Content Standards

The two main areas that were not covered in the textbooks that I found was solving inequalities at a 6^th grade level and graphing systems of equations at an 8^th grade level. Both of these concepts would be on the more difficult end of the curriculum for the grade level but should be included in the textbooks so that the teachers can correlate their units to the Common Core Standards while using the textbook. It is also important that the teachers challenge their students with new or more demanding material, specifically those students that may be above grade level in mathematics.

From kindergarten through 5th grade, it seems that most areas were covered in the textbooks that my peers located. Also, the curriculum seems to build from year to year quite well. Review is incorporated but the material also becomes more difficult as the student advances, as it should. The textbooks seem to mirror this growth and include information covering most of the Common Core Standards in these younger grade levels. However, one area that some of the textbooks seemed to fall short was including real world applications and a large amount of word problems compared to mostly straight computational problems.

Problem/Project Reflection

This assignment was very enjoyable to me and I think that I gained some valuable practice in planning and developing a great student project. After my group and I came up with our initial idea, we were able to make a five day plan that encompassed all of the material that would be needed to complete the project. I think that having the students do a project is an awesome way to have them apply material over many different math topics. Any time that math can be applied to a real life situation brings more excitement and appeal to the students so they will be more likely to remember what they have learned and take that information onto their next grade or even in their own lives. It was fun for me to work with my group to expand our idea and add details to make it like a real project that I could use someday!  

Error Analysis Reflection

Our work with decoding student work and deciphering what errors they were making in their work was a surprisingly difficult yet constructive task. Sometimes it was very challenging to figure out what mistake the student was making or the misconception they had, so it was good practice for us future teachers to analyze the various student work. However, that is only the first step. Next we will need to know what we can do to help the student to correct these errors and understand the topic. In class, we discussed how to accomplish this which I thought was a very beneficial conversation, It got me thinking about how in mathematics it is important for the teacher to allow room for multiple ways of solving a problem so that each student has a method in which the information clicks and so that they can be successful.

Assessments in Math Reflection

This class truly opened my eyes to a multitude of ways to assess mathematics that I had not previously experienced. One main facet in the math classroom that can be used as a form of informal assessment that I had not thought would be an important part of learning math is discussion. Whether it contains the whole class, small groups, or student pairs, allowing the students to communicate their ideas and listen to the suggestions of their peers is an important step in helping them to fully understand a concept. When it comes to more of a formal assessment, I think that mathematics tends to fall in the routine of paper and pencil testing. However, this class let me see that students can be even more thoroughly assessed through completing a project or solving a question that has multiple parts where the student is required to justify their answers as part of their evaluation. In class, we were even assessed in many ways such as through blogging, individual and group projects, as well as an interview. I am confident that I now have the knowledge and ability to continue to create authentic assessments for my future students as I have been doing throughout this class.  

Technology in Math Class Reflection

Technology is clearly an enormous aspect in today's classroom and will continue to expand and evolve as time goes by so it is important to continually integrate the use of various technologies in the classroom. In this class, I think that sharing Smart Board skills was a beneficial task for everyone. I not only learned more by presenting an ability of the board but learned many new skills from my peers as well. Also, the video aspects required for a couple of our group projects as well as my portfolio challenged me to broaden my normal skill set for presenting material and I specifically became more familiar with Jing. Lastly, our assignment that focused on finding and experimenting with math apps and applets was very useful to me. I found many programs that would have a strong purpose in my future classroom. due to their content and the fact that students enjoy working with technology.

Manipulative Reflection

Allowing students to work hands-on while learning gives them use of another sense that will be geared toward the learning experience besides just listening and watching the teacher. It will make the material they are learning more memorable as well as give them a visualization that they can refer back to when solving a problem. If the students are able to use manipulatives to obtain an answer to a question, it usually means that they have grasped the entire concept, not simply memorized an algorithm. The students will then also be able to apply their ability to draw representations to solve other problems which relates directly back to them practicing with the manipulatives.

Always requiring students to justify their work will allow the teacher to see if the use of manipulatives was helpful or influenced the student's reasoning behind solving a particular problem. Also, when students work in groups, it is important for each student to individually turn in a completed task so that they get practice writing down the steps or creating justifications for the answer that the group reached. The teacher needs to make sure to observe the students during group work to ensure that all members are participating and gaining valuable knowledge while working with their peers.

Curriculum Plan Reflection

I think that this assignment required me to apply many skills that I will be needed in my future career. Planning one lesson or even a unit is a task that I have completed numerous times but compiling a whole plan for one year, let alone three years was a whole other challenge. It forced us to work as a group to come up with creative ways to meet all the math standards from grades 6 to 8. I was proud of my group and I for our ability to use our resources and previous knowledge to come up with some great ideas for quarter and semester long projects for our students. Being able to work with a group and cooperate is an important life skill and will definitely apply to my teaching career as well so I was glad that we were able to collaborate throughout the whole process. Also, since these were standards that we had previously worked with for our Lesson Segment Assignment, all of our work fit together nicely and we were sort of experts in each of our designated math topics. Creating the video presentation was the aspect that I was most worried about, however I felt that it was an effective way to deliver our plan and our video ended up working out just fine. This assignment was a valuable experience and gave me a taste for how much work I can look forward to when I have a class of my own!

NAEP Student Analysis Reflection

This assignment made me realize how difficult evaluating math problems can be, especially when we are taking into account more than just a simple right or wrong answer. It is important to analyze the student work so that we can understand how they reached the solution whether it is correct or incorrect. For example, some students have grasped some skills necessary to solve a problem but need more assistance in another aspect of the problem. Also, we want students to get away from putting an answer and not explaining their reasoning because the further they may dive into math curriculum, the more important it becomes to understand why answers are true and how certain theories or algorithms work. 

My group was in charge of assessing the 8th grade level problem entitled "Marcy's Dots". Our group worked well together to decide which student work constituted each level of the rubric which was an important step so that we were all on the same page when deciding a score. When sharing our scores with the class, it seemed that there was agreement with our decisions and that our implications for the teacher made sense as well. I enjoyed hearing other ideas from the class and discussing each level of student work. 

When looking at other groups' NAEP problems, it seemed that some were much more difficult to grade than our group because of a complicated or not specific rubric. This definitely sparked discussion among the class as to which grades we thought that each piece of work deserved. It was difficult to reach a conclusion at times because people can interpret what the students have shown and what that implies that they know differently. Overall, I learned a lot about assessment during this project as well as the importance to instruct students at a young age to completely explain and justify their work with math problems.     

Rich Activity Reflection

I think that too often math becomes a subject in which students repeatedly complete computational problems out of a textbook, take notes from the whiteboard, and then take an exam. Although there is a time and place for these sorts of exercises, there is a big need for teachers to incorporate activities that require their students to take the knowledge that they have been building and apply it to a task or problem. Allowing students to explore in mathematics and come up with ideas on their own is critical in developing their problem solving skills and it may also give them more confidence in this subject area. 

The rich activity that my group decided would have worthwhile content as well as being enjoyable to middle school-aged students involved using ratios, proportions, and geometry to find the relationship between objects and their shadows. I was very disappointed that we were not able to complete the activity with the class due to bad weather, but I think that granting students the ability to go outside to complete a hands-on assignment is ideal. This type of activity also shows the students that what they are learning in the classroom relates directly to real life.  

Video Analysis #2

The teacher's overall theme of connecting back to "pictures being worth a thousand words" is a very interesting concept. In mathematics, drawing pictures to accompany the answer will show how much of the information the students understand on a deep level and what are simply facts that they have memorized and can recite when needed. I like that the teacher gives examples of drawings and representations when the students come up with the idea that multiplication as repeated addition so that they know what she means by "math pictures". 

During the discussion about multiplication, the idea of equal groups comes up. This is a key concept to multiplication and division so is important for the students to be familiar with the idea. In the teacher commentary, she explains that she kept trying to bring this notion back up during the discussion of division, since the students were struggling, but they were not connecting it back to grouping. The reason for this is probably because the students are not used to the mathematical language and therefore have a difficult time voicing their explanations of retrieving their solutions. So, even at a young age, students need to be introduced to and expected to use mathematical vocabulary to explain the reasoning behind their problem solving.

Since there is such a large amount of discussion and interaction with peers, it is critical that there is a very comfortable classroom environment in place so that all students feel that they can express their ideas safely. Being able to critique the work of others and taking corrections on your own work are crucial steps in finding a solution and group discussion. Rules should be put in place so that working together goes smoothly whether it is whole group, small group, or partner work. In the video, the class seems to have healthy conditions for collaboration, however when the student work shows that the small groups have a big impact on the students final answers. If they think that their peers have the correct answer, they may change theirs without being able to explain how the solution was reached. The teacher also had great control of the classroom and was able in incorporate fun attention-getters that keep the students engaged and attentive. 

I thought that the teacher's mental math question was well chosen because it was not too difficult but was not a fact that most students would have memorized; they would have to choose a strategy to think about to get to the answer. Moving to a word problem was a very sensible next step in the lesson because it allows the students to apply the knowledge that they have been building upon throughout the lesson. Posing this question also lends itself to connecting back to the theme of drawing mathematical pictures. The teacher sets up the problem so that there are not specific instructions and therefore lets the students use their own exploration to come up with an answer. When looking at the student work it seems that some students knew how to get the right answer, but their pictures do not align with their answer, so we know that these students just know the facts of the problem and need to continue gaining a deeper understanding of the topic. Lastly, including student reflection can help the teacher to place where each student stands in grasping the concept and what next step needs to be taken in the overall unit.  

Math Applets/App Review

Candy Factory Educational Game (Grades 6-8) 

https://itunes.apple.com/us/app/candyfactory-educational-game/id446248045?mt=8 

This app allows middle school aged students to practice partitioning and iterating involving fractions. The main concept is that the player is expected to complete a candy bar order for the customer. However, they will need to split up and copy parts or the candy bar to reach the correct size. The game is timed and and accuracy is monitored, however there is the chance to adjust your answer, considering the tasks become quite difficult to complete correctly on the first try. 

I think that app can be very beneficial to students, especially since fractions tend to be an area where extra practice and deepened understanding is vital. The tasks that are being asked force the students to look at fractions in a way that will help them when using improper fractions as well as multiplying and dividing fractions. The player receives immediate feedback when completing each challenge and is also given a summary of how they came to the correct answer which allows for reflection over the process of solving the problem.

Circle 0 (Grades 3-5)

http://nlvm.usu.edu/en/nav/frames_asid_122_g_2_t_1.html?open=instructions&from=grade_g_2.html 

This applet involves solving a puzzle using properties of numbers and operations. The player is given seven circles that are overlapping, so there are three parts in each circle. A few numbers are filled into various parts of the circle which cannot be moved. Then there are the remaining number (no extras) off to the side in which the player uses to fill in the empty spots. Each circle (so three numbers) need to add up to zero using the numbers given (there are positive and negative numbers).
multiple 

This task gets a student thinking about how numbers relate to one another. The challenge lies in the fact that there are only certain numbers available to complete the puzzle and the circles are interconnected so the placement of the numbers is important and causes the students to think ahead and use problem solving skills. Once one circle has reached zero it changes colors so that the player is aware of their progress and the puzzle lends itself to much adjusting and revising which is a good skill to embrace when completing math tasks.

Color Patterns (Grades K-2)

http://nlvm.usu.edu/en/nav/frames_asid_184_g_1_t_1.html?from=category_g_1_t_1.html

This applet gives the player practice with patterns by using color. A strand of colored dots appear on the screen, followed by multiple blank dots. There are all the possible colors including ones that are not in the pattern on the side of the screen. The player then clicks the empty circles and then the color that would continue the established pattern. 

Allowing young students to work with patterns involving color prepares them for intricate patterns involving numbers, and eventually formulas and functions. The applet gives the player the ability to check their answer and make corrections if necessary which can activate their problem solving skills. Completing the pattern also stimulates the child's attention to detail in which students need to be proficient when succeeding in furthering their understanding of math concepts. 
 

CCSSM Standards for Mathematical Practice and NCTM Process Standards

Problem Solving

CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

Students who go through the entire problem solving process will meet this standard. Problem solving occurs when the student determines what their method of solving a given problem will be, carries out this method, making any changes as necessary throughout the process, and comes to a conclusion that makes sense in the context of the problem. 

CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.


This standard relates to problem solving in that the student needs to decide which tool would be most helpful and beneficial in solving a problem. Also, with the use of tools comes some kinds of errors which will require the students to use their problem solving skills to decipher, recognize, and fix. Tools can be used to further their understanding of the problem as well.

Reasoning and Proof

CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.

Making sense of the problem is a vital step in finding the solution and this can usually be completely in various ways. The student should recognize that every problem has many methods to reaching the solution and how to choose a method that is best for the situation. This means taking the general problem and reaching a specific conclusion using the evidence found by using methods of reasoning.

CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.

There are many patterns in math and algorithms that can be used to solve math problems that students should be able to identify and apply to appropriate problems. They will use reasoning to relate problems to one another and check to make sure that their answer makes sense. They should also know the reasoning behind the patterns and why the algorithm works to deepen their understanding.

Communication

CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

Students should be able to create conjectures about a topic using prior knowledge and then dig deeper by looking at evidence to support or reject their claims. An important step in this process is the sharing of ideas with peers or teachers, and allowing them to critique the work. Then, the students will also look at the work of their peers and give their opinions. It is necessary for the students to communicate in a positive and professional way when completing these tasks so that everyone will be most successful.

CCSS.MATH.PRACTICE.MP6 Attend to precision.

Mathematical language is very precise and specific. It is important for students to communicate their answers and ideas using the correct math terminology so that others can understand what they are trying to express. This skill takes practice, so it is vital that students not only get practice in writing mathematical statements but verbally communicating their thoughts as well.

Connections

CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

As a whole, mathematics have countless connections in content and is continually building off of prior knowledge and skills. Also, many skills have shortcuts that students can discover after much practice. Investigating these shortcuts are great way for students to understand the skills on a deeper level and prepare themselves to move on to the next step in the curriculum.

Representation

CCSS.MATH.PRACTICE.MP4 Model with mathematics.

Utilizing graphs and charts are a real life skill that students should be able to complete. This is important because demonstrating the data that has been found and reaching a conclusion can be difficult to execute without a visual. Models are a way to do so in an organized and systematic manner. Students should also be able to interpret models that have already been created and analyze the results of the representation.

NCTM Process Standards

Problem Solving

Strong problem solving skills are important to not only exhibit in the math classroom but throughout all parts of life. The students should be able to come up with the correct strategy to solve a given problem and reflect on the process in which they used to come up with a solution.

Reasoning and Proof

Students should be able to come up with their own arguments and ideas about a topic using viable reasoning techniques and having evidence to support their conjecture. They will also be able to look at arguments of their peers or others and investigate the truth behind the claim.

Communication

In mathematics, there is a very specific and unique sort of communication that is expected. Students should be able to communicate mathematical ideas to their teachers and peers. They will also be able to analyze the reasoning of others and explain their opinions about the topic.

Connections

Nearly all topics in mathematics connect to one another and many build upon other concepts in math. It is important that all students understand this and can relate the topics together. This will deepen their understanding of mathematics as a whole.

Representations

The use of models whether it be graphs or three dimensional objects can be a great way to visualize the information. Students should be able to organize the information they know to create a representation, analyze all representations, and alter the information if necessary.  

Rich Activities and Groupworthy and Ideas About Math

Rich Activities

Rich activities occur when the student are required to not just memorize the material, but connect to the concepts on a deeper level. Planning is an important step in making sure that the activity has reached a high level of cognitive demand. Two ways that this can be done is tapping into prior knowledge and having the students explore multiple methods of solving the problem. Now, the teacher needs to find ways to support the students as they are completing the task. All students will work at various paces so the teacher must make sure that the students who are finding the activity easier can be presented with a challenge and additional help can be added for those students who are struggling. Making sure that the focus stays on the mathematical concepts of the lesson is also very important. Lastly, the discussion and sharing of the lesson should always take place. This will make sure that all students are on the same page and have a deeper understanding of the topic and can communicate their ideas regarding mathematics.

Groupworthy and Ideas About Math

There are many math skills that students should practice and have in their skill set as they are completing their schooling that will help them in not only math class, but solving real world problems. Equitable math teaching occurs when students are given the opportunity to learn to their fullest potential. Group work is a great way to accomplish this goal in the math classroom but norms need to be set up so that all students feel comfortable working with their peers. Also, all students should feel that they can succeed and that they are being challenged. Tasks should pose a question that can be solved using many different methods and can be related to multiple examples. When expecting group work with your students, there should be individual and group expectations that are clear so that each student knows their role. The students will be held accountable throughout their work with their group, whole class discussion, and personal work and they will be assessed formally at the close of the lesson that will be precise and based on work that the student and their group has completed.

CCSSM Standards of Mathematical Practice

CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

To create mathematical arguments, the students will use prior knowledge and definitions to come up with a conclusion about the topic. Then they will analyze various situations and find any counterexamples that may apply. This helps them to justify their findings and share their conclusions with others. They can then compare their ways of reasoning and arguing the case with other arguments and determine if there is a flaw in any of the logic and then explain the issue. Younger students can use diagrams to construct more general arguments that will be studied in depth in the later grades. All students can listen to the arguments of others, decide if they make sense, and ask clarification questions.

Reference: National Governors Association (2010). Common Core State Standards Initiative - Mathematics Standards. Retrieved from http://www.corestandards.org/

Application to PLC at Work (Grades 6-8)

This reading focuses on implementing the two main parts of the Mathematical Practice Standard constructing viable arguments and critiquing the reasoning of others at the middle school level and explains why these actions make math become more meaningful. Overall, the teacher should be a facilitator of the construction of the knowledge and students should spend more time creating their own conjectures. Also, less teacher to student interactions should take place and more student to student interactions should be implemented. To earn success however, the teacher must establish a social learning environment so that all the students feel safe to give and receive critiques about their work. The expectations should be clear and consist of students being expected to provide explanations, ask questions when attempting to make sense of other solutions, communicate any disagreement or if they are not understanding, and respect all ideas provided. The students should also understand that discussion in mathematics is important to the learning process, can be done verbally or orally, and is an expectation in the classroom. Some other norms that should be established are that every student is responsible for contributing their ideas, asking questions directed towards their peers before asking the teacher, and understanding that mistakes are not bad, simply a learning opportunity. Eventually, this type of environment will be natural and the teacher will not need to enforce as much direction with the students; they will know their expectations and the procedures. One important skill that the reading specifies that students should grasp at the middle school level is that they can distinguish correct and incorrect logic and recognize methods that can be used to do so. They should realize that counterexamples can be used to prove that a statement is false, however examples cannot prove that it is true. The geometry standards at this grade level provide a good base for introducing proofs and allowing the students to construct arguments within the topic. 

Reference: Briars, D. J., Austurias, H., Foster, D., and Gale, M. A. (2013). Common core mathematics in a PLC at work: Grades 6-8. Bloomington IN: Solution Tree Press.

Application to Connecting the Missing Words to the Common Core

In this article, the sixth grade teacher is striving to create a task that pushes her students to find new approaches to a solution. She has decided on an activity entitled "Missing Words". The students were given a page from a book with some of the words whited out with about three lines of text above and below the blank space, were asked to find out how many words are missing, and explain their reasoning. This task would require the students to access prior knowledge about estimation, means, medians, and modes. The teacher was very concerned about making sure that the task connected to the Standards for Mathematical Practice so there was a very helpful figure embedded into the article explaining how each standard was being met. The students met constructing viable arguments and critiquing the reasoning of others by completing a written description of the steps that they took to estimate how many missing words there are and justify their conclusions. In addition, after they reached their solutions individually, the students were put into small groups to verbally compare their answers and express their result with support. The article also explains how the process unfolded as the activity was being administered. The teacher made sure to ask students to share their various methods of finding the answers as they were working and asked clarification questions to make sure that the students had a deep understanding of the concepts and could explain their argument. In closing, the teacher asked if the class mean would be the exact number of words missing from the page. This gave the students one last opportunity to construct an argument and use all of the information that they learned during the lesson to provide a viable response.  

Reference: Kulbacki, A., & Wilburne, J. (2014, March). Connecting the missing words to the common core. Mathematics Teaching in the Middle School, 19, 430-436. Retrieved May 21, 2014, from nctm.org

CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.

The students need to make sense of the quantities and how they relate to each other to solve problems. They do so by mastering two concepts of abstract thinking; decontextualizing and contextualizing. Students will use their ability to think abstractly and use symbols to solve the problem when decontextualizing. Contextualizing occurs when the students pause and take a deeper look into the topic. Quantitative reasoning is not only using numbers and equations to compute an answer to a question, but also being able to use various methods to find the answer including different properties. 

Reference: National Governors Association (2010). Common Core State Standards Initiative - Mathematics Standards. Retrieved from http://www.corestandards.org/

Application to PLC at Work (Grades 6-8) 

This reading explains how vital mathematical reasoning is for students to grasp. Reasoning consists of thinking through their ideas carefully, looking at examples or alternatives, asking questions, and hypothesizing. We want students to be able to take specific situations and make them more abstract or general, interpret the results, and reverse the process. Mathematical reasoning is important because it gives students the ability to use the concepts and procedures they have learned in meaningful ways and helps them to retrieve forgotten knowledge. Using their ability to decontextualize and contextualize are vital in solving real world problems, however most times contextualizing is forgotten and all the focus is placed on the decontextulaizing process. Contextualizing can be accomplished by asking the students to interpret their solutions and then asking themselves if their solutions make sense. Some student to teacher interactions that lend themselves toward improving their reasoning skills are when the teacher poses probing questions and has discussions with the student about their hypothesis or procedure taken. Student to student interactions such as debates and explanations of their reasoning are also great for the development of reasoning skills. Ways in which teachers can promote discussion is to ask students to fix the problems a problem that has been solved incorrectly and ask students to elaborate on prior experiences to incorporate more math skills. Mathematical reasoning should be a continuous occurrence and expectation of the students in whole class, small group, and homework settings.  

Reference: Briars, D. J., Austurias, H., Foster, D., and Gale, M. A. (2013). Common core mathematics in a PLC at work: Grades 6-8. Bloomington IN: Solution Tree Press.

Application to Using KenKen to Build Reasoning Skills

KenKen puzzles use not only logical reasoning, but number and operations reasoning, and develops number sense. Completing the puzzles can also promote problem solving and communication. They can be solved in many ways such as using the X-wing strategy which is creating a list and eliminating possibilities, fault lines which creates smaller puzzles, parity, stacked, parallel, or orthogonal cages, and elimination. The class can explore all of the possible strategies while completing a large puzzle as a class. Asking the students to share any of their ideas will be beneficial to their reasoning skills. One activity that the article outlines is giving the students two blank puzzles and asking them to solve the puzzle using one of the strategies discussed, then they will give the other blank sheet to a partner ask explain their process so that their peer can solve the puzzle using the same strategy (and a different one than they used when completing their first KenKen). This increases discourse in the classroom and gets the students communicating about their methods of reasoning. 

Reference: Reiter, H., Thornton, J., & Vennebush, P. (2014, Dec. - Jan.). Using KenKen to build reasoning skills. Mathematics Teacher, 107, 341-347. Retrieved May 21, 2014, from nctm.org