6/22/2015 Unlocking the Power of Problem Solving 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) BCPS NCSM Baltimore County Goals for 2015-2017 • 25th largest in the U.S. •…making mathematics meaningful, relevant, and • 3rd largest in Maryland accessible for each and • 109,000+ students every student. • 110 Schools • Growing and supporting mathematics education leaders at all levels. www.bcps.org www.mathedleadership.org 1
6/22/2015 Handshakes Which problem solving strategies did you use? 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? Essential Questions • Why focus on Problem Solving? • What can I do to change the future of my students’ mathematical learning trajectory? 2
6/22/2015 Why Problem Solving? 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? Standards for Mathematical Practice 2. Reason abstractly and quantitatively 1. Make sense of problems and 3. Construct viable arguments and persevere in solving them. critique the reasoning of others 6. Attend to precision 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 Modeling and Seeing Structure Explaining Using Tools and Generalizing 3
6/22/2015 Why Problem Solving? 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. Poyla… S.O.L.V.E. • Read - Understanding the Problem – S tudy the Problem – O rganize the Facts • Plan - Devising a Plan – L ine up a Plan • Do - Carrying out the Plan – V erify your Plan with Action • Look Back - Looking Back – E xamine your Answer 4
6/22/2015 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? READ = Understand the Problem S tudy the Problem – Highlight the question. – Answer the question “What is the problem asking me to find?” 5
6/22/2015 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? This problem is asking me to find whether Tito or Luis is correct on who ate more pizza. READ = Understand the Problem O rganize the Facts – Identify each fact. – Eliminate unnecessary facts. – List all necessary facts. 6
6/22/2015 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 PLAN = Devise a Plan • L ine 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. 7
6/22/2015 PLAN : Problem Solving Strategies • Act it Out • Draw a Picture • Make a List • Make a Table/Graph • Look for a Pattern • Try, Check, Revise • Write an Equation • Use Reasoning • Work Backwards • Solve a Simpler Problem Multiple Representations • Verbal Verbal Graphic Representation Representation • Numerical • Graphic • Algebraic Numeric Algebraic Representation Representation 8
6/22/2015 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 DO = Carry Out the Plan • V erify your Plan with Action – Make an estimate. – Carry out your plan. 9
6/22/2015 V erify Your Plan with Action • Estimate your answer. I think they ate the same, so Luis • Carry out your plan. 1 1 1 3 1 Tito Luis 4 8 2 8 2 2 1 4 7 3 4 7 8 8 8 8 8 8 8 Luis is correct. Both boys ate the same amount. 1 Whole 1 1 __ __ 2 2 1 1 1 1 __ __ __ __ 4 4 4 4 1 1 1 _ _ 1 1 1 1 _ _ 1 _ _ _ _ 8 8 8 8 8 8 8 8 10
6/22/2015 Cheese Pepperoni Mushroom Tito ate 1/4 ate 1/8 ate 1/2 Luis ate 3/8 ate 1/2 Tito __ 1 1 __ 1 _ 4 2 8 = 7 1 1 1 _ 1 _ _ 1 1 _ _ _ 8 8 8 8 8 8 8 Luis _ 1 _ 1 _ 1 __ 1 2 8 8 8 = 7 1 _ 1 _ 1 _ 1 _ 8 8 8 8 8 LOOK BACK = Look Back and Reflect • E xamine your Answer – Does your answer make sense? – Is your answer reasonable? – Is your answer accurate? – Write your answer in a complete sentence. 11
6/22/2015 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 7 same; of a pizza. 8 12
6/22/2015 Poyla… S.O.L.V.E. • Read - Understanding the Problem – S tudy the Problem – O rganize the Facts • Plan - Devising a Plan – L ine up a Plan • Do - Carrying out the Plan – V erify your Plan with Action • Look Back - Looking Back – E xamine your Answer 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 13
6/22/2015 Why Problem Solving? 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) 14
6/22/2015 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 15
6/22/2015 http://www.mathedleadership.org 16
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