Developing a New Paper Structure For The G.C.E.(O/L) Mathematics Paper Using Graph Theory

Abstract

This study focuses on developing a new paper structure for the G.C.E.(O/L) Mathematics paper using Graph Theoretical concepts. The graph theoretical model is built based on the competency levels of the O/L (Ordinary Level) Mathematics syllabi and their weights. Centrality measures such as Weighted Degree Centrality, Betweenness Centrality, Authority and Hub are used to analyze this model. The open source software Gephi is used to do the analysis. The new paper structure is designed based on the dominancy of each competency and the suggested weight of each Mathematical Theme.

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International Journal of Scientific and Research Publications, Volume 10, Issue 8, August 2020 426
ISSN 2250-3153
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Developing a New Paper Structure For
The G.C.E.(O/L) Mathematics Paper Using Graph Theory
G.H.J. Lanel*, P.W.C.E. Pathirana**
* Department of Mathematics, University of Sri Jayewardenepura, Sri Lanka.
** Department of Mathematics, University of Sri Jayewardenepura, Sri Lanka.
DOI: 10.29322/IJSRP.10.08.2020.p10451
http://dx.doi.org/10.29322/IJSRP.10.08.2020.p10451
Abstract- This study focuses on developing a new paper structure for the G.C.E.(O/L) Mathematics paper using Graph Theoretical
concepts. The graph theoretical model is built based on the competency levels of the O/L (Ordinary Level) Mathematics syllabi and
their weights. Centrality measures such as Weighted Degree Centrality, Betweenness Centrality, Authority and Hub are used to
analyze this model. The open source software Gephi is used to do the analysis. The new paper structure is designed based on the
dominancy of each competency and the suggested weight of each Mathematical Theme.
Index Terms- G.C.E.(O/L) Mathematics, Graph theory, Centrality measures
1. INTRODUCTION
1.1 Background of Mathematics Education in Sri Lanka
The modern education system in Sri Lanka was introduced by the British in the 19th century. Since then, Mathematics has been taught
in schools in different levels and in different modules by either making it compulsory or optional. But in 1972, Mathematics was made
compulsory up to SSC (Secondary School Certificate) where nowadays call as G.C.E.(O/L). After several amendments in the
curricula, in 2006, the Ministry of Education (MOE) in Sri Lanka introduced the Competency Based Curriculum incorporating the 5E
learning cycle (5E- Engage, Explore, Explain, Elaborate and Evaluation). Parallel to the curriculum reformation, questions relating to
real life scenarios were added to the O/L paper from 2008. Mathematics taught in junior secondary (year 6 to 9) has many applications
in day to day activities. But most of the lessons covered in year 10 and 11 lay out a foundation for higher mathematics.
1.2 Current Stage of G.C.E.(O/L) Mathematics
According to statistics released by Department of Examinations (DOE) the failure rate from 2011 to 2015 was in between 40% to
45%. The pass rate has improved slightly after 2015, but still the failure rate is in between 30% to 40%. In 2017, 53.68% of students
have scored less than 40 marks in the O/L examination. High failure rate in the G.C.E.(O/L) Mathematics is a huge problem that the
educators and the authorities should look into. MOE has taken a number of steps to reduce the failure rate in O/L. Back in 1999, DOE
has had two examination papers at two levels of difficulty. Then in 2001, Multiple Book Option (MBO) was introduced by MOE.
Again in 2005, DOE conducted a mock examination for Mathematics. But all these initiatives did not last long. According to
Mampitiya (2014) the O/L results can be improved by a properly designed examination paper.
2. METHODOLOGY
The current mathematics syllabus is built on 31 competencies. When generating the graph theoretical model, these competencies were
taken as nodes and the links between each competency were represented by edges. These edges were given weights based on how
much they are connected with each other. The dominancy of each competency in the directed weighted graph was analyzed using
Gephi under the statistics; weighted degree, weighted in-degree, weighted out-degree, betweeness centrality, authority and hub. Points
were given for each competency (node) according to their value of the statistic. Most dominant competencies were identified and
percentage (suggested weight) of each mathematical theme was calculated based on Node Values. A new paper structure was
designed based on the suggested weights.

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3. RESULTS AND DISCUSSION
Figure 3.1: Graph Theoretical Model
Figure 3.1 is the graph obtained after considering competencies as nodes and the links between them as edges. It can be clearly seen
that Solids, Tessellation and Symmetry have no link with the other nodes. These three topics are taught in year 6 to 8 and the content is
not strong to be tested in the O/L exam. Therefore the above three nodes were removed from the further analysis.
Figure 3.2: Ranking using Weighted Degree Centrality
According to the weighted degree centrality Solving Equations is the dominant node. Geometry-Rectilinear and Real Numbers are the
next two important nodes. But on the other hand, Sets and Probability have only one link with the others. A node with high weighted
degree indicates that the node is having a lot of links or strong links with the other nodes. Therefore, topics like Solving Equations,
Geometry-Rectilinear and Real Numbers should be given more attention in teaching and learning process.

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Figure 3.3: Ranking using Weighted In-degree Centrality
When the weighted in-degree is considered, Solving Equations has the highest in-degree centrality. Number Patterns, Volume and
Time are some of the next few important nodes. Since no topics are applied in Real Numbers, Fractions, Changing the Subject,
Angles, Mass and Liquid Measures, they have no weighted in-degree. The questions that are set on these six topics can be addressed
using what is taught under that topic, because they have no incoming links. Questions on topics that are having high weighted in-
degree might be difficult for the students as they need to be familiar with all the topics linked into those topics.
Figure 3.4: Ranking using Weighted Out-degree Centrality
The node Real Numbers is dominant when the weighted out-degree is considered. The next two important nodes are Changing the
Subject and Geometry_ Rectilinear. It is possible to form individual questions on topics such as Number Patterns, Volume, Time,
Perimeter, Percentages, Interpretation of Data, Scale Diagrams and Loci Construction because the weighted out-degrees of those
nodes are zero. Studying thoroughly on topics having high weighted out-degree will be very much beneficial for the students as those
are applied in many other topics linked.

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Figure 3.5: Ranking using Betweenness Centrality
There are few nodes acting as bridges in this model. Among those Solving Equations is the most dominant node. Nodes acting as
bridges are important to merge topics when generating questions. The betweenness centrality of Solving Equations is very high
compared to other nodes. Therefore, students should be familiar with the lessons that are covered around Solving Equations. When the
three most dominant nodes are considered, it is noticeable that they belong to three different themes in the syllabus. Solving
Equations belongs to Algebra, Geometry_Rectilinear belongs to Geometry and Logarithm belongs to Numbers.
Figure 3.6: Ranking using Authority values
Nodes that contain useful information on a topic of interest are defined as authorities. But in this setting, a node having high authority
value emphasize that, that particular topic uses the information of many more topics. Solving Equations, Number Patterns and Volume
are the nodes that are having high authority values correspondingly. When the students study, they should practice more on topics
having high authority values. The paper setters can select these high authoritative topics to test not only on the selected topic but the
topics linked into those as well, rather than giving another question to test the linked topics. This can be used to reduce the number of
questions in a test paper.

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Figure 3.7: Ranking using Hub values
Hubs are nodes that tell where the best authorities can be found. The hub value of a node is high, if it is connected to a good authority.
According to figure 3.7, Real Numbers and Fractions are the two main hubs. Their hub values are high, not only because they have lot
of links but also because they are linked into good authorities such as Solving Equations, Number Patterns and Volume.
After analyzing the graph based on the six statistics that were selected, nodes were given points according to their value of the
statistic. After total points of each node was calculated, Solving Equations was identified as the most dominant node. Geometry_
Rectilinear, Logarithms, Number Patterns and Real Numbers lie next in the sequence. Students, teachers and paper setters can identify
the above topics as most powerful areas in the syllabus.
Questions in the O/L mathematics paper fall under six themes namely, Numbers, Measurements, Algebra, Geometry, Statistics and
Sets & probability. Therefore the percentages of the six mathematical themes were calculated based on the total node values.
Table 3.1: Percentages of the mathematics themes
Theme
Node value Total
Percentage
Numbers
226
25.22321
Measurements
188
20.98214
Algebra
287
32.03125
Geometry
125
13.95089
Statistics
51
5.691964
Sets & Probability
19
2.120536
Since the percentages of the themes Statistics and Sets & probability are both less than 10%, the two themes were combined and
called as Statistics.
Table 3.2: Comparison between the existing figures and the suggested figures
Theme
Percentage in the
existing structure
Suggested
Percentage
Numbers
22%
25%
Measurements
17%
21%
Algebra
19%
32%
Geometry
21%
14%
Statistics
21%
8%
According to table 3.2, the weight of Algebra should be increased up to 32% and it is a major change that should be done while
making the exam paper. Also the weight of Statistics should not be more than 8% and weight of Geometry should be decreased by
7%.

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Table 3.3: Existing Structure of the Mathematics paper
32 - Mathematics
Paper I
2 hours
100 marks
Paper II
3 hours
100 marks
Part A
Part B
Part A
Part B
25 short answer
questions
5 Structured questions
6 Structured questions
Should select 5 to answer
6 Structured questions
Should select 5 to answer
(25×2marks)
(5×10marks)
(5×10marks)
(5×10marks)
Questions from the entire
syllabus
No questions on Algebra
& Geometry
No questions on
Geometry
3 Questions on Algebra
3 Questions out of
Numbers,
Measurements, Statistics
and Probability
No questions on Algebra
3 Questions on
Geometry
3 Questions out of
Numbers,
Measurements, Statistics
and Probability
The table 3.3 gives an overview of the existing structure of the mathematics paper.
Table 3.4: Compulsory subjects with their time duration
Compulsory Subject
Time Duration for both
paper 1 & paper 2
Religion
3 hours
First Language *
3 hours
English
3 hours
Mathematics
5 hours
History
4 hours
Science
4 hours
*First language consists of a third paper which is a literature paper in which the duration is 2 hours.
According to the above table it is clear that students have to be occupied more on mathematics. This again shows that mathematics is
given more weight amongst the other subjects. So at least time duration for mathematics should be reduced up to 4 hours for the
weights of the subjects to be fairly distributed.
The proposed structure for the O/L mathematics paper is as follows.
Table 3.5: Suggested Paper Structure
Paper 1
Paper 2
100 marks
100 marks
2 hours
2 hours
50%
50%
Answer all questions
Answer all questions
Type of questions:
 Short answer
 Structured
Type of questions:
 Structured
 Un-Structured
Level of Complexity of the questions:
 Low
 Moderate
Level of Complexity of the questions:
 Moderate
 High
Distribution of Themes :
 Numbers – 25 marks
 Measurements – 21 marks
 Algebra – 32 marks
 Geometry – 14 marks
 Statistics – 8 marks
Distribution of Themes :
 Numbers – 25 marks
 Measurements – 21 marks
 Algebra – 32 marks
 Geometry – 14 marks
 Statistics – 8 marks
Both Short answer questions (SAQ) and Structured questions (SQ) should be included in the first paper to maintain the standard of the
paper, and in the second paper SQs and Unstructured questions (USQ) can be included, where USQs can be used to filter the best

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students to do Advanced Level Mathematics. Also the paper setters should be given the freedom to decide the number of questions
that can be put in the exam paper while maintaining the weights proposed in this article and maintaining the standard of the paper.
4. CONCLUSION
According to the centrality analysis Solving Equations is the most dominant competency. Geometry_Rectilinear and Logarithms are
next in order. Students can use the graph theoretical model used for the analysis to see how the topics are connected, how important
each topic is etc. and prepare themselves better for the exam. According to the analysis, more weight should be put on the theme
Algebra and one fourth of the paper should be based on the theme Numbers. It was also suggested to combine the themes Statistics
and Sets & Probability. The time duration of the mathematics paper was suggested to be reduced up to 4 hours.
In the proposed first paper, there should be short answer questions and structured questions with low and moderate complexity and
structured and unstructured questions with moderate and high complexity in the second paper. But both the papers should carry
questions on Numbers with 25 marks, Measurements with 21 marks, Algebra with 32 marks, Geometry with 14 marks and Statistics
with 8 marks. It should also be compulsory to answer all the questions.
REFERENCES
[1] An Investigation in to the Nature of the School Based Assessment Program Implemented in G.C.E.(O/L) Classes (2015), Department of Research and
Development, National Institute of Education.
[2] Freeman, L.C. (1977), A Set of Measures of Centrality Based on Betweenness. Sociometry, Vol.40, No.1, pp.35-41.
[3] G.C.E.(O/L) Examination- from 2016-paper structure and model papers (2016), Research and Development Branch, Department of Examinations.
[4] Gephi (2017), Available at https://gephi.org/about/
[5] International GCSE Mathematics Specification B_9-1. (2016), Pearson Education Limited.
[6] Mampitiya (2014), Mathematics Education – Past, Present and Future, J.E.Jayasuriya Memorial Foundation, Colombo.
[7] Mathematics - Essential Learning Concepts (2015), National Institute of Education
[8] Mathematics Framework for the 2019 National Assessment of Educational Progress, National Assessment Governing Board, U.S. Department of Education.
[9] Mathematics Grade 10 Text Book (2016), Educational Publications Department.
[10] Mathematics Grade 11 Text Book (2016), Educational Publications Department.
[11] Mathematics Grade 9 Text Book (2016), Educational Publications Department.
[12] Strengthening Mathematics Education in Sri Lanka(2011), Report No.43, South Asia Human Development Sector
AUTHORS
First Author – Dr.G.H.J.Lanel, Department of Mathematics, University of Sri Jayewardenepura, Sri Lanka, ghjlanel@sjp.ac.lk
Second Author –P.W.C.E. Pathirana, Department of Mathematics, University of Sri Jayewardenepura, Sri Lanka,
chala.eshanee@gmail.com
Correspondence Author – P.W.C.E. Pathirana, Department of Mathematics, University of Sri Jayewardenepura, Sri Lanka,
chala.eshanee@gmail.com