UPS √UPS √Delivering Effective Problem Solving Delivering Effective Problem Solving TechniquesTechniques

Presenter: Jesus A. Martinez

Professional Development Services

ABRAZO DAY 5 February 2, 2009

Bell WorkBell Work

Using two post it notes, write Using two post it notes, write your definition of what is an your definition of what is an exercise and a problem, then exercise and a problem, then post them on the T-Chart.post them on the T-Chart.

SUDOKUSUDOKUExercise or a Problem? Exercise or a Problem?

An An exercise exercise asks you to repeat a method asks you to repeat a method you learned from a similar example.you learned from a similar example.

A A problem problem is usually more complex than is usually more complex than an exercise, so it is harder to solve an exercise, so it is harder to solve because you do not have a preconceived because you do not have a preconceived notion about how to solve it.notion about how to solve it.

Components of a Successful Components of a Successful Problem SolverProblem Solver

Practice (Exposure)Practice (Exposure) PersistencePersistence Self MonitoringSelf Monitoring

PracticePractice

To develop students’ thinking skillsTo develop students’ thinking skills

To develop students’ abilities to select To develop students’ abilities to select and use problem-solving strategiesand use problem-solving strategies

To develop students’ abilities to find To develop students’ abilities to find correct answers to a variety of types of correct answers to a variety of types of problemsproblems

PersistencePersistence

To develop helpful attitudes and To develop helpful attitudes and beliefs about problem solvingbeliefs about problem solving

To develop students’ abilities to To develop students’ abilities to solve problems in cooperative solve problems in cooperative learning situationslearning situations

Self MonitoringSelf Monitoring

To develop students’ abilities to use To develop students’ abilities to use related knowledgerelated knowledge

To develop students’ abilities to monitor To develop students’ abilities to monitor and evaluate their thinkingand evaluate their thinking

Types of ProblemsTypes of Problems

One-Step Problems (Story Problems)One-Step Problems (Story Problems)

Multiple-step ProblemsMultiple-step Problems

Process ProblemsProcess Problems

Applied Problems (Situational Problems)Applied Problems (Situational Problems)

One –Step ProblemsOne –Step Problems

One-step problems provide experience in One-step problems provide experience in translating situations to a number translating situations to a number sentence involving addition, subtraction, sentence involving addition, subtraction, multiplication, or division. multiplication, or division. These are the familiar “story problems” These are the familiar “story problems” that have always been part of school of that have always been part of school of mathematics programs. The primary mathematics programs. The primary strategy used to solve one-step problems strategy used to solve one-step problems is is choose an operation.choose an operation.

Multiple-step ProblemsMultiple-step Problems

The difference between one-step The difference between one-step problem and multiple-step problems is problem and multiple-step problems is the number of operations needed to find the number of operations needed to find a solution. a solution.

The primary strategy used to solve The primary strategy used to solve multiple-step problems is to multiple-step problems is to choose choose the operationsthe operations..

Process ProblemsProcess Problems

Process problems are solved using such Process problems are solved using such strategies as guess-check-revise, draw strategies as guess-check-revise, draw a picture, make a table, look for a a picture, make a table, look for a pattern, work backward, solve a simple pattern, work backward, solve a simple problem, and make an organized list. problem, and make an organized list. The solution may involve some The solution may involve some computation, but these problems can computation, but these problems can not be solved by the student by simple not be solved by the student by simple choosing the operation or operations.choosing the operation or operations.

Applied ProblemsApplied Problems

Require the problem solver to collect Require the problem solver to collect data outside the problem statement. data outside the problem statement.

Many applied problems require the Many applied problems require the student to formulate a clear student to formulate a clear statement of the problem and sub-statement of the problem and sub-problems and identify assumptions problems and identify assumptions that need to be made and the data that need to be made and the data needed for finding a solution.needed for finding a solution.

Activities that Develop and Activities that Develop and Promote Problem Solving Promote Problem Solving SkillsSkills

Problem of the DayProblem of the Day Sharing solutions of Problem of the Week to the Sharing solutions of Problem of the Week to the

whole class (Group Activity)whole class (Group Activity) Summarizing and Paraphrasing the solution to a Summarizing and Paraphrasing the solution to a

math problem (Explain your reasoning)math problem (Explain your reasoning) Create real-life word problemsCreate real-life word problems

Group ActivityGroup Activity

Solve the problem that has been Solve the problem that has been assigned to your group using assigned to your group using METACOGNITION METACOGNITION think about your think about your thinking.thinking.

Follow directions for each problem.Follow directions for each problem.

Share your solution as a group.Share your solution as a group.

Problem # 1Problem # 1

Farmer Brown had ducks and cows. Farmer Brown had ducks and cows. One day noticed that the animals One day noticed that the animals had a total of 12 heads and 32 feet. had a total of 12 heads and 32 feet. How many of the animals were How many of the animals were ducks and how many were cows? ducks and how many were cows?

(Find the answer using two (Find the answer using two different ways) Explain.different ways) Explain.

Problem # 2Problem # 2Three jackals and three coyotes are on a trek across the Three jackals and three coyotes are on a trek across the Mokalani Plateau when they come to a river filled with Mokalani Plateau when they come to a river filled with carnivorous fish. carnivorous fish. There is a rowboat in sight, and the party decides to use There is a rowboat in sight, and the party decides to use it. However, the boat is too small for any more than two it. However, the boat is too small for any more than two of the group at a time. So they must transverse the river of the group at a time. So they must transverse the river in successive crossings. in successive crossings. There is one hitch, though: The jackals must not There is one hitch, though: The jackals must not outnumber the coyotes at any time, in any place. For outnumber the coyotes at any time, in any place. For example, if two jackals and only one coyote are together, example, if two jackals and only one coyote are together, the jackals will overpower, kill, and eat the coyote. It’s the jackals will overpower, kill, and eat the coyote. It’s okay to have an equal number of each, and it’s okay to okay to have an equal number of each, and it’s okay to have more coyotes than jackals in a given place-neither have more coyotes than jackals in a given place-neither situation poses a danger to the coyotes, and the coyotes situation poses a danger to the coyotes, and the coyotes do not pose a threat to the jackals. do not pose a threat to the jackals. So, the trick here is to use the one small rowboat, a lot of So, the trick here is to use the one small rowboat, a lot of sweat, and a little brainpower to ensure the coyotes’ sweat, and a little brainpower to ensure the coyotes’ safety while both groups cross the river.safety while both groups cross the river.(Use the Ac It out strategy) Explain.(Use the Ac It out strategy) Explain.

Problem # 3Problem # 3

A rectangle has an area measuring A rectangle has an area measuring 120 square centimeters. Its length 120 square centimeters. Its length and width are whole numbers of and width are whole numbers of centimeters. What are the possible centimeters. What are the possible combinations of length and width? combinations of length and width? Which possibility gives the Which possibility gives the smallest perimeter?smallest perimeter?

(Use a table/List) Explain.(Use a table/List) Explain.

Teacher Role in Problem Teacher Role in Problem SolvingSolvingProvide learning activities that create an Provide learning activities that create an

environmentenvironment

that engages students while facilitating theirthat engages students while facilitating their

development of mathematical concepts through:development of mathematical concepts through:

ObservationsObservations Questioning TechniquesQuestioning Techniques Modeling StrategiesModeling Strategies Focus in Math languageFocus in Math language

ObservationsObservations

Limit your observation to aspects of Limit your observation to aspects of performance and attitudeperformance and attitude

Select only few students at a time Select only few students at a time to observeto observe

Your observation should be focusedYour observation should be focused It is important that they learn not be It is important that they learn not be

distracted by your presencedistracted by your presence

Questioning TechniquesQuestioning Techniques

Stimulate mathematical thinkingStimulate mathematical thinking

Help the student to solve a problemHelp the student to solve a problem

Example of questions (Heuristics).Example of questions (Heuristics).

Focus on Math LanguageFocus on Math Language SymbolicSymbolic

This refers to mathematical notationThis refers to mathematical notation Content-SpecificContent-Specific

Technical words associated with abstract Technical words associated with abstract mathematical concepts and skills such as mathematical concepts and skills such as sum, addend, product, denominatorsum, addend, product, denominator

AcademicAcademic

Terms used in tests and discourse Terms used in tests and discourse language, such as determine, simplify, language, such as determine, simplify, predictpredict

Model the Most Common Model the Most Common Problem Solving StrategiesProblem Solving Strategies

Acting It OutActing It Out Drawing a PictureDrawing a Picture Making a TableMaking a Table Looking for a PatternLooking for a Pattern Making a ListMaking a List Making a ModelMaking a Model Breaking the problem down into smaller Breaking the problem down into smaller

problemsproblems

Group ActivityGroup Activity

Using the TAKS booklets at your table Using the TAKS booklets at your table identify the types of problemsidentify the types of problems

Share your findings Share your findings

What is problem Solving?What is problem Solving?

Is an extreme complex activity. It Is an extreme complex activity. It involves the recall of involves the recall of factsfacts, the use , the use of a variety of of a variety of skillsskills and and proceduresprocedures, , the ability to the ability to evaluateevaluate one’s own one’s own thinking and progress while solving thinking and progress while solving problems, and many other problems, and many other capabilitiescapabilities

Success in Problem Success in Problem Solving….Solving….

Very much depends on the Very much depends on the students’ interest, motivation, and students’ interest, motivation, and self-confidence.self-confidence.

Involves the coordination of Involves the coordination of knowledge, previous experience, knowledge, previous experience, intuition, attitudes, beliefs, and intuition, attitudes, beliefs, and various abilities.various abilities.

ClosureClosure

The ability to coordinate skills and The ability to coordinate skills and abilities for problem solving abilities for problem solving develops gradually over time. develops gradually over time. This means that the ability to get This means that the ability to get correct answers will develop more correct answers will develop more slowly than the skills and abilities slowly than the skills and abilities associated with problem solving.associated with problem solving.

ReferencesReferences

How to Evaluate Progress in Problem Solving, How to Evaluate Progress in Problem Solving, Charles, Randall; Lester, Frank; O’Daffer, Phares. Charles, Randall; Lester, Frank; O’Daffer, Phares.

National Council of Teachers of MathematicsNational Council of Teachers of Mathematics The Adventures of Johnny Bunko, The Last Career The Adventures of Johnny Bunko, The Last Career

Guide you’ll Ever Need by Daniel H. PinkGuide you’ll Ever Need by Daniel H. Pink Problem Solving Strategies, Crossing the River Problem Solving Strategies, Crossing the River

with Dogs by Ken Johnson and Ted Herrwith Dogs by Ken Johnson and Ted Herr Implementing Standards-Based Mathematics Implementing Standards-Based Mathematics

Instruction, A Case Book for Professional Instruction, A Case Book for Professional Development by Mary Kay SteinDevelopment by Mary Kay Stein

““We sometimes think of being good We sometimes think of being good at mathematics as an innate ability. at mathematics as an innate ability. You either have it or you do not, but You either have it or you do not, but the true is that being good at math the true is that being good at math is more about attitude….”is more about attitude….”

[Outliers by Malcolm [Outliers by Malcolm Gladwell]Gladwell]