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