{{Problem SolvingProblem Solving

Dr. Melanie Maxwell & Dr. Stephen SherDr. Melanie Maxwell & Dr. Stephen SherPythagoras Project Common Core SMP Event Pythagoras Project Common Core SMP Event July 29 – August 2, 2013 July 29 – August 2, 2013

Transitioning from Teacher Problem Solving to Student Problem Solving

Today you will be able to:Today you will be able to:Extend your understanding of Extend your understanding of problem solvingproblem solving..

Identify and apply stages of the problem solving Identify and apply stages of the problem solving processprocess

Identify and apply a variety of problem solving Identify and apply a variety of problem solving strategies.strategies.

Discuss and begin Discuss and begin planplan for implementing instruction to for implementing instruction to increase studentincrease student’’s problem skills and aptitude.s problem skills and aptitude.

ObjectivesObjectives

{{How do we want students to How do we want students to view problem solving?view problem solving?

Problem Solving Problem Solving Desired RealityDesired Reality

Common Core State Standards Mathematical PracticesCommon Core State Standards Mathematical Practices

MP MP 1 Make sense 1 Make sense of problems and of problems and persevere in solving them…persevere in solving them…

Students start by explaining to Students start by explaining to themselves the meaning of a problem themselves the meaning of a problem

and looking for entry points to its and looking for entry points to its solution. solution.

CCSS SMP GoalsCCSS SMP Goals

CCSS MP GoalsCCSS MP GoalsCommon Core State Standards Mathematical Common Core State Standards Mathematical PracticesPractices

Shift the paradigm Shift the paradigm from just just knowing algorithms for isolated knowing algorithms for isolated procedures procedures towards flexible skills flexible skills needed for real world problem needed for real world problem solving (MP 4- Modeling)solving (MP 4- Modeling)

Math to meet future problemsMath to meet future problems

NRC Mathematical Proficiencies NCTM Process NRC Mathematical Proficiencies NCTM Process Standards (2000)Standards (2000)

(Adding It Up, 2001)(Adding It Up, 2001)

CCSS Mathematical CCSS Mathematical PracticesPractices

What is “necessary for anyone to

learn mathematics successfully”

http://www.carnegielearning.com/webinarshttp://www.carnegielearning.com/webinars

Problem Solving Reasoning and Proof Communication Connections Problem Solving Reasoning and Proof Communication Connections RepresentationsRepresentations

Build new Build new

knowledgeknowledge

Solve problems Solve problems

From a variety From a variety

Of contextsOf contexts

Apply a varietyApply a variety

of Strategiesof Strategies

Monitor andMonitor and

Reflect on theReflect on the

ProcessProcess

NCTM Process StandardsNCTM Process Standards

FundamentalAspect of math

Make and investigate conjectures

Develop and evaluate math. conjectures

Select and use various types of reasoning and methods of proof

Organize and Organize and consolidate consolidate thinkingthinking

Communicate Communicate coherently coherently and clearly to and clearly to peers, and peers, and othersothers

Analyze and Analyze and evaluate the evaluate the thinking of thinking of othersothers

Use the Use the language to language to express math. express math. ideas ideas preciselyprecisely

Use connections between math. ideas

Math. ideas interconnect and build into a coherent whole

Recognize and apply math. in contexts outside of math.

Create and use representations to organize, record, and communicate math. ideas

Select, apply, translate between math. representations to solve problems

Model and interpret physical, social, and math phenomena

Making Sense -the first step in Problem Solving

Making Connections within and outside of mathematics

{{Now!Now!

How do we How do we bridge this gap?bridge this gap?

““ProblemProblem””A puzzling, yet intriguing situation for which A puzzling, yet intriguing situation for which there is no immediate, apparent solution, yet there is no immediate, apparent solution, yet there are multiple solution paths. (compiled there are multiple solution paths. (compiled

definition)definition)

Depth of Knowledge (DOK) perspective Depth of Knowledge (DOK) perspective Interaction of task and learnerInteraction of task and learner Varies by grade level, student, and conceptVaries by grade level, student, and concept

Consider a continuum of definitionsConsider a continuum of definitionsSimple Complex

Computation, including computation surrounded by words

Coordinating a variety of concepts, processes, and tasks.

George PólyaGeorge Pólya

Understand the ProblemUnderstand the Problem

Devise a PlanDevise a Plan

Carry out the PlanCarry out the Plan

Look Back Look Back

Activity: Activity:

Read Standard for Mathematical Practice 1 (SMP Read Standard for Mathematical Practice 1 (SMP 1).1).

Identify correlations between PólyaIdentify correlations between Pólya’’s research and s research and SMP 1.SMP 1.Problem Solving Problem Solving ProcessProcess

Problem Solving Problem Solving StrategiesStrategies• Draw a Diagram• Act It Out• Make a Table or Chart• Look for a Pattern• Make a Graph• Write an Equation• Try, Check, Revise• Make an Organized List• Use Reasoning• Work Backwards• Solve a Simpler Problem

Task: Each team member will solve the Task: Each team member will solve the problem using the PS strategy they pull problem using the PS strategy they pull from the bag.from the bag.

Reflect on the Focus Questions regarding Reflect on the Focus Questions regarding the strategy or strategies you used.the strategy or strategies you used.

PS Strategies PS Strategies Activity I

From Inside Mathematics

Sn

ail

Pace

Sn

ail

Pace

Which snail moves more quickly Which snail moves more quickly than the others? than the others?

Explain how you figured this out. Explain how you figured this out.

Focus Questions Focus Questions (Metacognition)(Metacognition)

How did you How did you startstart this strategy? this strategy? What What challengechallenges did you have applying this s did you have applying this

strategy?strategy? What are the What are the advantagesadvantages of your strategy for of your strategy for

this problem?this problem? What are the What are the disadvantagesdisadvantages of your strategy for of your strategy for

this problem? this problem? How does this strategy How does this strategy clarify the mathematicsclarify the mathematics

in this problem? in this problem? What did you need to What did you need to understandunderstand to be able to to be able to

applyapply your strategy? your strategy?

Discuss a focus question with insights Discuss a focus question with insights from each team member. (No from each team member. (No monopolizing the discussion.)monopolizing the discussion.)

Pick a different focus question, repeat.Pick a different focus question, repeat. Write one sentence summarizing a Write one sentence summarizing a

personal or team personal or team ““aha!aha!”” Whole group whip around, biggest aha!Whole group whip around, biggest aha!

DebriefingDebriefing

EQ: How do I determine which strategy to use to start solving a EQ: How do I determine which strategy to use to start solving a problem?problem? Do Do notnot solve the problems, just understand and plan. solve the problems, just understand and plan.

Choose Choose a problem.a problem.

Identify Identify the strategies you could use to start the the strategies you could use to start the problem. problem.

Explain Explain what about the problem made you think to what about the problem made you think to use each strategy.use each strategy.

Choose Choose another problem and repeat the processes. another problem and repeat the processes. Try to select a problem that would use different Try to select a problem that would use different strategies. strategies.

Goal: Articulate why and how specific Goal: Articulate why and how specific strategies help solve certain types of problems. strategies help solve certain types of problems. PS Strategies PS Strategies Activity II

{{How do we create opportunities for How do we create opportunities for students to develop process and students to develop process and strategy skills?strategy skills?

Teaching Teaching Problem Solving Problem Solving SkillsSkills

Solve using _____ strategy then discuss Solve using _____ strategy then discuss focus questions.focus questions.

Problem ClassificationProblem Classification Explicit direct instruction of PS strategies. Explicit direct instruction of PS strategies.

(How does it work?)(How does it work?) Consistent use of Think AloudConsistent use of Think Aloud Label processes and strategies students useLabel processes and strategies students use Discussing possible strategies (which ones, Discussing possible strategies (which ones,

why, how will you start) before solvingwhy, how will you start) before solving Sample problems for students to reference, Sample problems for students to reference,

““It is like the monkey problem.It is like the monkey problem.””

Ways to Build PS SkillsWays to Build PS Skills

One Method – PS WallOne Method – PS Wall

Climate is essentialClimate is essential Establish respectful & courteous talk, no Establish respectful & courteous talk, no

exceptions exceptions Everyone has a right to be heard and their Everyone has a right to be heard and their

ideas considered with respectideas considered with respect Everyone has the obligation to listen with Everyone has the obligation to listen with

respectrespect Respect for person does not mean you Respect for person does not mean you

agree with them. agree with them. ““I revised my thinking.I revised my thinking.”” ““I donI don’’t understand . . ..t understand . . ..”” ““Please clarify ….Please clarify ….””

Building a PS Building a PS EnvironmentEnvironment

Be less helpful, but be very supportive. Be less helpful, but be very supportive. Teacher guides information, but is not the Teacher guides information, but is not the

main source of information when problem main source of information when problem solvingsolving

Idea sharing poster (thanks to KH)Idea sharing poster (thanks to KH) Build studentsBuild students’’ work time (5 min + 5 min + ...) work time (5 min + 5 min + ...)

Remember the biomolecule of Problem Remember the biomolecule of Problem Solving?Solving?

QuestioningQuestioning Knowing mathematical objectiveKnowing mathematical objective Choosing good tasks Choosing good tasks Anticipating student solutions, yet being open to Anticipating student solutions, yet being open to

unanticipated solutions unanticipated solutions Building a PS Building a PS EnvironmentEnvironment

Problem solving is . . .Problem solving is . . .

What do you plan to implement to begin moving What do you plan to implement to begin moving from your current reality to a CC reality of from your current reality to a CC reality of students as engaged problem solvers?students as engaged problem solvers? What can you explore THIS year?What can you explore THIS year? How will you How will you beginbegin the year? (KISS) the year? (KISS)

Summary & Action Summary & Action PlanPlan