Inquiry Co-operation Model for Enhancing Junior High School Students' Mathematical Problem Solving...

Inquiry Co-operation Model for

Enhancing Junior High School Students'

Mathematical Problem Solving Ability

Heni Pujiastuti1, Yaya Sukjaya Kusumah

2, Utari

Sumarmo2, Jarnawi Afgani Dahlan

2

1 Universitas Sultan Ageng Tirtayasa

2 Universitas Pendidikan Indonesia

To cite this article:

Pujiastuti, H., Kusumah, Y. S., Sumarmo, U., & Dahlan, J. A. (2014). Inquiry co-operation

model for enhancing junior high school students' mathematical problem solving ability.

International Journal of Contemporary Educational Research, 1(1), 51-60.

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connection with or arising out of the use of the research material.

International Journal of Contemporary Educational Research

Volume 1, Number 1, January 2014, Page 51-60

Inquiry Co-operation Model for Enhancing Junior High School Students'

Mathematical Problem Solving Ability

Heni Pujiastuti1*

, Yaya Sukjaya Kusumah2, Utari Sumarmo

2, Jarnawi Afgani Dahlan

2

1 Universitas Sultan Ageng Tirtayasa 2 Universitas Pendidikan Indonesia

Abstract

This research aims to describe the enhancement and achievement of students' mathematical problem solving

ability (MPSA) as a result of Inquiry Co-operation Model (ICM) and Conventional Learning (CL)

implementation. This research used experimental method with pretest-posttest control group design. Population

of the research is Junior High School students in Serang City, Banten Province, Indonesia. The sample is eighth

grade students from two school levels classified as high and medium levels. Two classes are randomly selected

from each school, one class as the experimental group who received ICM and the other class as a control group

who received CL. The instrument used consists of MPSA test and observation sheets. The results of data

analysis using t-test and two-way ANOVA concluded that: (1) the enhancement and achievement of students'

MPSA who received ICM are better than those of students who received CL; (2) school level factors has

significantly affect toward enhancement of students' MPSA, but does not has significantly affect toward

achievement of students' MPSA; and (3) there are no interaction between learning model and school levels

toward achievement and enhancement of students' MPSA.

Key words: Inquiry co-operation model, Mathematical problem solving ability

Introduction

Problem solving is important ability to mastered by students. It is argued that students who are skillfull in

solving problems will also be skillfull in identifying a problem, selecting relevant information, composing,

analyzing, evaluating, and reflecting on the results. According to Nasution (2000), those skills will lead to

students’ intellectual satisfaction, enhance their intellectual potential, and encourage them to investigate

throughly a discovery activity.

National Council of Teachers of Mathematics (NCTM, 2003) stated that problem solving has been an integral

part of all mathematics learning so that it should not be an isolated part of mathematics program. Anderson

(2009) also asserted that problem solving is recognised as an important life skill involving a range of processes

including analysing, interpreting, reasoning, predicting, evaluating and reflecting. It is either an overarching

goal or a fundamental component of the school mathematics curriculum in many countries.

However, some results of the study found that students' MPSA is low, as mentioned by Kadir (2010) that the

MPSA of junior high school students is still low. This can be seen from the mean scores of MPSA test of

students who obtained a score of only 2.7 of maximal ideal score 10.

Preliminary studies involving 38 junior high school students (Pujiastuti, 2012) reported that the mean score of

MPSA test of students is only 6.5 and the highest score obtained by the students only reached 12, while the

maximal ideal score is 28. This result shows that the percentage of students' mean score was only 23.21% of the

maximal ideal score. In general, the results of the study concluded that junior high school students' MPSA is still

low.

The results of the studies described above illustrate that in general mathematics learning process so far only

develop the ability to think in such a low level which is very procedural in nature. In other words, the learning

* Corresponding Author: Heni Pujiastuti, [email protected]

52 Pujiastuti, Kusumah, Sumarmo, & Dahlan

processes that occur have not been able to develop students' mathematical thinking skills to a higher level. As

stated by Kesumawati (2010), the emphasis of teaching mathematic today is emphasized more on math

formulas, sample questions, and regular exercises. In this case, students only do exercises directly by using

formulas and algorithms that have been given. Through this way, the students are only trained to memorize

things they have learned before. As a result, the learning process creates mostly passive students with limited

knowledge to what is transferred by teachers.

These conditions lead to students' inability to solve various problems in which they are only able to resolve the

problem in accordance to the examples given by teachers. Schoenfeld (Even & Tirosh, 2003) in his study

revealed that students who have all the knowledge needed to solve a problem are often not able to use that

knowledge to solve problems that are not familiar to them. These findings indicate that the students have not

been able to apply important concepts in solving the problems being faced.

To overcome the students' inability in using the knowledge in solving problems, an effort that does not just

develop mathematical ability of a procedural nature is needed. The efforts should allow to train and develop

students' MPSA optimally. Therefore, it is indeed a need for a supportive learning and training aspects of the

capabilities in the learning process to be included.

Aspects of students' MPSA can be trained through learning Inquiry Co-operation Model (ICM). ICM is a

process of learning that emphasizes inquiry, discovery of a concept (knowledge), and problem resolution. The

principles of ICM is that the knowledge students gain is the result of investigations (findings) of the students

themselves.

Alr∅ and Skovsmose (2002) explain that ICM consists of eight components of learning process, namely: (1)

getting in contact, (2) locating, (3) identifying, (4) advocating, (5) thinking aloud, (6) reformulating, (7)

challenging, and (8) evaluating. The eight components are integrated each other and engage students to be more

active in their learning.

Students' involvement in ICM can be seen from the processes that occur at each components. In the component

getting in contact, a teacher presents a situation or mathematical problems related to the material being studied.

Then in locating component, each student learns how to express and write down their perspective (ideas or

opinions) to a given problems. This is followed by identifying the things that are necessary and are known from

the given problems (identifying).

Advocating component may arise when students are discussing and critiquing each other, giving advice and

when providing an alternative way to other students. Furthermore, each student is guided to be able to solve the

problem based on the identification result and the way they had planned (thinking aloud). In reformulating

component, students are guided to solve the problem in a different way or to make a conclusion with their

language. Once they understand simple problems, they are given a challenge (challenging) through the

provision of more complex problems. In the final stage, teacher does evaluation (evaluating) in order to

determine the quality of students' understanding in the concepts that have been studied.

Furthermore, ICM encourages students to engage actively in the process of investigation, to construct a concept,

and to resolve the problems through the guidance and direction of teachers. However, this does not necessarily

mean that the teachers provide students with information or answers of the problems they face. In this context,

the teachers only assist students when they really need it. In this way, the students have an opportunity to

express broader perspectives (ideas or opinions), to construct concepts, and to solve problems. In this situation,

it is possible that the teacher finds a different solution shared by the students.

Method

This study used experimental method with pretest-posttest control group design, as described below (Fraenkel,

1993).

R O X O

R O O

Notes: R = Random sampling

X = ICM

O = Pretest of MPSA = Posttest of MPSA (test of mathematical problem solving ability)

53

IJCER (International Journal of Contemporary Educational Research)

Population and Sample

The population of the research is Junior High School students in Serang City, Banten Province, Indonesia. The

sample is eighth grade students from two school levels classified as high and medium levels. Two classes are

randomly selected from each school, one class as the experimental group who received ICM and another class

as a control group who received CL. School category has been determined based on school accreditation level

which is valid until the year 2013. High and medium-level school, respectively, are schools that have

accreditation A and B.

In this research, one school was randomly selected from both the high and medium-level school. Furthermore,

two classes were randomly selected from all of the eighth grade students in the high-level school (SMP A) and

medium-level school (SMP B). One class was treated as experimental group and the other class as a control

group. Students in the experimental group received ICM while students in the control group received CL.

Instruments

The research instrument used is in the form of MPSA test and observation sheets. MPSA tests are given prior to

the learning activity (pretest) and after the learning activity (posttest). MPSA test used in this research is

analytical test that is intended to find out the students' ways of thinking in solving problem in order to make it

more clearly defined. This is in accordance to the opinion of Ruseffendi (1991) saying that one of the

advantages of analytical test is that we can see clearly the thinking process of the students through their answers

of the given problem.

MPSA test consists of 6 test items with a maximal ideal score of 38. The material of MPSA test was adjusted to

the subject matter of Mathematics in the second semester of 2012/2013 which refers to the curriculum. The

preparation of the test begins with a first lattice covering subject matter which measured the ability aspects,

indicators of MPSA, and the number of test items. Then, continued by preparing the MPSA test in accordance to

their respective indicators to measure along with the answer key and the scoring guidelines.

The achievement of students' MPSA is obtained by posttest scores of MPSA test. Criteria for achievement of

students' MPSA can be seen in Table 1.

Table 1. Category of achievement

Posttest score (X) Category

High

X % Medium

% Low

The magnitude of the enhancement of students' MPSA is calculated using the formula of normalized gain

developed by Meltzer (2002), as follows:

The results of the calculation of the gain were interpreted by using the classification gain of Hake (1999) and

can be seen in Table 2.

Table 2. Category of gain (g)

g Category

g 0.7 High

0.3 g < 0.7 Medium

g < 0.3 Low

Results

Description of the data


The mean scores of students' MPSA is obtained based on the pretest and posttest. Recapitulation of students'

MPSA test result can be seen in Table 3.

Table 3. Recapitulation of students' MPSA test result

School

Levels Group N

Means g

Pretest Posttest

High ICM 40 8.13 24.63 0.56

CL 41 8.29 19.61 0.39

Medium ICM 34 8.35 23.18 0.52

CL 37 8.22 17.03 0.31

Total ICM 74 8.23 23.96 0.53

CL 78 8.26 18.38 0.34

From Table 3 above it is known that the enhancement of students' MPSA from experimental group is greater

than the control group. Overall, the enhancement of experimental group students' MPSA is 0.53, while the

control group is only 0.34. In a high level schools, the enhancement of experimental group students' MPSA is

0.56, while the control group students' MPSA is only 0.39. In a medium level schools, the enhancement of

experimental group students' MPSA is 0.52, while the control group students' MPSA is only 0.31.

The achievement percentage of students' MPSA can be determined based on posttest scores. The achievement

percentage of students' MPSA can be seen in Figure 1 below.

Figure 1. The achievement of students' MPSA

From Figure 1, it is known that the achievement of students' MPSA from experimental group is greater than the

control group, both in terms of overall and every school level. In terms of overall, the achievement of

experimental group students' MPSA is 63.05% of the maximal ideal score, while the control group is only

48.37% of the maximal ideal score. In a high-level school, the achievement of experimental group students'

MPSA is 64.82% of the maximal ideal score, while the control group is only 51.61% of the maximal ideal score.

In a medium-level school, the achievement of ICM students' MPSA is 61% of the maximal ideal score, while

the CL students' is only 44.82% of the maximal ideal score.

Data Analysis

To determine whether the enhancement and the achievement of students' MPSA who received ICM is better

than those of students who received CL, further testing through statistical mean difference test is conducted.

Before doing the mean difference test, test of normality of distribution and homogeneity of data variance were

firstly conducted. The test of normality of distribution data using the Kolmogorov-Smirnov Z (KS Z) and

homogeneity of data variance using Levene's test.

55


The statistical test results of enhancement and achievement data of experimental and control groups students'

MPSA, both known to have a normal distribution data and homogeneous variance. Thus, the mean difference

test used is t-test. The summary of mean difference test results of MPSA enhancement data can be seen in Table

4.

Table 4. The summary of mean difference test results of MPSA enhancement

School Levels Group g t Sig. (5%) H0

High ICM 0.56 3.869 0.000 rejected

CL 0.39

Medium ICM 0.52 4.202 0.000 rejected

CL 0.31

Total ICM 0.53 5.688 0.000 rejected

CL 0.34

From the summary of mean difference test results of the enhancement of students' MPSA data in Table 4, it is

known that the probability value (significance), both in terms of overall and every school levels are smaller than

the significance level = 0.05, so that the null hypothesis is rejected. Thus, it can be concluded that the

enhancement of MPSA of ICM students better than those of CL students, both in terms of overall and every

school level categories (high and medium-level).

The summary of mean difference test results of MPSA achievement data can be seen in Table 5.

Table 5. The summary of mean difference test results of MPSA achievement

School Levels Group posttest t Sig. (5%) H0

High ICM 24.63 3.338 0.0005 rejected

CL 19.61

Medium ICM 23.18 3.663 0.000 rejected

CL 17.03

Total ICM 23.96 4.950 0.000 rejected

CL 18.38

The mean difference test results of the achievement of students' MPSA data in Table 5 shows that the

probability value (significance), both in terms of overall and every school level is smaller than the significance

level = 0.05, so the null hypothesis is rejected. Thus, it can be concluded that the achievement of MPSA

students who received ICM better than those of students who received CL, both in terms of overall and every

school level categories.

Interaction between Learning Model and School Levels toward Enhancement of Students' MPSA

The two-way ANOVA was used to determine the interactions between learning model and school levels toward

enhancement of students' MPSA. Before performing two-way ANOVA, the data was assured to have normally

distributed and has homogeneous variance. The summary of two-way ANOVA results are presented in Table 6.

Table 6. The summary of two-way ANOVA of MPSA enhancement

Source Type III Sum of

Squares df Mean Square F Sig.

Corrected Model 1.523a 3 0.508 12.580 0.000

Intercept 29.726 1 29.726 736.640 0.000

Learning Model 1.329 1 1.329 32.944 0.000

School Level 0.173 1 0.173 4.277 0.040

Learning Model *

School Level 0.017 1 0.017 0.431 0.513

Error 5.972 148 0.040

Total 37.329 152

a. R Squared = 0.203 (Adjusted R Squared = 0.187)


From Table 6 it can be seen that the learning model has a significant effect on the enhancement of students'

MPSA. This is indicated by the value of the probability (significance = 0.00) that is smaller than 0.05.

Similarly, school level factors have a significant effect on the enhancement of students' MPSA. This is indicated

by the value of the probability (significance = 0.040) is smaller than 0.05. Based on the enhancement of

students' MPSA, student in higher school level obtained higher enhancement of MPSA. The results of statistical

tests and the enhancement mean of students' MPSA, it can be concluded that the enhancement of students'

MPSA in high-level school are better than those of students in medium-level school.

In Table 6 above, it can also be seen that the probability value (significance) for the interaction between the

learning model and school levels is 0.513. The significance value is greater than 0.05, so the null hypothesis is

accepted. It can be concluded that there is no interaction between learning model (ICM and CL) and school

levels (high and medium) toward enhancement of students' MPSA. Graphically, the interaction between

learning model and school levels toward enhancement of students' MPSA can be seen in Figure 2.

Figure 2. Interaction between learning model and school levels toward enhancement of students' MPSA

Figure 2 shows that enhancement of MPSA in high and medium-level school students who received ICM is higher

than students who received CL. In ICM, the order of the enhancement mean of students' MPSA from the largest to

the smallest are high-level school students followed the medium-level school students. The same sequence

happened with the enhancement mean of students' MPSA who received CL. The similarity sequence suggests that

there is no interaction between learning model and school levels toward enhancement of students' MPSA.

The lack of interaction between learning model and school levels toward enhancement of students'MPSA also

can be seen from the difference in the enhancement mean. The difference in enhancement mean of students' MPSA

who received ICM and students who received CL in the high-level school are relatively the same as the medium-

level school.

Interaction between Learning Model and School Levels toward Achievement of Students' MPSA

To determine the interaction between model learning and school levels toward achievement of students' MPSA,

the two-way ANOVA was used. The summary of two-way ANOVA results are presented in Table 7.

57


Table 7. The summary of two-way ANOVA of MPSA achievement

Source Type III Sum of

Squares df Mean Square F Sig.

Corrected Model 1348.474a 3 449.491 9.427 0.000

Intercept 67371.101 1 67371.101 1.413E3 0.000

Learning Model 1177.845 1 1177.845 24.702 0.000

School Level 153.559 1 153.559 3.220 0.075

Learning Model *

School Level 12.156 1 12.156 0.255 0.614

Error 7057.045 148 47.683

Total 76069.000 152

a. R Squared = 0.160 (Adjusted R Squared = 0.143)

Table 7 above shows that the learning model has significant effect on achievement of students' MPSA because

probability value (significance 0.00) is smalless than 0.05. While the school level factors does not has a significant

effect on achievement of students' MPSA. This is indicated by the value of the probability (significance 0.075) is

greater than 0.05. Based on the achievement mean, high and medium-level school students have achievement mean

is relatively closely. Both the results of statistical tests and the achievement mean, it can be concluded that the

achievement of students' MPSA in high-level school same with students in medium-level school.

From Table 7 it can be seen also that the probability value (significance) for the interaction between learning

model and school levels is 0.614. Significance value is greater than 0.05, so the null hypothesis is accepted.

Therefore, it can be concluded that there is no interaction between learning model and school level toward

achievement of students' MPSA. Graphically, the interaction between learning model and school levels toward

achievement of students' MPSA can be seen in Figure 3.

Figure 3. Interaction between learning model and school levels toward achievement of students' MPSA

In Figure 3 it can be seen that ICM students in the high and medium-level school gain the achievement mean of

MPSA higher than those of CL students. The achievement mean of ICM students' MPSA sequence from largest to

smallest are high-level school students and medium-level school students. The same sequence occurs in the

achievement mean of CL students' MPSA. The similarity sequence suggests that there is no interaction between

learning model and school levels toward achievement of students' MPSA.

Figure 3 above also shows that the difference between the achievement mean of ICM and CL students' MPSA in

high-level school almost the same as the medium-level school. The similarity of the difference in the


achievement mean of MPSA can be a strong indication that there is no interaction between learning model and

the school levels toward achievement of students' MPSA.

Discussion

In terms of overall students, the enhancement of MPSA students who received ICM is 0.53, while the students

who received CL only reaches 0.34. Although both of the enhancements are categorized into medium level

according to the classification of Hake, but the statistical test results confirms that the enhancement of MPSA

students who received ICM is significantly better than those of students who received CL.

The enhancement of students' MPSA is also supported by its achievement. The achievement of students' MPSA

for overall of students who received ICM is 23.96 or about 63.05% of the maximal ideal score. While the

achievement of students' MPSA for overall of students who received CL is only 18.38, or approximately

48.37% of the maximal ideal score. The statistical test results concluded that the achievement of students'

MPSA who received ICM is better than students who received CL. This suggests that learning model has a

significant effect on students' enhancement and achievement MPSA.

The strong effect of learning model on the enhancement and achievement of students' MPSA can also be seen in

each category of school levels. The ehancement and achievement mean of MPSA ICM students for each school

level category is greater than the CL students. The ehancement of MPSA ICM students on high-level school is

0.56 which classified as medium and the achievement is 24.63 or about 64.82% of the maximal ideal score.

Meanwhile, the enhancement of MPSA CL students on high-level school is only 0.39 which classified as

medium and the achievement is 19.61 or about 51.61% of the maximal ideal score.

In the medium-level school, the enhancement of students' MPSA who received ICM 0.52 classified as medium

with the achievement of 23.18 or about 61% of the maximal ideal score. Meanwhile the enhancement of

students' MPSA who received CL is 0.31 classified as medium with the achievement for 17.03 or about 44.82%

of the maximal ideal score. By using t-test, it can be concluded that for all categories of school levels, the

enhancement and achievement MPSA ICM students are better than those of CL students. This suggests that

ICM is superior to learning in terms of developing the students' MPSA than CL in all categories of school

levels.

One of the findings in this study revealed that the school level factor has significantly affect toward

enhancement of students' MPSA. This means that school level factors contribute to determine the difference in

enhancement of students' MPSA. These findings can be seen from the the enhancement mean of students'

MPSA in high and medium-level school. Students in a high-level school obtain a much larger enhancement than

students in medium-level school. These findings suggest that the higher students' school level, the higher

enhancement of MPSA obtained by students. In other words, in terms of enhancement of MPSA, students in the

highlevel school more benefit from their lessons than those of students in medium-level school. These results

are consistent with findings in research reported by Kesumawati (2010) which revealed that the school levels

factors provide a strong effect on the enhancement of students' mathematical problem solving ability.

Another finding in this study revealed that suggest that the school level factor does not has significantly affect

toward achievement of students' MPSA. It means that the differences achievement of students' MPSA are not

influenced by school level factors. It can be seen from the the achievement MPSA of students in both of school

levels categories (high and medium). The achievement of MPSA students in high-level school is relatively the

same as the students in medium-level. All these may suggest that students at any school level get equally benefit

from ICM. Thus, ICM suitable for use in high and medium-level school.

The results of this study demonstrate that compared to the CL, ICM is better in facilitating the learning of

students in training and developing mathematical problem solving skills. This is possible because the ICM

students are required to be actively involved in investigating, studying, and finding their mathematical concepts

learned through situations or problems presented in the student worksheets. Bruner (Hudojo, 1998) suggested

that students should learn by active involvement in learning the concepts and principles in solving problems. In

this case, the teacher as a facilitator of learning, helping students to build knowledge of students who may be

found by solving the problem.

59


In ICM, students actively involved in the investigation, settlement of mathematical problems, and the discovery

of mathematical concepts, so that the mathematical concepts learned can be understood deeper and remembered

longer by the students. In general, ICM learning requires students to form small groups, and to discuss, to

investigate, to explore ideas, and to write down their perceptions related situations or matematical problems

presented in the worksheet. Furthermore, students are asked to present the results of focus group discussions in

class individually. Then the students are asked to complete some challenging questions and to solve the

problems in form of quizzes as learning evaluation.

This research notes that there was no significant interaction between learning model and school levels toward

enhancement and achievement of students' MPSA. This suggests that jointly the interaction between learning

model and school levels does not provide a significant effect on the differences of enhancement and

achievement of students' MPSA. The differences of enhancement and achievement of students'MPSA only due

to differences learning model factor. In other words, at any school levels categories, the enhancement and

achievement of MPSA students who received ICM better than those of students who received CL.

Conclusion

Referring to the results of research and discussion above, it can be concluded that the enhancement of students'

MPSA who received ICM is better than students who received CL. Overall, the enhancement of ICM students'

MPSA is classified as medium with the achievement percentage classified as medium. The enhancement of CL

students is relatively medium but the achievement percentage is classified as low.

For each category of school levels, the enhancement and achievement of MPSA ICM students are better than

those of CL students. In high-level school, the enhancement of MPSA ICM students classified as medium with

the achievement percentage is classified as medium. While the enhancement of MPSA CL students is relatively

medium but the achievement percentage is classified as low. In medium-level school, the enhancement of

MPSA ICM students is classified as medium with the achievement percentage is classified as medium. While

the enhancement of MPSA CL students is relatively medium but the achievement percentage is classified as

low.

This research shows that school level factors has significantly affect toward enhancement of students' MPSA,

but does not has significantly affect toward achievement of students' MPSA. This means that school level

factors contribute to give effect to the differences of MPSA enhancement that occur between ICM and CL

students. Based on the enhancement mean of students' of MPSA, it is known that the higher students' school

level, the higher the students' MPSA enhancement. It stands to reason because the students in a high-level

school generally have mathematical ability which is relatively higher than those of students in a medium-level

school. Therefore, one of the factors that must be considered by teachers or researchers to enhancing students'

MPSA is students' school levels factor.

In addition, known also that there is no interaction between learning model and school levels toward

achievement and enhancement of students' MPSA. Its means that jointly the interaction between learning model

and school levels does not provide a significant effect on the differences enhancement and achievement of

students' MPSA.

Recommendations

Situations or mathematical problems presented in the worksheet is the focus of investigation of the students

working in the groups. Teachers, in this case, are suggested to create situations or problems in accordance with

the level of mathematical thinking skills of the students so that learning materials can encourage all students to

think and actively involved in learning process.

One of the obstacles faced during teaching and learning process is limited learning time. To overcome this,

students should learn problems provided in the individual worksheets in advance at home. Thus, during a lesson

in class, students will be more familiar with the material to study, problems in the worksheet they still find

difficult. Students will understand what to do during the learning process in the classroom. Through this way, it

is expected that the discussion group will not be time consuming.


References

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Anderson, J. (2009). Mathematics curriculum development and the role of problem solving. ACSA Conference

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Meltzer, D.E. (2002). Addendum to: the relationship between mathematics preparation and conceptual learning

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http://www.physics.iastate.edu/per/docs/Addendum_on_normalized_gain.

Nasution, S. (2000). Berbagai pendekatan dalam proses belajar mengajar. Jakarta: Bumi Aksara.

NCTM. (2003). Program for initial preperation of mathematics specialists. Retrieved from

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Pujiastuti, H. (2012). Analisis kemampuan pemecahan masalah dan komunikasi matematis siswa SMP di kota

Serang. Preliminary Study Report: unpublished.

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