GRAPH THEORETICAL AND
NETWORKS APPROACH FOR THE
DEVELOPMENT OF A LEARNING
MODEL – A CASE STUDY
PROF. DR. P. K. SRIMANI
Former Chairman, Dept of Computer Science and Maths, BU, Director, R&D, B.U., Bangalore, India
profsrimanipk@gmail.com
ANNAPURNA S KAMATH*
Former H.O.D, MCA Department, Mount Carmel College, Director, Shishulok, 42, Satyashri, Sampigehalli,Jakkur,Bangalore 560064
anuskamath@hotmail.com
ABSTRACT
This paper deals with the graph theoretical approach for developing a framework for the Learning model used to optimise the Mathematical Pathway in children at the elementary level and verifying it by using Networks model. Data collected pertaining to the mathematical concepts a child needs to learn at elementary level [Class I to VII] is represented by using Concept Flow Graphs and are optimized by using graph theory techniques and algorithms by rearranging nodes as per the learning progression, partitioning the graphs into sub-graphs to represent levels of learning, optimizing the sub-sub-graphs using merging and elimination technique and identifying / marking the optional nodes. The design of the framework by using the graph theoretical approach is validated by the application of the Networks approach and this is used to design the Mathematical Pathway driver which is the core component of the Learning model. This approach is novel and the Learning model developed is highly accurate.
Keywords: Learning Model, Mathematical Pathway, Concept Flow Graphs, Graph optimisation techniques,
Network
1. Introduction
A Learning Model is an integration of the various components used for effective teaching and learning. It is a conceptualization of the learning process. Its purpose is to enhance learning. The Learning model contains the following components: i) The Pathway, Content to be taught as per learning progression ii)The Curriculum, the teaching methods and approaches iii) Implementation strategies iv)Assessment, Tracking student progress v) Identification of Learning abilities and vi)Feedback for refinement of the process. A Learning model can be developed for learning of any subject but in this research it is used specifically for learning of Mathematics. Among the subjects learnt at primary level the most interesting and challenging is Mathematics. It is a core subject with wide applications in all fields specially Engineering, Astronomy, Commerce, Industry and Research.
Mathematics, unlike other subjects has concepts that are very much interrelated yet some can be learnt in parallel. In the Mathematical Pathway the child has scope to make parallel progress in different independent paths and at times has to wait to finish a particular task before moving on to another as he is required to fulfill a prerequisite. This makes learning of Mathematics an interesting case study to apply the Learning Model to optimize the Pathway. A learning model developed for Mathematics with minimum changes can be adopted for any other Learning Pathway. The main objective of the present investigation is to develop an accurate and reliable framework for designing the Mathematical Pathway Driver an integral component of the learning model. The framework is the progressive pathway a child needs to adopt to learn mathematics effectively. Optimizing this pathway ensures that the learning is faster. A Graphical representation is the best way of representing the framework and this is done in detail in the present work. The outcome of the present investigation would have an impact both at the National and the International scenario.
is an ordered pair comprising a set V of vertices or nodes together with a set E of edges or lines. An edge is related with two vertices, and the relation is represented as unordered pair of the vertices with respect to the particular edge. This type of graph is undirected and simple. The vertices belonging to an edge are called the ends, endpoints, or end vertices of the edge. A vertex may exist in a graph and not belong to an edge. A directed graph or digraph is an ordered pair D = (V, A) with V, a set whose elements are called vertices or nodes, and A, a set of ordered pairs of vertices, called arcs, directed edges, or arrows. An arc a = (x, y) is considered to be directed from x to y; y is called the head and x is called the tail of the arc; y is said to be a direct successor of x, and x is said to be a direct predecessor of y. If a path leads from x to y, then y is said to be a successor of x and reachable from x, and x is said to be a predecessor of y. The arc (y, x) is called the arc (x, y) inverted. Graphs are represented graphically by drawing a dot or circle for every vertex, and drawing an arc between two vertices if they are connected by an edge. If the graph is directed, the direction is indicated by drawing an arrow. A graph drawing is a pictorial representation of the graph itself. There are several operations that produce new graphs from old ones, which might be classified into the following categories: Elementary operations called editing operations on graphs, which create a new graph from the original one by a simple, local change, such as addition or deletion of a vertex or an edge, merging and splitting of vertices; Graph rewrite operations replacing the occurrence of some pattern graph within the host graph by an instance of the corresponding replacement graph.; Unary operations, which create a significantly new graph from the old one.;Binary operations, which create new graph from two initial graphs.
Networks are techniques of solving multitude of Operations research problems. As much as 70% of real world mathematical programming can be represented by network related models.(Taha, 2004) In this work, the Mathematical Pathway is represented using a Network and optimization techniques are used on this. A Network consists of a set of nodes linked by arcs or branches. The notation for describing a network is (N,A) where N is the set of nodes and A is the set of arcs. Associated with network is some type of flow. The flow in a network is limited by the capacity of its arcs which may be finite or infinite. An arc is said to be directed or oriented if it allows positive flow in one direction and zero flow in the opposite direction. A directed network has all directed arcs. A path is a sequence of distinct arcs that join two nodes through other nodes regardless of the direction of flow. A path forms a cycle if it connects a node to itself through other nodes. A cycle is directed if it consists of a directed path. A connected network is such that every two distinct nodes are linked by at least one path. Critical path algorithm (CPM) and PERT (Program evaluation and review technique) provide the critical path that affect the performance.
1.1 Literature Survey
Methodologies for Teaching and Learning of Mathematics has been explored by many math educators and researchers for school children. Montessori (1967), Canny (1984), Papert (1991), Resnick et. al(1996) have explored the use of manipulatives. Piaget (1920), Fernald et al(1920), Bloom et al(1956), Dienes (1960), Simpson(1966), Gardner(1983), Kolb(1984), Sidhu(1995), Knsiley(2002) and Costu(2007) have proposed various learning theory and models to be adopted for making math learning effective. Ravalgia et al(1994), Mayer(1998),Banerjee et al(2007) have worked on computer assisted teaching. Work specifically in developing a Mathematical Pathway is by Sarama and Clements(2009) who have worked on progressions in learning of mathematics. In all the above works no Mathematical or computer based approach has been adopted to design the learning pathway. Annapurna(2011) has used Mathematical and Computer Based techniques to optimise the Mathematical pathway in children. Neural Network approach for optimising mathematical learning has been explored by Srimani and Annapurna(2012). In this work the authors have employed data mining and neural network approaches in order to study the performance analysis of learning model and an excellent justification has been found. The present investigation is carried out to build up an efficient framework for design of the Learning model using mathematical techniques.
2. Materials and Methods 2.1 Data Set Description
The data required for the Learning Model is the Mathematical contents that a child has to learn at the elementary level i.e.Preschool to Class 7 [Ages 0 to 12 years]. Data regarding the Mathematical content, Teaching practices, Homework, Difficulties, Gifted and remedial learning and Teacher training was collected. All learning setups were considered to ensure that the data was generic and reflective of various learning environments.
2.1.1Primary Data Source
2.1.2 Secondary Data Source
The main source was the National Curriculum Framework 2005, NCERT. It incorporates the Minimum Levels of Learning (MLL) in Mathematics. Internet was also a very important source as a lot of educational documents were available online. Textbooks, Syllabus of ICSE [Indian Certificate of Secondary Education], CBSE [Central Board of Secondary Education] and SSLC [Secondary School Leaving Certificate] obtained from schools which come under these boards provided the contents. Mathematical frameworks from across the globe including countries like USA, Singapore, Australia were studied and the same served as the data to keep the Learning model generic in nature .
2.1.3 Data Tabulation
Data collected was then categorized into different areas of mathematics. The basic branches of mathematics which are introduced at the elementary level are Arithmetic, Geometry, Algebra, Metrology, Statistics and Commercial Mathematics. About eighteen concepts were identified broadly: Whole numbers, Addition, Subtraction, Multiplication, Division, Fractions, Decimals, Number system, Length, Weight, Volume, Time, Money, Geometry, Mensuration, Algebra, Data Handling and Ratio & Proportion,
2.2 Methodology
2.2.1 Graph Theoretical Approach
The methodology employed for the analysis constitutes the following steps:
Step 1: The data collected from primary and secondary source was tabulated to create the Concept Tables. Step2: The concept tables were used to create the Concept Flow Graphs
Step3:T he concept flow graphs were optimised using Graph Theoretical Algorithms Step4: The optimised flow graphs were converted into Concept Framework Tables
Eighteen concept Tables one each for the concepts was tabulated based on the primary and secondary source of data. For each of the concepts, the topics covered were identified from the various syllabi and frameworks. It was tabulated in the form of a Concept Table (CT). The concept table reflects the various topics dealt with respect to a concept across classes/grades. The concept table has STEP: It is the number reference for a particular topic pertaining to a concept, TOPIC: These are a list of topics that pertain to the Concept to be learnt arranged in the order indicated by the various secondary sources of data.,MLL: It is the Class/Grade by which the concept needs to be mastered by a child. Here the class or grade is expressed as a range as they differ across the globe. The range starts with the earliest class/grade the topic is taught and ends with the latest class/grade.
To automate the progress tracking and assessment of Mathematical learning, the data had to be tabulated in an optimal form. Data had to be tabulated to adhere to the learning progression which is based on the developmental theories and mathematical thinking in children. The concept tables had to be divided into partitions representing different levels of learning for a concept so that identification of the learning level accomplished by a child becomes possible. Data had to be represented in a precise form that can be interpreted with clarity. The data should not have any inconsistency and redundancy.
The Concept flow graph[CFG] was designed using the Concept Table. The CFG is the graph representation of the Concept Table and gives a visual representation of the flow. Here each vertex is a concept and directed edge between the vertices shows the transition. Edges are drawn between vertices where a transition is possible. Based on the observations on mathematical learning and thinking the Concept Flow graph was optimized by ordering as per Learning progression, Partitioning, Merging and elimination.
Start.
Let N be the total number of nodes in the graph. Let i = node1 and j = node2. Compare nodes Ni with Nj.
If Ni must appear after Nj due to Learning progression then
Remove node Ni from its current position
Insert Ni after Nj.
j = i + 1 else let j = j + 1.
If j <= N then go to step 3 If i <= N then
i = i + 1, j = i + 1; go to step 3. Stop
Fig 1 Ordering Algorithm
The graph was then Partitioned based on the MLL. MLLs determine the levels of learning. Wherever the MLL tabulated in the CT were differing from the ones indicated by the developmental theories they were modified. As the graph is already adhering to the learning progression the nodes pertaining to a particular MLL are all connected serially. These nodes belonging to a particular MLL were grouped together to form a level of learning. They were nodes generally associated to a particular age group. The graph was partitioned to form sub-graphs to represent each level of learning. Each sub-graph corresponds to the Topics that need to be completed at one level of learning.
The graph was optimised by reducing its size using Merging and elimination algorithm (Fig 2). The sub-graphs were optimized by reducing the number of nodes. A learning objective is an outcome statement that captures specifically what knowledge, skills, attitudes learners should be able to exhibit following instruction. A learning objective was defined for a topic. Nodes having topics related to this learning objective were merged and nodes that were repetitive or redundant were eliminated using the merging and elimination algorithm.
Start.
Let the total number of nodes in a sub-graph be N. Let i = 1 and j = 2. Define the learning objective corresponding to node Ni
If Ni is redundant then delete Ni , N=N-1, Go to step 6.
Compare node Ni with Nj.
If nodes Ni and Nj contain topics that is a repetition or already addressed then
Nj is a redundant node and needs to be eliminated. Delete Nj. N= N-1
else
If Nj contains topic related to the same learning objective as Ni then
Merge Nj with Ni :
Copy contents of node Nj to node Ni
Delete node Nj. N = N-1.
else j = j + 1
If j <= N then go to step 5
If i <=N, i = i + 1, j = i + 1, go to step 3 Stop
Fig 2 Merging and Elimination Algorithm
The Optional nodes were then identified.. These are topics that are not mandatory at a particular level of learning. They can be skipped if a child is not able to complete it in the given time frame. These nodes were marked.
These are the learning objectives corresponding to a level of learning. MLL: It is the Class/Grade by which the concept needs to be mastered by a child. Here the class or grade is expressed as a range as they differ across the globe. The range starts with the earliest class/grade the topic is taught and ends with the latest class/grade.
2.2.2 Network Approach
A Mathematical Pathway (MP) Network is constructed using the Concept framework. The MP Network depicts the interdependencies between concepts. Each node is a Level of a concept. For each node, a set of Predecessor nodes are defined. These are the prerequisites that need to be fulfilled to start the level of learning represented by the current node. When a child initially starts with learning of numbers then the minimum requirement is that his sensory skills -Vision, Auditory and Tactile have developed. He has to be provided with ample stimulation to enrich his sensory experiences. Once he is ready he can start his journey through the Mathematical pathway.
The starting node of this Network is the first level of numbers and geometry as they have no prerequisites. Then on, each node is connected to other nodes to show the possible transitions. Using the Concept Framework the network is developed. Each level of learning is referenced using a node number. It contains the nodes of the network with their predecessors defined. The predecessors are the prerequisites. The minimum age and the maximum age that defines the MLL is also given.
The MP Network contains redundant precedence nodes. The MP Network can be optimized by deleting the redundant nodes in the predecessor set. A node in the predecessor list is redundant if any other member in the predecessor list is reachable from this node. A backtracking algorithm(Fig 3) is used to delete nodes.
START
Let the predecessor list be P;
R= P;
Let Q={}, i= 1; Let N = size of R;
Q = Q ∪ Predecessor set of Ri;
i= i+ 1; IF i<= N go to step 6; Q = Q – {start};
P = P- Q; R = Q;
If Q not equal to null set go to step 4. STOP
Fig 3 Backtracking Algorithm for deletion of redundant nodes
Applying this algorithm the Mathematical network is optimized. Traversing through the mathematical pathway is moving through the network. A child starts with Numbers and Geometry and then on the child progresses as per the prerequisite he has fulfilled. Every time the child completes a level of learning the network expands the learning options. Opportunities to learn the next set of concepts the child is eligible for are provided. The child makes transitions through a level of learning of a concept by answering the test corresponding to the learning objective represented by the state and if the child succeeds then the child moves ahead till the final state representing completion of the level is reached. If the child fails at any intermediate step then the child remains in that state till the concept is learnt and there is a readiness to take the test again. On completing a level of learning the control moves back to the network and the process continues till he reaches the final node of the Mathematical Pathway Network.
2.1.3 Network Techniques
distribution, average duration time D = a + 4m + b / 6 was computed. The variance v = (b-a/6)2 was also computed.
3 Experimental Results and Discussion
The entire mathematical pathway consists of 18 concepts. The graph theoretical approach was applied to all the concepts. Here the detailed process of obtaining the optimal CFG is explained with the CFG for Addition. The same is applicable to the rest of the concept flow graphs. The data collected from primary and secondary source was tabulated
into a Concept Table for each concept. Table 1 is the concept table for Addition. This is then converted into a flow graph. Fig 4 represents the Initial CFG for Addition Concept Table. Here there are a total of 36 nodes labeled from A1 to A36.
Fig. 4 Initial CFG for Addition
Applying the algorithm for ordering this graph as per learning progression we obtain the graph represented by Fig 5 Here nodes A7, A8, A14, A15, A17, A21, A22, A27, A28 are reordered.
This graph is now portioned into sub-graphs. In Fig 6 each sub-graph represents a level of learning for addition concept. There are five levels of learning for the addition concept. Each of the sub-graphs is optimized using the merging and elimination algorithm to obtain the graph represented in Fig 7. The following nodes are merged: A2 and A3, A5 and A6, A11 and A12, A14 and A15, A17 and A19, A21 and A22, A27 and A28, A31 and A34, A32 and A35, A33 and A36. The following nodes are deleted as they are redundant: A8, A30.
Fig.6 Addition CFG after Partitioning
Fig. 7 Addition CFG after applying Merging and Elimination algorithm
performed on the CFG’s the node labels would no more indicate the position of the node in the graph. To make it easy to refer to the graph all the nodes are relabeled.
Fig. 8 Addition CFG with Optional Nodes
The label contains Concept Code, Level number and Node number. The format of the label is <Concept Code><Level Number> - <Node Number>. Here concept code is a unique code used to represent the concept. Level number refers to the level of learning, the sub-graph represents and the node number is the position of the node in the new optimized graph. e.g the first node of an addition level 0 graph would be labeled A0-1, here A is the Concept code for addition 0 represents that it is level 0 graph and 1 represents the first node in this sub-graph. This node refers to the first node in the level 0 sub-graph of addition CFG. Then on they are labeled A0-2, A0-3 and so on. The level 1 node would have the label A1-1, A1-A0-2, A1-3 and so on. Table 2 is the Concept Framework for the Addition concept having four levels of learning. This is the optimal form obtained from the Concept Table of addition.
Table 2 Addition CFT LEVEL 0
STEP PREREQUISITE LEARNING OBJECTIVE MLL
A0-1 N0 Increase: Learns concept of One more P-1
A0-2 A0-1
Joining is adding: Joining of collections to make bigger collections. Learns joining together or putting together is adding. . Learns the vocabulary ‘add’, ‘addition’, ‘plus’, ‘sum’, ‘combine’, ‘altogether’, ‘put with’. Uses collections of real objects or picture representation to add two numbers.
P-1
A0-3 A0-2 Addition: Uses number line to Add two numbers whose sum does not exceed 9.
Uses plus symbol ‘+’. P-1
A0-4 A0-3
Addition Table: Makes single digit number pairs whose sum does not exceed 9. Mentally adds a pair of number whose sum does not exceed 9. Learns adding zero to a number does not change its value.
P-1
A0-5 A0-4
Application: Solves one step problems of real life on single digit addition presented through pictures and verbal description and writes number sentences using + and =
P-1
LEVEL 1
STEP PREREQUISITE LEARNING OBJECTIVE MLL
A1-1 N1,A0 Addition T able: Builds the cayley addition table for sums up to 10. Mentally adds a
pair whose sum does not exceed 10. 1-2
A1-2 A1-1 Addition: Adds two digit numbers by drawing representations of tens and ones
without and with regrouping. 1-2
A1-3 A1-2
Addition without carry: Adds 2-3 two digit number without carry and the sum not exceeding 99. Adds both vertically and horizontally. Understands order /commutative property of addition. Learns vocabulary augends, addend and sum.
1-2
A1-4 A1-3 Addition with carry: Adds 2-3 two digit number with carry from ones to tens and
the sum not exceeding 99. 1-2
A1-5 A1-4 Estimation: Estimates the result of addition and compares the result with another given number.
A1-6 A1-5
Application: Solves 1-2 step problems of real life on two digit addition presented through pictures and verbal description and can write number sentence for the problem. Describes orally the situations that corresponds to the given addition facts.
LEVEL 2
STEP PREREQUISITE LEARNING OBJECTIVE MLL
A2-1 N2,A1 Addition Table: Extends cayley table based on tens to hundreds. Adds two digit numbers and doubles two digit numbers mentally
2-3
A2-2 A2-1 Addition without carry: Adds 2-3 three digit numbers without carry and the sum not exceeding 999. Reads and writes the number name for the sum. Adds numbers by writing them vertically. Uses place value in standard algorithm of addition.
2-3
A2-3 A2-2 Addition with carry: Adds 2-3 three digit numbers with carry from a) ones to tens and b) tens to hundreds c) ones to tens and tens to hundreds provided the sum does not exceed 999.
2-3
A2-4 A2-3 Estimation: Estimates sum of two given numbers. 2-3
A2-5 A2-4 Application: Solves 1-2 step problems of real life on three digit addition presented through pictures and stories by writing number sentences using ‘+’ and ‘=’. Frames problems for addition facts.
2-3
LEVEL 3
STEP PREREQUISITE LEARNING OBJECTIVE MLL
A3-1 N3,A2 Addition Table: Extends cayley addition table to thousand. Adds multiples of 10 and 100 mentally
3-4
A3-2 A3-1 Addition without carry: Adds 2-3 four digit number without carry and the sum not exceeding 9999. Adds both vertically and horizontally.
3-4
A3-3 A3-2 Addition with carry: Adds 2-3 four digit number with carry and the sum not exceeding 9999.
3-4
A3-4 A3-3 Estimation: Estimates sums of given numbers and verifies using
approximation.
3-4
A3-5 Reading, Comprehension, A3-4
Application: Solves 1-2 step problems of real life on four digit addition by writing number sentences using ‘+’ and ‘=’. Frames word problems.
3-4
LEVEL 4
STEP PREREQUISITE LEARNING OBJECTIVE MLL
A4-1 N4,A3 Addition without carry: Adds 2-3 three five to nine digit numbers without carry and the sum not exceeding 99 crores.
4-6
A4-2 A4-1 Addition with carry: Adds 2-3 five to nine digit numbers with carry and the sum not exceeding 99 crores.
4-6
A4-3 Reading, Comprehension,
A4-2
Application: Solves 1-2 step problems of real life on five to nine digit addition by writing number sentences using ‘+’ and ‘=’.
4-6
Using the same technique CFT for all the eighteen concepts was developed. All of them were in their optimal form. Now each level of a concept was considered as a node and a Mathematical Network to represent the Mathematical Pathway is designed. Table 3 shows the class and age mapping of the MLL. Table 4 is the Mathematical Pathway Network. This is a tabular representation of the Mathematical Pathway Network. Here N-Whole Numbers, A-Addition, S-subtraction, A- Addition, M- Multiplication, D-Division, F-Fractions, DC-Decimals, NS – Number System, ML-Measurements length, MW-Measurements Weight, MC-Measurements Capacity, MT-Measurements Time, MY- Money, G – Geometry, ME-Mensuration, AG – Algebra, DH – Data Handling, RP – Ratio and Proportion. The numbers suffixed represent the various levels of learning for each of the concepts. The redundant nodes identified using the backtracking algorithm are marked in bold the Table 4. The nodes are deleted and the nodes labels are replaced with node numbers to obtain the optimal MP Network shown in Table 5.
Table 3 Class-Age Mapping of the MLL
CLASS AGE PRESCHOOL 2
CLASS 1 6
CLASS 2 7
CLASS 3 8
CLASS4 9 CLASS 5 10
Table 4 The Mathematical Pathway Network
LEVEL OF LEARNING
PREREQUISITE NODE
MIN AGE
MAX AGE
START
N0 2 6
N1 N0 6 8
N2 N1 7 9
N3 N2 8 10
N4 N3 9 12
A0 N0 2 6
A1 N1 A0 6 8
A2 N2 A1 7 9
A3 N3 A2 8 10
A4 N4 A3 9 12
S0 A0 2 6
S1 A1 S0 6 8
S2 A2 S1 7 9
S3 A3 S2 8 10
S4 A4 S3 9 12
M0 A0 2 8
M1 A1 M0 6 9
M2 A2 M1 7 10
M3 A4 M2 8 11
M4 A4 M3 10 12
D0 S0 M1 2 6
D1 S0 M1 D0 6 9
D2 S2 M1 D1 7 9
D3 S4 M1 D2 8 10
D4 S4 M1 D3 9 12
F0 N0 D1 2 9
F1 N1 A1 S1 M2 D2 F0 9 12
F2 F1 10 13
DC0 N1 M2 F1 8 11
DC1 DC0 9 12
DC2 M3 DC1 11 13
NS0 N4 A4 S4 M4 D4 10 12
NS1 NS0 10 13
NS2 NS0 DC1 11 13
NS3 NS1 11 13
ML0 N0 2 8
ML1 N2 A2 S2 ML0 7 10
ML2 M2 D2 F1 DC1 ML1 9 11
MW0 N0 2 8
Table 4 The Mathematical Pathway Network (Continued)
LEVEL OF LEARNING
PREREQUISITE NODE
MIN AGE
MAX AGE
MW2 N2 D2 F1 DC1 MW1 9 11
MC0 N0 2 6
MC1 N2 MC0 7 9
MC2 A2 S2 MC1 8 10
MC3 M2 D2 F1 DC1 MC2 9 11
MT0 N1 2 6
MT1 MT0 6 9
MT2 A1 S1 M2 D2 MT1 8 10
MT3 D3 MT2 9 11
MY0 N1 A1 S1 2 8
MY1 MY0 6 9
MY2 DC1 MY1 8 11
G0 2 6
G1 N0 G0 6 8
G2 ML1 G1 7 9
G3 G2 8 10
G4 G3 9 11
G5 N2 ML1 G4 10 13
G6 G5 11 13
G7 G6 11 13
ME0 G2 8 11
ME1 M2 D2 F1 ME0 10 13
ME2 ME1 11 13
AG0 N0 2 7
AG1 N2 AG0 6 8
AG2 AG1 10 12
AG3 A1 S1 AG2 11 13
AG4 NS3 AG3 11 13
DH0 N1 ML1 6 8
DH1 G2 DH0 7 11
DH2 A3 S3 M3 D3 DH1 10 12
DH3 DH2 11 13
RP0 NS2 10 12
RP1 DC1 RP0 10 13
RP2 RP1 11 13
FINAL
Table 5 Optimal Mathematical Pathway Network
NODE NO. LEVEL OF LEARNING PREREQUISITE NODE MIN AGE MAX AGE
1 START
2 N0 1 2 6
3 N1 2 6 8
4 N2 3 7 9
5 N3 4 8 10
6 N4 5 9 12
7 A0 2 2 6
8 A1 3 7 6 8
9 A2 4 8 7 9
10 A3 5 9 8 10
11 A4 6 10 9 12
12 S0 7 2 6
13 S1 8 12 6 8
14 S2 9 13 7 9
15 S3 10 14 8 10
16 S4 11 15 9 12
17 M0 7 2 8
18 M1 8 17 6 9
19 M2 9 18 7 10
20 M3 11 19 8 11
21 M4 20 10 12
22 D0 12 18 2 6
23 D1 22 6 9
24 D2 14 23 7 9
25 D3 16 24 8 10
26 D4 16 25 9 12
27 F0 23 2 9
28 F1 19 24 27 9 12
29 F2 28 10 13
30 DC0 28 8 11
31 DC1 30 9 12
32 DC2 20 31 11 13
33 NS0 21 26 10 12
34 NS1 33 10 13
35 NS2 33 31 11 13
36 NS3 34 11 13
37 ML0 2 2 8
38 ML1 14 37 7 10
39 ML2 31 38 9 11
Table 5 Optimal Mathematical Pathway Network(Continued)
NODE NO. LEVEL OF LEARNING PREREQUISITE NODE MIN AGE MAX AGE
41 MW1 14 40 8 10
42 MW2 31 41 9 11
43 MC0 2 2 6
44 MC1 4 43 7 9
45 MC2 14 44 8 10
46 MC3 31 45 9 11
47 MT0 3 2 6
48 MT1 47 6 9
49 MT2 19 24 48 8 10
50 MT3 25 49 9 11
51 MY0 13 2 8
52 MY1 51 6 9
53 MY2 31 52 8 11
54 G0 1 2 6
55 G1 2 54 6 8
56 G2 38 55 7 9
57 G3 56 8 10
58 G4 57 9 11
59 G5 38 58 10 13
60 G6 59 11 13
61 G7 60 11 13
62 ME0 56 8 11
63 ME1 28 62 10 13
64 ME2 63 11 13
65 AG0 2 2 7
66 AG1 4 65 6 8
67 AG2 66 10 12
68 AG3 13 67 11 13
69 AG4 36 68 11 13
70 DH0 38 6 8
71 DH1 56 70 7 11
72 DH2 20 25 71 10 12
73 DH3 72 11 13
74 RP0 35 10 12
75 RP1 31 74 10 13
76 RP2 75 11 13
77 OTHER 29 32 61 64 69
78 MEASURE 39 42 46 50 53
79 FINAL 73 76 77 78
Table 6 shows the computations applying Triangular time distribution on the network and Table 7 shows the computations applying Beta time distribution. Using the CPM/PERT technique the critical path is obtained. It is represented by the Gantt chart in Fig. 9
Table 6 Computations of CPM/PERT using Triangular Time distribution
ID Task Name Predecessors
(Enter one ID per cell)
O
(min)
M
(most likely)
P
(max)
Duration
(exp. time)
ES EF LS LF Slack
1 START 0.00 0.00 0.00 0.00 0.00 0.00
2 N0 1 3 4 6 4.33 0.00 4.33 0.00 0.00 0.00
3 N1 2 0.5 0.75 1 0.75 4.33 5.08 0.00 0.00 0.00
4 N2 3 0.5 0.75 1 0.75 5.08 5.83 0.00 0.00 0.00
5 N3 4 0.5 0.75 1 0.75 5.83 6.58 0.00 0.00 0.00
6 N4 5 0.5 0.75 1 0.75 6.58 7.33 0.00 0.00 0.00
7 A0 2 0.5 0.75 1 0.75 4.33 5.08 0.00 0.00 0.00
8 A1 3 7 0.5 0.75 1 0.75 5.08 5.83 0.00 0.00 0.00
9 A2 4 8 0.5 0.75 1 0.75 5.83 6.58 0.00 0.00 0.00
10 A3 5 9 0.5 0.75 1 0.75 6.58 7.33 0.00 0.00 0.00
11 A4 6 10 0.5 0.75 1 0.75 7.33 8.08 0.00 0.00 0.00
12 S0 7 0.5 0.75 1 0.75 5.08 5.83 0.00 0.00 0.00
13 S1 8 12 0.5 0.75 1 0.75 5.83 6.58 0.00 0.00 0.00
14 S2 9 13 0.5 0.75 1 0.75 6.58 7.33 0.00 0.00 0.00
15 S3 10 14 0.5 0.75 1 0.75 7.33 8.08 0.00 0.00 0.00
16 S4 11 15 0.5 0.75 1 0.75 8.08 8.83 0.00 0.00 0.00
17 M0 7 0.5 0.75 1 0.75 5.08 5.83 0.00 0.00 0.00
18 M1 8 17 0.5 0.75 1 0.75 5.83 6.58 0.00 0.00 0.00
19 M2 9 18 0.5 0.75 1 0.75 6.58 7.33 0.00 0.00 0.00
20 M3 11 19 0.5 0.75 1 0.75 8.08 8.83 0.00 0.00 0.00
21 M4 20 0.5 0.75 1 0.75 8.83 9.58 0.00 0.00 0.00
22 D0 12 18 0.5 0.75 1 0.75 6.58 7.33 0.00 0.00 0.00
23 D1 22 0.5 0.75 1 0.75 7.33 8.08 0.00 0.00 0.00
24 D2 14 23 0.5 0.75 1 0.75 8.08 8.83 0.00 0.00 0.00
25 D3 16 24 0.5 0.75 1 0.75 8.83 9.58 0.00 0.00 0.00
26 D4 16 25 0.5 0.75 1 0.75 9.58 10.33 0.00 0.00 0.00
27 F0 23 0.5 0.75 1 0.75 8.08 8.83 0.00 0.00 0.00
28 F1 19 24 27 0.5 0.75 1 0.75 8.83 9.58 0.00 0.00 0.00
29 F2 28 0.5 0.75 1 0.75 9.58 10.33 0.00 0.00 0.00
30 DC0 28 0.5 0.75 1 0.75 9.58 10.33 0.00 0.00 0.00
31 DC1 30 0.5 0.75 1 0.75 10.33 11.08 0.00 0.00 0.00
32 DC2 20 31 0.5 0.75 1 0.75 11.08 11.83 0.00 0.00 0.00
33 NS0 21 26 0.5 0.75 1 0.75 10.33 11.08 0.00 0.00 0.00
34 NS1 33 0.5 0.75 1 0.75 11.08 11.83 0.00 0.00 0.00
35 NS2 33 31 0.5 0.75 1 0.75 11.08 11.83 0.00 0.00 0.00
36 NS3 34 0.5 0.75 1 0.75 11.83 12.58 0.00 0.00 0.00
37 ML0 2 0.5 0.75 1 0.75 4.33 5.08 0.00 0.00 0.00
38 ML1 14 37 0.5 0.75 1 0.75 7.33 8.08 0.00 0.00 0.00
39 ML2 31 38 0.5 0.75 1 0.75 11.08 11.83 0.00 0.00 0.00
40 MW0 2 0.5 0.75 1 0.75 4.33 5.08 0.00 0.00 0.00
41 MW1 14 40 0.5 0.75 1 0.75 7.33 8.08 0.00 0.00 0.00
42 MW2 31 41 0.5 0.75 1 0.75 11.08 11.83 0.00 0.00 0.00
43 MC0 2 0.5 0.75 1 0.75 4.33 5.08 0.00 0.00 0.00
Table 6 Computations of CPM/PERT using Triangular Time distribution(Continued)
ID TASK NAME
PREDECESSORS O(MIN)
M(Mos t
Likely) P
DUR ATI
ON ES EF LS LF SLAC
K 45 MC2 14 44 0.5 0.75 1 0.75 7.33 8.08 0.00 0.00 0.00
46 MC3 31 45 0.5 0.75 1 0.75 11.08 11.83 0.00 0.00 0.00
47 MT0 3 0.5 0.75 1 0.75 5.08 5.83 0.00 0.00 0.00
48 MT1 47 0.5 0.75 1 0.75 5.83 6.58 0.00 0.00 0.00
49 MT2 19 24 48 0.5 0.75 1 0.75 8.83 9.58 0.00 0.00 0.00
50 MT3 25 49 0.5 0.75 1 0.75 9.58 10.33 0.00 0.00 0.00
51 MY0 13 0.5 0.75 1 0.75 6.58 7.33 0.00 0.00 0.00
52 MY1 51 0.5 0.75 1 0.75 7.33 8.08 0.00 0.00 0.00
53 MY2 31 52 0.5 0.75 1 0.75 11.08 11.83 0.00 0.00 0.00
54 G0 1 0.5 0.75 1 0.75 0.00 0.75 0.00 0.00 0.00
55 G1 2 54 0.5 0.75 1 0.75 4.33 5.08 0.00 0.00 0.00
56 G2 38 55 0.5 0.75 1 0.75 8.08 8.83 0.00 0.00 0.00
57 G3 56 0.5 0.75 1 0.75 8.83 9.58 0.00 0.00 0.00
58 G4 57 0.5 0.75 1 0.75 9.58 10.33 0.00 0.00 0.00
59 G5 38 58 0.5 0.75 1 0.75 10.33 11.08 0.00 0.00 0.00
60 G6 59 0.5 0.75 1 0.75 11.08 11.83 0.00 0.00 0.00
61 G7 60 0.5 0.75 1 0.75 11.83 12.58 0.00 0.00 0.00
62 ME0 56 0.5 0.75 1 0.75 8.83 9.58 0.00 0.00 0.00
63 ME1 28 62 0.5 0.75 1 0.75 9.58 10.33 0.00 0.00 0.00
64 ME2 63 0.5 0.75 1 0.75 10.33 11.08 0.00 0.00 0.00
65 AG0 2 0.5 0.75 1 0.75 4.33 5.08 0.00 0.00 0.00
66 AG1 4 65 0.5 0.75 1 0.75 5.83 6.58 0.00 0.00 0.00
67 AG2 66 0.5 0.75 1 0.75 6.58 7.33 0.00 0.00 0.00
68 AG3 13 67 0.5 0.75 1 0.75 7.33 8.08 0.00 0.00 0.00
69 AG4 36 68 0.5 0.75 1 0.75 12.58 13.33 0.00 0.00 0.00
70 DH0 38 0.5 0.75 1 0.75 8.08 8.83 0.00 0.00 0.00
71 DH1 56 70 0.5 0.75 1 0.75 8.83 9.58 0.00 0.00 0.00
72 DH2 20 25 71 0.5 0.75 1 0.75 9.58 10.33 0.00 0.00 0.00
73 DH3 72 0.5 0.75 1 0.75 10.33 11.08 0.00 0.00 0.00
74 RP0 35 0.5 0.75 1 0.75 11.83 12.58 0.00 0.00 0.00
75 RP1 31 74 0.5 0.75 1 0.75 12.58 13.33 0.00 0.00 0.00
76 RP2 75 0.5 0.75 1 0.75 13.33 14.08 0.00 0.00 0.00
77 OTHER 29 32 61 6 4
6
9 0 0 0 0.00 11.83 11.83 0.00 0.00 0.00
78
MEASU
RE 39 42 46 5 0
5
3 0 0 0 0.00 11.83 11.83 0.00 0.00 0.00
79 FINAL 73 76 77 7
8 0.00 14.08 14.08 14.08 14.08 0.00
Table 7 Computations of CPM/PERT using BetaTime distribution
ID Task Name Predecessors
(Enter one ID per cell)
O
(min)
M
(most likely)
P
(max)
Duration
(exp. time)
ES EF LS LF Slack
1 START 0.00 0.00 0.00 0.00 0.00 0.00
2 N0 1 3 4 6 4.17 0.00 4.17 0.00 0.00 0.00
3 N1 2 0.5 0.75 1 0.75 4.17 4.92 0.00 0.00 0.00
4 N2 3 0.5 0.75 1 0.75 4.92 5.67 0.00 0.00 0.00
5 N3 4 0.5 0.75 1 0.75 5.67 6.42 0.00 0.00 0.00
6 N4 5 0.5 0.75 1 0.75 6.42 7.17 0.00 0.00 0.00
7 A0 2 0.5 0.75 1 0.75 4.17 4.92 0.00 0.00 0.00
8 A1 3 7 0.5 0.75 1 0.75 4.92 5.67 0.00 0.00 0.00
9 A2 4 8 0.5 0.75 1 0.75 5.67 6.42 0.00 0.00 0.00
10 A3 5 9 0.5 0.75 1 0.75 6.42 7.17 0.00 0.00 0.00
11 A4 6 10 0.5 0.75 1 0.75 7.17 7.92 0.00 0.00 0.00
12 S0 7 0.5 0.75 1 0.75 4.92 5.67 0.00 0.00 0.00
13 S1 8 12 0.5 0.75 1 0.75 5.67 6.42 0.00 0.00 0.00
14 S2 9 13 0.5 0.75 1 0.75 6.42 7.17 0.00 0.00 0.00
15 S3 10 14 0.5 0.75 1 0.75 7.17 7.92 0.00 0.00 0.00
16 S4 11 15 0.5 0.75 1 0.75 7.92 8.67 0.00 0.00 0.00
17 M0 7 0.5 0.75 1 0.75 4.92 5.67 0.00 0.00 0.00
18 M1 8 17 0.5 0.75 1 0.75 5.67 6.42 0.00 0.00 0.00
19 M2 9 18 0.5 0.75 1 0.75 6.42 7.17 0.00 0.00 0.00
20 M3 11 19 0.5 0.75 1 0.75 7.92 8.67 0.00 0.00 0.00
21 M4 20 0.5 0.75 1 0.75 8.67 9.42 0.00 0.00 0.00
22 D0 12 18 0.5 0.75 1 0.75 6.42 7.17 0.00 0.00 0.00
23 D1 22 0.5 0.75 1 0.75 7.17 7.92 0.00 0.00 0.00
24 D2 14 23 0.5 0.75 1 0.75 7.92 8.67 0.00 0.00 0.00
25 D3 16 24 0.5 0.75 1 0.75 8.67 9.42 0.00 0.00 0.00
26 D4 16 25 0.5 0.75 1 0.75 9.42 10.17 0.00 0.00 0.00
27 F0 23 0.5 0.75 1 0.75 7.92 8.67 0.00 0.00 0.00
28 F1 19 24 27 0.5 0.75 1 0.75 8.67 9.42 0.00 0.00 0.00
29 F2 28 0.5 0.75 1 0.75 9.42 10.17 0.00 0.00 0.00
30 DC0 28 0.5 0.75 1 0.75 9.42 10.17 0.00 0.00 0.00
31 DC1 30 0.5 0.75 1 0.75 10.17 10.92 0.00 0.00 0.00
32 DC2 20 31 0.5 0.75 1 0.75 10.92 11.67 0.00 0.00 0.00
33 NS0 21 26 0.5 0.75 1 0.75 10.17 10.92 0.00 0.00 0.00
34 NS1 33 0.5 0.75 1 0.75 10.92 11.67 0.00 0.00 0.00
35 NS2 33 31 0.5 0.75 1 0.75 10.92 11.67 0.00 0.00 0.00
36 NS3 34 0.5 0.75 1 0.75 11.67 12.42 0.00 0.00 0.00
37 ML0 2 0.5 0.75 1 0.75 4.17 4.92 0.00 0.00 0.00
38 ML1 14 37 0.5 0.75 1 0.75 7.17 7.92 0.00 0.00 0.00
39 ML2 31 38 0.5 0.75 1 0.75 10.92 11.67 0.00 0.00 0.00
40 MW0 2 0.5 0.75 1 0.75 4.17 4.92 0.00 0.00 0.00
41 MW1 14 40 0.5 0.75 1 0.75 7.17 7.92 0.00 0.00 0.00
42 MW2 31 41 0.5 0.75 1 0.75 10.92 11.67 0.00 0.00 0.00
43 MC0 2 0.5 0.75 1 0.75 4.17 4.92 0.00 0.00 0.00
44 MC1 4 43 0.5 0.75 1 0.75 5.67 6.42 0.00 0.00 0.00
45 MC2 14 44 0.5 0.75 1 0.75 7.17 7.92 0.00 0.00 0.00
46 MC3 31 45 0.5 0.75 1 0.75 10.92 11.67 0.00 0.00 0.00
Table 7 Computations of CPM/PERT using BetaTime distribution(Continued)
ID Task Name Predecessors
(Enter one ID per cell)
O
(min)
M
(most likely)
P
(max)
Duration
(exp. time)
ES EF LS LF Slack
48 MT1 47 0.5 0.75 1 0.75 5.67 6.42 0.00 0.00 0.00
49 MT2 19 24 48 0.5 0.75 1 0.75 8.67 9.42 0.00 0.00 0.00
50 MT3 25 49 0.5 0.75 1 0.75 9.42 10.17 0.00 0.00 0.00
51 MY0 13 0.5 0.75 1 0.75 6.42 7.17 0.00 0.00 0.00
52 MY1 51 0.5 0.75 1 0.75 7.17 7.92 0.00 0.00 0.00
53 MY2 31 52 0.5 0.75 1 0.75 10.92 11.67 0.00 0.00 0.00
54 G0 1 0.5 0.75 1 0.75 0.00 0.75 0.00 0.00 0.00
55 G1 2 54 0.5 0.75 1 0.75 4.17 4.92 0.00 0.00 0.00
56 G2 38 55 0.5 0.75 1 0.75 7.92 8.67 0.00 0.00 0.00
57 G3 56 0.5 0.75 1 0.75 8.67 9.42 0.00 0.00 0.00
58 G4 57 0.5 0.75 1 0.75 9.42 10.17 0.00 0.00 0.00
59 G5 38 58 0.5 0.75 1 0.75 10.17 10.92 0.00 0.00 0.00
60 G6 59 0.5 0.75 1 0.75 10.92 11.67 0.00 0.00 0.00
61 G7 60 0.5 0.75 1 0.75 11.67 12.42 0.00 0.00 0.00
62 ME0 56 0.5 0.75 1 0.75 8.67 9.42 0.00 0.00 0.00
63 ME1 28 62 0.5 0.75 1 0.75 9.42 10.17 0.00 0.00 0.00
64 ME2 63 0.5 0.75 1 0.75 10.17 10.92 0.00 0.00 0.00
65 AG0 2 0.5 0.75 1 0.75 4.17 4.92 0.00 0.00 0.00
66 AG1 4 65 0.5 0.75 1 0.75 5.67 6.42 0.00 0.00 0.00
67 AG2 66 0.5 0.75 1 0.75 6.42 7.17 0.00 0.00 0.00
68 AG3 13 67 0.5 0.75 1 0.75 7.17 7.92 0.00 0.00 0.00
69 AG4 36 68 0.5 0.75 1 0.75 12.42 13.17 0.00 0.00 0.00
70 DH0 38 0.5 0.75 1 0.75 7.92 8.67 0.00 0.00 0.00
71 DH1 56 70 0.5 0.75 1 0.75 8.67 9.42 0.00 0.00 0.00
72 DH2 20 25 71 0.5 0.75 1 0.75 9.42 10.17 0.00 0.00 0.00
73 DH3 72 0.5 0.75 1 0.75 10.17 10.92 0.00 0.00 0.00
74 RP0 35 0.5 0.75 1 0.75 11.67 12.42 0.00 0.00 0.00
75 RP1 31 74 0.5 0.75 1 0.75 12.42 13.17 0.00 0.00 0.00
76 RP2 75 0.5 0.75 1 0.75 13.17 13.92 0.00 0.00 0.00
77 OTHER 29 32 61 64 69 0 0 0 0.00 11.67 11.67 0.00 0.00 0.00
78 MEASURE 39 42 46 50 53 0 0 0 0.00 11.67 11.67 0.00 0.00 0.00
0 5 10 15
START N0 N1 N2 N3 N4 A0 A1 A2 A3 A4 S0 S1 S2 S3 S4 M0 M1 M2 M3 M4 D0 D1 D2 D3 D4 F0 F1 F2 DC0 DC1 DC2 NS0 NS1 NS2 NS3 ML0 ML1 ML2 MW0 MW1 MW2 MC0 MC1 MC2 MC3 MT0 MT1 MT2 MT3 MY0 MY1 MY2 G0 G1 G2 G3 G4 G5 G6 G7 ME0 ME1 ME2 AG0 AG1 AG2 AG3 AG4 DH0 DH1 DH2 DH3 RP0 RP1 RP2 OTHER MEASURE FINAL YEARS:
Fig. 9 Gantt chart obtained from PERT/CPM on MP NETWORK
3. Conclusion
In the present study Graph theoretical and Network approaches are used to design the framewrook for the learning model associted with learning of mathematics for school children at elementary level [std I to std VII]. Data Collected pertaining to the Mathematical pathway from primary and secondary source was represented using Concept Tables and was analyzed using Concept Flow Graphs. The CFG’s were optimized using graph theory concepts by rearranging nodes as per the learning progression, partitioning the graphs into sub-graphs to represent levels of learning, optimizing the sub-graphs using merging and elimination technique and identifying and marking the optional nodes. Nodes were relabeled and the interdependencies between concepts were identified. The optimal CFG’s were used to create the Concept Framework with Concept Framework Tables. The Concept Framework so obtained is in its optimal state and accurate as it evolved through a mathematical process. In totality there were 18 concepts with each concept having topics. In the experiments the addition graph which had about 36 nodes was optimised to have 24 nodes. Similarly all concept graphs were optimised which in turn optimised the Mathematical Pathway.
The real life problem is now in the form of a mathematical model and is directly used to design the automata machine. Using this framework the learning model was developed using theory of automata and databases generated using this were used for the performance analysis. The performance analysis constituted the DM techniques and Neural Network approach. This is a novel approach and first of its kind where concepts of discrete mathematics has been applied for learning of mathematics.
The MP Network links all the concepts and helps a child traverse through the entire Mathematical Pathway. It is a model which is adopted for developing the automated Progress tracking module. The critical path obtained using CPM and PERT techniques, represented by the Gantt chart shows that all activities are critical in the Mathematical Pathway. There is no slack time. Hence learning is continuous. The CPM/ PERT techniques using a beta time distribution estimated the total number of years required to traverse the Mathematical Pathway as 13.92 years and the triangular distribution estimated it as 14.08 years. The desired duration of completion was 13 years. Since the critical path determines the worst case scenario the Mathematical Pathway can be traversed on time by average learners and faster by gifted learners. This is well within our learning goals of being able to complete it in Class 7. So in the worst case a child can finish the learning at the elementary level by 14 years.
Acknowledgement
One of us, Annapurna Kamath is grateful to the Bangalore University for providing the research facilities to carry out the research program.
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