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Developing Students' Mathematical Critical Thinking Through Problem

Solving Activities in Mathematics Classroom

ByI Gst. Putu Sudiarta

Department of Mathematics Education

Faculty of Mathematics and Science Education

IKIP Negeri Singaraja

ABSTRACT

The Indonesian's new national mathematics curriculum based oncompetency (KBK) addresses the importance of children's problem solvingactivities as an aid for improving mathematical performance, which includes

critical thinking and understanding in the mathematical or non-mathematical

context. Unfortunately, KBK's document does not adequately explain this twoimportant notions; problem solving and critical thinking in mathematics learning.

On the other hand, the notion of 'problem solving' and 'critical thinking' itself, even

in research of mathematics education is still confusing. This article discusses

what is meant by problem solving in mathematics classroom and the effect ofricher problem solving activities to develop critical thinking in mathematics

learning. Working definition concerning the term of problem solving and critical

thinking in mathematics teaching and learning is constructed, and criteria toexamine their implementation in mathematics classroom are suggested.

Key Words: mathematical problem solving, mathematical competency,mathematical critical thinking.

ABSTRAK

Kurikulum Berbasis Kompetensi (KBK) untuk mata pelajaran matematika

menekankan pentingnya kegiatan pemecahan masalah dalam pembelajaran________________ Jurnal Pendidikan dan Pengajaran IKIP Negeri Singaraja, No. 2TH. XXXVII April 2004

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matematika, untuk meningkatkan performance matematis siswa, terutama dalam

berpikir dan mengerti secara kritis, baik dalam konteks matematis maupun non

matematis. Namun dalam dokomen KBK tidak ditemukan penjelasan secaramemadai mengenai makna konkret dari istilah pemecahan masalah dan berpikir

kritis dalam matematika. Di samping itu, dalam penelitian pendidikan matematikaitu sendiri, kedua istilah ini masih sering membingungkan. Artikel ini membahas

pengertian, apa yang sesungguhnya dimaksud sebagai pemecahan masalah dan

berpikir kritis dalam pembelajaran matematika, dan bagaimana pengaruh kegiatan

pemecahan masalah untuk mengembangkan kemampuan berpikir kritis dalammatematika. Dalam artikel ini akan dibahas pula definisi kerja untuk pengertian

pemecahan masalah dan berpikir kritis dalam pembelajaran matematika, serta

kriteria untuk mengidentifikasi dan menguji penerapannya di dalam pembelajaranmatematika.

Kata kunci: pemecahan masalah matematis, kompetensi matematis, berpikir kritismatematis

1. Introduction

In the last three decades, the school curriculum in Indonesia has been

changed four times (curriculum 1975, 1984, 1994 and 2002). Each curriculum

used a different approach and each one was promised as a nice ideal curriculum.

Curriculum 1984 for example focused on Pupils Active Learning, and evencurriculum 1994 advocated active learning with problem solving. But, in fact, the

change from one curriculum to another did not result in any significant educational

improvement. One can mention many reasons for the lack of significant

improvement (see for example Sudiarta, 2004). Since 2001 again a new school

curriculum, namely curriculum based on competency or "Kurikulum Berbasis

Kompetensi"(KBK) has been developed. This KBK may be named as 'curriculum

2004' and is planed to be launched still in this year. KBK deals with some

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important issues concerning the change of educational paradigm, especially

concerning the re-orientation on the concept of teaching and learning as well as on

the assessment procedure of students' achievement. It particularly advocates that

mathematics teaching and learning has to move away from a mechanical view of

mathematics (traditional teacher-centered pedagogy in favor of student-centered

approaches), to one with an emphasis on problem solving of contextual problem,

and on reasoning and communicating mathematically with others. In fact, more

than three decades children have been mainly taught algorithmic formulae. The

school children have been also repeatedly practiced such an approach, in tasks that

have been removed from meaningful contexts and with an expectation of working

individually.KBK is hoped to bring a comprehensive reform in to the school, that is, a

continuous reform that touches all aspects of teaching and learning in the

classroom. Through this reform school children are now expected (1) to work on

problem solving, to solve contextual problems which nearly related to the real-

world problem situations, in order to develop students' critical thinking and depth

understanding in mathematics, and (2) to work cooperatively with others to solve

problems, to trial strategies, and to find effective strategies that work for them. In

research on mathematics education these two issues, mathematical problem

solving as well as mathematical reasoning and communication, are popularly

referred to as mathematical competencies (see for example NCTM, 1989). But, in

fact, many teachers often do not have a clear idea what is meant by these two

terms. This article addresses one of these two importance issues, that is, problem

solving and its potential to develop students' critical thinking and understanding in

mathematics. Important questions concerning this issue may be formulated as

follows:

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(1) What does it mean mathematical problem solving in the mathematics

classroom, and how to develop its instructional environment?

(2) What is meant by critical thinking in mathematics, and how to examine

whether mathematics classroom environments support the achieving of

students' critical thinking ability?

In this article working definition concerning the term of problem solving

and critical thinking in mathematics is constructed, and a criteria to examine their

implementation is also suggested.

2. Discussion

2.1 Mathematics as Problem SolvingThe Indonesian's new national mathematics curriculum based on

competency (KBK) has clearly stated that mathematics instruction must focus on

problem solving of mathematical problems, by means of mathematical real-world

problem. Those mathematical problems may include both 'closed' problems which

have a unique solution and also a unique procedure, and what so called 'open-

ended' problems which may have some reasonable solutions with some possible

reasonable procedures to achieve solutions. This can be seen on the following

quotation:

"Pendekatan pemecahan masalah merupakan fokus dalam pembelajaranmatematika, yang mencakup masalah tertutup, mempunyai solusi tunggal,terbuka atau masalah dengan berbagai cara penyelesaian"

(Depdiknas,Draf Final Kurikulum SMP dan Madrasah Tsanawiyah Mata

Pelajaran Matematika, 2003:4).

On one hand, it must be clearly stated that this issue is obviously not new.

Polya, the well known father of the mathematical problem solving had proposed

the problem solving method to teaching mathematics since 1940's. In the other

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hand, it needs a further explanation and justification what the problem solving in

mathematics instruction really means, and how teachers can design and determine

environments that support problem-solving activities in mathematics classroom. In

the KBK's document these issues are unfortunately not adequately thought yet.

Indeed, Indonesian teachers may now refers to some related words concerning the

mathematical problem solving, such as 'contextual teaching and learning (CTL)' or

'Indonesian Realistic Mathematics Education (IRME)', but this increasingly used

terminology is far from enough to make a clear and precise explanation about the

definition and 'concretization' of problem solving approaches in Indonesian

mathematics classroom. For that reasons, in this article it is useful to take a look at

the mathematics curriculum development in the U.S. initiated by the NationalCouncil of Teacher Mathematics (NCTM) and to point out the concern of

mathematical problem solving related to those proposed by KBK in Indonesian

mathematics curriculum development.

In the U.S. the NCTM has released in 1989 the Curriculum and Evaluation

Standards for School Mathematics, referred to as 'The Standards'. This is an effort

to empower students to become critical consumers of information. NCTM

emphasized that the society has been transformed into an 'Information Age', where

'knowing how to access and use information' is more important than 'memorizing'

information. NCTM declared that a climate should be established in the classroom

that places critical thinking in the heart of instruction (NCTM, 1989). The

proposed changes are an effort to promote mathematical literacy, including the

ability to apply mathematical ideas to problem situations and work with others to

set up and solve problems (Confrey et al., 1990;1991). The U.S. National Research

Council (1989) also reported the necessity to strengthen the U.S. mathematics

curriculum, so that it can empower all students to become actively thinking

citizens in a changing society driven by advances in technology.

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According to the NCTM's proposal, the mathematics curriculum should

include the refinement and extension of methods of mathematical problem solving

so that all students can:

(1) use, with increasing confidence, problem-solving approaches to investigate

and understand mathematical content;

(2) apply integrated mathematical problem-solving strategies to solve problems

from within and outside mathematics;

(3) recognize and formulate problems from situations within and outside

mathematics;

(4) apply the process of mathematical modeling to real-world problem

situations.In contras to KBK, the NCTM has distinguished two 'standards' of school

mathematics, namely, the content strands (content standards) and the process

standards. Mathematics as Problem Solving belongs to one of the fourprocess

standards proposed by NCTM. The others are mathematics as Communication,

Reasoning and as Connections. Each of these process standards proposed by

NCTM is actually referred to as 'mathematical competencies' to be mastered by the

students after finishing mathematics instruction. From this point of view, KBK

seems to lack of issuing and advocating the domain of mathematical competencies

that students should master after following mathematics instruction. Furthermore,

the substance of school mathematics reform proposed by KBK focusses mainly on

the achieving students' competencies, without unfortunately making a clear and

operational definition of what students mathematics competencies to be achieved

are. This is a very critical point that should be addresed very clearly. In this

respect, a sugestion may be made to consider and adopt the four 'process

standards' proposed by NCTM as a considerable definition to students

mathematical competencies.

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In this context a clear distiction concerning the the notion of problem

solving must be made. Since the above notion of mathematical problem solving is

not just a matter of instructional methods, in fact, it is not an ordinary

instructional method comparable with some teaching and learning methods

adopting from psychology, such as cooperative learning, discovery learning

methods, etc. It represents mathematical students' competencies which must be

achieved after finishing some processess of mathematics learning.

2.2 Problem Solving to Develop Critical Thinking

After the notion of mathematical problem solving is discussed, it is now

important to address what the roles of problem solving activities in relating withthe students' thinking really are. It seems commonly to accept that problem

solving promote understanding. Some literatures also describe that problem

solving activities can develop critical thinking, but what the critical thinking in

mathematics learning do really means, needs a clear description. At least, a

domain specific definition of critical thinking must be discussed in order to draw

connections with research and implications for research in mathematics education.

First, critical thinking in mathematics education may be epistemologically

different from critical thinking in other domains (Craver, 1989; Ennis, 1989;

Kuhn, 1999). McPeck (1981) argued that critical thinking varies from field to

field because different situations constitute good reasons for various beliefs. For

example, in mathematics, Ennis (1989) claimed that the domain has different

criteria for good reasons from most other fields, because mathematics accepts only

deductive proof, whereas most fields do not even seek it for the establishment of a

final conclusion. However, technology in mathematics may close this

differentiation because inductive mathematical explorations using computer

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software can be used to give intuitive understanding and justify a mathematical

argument.

Although most of the research literature addressing critical thinking in

mathematics does not define the term, and leaves the assumption that the readers

understand what critical thinking means, in this article some of the descriptions

associated with critical thinking in mathematics are discussed. Krotetski (1976),

Lester et al (1989), and Paul et al (1993) explained that one factor in determining

mathematical giftedness in the twenties, was the presence of critical thinking that

is the ability to abandon an erroneous train of thought. He also associated the term

with solving problems that have multiple solutions as a measure of flexible

thinking.Pascarella and Terenzini's definition (1991) has implications for statistical

reasoning when stating that critical thinking is, the individual's ability to interpret,

evaluate, and make informed judgments about the adequacy of arguments, data,

and conclusions. Resnick (1987;1988) connected high levels of reading skills with

mathematical thinking that incorporates alternate and simplifying strategies for

solving non-routine problems. O'Daffer and Thornquist (1993) constructed a

working definition for critical thinking in mathematics based on a synthesis of

research, stating that; critical thinking is the process of effectively using thinking

skills to help one make, evaluate, and apply decisions about what to believe or do.

Furthermore, they suggested that the process of critical thinking can involve

understanding the situation; dealing with and going beyond evidence/data/

assumptions; and stating, supporting, and applying conclusions, decisions, and

solutions. Ironically, this definition, processes, and a large portion of their

literature review on this topic are based on research outside of the field of

mathematics. They also incorporated mathematical reasoning and proof as

connected elements.

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Obviously, one might think of mathematical reasoning as a component of

critical thinking, if he defines mathematical reasoning as, a part of mathematical

thinking that involves forming generalizations and drawing valid conclusions

about ideas and how they are related. One may also consider mathematical proof

as another component of critical thinking.

Schoenfeld's work (1982;1985;1989) in mathematical problem solving

contained another component of critical thinking. It implies that higher ordered

mathematical thinking is displayed when students use their metacognitive

strategies to determine if they should apply an algorithm to a particular situation.

For example, Schoenfeld (1989) reported dismal results when asking students to

solve, "There are 26 sheep and 10 goats on a ship. How old is the captain?" Astudent using critical thinking skills should realize that there is insufficient

information to solve the problem. However, over 75% of the elementary students

claimed to have solved the problem using some form of arithmetic. That means

that those students tend to believe that all mathematical problems have a unique

solution and must be solved by applying some necessary arithmetic form. This fact

indicates that the students lack of critical thinking skills

The National Council of Teachers of Mathematics (1989) encapsulated

critical thinking as a primary issue related to mathematical reasoning. Their vision

suggests that children need to know that 'being able to explain and justify their

thinking'is important and 'how a problem'is solved is as important as its answer.

At a variety of grade levels, mathematical reasoning, according to NCTM (1989),

should be emphasized so that students can:

(1) draw logical conclusions about mathematics;

(2) use models, known facts, properties, and relationships to explain their

thinking;

(3) justify their answers and solution processes;

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(4) use patterns and relationships to analyze mathematical situations;

(5) believe that mathematics makes sense;

(6) recognize and apply deductive and inductive reasoning;

(7) understand and apply reasoning processes;

(8) make and evaluate mathematical conjectures and arguments;

(9) validate their own thinking;

(10) appreciate the pervasive use and power of reasoning as part of mathematics;

(11) formulate counterexamples;

(12) follow logical arguments;

(13) judge the validity of arguments;

(14) construct simple valid arguments;(15) construct proofs for mathematical assertions (1989, p.81)

It is important to note that NCTM recognizes that mathematical reasoning

is not only based on ability, but also on elements of the affective domain such as

believing and appreciating. Furthermore, many of these desired outcomes often

take place in problem solving or proving environments, illustrating that reasoning

in mathematics is dependent on and not mutually exclusive from its context.

Therefore, the NCTM suggests that problem solving, proving, and mathematical

reasoning contribute to critical thinking in mathematics, but the extent of their

contributions to this domain are still unclear.

Based on the above literature analysis, a constructive suggestion may be

addressed to KBK. Since KBK advocates the problem solving based mathematics

learning and teaching that has an orientation on the achieving of the students'

mathematical critical thinking, it is a need to develop a working definition of

critical thinking in mathematics. It is also very important to design criteria that can

be used to determine whether a mathematics classroom environment does facilitate

students' problem solving activities that promote critical thinking. Based on a

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synthesis of the ideas discussed and literature reviewed earlier in this article,

following definition may be suggested to be considered as a spotlight to the issue

of problem solving advocated by KBK. Critical thinking in mathematics

instruction (mathematical learning and teaching) should be referred to as the

ability and disposition to incorporate prior knowledge, mathematical reasoning,

and cognitive strategies to generalize, prove, or evaluate unfamiliar mathematical

situations in a reflective manner.

Based on this definition, conditions for critical thinking in mathematics

classroom must include, for example:

(1) an unfamiliar situation where an individual does not immediately understand

the mathematical concept or know how to determine the solution of aproblem

(2) use of prior knowledge, mathematical reasoning, and cognitive strategies

(3) either a generalization, proof, and/or evaluation

(4) reflective thinking that involves communicating a solution thoughtfully,

making sense about the reasonableness of an answer or an argument,

determining alternate ways to explain a concept or solve a problem, and/or

generating extensions for further study.

These criteria can be used to identify mathematics classroom, whether it

explicitly and implicitly incorporates critical thinking in mathematics learning, as

well as help determine environments that support critical thinking in mathematics

instruction.

Instructional design for mathematics classroom which oriented on problem

solving approach to develop students' critical thinking can be adopted from

Schoenfeld (1992;1985;1989) and Kauchak et al (1998). A modification of

mathematics instruction based on problem solving approach has been proposed by

Sudiarta (2003b) as follows:

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(1) Problem identification and formulation

In this phase the instruction should enable the students to construct

conceptually an understanding of the mathematical concepts related to the

task. The students should able to use all necessary information and/or data

from the task and then change the task into complete mathematical ideas.

(2) Determining processes and strategies

In this phase the students should thinks critically in order to choose

reasonable strategies that can work, and then carry out the strategies

chosen. The student should able to complete mathematical skills / strategies

that fit the task, for example, to use clearly pictures, models, diagrams,

and/or symbols to solve the task.(3) Reasoning and Communication

This is a very important stage to give the students time to reason. The

students should be challenged to explain the reasoning (the "why") at each

step, using pictures, symbols, and/or vocabulary. They also have to reason

clearly their thinking behind each step, and to present their whole work in

a logical and coherent manner.

(4) Evaluate and Interpret Reasonableness

After solving the task the students should able to evaluate the whole steps

have been made to construct solution. They should review the work and

show clearly why the solution is reasonable in relation to the task.

3. Conclusion

No doubt, that problem solving activities in mathematics class room should

be adequately facilitated, since its potensial to promote students' critical thinking

and depth understanding in mathematics learning. But a clear idea concerning the

notions, both of problem solving and critical thinking should firstly addresed.

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Secondly, the most important concerns are then how to develop its instructional

class room environment for problem solving, and also how to examine whether

mathematics class room environments support the achieving of students' critical

thinking ability.

In this respect, KBK seems to lack of issuing and providing an adequte

answer related to those questions. Since there is an evidence in KBK's document

that, the substance of school mathematics reform proposed by KBK focusses

mainly on the achieving students' competencies, without unfortunately making a

clear and operational definition of what students mathematics competencies to be

achieved are. This is a very critical point that should be addresed very clearly. In

this point a sugestion has been made. KBK should consider or might adopt thefour 'process standards' proposed by NCTM as a working definition to students

mathematical competencies. In this context a clear distiction concerning the the

notion of problem solving have been made. It should also be stated clearly that

problem solving is now a mathematical competence. Problem solving is not just a

matter of instructional methods, in fact, it is not an ordinary instructional method

comparable with some teaching and learning methods adopting from psychology,

such as cooperative learning, discovery learning methods, etc. It represents

mathematical students' competencies which must be achieved after finishing some

processess of mathematics learning.

Based on the literature analysis, there is another constructive suggestion

may be addressed to KBK. Since KBK advocates the problem solving based

mathematics learning and teaching that has an orientation on the achieving of the

students' mathematical critical thinking, it is a need to develop a working

definition of critical thinking in mathematics. It is also very important to develop

criteria that can be used to determine whether a mathematics classroom

environment does facilitate students' problem solving activities that promote

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critical thinking. Based on a synthesis of the ideas discussed and reviewed earlier

in this article, following definition may be suggested to be considered as a

spotlight to the issue of problem solving advocated by KBK. Critical thinking in

mathematics instruction (mathematical learning and teaching) should be referred

to as the ability and disposition to incorporate prior knowledge, mathematical

reasoning, and cognitive strategies to generalize, prove, or evaluate unfamiliar

mathematical situations in a reflective manner.

Based on this definition, conditions for critical thinking in mathematics

classroom must include, for example:

(1) an unfamiliar situation where an individual does not immediately understand

the mathematical concept or know how to determine the solution of aproblem.

(2) use of prior knowledge, mathematical reasoning, and cognitive strategies

(3) either a generalization, proof, and/or evaluation reflective thinking that

involves communicating a solution thoughtfully, making sense about the

reasonableness of an answer or an argument, determining alternate ways to

explain a concept or solve a problem, and/or generating extensions for

further study.

These criteria can be used to identify the mathematics classroom

environment, whether it explicitly and implicitly incorporates critical thinking in

mathematics learning, as well as help determine environments that support critical

thinking in mathematics instruction.

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