NURIN IRDINA BINTI ISMAIL_PROJECT REPORT

INSTITUT PENDIDIKAN GURU
KEMENTERIAN PENDIDIKAN MALAYSIA

KAMPUS IPOH, 31150 HULU KINTA
PERAK DARUL RIDZUAN

MTES3023
MATHEMATICAL REASONING – PROJECT REPORT

Nama : NURIN IRDINA BINTI ISMAIL
No. K/P : 021029070372
Angka Giliran : 2021242340204
Program : PISMP
Ambilan : JUN 2021
Unit : W12
Nama Pensyarah : MADAM ROSMAH BINTI RAMLI
Tarikh Hantar : 06.08.2021

PENGAKUAN PELAJAR
Saya mengaku bahawa tugasan kerja kursus ini adalah hasil kerja saya sendiri kecuali
nukilan dan ringkasan yang setiap satunya saya jelaskan sumbernya.

Tandatangan Pelajar:___________nurin__________Tarikh : ______06.08.2021_______

PEMERIKSA MODERATOR

Markah Markah

Tandatangan Tandatangan

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Tarikh Tarikh

PENGESAHAN PELAJAR
Saya mengesahkan bahawa maklum balas yang diberikan oleh pensyarah telah saya
rujuki dan fahami.

Tandatangan Pelajar:__________________________Tarikh : _____________________

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LIST OF CONTENT

CONTENT PAGE

APPRECIATION…………………………………………………………………………3

1.0 INTRODUCTION…………………………………………………………………….4

1.1 EXAMPLE FOR NON-ROUTINE PROBLEM…………………………………….4

2.0 METHOD……………………………………………………………………………..4

3.0 RESULT……………………………………………………………………………...5

3.1 PHASE 1: UNDERSTAND THE PROBLEM………………………………….….5

3.2 PHASE 2: DEVICE A PLAN……………………………………………………….5

3.3 PHASE 3: CARRY OUT THE PLAN……………………………………………...6

3.4 PHASE 4: LOOK BACK……………………………………………………….……6

4.0 CONCLUSION……………………………………………………………………….7

5.0 LEADERSHIP SKILLS……………………………………………………………...7

6.0 ORGANIZATION…………………………………………………………………….7

REFERENCES…………………………………………………………………………...9

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APPRECIATION
Bismillahirrahmanirrahim.

Alhamdulillah, as much as gratitude is extended to the divine dignity with the overflow
of grace and the blessings of time, life and energy bestowed on me, I can also complete the
assignment for this subject code (MTES3023) successfully.

On this occasion, I am honored to present a million thanks to Madam Rosmah binti
Ramli for placing his full trust in me to fulfill the task entrusted to me.

I would also like to thank all the parties who did not stop in their efforts to help complete
this task, especially to my lecturer Madam Rosmah bin Ramli for his discretion in providing
guidance throughout the course of this task. Apart from that, I would also like to thank my
comrades-in-arms for their help and cooperation in realizing the effort to complete this task
successfully.

As the final speaker, I address this speech to all parties who have been involved in the
success of this task either directly or indirectly. All the help they have given is very much
appreciated because without their help and support all these tasks may not be possible.
Thank you.

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1.0 INTRODUCTION

Problem solving can basically be seen as a process, a step, a method or the means
taken by a person to overcome problems in daily life. Problem solving is also defined as a
cognitive process that involves the use of information in an effort to find the appropriate path
to achieve a goal. According to Ma’rof and Haslinda. (2003), problem solving is an effort to
find or a reasonable path to achieve a goal. While Azlena and Munir. (2004) said that problem
solving encompasses all efforts looking for ways or strategies that can result in unfulfilled
missions.

Non-routine problems are among the types of problems that involve many processes
solutions than routine problems that only solving the simple math. Non-routine problem solving
requires higher order thinking skills. It can also be categorized as a skill and procedure level
high that can only be obtained after skills in problem solving of the routine type mastered or
basic concepts and skills in Mathematics learned. Furthermore, Mohini (2008) said that
problems are not routine involves a complex problem -solving process when compared to
skills simple algorithm. Mathematical solutions cannot be memorized but it is requires the
implementation of a structured set of activities along with strategic planning and means
accordingly. In connection with that, the achievement of solving the problem can be obtained
if metacognitive aspects are included together.

1.1 EXAMPLE FOR NON-ROUTINE PROBLEM

Aziera went to school with a little pocket money. While waiting on the bus, she had paid her
friend's debt of RM2. During the recess, she had spent RM3 on lunch. After school, Aziera
and her friend shared their money to pay equally for their meal. Their meal’s price is RM6. On
the way home, Aziera's neighbor paid RM5 on the wages of uprooting the grasses in his yard
last week. Upon reaching home, Aziera found that the money in her pocket was only RM8.
How much was Aziera's pocket money when she went to school this morning?

2.0 METHOD

There are several problem-solving models that are often used in mathematics
education such as the Lester Model (1975), the Mayer Model (1983), the Polya Model (1973)
and the Schoenfeld Model (1985). Nevertheless, some mathematicians have suggested a
more effective and systematic problem-solving strategy is Polya (1957). Therefore, we will use
the Polya Model to solve the above non-routine problems. The Polya model was founded by

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George Polya (1957). He suggested that in determining the ability of students to solve
problems, they should be able to master these four phases, namely the phase of
understanding the problem, the phase of planning strategies, the phase of implementing
strategies and the phase of reviewing answers. In the second phase, which is the strategy
planning phase, there are 12 ways to use as a strategy to solve problems. Among them are
solving a simpler problem, guessing and checking, drawing a diagram, looking for a patterns,
making tables, working backward, identifying the sub-objectives, acting it out, using direct
reasoning, using indirect reasoning and using algebra/equations (Pumadevi, S., et al. 2016).
These steps are considered as a metacognitive process in solving non-routine mathematical
problems conducted in the study by the researcher. So, I chose the working backwards to
solve the non-routine problem above. However, According to Charles et. al. (1997), the
implementation of these steps in the Polya solution process affects our ability in solving non-
routine mathematical problems as well as involves attitudes towards problem solving.

3.0 RESULT

3.1 PHASE 1: UNDERSTAND THE PROBLEM

There are various ways to understand the problem namely: 1) identification of variables
concerning with the problem; 2) relationship between variables that have been determined
and 3) variables needed through studies or answers. From the example of the non-routine
problem above:-

*She had paid her friend's debt of RM2 = -RM2

*She had spent RM3 on lunch = -RM3

*Aziera and her friend shared their money to pay equally for their meal. Their meal’s price is

RM6 = -( 6 × 12) = -RM3

* Aziera's neighbor paid RM5 = +RM5

* The money in Aziera’s pocket when reaching home = RM8

*How much was Aziera's pocket money when she went to school this morning?

3.2 PHASE 2: DEVICE A PLAN

There are some aspects to be prepared in making a plan to solve a problem, namely: 1)
choose stages in accordance with the obtained information on the problem to solve; 2) make
an appropriate diagram, and this might help to determine appropriate step in solving the
problem; 3) make an analogy, as an effort to determine an appropriate strategy, approach and
method by making analogy with the relatively similar problems, since different problems need

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different approaches and not each strategy, approach and method might be used to solve all
problems. From the example of no-routine problem above:-
*Use the working backwards strategy. ‘Working backwards’ is “we start from what is required
and assume what is sought as already found”, or “from what antecedent the desired result
could be derived”. In working back-wards, it is often required to reverse the operations as in
finding the inverse of a function (Ajay Ramful, 2015).

*So, Aziera's pocket money when she went to school this morning is considered as
‘unknown’ or ‘ ’.

3.3 PHASE 3: CARRY OUT THE PLAN

Understanding a problem, and then making a good plan to solve the problem will not be useful
if it has not been implemented. An effort to show that the problem solving is suitable for solving
the problem is by implementing the problem solving in line with the chosen approach, strategy
and model.
* From Aziera’s money when reaching home (RM 8), minus the money that Aziera gets from
her neighbour (-RM 5)

*Then, plus the money that Aziera and her friend shared to pay equally for their meal
+(RM6×1/2)

*Next, plus the money that Aziera had spent on lunch (+RM3)

*Lastly, plus the money that Aziera had paid her friend's debt (+RM2)

*Aziera's pocket money when she went to school this morning =
1

= 8 − 5 + (6 × 2 ) + 3 + 2
= 8−5+3+3+2

= 11

3.4 PHASE 4: LOOK BACK

Anything which human being made sometimes is planned, sometimes not; it is also the case
in implementing a plan. An effort should be made in solving a problem to review the obtained
answers. The activity might be done by using the answer through inverse method, so that it
can be seen whether the answer is really appropriate with the one expected from the problem.
Aziera’s Pocket Money - Money Spent + Money Received = Balance

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RM 11.00 – ( RM 2.00 + RM 3.00 + RM 3.00 ) + RM 5.00 = ?
RM 11 – RM 8.00 + RM 5.00 = ?
RM 3.00 + RM 5.00 = RM 8.00  Balance of Aziera's pocket money (same value as the

example for non-routine problem above)

∴ The final answer = RM 11 is correct.

4.0 CONCLUSION

In conclusion, the choice of problem solving strategy depends a lot on the type of
problem we want to solve. In mathematics, we must have the high-order thinking skills (HOTS)
to apply knowledge, skills and values in reasoning and reflection to solve problems, make
decisions, innovate and be able to create something. Ball and Bass. (2003) state that
“mathematical reasoning is no more than a basic skill”. The latter implies that reasoning can
be found in all levels of mathematical understanding. The overall assumption in this study is
that mathematical reasoning can be used at all levels of difficulty in solving non-routine tasks.
In fact, various terms can be used to describe what is reasoning. Among the commonly used
terms are critical thinking, higher-order thinking, logical reasoning or simply reasoning.

5.0 LEADERSHIP SKILLS

Reasoning skills are usually used in problem solving processes and they refer to a
person's cognitive abilities that involved thinking systematically and abstractly. Higher-level
thinking skills are increasingly important and necessary for human capital development for the
21st century. In order to prepare the people in that direction, changes to the level of thinking
of students in schools must be implemented immediately.

One of the subjects that prioritizes this level is Mathematics. It is a process in which a
student is able to solve a problem by simply using more systematic methods to make his work
more robust and beautiful. Mathematical reasoning is just an opportunity for students to apply
mathematical knowledge in their daily lives. In this case, teachers should play their role in
shaping the nature of higher-level thinking among students. Teachers need to be more
creative in providing more challenging activities to students as a basis in producing students
who are able to reason logically and systematically which eventually they are able to compete
in the increasingly challenging world of work.

6.0 ORGANIZATION

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The 21st century has seen developments in technology and economics globally, in
turn changing the types and characteristics of employment. Jobs on this age involves “learning
organizations” that need to adapt quickly in response to a changing environment. The most
important asset in this "learning organization" involves experience and a network, rather than
a big building filled with goods and physical resources. Therefore, the experience in
completing this task allows me to solve problems in life.

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REFERENCES
Ajay Ramful. (2015). Reversible reasoning and the working backwards problem solving

strategy. The Australian Mathematics Teacher, 71(4), 1-5.
https://files.eric.ed.gov/fulltext/EJ1093107.pdf
Azlena Zainal dan Munir Shuib. (2004). Meningkatkan Potensi Minda. PTS Publications &
Distributors.
Ball, D. & Bass, H. (2003). Making mathematics reasonable in school.
Charles, R., Kester, F. & O’Daffer, P. (1997). How to Evaluate Progress In Problem Solving.
National Councils Of Teachers Of Mathematics
Ma’rof Redzuan dan Haslinda Abdullah. (2003). Psikologi. Universiti Putra Malaysia.
Mohini, M. (2008). Proses Tingkah Laku Metakognitif dalam Penyelesaian Masalah
Matematik. Projek Sarjana Muda. Universiti Teknologi Malaysia.
Polya, G. (1957). How to solve it: a new aspect of mathematical method. Princ Eton University
Press.
Pumadevi, S., Shuib, A. H., Shamsiah, A. S., Syed Azman, S. I., Nur Syamsila, M. H., (2016).
Nombor dan Struktur Nombor. Oxford Fajar Sdn. Bhd.

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