Routine Problem Solving

From the curricular point of view, routine problem solving involves using at least

one of the four arithmetic operations and/or ratio to solve problems that are practical in

nature. Routine problem solving concerns to a large degree the kind of problem solving

that serves a socially useful function that has immediate and future payoff. Children

typically do routine problem solving as early as age 5 or 6. They combine and separate

things such as toys in the course of their normal activities. Adults are regularly called

upon to do simple and complex routine problem solving.

Here is an example.

A sales promotion in a store advertises a jacket regularly priced at $125.98 but now

selling for 20% off the regular price. The store also waives the tax. You have $100 in

your pocket (or $100 left in your charge account). Do you have enough money to buy the

jacket?

As adults, and as children, we normally want to solve certain kinds of problems (such as

the one above) in a way that reflects an ‘Aha, I know what is going on here and this is

what I need to do to figure out the answer.’ reaction to the problem. We do not want to

guess and check or think backwards or make use of similar strategies. Invariably,

solving such problems involves using at least one of the four arithmetic operations

(and/or ratio). Being good at doing arithmetic (e. g. adding two numbers: mentally, by

pencil and paper, with manipulatives, by punching numbers in a calculator) does not

guarantee success at solving routine problems. The critical matter is knowing what

arithmetic to do in the first place. Actually doing the arithmetic is secondary to the matter.

A mathematics researcher interviewed children about how they solve routine problems.

One boy reported his method as follows: If there were two numbers and they were both

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big, he subtracted. If there was one large and one small number, he divided. If it did not

come out even, he multiplied. The other interesting aspect of all of this is that the child

had done quite well at solving routine problems throughout his school career. What does

this say about teaching practice? What does this say about assessing what children

understand?

Is the case of the boy an isolated incident or is it the norm? Unfortunately, research tells

us that it is likely the norm. Not enough students and adults are good at solving routine

problems. Research also tells us that in order for students to be good at routine problem

solving they need to learn the meanings of the arithmetic operations (and the concept of

ratio) well and in ways that are based on real and familiar experiences. While there are

only four arithmetic operations, there are more than four distinct meanings that can be

attached to the operations. For example, division has only one meaning: splitting up into

equal groups. Subtraction, on the other hand, has at least two meanings: taking away

something away from one set or comparing two sets .

Once students understand the meaning of an arithmetic operation they have a powerful

conceptual tool to apply to solving routine problems. The primary strategy becomes

deciding on what arithmetic operation to use. That decision cannot be made in the

manner done by the boy of the research anecdote. The decision should be made on the

basis of IDENTIFYING WHAT IS GOING ON IN THE PROBLEM. This approach

requires understanding the meanings of the arithmetic operations.

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The research evidence suggests that good routine problem solvers have a repertoire of

automatic symbol-based and context-based responses to problem situations. They do

not rely on manipulating concrete materials, nor on using strategies such as 'guess and

check' or ‘think backwards’. Rather, they rely on representing what is going on in a

problem by selecting from a limited set of mathematical templates or models. Refer to

Using arithmetic operation meanings to solve routine problems for details.

Solving routine problems should at some point involve solving complex problems.

Complexity can be achieved through multi-step problems (making use of more than one

arithmetic operation) or through Fermi problems. It is advisable to do both.

Non-routine Problem Solving

Non-routine problem solving serves a different purpose than routine problem solving.

While routine problem solving concerns solving problems that are useful for daily living

(in the present or in the future), non-routine problem solving concerns that only indirectly.

Non-routine problem solving is mostly concerned with developing students’

mathematical reasoning power and fostering the understanding that mathematics is a

creative endeavour.

From the point of view of students, non-routine problem solving can be challenging and

interesting. From the point of view of planning classroom instruction, teachers can use

non-routine problem solving to introduce ideas (EXPLORATORY stage of teaching); to

deepen and extend understandings of algorithms, skills, and concepts (MAINTENANCE

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stage of teaching); and to motivate and challenge students (EXPLORATORY and

MAINTENANCE stages of teaching). There are other uses as well. Having students do

non-routine problem solving can encourage the move from specific to general thinking;

in other words, encourage the ability to think in more abstract ways. From the point of

view of students growing to adulthood, that ability is becoming more important in today’s

technological, complex, and demanding world.

Non-routine problem solving can be seen as evoking an ‘I tried this and I tried that, and

eureka, I finally figured it out.’ reaction. That involves a search for heuristics (strategies

seeking to discover). There is no convenient model or solution path that is readily

available to apply to solving a problem. That is in sharp contrast to routine problem

solving where there are readily identifiable models (the meanings of the arithmetic

operations and the associated templates) to apply to problem situations.

The following is an example of a problem that concerns non-routine problem

solving.

Consider what happens when 35 is multiplied by 41. The result is 1435. Notice that all

four digits of the two multipliers reappear in the product of 1435 (but they are

rearranged). One could call numbers such as 35 and 41 as pairs of stubborn numbers

because their digits reappear in the product when the two numbers are multiplied

together. Find as many pairs of 2-digit stubborn numbers as you can. There are 6 pairs

in all (not including 35 & 41).

Solving problems like the one above normally requires a search for a strategy that seeks

to discover a solution (a heuristic). There are many strategies that can be used for

solving unfamiliar or unusual problems. The strategies suggested below are teachable to

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the extent that teachers can encourage and help students to identify, to understand, and

to use them. However, non-routine problem solving cannot be approached in an

automatized way as can routine problem solving. To say that another way, we cannot

find nice, tidy methods of solution for all problems. Inevitably, we will be confronted with

a situation that evokes the response; “I haven't got much of a clue how to do this; let me

see what I can try.”

Example for routine and non-routine problem solving

Problem 1

My mom gave me 35 cents. My father gave me 45 cents. My grandmother gave me 85

cents. How many cents do I have now?

Problem 2

Place the numbers 1 to 9, one in each circle so that the sum of the four numbers along

any of the three sides of the triangle is 20. There are 9 circles and 9 numbers to place in

the circles. Each circle must have a different number in it.

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In problem 1, you need to figure out that you need to add. Understanding addition as

modeling a ‘put together’ action helps you realize that.

In problem 2, you are told to add by the word ‘sum’. Understanding addition as

modeling a ‘put together’ action does not help you with solving problem 2. Being good at

arithmetic might help you a bit, but the matter really concerns a search for strategies to

apply to the problem. Guess and check is a useful strategy to begin with.

Polya’s Problem Solving Model

Background of George Polya(1887 – 1985)

George Polya was a Hungarian who immigrated to the United States in 1940. His major

contribution is for his work in problem solving.

Growing up he was very frustrated with the practice of having to regularly memorize

information. He was an excellent problem solver. Early on his uncle tried to convince him

to go into the mathematics field but he wanted to study law like his late father had. After

a time at law school he became bored with all the legal technicalities he had to

memorize. He tired of that and switched to Biology and the again switched to Latin and

Literature, finally graduating with a degree. Yet, he tired of that quickly and went back to

school and took math and physics. He found he loved math.

His first job was to tutor Gregor the young son of a baron. Gregor struggled due to his

lack of problem solving skills. Polya (Reimer, 1995) spent hours and developed a

method of problem solving that would work for Gregor as well as others in the same

situation. Polya (Long, 1996) maintained that the skill of problem was not an inborn

quality but, something that could be taught.

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He was invited to teach in Zurich, Switzerland. There he worked with a Dr. Weber. One

day he met the doctor’s daughter Stella he began to court her and eventually married

her. They spent 67 years together. While in Switzerland he loved to take afternoon walks

in the local garden. One day he met a young couple also walking and chose another

path.

He continued to do this yet he met the same couple six more times as he strolled in the

garden. He mentioned to his wife ?how could it be possible to meet them so many times

when he randomly chose different paths through the garden?.

He later did experiments that he called the random walk problem. Several years later he

published a paper proving that if the walk continued long enough that one was sure to

return to the starting point.

In 1940 he and his wife moved to the United States because of their concern for Nazism

in Germany (Long, 1996). He taught briefly at Brown University and then, for the

remainder of his life, at Stanford University. He quickly became well known for his

research and teachings on problem solving. He taught many classes to elementary and

secondary classroom teachers on how to motivate and teach skills to their students in

the area of problem solving.

In 1945 he published the book How to Solve It which quickly became his most prized

publication. It sold over one million copies and has been translated into 17 languages. In

this text he identifies four basic principles.

George Polya was a Hungarian who immigrated to the United States in 1940. His major

contribution is for his work in problem solving.

7

Growing up he was very frustrated with the practice of having to regularly memorize

information. He was an excellent problem solver. Early on his uncle tried to convince him

to go into the mathematics field but he wanted to study law like his late father had. After

a time at law school he became bored with all the legal technicalities he had to

memorize.

He tired of that and switched to Biology and the again switched to Latin and Literature,

finally graduating with a degree. Yet, he tired of that quickly and went back to school and

took math and physics. He found he loved math.

His first job was to tutor Gregor the young son of a baron. Gregor struggled due to his

lack of problem solving skills. Polya (Reimer, 1995) spent hours and developed a

method of problem solving that would work for Gregor as well as others in the same

situation. Polya (Long, 1996) maintained that the skill of problem was not an inborn

quality but, something that could be taught.

He was invited to teach in Zurich, Switzerland. There he worked with a Dr. Weber. One

day he met the doctor’s daughter Stella he began to court her and eventually married

her. They spent 67 years together. While in Switzerland he loved to take afternoon walks

in the local garden. One day he met a young couple also walking and chose another

path. He continued to do this yet he met the same couple six more times as he strolled

in the garden. He mentioned to his wife ?how could it be possible to meet them so many

times when he randomly chose different paths through the garden?.

He later did experiments that he called the random walk problem. Several years later he

published a paper proving that if the walk continued long enough that one was sure to

return to the starting point.

8

In 1940 he and his wife moved to the United States because of their concern for Nazism

in Germany (Long, 1996). He taught briefly at Brown University and then, for the

remainder of his life, at Stanford University.

He quickly became well known for his research and teachings on problem solving. He

taught many classes to elementary and secondary classroom teachers on how to

motivate and teach skills to their students in the area of problem solving.

In 1945 he published the book How to Solve It which quickly became his most prized

publication. It sold over one million copies and has been translated into 17 languages. In

this text he identifies four basic principles.

Polya’s First Principle: Understand the Problem

This seems so obvious that it is often not even mentioned, yet students are often

stymied in their efforts to solve problems simply because they don’t understand it fully, or

even in part. Polya taught teachers to ask students questions such as:

Do you understand all the words used in stating the problem?

What are you asked to find or show?

Can you restate the problem in your own words?

Can you think of a picture or a diagram that might help you understand the

problem?

Is there enough information to enable you to find a solution?

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Polya’s Second Principle: Devise a plan

Polya mentions (1957) that there are many reasonable ways to solve problems. The skill

at choosing an appropriate strategy is best learned by solving many problems. You will

find choosing a strategy increasingly easy. A partial list of strategies is included:

Guess and check

Make and orderly list

Eliminate possibilities

Use symmetry

Consider special cases

Use direct reasoning

Solve an equation

Look for a pattern

Draw a picture

Solve a simpler problem

Use a model

Work backward

Use a formula

Be ingenious

Polya’s third Principle: Carry out the plan

This step is usually easier than devising the plan. In general (1957), all you need is care

and patience, given that you have the necessary skills. Persistent with the plan that you

have chosen. If it continues not to work discard it and choose another. Don’t be misled,

this is how mathematics is done, even by professionals. Polya’s Fourth Principle: Look

back

Polya mentions (1957) that much can be gained by taking the time to reflect and look

back at what you have done, what worked and what didn’t. Doing this will enable you to

predict what strategy to use to solve future problems.

George Polya went on to publish a two-volume set, Mathematics and Plausible

Reasoning (1954) and Mathematical Discovery (1962). These texts form the basis for

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the current thinking in mathematics education and are as timely and important today as

when they were written. Polya has become known as the father of problem solving.

Polya’s fourth Principle: looking back the problem

This step is to check for the plan :

Does answer reasonable?

Do the steps correct?

Is there another way to solve the problem?

Extend the problem.

Make up another problem of the same type.

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Comparison between Routine and Non Routine Problem

Routine Non routine

Stresses the use of sets of known or prescribed procedures (algorithms) to solve problems.

Stresses the use of heuristics which do not guarantee a solution to a problem but provide a more highly probable method for solving problems.

Strength: easily assessed by paper-pencil tests.

Strength: most relevant to human problem solving.

Weakness: least relevant to human problem solving.

Weakness: least able to be assessed by paper-pencil tests.

Static• fixed, known goal and known element

Active• fixed goal(s) with changing elements.• Changing or alternative goal(s) with fixed elements.• Changing or alternative goal(s) with changing elements.

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Non Routine Problem 1:

Question 1:

James won 10 trophies in drawing competition and Carman won 6 trophies. How many

trophies that James have than Carman?

Strategy 1 : Diagram

Strategy 2 : Multiplication Steps

James=x ; x=10

Carman=y ; y= 6

z: the trophies that James have than Carman

So, z= x – y

= 10 – 6

= 4

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Similar question :

Megan has 5 stickers. Randy has 12 stickers. How many more stickers does Randy

have than Megan?

The best strategy is Multiplication strategy. The reasons are:

1. Save time. We need spend a lot of time to draw a diagram.

Strategy: Multiplication steps

Randy=x x=5

Megan=y y=12

So,12-5 = 7

Answer: Randy has 5 more stickers than Megan.

Question 2 :

The Lachance family must drive an average of 250 miles per day to complete their

vacation on time. On the first five days, they travel 220 miles, 300 miles, 210 miles, 275

miles and 240 miles. How many miles must they travel on the sixth day in order to finish

their vacation on time?

Strategy 1 (diagram)…

1st day 2nd day 3rd day 4th day 5th day 6th day

220 miles 300 miles 210 miles 275 miles 240 miles ? miles

250 miles x 6 days = 1500 miles

So, the distance on the sixth day is = 1500 – 220 – 300 – 210 – 275 – 240

= 255 miles

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Strategy 2( table ) …

The distance on the sixth day =1500- 1245

= 255 miles

Why I chose diagram?

This is because the diagram is more easy to understand for primary student.

Furthermore, draw a diagram is more save time if compare with draw a table. Student

can have more time to do the other question.

Similar question :

Gini's test scores are 95, 82, 76, and 88. What score must she get on the fifth test in

order to achieve an average of 84 on all five tests?

Strategy 1(diagram)

95 82 76 88

?

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84 x 5 = 420

So, the score of the fifth test = 420 – 95 – 82 – 76 – 88

= 79

Question 3 :

The villagers were building a bridge. While working under the bridge Rodney could see

only the legs of those walking by. He counted 10 legs in one group. What combination of

sheep and children could have been in that group?

Strategy 1 : Guess and check.

Try 3 people and 2 sheep.

6 legs + 8 legs = 14 legs

I can use That's too many legs.

Strategy 2 : Table

People Legs Sheep Legs Total

1 2 2 8 10 ok

2 4 2 8 12 not ok

3 6 1 4 10 ok

4 8 1 4 12 not ok

Answer:

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There are two possible answers: 1 person and 2 sheep, or 3 people and 1 sheep.

Similar question :

Prince Carl divided 15 stone games into two piles: games he owns and games his

brother owns. He owns 3 more games than his brother. How many games does his

brother own?

Why I choose table ?

I choose table to solve the problem because it is more easy to understand and save

time. By using the table, I can more easy explain to the student if compare with using

guess and check.

Solve using a table.

Prince Carl His brother Total

11 8 19 not ok

10 7 17 not ok

9 6 15 ok

Prince Carl owns 9 games. His brother owns 6 games.

That's a total of 15 games.

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