The document is a student's short coursework on problem solving in mathematics. It includes: - Discussions of routine and non-routine problems, and Polya's 4-stage model for problem solving. - Two sample math word problems that the student solves by applying Polya's model: drawing diagrams to understand and devise a plan for the first, and working backwards for the second. - A reflection on strategies used, including checking answers through tables and reverse operations. The student gains experience applying problem-solving approaches to self-created math questions.

1. INSTITUT PENGURUAN KAMPUS ILMU KHAS
SHORT COURSEWORK
PRA PERSEDIAAN IJAZAH SARJANA MUDA PERGURUAN
NAME : ANIS SYAFIQAH BINTI MOHAMED NOOR
GROUP/ UNIT : J2.1 ( PEND. KHAS ( BM, KS, PJ )
SUBJECT : MT 2310 D1
NAME OF LECTURER : PUAN AMINAH HJ.SAMSUDIN
DATE OF SUBMISSION : 2 APRIL 2010

2. Contents
1.0 What is mathematics problem? .......................................................
2.0 Problem solving in mathematic.........................................................
3.0 Types of problems..............................................................................
3.1 Routine problem....................................................................
3.2 Non-rotine problem...............................................................
4.0 Polya’s modle...................................................................................
4.1 Understand the problem.......................................................
4.2 Devise a plan........................................................................
4.3 Carry out the plan..................................................................
4.4 Look back..............................................................................
5.0 Question 1..........................................................................................
5.1 Understand the problem........................................................
5.2 Devise a plan.........................................................................
5.3 Carry out the plan..................................................................
5.4 Look back..............................................................................
6.0 Question 2.........................................................................................
6.1 Understand the problem.........................................................
6.2 Devise a plan.........................................................................
6.3 Carry out the plan..................................................................
6.4 Look back...................................................................................
7.0 Reflection ............................................................................................
8.0 References ............................................................................................
9.0 Lampiran ....................................................................................................

3. 1.0What is mathematics problem?
(Kantowski 1977)
Problem develops when students are given mathematics question which they cannot answer
directly or cannot apply their acquired knowledge or given information within a very short
time interval.
(Hayes 1978)
He stated that students are said to face mathematics problem when they try to solve it, but
cannot find ways to achieve to goal directly.
10.0
Problem solving in mathematic.
Problem solving in mathematics is an organised process that needs to achieve the goal of
problem. The aim of problem solving is to overcome obstacles set in the problem. In addition
to overcome the obstacles, students need to analyze the information given, decide and use
much kind of strategies and methods to solve the problems.
Brownell (1942) says
...‘problem solving refers only to perceptual and conceptual tasks, the nature of which the
subject by reason of original nature of previous learning, or of organization of the task, is
able to understand, but for which at the time he knows no direct means of satisfaction. The
subject experiences perplexity in the problem situation, but he does not experiences utter
confusion...problem solving becomes the process by which the subject extricates himself from
his problem’

4. 3.0 Types of problems.
There are two types of problems. Problems can be divided to a routine and non-routine
problem.
3.1 Routine problem.
Routine problem is a problem that can be easily solve f problem. Routine problems consist
of a direct question where is problem solver can answer the question directly without use all
sort of strategy. Besides that, in a routine problem, problem solver can initially know the
method to solve the problem. Routine problems sometime can be a non-routine problem for
certain person. For example, 678 ×25 = ___ is a non- routine problem for primary student
in standard one. It is because, they does not know the procedure for multicolumn
multiplication. Routine problem are those that merely involved an arithmetic operation. These
arithmetic operation consist of a several characteristics. The arithmetic operation should
present a question to be answered, gives facts or number to use, and can be solved by direct
application of previously learned algorithms and basic task is to identify the operation
appropriate for solving problem.
3.2Non-rotine problem.
Non- routine
problems are totally different from routine problems. In fact,non-routine
problem is a kind of unique problem solving which requires the application of skills, concept
or principle which have been learned and mastered. The question for non-routine problem is
not directly as routine problems.It is also meant that whenever we are facing an unusual
problem or situation which we don’t know the procedures to solve it. Besides that, nonroutine problem stresses the uses of
heuristics which a problem solver need to learn by
discovering many things for themselves. Heuristics in non-routine problem do not guarantee a
solution but provide a more highly probable method for solving problems. In fact non-routine

5. problems can be divided into two, static and active. Static mean fixed which problem solver
known goal and known element. Active can be simply to three element which have fixed
goal with changing elements, fixed element with alternative goal and changing elements with
alternative goal. Non-routine problem also serves a different purpose than routine problem
solving. Furthermore, non-routine problem are useful for daily life (in the present or in the
future). Non - routine problemmostly concerned with developing students’ mathematical
reasoning power. It also fostering the understanding that mathematics is a creative endeavor.
Non - routine can be seen as evoking an ‘I tried this and I tried that, and eureka, I finally
figured it out.’ reaction. In non-routine problem 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.
4.0 Polya’s modle
In mathematics have a model called Polya’s model that use for solve the problems solving.
George Polya is the one who successfully established this model in 1957. According to him,
problem solving in mathematics could be implemented in four stages as shown below:
1) Understand the problem
The initial step in problem solving is the student need to understand the problem in hand. The
student needs to identify:
What is given, what are the entities, numbers, connections and values involve?
What are they looking for?

6. To understand complex problem students need to post questions, relate it to other
similar problem, focus on the important parts of the problem, develop a model and
draw a diagram.
2) Devise a plan
Students need to identify the following aspects when devising a strategy:
What are the operations involved?
What are the heuristic/ strategy/ algorithm required?
There are several heuristics/ strategies that students can apply in problem solving and they
are:
Guess and trial & error
Develop a model
Sketch a diagram
Simplify the problem
Construct a table
Experiment and simulation
Working backward
Investigate all possibilities
Identify sub goal
Making analogy
3) Carry out the plan.
What is to be conducted depends on what has been planned earlier and that includes:
Interpreting the information given into mathematical form

7. Carry the strategies that have chosen with calculation and processes.
Checking every step use.
4) Look back
Check all the calculations involved
Check again all the important information that has been given
Consider whether the solution is logical
Look other alternative solution
Read the question again and make sure the question has really been answered.

8. 5.0Question 1
 At the bus station have 7 girl that wait for the bus. Each girl carries 7 bags. In each
bag have 7 cats. Each cat has 7 kittens. Hence, calculate the total of legs there.
1) Understand the problems
Information given:
 There are 7 girls wait for bus at the bus station.
 Each of them carries 7 bags that contain 7 cats.
 Each cat has 7 kittens.
 Find out the total of legs.
What question asks for? :
 How much the total legs at all?
 How much cats legs?
 How much kitten legs?
 How much girl legs?
2) Devise a plan
 Draw a diagram.
3) Carry out the plan
 1 person carries 7bags.
 1 begs have 7 cats.
 1 cat has 7 kittens.

9. cat
beg
7
kitten
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7

10. We could actually continue drawing the tree diagram until we had show finish shows the
branches. However, this would be complicated to draw and might not very accurate instead
let’s draw the simple one diagram to describe the relationship between the layers and
branches as shown in the next diagram.
Girl
Cat
Kitten

11. Kitten
1
2
3
4
5
6
7
cats
7 × 8 = 56 ( cats and kitten in one bag )
Total beg seven ( 7) :
Hence, 56 × 7 beg = 392 ( Kitten and cats in
7 beg for one person )
392
392
392
392
392
392
392

12. 1 person carry 392 cat and kitten.
Hence, 7 person carry how much cat and kitten at that time???
Alternative one use addition:
392 + 392 + 392 +392 + 392 + 392 + 392 = 2744 cats and kitten
Alternative two use multiplication:
1 person = 392 total of cats and kitten
7 person =???????
= (392 × 7) ÷ 1
= 2744 total of cats and kitten.
The questions ask how much the total leg at all?
........... Hence,
1 girl = 2 legs = 1 pairs of legs.
1 cat and kitten = 4 legs = 2 pairs of legs.
Total person that wait for the bus is 7
Hence, 7 × 2 = 14 = 7 pairs of legs of that girl.
Total cats and kitten = 2744 × 4 legs = 10976 legs = 5488 pairs of legs.
 The total legs at the bus stop = the total legs of kitten and cats + the total legs of
person.
= 10976 + 14 = 10990 legs.

13. The answer: 10990 legs.
4) Look back
To solve this question we need to calculate the total amount of legs for cat, kitten and a
girl that carries 7 begs each . When we look back we need to pay attention on this. It is
because went someone do this question they might be do a mistakes. For example they might
ignore the girls leg that carries that beg.
To check whether this answer correct or not we can use table as shown below:
Bag
Cat
Kitten
The total of cat
and kitten
1
7
49
56
2
14
98
112
3
21
147
168
4
28
196
224
5
35
245
280
6
42
294
336
7
49
343
392
To get the number of kitten we need to use multiplication. In addition, we need to time
the number of cats with 7. We need to time it with 7 because each cat has 7 kittens at all. This
concept applies same to when we wanted to calculate the number of cat. It is because in each

14. bag also have 7 cat. But we use addition when we want to calculate the total number of cat
and kitten.
No of girl
No. Of beg
No. Of cat
No. Of kitten
The total number of cat and kitten
1
7
49
343
392
2
14
98
686
784
3
21
147
1029
1176
4
28
196
1372
1568
5
35
245
1715
1960
6
42
294
2058
2352
7
49
343
2401
2744
Hence, from the total of cat and kitten we can calculate the total legs at the bus station by
times 2744with 4. We times that total with 4 because one cat or kitten has 4 legs. However,
we also need to calculate the total number of girl legs. As at the bus station has 7 people, we
need to time 7 with 2 that represent to the total legs that 1 person have.
....... 2744×4 = 10976
.......7×2
= 14
......10976 + 14 = 10990

15. 6.0 Question 2:
Zaki wanted to know the age of a tiger at the zoo. The zoo keeper told Zaki that if he added
10 years to the age of the tiger and then doubled it, the tiger would be 90 years old. How old
is the tiger ?
1) Understanding the Problem.
Information given:
Did the zoo keeper tell Zaki the tiger's age? (no)
What was the last thing the zoo keeper did to the tiger's age? (He doubled it.)
What was the first thing the zoo keeper did to the tiger's age? (He added 10.)
What question ask for?
How old is the tiger?
2) Devise a plan
Work Backwards
3) Carry out the plan
By using work backwards strategy I need to takes the opposite turn to solve these question. I
need to start to solve this problem with the end result of the problem (90), and I need to carry
the action backward to find conditions at the beginning.

16. Start with 90, the final number given by zoo keeper.
Then, divide it by 2 to get the number that was doubled.
........... 90 ÷ 2 = 45.
Subtract the answer with 10 to get the age of the tiger before 10 years was added
........... 45 - 10 = 35
Answer: The tiger was 35 years old.
Look back.
As the answer has been found , I can checked by starting with that answer and carrying the
action through from start to finish. This is the one strategy that ‘‘ advertise’’ itself by stating
the end conditions of the problem and asking to find starting conditions. The steps showed as
below:
If you double a number and get 10, what number did you double? (5)
operation did you use to get 5?
Theoperation that use it division.
........... 10 ÷ 2 = 5
What

17. The zoo keeper doubled a number and got 90.
What operation could you use to get the number he doubled?
Hence, we can use division by divide the number with 2
.......... 90 ÷ 2 = 45
Is the tiger 45 years old?
....... no, it is because zoo keeper had add 10 to the tiger’s age
What did the zoo keeper do before he doubled the tiger's age?
........ He added 10 to the tiger's age.
Which operation would you use to find out how old the tiger is?
....... To find out the tiger age is we need to use subtraction by subtrac 45 with 10. Then the
answer will come out.
The answer is the tiger is 35 year’s old
Others ways to check the answer:
We can check this answer by calculate back the real answer 35 year’s old until we get 90.
The question stated that zoo keeper had add 10 year’s to the tiger’s age and double it that
makes the tiger’s age 90 year’s old.
Hence, to check the answer we need to add 35 with 10 and multiply it by 2. The answer for
this question correct when we can get 90 as the answer for the addition and multiplication
processes.

18. Calculation:
...... 35 + 10 = 45
...... 45 × 2 = 90
Conclusion: The answer for this question is correct. The tiger’s age is 35 year’s old.

19. Reflection
Based on this short coursework, I need to create a question regarding to non-routine
problems. Then I need to apply the polya’s model by use strategies that I have learn before.
To solve that question that I has created I used draw a diagram and working backwards
strategies. I choose these two strategies because I think these two strategies are suitable and
easy to apply on the question that I have created.
For the first question many strategies can be apply on such as logical reasoning, find a
pattern and make a table. However, I think the most suitable ways to solve this question is
draw adiagram. It is because I can clearly imagine the situation on that time. Hence, it will
make easy for me to make a calculation during solve this problem. Furthermore, it also can
avoid me for being careless when solve this question. However,this strategy alsohas a
weakness behind the advantages. For example, this strategy cannot be use by the person who
has low imagination as her or his cannot imagine the situation clearly as guided by the
question.
For the next question, I choose to use working backwards. It is because by refer to
this question I only have information about the end result of the problem and I need to find
the condition at the beginning refers to the clue that had given. Working backward is the one
strategies that can be use to solve the situation that only have the end result.
Then, I used make a table and reverse operation as strategies to check the answer for
both questions that I have created. I have check the first question by used the table. It is
because I can detect directly if I make a mistake on that question. For the second question I
just use the reverse operation to check the answer.

20. I feel very excited during do this assignment. It is because I have a chance to create a
question by my own. By doing this short coursework I arise much knowledge on how to solve
problems. I also know how to apply polya’s model in addition to solve non – routine
problems. As a result of that I can easily solve the mathematics problems in a short period.

21. References
Alfred S. Posamentier & Stephen Krulik.2009.Problem Solving in Mathematics Grades 3-6,
Powerful strategies to Deepen Understanding.United States of America:Corwin A Sage
Company.
Mok Soon Sang.2004. A Primary Education Course in Mathematics for Post Graduated
diploma (K.P.L.I).Kuala Lumpur: Kumpulan BudimanSdn.Bhd.
Mayer, Richard E. 1992. Thinking, Problem Solving, Cognition, 2nd edition. New York:
Freeman.
Noor Shah Saad&Sazelli Abdul Ghani.2008.Teaching Mathematics in secondary school:
Theories and Practices. Kuala Lumpur: UniversitiPendidikan Sultan Idris.
Mok Soon Sang. 2003. A Mathematics Course for Diploma of Education
( Semester 2 & 3).Kuala Lumpur:KumpulanBudimanSdn.Bhd.
Paul Lau Ngee Kiong.2004. Mathematics Education: Exploring Issues to Improve
Performance.Sarawak:PersatuanPerkembanganProfesionalismePendidikan Sarawak
(
PROFES).
TiadaNamaPengarang.The use of diagram in solving non routine.Dilayaraipada 24 March
2010 di URL http://www.emis.de/proceedings/PME28/RR/RR103_Pantziara.pdf.
TiadaNamaPengarang.Learning to solve non-routine problems. Diayaripada 24 March 2010 di
URL http://ilkogretim-online.org.tr/vol6say1/v6s1m5.pdf.
Tiada Nama Pengarang. Wikipedia. Dilayari pada 24 March 2010 di URL
http://wiki.answers.com/Q/Definition_of_routine_and_non_routine