BCPS NCSM Baltimore County Goals for 2015-2017 25th largest in - - PDF document

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• Dr. John W. Staley

Email: jstaley@bcps.org Twitter: @jstaley06 Director Mathematics PreK-12, Baltimore County Public Schools President, National Council of Supervisor of Mathematics (NCSM)

Unlocking the Power of Problem Solving BCPS

Baltimore County

• 25th largest in the U.S.
• 3rd largest in Maryland
• 109,000+ students
• 110 Schools

www.bcps.org

NCSM

Goals for 2015-2017

• …making mathematics

meaningful, relevant, and accessible for each and every student.

• Growing and supporting

mathematics education leaders at all levels.

www.mathedleadership.org

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The students in your class have a practice of greeting each other with a “handshake”. If there were 22 students in class on the last day of school, how many “handshakes” were there?

Handshakes

Which problem solving strategies did you use?

Essential Questions

• Why focus on Problem Solving?
• What can I do to change the future
• f my students’ mathematical

learning trajectory?

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TEKS Process Standards

Students will…

1.use a problem-solving model… 2.select appropriate tools… 3.effectively communicate mathematical ideas,… 4.use mathematical relationships to generate solutions…. 5.analyze mathematical relationships… 6.display, explain, or justify…

Why Problem Solving?

• 2. Reason abstractly and quantitatively
• 3. Construct viable arguments and

critique the reasoning of others

• 4. Model with mathematics
• 5. Use appropriate tools strategically
• 7. Look for and make use of structure
• 8. Look for and express regularity in

repeated reasoning

Overarching habits of mind of a productive mathematical thinker

Reasoning and Explaining Modeling and Using Tools Seeing Structure and Generalizing

• 1. Make sense of problems and

persevere in solving them.

• 6. Attend to precision

Standards for Mathematical Practice

Why Problem Solving?

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TEKS Process Standard 1

Students will use a problem-solving model that incorporates…

• analyzing given information,
• formulating a plan or strategy,
• determining a solution,
• justifying the solution, and
• evaluating the problem-solving process and the

reasonableness of the solution.

Why Problem Solving?

• Read - Understanding the Problem

– Study the Problem – Organize the Facts

• Plan - Devising a Plan

– Line up a Plan

• Do - Carrying out the Plan

– Verify your Plan with Action

• Look Back - Looking Back

– Examine your Answer

Poyla… S.O.L.V.E.

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Tito and Luis are stuffed with pizza! Tito ate one-fourth of a cheese pizza. Tito ate

• ne-eighth of a pepperoni pizza. Tito ate
• ne-half of a mushroom pizza. Luis ate

three-eights of a cheese pizza. Luis ate the

• ther half of the mushroom pizza. All the

pizzas were the same size. Tito says he ate more pizza than Luis because Luis did not eat any pepperoni pizza. Luis says they each ate the same amount of pizza. Who is correct?

Study the Problem –Highlight the question. –Answer the question

READ = Understand the Problem “What is the problem asking me to find?”

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Tito and Luis are stuffed with pizza! Tito ate

• ne-fourth of a cheese pizza. Tito ate one-

eighth of a pepperoni pizza. Tito ate one-half of a mushroom pizza. Luis ate three-eights of a cheese pizza. Luis ate the other half of the mushroom pizza. All the pizzas were the same

• size. Tito says he ate more pizza than Luis

because Luis did not eat any pepperoni pizza. Luis says they each ate the same amount of

• pizza. Who is correct?

This problem is asking me to find whether Tito or Luis is correct on who ate more pizza.

Organize the Facts –Identify each fact. –Eliminate unnecessary facts. –List all necessary facts.

READ = Understand the Problem

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Tito and Luis are stuffed with pizza! Tito ate one- fourth of a cheese pizza. Tito ate one-eighth of a pepperoni pizza. Tito ate one-half of a mushroom

• pizza. Luis ate three-eights of a cheese pizza. Luis

ate the other half of the mushroom pizza. All the pizzas were the same size. Tito says he ate more pizza than Luis because Luis did not eat any pepperoni pizza. Luis says they each ate the same amount of pizza. Who is correct? Show all your mathematical thinking.

Cheese Pepperoni Mushroom Tito ate 1/4 ate 1/8 ate 1/2 Luis ate 3/8 ate 1/2

• Line up a Plan

–Identify operation(s) and/or a problem solving strategy. –Write out the plan in words or as a diagram according to the strategy selected.

PLAN = Devise a Plan

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• Try, Check, Revise
• Write an Equation
• Use Reasoning
• Work Backwards
• Solve a Simpler Problem

PLAN: Problem Solving Strategies

• Act it Out
• Draw a Picture
• Make a List
• Make a Table/Graph
• Look for a Pattern

Multiple Representations

• Verbal
• Numerical
• Graphic
• Algebraic

Verbal Representation Algebraic Representation Numeric Representation Graphic Representation

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Cheese Pepperoni Mushroom Tito ate 1/4 ate 1/8 ate 1/2 Luis ate 3/8 ate 1/2

• Write in words what your plan of action will be.
• Add all of the pizza Tito ate to get a total.
• Add all of the pizza Luis ate to get a total.
• Compare the totals to see who is correct.
• Choose an operation or operations.
• Addition and Comparing
• Verify your Plan with Action

–Make an estimate. –Carry out your plan.

DO = Carry Out the Plan

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Verify Your Plan with Action

• Estimate your answer.

I think they ate the same, so Luis

• Carry out your plan.

2 1 4 8 8 8   

Tito Luis

1 1 1 4 8 2   

3 1 8 2   3 4 8 8   7

8 7 8

Luis is correct. Both boys ate the same amount.

__ 4 1 __ 4 1 __ 4 1 __ 4 1 _ 8 1 1 Whole __ 2 1 __ 2 1 _ 8 1 _ 8 1 _ 8 1 _ 8 1 _ 8 1 _ 8 1 _ 8 1

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Cheese Pepperoni Mushroom Tito ate 1/4 ate 1/8 ate 1/2 Luis ate 3/8 ate 1/2 Tito

__ 4 1 _ 8 1 __ 2 1 _ 8 1 _ 8 1 _ 8 1 _ 8 1 _ 8 1 _ 8 1

= 7 8

__ 2 1 _ 8 1 _ 8 1 _ 8 1 _ 8 1 _ 8 1 _ 8 1 _ 8 1

= 7 8

Luis

• Examine your Answer

–Does your answer make sense? –Is your answer reasonable? –Is your answer accurate? –Write your answer in a complete sentence.

LOOK BACK = Look Back and Reflect

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Examine Your Results

Does your answer make sense? (Compare your answer to question.) Yes, because we are looking for whether Tito or Luis are correct on who ate more pizza. Is your answer reasonable? (Compare your answer to the estimate.) Yes, because it is close to my estimate that both boys ate the same amount.

Is your answer accurate?

(check your work.) Yes. Write your answer in a complete sentence. Luis is correct because they each ate the same; of a pizza.

7 8

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• Read - Understanding the Problem

– Study the Problem – Organize the Facts

• Plan - Devising a Plan

– Line up a Plan

• Do - Carrying out the Plan

– Verify your Plan with Action

• Look Back - Looking Back

– Examine your Answer

Poyla… S.O.L.V.E.

S.O.L.V.E….

• is not a problem-solving strategy; it is a

problem-solving thought process.

• use to enter into a problem, interact with a

problem, and get out of a problem accomplishing some or all of the goals of the problem.

• can be used with any type of problem.
• can be used to highlight test-taking and

problem-solving strategies.

National Training Network, http://www.ntnmath.com

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Research Recommendations

Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools

• Recommendation 3. Instruction during the

intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

WHAT WORKS CLEARINGHOUSE, April 2009 http://www.rti4success.org/sites/default/files/rti_math_pg_042109.pdf Why Problem Solving?

Research Recommendations

Improving Mathematical Problem Solving in Grades 4 Through 8

• Recommendation 2. Assist students in

monitoring and reflecting on the problem- solving process.

• Recommendation 3. Teach students how to

use visual representations.

WHAT WORKS CLEARINGHOUSE, May 2012 http://ies.ed.gov/ncee/wwc/pdf/practice_guides/mps_pg_052212.pdf http://ies.ed.gov/ncee/wwc/practiceguide.aspx?sid=16 (supporting videos)

Why Problem Solving?

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Mathematics Teaching Practices

• 1. Establish Mathematical Goals to Focus Learning
• 2. Implement tasks that Promote Reasoning and

Problem Solving

• 3. Use and Connect Mathematics Representations
• 4. Facilitate Meaningful Mathematics Discourse
• 5. Pose Purposeful Questions
• 6. Build Procedural Fluency from Conceptual

Understanding

• 7. Support Productive Struggle in Mathematics
• 8. Elicit and Use Evidence of Student Thinking

Implement tasks that Promote Reasoning and Problem Solving

Teacher Actions

• Opportunities for exploring and solving problems that

build on and extend their current mathematical understanding

• Selecting tasks that have multiple entry points

through the use of varied tools

• Posing tasks on a regular basis
• Support without taking over
• Encourage varied approaches and strategies

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http://www.mathedleadership.org

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Essential Questions

• What can I do to change the future
• f my students’ mathematical

learning trajectory?

• Why focus on Problem Solving?

Resources

• Principles and Standards, (NCTM, 2000)
• S.O.L.V.E. videos, National Training Network, http://www.ntnmath.com/
• Improving Mathematical Problem Solving in Grades 4 Through 8 WHAT

WORKS CLEARINGHOUSE, May 2012 http://ies.ed.gov/ncee/wwc/pdf/practice_guides/mps_pg_052212.pdf

• http://ies.ed.gov/ncee/wwc/practiceguide.aspx?sid=16 (supporting videos)
• Assisting Students Struggling with Mathematics: Response to Intervention

(RtI) for Elementary and Middle Schools (WHAT WORKS CLEARINGHOUSE, April 2009, http://www.rti4success.org/sites/default/files/rti_math_pg_042109.pdf)

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• Dr. John W. Staley

Email: jstaley@bcps.org Twitter: @jstaley06

Unlocking the Power of Problem Solving