The Application of Discovery Learning Method and Small Group

Discussion in PAM – 472 Topics in Combinatorial Mathematics II

Lyra Yulianti

and Syafruddin

Department of Mathematics, Andalas University, Padang, Indonesia

Keywords: Discovery Learning, Small Group Discussion.

Abstract: This research concerns classroom action research conducted in the academic year 2017 – 2018, involving

undergraduate students who took the Topics in Combinatorial Mathematics II Course, an eighth semester

course in Department of Mathematics, Faculty of Mathematics and Natural Science in Andalas University. In

this research, we combined two learning methods, namely Discovery Learning and Small Group Discussion

to increase the ability of the students to understand the course material. By comparing the final grade in the

academic years 2016 – 2017 with 2017 – 2018, we found that this combination of the methods successfully

improved grades.

1 INTRODUCTION

The course PAM 472 Topics in Combinatorial

Mathematics II is an elective course in the

Combinatorial Mathematics Research Group. The

course is a 3 hour a week course (3 SKS) at level IV

(semester VIII). The prerequisite courses for this

elective course are two other elective course taught by

the Combinatorial Mathematics Research Group, i.e.

PAM 271 Introduction to Graph Theory, given at

level II, semester III, and PAM 272 Discrete

Mathematics (given at level II, semester IV).

Another course related to Topics in Combinatorial

Mathematics II is PAM 471 Topics in Combinatorial

Mathematics I, which is given at level IV (semester

VII), but this course is not a prerequisite for students

who will take the Topics in Combinatorial

Mathematics II course because both courses appeared

in the 2015 – 2016 academic year. Topics in

Combinatorial Mathematics II is not a continuation of

Topics in Combinatorial Mathematics I.

In Topics in Combinatorial Mathematics II, we

focus on understanding some of the latest results in

the field of graph theory, namely (a) the metric

dimension, (b) the partition dimension and (c)

locating chromatic-number of a graph.

We provide several definitions, theorems, and

their proofs, as well as detailed explanations through

examples. The course materials are some recent

articles related to the topics given, as well as some

lecture handouts which contain summaries of articles

in the previously mentioned topics (a), (b) and (c).

After attending this course, the students are

expected to have a strong understanding of the

concepts of the metric dimensions, the partition

dimensions and the locating-chromatic number of a

given graph. Furthermore, the students are expected

to be able to use the concepts required to determine

the metric dimensions, partition dimensions and

location chromatic numbers of a given graph

themselves. It is expected that students can think

critically, analytically and innovatively, structure

arguments logically, and be able to communicate

their thoughts systematically, be able to work

together and adapt themselves to other students in the

group and conduct some good discussions.

In the academic year 2016 – 2017, fifteen students

took the course Topics in Combinatorial Mathematics

II. In that semester, the lecturer applied the Small

Group Discussion (SGD) method as follows.

Students were divided into five groups, where the

students themselves determined members of each

group. After the basic concepts of each topic was

provided by the lecturer, the lecturer gave

assignments to each group to be presented in the next

meeting. Each group was directed to search for an

article related to the topics discussed, in international

or national journals, and then give a presentation of

their understanding of the article. The lecturer chose

the presenting group randomly so that each group had

to be well prepared for each presentation assignment.

Yulianti, L. and Syafruddin, .

The Application of Discovery Learning Method and Small Group Discussion in PAM – 472 Topics in Combinatorial Mathematics II.

DOI: 10.5220/0008679400570063

In Improving Educational Quality Toward International Standard (ICED-QA 2018), pages 57-63

ISBN: 978-989-758-392-6

During the group presentation the lecturer acted

as a facilitator and moderator for the discussion. The

lecturer assessed the presenting group on their

understanding of the article material. The lecturer

also assessed the attitudes and presentation technique.

The presentation material was then collected after the

presentation was complete.

The lecturer provided assessments to students

from non-presenter groups, based on their activeness

in responding to the first group's presentation.

Assessment was made of all groups when they

presented their major assignments in the last three

weeks of the semester. This presented the results of

the group's determination of (a) the metric dimension,

(b) the partition dimension or (c) the location

chromatic number of a graph chosen by each group.

The lecturer provided assessment criteria to

measure the student’s learning outcomes as listed in

Table 1.

Table 1. Assessment Criteria

Assessment Components

Bobot

(%)

Assessment of Results

Mid-test

30 %

Final-test

30 %

Assessment of Process

The ability to think critically and

logically

20 %

Analytical ability

10 %

Ability to cooperate in teams

10 %

TOTAL

100 %

The distribution of the final grade in the 2016 – 2017

academic year is given in Figure 1.1.

Fig.1 The Final Grade of PAM 472 in 2016 – 2017

Academic Year.

From Figure 1.1 it can be seen that the distribution of

the students’ final grades is not satisfactory, because

10 out of 15 students scored less than B+. The lecturer

thought that one reason for the unsatisfactory results

was the inappropriate application of learning

methods. Therefore, in the 2017–2018 academic year,

we combined two learning methods, the Small Group

Discussion (SGD) method and the Discovery

Learning (DL) method. We hoped that the

combination would increase the students’ level of

understanding of the course materials and increase

their final grades eventually.

2 THE SMALL GROUP

DISCUSSION AND DISCOVERY

LEARNING

This section contains brief definitions of SGD and the

DL methods.

2.1 Small Group Discussion

SGD is a process of learning that takes place when

students work together in groups of 4 – 5.

It is a

learning method that spurs student activity. The

lecturer presents the course materials, and then the

issues to be discussed are given presented as a whole.

Next, the problem is divided into several sub-

problems to be solved by each group. After discussion

in the group, representatives of each group present the

results of the discussion.

Meo (2013) stated that over the last four decades,

SGD has achieved an admirable position in education

and is well-liked as a means of encouraging students

and enhances the process of deep learning. SGD

increases student interest and retention of knowledge,

enhances the transfer of concepts to novel issues,

students' critical skills, teamwork ability, self-

directed learning, communication skills, and student-

faculty and peer-peer interaction. It provides an

opportunity for articulating thoughts and formulating

views, and also provides a chance for the students to

monitor their learning and gain experience of self-

direction and independence from the instructors.

2.2 Discovery Learning

DL is a teaching method that governs teaching in such

a way that students gain knowledge that they have not

previously known not directly though instruction but

partially or wholly by themselves. In DL, activities or

learning are designed in such a way that students can

discover concepts and principles through their mental

processes. In finding concepts, students observe,

ICED-QA 2018 - International Conference On Education Development And Quality Assurance

classify, make guesses, explain, and draw conclusions

to construct concepts or principles for themselves.

Bruner (1961) is often credited with originating

discovery learning in the 1960s. He argued that

practice in discovering for oneself teaches one to

acquire information in a way that makes that

information more readily viable in problem-solving.

This philosophy later became the basis of the

discovery learning movement of the 1960s. The

mantra of this philosophical movement suggests that

we should 'learn by doing'.

The DL method is defined as a teaching procedure

that emphasizes teaching manipulation of objects

before reaching generalization. Discovery carried out

by students to find a concept or principle. It is a series

of mental processes that enables students to assimilate

a concept or principle. The mental processes in

question include: observing, digesting,

understanding, classifying, making assumptions,

explaining, measuring, and making conclusions. With

these technique students are allowed to find out things

by themselves using their own mental processes. The

lecturers only guide and provide instructions. It

involves students in brainstorming, discussing,

reading by themselves and trying things out, so that

they can learn by themselves.

As the lecturer only acts as a mentor and

facilitator to direct students to find concepts,

propositions, procedures, algorithms on their own DL

is a teaching method that focuses on student activities

in learning.

2.3 Class Action Research Parameters

This research, uses the following parameters.

a) Student Learning Outcomes Results

Learning outcomes are measured by (i) the questions

posed by the lecturer to the presenting groups and to

the listening groups, (ii) the mid-test and final-test

results which measure the competency level of the

students against the semester learning plan objectives.

In (i), the competencies measured are psychomotor

abilities, while in (ii), the competencies measured are

cognitive and affective abilities. The results are

processed in order to obtain grades at the end of the

semester.

b) Student response to the implementation of the

learning method

To find out whether the students consider the learning

method to be effective, the lecturer conducted a

student survey at the end of the semester. This survey

data illustrated how students responded to the

learning methods and helps direct any follow up

action based on the findings in this class research.

3 RESEARCH METHODOLOGY

This research was conducted in class within one

semester, in the 2017 – 2018 academic year, and

involved students who took Topics in Combinatorial

Mathematics II courses. This research is qualitative

research, conducted by observing students and

participating in class actions. Researchers acted as

observers and students as observed objects. In this

research, we combined the Discovery Learning and

Small Group Discussion methods to increase the

ability of the students to understand the course

material. A comparison of the final grade of this

course with the same course run in the previous year

was used to determine the success of the method.

4 RESULTS AND DISCUSSIONS

In the academic year 2017 – 2018, there were twenty

students in Topics in Combinatorial Mathematics II

class. The lecturer applied the combination of Small

Group Discussion (SGD) method and Discovery

Learning as follows. Students were divided into 7

(seven) groups, where members were assigned to

each group randomly and stayed in the same group

until the end of the semester. The course materials

were similar to the previous academic year: (a) the

metric dimension, (b) the partition dimension and (c)

the locating-chromatic number of a graph.

After the basic concepts of the material in each

topic was explained the lecturer gave assignments to

each group that was to be presented in the next

meeting. Each group was given a different type of

graph and was assigned to determine the value of (a)

the metric dimension, (b) the partition dimension, and

themselves. The presentation was carried out by one

group for 40 minutes; every student in the group was

required to play a role in the presentation. After the

presentation, a discussion was held regarding the

presented material.

As in the previous academic year, the lecturer

acted as a facilitator and moderator during the

presentations and in the discussion. The lecturer

assessed the presenting group based on (i) their

understanding of the material that they found and (ii)

the attitudes and presentation technique of the group.

The lecturer observed the ability of the presenters to

cooperate in teams, their logical arguments, and their

analytical skills. The lecturer also provided

assessments to students in non-presenter groups,

The Application of Discovery Learning Method and Small Group Discussion in PAM – 472 Topics in Combinatorial Mathematics II

based on their activeness in responding to the first

group's presentation.

Presentation assignments began in the fifth

meeting. Reference materials are left up to students,

but the primary references are Chartrand (1998; 2000;

2002). All basic definitions and notations in graph

theory used in this class are taken from Meo (2013).

Table 2, lists the presentation topics given to

every group. These are similar to those given in the

2016 – 2017 academic year.

Table 2: Group Presentation Material.

Week

Material

Group

On the metric dimension of some

graphs

On the partition dimension of some

connected graphs

On the partition dimension of some

disconnected graphs

III

On the locating chromatic number of

some connected graphs

On the locating chromatic number of

some connected graphs

On the locating chromatic number of

some disconnected graphs

On the locating chromatic number of

some disconnected graphs

VII

On the metric dimension, partition

dimension and locating a chromatic

number of some connected graphs

I, II,

III, IV

On the partition dimension and

locating a chromatic number of some

disconnected graphs

V, VI,

VII

There were also additional tasks given to every group

to be finished during the meeting. If the assignment

was not completed in class, then the assignment was

used as homework. The lecturer observed the

students' ability to work together in teams, think

critically and analyze problems.

In table 3, we list the tasks given to every group.

These are similar to those given in the 2016 – 2017

academic year. The assessment rubric is displayed in

table 4.

Table 3: Tasks.

Task

Due

date

Give some examples of graphs G and

H that fulfill the condition G, H,H ,G

meeting

Find the metric dimension of graphs

, G

, and G

meeting

Find the metric dimension of F

, F

and F

, where G is an arbitrary graph

on n vertices

meeting

Find the partition dimension of cycle

and wheel W

meeting

Find the partition dimension of F

, F

and F

, where G is an arbitrary graph

on n vertices

meeting

Find the partition dimension of

disconnected graphs kP

, K

1,n t

for some k, t, and m

meeting

Find the locating chromatic number of

graphs F

, F

and F

, where G is an

arbitrary graph on n vertices

meeting

Find the locating-chromatic number

of disconnected graphs kP

, K

1,n

for some k, t, and m

meeting

ICED-QA 2018 - International Conference On Education Development And Quality Assurance

Table 4: Assessment Rubric.

Grade

Score

Performance Indicator

Poor

No clear discussion is written

(included in this category, the

students that did not collect their

tasks)

Less

21 –

There are discussions put forward,

but only in small parts, only what is

written in the textbook

Standard

41 –

The discussion presented was clear

enough, covering the entire task

order, but less innovative (in the

sense of only translating textbooks)

Good

61 –

The discussion is quite clear,

covers the whole, but not too broad

(in a sense, there should be more

than just translating textbooks)

Very

Good

The discussion is clear, covers the

whole, innovative and broad

Figure 3 – Figure 4.10 compares presentation, tasks,

mid-test, final test, and the final grade results between

2016 – 2017 and 2017 – 2018 academic years.

Figure 3: Presentation Assessment in 2016 – 2017

Academic Year.

Figure 4: Presentation Assessment in 2017 – 2018

Academic Year.

From Figure 3 - Figure 4, it can be seen that the

average grade for student presentations in the 2017 –

2018 academic year was lower than in the 2016 –

2017 academic year. After being evaluated at the end

of the semester, the problem occurred because the

students focused more on the more massive

presentation material, with a narrow preparation time

of only one week, because they have to find the metric

dimensions, partition dimensions and to locate

chromatic numbers of a new graph chosen by

themselves rather than by reading the papers. In the

future, it is planned that the lecturer would give the

presentation material two or three weeks before. It is

hoped with longer preparation time; students can

prepare their presentations better.

Figure 5: Tasks Assessment in 2016 – 2017 Academic

Year.

Figure 6: Tasks Assessment in 2017 – 2018 Academic

Year.

Figure 5 – Figure 6 show that the average grades for

of student tasks in the 2017 – 2018 academic year is

higher than in the 2016 – 2017 academic year. After

being evaluated at the end of the semester, the

students said that they felt challenged to find

something new, namely the metric dimension,

partition dimension and locating a chromatic number

of new graphs, and always eager to do all the

assignments given.

The Application of Discovery Learning Method and Small Group Discussion in PAM – 472 Topics in Combinatorial Mathematics II

Figure 7: Mid Test Assessment in 2016 – 2017 Academic

Year.

Figure 8: Mid Test Assessment in 2017 – 2018 Academic

Year.

From Figure 7 – Figure 8, the average value of the

mid-test exam in the 2017–2018 academic year was

higher than that of the 2016 –2017 academic year.

After being evaluated at the end of the semester, the

students said that because they are pushed to find the

metric dimension, partition dimension and locating a

chromatic number of graphs they chose themselves,

and then explained their discovery in their

presentation they understood the course material

much better.

Figure 9: Final Test Assessment in 2016 – 2017 Academic

Year.

Figure 10: Final Test Assessment in 2017 – 2018 Academic

Year.

From Figure 9 – Figure 10, it can be seen that the

average value of the final test exam in the 2017 –

2018 academic year was higher than that of the 2016

– 2017 academic year. After being evaluated at the

end of the semester, the students said, as with their

mid-tests, it was because they were pushed to find the

metric dimension, partition dimension and to locate a

chromatic number of some graphs they chose

themselves, and then explained their discovery in

their presentation they understood the course material

much better.

Figure 11: Final Grade in 2016 – 2017 Academic Year.

Figure 12: Final Grade in 2017 – 2018 Academic Year.

ICED-QA 2018 - International Conference On Education Development And Quality Assurance

Figure 11 – Figure 12, compares the final grade in

the 2017 – 2018 academic year with the final grade in

2016 – 2017 academic year. The final score was

obtained from 20% presentation assessment, 20%

task and activeness assessment, 30% mid-test score

and 30% final test score, according to the assessment

criteria in Table 1.1. It can be seen as the percentage

of students with a final grade of less than B+

decreased from 66.67% (10 out of 15 students) to

35% (7 out of 20 students).

5 CONCLUSION

In this research, we combined two learning methods,

the Discovery Learning and Small Group Discussion.

We aimed to increase the ability of the students to

understand the course material. By comparing the

final grade in the academic year 2016 – 2017 and

2017 – 2018, we found that the combination of the

methods was successful as is evidenced by the

decreased percentage of students with a final grade

less than B+.

REFERENCES

Bondy, J.A, Murty, U.S.R, 2010, Graph Theory with

Applications, the Macmillan Press Ltd., New York

Bruner, J. S, 1961, The act of discovery, Harvard

Educational Review. 31(1): 21 – 32

Chartrand, G., L. Eroh, M.S Johnson, O.R. Oellerman,

2000, Resolvability in Graphs and the Metric

Dimension of a Graph, Discrete Appl. Math., 105: 99 –

113

Chartrand, G., Erwin, D., Henning, M., Slater, P., Zhang,

P., 2002, The Locating-chromatic number of a graph,

Bull. Inst. Combin. Appl 36: 89 – 101

Chartrand, G., Salehi, E., Zhang, P., 1998, On the partition

dimension of graph, Cong. Numer. 130: 157 – 168

Meo, S.A., 2013, Basic Steps in Establishing Effective

Small Group Teaching Sessions in Medical Schools,

Pak. J. Med. Sci 29(4) : 1071 – 1076

Moore, R., Lopes, J., 1999. Paper templates. In

TEMPLATE’06, 1st International Conference on

Template Production. SCITEPRESS.

Smith, J., 1998. The book, The publishing company.

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The Application of Discovery Learning Method and Small Group Discussion in PAM – 472 Topics in Combinatorial Mathematics II