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“The essence of mathematics is not to make simple things complicated, but to make complicated things simple.” Stan Gudder### Transitioning To CCSSM

Grades K-2

Norms

- Listen to others.
- Engage with the ideas presented.
- Ask questions.
- Reflect on relevance to you.
- Next, set your learning into action.

Math is fun…

COUNCIL OF CHIEF STATE SCHOOL OFFICERS (CCSSO)

&

NATIONAL GOVERNORS ASSOCIATION

CENTER FOR BEST PRACTICES

(NGA CENTER)

JUNE 2010

Common Core Development

- As of now, most states have officially adopted the CCSS
- Final Standards released June 2, 2010, atwww.corestandards.org
- Adoption required for Race to the Top funds
- Florida adopted CCSS in July of 2010

Common Core Mission Statement

The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.

Characteristics

- Fewer and more rigorous standards
- Aligned with college and career expectations
- Internationally benchmarked
- Rigorous content and application of higher-order skills
- Builds on strengths and lessons of current state standards
- Research based

Intent of the Common Core

- The same goals for all students
- Coherence
- Focus
- Clarity and Specificity

Eight Mathematical Practices

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

What does a teacher need to do to ensure the implementation of:

- Content standards?
- Practice standards?

Design

and Organization

Focal points at each grade level

Each grade level addresses specific “critical areas”

Conclusion: The Promise of Standards

These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms. It is time to recognize that standards are not just promises to our children, but promises we intend to keep.

Mathematical Practices

Let’s Dig Deeper!

Video

- Teaching Channel- MP 1

Questions to Consider

- How will open-ended problem solving prepare students for future math classes?
- How will the addition of speaking and listening standards shape classroom activities?
- How does analyzing lessons for Common Core alignment help this group of teachers with their own practice?

Einstein is quoted to have said :

“if he had one hour to save the world he would spend fifty-five minutes defining the problem and only five minutes finding the solution”.

What is Problem Solving?

Problem solving is a process and skill that you develop over time to be used when needing to solve immediate problems in order to achieve a goal.

University of South Australia

Metacognition

- Several research studies have concluded that metacognitive processes improve problem solving performance.

(Artzt & Armour-Thomas, 1992; Goos & Galbraith, 1996; Kramarski & Mevarech, 1997)

- Metacognitionis also believed to help students develop confidence to attempt authentic tasks (Kramarski, Mevarech, & Arami, 2002), and to help students overcome obstacles that arise during the problem-solving process (Goos, 1997; Pugalee, 2001; Stillman & Galbraith, 1998).

Cognitive and Metacognitive Aspects of Mathematical Problem Solving: An Emerging Model

by AsmamawYimer and NeridaF. Ellerton

What is metacognition?

Metacognition is defined as "cognition about cognition", or "knowing about knowing." It can take many forms; it includes knowledge about when and how to use particular strategies for learning or for problem solving.

Wikipedia

Categories of Cognitive and Metacognitive behaviors

- Engagement: Initial confrontation and making sense of the problem.
- Transformation-Formulation: Transformation of initial engagements to exploratory and formal plans.
- Implementation: A monitored acting on plans and explorations.
- Evaluation: Passing judgments on the appropriateness of plans, actions, and solutions to the problem.
- Internalization: Reflecting on the degree of intimacy and other qualities of the solution process.

Cognitive and Metacognitive Aspects of Mathematical Problem Solving: An Emerging Model

by AsmamawYimer and NeridaF. Ellerton

Problem solving and metacognition

“Without metacognitive monitoring, students are less likely to take one of the many paths available to them, and almost certainly are less likely to arrive at an elegant mathematical solution.”

Cognitive and Metacognitive Aspects of Mathematical Problem Solving: An Emerging Model

by AsmamawYimer and NeridaF. Ellerton

Problem Solving

Mathematical problem solving is a complex cognitive activity involving a number of processes and strategies.

Problem solving has two stages:

problem representation

problem execution

Successful problem solving is not possible without first representing the problem appropriately.

Appropriate problem representation indicates that the problem solver has understood the problem and serves to guide the student toward the solution plan. Students who have difficulty representing math problems will have difficulty solving them.

Math Problem Solving for Upper Elementary Students with Disabilities

by Marjorie Montague, PhD

Visualization

A powerful problem-solving strategy…

Math Problem Solving for Upper Elementary Students with Disabilities

by Marjorie Montague, PhD

Problem Solving

READ the problem for understanding.

PARAPHRASE the problem by putting it into their own words.

VISUALIZE or draw a picture or diagram.

HYPOTHESIZEby thinking about logical solutions.

ESTIMATE or predict the answer.

COMPUTE.

CHECK.

Math Problem Solving for Upper Elementary Students with Disabilities

by Marjorie Montague, PhD

Instructional Procedures

The content of math problem solving instruction are the cognitive processes and metacognitive strategies that good problem solvers use to solve mathematical problems.

~Marjorie Montague

Math Problem Solving for Upper Elementary Students with Disabilities

by Marjorie Montague, PhD

Problem Solving

Effective instructional procedures for teaching math problem solving!

Math Problem Solving for Upper Elementary Students with Disabilities

by Marjorie Montague, PhD

Instructional Procedures

- Explicit Instruction
- Sequencing and Segmenting
- Drill-repetition and Practice-review
- Directed Questioning and Responses
- Control Difficulty or Processing Demands of the Task
- Technology
- Group Instruction
- Peer Involvement
- Strategy Cues
- Verbal Rehearsal
- Process Modeling
- Visualization
- Role Reversal
- Performance Feedback
- Distributed Practice
- Mastery Learning

Math Problem Solving for Upper Elementary Students with Disabilities

by Marjorie Montague, PhD

Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them.

- Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

MP1: Make sense of problems and persevere in solving them. K-2

- Kindergarten: In Kindergarten, students begin to build the understanding that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? or they may try another strategy.
- 1st Grade: In first grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They are willing to try other approaches.
- 2nd Grade: In second grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. They may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, ―Does this make sense? They make conjectures about the solution and plan out a problem-solving approach.

Adapted from Arizona Department of Education Mathematics Standards-2010

Problem scenario

Ms. McCrary wants to make a rabbit pen in a section of her lawn. Her plan for the rabbit pen includes the following:

- It will be in the shape of a rectangle.
- It will take 24 feet of fence material to make.
- Each side will be longer than 1 foot.
- The length and width will measure whole feet.

Rabbit’s pen

Draw 3 different rectangles that can each represent Ms. McCrary’s rabbit pen. Be sure to use all 24 feet of fence material for each pen.

New scenario

Ms. McCrary wants her rabbit to have more than 60 square feet of ground area inside the pen. She finds that if she uses the side of her house as one of the sides of the rabbit pen, she can make the rabbit pen larger.

- Draw another rectangular rabbit pen.
- Use all 24 feet of fencing for 3 sides of the pen.
- Use one side of the house for the other side of the pen.
- Make sure the ground area inside the pen is greater than 60 square feet.

Standard for Mathematical Practice 4:Model with mathematics

- Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflecton whether the results make sense, possibly improving the model if it has not served its purpose.

MP4: Model with mathematics. K-2

- Kindergarten: In early grades, students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed.
- 1st Grade: In early grades, students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed.
- 2nd Grade: In early grades, students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed.

Adapted from Arizona Department of Education Mathematics Standards-2010

MACC.1.NBT Number and Operations in Base Ten

- MACC.1.NBT.3 - Use place value understanding and properties of operations to add and subtract.
- MACC.1.NBT.3.4 - Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Explanations and examples:

- Students extend their number fact and place value strategies to add within 100. They represent a problem situation using any combination of words, numbers, pictures, physical objects, or symbols. It is important for students to understand if they are adding a number that has 10s to a number with 10s, they will have more tens than they started with; the same applies to the ones. Also, students should be able to apply their place value skills to decompose numbers. For example, 17 + 12 can be thought of 1 ten and 7 ones plus 1 ten and 2 ones. Numeral cards may help students decompose the numbers into 10s and 1s.

Arizona Department of Education: Standards and Assessment Division

43 + 36

- Student counts the 10s (10, 20, 30…70 or 1, 2, 3…7 tens) and then the 1s

Arizona Department of Education: Standards and Assessment Division

28 +34

- Student thinks: 2 tens plus 3 tens is 5 tens or 50. S/he counts the ones and notices there is another 10 plus 2 more. 50 and 10 is 60 plus 2 more or 62.

Arizona Department of Education: Standards and Assessment Division

45 + 18

- Student thinks: Four 10s and one 10 are 5 tens or 50. Then 5 and 8 is 5 + 5 + 3 (or 8 + 2 + 3) or 13. 50 and 13 is 6 tens plus 3 more or 63.

Arizona Department of Education: Standards and Assessment Division

29 +14

- Student thinks: “29 is almost 30. I added one to 29 to get to 30. 30 and 14 is 44. Since I added one to 29, I have to subtract one so the answer is 43.”

Arizona Department of Education: Standards and Assessment Division

Mathematical Practices 1 and 4

- Find a partner and discuss some of the best practices to foster the development of mathematical practices in the classroom.

MP1- Make sense of problems and persevere in solving them.

- Young children are eager for challenges and are problem solvers by nature. A challenge for elementary teachers is to help children maintain their enjoyment for engaging with problems. Teachers can help their students explore, investigate, and persevere in solving problems by creating a nurturing classroom environment. It is important for teachers to convey that everyone can learn math and that it takes active effort and thinking to do so.It is also important for teachers to convey that by thinking hard, we can actually increase our intelligence. Research on motivation indicates that supporting autonomy, competence, and relatedness supports internal motivation and leads to better outcomes than environments that are experienced as highly controlling. Elementary school teachers often want to make mathematics “fun” for students and shelter them from the difficulty of learning mathematics, which frequently leads to activities that have little mathematical substance.

cbmsweb.org

MP4 - Model with mathematics.

- In elementary school, modeling with mathematics often involves writing an equation for a situation and then solving the equation to solve a problem about the situation. Students also model with mathematics when they draw a quadrilateral to show a route that started and ended at the same location and had four turns. At elementary school, modeling with mathematics is often mathematizing—which means focusing on the mathematical aspects of a situation and formulating it in mathematical terms. For example, students may notice shapes in objects around them, such as triangular bracing in chairs or quadrilaterals in collapsible gates. Teachers also need to help students notice math in the world around them.

cbmsweb.org

Make sense of problems and persevere in solving them.

Wake County Public Schools

Model with mathematics

Wake County Public Schools

Developing Processes andProficiencies in Mathematics Learners

- There are eight mathematical practices. They are based on the National Council of Teachers of Mathematics’(NCTM) Process Standards (NCTM, 2000) and the National Research Council’s (NRC) Strands of Mathematical Proficiency (NRC, 2001). GO Math! supports the standards for mathematical practice through several specific features of the series:

Dr. Juli K. Dixon

Go Math! Florida

- Lessons focused on depth of content knowledge
- Unlock the Problem sections to begin lessons
- Math Talk questions prompting students to use varied strategies and to explain their reasoning
- Support for manipulative use and drawings directly on the student pages
- Prompts that lead students to write their own problems or to determine if the reasoning of others is reasonable
- Real-world problems that encourage students to develop productive dispositions.

Dr. Juli K. Dixon

Contact information:

- Maria Teresa Diaz-Gonzalez, District Instructional Supervisor
- [email protected]
- 305-995-2763
- Maria Campitelli, District Curriculum Support Specialist
- [email protected]
- 305-995-2927
- Isis Casares, District Curriculum Support Specialist
- [email protected]
- 305-995-7280

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