transitioning to ccssm grades 3-5 “the essence of mathematics is not to make simple things...
TRANSCRIPT
TRANSITIONING TO CCSSM
Grades 3-5
“The essence of mathematics is not to make simple things complicated, but to make complicated things simple.” Stan Gudder
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…
M-DCPS
3 4 5
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, at
www.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
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Eight Mathematical Practices
Content standards? Practice standards?
What does a teacher need to do to ensure the implementation of:
Design and
Organization
Focal points at each grade level
Each grade level addresses specific “critical areas”
MACC.3.NF.1.1
Math Common Core
Grade level
Domain
Cluster
Standard
New Florida Coding for CCSSM:
K-5 Content Domains, CCSSM
Common Core Leadership in Mathematics Project
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!
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)
Metacognition is 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 Asmamaw Yimer and Nerida F. Ellerton
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.
What is metacognition?
Wikipedia
•Categories of Cognitive and Metacognitive behaviors
1. Engagement: Initial confrontation and making sense of the problem.
2. Transformation-Formulation: Transformation of initial engagements to exploratory and formal plans.
3. Implementation: A monitored acting on plans and explorations.
4. Evaluation: Passing judgments on the appropriateness of plans, actions, and solutions to the problem.
5. 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 Asmamaw Yimer and Nerida F. Ellerton
Cognitive and Metacognitive Aspects of Mathematical Problem Solving: An Emerging Model
by Asmamaw Yimer and Nerida F. 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.”
Problem Solving
Mathematical problem solving is a complex cognitive activity involving a number of processes and strategies.
Problem solving has two stages: 1. problem representation2. 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
Math Problem Solving for Upper Elementary Students with Disabilities
by Marjorie Montague, PhD
A powerful problem-solving
strategy…
What do good problem solvers do?
Math Problem Solving for Upper Elementary Students with Disabilities
by Marjorie Montague, PhD
Problem Solving
1. READ the problem for understanding.
2. PARAPHRASE the problem by putting it into their own words.
3. VISUALIZE or draw a picture or diagram.
4. HYPOTHESIZE by thinking about logical solutions.
5. ESTIMATE or predict the answer.
6. COMPUTE.7. CHECK.
Math Problem Solving for Upper Elementary Students with Disabilities
by Marjorie Montague, PhD
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
Instructional Procedures
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
Part A
A restaurant makes a special seasoning for all its grilled vegetables. Here is how the ingredients are mixed:
1/2 of the mixture is salt
1/4 of the mixture is pepper
1/8 of the mixture is garlic powder
1/8 of the mixture is onion powder
When the ingredients are mixed in the same ratio as shown above, every batch of seasoning tastes the same.
Study the measurements for each batch in the table. Fill in the blanks so that every batch will taste the same.
The Charles A. Dana Center at the University of Texas at Austin and Agile Mind, Inc.
The Charles A. Dana Center at the University of Texas at Austin and Agile Mind, Inc.
Answers
Part B
A restaurant makes a special seasoning for all its grilled vegetables. Here is how the ingredients are mixed:
1/2 of the mixture is salt
1/4 of the mixture is pepper
1/8 of the mixture is garlic powder
1/8 of the mixture is onion powder
The restaurant mixes a 12-cup batch of the mixture every week. How many cups of each ingredient do they use in the mixture each week?
The Charles A. Dana Center at the University of Texas at Austin and Agile Mind, Inc.
Answers
The Charles A. Dana Center at the University of Texas at Austin and Agile Mind, Inc.
Rubrics
The Charles A. Dana Center at the University of Texas at Austin and Agile Mind, Inc.
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. 3-5
3rd Grade In third grade, students know 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. Third graders 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 listen to the strategies of others and will try different approaches. They often will use another method to check their answers.
4th Grade In fourth grade, students know 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. Fourth graders 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 listen to the strategies of others and will try different approaches. They often will use another method to check their answers.
5th Grade In fifth grade, students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, ―What is the most efficient way to solve the problem?, ―Does this make sense?, and ―Can I solve the problem in a different way?.
Adapted from Arizona Department of Education Mathematics Standards-2010
Video
Teaching Channel- MP 1
Questions to Consider How does the graphic organizer help
scaffold problem solving for students? Why does Ms. Saul choose to have students
work alone without help? How do "Heads Together Butts Up" and
"Student-led Solutions" contribute to the class culture around problem solving?
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 reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
MP4: Model with mathematics. 3-5
3rd Grade Students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart, list, or graph, 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. Third graders should evaluate their results in the context of the situation and reflect on whether the results make sense.
4th Grade Students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, making a chart, list, or graph, 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. Fourth graders should evaluate their results in the context of the situation and reflect on whether the results make sense.
5th Grade Students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, making a chart, list, or graph, 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. Fifth graders should evaluate their results in the context of the situation and whether the results make sense. They also evaluate the utility of models to determine which models are most useful and efficient to solve problems.
Adapted from Arizona Department of Education Mathematics Standards-2010
MACC.3.NF Number and Operations - Fractions
(Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.) MACC.3.NF.1 Develop understanding
of fractions as numbers. MACC.3.NF.1.1: Understand a fraction
1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Some important concepts related to developing understanding of fractions include:
Example Non-example
Understand fractional parts must be equal-sized
These are thirds These are NOT thirds
Arizona Department of Education: Standards and Assessment Division
Some important concepts related to developing understanding of fractions include: The number of equal parts tell how many make a whole As the number of equal pieces in the whole increases, the
size of the fractional pieces decreases The size of the fractional part is relative to the whole
The number of children in one-half of a classroom is different than the number of children in one-half of a school. (the whole in each set is different therefore the half in each set will be different)
When a whole is cut into equal parts, the denominator represents the number of equal parts
The numerator of a fraction is the count of the number of equal parts ¾ means that there are 3 one-fourths Students can count one fourth, two fourths, three fourths
Arizona Department of Education: Standards and Assessment Division
Some important concepts related to developing understanding of fractions include: Students express fractions as fair sharing, parts of a
whole, and parts of a set. They use various contexts (candy bars, fruit, and cakes) and a variety of models (circles, squares, rectangles, fraction bars, and number lines) to develop understanding of fractions and represent fractions. Students need many opportunities to solve word problems that require fair sharing.
To develop understanding of fair shares, students first participate in situations where the number of objects is greater than the number of children and then progress into situations where the number of objects is less than the number of children.
Arizona Department of Education: Standards and Assessment Division
Examples:
Four children share six brownies so that each child receives a fair share. How many brownies will each child receive?
Six children share four brownies so that each child receives a fair share. What portion of each brownie will each child receive?
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!
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
Representing
Skill Practice
Problem Solving . . .
Problem Solving Strategy~ Act It Out
Problem Solving Strategy~ Act It Out
Problem Solving Strategy~ Act It Out
Problem Solving Strategy~ Act It Out
Contact information:
Maria Teresa Diaz-Gonzalez, District Instructional Supervisor
Maria Campitelli, District Curriculum Support Specialist
Isis Casares, District Curriculum Support [email protected]
305-995-7280