MAKING PROBLEM SOLVING LESS PROBLEMATIC

Prepared by SER Literacy & Numeracy Lead Coaches - 2013

What is problem solving?

The term ‘problem’ and ‘problem solving’ occur in many subject areas, however, is most commonly associated with mathematics.

‘Solving a problem is finding the unknown means to a distinctly conceived end…to find a way where no way is known off-hand. For a question to be a problem, it must represent a challenge that cannot be resolved by some routine procedure. Problem solving is a process of accepting a challenge and striving to resolve it.’ (George Polya – ‘the father of problem solving’ 1945)

• An interesting and enjoyable way to learn mathematics.

• An opportunity to learn mathematics with greater understanding.

• Produces positive attitudes towards mathematics.

• Teaches varied ways of thinking, flexibility and creativity.

• Teaches general problem solving skills generally applicable in other KLAs and life.

• Encourages cooperative skills.

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Why teach problem solving?

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The Importance of Problem Solving

‘With exposure, experience, and shared learning, children will develop a repertoire of problem-solving strategies that they can use flexibly when faced with new problem-solving situations.’

Burris, Anita (2005). Understanding the Math You Teach. Merrill Prentice Hall, ISBN 0-13-110737-2.

Australian Curriculum

Why do children have difficulties with PROBLEM SOLVING?

• When solving problems students will need to know general strategies and techniques that will guide the choice of which skills or knowledge to use at each stage in problem solving.

• When a problem has ‘a new twist’ students cannot recall how to go about it – this is when general strategies are useful in providing possible approaches that may lead to a solution.

• Some children find it difficult to think of ideas and strategies. A brain-storming session might help students reflect on problems they have previously solved.

The misconceptions that students have, are a problem with the understanding proficiency strand rather than with fluency.  When students don't understand something we can't fix it by just doing more of the same. 

Approaches to teaching problem solving …

The approaches teachers use:

Teaching for problem solving - knowledge, skills and understanding (the mathematics)

Teaching about problem solving - experience-based techniques for problem solving, learning, and discovery that give a solution which is not guaranteed

to be optimal and behaviours (the strategies and processes)

Teaching through problem solving - posing questions and investigations as key to learning new mathematics (beginning a unit of work with a problem the students cannot do yet)

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Procedural ( Closed) and Open-ended Problems

Procedural problems: One- or two-step simple word problems

Open-endedproblems:

Problems that require mathematicalanalysis and reasoning;Open-ended problems

can be solved in more than one way, and can have more than one solution.

A typical conventional classroom task is:a

Find the perimeter of this rectangle. 10 cm

4cm

a

A corresponding open-ended task is:a

The perimeter of a rectangle is 28cm. What might be the length and width of the rectangle?

Procedural or Open-ended?

(cited in NAPLAN Numeracy 2009 by Bob Wellham K-12 Mathematics Consultant, Swansea)

1. Reading Can students read the words of the problem?

2. Comprehension Can students understand the meaning?

3. Transformation Can students determine a way to solve the problem?

4. Process Skills Can students do the mathematics?

5. Encoding Can students record and interpret their answer?

Newman’s analysis of children’s problem solving errors

So what do we do?

Help students to ………

- read the questions

- comprehend what they read

-provide strategies that aid understanding

-teach students to check their answers are

reasonable.

In multiple choice questions – many students

guess or are mislead.

How can we do this?

• Teach a problem solving model explicitly

• Increase the mathematical metalanguage used

• Train students to explain how they get their

answers

• Scaffold the teaching of the problem solving

strategies

• Assess for differentiation of learning

• Explicit teaching of concepts

George Polya has had an important influence on problem solving in mathematics education. He stated that good problem solvers tend to forget the details and focus on the structure of the problem, while poor problem solvers do the opposite.

Four-Step Process:

1. Understand the problem (See)2. Devise a plan (Plan)3. Carry out the plan (Do)4. Look back (Check)

Polya’s Model – Problem Sovling

Four Step Process

Newman’s Analysis aligned with Polya’s model

Step 1 – SEE: Understand the problem

Students can be overwhelmed just by reading a problem. At this point group discussion is beneficial.

Questions that may help you lead students to an understanding of the problem:

•What are you asked to find or show?•What information is given? (valuable/useless?)•Has anyone seen a problem like this before?•Can you restate the problem in your own words?•What conditions/operations apply?•What type of answer do you expect/Can you give an estimate?•What units will be used in the answer?

Visualisation

Is an important way to read and understand the problem.Do I see pictures in my mind?How do they help me understand the situation?Imagine the situationWhat’s going on here?

Drawing pictures/models: Using models to visualise/understand the problem

Problem:

Sally had some stickers. After she gave her best friend 280 stickers she had 754 stickers left. How many stickers did Sally have in the first place?

Model

drawing

Sally had 1034 stickers in the first place.

1. Is this problem similar to a problem you’ve done before? (Can you use the same method or part of the previous plan?)

2. What strategy would help you solve it? (Students can best learn skills at choosing an appropriate strategy by exposure to many different problems.)

3. Are the units consistent?

Step 2- PLAN: Devise a Plan

Problem-solving strategies include:

• Draw a picture or diagram

• Act it out

• Use concrete materials/make a model

• Look for a pattern

• Make an orderly list or table

• Work backwards

• Use direct/logical reasoning

• Solve a simpler problem

• Predict and test (guess & check)

• Write an equation/use a formula

• Eliminate possibilities

Make a Drawing or Diagram

• Stress that there is no need to draw detailed pictures.

• Encourage children to draw only what is

essential to tell about the problem.

• Stress that other objects may be used in the place of the real thing.

• The value of acting it out becomes clearer when the problems are more challenging.

Act It Out

Use concrete materials/make a model

• Students use materials to help them discover the relationship they need to see, to lead them to a solution.

• Using concrete materials may make the problem easier to see.

Look for a Pattern

• This involves identifying a pattern and

predicting what will come next.• Often students will construct a table, then

use it to look for a pattern.• Explain to the students that:o They can look at a series of shapes, colours or

numbers to see if you can find a pattern

o The pattern should repeat

o The pattern is not always obvious•  

PROBLEM: Restaurants often use small square tables to seat customers. One chair is placed on each side of the table. Four chairs fit around one square table.Restaurants handle larger groups of customers by pushing together tables.Two tables pushed together will seat six customers. How many people will 4 tables pushed together seat?

PROBLEM: Restaurants often use small square tables to seat customers. One chair is placed on each side of the table. Four chairs fit around one square table.Restaurants handle larger groups of customers by pushing together tables.Two tables pushed together will seat six customers. How many people will 4 tables pushed together seat?

Number of tables

Number of people

1 4

2 6

3 8

4 ?

Example: Look for a pattern using a table

Construct a Table orOrganised List

• This is an efficient way to classify or order a large amount of data.

• An organised list provides a systematic way to record computations.

The letters ABCD, can be put into a different order: DCBA or BADC. How many different combinations of the letters ABCD can you make? To answer this question students may choose to make a list. By making a SYSTEMATIC list, students will be able to see every possible combination.

Work Backwards

• This strategy seems to have limited application opportunities, however, is a powerful tool when it can be used.

• Some problems are posed in such a way that students are given the final conditions of an action and are asked about something that occurred earlier.

Example of Working Backwards

Direct /Logical Reasoning

Explicitly teach students:

•To tackle the problem step by step

•That each piece of information is a piece of the puzzle, put all pieces together to find the solution

•To read each clue thoroughly and work by a process of elimination

•When the first plan (strategy) is unsuccessful, to try another one

Solve a Simpler or Similar Problem

• Some solutions are difficult because the problem contains large numbers or complicated patterns.

• Sometimes a simpler representation will show a pattern which can help solve a problem.

Teach the students to:· Set aside the original problem and work through a simpler related problem· Replace larger numbers with smaller numbers to make calculations easier, then apply same method of solving it to the original problem· Look for a pattern that may be emerging

Teach the students to:· Set aside the original problem and work through a simpler related problem· Replace larger numbers with smaller numbers to make calculations easier, then apply same method of solving it to the original problem· Look for a pattern that may be emerging

Predict & Test (Guess and Check)

• This strategy does not include “wild” or “blind” guesses.

• Students should be encouraged to incorporate what they know into their guesses - a logical guess.

• The “Check” portion of this strategy must be stressed.

• When repeated guesses are necessary, using what has been learned from earlier guesses should help make each subsequent guess better and better.

Write an equation/use a formula

• Students can sometimes make sense of a problem by changing the written problem to a number sentence.

Eliminate possibilities

• Eliminating possibilities is a strategy where students use a process of elimination until they find the correct answer.

• This is a problem-solving strategy that can be used in basic math problems or to help solve logic problems.

• At this stage students need to follow the relevant steps to solve the problem.

• They need to learn how to check each step of the solution and ensure the accuracy of the computation as they implement their chosen problem solving strategy.  They need to be explicitly taught how to keep an accurate record of each step and how to avoid making careless mistakes and computational errors.

Step 3 – DO: Carry out the plan

1. Does my answer make sense?

2. Check the mathematics by working backwards, estimating, showing that the answer is reasonable, or doing the problem another way.

3. Have I answered the question?

4. Have I learned anything new from solving this problem?

Step 4 – CHECK: Look Back

Polya’s Model in action

Maddison, Bella, Tia and Talia all exchanged Valentine’s Day cards. How many Valentine’s Day cards were exchanged?

See

Understanding the Problem

Do I understand all of the words?

Exchange = swap

Valentine = loved one

Many = more than one

What am I asked to do?

I am asked to find out how many cards were exchanged between the 4 friends.

Can I restate the problem in my own words?

Four friends want to give each other a card. How many cards would be swapped if they were each to get a card from one another?

Can I think of a picture or diagram

that might help me

understand the problem?

Four Friends

Is there enough information to enable

me to solve the problem?

Yes!

Plan

Devise a plan

ListMaddison Bella-Tia-Talia

Bella Tia-Talia-Maddison

Tia Talia-Maddison-Bella

Talia Maddison-Bella-Tia

Diagram Maddison

Bella

Tia

Talia

Act it outThree friends and I could stand in a

circle. Each of us would hold enough pieces of paper to give one another one each. Then I

could count how many pieces of paper we held altogether.

Look for a PatternMaddison would need ***

Bella would need ***

Tia would need ***

Talia would need ***

SO… ***+***+***+*** = 12

Solve a Simpler ProblemIf Maddison was to give a card to each of her three friends, She would need 3 cards.

SO…There are four friends, each needing 4 cards.

SO…

4 X 3 = 12

Write an Equation4 friends each need to give out 3

cards.

SO…

4 X 3 = 12

OR…

3 + 3 + 3 + 3 = 12

Problem Solving Think boardWhat is the question asking?What information is important?

What strategy will you use to solve the problem?

Attack the problem!Do the maths!

Write your answer as a complete sentence. DO