strategies we learning in ETE 107 and showing how the groups are separated using a picture while we divide. This idea would really help students to make sense of what is being divided and into how many groups, etc.

I think one of the things I liked most about this video was how the teacher had the students build off the idea of comparing operations. As she compared operations throughout the entire lesson, she also continued to show the students all of the various strategies to solve the problems using mental math. This included using the pictures to show grouping. I think that this lesson was very well thought out and would help many students to deepen their understanding of multiplication and division, as well as helping them to be able to better explain their answers and thought processes for future problems.

Math Applets

Applet #1: Alien Angles

Math Playground
URL: http://www.mathplayground.com/alienangles.html

Angle Aliens supports Grade 4 Common Core Math Standards in Measurement and Data.

"Friendly aliens are traveling to a new space colony on Planet Geometry. Unfortunately, some aliens have lost their way. You have been placed in charge of the rescue mission. It is your responsibility to set the correct angle on the rescue launcher. The computer will generate the launch angle. To be successful, you must estimate the angle on the rescue launcher within 5 degrees. There are 10 aliens to find in all. Good luck with your mission."

This would be a great applet for 3-5 graders studying angles in Geometry whether it is just now being introduced, or whether it is used for a review game. First of all, I believe students would respond well to this applet because of the nature of the game being about aliens and a rescue mission. This will help to keep the students' interest while they are practicing their Geometry skills. As far as their Geometry skills go, students will be able to begin to recognize what certain angle measurements look like in terms of drawings or figures. For example, a student will begin to see that a 90 degree angle takes an "L" shape, or that 180 degrees lies flat. This will also help build their estimation skills when working with angles, as they are asked to provide an estimate of a certain angle by drawing the lines themselves just from remembering what that angle may look like. I would definitely use this applets in the classroom with beginning and more advanced Geometry students.

Applet #2: Platform Scale Subtraction

Visual Fractions
URL: http://www.visualfractions.com/scale/platformscale3.htm

Platform Scale Subtraction supports Grade 5 Common Core Standards in Measurement and Data.

In this applet, you are given a virtual "scale" on the screen and are shown 5 "Gregs" (simply the name of the character that each student is weighing. The instructions are as followed: 

"Press the <Start> button when the program begins. Five Gregs will appear and one of the Gregs will drop down to be weighed. You can weigh Greg by moving the larger and smaller weights to the right or left on the graduated bar. 

The bottom (larger) weight shows units and the top(smaller) weight shows hundredths of a gram. Move the weights until the bar is level. 

The red pointer on the right should be on the center mark when you arrive at Greg's weight.

When you arrive at Greg's weight a plunger will gently push him to the right. You will be prompted to type the difference between the remaining gregs and the Greg you weighed. Type the difference into the edit box and press the <Submit> button. If the amount you submitted is correct, another Greg will land onto the scale. Continue until all Gregs are weighed.

After five Gregs are weighed, you will have a chance to weigh more Gregs by pressing the <Start> button or you can press the <rReport> button to report the time it took to weigh five Gregs.
The program will keep track of the total Gregs you have weighed and the time you took to weigh them.

A challenge - See if you can find the differences mentally."

This applet would be very useful in class to bring a visual representation for the students to see as you are discussing weight/weight changes and measurement. With this applet, students are able to actually see the scale, play around with different measurements, and put their knowledge to the test as they manually set the scale to difference weights and subtract them from the original. I really liked how this applet was set up when I first discovered it because of how many different skills it allows the students to work on at one time. However, one of my complaints about this site is that I can see it being somewhat tricky for young students to navigate or make use of the game. It would take some time introducing this applet to the students, and it may also take time for them to get used to this idea and be able to use it on their own. I would recommend using this activity on the Smart Board and as a whole class activity to make the most of your time and for students to reap the full benefit of this applet.

App #1: Splash Math

iTunes Store
URL: https://itunes.apple.com/us/app/3rd-grade-splash-math-games./id449564960?mt=8 

Math Splash supports Grade 3 Common Core Math Standards.

Splash Math is an app for 3rd grade classrooms that includes a variety of fun and interactive math problems aligned to Common Core Standards. "The app reinforces math concepts with self-paced and adaptive practice anytime, anywhere (works on iPhone, iPad, laptops, and desktops)."

This program is currently being used by more than 10 million teachers and students, and many teachers have given this app great reviews such as this one:

"I’m very impressed with the thoroughness of content in Splash Math. I really like the workbook concept and it works well here. This is most definitely a well thought out, well designed, educational and entertaining Math app."

Splash Math has various key features such as:

• self-paced math programs
• explanations for wrong answers
• scratchpad for rough work
• virtual rewards and games
• monitors progress with a real-time progress dashboard
• progress synced across multiple iPhones, iPads, laptops, and desktops
• HD graphics and sound effects 

Likewise, Splash Math covers the various topics needed to align with the Common Core Standards:

"1. Place Value - Ones; Tens; Hundreds; Thousands; Ten thousands; Expanded and word forms
2. Number Sense - Compare; Order and round numbers 
3. Addition – Two and three digit addition and regrouping
4. Subtraction – One, Two and Three digit subtraction and regrouping; Subtract Across zeroes
5. Four Digit Addition – Two and three digit addition and regrouping with four digits
6. Four Digit Subtraction – Two and Three digit subtraction and regrouping with four digits; Subtract Across zeroes
7. Multiplication Facts - Properties of multiplication; Multiply by 0 to 10 
8. Division Facts- Properties of division; Divide by 2 to 10 
9. Fractions – Identify fractions, Model fractions; Equivalent fractions; Compare fractions
10. Time - Read and set time; Elapsed time 
11. Measurements and Data - Measuring length; Units of length, Capacity, Weight; Data on Line plots and Bar graphs; Perimeter; Area
12. Geometry - Triangles; Quadrilaterals
13. Decimals - Tenths; Hundredths 
14. Multiplication - Multiply multiples of 10; Estimate products; Multiply 2, 3 digit numbers 
15. Division - Divide multiples of 10, 100; Estimate quotients; 2/3 digit quotients 
16. Money - Add and Subtract money; Multiply and Divide money"

All in all, I think this is a WONDERFUL app for the classroom. First of all, this app is eye appealing, fun, and interactive for all students. It allows students to be entertained while working through the math skills necessary for their grade level. I also love that this app is self-paced and students can work to the best of their ability without being rushed. This app also gives great explanations for wrong answers so that students can reflect on the feedback. Another huge plus for this app is how it allows students to sync and track their progress through various devices. This is a great way for students to keep practicing at home from where they left off in class. This also could be a helpful benefit for teachers as they could give a fun homework assignment of completing some Splash Math activities at home. Lastly, this app is wonderful just due to the sheer amount of informations and topics that it covers.  

A Model for Understanding Understanding in Mathematics

When I first began reading this article, it interested me right away because of its topic of “understanding understanding.” This is something that we talk a lot about in all of our classes as we progress through the education program, and something that I think a lot about in the actual classroom. A concept that was new to me in this article was talking about teachers using different “moves” while teaching, and how those moves can vary when teaching concepts and ideas. Another thing that I liked in the article was how it helped get this point across by giving examples of what kinds of questions a teacher will ask her students to check for understanding. In the closing of the article, they pose the questions: “Can effective instructional sequences be described in terms of moves? Do moves help structure knowledge in ways most readily grasped by the learner?” That is something that I would even like to research more on my own.

Rich Task Reflection

I learned a lot working with my group for this project. One of the things we really focused our attention on at first was getting to know what exactly should constitute a "rich task." Once we were sure we've wrapped our head around the idea, we shared ideas that we could use for teaching our lesson. Something that I was surprised to learn was that once we had selected a lesson, how difficult it would be to adapt the lesson to fit our class's needs. It is something valuable to remember, though, especially when thinking about how lessons and projects will need to be adapted through the various classes.

It was also very interesting watching other groups do their lessons and also very helpful. There were a lot of tips that I was able to take from how the other lessons were presented, how they decided to assess students, and what kind of reflection questions were asked after the lesson was completed. I thought it was interesting that both of the other groups decided to use a worksheet to follow along with their lessons, however, they used the worksheets in a way that was actually helpful to the students and not just as a time filler.

CCSM Reflection

This project really served to help me learn a lot more about the standards. Our group worked the closest with the standards of Attend to Precision and Look for and Make Use of Structure. Sure, doing the initial research on these standards was beneficial, but I think I learned the most about them in depth when we made our presentations and really got into discussing them together as a group. This also helped me to look at the standards as more of a usable structure instead of just reading through them without full understanding.

Since I was not in class on the day of presentations, I went back and listened to the Prezis and Jings from home. It was very interesting to see how the other groups went about presenting their particular standard, and especially interesting to see what individuals had to say about each one. This is why I love the idea of presenting our work whenever we do some in depth research; it gives us the opportunity to understand how others are interpreting our group work.

A Blizzard of Value

This article begins with the teacher talking about the background for why she decided to use this idea for problem/project. She had seen many of her students over a break at their local Dairy Queen, and she came up with the question: Which Blizzard size is the best value? Working with this problem, her students were given the opportunity to work on modeling with mathematics. NCTM also mentioned that this task "supports learning mathematics content, promotes engagement in mathematical practices, and fosters mathematical communication among students." To begin the problem, students in the class came to the conclusion that they think the large Blizzard would be the best value, based on prior experience. The class self-selected groups of 3 or 4, and the teacher distributed the problem:

Students went about the problem using different approaches. For example: some students used a graph to model the mathematical elements. Other students used pictures to show their calculations of the diameter of the bottoms of each sized cup. Lastly, other students created a "tabular" approach and made a table including each element of all sizes of Blizzards to compare and contrast. The teacher here was able to remind students, then, that there could be many different ways to go about figuring out the problem and problems similar to this. 

I would love to use a problem like this in my classroom because it ties in so many good aspects together. For one, it would be great to use this problem or something similar because the students will be able to connect to it very well and realize that it is a real life situation that they might come across. Another great reason this would be useful in class is the fact that you can show many different ways to figure out the problem that all students will be able to try and find the best way for them. 

Word Problem Clues Video

I loved the first clip of this video right from the start. I think it was great that they pulled together a group of so many educators to discuss a common issue this second grade teacher was seeing in her classroom. I thought this was a good process to go through to get more insight into what the teacher's thinking process was while coming up with this reengagement lesson, and it will also benefit everyone in the end because they will be able to give her more feedback after the lesson is completed.

What I really liked about how the beginning of the lesson was going was how she had larger examples on posters of some of the students' work to show the students as they sat down in front of them. This way, the students were able to see many different ways that other students figured out the word problems. The only thing that I didn't like about this part of the lesson was that I think Ms. Lewis did spend a little too much time on the previous examples and it was sometimes unorganized discussion with the students about each problem. If she would have organized this section a little more, there would have been more time at the end for the students to be correcting their papers and talking with their classmates.

Another thing that I really liked was how Ms. Lewis constantly encouraged the students to use correct vocabulary when speaking about the word problems. We can see this connect back to the standard of Attending to Precision, where the goal is to be able to use and understand correct mathematics vocabulary and be able to communicate your work clearly to others.

The final debriefing section was a wonderful way for Ms. Lewis to reflect on her lesson and talk with the group about what she thought went well, and what she would have liked to have done differently. The ending clip here also brought to my attention how affective only watching this series of video clips was. It brought to my attention the many ways that we can get students involved in learning math processes as an entire class in the way that they shared with partners and discussed different strategies. It also brought to my attention that even the younger students can learn to self-correct and are able to explain their thought process when given the opportunity.

CCSSM Article: Standards of Mathematical Practice 6 & 7

The article that I found in the NCTM Journals really helps to explain some of what our standard, Attend to Precision, is all about. The title of the article is The Language of Mathematics, and can be found here: http://www.nctm.org/Publications/Teaching-Children-Mathematics/2015/Vol21/Issue9/The-Language-of-Mathematics/.

Bruun, F., Diaz, J., & Dykes, V. (2015). The Language of Mathematics. Teaching Children
Mathematics, 21(9). Retrieved from http://www.nctm.org/Publications/Teaching-Children-Mathematics/2015/Vol21/Issue9/The-Language-of-Mathematics/

Right from the beginning, this article starts to explain the importance of learning how to communicate Mathematics, and the struggle it poses to learn how since the learning of this is largely limited to school. "Principles and Standards for School Mathematics (NCTM 2000, p. 60) states that students who have opportunities, encouragement, and support for speaking, writing, reading, and listening in math classes benefit because 'they communicate to learn mathematics, and they learn to communicate mathematically,' which helps children be successful in math class" (Bruun, et al. 2015). This is the statement that really sparked my interest in following along with the rest of the article to see that it really goes hand in hand with the communication aspect of attending to precision. 

We then get to see how two graduate students use two different methods to teach math vocabulary to their 4th grade students. Method 1 was all about using journal writing and peer discussion. The journals allow students to explain their thought processes and make connections. They could share their journals to check for accuracy as well as to provide an aid for working through meanings of words. Method 2 focused on the modified Frayer model. Putting graphic organizers to use, students have the opportunity here to employ the reading strategy of visualization and drawing pictures of the math vocabulary, which should ultimately help them to better understand and connect with the text/vocabulary. 

After reading about these 2 different methods, the article gives us some examples of visuals we would see the students complete using the modified Frayer model. It is noted that the graphic organizers took students no more than 20 minutes to complete, and, in my opinion, this method has the opportunity to work wonders in the classroom. It gives all students a chance to learn the material in there own way since it connects visual, oral, and hands-on learning. 

In the end, after research had been conducted, it was confirmed that both of the graduate students' methods had a positive effect in the classroom. Students had learned and developed a conceptual understanding of the math vocabulary.

Standards

While reading through the CCSSM Standards, I mostly focused on Attend to Precision and Look for and Make Use of Structure, as those were the two main standards my group would be working with. There were two main points from Attend to Precision that really stuck out to me. One was how important it is for students to be able to communicate their work to other students/colleagues. The second most important part of this standard was seeing how communicating work should vary through grade levels. For example, younger students should be able to state clear definitions and meanings, while older students should then be able to examine these claims and use them explicitly. There were also two important factors that stuck out to me from Look for and Make Use of Structure. One was how even younger students can start to realize that their is structure in their mathematics. For example, if a student's flashcard had the problem 3+4=7, they can also realize that 7-4=3. Secondly, with older students, they can begin to visualize the structure in more advanced algebraic equations, making it ultimately easier for them to begin and end a problem with many different components.