Ill-Structured Mathematical Problems to Develop

Creative Thinking Students

Abdillah

, Ajeng Gelora Mastuti

and Muhajir Abd. Rahman

Jurusan Pendidikan Matematika, IAIN Ambon, Indonesia

Jurusan Pendidikan Agama Islam, IAIN Ambon, Indonesia

Keywords: Ill-Structured Mathematical Problems, Development Creative.

Abstract: One of the main challenges facing lecturers is preparing their students to face the transition from acquiring

knowledge of problem-solving skills to the challenges they will face after graduation. This study presents

ill-structured mathematical problems (ISMP) to develop students' creative thinking. This research was

conducted by giving a two-point ISMP pre test to 130 students in the fourth and sixth semester of the

Mathematics Study Program and Mathematics Education on two state Islamic religious college campuses.

Researchers evaluate and categorize students' abilities based on indicators of ISMP solving ability.

Furthermore, six research subjects were selected from students who had the potential to think creatively.

The results show that the presentation of ISMP is useful for developing students' creative thinking. In

solving ISMP Students create diverse and correct answers, done in many different ways, and create a variety

of different answers and correct. The development of creativity of students looked after given ISMP second,

with the increasing variety and correct answers, the emergence of a variety of different ways, and creation a

variety of different answers and correct. This happens because the ISMP has several solutions, has a specific

context and complex situations; and in accordance with everyday life so that students feel they experience

the problem.

1 INTRODUCTION

There is several research results that show the

problems related to knowledge gained by students in

learning (Daniels, et al., 2007, Akinmola, 2014).

Many teachers in the learning process provide quite

structured problems. However, problems in the real

world usually faced with ill-structured problems. We

argue that if only using a well-structured problem as

an example of the learning leads students not to be

prepared for the problems they will face in their

professional lives (Daniels, et al., 2007).

Furthermore, the demands of the new century (21st

century) require all students to gain an

understanding of concepts, skills with positive skills

and attitudes in mathematics if they want to succeed

(Akinmola, 2014). This shows the importance of the

knowledge gained by students in learning to achieve

success in solving problems in real life.

There are many research results that show the

importance of problem-solving (Akinmola, 2014,

Bradshaw & Hazel, 2017, Yu, et al., 2015).

Revealed that the importance of problem-solving

skills through learning and application of

mathematics appear in everyday life and at work

(Akinmola, 2014). Scientific and technology-based

problem solving for workers requires a strong

foundation of mathematical knowledge.

Mathematical problem solving has become an

important aspect (Bradshaw & Hazel, 2017).

Therefore it is important for us to equip students

with the skills needed to solve problems. Highlight

that "problem-solving is specific high-level

procedural knowledge (Yu, et al., 2015).

Some opinions about Ill-structured problems are

presented by Abdillah & Mastuti (2018), Jonassen

(1997), Kitchener (1983), Voss & Post (1988), and

Wood (1983). Ill-structured problems presented to

students are problems that involve unknown

elements, have several concept relationships, several

solutions, solution pathways that require a person to

express personal opinions because of their unique

interpersonal activities (Abdillah & Mastuti, 2018).

Ill-structured problems is vague, as well as goals,

seemingly unclear. In addition, constraints are also

not clearly stated (Voss & Post, 1988). The same

Abdillah, ., Mastuti, A. and Rahman, M.

Ill-Structured Mathematical Problems to Develop Creative Thinking Students.

DOI: 10.5220/0008516700280033

In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 28-33

ISBN: 978-989-758-407-7

thing was expressed by Wood (1983) that ill

structured problems seem unclear because the

problem or one of the elements of the problem is

unknown. In this case, the ill structured

mathematical problems are the types of ill structured

problems faced in the practice of everyday life,

containing mathematical content, involving

unknown elements, having several concepts, several

solutions, pathways of solutions that require

someone to express a personal opinion because it is

related to unique human interpersonal activities.

Ill-structured problems arise from specific

contexts, have the following characteristics: first,

aspects of the situation are not concrete; second, the

problem is not well defined; third, this problem is

based on real-life situations and has openness; and

finally, complex situations are presented (Hong &

Kim, 2016). Ill-structured problems as authenticity,

complexity, and openness are the properties of Ill-

structured problems (Kim, et al., 2011). Authenticity

means that it is in accordance with everyday life,

with math homework or problems that describe real

life outside of school (Palm, 2008). A problem can

be said to have authenticity if the problem covers the

context of everyday life and is relevant enough to

deduce an integral part of the actual situation.

In terms of complexity, Jonassen (1997)

considers that the attributes of complexity contain:

the uncertainty of concepts, rules, and principles

needed to solve problems, or how the problem is

organized. The relationship between concepts and

rules and principles do not set. In terms of openness,

Jonassen (1997) said: first, some evaluation criteria

must exist to solve problems; second, the clarity of

the purpose of the problem is not presented; third,

students must express personal opinions and beliefs

about the problem; fourth, it is recommended that

students judge and maintain problems. Shin, et al.

(2003) says that the nature of openness allows

students to place various interpretations of problem-

solving and to justify their interpretation.

2 METHOD

This research data was obtained from the granting of

two ISMP pretest points to 130 students in the fourth

and sixth semester of the Mathematics Study

Program and Mathematics Education from two state

Islamic religious college campuses, in two provinces

in Indonesia. Researchers evaluate and categorize

students' abilities based on indicators of ability to

solve structured problems. Furthermore, six research

subjects were selected from students who had the

potential to think creatively. The selection of

research subjects is based on the quantity of

achievement of indicators of structured problem-

solving abilities conducted by the subjects in

completing the ISMP pretest. In addition,

researchers also pay attention to the results of direct

observations related to the verbal communication

skills of prospective subject students.

The approach used in this study is to use a

qualitative approach with a type of descriptive-

explorative research (Miles & Huberman, 1984).

When viewed from the purpose of this study is to

produce a description of the development process of

students' creative thinking in the face of their

professional life through the provision of ISMP. To

reveal or obtain a description of the subject's

thinking process in completing the ISMP, the

researcher applies the think-aloud technique to the

subject. The researcher tries to conduct a thorough

and in-depth examination (by exploring) the subject

about what is done, written, spoken, body

movements, or even what they think when

completing the ISMP. Therefore researchers act as

key instruments, their existence is absolutely

necessary and cannot be represented by others or

with something else (Creswell, 2013). In terms of

obtaining data, researchers used assistive

instruments such as ill-structured mathematical

problems (ISMP), audio and audiovisual recording

devices (Handy cam) as supporting instruments. The

ISMP used is as follows:

1) The price of one shirt in Toko A (Store I) is

Rp. 5,000, - more expensive than the price of

one shirt in Toko B (Toko II). Shop B gives a

10% discount for the purchase of each shirt.

Shop A gives a special price, that is, if

someone buys more than one shirt, they will

get a 40% discount on the second purchase of

each shirt. If you want to buy 3 clothes, then

how do you get the cheapest purchase costs?

Give an explanation about buying clothes in

both stores!

2) There are two shoe stores, namely Toko S

(Shop I) and Toko P (Toko II). To attract

buyers, each store has its own way of attracting

buyers' attention. Shop P gives a 15% discount

on the purchase of each pair of shoes. Toko S

gives a 35% discount on the second purchase

of each pair of shoes, but the price of a pair of

shoes at Toko S is Rp. 6,500, - more expensive

than the price of a pair of shoes in Shop P. If

you want to buy 3 pairs of shoes, to pay the

cheapest purchase price explanation of shoe

purchases in both stores.

Ill-Structured Mathematical Problems to Develop Creative Thinking Students

3 RESULTS AND DISCUSSION

The following is a discussion about ill-structured

mathematical problems to develop students' creative

thinking. The problem presented contains two

important things: first, there is no standard

procedure used to solve the problem; the second

raises a dilemma in the form of a choice between

buying three pairs of shoes or clothes in Toko I,

three pairs of shoes or in a shop II, two pairs of

shoes or clothes in Toko I and one pair of shoes or

clothes in Toko II, or one pair of shoes or clothes in

the Shop I and two pairs of shoes or clothes in Toko

II; the third purchase problem in Toko I is not well

defined; fourth, problems in both stores are based on

real-life situations and have openness; and fifth, a

complex situation is presented, how to make the

cheapest purchase costs?.

Related to the solution of ill-structured problems

Kitchener (1983) argues that ill-structured problems

have several solutions or possible solutions. Another

possibility is that there is no solution at all, that is,

there is no agreement between problem solvers or

not based on consensus on the right solution if done

by the group. So according to Jonassen (1997) in the

process of solving ill-structured problems, one must

make judgments about problems and can maintain

their opinions. As a result, one must express

personal opinions or beliefs about the problem

(Abdillah, et al., 2017). Thus the solution to

students' structured mathematical problems in this

discussion is the unique interpersonal activity of

students known at the time of think aloud. Other

supporting data is the result of clarification at the

time of the interview.

Furthermore, Ill-structured problems are a type

of problem encountered in the practice of everyday

life, so this problem usually raises the dilemma of

choice (Jonassen, 1997). Because Ill-structured

problems are not only limited by the content domain

learned in class, the solution is not convergent. This

problem also allows requiring the integration of

multiple content domains. Have many alternative

problems to solve (Abdillah, et al., 2017).

Furthermore, Abdillah et al. (2016) argued that in

exploring decision making one would need a theory

as a guide. The student's decision in choosing

several choices made is an alternative action from a

series of actions or strategies made by students.

These actions can be done intuitively, analytically or

interactively. However, because this problem lies in

the practice of everyday life, it is far more

interesting and meaningful for students to define

problems and determine whether information and

skills are needed to help overcome the problems at

hand.

As a result of the problem of ill-structured

mathematical problems that are given repeatedly to

the student, the student's creative process is revealed

and develops when solving the problems presented.

The development of student creativity, as seen from

the problem-solving strategy plan for ISMP (1),

students use guess and test strategies in completing

with the direct substitution of the equations made. In

ISMP (2), in the planning phase of completion,

students first organize and represent the data and

then make an equation to list alternative equations

that might occur.

When viewed from creative thinking component,

students' thinking in solving the problems of ill

structured mathematical problems presented by the

researcher is fulfilling the criteria of creative

thinking (Silver, 1997). Silver (1997) explains that

to assess the ability to think creatively children and

adults can be done using "The Torrance Test of

Creative Thinking (TTCT)". The three components

used to assess the ability to think creatively through

TTCT are fluency, flexibility and novelty. The

following describes the fulfillment of the three

components experienced by students.

The first component, namely fluency, by making

diverse and correct answers to solving problems.

This can be seen in students when making three

alternative solutions, namely (a) total expenditure in

Toko I is equal to total expenditure in Toko II, so

that they can make choices freely, (b) buy all clothes

in Toko I, (c) buy all clothes in Store II. Structurally,

the model of students' creative process in solving

problems of ill-structured mathematical problems in

fluency components can be seen in Fig. 1., in the

green dotted line.

Events make diverse and correct answers in

solving the structured mathematical problems

carried out by students in line with what was

expressed, that this happens because students must

develop their representational and strategic fluency

(Silver, 1997). In addition, they consider the unclear

situation of the problems they face, then they solve a

number of problems, produce solutions for each of

the different problems.

The second component of flexibility is students

solve problems in different ways. This can be seen

from the following explanation:

1) Students unravel that each shirt in store B gets a

10% discount, so that the total expenditure in

store A is equal to total expenditure in store B,

then Store A must apply the first clothes

purchase at normal prices, second clothes

ICMIs 2018 - International Conference on Mathematics and Islam

purchase at 40% discount, and purchase of third

clothes 44% discount.

2) Students examine that the choice will fall on

store B, which is choosing to buy all the clothes

in store B if the third shirt in store A does not

get a discount. In accordance with the editor of

the mathematical problem, the third shirt in

store A has no information, so if the price of

one shirt in store A is 25,000, then the first shirt

purchase is the normal price of 25,000, the

second purchase of clothes gets a 40% discount

of 15,000, and the purchase of the third shirt is

back again to the normal price of 25,000. Means

the total expenditure in store A is 65,000 if the

purchase of the third shirt does not get a

discount. So it's more expensive than buying

clothes all at B.

3) Students examine that to choose to buy all the

clothes in store A, the condition is that the third

shirt in store A must apply a 44% discount and

above. Because the application of a 44%

discount for the purchase of a third shirt in store

A will cause the total expenditure in store A

equal to the total expenditure in store B. Thus if

store A applies a discount above 44% for the

purchase of a third shirt, then the total

expenditure will be obtained in the store A is

smaller or cheaper than total expenditure in

store B.

Student events solve ill-structured mathematical

problems in a variety of different ways because they

are complex and unstructured problems, thus

providing opportunities and opportunities for

students to display various methods of solution

(Silver, 1997). Structurally, the student's creative

process model in solving ill-structured mathematical

problems in the flexibility component can be seen in

Figure 1, in the orange dashed line.

The third component, novelty makes various

answers that are different and correct in solving

problems. This can be known from the students'

strategies in solving problems. Looking for a third

shirt discount in store A so the results can be

accumulated with the purchase of second and third

clothes, then compared with the purchase of three

clothes in store B. Therefore students make various

different and correct answers. This happens because

ill-structured mathematical problems result from real

life experience and require the integration of various

variables in particular contexts, the resulting solution

will require integration in several content domains

(Jonassen, 2004). Furthermore, the importance of

solving this type of problem is to understand

solution to change and generate new solutions

Figure 1: (a) Structure of the process of student creativity

in Ill-Structured Mathematical problems in components of

fluency, flexibility, and novelty, (b) Structure of the

process of development of student creativity in Ill-

Structured Mathematical problems in components of

fluency, flexibility, and novelty.

: Information I on ISMP

: Information II on ISMP

: Information III on ISMP

: Information IV on ISMP

: relationship between two information on ISMP

: the structure changes in the process of completion

: alternatif solusi I

: alternatif solusi II

: alternatif solusi III

: processed information I

: processed information II

: processed information III

: fluency

: flexibility

: novelty

Solution

ofI

Solution

of II, III

Solution

of IV

: equation I

: equation II

: equation III

: equation IV

(b)

progres

Problem

structure

Alternatives

of solusi

(a)

Problem structure

Alternatives

of solution

Solution

of I

Solution

of II

Solution

of III

Ill-Structured Mathematical Problems to Develop Creative Thinking Students

(Mayer & Wittrock, 2006). Thus in solving ill

Structured Mathematical Problems, students must

have cognitive skills that include creative thinking

(including different and convergent thoughts),

tolerance for new things, and cognitive flexibility.

In Figure 1a, the process structure of Student

Creativity in Completing Structured Mathematical

Problems in Components of Fluency, Flexibility,

and Novelty. In figure 1, the structure changed from

the figure (a) to figure (b) shows the development of

creativity related to the development of creative

thinking students in completing ill-structured

mathematical problems. This is in line with several

research results (Collins, 2014; Jaarsveld &

Lachmann, 2017; Silver, 1997; Ulger, 2018, Yu et

al., 2015). Ill-structured problems are important and

can develop creative thinking because it allows

individuals to re-imagine problems, generate new

solutions, and reconstruct ideas (Collins, 2014). The

development of the components of students' creative

thinking because they complete ill-structured

problems tends to be driven by open problems

expressed in a way that allows the creation of

specific targets and perhaps some appropriate

solutions (Silver, 1997). Furthermore, the activity of

formulating problems in the thinking process will

improve and complement individual understanding

of the principles relevant to the task (Jaarsveld &

Lachmann, 2017).

4 CONCLUSIONS

The provision of ISMP is useful for developing

students' creative thinking. In solving ISMP students

make diverse and correct answers, carried out in a

variety of different ways, and make different and

correct answers. The development of student

creativity is evident after the second ISMP is given,

namely increasing the variety of answers and

correct, the emergence of various different ways,

and making various different and correct answers.

This happens because the ISMP has several solution

paths, has a specific context and complex situations;

and in accordance with everyday life so that students

feel they experience the problem.

ACKNOWLEDGEMENTS

We thank the LP2M IAIN Ambon for facilitating the

implementation of this research.

We thank the mathematics students at UIN

Alauddin and the mathematics education students at

IAIN Ambon who gave a huge support to this

research.

We thank our friends Muhammad Ridwan,

Nursalam, and friends who helped in every process

of this research.

REFERENCES

Abdillah& Mastuti, A. G., 2018. Munculnya Kreativitas

Siswa Akibat Ill Structured Mathematical Problem.

Matematika dan Pembelajaran, 6(1), pp. 48-59.

Abdillah, et al., 2016. The Students Decision Making in

Solving Discount Problem. International Education

Studies, 9(7), pp. 57-63.

Abdillah, Nusantara, T., Subanji & Susanto, H., 2017.

Proses berpikir siswa dalam menyelesaikan ill

structured problem matematis. Malang, s.n.

Akinmola, E. A., 2014. Developing Mathematical

Problem Solving Ability: a Panacea for a Sustainable

Development in The 21st Century. International

Journal of Education and Research, 2(2), pp. 1-8.

Bradshaw, Z. & Hazel, A., 2017. Developing problem-

solving skills in mathematics: a lesson study.

International Journal for Lesson and Learning

Studies, 6(1), pp. 32-44.

Chi, M. T. H. & Glaser, R., 1985. Problem solving ability.

In: R. J. Sternberg (Ed.), Human abilities: An

information-processing approach. New York: W.H.

Freeman & Co., pp. 227-250.

Collins, R. H., 2014. the effect of an extended wilderness

education, Utah: Department of Parks, Recreation, and

Tourism, The University of Utah.

Creswell, J. W., 2013. Research Design: Qualitative,

Quantitative, and Mixed Methods Approaches. 3rd

Edition. Los Angeles: Sage Publications, Inc..

Daniels, M., Carbone, A., Hauer, A. & and Moore, D.,

2007. Ill-Structured Problem Solving in Engineering

Education. Milwaukee, WI, s.n.

Hong, J. Y. & Kim, M. K., 2016. Mathematical

Abstraction in the Solving of Ill-Structured Problems

by Elementary School Students in Korea. Eurasia

Journal of Mathematics, Science & Technology

Education, 12(2), pp. 267-281.

Jaarsveld, S. & Lachmann, T., 2017. Intelligence and

Creativity in Problem Solving: The Importance of Test

Features in Cognition Research. Front. Psychol,

8(138), pp. 1-12.

Jonassen, D. H., 1997. Instructional design models for

well-structured and III-structured problem-solving

learning outcomes. Educational Technology Research

and Development, 45(1), p. 65–94.

ICMIs 2018 - International Conference on Mathematics and Islam

Jonassen, D. H., 2004. Learning to solve problems : an

instructional design guide. San Francisco: Pfeiffer An

Imprint of Wiley.

Kim, M. K., Lee, J., Hong, J. Y. & Kim, E. K., 2011.

Study of ‘Ill-Structured’ status from mathematics

problems in elementary school textbooks. Journal of

Learner-Centered Curriculum and Instruction, 11(2),

pp. 1-21.

Kitchener, K., 1983. Cognition, metacognition, &

epistemic cognition: A three-level model of cognitive

processing. Human Development, 26(4), pp. 222-232.

Mayer, R. E. & Wittrock, 2006. Problem Solving. In: P. A.

Alexander & Winne, P. H. (Eds.), Handbook of

Educational Psychology. Mahwah, New Jersey:

Lawrence Erlbaum Associates., pp. 287-303.

Miles, M. B. & Huberman, A. M., 1984. Qualitative Data

Analysis: A Sourcebook of New Methods. California:

SAGE publications Inc.

Palm, T., 2008. Impact of authenticity on sense making in

word problem solving.. Educational Studies in

Mathematics, 67(1), pp. 37-58.

Shin, N., Jonassen, D. H. & McGee, S., 2003. Predictors

of well-structured and ill-structured problem solving

in an astronomy simulation. Journal of Research in

Science Teaching, 40(1), pp. 6-33.

Silver, E. A., 1997. Fostering creativity though instruction

rich mathematical problemsolving and problem

posing. International Reviews on Mathematical

Education, 29(3), p. 75–80.

Ulger, K., 2018. Te Eﬀect of Problem-Based Learning on

the Creative Tinking and Critical Tinking Disposition

of Students in Visual Arts Education. Interdisciplinary

Journal of Problem-Based Learning, 12(1).

Voss, J. F. & Post, T. A., 1988. On the solving of ill-

structured problems. . In: Chi, M. T. H.; Glaser, R. ;

M. J. Farr (Eds.) The nature of expertise. Hillsdale:

NJ: Lawrence Erlbaum.

Wood, P., 1983. Inquiring systems & problem structures:

Implications for cognitive development. Human

Development, 26(5), pp. 249-265.

Yu, K.-C., Fan, S.-C. & Lin, K.-Y., 2015. Enhancing

students’ problem-solving skills through context-based

learning. International Journal of Science and

Mathematics Education, 13(6), p. 1377–1401.

Ill-Structured Mathematical Problems to Develop Creative Thinking Students