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]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: August, 2019

**Hnin Hnin Khaing & Yee Yee Htun**

Lecturer, Department of Engineering Mathematics, TU (Maubin), Ayarwaddy Division

Lecturer, Department of Engineering Mathematics, TU (Hmawbi), Yangon Division

Myanmar

Journal Full Text PDF: Implementation of Numerical Solutions for Nonlinear Equations using MATLAB.

**Abstract**

Problems that most frequently encountered are nonlinear equations in sciences and engineering problems. In this paper, we will focus on MATLAB solutions to nonlinear equations by studying various methods. In this paper, the numerical methods for solving nonlinear equations using MATLAB can be carried out. This present the most widely used iterative methods for nonlinear equations and MATLAB features for finding numerical solutions. The numerical examples are considered and implemented in this paper.

**Keywords:** nonlinear equations, MATLAB, numerical methods, iteratives methods.

**1. INTRODUCTION**

Compare to linear algebraic equations, most frequently encountered are nonlinear equations in sciences and engineering problems. Solving nonlinear equations could be computationally expensive; therefore, the solving of approximate linear equations was indispensable especially in the early times when the computers were not powerful enough. Today, with the rapid development of computing technology, solving directly nonlinear equations is becoming increasingly important. In this paper, various methods of solving nonlinear equations problems are studied and we will focus on MATLAB solutions to nonlinear equations. The three methods of solutions to nonlinear algebraic equations will be presented in this technical approach paper. The graphical method for nonlinear equations with one and two unknown variables can be analysis with polynomial equations. Numerical solutions to nonlinear equations and nonlinear matrix equations can also be implemented in this paper. (Dingyü Xue, 2009).

Graphical and numerical methods will be presented and the simplest solution is implemented using MATLAB. In this paper, the linear programming, quadratic programming and general nonlinear programming will be studied and MATLAB-based solutions will be carried out. As a procedure, the theory background of Bisection Method, Secant Method and Newton Raphson Method of solving nonlinear equations problems are studied and carried out. Then, to be approached to simplest and fast way, MATLAB instructions relative to nonlinear equations solving process are also studied and tested. Actually, various possible problems are applied to three methods and tested. But, only the main points or examples are expressed in this paper as a portion of Mathematical approach to Engineering Problems.

**2. NUMERICAL METHODS FOR NONLINEAR EQUATIONS**

2.1 Bisection method

The idea behind the Intermediate Value Theorem can be stated: when we have two points connected by a continuous curve, as shown in Picture 1:

• one point below the line

• the other point above the line

• then there will be at least one place where the curve crosses the line.

Picture 1(a). Intermediate values Picture 1(b). Bisection Algorithm

(www mathsisfun com) (www codewithc com)

The first step in iteration is to calculate the mid-point of the interval [ a, b ]. If c be the mid-point of the interval, it can be defined as: c = ( a+b)/2. The function is evaluated at ‘c’, which means f(c) is calculated.

• f(c) = 0 : c is the required root of the equation.

• f(b) * f(c) > 0 : if the product of f(b) and f(c) is positive, the root lies in the interval [a, c].

• f(b) * f(c) < 0 : if the product of f(b) and f(c) is negative, the root lies in the interval [ b, c].

In the second iteration, the intermediate value theorem is applied either in [a, c] or [ b, c], depending on the location of roots. And then, the iteration process is repeated by updating new values of a and b. Picture 2 shows how iteration done in bisection method.

Picture 2. Iteration Process in Graph

2.2 Secant Method

This method uses two initial guesses and finds the root of a function through interpolation approach. For each successive iteration, two of the most recent guesses can be used as two most recent fresh values to find out the next approximation. Features of Secant Method can be shortly expressed as:

• No. of initial guesses – 2

• Type – open bracket

• Rate of convergence – faster

• Convergence – super linear

• Accuracy – good

• Approach – interpolation

• Programming effort – tedious

The Procedure of Secant Method can be shown as flowchart as in Picture 3 compare with Newton Raphson Method and this will give the algorithm to implement in MATLAB.

(a) Secant Method Algorithm (b) Newton Raphson Method Algorithm

Picture 3. Comparison of Secant Method and Newton Raphson (www codewithc com)

2.3 Newton Raphson Method

The theoretical and mathematical background behind Newton-Raphson method and its MATLAB program (or program in any programming language) is approximation of the given function by tangent line with the help of derivative, after choosing a guess value of root which is reasonably close to the actual root. The x- intercept of the tangent is calculated by using elementary algebra, and this calculated x-intercept is typically better approximation to the root of the function. This procedure is repeated till the root of desired accuracy is found. Lets now go through a short mathematical background of Newton’s method. For this, consider a real value function f(x) as shown in the Picture 3(c).

Picture 3(c). Newton’s Method (Dingyü Xue, 2009)

Let’s try the example problem using Newton-Raphson method, solving it numerically. The function is to be corrected to 9 decimal places. For a given function: x3−x−1 = 0, which is differentiable,

**3. MATLAB FUNCTIONS FOR SOLVING NONLINEAR EQUATIONS**

3.1 Function “roots”

The Syntax : r = roots(p) returns the roots of the polynomial represented by p as a column vector. Input p is a vector containing n+1 polynomial coefficients, starting with the coefficient of xn. A coefficient of 0 indicates an intermediate power that is not present in the equation. This function can study as shown in Picture 4(a). For example: p = [3 2 -2] represents the polyno-mial 3×2+2x−2.The roots function solves polynomial equations of the form p1xn + …+ pnx + pn+1 =0. Polynomial equations contain a single variable with nonnegative exponents.

(a) roots() (b) fzero()

Picture 4. Study on MATLAB Functions

3.2 Function: “fzero”

Root of nonlinear function (fzero) can be write in syntax as:

3.3 Function: “inline”

The constructing of inline object can be scriptable in MATLAB as the fucncitons: inline (expr), inline (expr,arg1,arg2,…) or inline (expr,n). “inline (expr)” constructs an inline function object from the MATLAB expression contained in “expr”. The input argument to the inline function is automatically determined by searching expr for an isolated lower case alphabetic character, other than i or j, that is not part of a word formed from several alphabetic characters. If no such character exists, x is used. If the character is not unique, the one closest to x is used. If two characters are found, the one later in the alphabet is chosen. “inline (expr,arg1,arg2,…)” constructs an inline function whose input arguments are specified by arg1, arg2,…. Multicharacter symbol names may be used. “inline (expr,n)” where n is a scalar, constructs an inline function whose input arguments are x, P1, P2, … .

Three commands related to inline allow you to examine an inline function object and determine how it was created. “char(fun)”converts the inline function into a character array. This is identical to “formula(fun)”. “argnames(fun)” returns the names of the input arguments of the inline object fun as a cell array of character vectors. “formula(fun)” returns the formula for the inline object fun.A fourth command “vectorize(fun)” inserts a . before any ^, * or /’ in the formula for fun. The result is a vectorized version of the “inline” function. (www mathworks com)

3.4 Function “num2str”, “abs”, “plot”

The simple functions of “num2str” which convert number to string, “abs” wich convert the valuse to be absolute or real only and “plot” which make data to graph or curve are also studied and apply in solution of nonlinear equations of MATLAB. Picture 5 shows example expression of these functions which can be studied by help of MATLAB.

(a) num2str() (b)abs()

(c) Plot()

Picture 5. Study on MATLAB Functions

**4. IMPLEMENTATION RESULTS**

4.1 M-Scripts for Three Methods

Input data are made as variable for any nonlinear equations, not only by getting answer, plotting graph process was followed at the end of program. The m-script for each method was carried out as in the following Table 1:

Table 1. Implemeting Codes of MATLAB

4.2 Simulation Results

The simulation results are shown in Picture 6 to Picture 8. We can study, analyze, compare and solve fastly and simply with our implemented MATLAB program for any nonlinear equations.

(a) m-file Running Results (b) Error Plot

Picture 6. Bisection Method Results

(a) m-file Running Results (b) Error Plot

Picture 7. Secant Method Results

(a) m-file Running Results (b) Error Plot

Picture 8. Newton Raphson Method Results

Picture 9. Results Comparison for ( )

**5. DISCUSSION, CONCLUSION AND RECOMMENDATION**

The Implementatin of Numerial Solutions for Non Linear Equations are sucessfully done in this research paper using MATLAB. By this practicle approach, we can easily compare and anlyse any nonlinear equations which are mostly represented in real- world Engineering process. How we can compare and used each method of Bisection, Secant and Newton Raphson’s can be express as shown in Picture 9 for a given same nonlinear equation of . We can proof that Bisection Method was simplest form. For Secant Method, it is faster than other numerical methods, except the Newton Raphson method and there is no need to find the derivative of the function as in Newton-Raphson method. If the function is not differentiable, Newton’s method cannot be applied. By this research paper, we can clearly analyze how nonlinear equations are easily be solved using various method by the help of MATLAB.

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]]>The post Experimental Calculation of Voltage Generated by Numbers of Atom When Photon Incident and Cross-section Area of Single Photon appeared first on Zambrut.

]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: August, 2019

**Saddam Husain Dhobi, Ram Chandra Pangeni, Kishori Yadav & M. D. Jahangeer Rangrej**

Science and Techonology, Tribhuvan University (Physics), Kathmandu, Province-3, Nepal

Journal Full Text PDF: Experimental Calculation of Voltage Generated by Numbers of Atom When Photon Incident and Cross-section Area of Single Photon.

**Abstract**

The main work of this research work is to calculate voltage generated by incidence of red photon on the surface atoms of poly crystalline silicon base solar panel. This is calculated by passing red leaser photon through aluminum pinhole of different diameters and then incidence on the solar panel pinhole whose surfaces closed by highly absorption materials i.e. whole panel is cover except the pinhole of desire diameter. On taking the different data i.e. voltage vs incidence photon through pinhole we find that on decrease the diameters of pinhole of aluminum plate for fixed pinhole of solar panel the voltage also decrease and vice-versa. On the other hand, we also calculate voltage generated by the number of atoms on which photon incidence in certain cross-section area. These data are listed on observation table1 and with the help of these data we are able to calculated the average cross-section area of red photon.

**Keywords:** Aluminium Pinhole, Poly Crysatlline Silicion, Photon, Cross-section area, Red Photon.

**1. Introduction**

The phenomena which is used to find the voltage across the atoms or voltage generate by atom when photon is incidence it. When photon incidence on the surface of atoms it ejected electron from the atom after distribution of photon energy. In this experiment, photon energy are able to ejected the electron from the atom which is bounded or correlated with other atoms. This phenomena can also called photoelectric effect.

In this work we are trying to introduce the the energy lose or distribution by a single photon energy which are unable to eject the electron from the atom is divided into numbers of packet energy. Due to the division of these numbers of packet energy of single photon, the work function is greater then theses packet energy generated by single incidence photon. This phenomena is only seen when photon is incidence on an atom expect normally i.e. the division of single photon is only experience when photon is incidence with certain angle expect perpendicular.

This is observed when we focus for the perpendicular incidence the photon on the pinhole of solar panel. Because during the focusing for perpendicular incidence of photon on the solar panel considerable surface the voltage obtained from multi meter is less than voltage obtained at perpendicular incidence of photon on considerable surface.

Figure 1: Sketch of different component of solar panel and circuit connection to obtained Voltage.

Some part of our experiment contain this whole system of figure 1. and more extra are design to obtained the data which is shown in experimental set up. The short description of different component of solar panel are given below:

1.2 Electrons & Hole: An intrinsic or pure silicon crystal at room temperature has sufficient heat or thermal energy for some valence electrons to jump the gap from the valence band into the conduction band and hence becoming free electrons. These free electrons are called conduction electrons. When an electron jumps to conduction band, a vacancy is left in valence band within silicon crystal. These vacancy is called a hole. For every electron raised to the conduction band by external energy, there is one hole left in the valence band, creating what is called an electron-hole pair. Recombination occurs when a conduction-band electron loses energy and falls back into a hole in the valence band

.3 Depletion Layer: The free electrons in the n-region are randomly drifting in all directions. At the instant of the pn junction formation, the free electrons near the junction in the n region begin to diffuse across the junction into the p region where they combine with holes near the junction. When pn junction is formed, the n region loses free electrons as they diffuse across the junction. This creates a layer of positive charges near the junction. As the electrons move across the junction, the p region loses holes as the electrons and holes combine, which creates a layer of negative charges near the junction. These two layers of positive and negative charges form the depletion region. As electrons continue to diffuse across the junction, more and more positive and negative charges are created near the junction as the depletion region is formed. A point is reached where the total negative charge in the depletion region repels any further diffusion of electrons into the p region and the diffusion stops. In other words, the depletion region acts as a barrier to the further movement of electrons across the junction.

1.4 N-type Silicon: To increase the number of conduction-band electrons in intrinsic silicon, pentavalent impurity atoms are added. These are atoms with five valence electrons such as arsenic, phosphorus (P), bismuth (Bi), and antimony (Sb). Each pentavalent atom forms covalent bonds with four adjacent silicon atoms. Four of the antimony atom’s valence electrons are used to form the covalent bonds with silicon atoms, leaving one extra electron. A conduction electron created by this doping process does not leave a hole in the valence band because it is in excess of the number required to fill the valence band. The electrons are called the majority carriers in n-type material. Holes in an n-type material are called minority carriers.

1.5 P-type Silicon: To increase the number of holes in intrinsic silicon, trivalent impurity atoms are added. These are atoms with three valence electrons such as boron, indium, and gallium. Each trivalent atom forms covalent bonds with four adjacent silicon atoms. All three of the boron atom’s valence electrons are used in the covalent bonds. Because the trivalent atom can take an electron, it is often referred to as an acceptor atom. The number of holes can be carefully controlled by the number of trivalent impurity atoms added to the silicon. A hole created by this doping process is not accompanied by a conduction electron. The holes are the majority carriers in p-type material. Conduction-band electrons in p-type material are the minority carriers.

1.6 Anti Reflection Coating: To reduces the reflection of light from the surface of the solar cell, to further reduce the reflection of incoming radiation from sun in order to maximize the absorption of light, a silicon nitride film (SiNx) or other such properties material, which acts as Anti reflection coating, is deposited by plasma enhanced chemical vapor deposition or any other method on the front surface of the solar cell. This film serves as a passivating layer for the front surface of the solar cell in addition to serving as an anti-reflection coating. This film must be optimized to absorb the majority of incoming radiation as well as passivate the surface satisfactorily. The target thickness of anti refelcting coating for baseline cells is set at 780nm1.

1.7 Metallic conducting Strips: Continuous efforts to develop new materials and modeling techniques for solar cells are being made in order to produce new photovoltaic devices with improved electrical performances. In addition to the new semi conducting materials, solar cells consist of a top metallic grid or other electrical contact to collect electrons from the semiconductor and transfer them to the external load. In a solar cell operating under the normal conditions, even a small deviation from the optimum power condition can cause a loss of conversion efficiency2.

**2. Review**

The rays of light neither mutually color each other, nor mutually illuminate each other, nor mutually impede each other in any way. This is just like one physical motion’s not impeding another as study by Kepler and first observe the scattering of photons by photons in an experiment seems to have been undertaken in 1928 in the Soviet Union by S. I. Vavilov. In the experiment, no experimental sign of photon-photon collisions was found and Hughes and Jauncey give as bound for the cross section.

When light and sound simultaneously pass through a medium, the acoustic phonons of the sound wave scatter the photons of the light beam. This scattering of light from acoustic modes is called Brillouin scattering. A particularly interesting effect of Brillouin scattering has to do with the frequency of the scattered light. An incident photon can be converted into a scattered photon of slightly lower energy, normally’ propagating in the backward direction, and a phonon. For a Stokes process, where a phonon is generated, the frequency of the scattered light is decreased; for an anti-Stokes process, where a phonon is annihilated, the frequency of the scatted light is increased. Increased frequency, by the equation E = hv, means increased photon energy. The difference between the energy of the scattered photon and the incident photon is called the Brillouin shift4.

QED is normally used to describe the scattering of electrons. It is recognized in QED, the quantum mechanical formalism describing the scattering of photons would be different than the scattering of electrons as the cross sections for scattering or their interaction coefficients with the lattice would be different. In describing diffraction consistent with the conventions of QED, we adopt a momentum representation for the Coulomb potential associated with the lattice. The most probable values for the magnitude of the y-momentum of the virtual photons associated with the scattering potential are integral multiples of h/2d. On recognize these energies are the eigenvalues of the photon standing wave eigenfunctions for the particle-in-a-box problem in quantum mechanics where the length of the box is d. The scattering probability distribution observed with photon diffraction is derived from a function of the y-momentum exchanged from the scattering by virtual photons of the lattice summed over the probabilities or densities of the virtual photons with the different momentum values5.

Comparing spectra, the Brillouin shift is much smaller than the Raman shift because the velocity of acoustic waves is much less than the velocity of light. This was already known from the spectrum of spontaneous scattering, where the Raman process gives a much larger shift than Brillouin scattering. Parametric processes require conservation of both energy and momentum. Stimulated Brillouin scattering occurs when a beam of laser light generates a parametric process that simultaneously produces an exactly retrorefected Stokes beam and an acoustic wave traveling in the forward direction. Energy conservation requires that the Stokes beam frequency is reduced from the laser frequency by the frequency of the acoustic wave. Historically the term “Stokes was named for Sir George Gabriel Stokes, who in 1852 described the change in wavelength of fuorescence, which is always at lower photon energy than the incident light. When Raman scattering was discovered, similar shift to lower photon energy was called Stokes light. When Raman scattering was discovered to have weak signals at shorter wavelength than the incident light, this was called “anti-Stokes” light6.

One or several scattering processed Rayleigh scattering, Brillouin scattering, and Raman scattering, can occur due to the interaction of an incident wave with a medium. When the intensity of light is low the resulting scattering process will be spontaneous. However, when the incident intensity reaches a certain threshold, stimulated scattering will be observed with a strong interaction between light fields and matter. In Brillouin scattering is both stimulated and spontaneous, a pump photon at a frequency up produces an acoustic phonon and a red-shifted. The energy and momentum conservation requirements on Stokes and anti-Stokes Brillouin scattering are as follows. Slow light, the propagation of an optical pulse at a very low group velocity, is of interest for enhancing the interaction of light and material and to provide higher controllability of the gain spectrum bandwidth. SBS has been studied in a variety of gas, liquid, and solid media for different applications, and SBS capabilities vary greatly depending on the choice of gain medium. Thus overall potential for SBS applications is broad, selection of the right gain material for a particular application is critical. In practice, many more parameters need to be taken into consideration when selecting a medium, including the transmittance at the wavelength of interest, gain coefficient, generation and damage threshold, Brillouin frequency shift and line width, environmental sensitivity, toxicity, as well as available size7.

The molecular theory of matter starts with quantum mechanics and statistical mechanics. According to the quantum mechanical Heisenberg Uncertainty Principle, the position and momentum of an object cannot be determined simultaneously and precisely. The Heisenberg Uncertainty Principle helps determine the size of electron clouds, and hence the size of atoms. Heisenberg’s Uncertainty Principle applies only to the subatomic particles like electron, positron, photon, etc. It does not forbid the possibility of nanotechnology, which has to do with the position and momentum of such large particles like atoms and molecules. This is because the mass of the atoms and molecules is quite large, and the quantum mechanical calculation by the Heisenberg Uncertainty Principle places no limit on how well atoms and molecules can be held in place8. The atomic radius is taken as half of the inter atomic distance in a crystalline state. The atomic radius of elements and the relationship with their position in the periodic table, together with the numerical values of atomic radius9.

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]]>The post Case Study of Shortest Path Algorithms and Implementation using MATLAB appeared first on Zambrut.

]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: July 25, 2019

**Yee Yee Htun**

Ph.D (Applied Maths), Department of Engineering Mathematics

TU (Hmawbi), Hmawbi Township, Yangon Division, Myanmar

Journal Full Text PDF: Case Study of Shortest Path Algorithms and Implementation using MATLAB.

**Abstract**

Shortest path problems are among the most studied network flow optimization problems withinteresting application across a range of fields. In this paper, three shortest path algorithms arediscussed via Dijkstra’s Algorithm (one to all pairs of nodes), Floyd Warshall’s Algorithm (all to allpairs of nodes) and Linear Programming Problems (LPP). These algorithms are also solved usingMatlab software, which gives quick results for larger nodes. By this research, we can sucessfully study how many ways to find shortest path. Graph technique in Matlab can also be applied to be simply solved the shortest path problems. The application of Direct Graph and Undirect Graph of shortest path was implemented for the route of ferry bus, North Dagon township to TU (Hmawbi).

**Keywords:** Shortest path, Dijkstra’s Algorithm,Floyd Warshall’s Algorithm, Linear Programming Problems, Matlab software & Direct Graph.

**1. INTRODUCTION**

Finding the shortest path is an important task in network and transportation related analysis. Shortestdistance problems are inevitable in road network applications, such as city emergency handling anddriving system, where optimal routing has to be found. Therefore, network optimization has alwaysbeen the heart of operational research. Also, as traffic conditions of a city change from time to time,there could be a huge amount of request occurring at any moment, for which an optimal path solutionhas to be found quickly. Hence, efficiency of an algorithm is very important to determine the shortestroutes are between nodes in a network.

There are many algorithms that can be used to determine the shortest route between two nodes in anetwork. In this paper, two standard algorithms Dijkstra’s algorithm and Floyd Warshall’salgorithm are discussed and also solved using Matlab software. The linear programming formulation of shortest route problem solved using (0-1) binary integer programming technique is also discussed.

The dual of formulated linear programming and shortest route problem solved by algebraic method is demonstrated for small number of nodes, as it is difficult to solve for large number of nodes. In such cases, Matlab software can be the best choice. Further, the shortest distance and shortest routedetermined using Complementary Slackness Theorem.(Dr. Roopa, K.M., 2013).

**2.GRAPH IN MATLAB**

A directed graphwith four nodesand three edges is as shown in Picture 1 in Matlab. Graph theory functions in the toolbox apply basic graph theory algorithms to sparse matrices. A sparse matrix represents a graph, any nonzero entries in the matrix represent the edges of the graph, and the values of these entries represent the associated weight (cost, distance, length, or capacity) of the edge. Graph algorithms that use the weight information will cancel the edge if a NaN or an Inf is found. Graph algorithms that do not use the weight information will consider the edge if a NaN or anInf is found, because these algorithms look only at the connectivity described by the sparse matrix and not at the values stored in the sparse matrix.(matlabexpo.co.kr, 2016).

Picture 1. Direct Graph example

Sparse matrices can represent four types of graphs:

1) Directed Graph — Sparse matrix, either double real or logical. Row (column) index indicates the source (target) of the edge. Self-loops (values in the diagonal) are allowed, although most of the algorithms ignore these values.

2) Undirected Graph — Lower triangle of a sparse matrix, either double real or logical. An algorithm expecting an undirected graph ignores values stored in the upper triangle of the sparse matrix and values in the diagonal.

3) Direct Acyclic Graph (DAG) — Sparse matrix, double real or logical, with zero values in the diagonal. While a zero-valued diagonal is a requirement of a DAG, it does not guarantee a DAG. An algorithm expecting a DAG will not test for cycles because this will add unwanted complexity.

4) Spanning Tree — Undirected graph with no cycles and with one connected component.There are no attributes attached to the graphs; sparse matrices representing all four types of graphs can be passed to any graph algorithm. All functions will return an error on nonsquare sparse matrices.

2.1 Finding the Shortest Path in a Directed Graph

1) Create and view a directed graph with 6 nodes and 11 edges.

2) Biograph object with 6 nodes and 11 edges.

3) Find the shortest path in the graph from node 1 to node 6.

4) Mark the nodes and edges of the shortest path by coloring them red and increasing the line width.

This example Graph results can be shown as in Picture 2 (a).

2.2 Finding the Shortest Path in an Undirected Graph

1) Create and view an undirected graph with 6 nodes and 11 edges.

2) Biograph object with 6 nodes and 11 edges.

3) Find the shortest path in the graph from node 1 to node 6.

4) Mark the nodes and edges of the shortest path by coloring them red and increasing the line width.

This example Graph results can be shown as in Picture 2(b).

(a) (b)

Picture 2. Direct Graph and Undirect Graph in Matlab for 6 nodes(www.mathwork.com)

**3. SHORTEST PATH ALGORITHMS**

3.1 Dijkstra’s Algorithm

Dijkstra’s algorithm considers two sets:

i) set P, which at any specific point consists of all the nodesthat were encountered by the algorithm;

ii) set S, a precedence set, which at any specific pointconsists of the precedent node for each node in the network. Apart from these sets, the algorithmutilizes the following distance information.

qij, for i, j=1, 2, 3, 4…n and i≠j, denote the weight of the directed edge (arc) from vertex i to vertex j.

If there is no arc from i to j, then qij, is set to be infinity.tj, for j=1, 2, 3, 4…n and j≠s where s is the start index. Also,

tj = q1j for j=2,3,4…n (1)

In each iteration, the sets P and S as well as the set of all tj, for j=1, 2, 3, 4…n and j≠s, that are outputfrom the previous iteration are taken as inputs. Initially P = {s}. S is a set of size n populated with

i) 0 if tj = infinity, ii) s if tj = finite value

The steps involved in each iteration for finding the shortest distance are summarized below:

Step 1: Identify minimum among the computed tj values. Let tk be the minimum.Add k to the set P.

Step 2: Now P = {1, k}. For each of the nodes not in P and with finite qkj, for j=1, 2, 3, 4…n and j ∉

P, recalculate tj using the below expression:

tj = min{ tj, tk + qkj } (2)

Only if (tk + qkj) < tj, then update the jth entry in S to k. Continue the iterations until the end node, e, is added to the set P.Similarly, the steps to trace the shortest path between nodes s and e, using Dijkstra’s algorithm aregiven below:

Step 1: Take node e as the last node in the shortest path

Step 2: Find the eth entry in the set S, let this be x. Add this prefix node x to the partially constructed

shortest path.

Step 3: Check whether x is equal to s. If so, go to Step 4; else go to Step 3.

Step 4: The required shortest path from node s to node e is thus constructed.

Picture3 shows an example of using Dijkstra’s Algorithm.

Picture 3. Example of Dijkstra’s Algorithm

3.2Floyd Warshall’s Algorithm:

Floyd Warshall’s Algorithm is a graph analysis algorithm to find the shortest route between any twonodes in a network with positive or negative edge weights with no negative cycle. This algorithm usesthe dynamic programming technique to solve the shortest path problem between all pairs of nodes (allto all) in a directed network. It represents the network as a square matrix with n-rows and n- columnsand at the end of the algorithm each (i,j) of the matrix gives the shortest distance from node i to node j.If there is a direct link between node i to node j, then the value at (i,j) is finite, otherwise it is infinite,i.e, d (i,j)= ∞.

The steps to trace the shortest path between two nodes, say i and j using Floyd-Warshall’s

algorithm are given below:

Step 1: Take node j as the last node in the shortest path.

Step 2: Find the value S [i, j] from the precedence matrix Sn, let it be x. Add this Prefix node x t

partially constructed shortest path.

Step 3: Check whether x is equal to i. if so, go to step (4); else, set j = x and go to the step 3.

Step 4: The required shortest path from node i to node j is constructed.

Picture 4. Shortest Route Network

To determine the shortest distance and shortest paths between all pairs of nodes in atransportation network as shown in Picture 4, using Iteration (3): Set k=3. Consider third column and third row of D3 as pivot column and pivot rowrespectively. Except d (1, 3), all the entries in the pivot column are infinity and also except d (3, 5) andd (3, 7), all the entries in the pivot row are infinity. Further, apply transitivity property to obtain thefollowing results:

(i) Since, d(1,4)=17 , d(1,5)=14 and d(1,7)=32 . So, d (1,7) = 32 cannot be improved .

(ii) Set precedence matrix S2 as S (1, 4) = 3, S (1, 5) = 3. The changes are as shown in the

matrix D3and S3.

Continuing in this way, the final matrix in the last iteration where none of the entries in the d (i,j) canbe improved by transitivity property, because all the elements in the last row are infinity.Finally, the shortest distance between any two nodes is determined from the matrix D7 as shown in Picture 5.

Picture 5. Iteration 7 Results ( Dr. Roopa, K.M., 2013).

3.3 Algebraic Method for solving the dual Linear Programming Problem

The dual linear programming problem can also be solved using algebraic method for only small

number of variables. However, solving the above dual Linear Programming Problem throughalgebraic method, by introducing slack variables which gives better result compared to any othersoftware packages. By using the first and final tableaus of algebraic method , the dual problem can also be solved using Matlab software. For example, if the solutions obtained from Matlab software aregiven below:

y1 = –10.4979, y2 = 3.6415, y3 = –0.4979, y4 = 6.5021, y5 = 3.5021, y6 = 5.5021, y7 = 11.5021

The value of Z = 22 gives the shortest distance from node 1 to node 7. By considering, the solutions thatsatisfy the above constraints the following routes: 1-3, 3-4, 3-5, 4-7, 5-7, 5-6 and 6 -7 are obtained. Fromthese sequence of routes 1-3, 3-4, 4-7, the shortest route 1—3—4—7, which is of distance 22 units fromnode 1 to node 7 is traced. Similarly, other alternate shortest routes that can be obtained are: 1—3—5—7and 1—3—5—6—7 respectively.

The shortest route can also be determined using Complementary Slackness Theorem . As the

sequence of routes 1-2, 3-2, 2-7, 2-4, 4-6 and 4-5, do not satisfy the constraints in the dual problem, fromthe Complementary Slackness Theorem it follows that x12=x32 = x24 =x27=x45 =x46 = 0.

Substituting these variable values in the primal problem, the following systems of equations are obtained:

x13 = 1; x13- x34 –x35 =0;x34-x47=0;x35-x56 – x57 =0;x56–x67 =0; x47+x57 +x67 =1 (3)

By solving above system of linear equations using Gauss Elimination Method, the system in echelon formbecomes:

x34+ x35 = 1; x35+ x47 =1; x47+x56 + x57 =1; x47+x57 +x67 =1 (4)

In the above system of equations, there are 4 equations (r=4) with 6 unknowns (n=6) and two freevariables (x35, x56,). Hence, the possible choices are: (0,0),(0,1),(1,0),(1,1). Each of these possible choicesmay or may not be the solution points because the dependent variables have the restriction, xij = 0 or 1.( Dr. Roopa, K.M., 2013).

**4.IMPLEMENTATION OF SHORTEST PATH**

(BETWEEN NORTH DAGON-MAWATA BUS STOP AND TU (HMAWBI))

The case was assigned as to find shortest path between Mawata Bus Stop of North Dagon township and TU(Hmawbi). By solving this case, the author can apply low cost and minimum time to arrive TU(Hmawbi) from her home. There are 9 main Bus Stations as focal points. But there are a lot of Bus Stops Between each focal Bus Stations. Unit are per miles to be considered. The m-script of this case is shown in Picture 6.

Table-1 Nodes Assignment

No. BUS STOPS NODE

1 Ma wa ta 1

2 Bay lie 2

3 Aung migalar highway station 3

4 8 miles 4

5 Saw bwar gyi kyone 5

6 Htaut kyant 6

7 Toll gate 7

8 Hmawbi market 8

9 TU Hmawbi 9

Picture 6. M-script of Implementation Case

**5. RESULTS**

As a result of implementing shortest path in Matlab, author can make effort of cost and time to go to office from home everyday. The Resut is to use Bus No (99) between Mawata Bus stop to Sawbwargyi Goan Bus stop, and again to TU(Hmawbi) with Bus No (37). The minimun cost for a route is 700 MMK. The results from Matlab program are shown in Picture 7 (a) and (b).

(a) Direct Graph (b) Undirect Graph

Picture 7. Implementation Results

**6. CONCLUSION AND RECOMMENDATION**

This paper has presented the results of implementing or application of shortest path algorithms in real case. Effectiness of Matlab software can also be proved in this paper. All three algorithms of shortest path finding methods are studied and compared. It is evident that Dijkstra’s algorithm takes a relatively lesser time than Floydsand Binary integer programming in finding shortest route. However, Dijkstra’s algorithm is the betteroption for identifying the shortest path in larger networks such as railway, water, power distribution andgas pipeline networks. For simplicity, the author mainly applied Direct Graph methods that used Dijkstra’s algorithm in background of Matlab. This research paper carried out mainly case study of three shortest path finding concepts and analysis with real world application.

**7. REFERENCES**

1. Dr. Roopa, K.M., Apoorva, H.R1., Srinivasu,V.K.2 and Viswanatah, M.C 3,“A Study on Different Algorithms for Shortest Route Problem”, International Journal of Engineering Research & Technology (IJERT) ,Vol. 2 Issue 9, September,2013.

2. Algorithms in Java, Chapter 21,http://www.cs.princeton.edu/introalgsds/55dijkstra.

3. Floyd, Robert W. (June 1962). “Algorithm 97: Shortest Path”. Communications of the ACM 5 (6): 345. doi:10.1145/367766.368168 (http:/ / dx. doi. org/ 10. 1145/ 367766. 368168).

4. http://www.matlabexpo.co.kr, 2016.

5. http://research.microsoft.com/users/goldberg.

6. http://www.mathwork.com

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]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: July 23, 2019

**Maman Budiman, Dwi Gunarto & Randy Asmuni**

Pancasakti University, Tegal

Widya Dharma University, Klaten

Indonesia

Journal Full Text PDF: Method Development Model Consensus on Analytic Hierarchy Process.

**Abstract**

This paper propose a model of consensus on the method of AHP (Analytic Hierarchy Process) to decision-making group. The consensus method can provide information about the level of agreement and disagreement, and disagreement range of individuals and groups, the structure of the cluster, the object identification on decision “problematic” and a marginal opinion. Formation of the consensus made by Delphi method approach, discussion (exchange of information) is performed to obtain a homogeneous opinion. Rules do when the termination discussion found more homogeneous the opinion of 66.6 percent, or based on time constraints.

**Keyword:** Analytic Hierarchy Process, consensus & homogeneous.

**I. INTRODUCTION**

Group decision-making process in general produces a complex decision. Each party involved in the decision making process, can have the values and principles are different. The difference in value or principle of life, can make a difference / conflict of goals / interests, which can cause differences in preferences between the parties involved, the situation is called a conflict. In conflict situations, ideally one party does not impose desires (strategy) to achieve his own wishes, but must be willing and able to work together (win-win) through negotiation.

The process of cooperation is based on the intention sincere and open, will achieve mutually beneficial results, even can give a synergy effect, so that the results will be obtained by each party will be better than the process that is competitive (win-loose), Putro and Tjakraatmadja [1].

The purpose of group decision making occurs when the owner of the decision (decision) subjectively included as participants in decision-making, which in some cases the participants did not know all the activities as a whole, Brugha [2]. In fact, to be able to describe the problem well in the evaluation, a decision, a decision maker needs to consider the opinions of others who understand the problem.

This paper propose a model of consensus on the method of AHP (Analytic Hierarchy Process) to decision-making group. The model developed is the development of Techniques for Analyzing Consensus Relevant Data / ACRD developed by, Ngwenyama [3].

**II. LITERATURE REVIEW**

Forman and Peniwati [4], said the process of synthesis by the AHP method can be done in one of two ways:

• Aggregation of individual assessment method (the aggregation of individual judgment / AIJ), or

• Method of aggregation of individual priorities (the aggregation of individual priorities / AIP).

For AIJ method it is assumed that each member of the group acts as a single entity and can no longer act on the individual’s identity. The assessment results of each individual in the group are aggregated with the geometric mean method. For this case, the Pareto principle is not relevant so it does not need to be considered.

In the AIP method, it is assumed that the group members act as individuals who are independent of the other group members. The priority sequence produced by each individual in the group are aggregated with the average method geometric or arithmetic average, and both of them do not violate the Pareto principle.

In producing the preferences of individual preference groups, to consider the fulfillment of the Pareto principle. According to the Pareto principle, if the two alternatives, the alternatives alternatives a1 and a2, compared and each individual group members preferred alternative compared to alternative a1 a2, then the group should prefer an alternative rather than an alternative a1 a2.

Zahir [5], stating that in a large group there are various possible patterns of thinking, namely:

a. All members of the group have the same thought

b. The views of members vary but they are a member of a coherent homogeneous group

c. There are several clusters or homogeneous sub-groups within a large group.

Homogeneous group does not demand identical preferences of each individual group members. In a homogeneous group, each individual opinion remains varied, but has a “similarity”.

The similarity of this argument can be seen by comparing the value of the cosine angle formed by a pair of individual preferences of group members with a limit of homogeneity (). If the value of the cosine angle formed by a pair of individual preference greater than or equal to the limit value homogeneity, Then a couple of individuals can be said to have a “similarity”.

2.1 Techniques for Analyzing Consensus Relevant Data/ ACRD (Ngwenyama et. Al., 2006)

The decision making process in computer-supported group allows anonymous process takes place with the help of a facilitator as a steering discussions. To generate information for the facilitator, [3] proposed a number of techniques and approaches for analyzing data group preferences in the decision process.

Data analysis techniques relevant to the consensus (the techniques for analyzing the consensus of relevant data / ACRD) proposed by [3], is utilizing the preference data individually produced by each decision maker (value and sequence) in a group against a set of alternative decisions using AHP.

The analysis was performed based on the similarity of individual partner preferences of group members, which is indicated by the cosine of the angle is greater than or equal to the value limits of agreement.

The disagreement between a pair of individuals reached if the cosine angle value less than or equal to the limit value of disagreement.

Value, which probably is 0985 (cosine angle 10o) and value which probably is 0.966 (cosine angle 15o). Rationalization of making these angles is that the largest possible angle between two vectors weight is 90o. So 0o 10o-90o on a scale equivalent to 1 on a scale of 1-9, and 15o equivalent to 1.5 on a scale of 1-9.

Conceptually, this approach to support this ACRD techniques can be divided into three stages, namely:

a. The pre-evaluation stage

b. Stage generate preferences

c. Phase data analysis and reporting

Pre-evaluation stage includes the selection of alternatives for evaluation and determination of the evaluation criteria, as well as the delimitation agreement to define the rules of the termination of the decision-making process, such as the achievement of an allocated time interval or a certain level of agreement on the issue of (partial or full).

While on stage do the sorting preference produce alternative and presentation of the data comparison using AHP.

At the stage of data analysis and reporting analyzing the preference data performed by the decision makers to identify their position in group decision-making process. In this phase also the identification of possible coalitions, identify problematic decision alternatives, and identification of key individuals who have a preference position that enables the negotiation of consensus.

This approach allows the facilitator in the group to assess the level of group consensus at every stage of the group, so that the resulting information can help the facilitator to negotiate the formation of a consensus within the group. Consensus can be achieved when the consensus map has been identified that describe the preferences that can be accepted by the group.

In an ideal situation, should be reached complete agreement within the group. But in general, taking into account the differences of opinion, this is not possible, so it takes the rules of termination.

Individual indicators expressed by the Consensus Individual Vector (ICVt), which is used to identify individuals who have a good level of agreement with other group members and do not have barriers that make it difficult. The key individuals have the greatest ICVt value (ICVtmaks). Individual keys are used to facilitate the formation of a group consensus.

2.2 Cluster algorithm Zahir (2009)

For medium-sized groups (intermidiate-sized group) or a group of large-sized (large group), group homogeneity can not be guaranteed or achieved, so [5] proposed a clustering algorithm based on the method VAHP (the Vector Space Formulation of the Analytic Hierarchy Process).

By using this algorithm, in a group consisting of N members can be formed each cluster homogeneous, where 1 N,

Clusters are naturally determined by the value of cluster membership boundary (). The value of cluster membership boundary () Varies depending on the type of problem. This cluster membership limit values set by agreement of members of the group.

To determine the membership of a decision-maker to a cluster, cosine of the angle between the weight vector of the decision makers and the resultant weight vector of all decision makers in the cluster than the cluster membership boundary value ().

If the cosine of the angle between the weight vector of the decision makers and the resultant weight vector is greater than or equal to the limit value of cluster membership (), Then the decision maker was elected to be a member of the cluster.

Vice versa, if the cosine of the angle formed between the decision makers of the weight vector and the resultant weight vector is smaller than the limit value of cluster membership, the decision was delayed to have become a member of the cluster.

This delayed decision makers should wait for a re-elected into the cluster until there is another group of decision makers who are elected cluster. If not selected, the pending decision makers have to wait to be placed in another cluster.

The formation of clusters in the algorithm, [5] using a Monte Carlo simulation.

**III. METHOD DEVELOPMENT MODEL**

Systematics design modeling cycle as a model to follow;

As an initial step in the development of the model is done the problem definition. Furthermore, based on the definition of the problem formulated a conceptual model that shows the relationship between the variables that determine the behavior of the model. This model includes verbal model which only outlines the relationship issue, a system, and the purpose of the study.

Objective studies provide indications of performance to be achieved and provides a framework of conceptual models that form the expected performance. To operationalize the conceptual model of symbolization and determination of quantitative rules. Idealization and simplification linkage model variables referred to as the characterization phase models. Model formulation conducted as early development of formal models that show the size of the model performance as a function of the variables of the model.

In the formulation of the model used teleologik principle (review the modeling purposes) for memfungsionalkan attributes by looking at the destination (Teleos) of the system. Through a systems approach, the existence of the system and its environment can be understood by knowing the elements of the system, the relationship between elements and attributes of each element.

Environment system is a collection of objects outside the limits (boundaries) system that affects (affected) systems.

After the initial formulation of the model is complete, then the model’s ability to reproduce the properties and behavior of the real system testing. In this case the testing is based on three criteria to evaluate the model, namely:

a. tested the suitability of the model behavior with the behavior of the real system represents

b. testing the structure between the model variables.

c. estimates for variables, testing the availability of estimated values for key variables.

**IV. DEVELOPMENT MODEL CONSENSUS ON AHP METHOD**

For group decision making, decision-making is done by a group of individuals who are considered worthy to determine a decision. The decision group is considered better than individual opinions. Assessment conducted by many participants will be possible to produce a different opinion from one another, Anonymous [6].

Preferences group with AHP method is generated by synthesizing the preference of each individual group members. However, to carry out the synthesis of individual preferences, there are prerequisites that its decision-making should be ensured homogeneous [5].

Analysis of the achievement of group consensus can be done with attention to the “sameness” of individual partner’s opinion partisan group members / respondents, the opinions of those individuals can be incorporated into a homogeneous clusters,

Formation of the cluster structure can be made by utilizing the information on individual agreements, the level of individual disagreements and disagreements range of individuals who will be included in a cluster. The resulting cluster structure can provide information to identify the possible presence of a marginal opinion of individual members of the group, which can be used as a reference for the group members to obtain the consensus of the group.

The structure of the cluster can be used to identify objects either problematic decision criteria and alternatives, taking advantage of the aggregate value of the cosine on that object. Based on this information, for objects that have a low weight, do a study on the possibility of the removal of that element from the process of decision analysis.

4.2 Formulation Model

4.2.1 Analysis of Consensus in Group Decision Making

Development of consensus methods made to the development of group decision based on concepts that have been previously known. Development of the consensus method is divided in four main stages, namely:

a. Pre-evaluation

At this stage an agreement to establish the boundaries that form clusters. Objects decisions made for each criteria and alternatives. While the agreement stipulated limit is the value of the deal retang individuals in the group, The value of disagreement, And the limit value of cluster membership, And the rules of termination, either a time limit or level of consensus reached.

b. Generate preferences

At this stage, the individual preferences of each member of the group of objects decisions made based on the method of AHP, including testing for consistency.

c. Data analysis

At this stage, the determination of the value of the aggregation of individual preferences into preference groups, the analysis of the level of agreement and disagreement individuals, as well as the formation of cluster structures.

d. Formation of consensus

Formation of the consensus made Delphi method.

4.2.2 Preferences group

Preferences group is the value of “average” individual preferences. preferences group AIP is calculated based method [4], it is assumed that the group members act as individuals who are independent of the other group members. The priority sequence produced by each individual in the group are aggregated with the average method arithmetic average.

… (1)

Where:

= Weighting preferences to-element group i

= Weight of individual preferences to-element i

n = Number of individuals in the group

To see the relationship between the value of individual preferences with the preferences of the group is determined based on the amount of the angle, To facilitate the analysis is then performed the conversion value the value cos, Under the condition:

– cos approaches a value of 1, indicating strong agreement between the preferences of individuals with a preference group

– cos approaching a value of 0, indicating a weak agreement between individual preferences with the preferences of the group

Figure 1. Individual preferences and the preferences of the group in Vector Spaces

For the purposes of data analysis, carried out the determination of the level of agreement and disagreement individuals, as well as the formation of cluster structures.

4.2.3 Range of agreement

The range of agreements taken in this paper is the value ranges of the agreement Of 1.0 on a scale of AHP, or the difference 10oin a vector space. The basis used to determine the range of the deal is the rationalization of making these angles is that the largest possible angle between two vectors weight is 90o. So 0o 10o-90o on a scale equivalent to 1 on a scale of 1-9, and 15o equivalent to 1.5 on a scale of 1-9, [3].

4.2.4 Value limits of agreement and disagreement

Agreements limit value set at (For a strong agreement) are set to the value of 0985 (cosine angle 10o) and (For strong disagreement) is determined by the value of 0.966 (cosine angle 15o), the members of the group said to have a strong agreement if the cosine smaller than the value and a strong disagreement if cosine greater than,

4.2.5 Formation of the cluster structure

The structure of the cluster using cluster membership delimitation. Ordinance on the development of consensus analysis in this paper uses membership limits as big as 10o. Cluster structure was formulated on the development of the model are presented in Table 1.

Table 1 Cluster structure

clusters restriction limitation cosine

Cluster-1 0 ° – 5 ° 1000-0996

Cluster-2 5o – 15o 0996-0966

Cluster 2 ‘

Cluster 3rd 15o – 25o 0966-0906

Cluster to-3 ‘

Cluster 4th 25o – 35o 0906-0819

4th cluster ‘

Cluster 5th 35o – 45o 0819-0707

Cluster to-5 ‘

Cluster 6th 45o – 55o 0707-0574

Cluster 6th ‘

Cluster 7th 55o – 65o 0574-0423

Cluster 7th ‘

Cluster 8th 65o – 75o 0423-0259

Cluster 8th ‘

Cluster 9th 75o – 85o 0259-0087

Cluster 9th ‘

Cluster 10th 85o – 90o 0087-0000

Cluster 10th ‘

Information:

(A) Cluster to-n indicates the direction of the horizontal axis

(B) Cluster to-n ‘indicates the direction of the vertical axis

As an illustration, if the preference groups showed a weight of 0.5 for the ith element and 0.5 for element j, then the cluster structure for this problem is shown in Figure 2.

Figure 2 Individual preferences in space Cluster Group

4.2.6 Coherence individual preferences

To identify individual coherence, the analysis is done by calculating the coherence of individual preferences ( ) based on the average value of cos for each element based on individual preferences to-i

… (2)

Where:

= to-individual coherence i

= Angle formed between the preferences of individuals and groups

n = Number of comparisons related preferences to-element i

Elements that have values 0.966 () Indicate that the individual is “problematic”.

4.2.7 Coherence elements

To identify objects problematic decisions, the analysis done by calculating coherence elements ( ) based on the average value of cos to each individual’s preferences with regard to the i-th element.

Where:

= coherence to-element i

= Angle formed between the preferences of individuals and groups related to the i-th element

n = Number of comparisons related preferences to-element i

m = The number of individual members of the group

Elements that have values 0.966. Indicate that the element (object decision) are “problematic”. Objects problematic decision is a decision if the object is removed from the preference vector will increase the aggregate value of coherence elements.

4.2.8 Termination Rule

Rules set before the termination discussion proper assessment of the decision object. This termination rule can be in the form of a certain level of agreement or a certain time limit which if achieved by the group, then the group should do the synthesis of individual decisions even though the level of agreement has not been reached.

The level of agreement of 0.66 or more indicates that there is a majority in the group (more than two-thirds of the members of the group have the same level of agreement). This means that members of the group had an order of preference are very close. However, if the level of agreement (individuals who are in cluster 1) is less than 0.66, then the group repeats the stages of generating preference.

4.2.9 Formation of Consensus

Formation of the consensus on this model using the rules of the Delphi method. Discussion (exchange of information) was first performed by distributing a marginal opinion on group preferences. The second iteration is done by distributing the opinion that had the most powerful closeness value to the preference group formed in the previous stage (after iteration 1).

If consensus is not formed after two iterations, the facilitator identifies the object decision “problematic”, then held discussions with the members of the group about the possibility of the removal of the element. The next iteration is done and stop until a homogeneous common preference or based on time constraints.

The method developed consensus on this method can provide information about the level of agreement and disagreement, and disagreement range of individuals and groups, the structure of vector preference clusters and each cluster, objects mengidentifikansi decision “problematic” and a marginal opinion.

**V. CONCLUSION**

Consensus method developed in this paper can provide information about the level of agreement and disagreement, and disagreement range of individuals and groups, cluster structure, coherence individual preferences, and the coherence of the object elements of the decision. This analysis provides decision object information “problematic” and that the marginal individual opinion. Formation of the consensus made by Delphi method approach.

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]]>The post Increasing Problem Solving Ability and Motivation Learning Through Grand of Math Teacher Comments on Results Homework (PR) appeared first on Zambrut.

]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: June 14, 2019

**Agustinus Setiawan, Natha Fransiska & Nadine Feronica**

Indraprasta PGRI University, South Jakarta

Atma Jaya Catholic University, South Jakarta

Indonesia

Journal Full Text PDF: Increasing Problem Solving Ability and Motivation Learning Through Grand of Math Teacher Comments on Results Homework (PR).

**Abstract**

Mathematical problem solving ability and student motivation Dipawangi Cianjur SDN is considered still low, and the low level of participation of students with homework, the reason for the emergence of this study. This study used mixed methods Embedded Design. The population is all students of SDN Dipawangi with grade samples VA and VB. The instrument used was a test and non-test instrument. The test instrument in the form of question pretest and posttest, while the non-test questionnaire, observation sheets, and interviews. This research resulted in several conclusions:1) The mathematical problem solving ability of students treated teachers commenting on the results of PR is not better than students who did not receive such treatment; 2) there is no difference in mathematical problem solving abilities increase significantly, between students who received treatment commenting on the results of PR teachers and students who did not receive such treatment; 3) The students’ motivation treated teachers commenting on the results of PR better than students who did not receive such treatment; 4) there is a positive correlation between mathematical problem solving ability of students with student motivation.

**Keywords:** Comments Teacher, Homework (PR), Troubleshooting, Motivation.

**1. Preliminary**

Education has an important role in the life of the nation in an effort to create quality human resources. Basic education is the beginning for further education, and is an integral part-kan of the overall national education system. To improve the quality of education, the government has run to-wins the 9-year basic education, 6 years at the elementary level and three years at junior high school level. Primary education provides basic supplies to the students, to be able to develop lif-THEIR and ready to follow further education.

Mathematics, which is the basis for every discipline, needs to be given to all students from primary schools. Students are provided with the ability to think logically, analytical, systematic, critical, problem solving, creativity, and ability to cooperate. These competencies necessary for students to have the ability to acquire, manage, and use information in order to survive in a state that is always changing, uncertain and competitive.

Some problems are quite disturbing to teachers and parents, among them about homework (PR). Does having a PR or no PR? How long the student is expected to learn at home? At what age, class and where to start penuga ladder-san PR? Can students be successful in achieving a good achievement without PR? Is it worth the time to check homework and teachers write comments on students’ homework has been completed? It would be very desirable to answer all these questions for all subjects and all levels of schooling.

Students tend to be menyele-saikan tasks (including PR) and improve the quality of their learning when they get consistent feedback and constructive (Paulu, 1995:18). Arends (2012:232) states that the feedback provided can be teacher comments on the results of PR students. The provision of these comments can be given in writing or oral (verbal). According Ghandoura (1982:80), the students were treated writing comments on the results of PR, was obtained a score higher than students who are not given such treatment

Beutlich (2008:11) states that homework (PR) has varying degrees to their effectiveness. It is important for teachers to know what elements are doing homework (PR) to be more effective for students. Two factors contribute to the effectiveness of homework that student motivation and parental involvement.

Based on the reviews and the above phenomenon, the author felt the need to do research with the title “Upgrading Pemeca-han and Motivation Mathematical Problems Through a Master’s Comments on Results Homework (PR)”. It is expected that the problem solving and student motivation can be increased with treated teachers commenting on the results of PR.

**2. Homework**

Homework (PR) is a specific task or job either written or oral that must be done outside school hours (especially at home). PR deals with subjects that have been submitted by teachers to improve the mastery of concepts or skills and provide development. PR done by the students and checked by the teacher (Cooper, 1989:1)

Work assignments (homework) is very important in the defense-jarkan students at home and there is no direct communication between teachers, students, and parents. Therefore, using the strategy of estab-belajaran homework (homework) given by teachers in schools, as a support to maximize student learning outcomes, as well as the attention of the parents also become supporters (Paulu, 2006:1).

Homework is not just about academic values that would be obtained at the school. This is consistent with the opinion Arends (2012:45), that PR can be a means for social communication among students, and the source of the interaction between the students and their parents.

Arends (2012:312) suggest guidelines for assigning homework is as follows:

a. is an interesting, potentially fun and make sure students understand their duties.

b. give students a challenging homework and convince them to complete successfully.

c. provided with a frequency that often and little, rather than rare but significant amounts.

d. inform parents about the level of involvement is expected of them.

e. make clear rules on deadlines and other details necessary things.

The Apostle Paul (1998:16) also established guidelines on how long students should spend the time to do homework per day. The guidelines are as follows:

a. grades 1-3 :< 20 minutes

b. grade 4-6 :20-40 minutes

c. grade 7-9 :< 2 hours

d. 10-12 class :1 ½ – 2 ½ hours

**3. Feedback**

According to Slavin (1997:80), feedback or feedback is information about the results of the efforts that have been made student learning. Another definition of feedback is also conveyed by the Arends (1997:62), that the information given to students about their performance; for example on the knowledge they gained from learning

Arends (2012:232) argues that teachers can provide feedback to students in various ways, such as verbal, video or sound recording, testing, or through written comments. The guidelines are quite important about the feedback is as follows:

1) provide feedback as soon as possible after exercise.

2) strive for specific dsan clear feedback.

3) feedback is aimed directly at behavior.

4) maintain proper feedback to the developmental level of students.

5) give praise and feedback on the correct performance.

6) if it gives negative feedback, should be shown how to do it right.

7) help students concentrate on process and not results.

**4. Mathematical Problem Solving Ability**

According Duncker (Adams, 2007:16), a problem arises when the living creatures have a purpose, but do not know how to achieve that goal. Based on the structure, Reitman (Adams, 2007:17) states that the problem can be divided into two types, namely:(1) The problem defined perfectly (well-defined) or closed matters and (2) the problem is defined as a weak (ill-defined) or problems. While based on the context Carpenter and Gorg (Prabawanto, 2013:19) mengiden-tifikasi problem becomes:(1) a mathematical problem related to the real world (outside of mathematics) and (2) the problem mathematically pure (pure mathematical problems) are attached as a whole in mathematics.

Mathematical problem solving ability is very dependent on the problems that exist in mathematics. Therefore, the need for a discussion of mathematical problems (Prabawanto, 2009:54).

Arifin (Kesumawati, 2010:38) reveals troubleshooting indicators, namely (1) the ability to understand the problem; (2) the ability to plan problem solving; (3) ability to perform the work or calculations; and (4) the ability to perform inspection or pengece-kan back.

Students can succeed in solving the problem, if teachers are confident in completing various types of mathematical problems and was able to teach a variety of skills needed (Yee, 2009:54).

According to Caballero (2011:282), when students solve problems, they are often an adventure with feelings and emotions that cause tension during the search for solutions / strategies, to find a solution to these problems. This could have led to an interest, or even vice versa, hampered by negative emotions that trigger anxiety.

According to Rachmat (2001:80) there are four factors that influence the problem solving process:(1) motivation; (2) beliefs and attitudes are wrong; (3) a habit; (4) emotion.

**5. Motivation**

Motivation is generally defined as a state of the self that can generate, directing and maintaining behavior (Woolfolk, 2009:186). Motivation is not observed directly, but rather inferred from some clues as verbalization, choices tasks and activities directed at a specific destination. Motivation is a clear concept that helps us to understand why people behave the way they do (Schunk, 2012:346).

Waege (2009:85) states that the motivation of students can be shown in the consciousness (cognition), emotions (emotion) or habits (behavior). For instance, the motivation of students to get a good performance in mathematics can be shown in the excitement (emotion), if you get a high score on a test. Motivation of students can be shown by the study (behavior) to face a test, as well as in new learning concept (cognition) when studying for a test.

According to Woolfolk (2009:188), teaching can create intrinsic motivation by linking student interest and competencies that support growth. If the teacher has always stressed the intrinsic motivation to energize all of their students, he will be disappointed. Teachers should encourage and foster intrinsic motivation, extrinsic motivation while ensuring that support learning.

The following points indicate that the feedback (feedback) mem-possess strong relationships with student motivation:

a. To increase motivation to learn, the important thing to remember when giving feedback teachers, especially the negative ones is a sense of security (comfortable) students. Teachers should blow-Give negative feedback with warmth, hospitality, and far from being mocking or condescending. So students still comfortable despite getting a correction or negative feedback (Arends, 1997:160).

b. Moreover, according to Kulik (Slavin, 1997:32), so that feedbackcan provide motivation to the students, then the feedback should be given-right with a clear and specific. It is important for all levels of student development, especially for lower grade students.

c. Kulhavy and Stock (1989:280) states that feedback specific informational and motivational (moti-vasi improve student learning).

d. Clifford (1990:23) states that once a negative feedback-pun can enhance children’s learning motivation, origin focuses on the desired performance of teachers (not to the inability of students in general).

**6. Method**

The method used is the method mix (mixed method) model of Embedded Design. This method combines qualitative and quantitative methods together. In this model, there are methods of primary and secondary methods. Researchers chose quantitative methods as the primary method. And as a secondary method, researchers used qualitative data obtained from instruments of observation and interviews, in order to describe the learning process of students’ motivation to learn.

The study involved two samples of equivalent grade categories, namely, the experimental class and control class. The sample classes are formed using an existing class. Both the experimental class and the control class were not chosen at random (Sugiyono, 2015:118). In the experimental group was given treatment teacher comments on the results of homework (PR) students, while the control class is not given treatment teacher comments on the results of homework (PR) students.

The design of this study using nonequivalent control group design (Ruseffendi, 2005:52) the following:

by:

X = Giving teacher comments on the results of homework (PR) students

O = Pretest / posttest

**7. Population and Sample**

The population in this study were all students of SDN Dipawangi District of Cianjur of Cianjur Regency. Based on the understanding of researchers, students at this school have problems in problem-solving ability and motivation to learn. The samples in this research were two samples taken at random from the population. One class of samples taken serve as the experimental group, while the other class as the control class. Randomized class is a class V.

The test instrument used in this study is a test instrument and nontes. The test instrument consisted of five questions that have been tested explanation that hasthe validity, reliability, difficulty index, distinguishing as follows:

**8. Results and Discussion**

Based on analysis of mathematical problem solving ability test, the value of significance (2-tailed) was 0,311. Because Sig. (2-tailed)> α, then H0 is accepted. So that means an increase in the abi-pared with students’ mathematical problem solving treatment given teacher commenting on the results of homework (PR) is no better than students who were not given the treatment, in terms of the whole student.

Based on the results of questionnaire data calculation motivation to learn through SMI method (Method of Successive Interval) values obtained significance (2-tailed) was 0,287. Because Sig. (2-tailed) > α, then H0 is accepted. This means that the average final grade students’ learning motivation experimental and control group did not differ significantly. It can be concluded, given the treatment of student motivation teacher commenting on the results of homework (PR) is no better than students who are not given such treatment.

Through observation and interviews, researchers discovered facts on the ground that that provision of teacher comments on the results of PR can increase students’ motivation. This is in accordance with what was presented by Orsmond (Muir, 2006:26), that motivation can be generated from the feedback of written comments provided by the teacher.

But there are obstacles when mela-kukan treatment to provide written comments on the results of PR student, that teacher must provide additional time to write comments in the form of a correction, a word of praise or encouragement at every PR students. Moreover, if the teacher is assigned to the class of 40 students as the number of students in primary schools in general.

Based on the analysis on the correlation of test data that has been done, it is stated that there is no significant correlation between the abi-pared with mathematical problem solving and motivation of students. It bertenta-ngan with research results obtained Callard (2009:1), that if the student has the ability pemeca-han that will either lead to a high learning motivation.

**9. Conclusion**

Based on the results of data processing and the findings obtained in this study, obtained some conclusions as follows:

a. Mathematical problem solving ability of students treated the provision of teacher comments on homework better results than students who did not receive such treatment.

b. No difference-tan peningka mathematical problem solving ability significant, between students who received treatment commenting on the results of PR teachers and students who did not receive such treatment.

c. Student motivation treated teachers commenting on the results of PR better than students who did not receive such treatment.

d. There is no positive correlation between mathematical problem solving ability of students with student motivation.

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]]>The post Impact of Two Contrasting Vermicomposts on the Fertility Status of a Sandy Soil appeared first on Zambrut.

]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: June 5, 2019

**Nweke, I. A., Ejinkonye, C. & Ogugua, U.V.**

Department of Soil Science Chukwuemeka Odumegwu Ojukwu University, Nigeria

College of Agriculture and Environmental Science, South Africa University, South Africa

Journal Full Text PDF: Impact of Two Contrasting Vermicomposts on the Fertility Status of a Sandy Soil.

**Abstract**

Vermicompost contains chemical nutrients of which has a positive effect on soil and crop life cycle. The study investigated the effect of fruit and vegetable vermicomposts respectively on the chemical characteristics of a sandy soil. The results of the study reveals that the vermicomposts studied increased the macro-nutrient (P, N, Ca Mg K and Na) contents of the sandy soil. The pH of the soil which was very acidic was increased to alkaline (9.09) by vegetable vermicompost (VGC) and neutral level (7.07) by fruit vermicompost (FVC). The OC and MC content of the soil showed elevated levels which were almost double the value recorded in the initial soil before vermicompost treatment. The two vermicomposts mediated over two fold increase in electrical conductivity, 251% for FVC and 100% for VGC respectively. When the two vermicomposts are compared the effect of VGC were more effective on pH, N, Ca, Mg, K, Na and MC of the soil compared with FVC results that was more effective on P, OC and EC contents of the sandy soil. The findings of the present study reveal that the vermicompost from biodegradable wastes has a great future for poor resource farmers and food production generally in the study zone. Hence soil prone to excessive leaching and erodible like sandy soil will no longer be a barrier to crop production when amended with vermicompost as its fertility status will be increased and erodibility greatly reduced.

**Keywords:** Electrical conductivity, exchangeable bases, organic matter, available phosphorous, sandy soil.

**1. Introduction**

Sandy soils pose many challenges to agricultural productivity in humid tropical climate and areas where there are seasonal hot dry climate. They are noted to have low water and nutrient holding capacity due to their low organic matter content and cation exchange capacity (CEC). Their plant available water according to the report of Allen (2007) is ≤ 50 -110mm per meter of soil couple with the fact that due to high soil temperatures in the tropics, soil organic carbon is rapidly lost (Jabbagy and Jackson 2000). The storage capacity for carbon of sandy soil is typically less than 1% because of the low potential to protect carbon from microbial activity (Six et al., 2006). The actual soil carbon content is however much lower than this due to low plant productivity, thus low carbon input rates. Farmers reliant on sandy soils need carefully designed and well integrated water and nutrient management system to increase their productivity and reduce adverse effects on ground water and soil acidity. The fertility of sandy soil can be upgraded through the help of alternative measure such as the use of vermicompost.

Vermicompost is finely divided mature organic matter with high level of plant nutrient availability increased surface area, aeration and drainage, microbial activity and water hold capacity, high porosity etc stabilized by interactions between earthworms and microorganisms (Nweke 2013; Aracon et al., 2008). Soils amended with vermicompost have the capacity according to the works of the following authors, Atiyeh et al. (2002), Arancon et al. (2003), Posstma et al. (2003), Perner et al. (2006), Mba and Nweke (2009) to improve soil moisture, soil aggregation, CEC, higher level of plant growth hormones and humic acids, higher microbial population and activity and less root pathogen or soil borne diseases as well as overall improvement in crop growth and yield (Arancon, et al., 2004; Nweke 2016). This study therefore, attempts to examine the influence of two contrasting vermicomposts on the fertility status of a sandy soil.

**2. Materials and Methods**

**2.1 Study Environment**

The experiment was set up at Faculty of Agriculture Chukwuemeka Odumegwu Ojukwu University Igbariam Campus Anambra State. The area is located between the Latitude 5’40 and 6’45N and longitude 6’40 to 7’20E.

**2.2 Collection of Materials**

The soil sample that was used for the experiment was collected from the Faculty of Agriculture Chukwuemeka Odumegwu Ojukwu University Igbariam campus. The soil was collected using shovel 15-20cm deep after scraping off 0-5cm from the Faculty of Agriculture premises in a plastic container. After the collection of the soil sample, dirty particles, stones and hard clods where carefully removed, aim is to ensure of fine silt before using it for the experiment. 300g of soil was weighed using weighing balance, 10 polythene bags was brought the 10 polythene bags was divided into 2, each 5 polythene bag contained 300g of sandy soil mixed with 50g fruit vermicompost (FVC) and the other 5 bags containing same measurement 300g of soil sample was also mixed with 50g vegetable vermicompost (VGC). The experiment was incubated for 2 months. The chemical properties of soil before incubation are recorded in Table 1. At the end of the study an aliquot of the sample was used to analyze for the chemical properties of soil based on the principles of Black (1965)

Table 1. Chemical properties of sandy soil before incubation with vermicompost

**2.3 Data analysis**

Data generated were subjected to T-Test analysis and mean values were compared using LSD at 5% alpha level.

**3. Result**

The nutrient content of the soil sample before incubation with the vermicompost showed that the chemical properties tested were at their lowest level (Table 1). The pH of the soil was 4.48; the available phosphorous (P) content of the soil was 10.80mgkg-1. The values of total nitrogen 0.03% (TN), organic carbon 0.49% (OC), and exchangeable bases (Ca2+, Mg2+, K+, and Na+) were generally low. The exchangeable acidity (EA), electrical conductivity (EC) and moisture content (MC) were 0.72 cmolkg-1, 110.00 µscm-1 and 16.78 % respectively. The soil contains low level of major nutrient elements. Hence the studied soil is considered poor in these essential plant nutrients.

The result presented in Table 2 showed that the mean value of soil pH measured in water (H2O) for VGC was 9.09 of which is alkaline, while FVC value obtained recorded neutral pH. The application of fruit vermicompost and vegetable vermicompost that was used to amend sandy soil respectively showed that there was great increase in the mean pH value of VGC and FVC. The available P value recorded from Table 2 in FVC was 0.82mgkg-1 higher than the value obtained from VGC. The value obtained from percentage nitrogen (N) from FVC showed that fruit vermicompost that was added to the soil sample (sandy soil) recorded low value compared to the VGC vermicompost. Organic carbon (OC) content showed that there was slight increase in FVC with the value of 0.042% while in the VGC the value obtained were low.

The Ca value from the result in Table 3 showed that both vegetable vermicompost and fruit vermicompost had great impact on the sandy soil. Mg of value 9.44cmolkg-1 recorded in VGC showed that the application of the vermicompost added to the soil sample contributed positively in Mg content of VGC than in FVC. The available K value in both FVC and VGC was low but FVC showed little change in value. For the Na; the value showed that there was slight increase in VGC than FVC. From Table 3 the study showed that the mean value of soil EA in FVC is greater than the value in VGC. The highest EC value was obtained from VGC (Table 3). Percentage moisture content (MC) result in the study showed higher value in VGC compared to FVC in the soil sample (sandy soil).

Table 2 Effect of two contracting vermicompost on sandy soil

Table 3 Effect of two contracting vermicompost on the parameters of sandy soil

**4. Discussion**

The studied soil is acidic in reaction according to the ratings of USDA-SCS (1974) and Chude et al. (2012) who considered soils of pH 4.8-5.1 to be strongly acidic in reaction. The low levels of exchangeable bases (Ca, Mg, K and Na) of the studied soil which are below their critical levels of 2.0-5.0 cmol+kg-1 moderate (Ca), 3.0-8.8 cmol+kg-1 very low to high (Mg), 2.0 cmol+kg-1 (K) (USDA, 1986) indicated that the studied soil is of low base status. This suggests appropriate amendment to provide the deficit between the inherent basic nutrients status, the amount removed by the crops and leaching losses for good crop performance. Since most of the parameters are at their lowest levels it is expected that the studied soil will benefit from vermicompost treatment because vermicompost is known to influence soil parameters positively. The implication of this is that appropriate soil amendment should be practised to realise optimum production capacity of the soil.

The findings from the study showed that there was great effect of FVC and VGC on the soil studied. This was in agreement with the report of (Lazcano et al., 2010). An increased pH range between7.07-9.09 seems to encourage mineralization of plant available nutrient observed in the study through the assistance of microbial decomposition of the vermicomposts. This invariable lead to increased EC status of the soil which could be due to reduced permeability and leaching of the soluble salts. The high OC recorded in the incubated soil visa vie the initial soil may be due to decomposition of OC compound from vermicompost. Soils with low OC have been reported to have low ability to hold cations in the exchangeable forms (Krasilinikoff et al., 2002). Increased moisture content (MC) in vermicompost amended sandy soil probably may be attributed to aggregation of the soil particles by the actions of microorganisms in the vermicompost which provide cementing action between the soil particles. Parthasarathi et al. (2008) observed that composted and worm-worked sludge increased the available soil moisture of a sandy loam soil from 10.5% to 54.4% and 31.6% respectively. In comparison of the two vermicomposts the impact of VGC on the properties of sandy soil was more effective compared to the FVC. Vermicompost emerges as one of the most feasible alternative techniques compared to conventional aerobic composting. This process is not only rapid, easily controllable, cost effective, energy saving, and zero waste, but also accomplishes the most efficient recycling of organic waste and nutrient.

**5. Conclusion**

From the findings of the study VGC performed better than the FVC in nutrient release content. The use of organic material has long been recognized in agriculture as beneficial for plant growth, yield, maintenance of soil fertility and soil amendment. Since worm worked vegetable waste and fruit waste are all use as organic manure which is chemically free and environmental safe, cheap, effortless and affordable, it is advised that farmers should make use of vermicompost in the production of crops and soil amendment.

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]]>The post An Appraisal of Mathematics Content Knowledge Learnt and Implications in the Teaching of Mathematics in English Speaking Primary Schools appeared first on Zambrut.

]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: June 3, 2019

**Forteck Aloysius Betangah**

University of Buea

South West Region, Cameroon, Central Africa

Journal Full Text PDF: An Appraisal of Mathematics Content Knowledge Learnt and Implications in the Teaching of Mathematics in English Speaking Primary Schools (Studied in South West of Cameroon, Central Africa).

**Abstract**

The study appraised mathematics content knowledge learnt and implications in the teaching of mathematics in English speaking primary schools in South west region of Cameroon. The survey research design was used on a target population of 31 mathematics teacher trainers and 6482 primary school teachers. 31 mathematics teacher trainers and 45 primary school teachers formed the accessible population. The instruments used to collect the data were a questionnaire made up of 40 close ended items. Validity and reliability of the instruments were insured with the aid of a primary school head teacher, a mathematics teacher trainer, and the supervisors. The researcher used the direct delivery technique to collect data from 31 mathematics teacher trainers on the questionnaire, 5 mathematics teacher trainers on one interview, and 45 primary school teachers on another interview. All the mathematics teacher trainers were from primary school teacher training colleges located in South West Region, while all the primary school teachers were from primary schools located in the South West Region. Data were analysed descriptively using frequencies and percentages, and also inferentially using Spearman Rank correlation. The results showed that: The mathematics content knowledge learnt in PSTTC has no significant impact on the teaching of mathematics in English Speaking Primary Schools in South West Region. Based on the findings, it is recommended that: More time should be allocated for the teaching/learning of mathematics in PSTTC, the mathematics syllabus for PSTTC should align with that of English Speaking Primary Schools, the weighting of mathematics in PSTTC should be increased so as to motivate student teachers to learn mathematics and the teaching of mathematics. Further research could be carried out on the strategies that could be employed in PSTTCs and during in-service training of primary school teachers, that would enable student teachers and primary school teachers develop positive attitudes and enthusiasm towards mathematics and the teaching of mathematics.

**Keywords:** Mathematics content knowledge, teaching of mathematics, learnt and implications.

**1. INTRODUCTION**

Mathematics is a compulsory subject in primary and secondary schools, as well as in primary school teacher training colleges (PSTTCs) in Cameroon. It is ‘the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them’ (McIntosh, 2013 p. 883). Ali (2013) opines that mathematics is an international language, a way of thinking and organizing a logical proof and it is the subject that is recognized as the mother of all learning with other subjects deriving their concepts from it, in both arts and sciences.

According to Ali (2013), mathematics is regarded as the queen of all sciences such as chemistry, physics, biology and economics, reason why any individual who is competent in mathematical sciences, can equally have the ability to do any other course. Close (2006) says that mathematics facilitates the study of academic subjects especially in the physical and social sciences, problem solving in our personal, educational, and occupational lives, and for studying and making sense of the world around us. Ali (2013) posits that mathematics can be used to determine whether an idea is true or not, or at least, whether it is probably true as a way of thinking, since it gives insight into the power of human mind and becomes a challenge to intellectual curiosity. Ali (2013) adds that mathematics is used in handling money, measurement in fashion and carpentry, as well as in technical economics. MINEDUB (2018) states that mathematics is a creative and highly inter-connected subject that is essential to everyday life, critical to science, technology, and agriculture and engineering, and also necessary for financial literacy and most forms of employment. According to MINEDUB (2018), mathematics develops logical and inferential thinking, as well as the ability to deduce and visualize in time and space. Mathematics according to Maliki, Ngban, and Ibu (2009) is described as a subject that affects all aspects of human life at different degrees (p. 131). According to The National Mathematics Advisory Panel (2008), mathematics is used throughout our daily lives. In Cameroon, mathematics is a prerequisite for admission into some professional programmes such as medicine, engineering, accountancy, agriculture and banking. It also forms part of the study of single subjects like physics, chemistry, economics, biology and geography. Generally, people in all works of life make use of some knowledge of numeracy either consciously or unconsciously. Therefore, the importance of mathematics cannot be over emphasized.

According to Close (2006, p. 53), “mathematics is a key subject in school curricula.” Therefore acquiring quality mathematics education requires quality primary education. There have been calls for primary education to be of desired quality in Cameroon and perhaps other parts of the world. Reason why one of the major concerns of the Growth and Employment Strategy Paper in Cameroon (GESP) (2010, p. 50) is to “Encourage quality primary education for all and nationwide.” In addition, one of the reforms envisaged with regard to Vision 2035 is “quality basic education” (GESP, 2010, p.74).

Apart from quality primary education, section 9 of Law No 98/004 of 14 April 1998 which laid down guidelines for education in Cameroon stipulates that primary education shall be compulsory for children of school going age to acquire basic literacy, numeracy and survival skills. Other educational programs such as the Millennium Development Goals (MDG) (2000) and the World Declaration on Education for All (1990) also emphasize the need for all children of school-going age to acquire primary education. The Draft Document of the Sector Wide Approach on Education which reflects a common and coherent vision of education in Cameroon (2005, p. 27) looks at primary school as “the major system of training, to which the state has the objective of providing a solid base for continuous training for the Cameroonian chi

**1.1 STATEMENT OF THE PROBLEM**

The mathematics curriculum for Primary School Teacher Training Colleges (PSTTCs) prescripts mathematics content and pedagogic knowledge. It therefore requires of student-teachers adequate competency in the teaching and learning of mathematics. Upon graduation, student-teachers are expected to use the mathematics content and pedagogic knowledge as well as the skills and attitudes acquired during their training, to teach mathematics in primary schools.

However, literature, experience and statistics show that percentage pass in mathematics in English Speaking Primary Schools in South West Region has remained low irrespective of the class, level and the type of school (public, confessional or lay-private). Poor performance in mathematics may result in pupils developing a negative attitude towards mathematics which they may carry along to higher classes and post-primary institutions. Pupils’ poor performance in mathematics could hinder them from studying subjects that have links with mathematics like economics, chemistry, and physics, as well as hinder them from studying professions that make use of mathematics like medicine, engineering and accounting. Generally, their use of numeracy would likely be hindered by their poor performance in mathematics. Pupils’ inability to study professions and subjects that have links with mathematics implies that by 2035 that Cameroon would be expected to emerge, there would be relatively fewer English Speaking Cameroonians as engineers, medical doctors, accountants and architects. This would likely slow down the rate at which Cameroon would emerge by 2035.

Primary school teachers are the main implementers of the mathematics curriculum for primary schools. Primary school teachers are also the guarantors of the quality of mathematics education in primary schools. Pupils’ poor performance in mathematics would suggest amongst other reasons that the teaching of mathematics in English speaking Primary Schools in South West Region still lags behind despite reforms in the mathematics syllabuses for PSTTCs in Cameroon.

Teachers’ behaviour in mathematics classrooms probably affects pupils’ performance in mathematics. The improvement in the quality of mathematics teaching is likely to succeed only if there is an adequate supply of suitably qualified mathematics teachers. It is against this backdrop that the researcher observes that there is a problem and wants to find out the impact that mathematics curriculum for Primary School Teacher Training Colleges has in the teaching of mathematics in primary schools, with the hope that findings from this study would help in improving the performance in mathematics in English speaking Primary Schools in the South West Region.

Objective, This study aims at investigating the extent to which Mathematics content knowledge learnt in PSTTC has an impact on the teaching of mathematics in primary schools.

Research Question, To what extent does the mathematics content knowledge learnt in PSTTC have an impact in the teaching of mathematics in primary schools?

**1.2 BACKGROUND**

In Cameroon indigenous education, those who played the role of teachers were parents, elders and members within age groups. According to Atayo (2000), they inculcated survival values which centred on man’s basic needs such as food, drink, health and sex. They also inculcated trans-survival values which touched directly on the quality of life. Ojong (2008) opines that teaching was done following the principle of Cameroonian indigenous education such as Functionalism. The survival values made use of some knowledge of numeracy in one way or the other. For example, counting was done in the vernacular. According to him, this form of indigenous education was functional because the curriculum was learnt and immediately applied in society. The curriculum was not written but was organised in sequence to fit the expectations of the different developmental stages recognized by the culture (Nsamenang, 2005). For example, after a child had learnt how to count in the vernacular, he/she applied it in number of grains planted in the farm, the quantity and time to take medication, as well as the number of wives and the spacing of children. The children developed interest in cooperative study, independent study and group work especially as the teaching methods were observation, imitation, and participation.

During the colonial period, primary teacher education in Cameroon was provided by missionary bodies and the government at different times, with each having a particular role, and addressing a particular population of student-teachers (Tchombe and Agbor, 2007). The earliest kind of teacher education in Cameroon appeared in the training of men to teach Christian doctrines of various religions by the late 19th century (Tchombe, 2000). Their efforts were geared towards evangelisation and training individuals to fulfil their roles as catechists, interpreters and teachers. That is why earlier missionaries trained teachers and leaders only in the basic 3Rs namely, writing, reading and arithmetic. There was no defined curriculum. However, arithmetic enabled teachers and leaders to trade, while writing and reading enabled them to read the bible. For example, by 1885 and 1907, Alfred Saker (a Baptist missionary) and the Roman Catholic Mission respectively provided such training in Douala (Gwei, 1975).

Gwei (1975) says that formal teacher training started in 1925, with the opening of a normal class at Government Primary School Victoria. The school had a secondary department that opened in 1924. Graduates from the normal class were awarded a third-class teacher certificate. Contract pupil teachers and those who passed out of class two of the secondary department were trained as teachers for two years. Ojong (2008) opines that the first formal teacher training institution in the country was opened in 1932. This was the normal class that eventually metamorphosed into the Government Teacher Training College (GTTC) in Kake, in the outskirt of Kumba. The students who were being trained at the Kake Teacher Training Centre were awarded the Grade III Teacher Certificate and prepared for teaching in the lower primary school. The curriculum included subjects such as principles and practice of education, general methods, school organisation and management, physical education, and child study as well as in the areas of pedagogy, didactics, school administration, educational psychology, environmental education, agriculture, and arts and craft. These subjects equipped the teachers with the necessary knowledge, skills and attitudes which enabled them communicate and write letters.

According to Tchombe (2000), the second phase of teacher training was categorized into two main stages. The first stage colleges were Elementary or Grade III colleges. Pupil-teachers, “C” teachers and able standard six and latter class seven pupils were admitted and underwent a three-year course which qualified them to teach in infant and junior primary schools. Students who succeeded at the completion of the course were awarded the Teachers’ Grade III Certificate while the unsuccessful ones were designated as Grade III CTR, that is, Grade III trained but uncertificated. The second stage colleges were Higher Elementary or Grade II Colleges. These colleges admitted Grade III Certificated Teachers and Secondary School Leavers and offered a two-year course leading to the Teacher’ Grade II Certificate. Those who failed at the end of the course were regarded as Grade II CRT, that is, Grade II trained but uncertificated. Tchombe in Ndongko and Tambo (2000) opines that in some cases, a four-year course was undertaken, leading to the award of a Grade II teacher’s certificate. The curriculum included subjects such as principles and practice of education, general methods, school organization and management, physical education, child study and other primary school subjects like arithmetic. The curriculum for arithmetic prepared the teachers to teach arithmetic either in the junior or senior primary school, depending on the grade of the teacher. For example the arithmetic curriculum for pupil-teachers, “C” teachers and able class seven pupils enabled them to teach arithmetic in the junior primary classes.

In 1944, the Teacher Training College in Kake was moved to its permanent site in Kumba. Since that period, more teacher training colleges have been opened by government and private agencies for both Grades III and II courses and in latter cases, for Grade I. For example, the confessional teacher training institutions started in 1944 with the Catholics opening a teacher training college in Bambui. However, mission teacher training colleges were more concerned with character formation than with intellectual development (Tchombe in Tambo, 2000). Ojong (2008) argues that between 1944 and 1957, the curriculum of the teacher training colleges included the following subjects: Arithmetic, English Language, Principles and Methods of Education and Practical Teaching, Oral English, Practical and Theoretical Rural Science, Hand Work, Physical Education, Teaching Aid and Classroom Exposition. The curriculum for females added the following: Domestic Science – Needle Work, Cookery, Child Care and Hygiene. The curriculum for arithmetic enabled teachers to teach primary school mathematics in order to prepare Cameroonians to succeed the colonial masters, especially in jobs that made use of some knowledge of numeracy.

Tchombe (2010) opines that by the early 1960s, twelve teacher training colleges were opened in English-Speaking Cameroon; one by the government and the rest by confessional agencies notably Catholic, Presbyterian, and Baptist. Of the twelve teacher training colleges, five were located in the North West Region, and seven in the South West Region. worthy of note is the fact that the lone government owned teacher training college was located in the South west Region. Seemingly, the increased number of teacher training colleges did not only increase the number of trained teachers but also increased the possibility for the curriculum of arithmetic to be learnt by many more teachers.

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]]>The post Architectural Design Pemaralelan (Parallel) for Finding the Shortest Path Algorithm Dijkstra appeared first on Zambrut.

]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: June 1, 2019

**Ponco Rahmad Hadi, Sugiawan Sulastro & Andika Irawan**

Department of Mathematics, Adhi Tama Institute of Technology (ITATS), Surabaya

Department of Engineering & Education, Ma Chung University, Malang

Departement of Informatics, Wijaya Kusuma University, Surabaya

Indonesia

Journal Full Text PDF: Architectural Design Pemaralelan (Parallel) for Finding the Shortest Path Algorithm Dijkstra.

**Abstract**

Pemaralelan architectural design is one of the important stages in parallel computing. This phase is intended that the complexity of computing and communications can be efficient. This paper is a study of architectural design pemaralelan looking Dijkstra Shortest Path Algorithm. This design aspects are reviewed by both the computational complexity analysis of algorithms and communication

**Keywords:** Design pemaralelan architecture, parallel computing, computational complexity and Shortest Path.

**1. INTRODUCTION**

Solution to seek Shortest Path by Dijkstra’s algorithm has been widely investigated in a sequential manner the experts [McHugh, 1990]. As is known aspect sequential programming run into obstacles include the limitations of the data transfer, and the calculation speed limitations [Lewis, 1992]. With the development of hardware and software technology today, an alternative that has been developed is a problem solving approach to process in parallel. Generally expected to improve performance and efficiency in dealing with a problem. Moreover, the part of the user also wanted a quick problem resolution system and can solve the problem much bigger and komplikated [Lewis, 1992], [Kumar, 1994].

Similarly to the solution of problems with the shortest path Dijkstra’s algorithm requires the completion of a problem with the parallel process approach, when sequentially solution approach has not been able to provide a quick solution and are faced with a number of much larger vertex and complicated.

In parallel programming involving multiple processors, the next load of problems distributed to various processors. By involving many processors it will have an impact on the communication aspect. Issues important to note is the communication process to keep a low-overhead.

Issues that will have an impact on a variety of issues, among others include setting and synchronization of computer architecture, communication and data transfer processes, and methods of parallels [Lewis, 1992], [Kumar, 1994].

This paper examines the parallels architectural design for the shortest path problem with Dijkstra’s algorithm, in order to obtain efficiency and increased speedup compared with a sequential manner.

**2. PARALLEL COMPUTING FOR DIJKSTRA SHORTEST PATH ALGORITHM**

Glance directed graph with non-negative weights G = (V, E), the shortest path problem with single-source is to find the shortest path from a vertex vV to all other vertex in V. A shortest path from u to v is the path with minimum weight. Besides finding the shortest path from a single vertex v to each vertex to another, may also to find the shortest path between every pair of points. Formally, each pair shortest path problem is to find the shortest path between all pairs of vertex vI, vjV such that ij. For a graph with n vertex, its output is a matrix of nxn size of D = d (i, j) such that dI, j is the cost of the shortest path from vertex to vertex vj vI.

Weights can represent time lines, costs, penalties, damages, or some other quantity cumulatively.

Dijkstra’s algorithm to find the single-source shortest paths from a single vertex s, done in increments seeking the shortest path from s to another vertex in G and always choose an edge to a vertex closest enclosed, with complexity (n2). The search algorithm is being all-pairs shortest path from one vertex to all other vertex, for all couples with a single-source algorithm executing on each processor, for every vertex v. This algorithm requires complexity (n3).

The following program segment shows Sequential Algorithm for Shortest Paths Dijkstra’s Single Source [Brassard, 1996]. In this procedure for each vertex u(V-VT), put l [u], as a minimal cost to reach vertex u from vertex s where the vertex-vertex is in VT.

1. Procedure DIJKSTRA-SINGLE-SOURCE-SP(V,E,w,s)

2. Begin

3. VT={s};

4. For all v(V-VT) do

5. If(s,v) exists set l[v]=w(s,v);

6. Else set l[v]=;

7. While VTV do

8. Begin

9. Find a vertex u such that l[v] = min { l[v] | v(V-VT) };

10. VT=VT{u};

11. For all v(V-VT) do

12. L[v] = min { l[v], l[u] + w(u,v) };

13. Endwhile

14. End DIJKSTRA-SINGLE-SOURCE-SP

**3. ARCHITECTURE PARALLEL COMPUTING FOR DIJKSTRA SHORTEST PATH ****ALGORITHM**

According to [Kumar, 1994], architectural models chosen for the implementation of the parallelism must be adjusted with the processor and hardware, in order to create process efficiencies. This should be taken into account, because it is not impossible that this communication problem, will be much more complex than the problem of architecture, and it is often overlooked in the performance calculation. Then, from the aspect of software (operating systems, compilers) can be done dynamically (detecting system itself) or static (programmer must specify the location keparalelannya) [Chaudhuri, 1992].

Furthermore, to obtain optimum results in addition to the design of parallel algorithms that right, must also consider the cost of communication, because sometimes the complexity of communication is higher than the computational complexity, or the time taken to set up data between the processor are higher than the time to process the data manipulation [Quinn, 1987 ]. It is also worth noting computer architecture, it is important for the process of synchronization between processors and processing.

**3.1. Parallel Architecture for Single-Source Shortest Paths Dijkstra Algorithm**

Parallel to this problem formulation principle is iteration. Each iteration seeking a vertex with minimal achievement of a vertex origin, between the vertex-vertex unvisited connected directly to a vertex already dukunjungi. This achievement allowed to select more than one vertex, if there are more than two choices already visited berhungan directly with unvisited vertex then been the closest distance. Weighted adjacency matrix partitioned by usingblock-striped mapping,

Architecture so that each processor is assigned sequentially p n / p columns of a matrix of weighted adjacent matrix, and calculate the value of n / p on the array l.

**3.2. Parallel Architecture All-pairs Dijkstra Shortest Path Algorithm**

Architectural design for all-pairs problem Dijkstra shortest path can diparalelisasikan in two different ways.

a. Source-partitoned Formulation

Formulation partitioned parallel source of Dijkstra’s algorithm using n processors. Each pIlooking shortest vI shortest path from vertex to all other vertex by executing a sequential algorithm Dijkstra from single-source shortest path. Thus there is no inter-processor communication process.

Thus, the parallel execution of this formulation is Tp= (n2). Communication processors like no, this is a parallel formulation excellence. However, this is not true, because the algorithms used for n processors. Furthermore, the function isoefisiensi for konkurensinya process is (p3), where this is the average functionality isoefisiensi this algorithm. If the number of processors available to resolve this problem is small (that n=(p)), then this algorithm has a good performance. However, if the number of processors is greater than n, another algorithm usually adopts this algorithm because its scalability is very small.

b. Source-Parallel Formulation

The main problem with source-partitioned formulation formulation is that if you only use n processors will happen busy at work. Source parallel formulation is the same as the source partitioned formulation, except that the single-source algorithm running on a separate processor subset.

In particular, p (= 16) processors are divided into n (= 4) partitions, each with p / n processors (this formulation is emphasized only if p > n). Each of these n single-source problem solved by the shortest path from n partitions. In other words, the first problem pemarelan all-pairs shortest path by assigning each vertex to a portion of a collection of processors, and single-source pemaralelan algorithm using p / n processors for menyelsaikannya. The total number of processors used efficiently with this formulation is (n2).

**3.3. Evaluation of Computing and Communication Overhead**

Assume that the architecture built have p processors with mesh structure, such that p is multiplication in n. mesh structure pxp partitioned into n submesh that each measuring (P / n) x(P / n).

Furthermore, single-source algorithm executed on every submesh, the parallel execution time is Tp=(n3/p)+ ((np)), where the computing time (n3/p) and time of communication ((np)). Medium sequential execution time is W=(n3), then Speedup and Efficiencies are (n3) / { (n3/p)+ ( (np) ) }, and 1 / { 1 + (p1.5/n2.5) }.

From these results it appears that the formulation is a minimal fee p1.5/n2.5=(1). Furthermore, these formulations can be used to be (n1.66) efficient processor. This shows the case isoefisiensi for communication is (p1.8).

As for the parallel architecture Dijkstra formulations for all couples, it seems that in the formulation of the source partitioned no communication, using a processor numbering no more than n processors, and solve problems in time (n2). As a contrast, the formulations used to n1,66 source parallel processors, have time (overhead) communication, and resolve the problem in time (n1.33) when used as n1,66 processor. Thus, the formulation further exploit the parallel source than the source-partitioned parallels.

**4. CONCLUSION**

Dijkstra Algorithm Architecture for Single-Source Shortest Paths. This requires each processor is assigned sequentially p n / p columns of a matrix of weighted adjacent matrix, and calculate the value of n / p on the array l. In the single algorithm Source-Parallel Formulation executed on an architecture where each submesh, the parallel execution time is the sum of computing time (n3/p) with a time of communication ((np)), is Tp=(n3/p)+ ((np)). It also shows isoefisiensi function for communication is (p1.8), with isoefisiensi for concurrent process is (p1.5).

Speedup for architectural model of Source-partitoned Formulation is (n3)/(n2) and its efficiency is (1), where there is no communication overhead. It is not a parallel formulation excellence, because when using n processors, obtained isoefisiensi function for concurrency process of (p3).

In two parallel architecture model for all couples, for the formulation of the source partitioned no communication, when using a processor numbering no more than n processors, and solve problems in time (n2). While the source parallel formulations using up n1,66 processors, have time (overhead) communication, and resolve the problem in time (n1.33) when used as n1,66 processor.

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]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: May 30, 2019

**G. Mpofu & M. Mpofu**

Ka-Zakhali High School, P.O. Box 3229, Manzini, M200, Eswatini

University of Eswatini, Luyengo Campus, P.O. Luyengo, M205, Eswatini

Eswatini/ Swaziland

Journal Full Text PDF: A Motivating Tool in the Teaching and Learning of Mathematics (Zimbabwe Indigenous Games).

**Abstract**

This survey study explored the Mathematics embedded in the indigenous games of the Karanga people of Zvishavane District, in Zimbabwe, so as to bridge the gap that exists between school Mathematics instruction and the learners’ home life. The Karanga games such as nhodo, tsoro, pada, bhekari, hwishu are rich in Mathematical concepts. When tapped, this Ethno-Mathematics can mitigate in the Mathematics phobia existing amongst learners. Number systems and sequences, geometry, transformations and constructions, for instance, were seen to be embedded in these Karanga games and even artefacts. Eight secondary schools, 15 Mathematics teachers, 65 secondary school Mathematics learners, were selected to participate in the study. Quota sampling technique was employed to select the samples. From the study, it emerged that 86.7% of the learners and 80% of Mathematics teachers affirmed that cultural games can be used to improve the learning and teaching of Mathematics. It was also found that learners and Mathematics teachers under study agreed to the integration of Karanga ethnic games into the Mathematics curriculum to inculcate interest for Mathematics in the learners. One other finding was that the games will help in improving positive attitude in learners towards Mathematics. These games were also found to be a powerful tool in removing stigma generally associated with Mathematics that it is abstract, difficult, and monstrous and a reserve for the gifted learners only. This study is expected to finally inform the Mathematics classroom practice, to reduce the dreaded fear and stigma associated with the Mathematics.

**Keywords:** Ethno-Mathematics, Embed, Karanga, Indigenous Games.

**1. Introduction**

Learners in general, ostensibly shun Mathematics for being difficult. Skemp (2006) reveals that the failure rate in Mathematics in Zimbabwe schools, especially in less resourced rural schools, is unacceptably high. The primary and secondary school system is falling short in supplying enough students into tertiary training institutions for those programmes that require Mathematics as a pre-requisite. The Ordinary (“O”) and Advanced (“A”) Level examination results analysis shows, generally, that very few students are passing Science and Mathematics (Kusure and Basira, 2012). Chacko (2004) concurs with Kusure and Basira (2012) that the national pass rate oscillates between 17% and 25%, against a policy that set Mathematics pass grade a prerequisite for admission to most tertiary institutions as well for employment.

This being the case, one wonders why learners have negative attitude towards Mathematics and fail it so badly. One of Chacko (2004)’s research findings was that entry into most jobs in Zimbabwe is based on a good pass in Mathematics, yet at secondary school level, learners seem to drift away from the subject. Chacko (2004) stressed that negative attitude towards Mathematics is not innate in the learners, but is deliberately developed over a period of time and affects achievement in the subject and vice versa. Chauraya (2010), in his study, established that attitude towards a subject is positively correlated to performance in that particular subject.

Chauraya (2010) and Chacko (2004) point towards the fact that all children have an innate desire to learn, but it is how this learning process is presented to them that will make them like or dislike Mathematics. Colgan (2014) says similar ideas, when she stated that teachers are well placed to improve student achievement and attitude by re-orienting their attention to resource creativity utilisation and strategic methodologies that “pique students’ motivation, emotion, interest and attention”. Mathematics is a common thread embedded within cultural activities. When Mathematics is linked to people’s way of life, it is called Ethno-Mathematics (Tun, 2014). Tun (2004) further defines Ethno-Mathematics as the study of mathematical ideas of non-literate people. Ethno-Mathematics is hereby seen as the study of the relationship between Mathematics and culture. It examines a diverse range of ideas including mathematical models, numeric practices, quantifiers, measurements, calculations, and patterns found in culture, as well as education policies and pedagogy regarding Mathematics education (Kusure & Basira 2012).

This state of mathematical affairs is a result of, among other factors, misconceptions, by most Mathematics learners, that Mathematics is an academic activity restricted within the four classroom walls, that the discipline is difficult and consequently taken as a frightening abstract ‘monster’. Little is known by these learners that the games they play daily at home are rich in the mathematical concepts that can be schemas onto which school Mathematics learning can be build. Mathematics provides an effective way of building mental discipline and encourages logical reasoning and mental rigor and is a passage to understanding many other subjects (Tshabalala & Ncube, 2012). According to Mupa (2015), posits that the ability to demonstrate that Mathematics can contribute towards success, may give all pupils, before leaving school, some realization of Mathematics’ inevitable intrinsic value for this success.

Mathematics is a practical subject, where the learner physically ‘does’ Mathematics all the time they will be engaged in Mathematics (Kusure & Basira, 2012). This calls for a force that accelerates this action. Games have been found to play this part quite well, as was stated by Larson (2002) argues that Mathematics needs a lot of practice in the form of written work, mostly found in “carefully designed Mathematics games or activities that reinforce concepts and skills” Ogunniyi and Ogawa, (2008) say ‘Indigenous’ implies belonging to or originating in an area, or naturally living, growing or produced in an area. However, demographic characteristics, like migration, culture diffusion, social dynamism, globalisation as an aftermath of technological advancement, render it complex to think of indigenous knowledge as an absolute phenomenon. Urbanisation and technological advancement have seriously impacted on indigenous knowledge systems (Mutema, 2013). Mutema alludes to the World Bank’s observation that much indigenous knowledge are at risk of becoming extinct because of rapidly changing natural environments, fast pacing economic, political and cultural changes on a global scale.

This is one of the motivations for this research; to unveil the indigenous Mathematics embedded in the Karanga traditional or cultural children’s games and artefacts. Success in this unveiling will provide an alternative means of motivating learners to acknowledge the beauty and vital role of Mathematics and Science in everyday real-life problem-solving and decision-making, for sustainable development. Kuphe (2014) reiterates that although the attributes above are an important guide in identifying knowledge and practices that qualify as indigenous, the definition, however is silent on how long is ‘ancient times’. Population migration in the last half of the twenty-first century has seen massive erosion of those unique attributes that hold a people together as an indigenous entity. The writers cited above collectively express that games used in the classroom in teaching Mathematics have potential to influence paradigm shifts in the teaching and learning of Mathematics. These are some of the implications of games to the learners, teachers and other interested parties: Communication improves as learners share and play the games with family members at home. This interaction impacts on the family members to like Mathematics and then encourage those who are still in school to take Mathematics seriously, without being coerced. Thus Mathematics becomes everyone and every where’s business (SANC, 2012).Learning of Mathematics through games is made interesting, meaningful and attractive, not only to the learner, but to the Maths educator too, by considering learner-teacher relations, learner motivation techniques, learner-learner interaction, content mastery by teacher, defining relevance of maths in real life’

Research findings from numerous researchers indicate that Mathematics is an indispensable aorta of society that hinges culture and school. Culture is a kind of an informal school, where all subjects are learnt by discovery as well as trial and error methods, with Mathematics being the major. Consequently, Ethno-Mathematics has been recommended by several Mathematicians in a bid to mitigate and ameliorate deteriorating attitude towards Mathematics (Mutema, 2013). ( hence, the need to carry out this study to establish particular games and the Mathematical concepts embedded in them.

**1.2 Research Questions**

The research sought to answer to the following questions:

a. Which indigenous games are played by the participants in this study?

b. Which Mathematical concepts are embedded in the indigenous games identified?

**2. Methodology**

The interpretative approach framework has been adopted which helped to generate data from a social practical context in a fresh way rather than controlling previous theory (Merria, 2009). In seeking the answers for research, the investigator who follows interpretive paradigm used the research participants’ experiences and perceptions to come up with own view point using gathered data. Creswell, (2009) states that an interpretative research approach is an inquiry that combines or associates both qualitative and quantitative forms of research designs. It involves the mixing of both qualitative and quantitative approaches in a study. This approach plays a role in making it easier for the researcher to understand and coin new meaning of Mathematics-culture relationship in education. This study engages in a research bordering around ethno-Mathematics, which could transcend Mathematics from the traditional monotonous classroom ‘drama’ to the appreciation of Mathematics as a life-long natural phenomenon embedded in humans’ daily activities. The Karanga people’s artefacts, games and practices are rich in ethno-Mathematics embodied informally and intuitively within them.

**2.1 Research Design**

This research study employed the descriptive survey research design. Locklear (2012) states that a descriptive study is usually applied to samples in the range 20% to 30% of the population from which the sample is taken. Kothari (2004) argues that a descriptive survey is concerned with describing, recording, analysing and interpreting conditions under study. He further posits that this technique resembles a laboratory research, where the scientist observes and then interprets what has been observed. The sample proportion and nature of the study qualified the researcher to apply the descriptive survey design on the 8 schools in Zvishavane, with its 16 teachers and 65 learners to enable a more in-depth study of ethno Mathematics embedded in the culture of the Karanga people.

**2.2 Population**

The study population comprised 27 secondary schools, about 54 Mathematics teachers and more than 9000 students altogether in Zvishavane District. It is from this population that the research sample was taken for the study, while 16 individuals from across the Zvishavane community were involved in the Karanga cultural games observations and interviews. This district is mainly Karanga community, and is one of the regions with very poor performance in Mathematics in schools in the country.

**2.3 Sampling and Sampling Techniques**

The study sample for this research had 8 secondary schools, 15 Mathematics teachers and 65 Form 4 Mathematics learners sampled from the population. Simple random sampling technique was applied to select secondary schools from the schools list provided at the district education office. Quota sampling method was employed in selecting teachers and learners. The researcher was allowed access to the staff list at district education offices for the purposes of this study. There was also a list of all the district Form 4 learners which had been provided by schools for the purposes of examination electronic registration. It was from these lists that quota sampling technique was used to ensure ratio gender representation.

**2.4 Instrumentation**

To capture the desired data for the study, questionnaires, interviews and observations were designed to gather information to cover a number of variables, such as the current performance in Mathematics, evidenced by the pass rates figures obtained from the statistics kept at the education offices. A questionnaire consists of a multiple choice or closed and open-ended questions, whereby the closed ones require the “Yes” or “No”, “Agree”, “Disagree” type of questions. The open-ended questions give room for the respondent to express their opinion freely (Creswell, 2009), although research subjects can abuse this freedom and give irrelevant information.

**2.5 Validity and Reliability Issues**

This research, as was alluded to earlier, takes a mixed method research design, with more inclination to qualitative design. Cohen (2000) posits that subjectivity of respondents’ opinions, attitudes and perspectives render the data somewhat biased, as is always the characteristics of qualitative research design. Reliability relate to whether the research results are consistent, or can be trusted in relation to the data collected (Cohen, 2000). As described by Atebe (2008), “reliability simply means dependability, stability, consistency and accuracy” of a methodology applied to gather research data, while validity, on the other hand, is based on the authenticity of the data collected. To address the issues of validity and reliability, in this study, the researcher made an attempt to record data in form of pictures, of children observed playing Karanga games and those few artefacts that were seen. The explanations, descriptions, diagrams, were generated from the participants, rather than from the researcher’s own pre-conceived ideas.

**2.6 Ethical consideration**

Ethics are principles that guide researchers when conducting their researches (Chiromo, 2010). These are behaviours and understandings expected as per group of people, animals, plants or professional code. These are principles that guide the researcher to determine what is right and what is wrong when carrying a study, so as to protect the participants or subjects of the study from harm. The researcher, therefore, informed the participants that there would not be direct benefit for participation and that participation was entirely voluntary and that they could decide to withdraw from the study at any time. Questionnaires assured respondents that the information they gave was to be used for this study only and treated with strictest confidentiality. Thus, they were encouraged not to give their identities. Written permission to conduct the study was sought from the Ministry of Primary and Secondary Education and was granted.

**3. Findings and Discussion**

The results are presented as indigenous games played in the area under study and Mathematical concepts embedded these indigenous games.

**3.1 Bhekari Game**

Bhekari is game which is more like cricket though cricket is apparently an advanced and modernized form of this game. This game is played by two teams with at least three players at a time, where two would be targeting to hit the player with a ball. The playing team runs round the whole box along the path shown by the arrows, in a clockwise direction, starting from Box P. The other team sets two players, one at A, the other at B, who aim to beat the players of the opposing team with a ball, as they attempt to run across the Danger Zone. If a player is hit by the ball, they are automatically eliminated from the game. A player is not supposed to be hit by the ball when in a Number Box or on a Free Zone. These numbers represent game levels.

Figure 1: The Bhekari playing court

The game aimed at players successfully evading being hit by the ball whenever they ran across the danger zones until they get to the Centre circle (Game Over Zone). The player would also keep correct count of the numbers in each of the boxes. Every time the opposite team would shout “Bhekari?” all the players would be expected to shout back the number of the box in which they are. If one says out a wrong number, they are eliminated. All players end up knowing the numbers in each set. If one of the members of the playing team catches the ball, the eliminated member comes back into the game and starts at P. If any one player manages to get to 25, then the whole team wins and they start all over again in Box P, but if they are all hit by the ball before any one player reaches the centre circle, it will be the other wins the turn to run round. While two players will be aiming at hitting the opponent team players, who attempt to cross the danger zone, the rest of the players (if any) would be scattered all around the playing field to catch stray balls. This is meant to avoid giving much time to the running players to do many rounds and also to monitor the counting down to 25 the winning box. Any cheating resulted in elimination from the game.

**3.1.1 The Mathematical concepts derived from Bhekari**

The bhekari game embeds much aspects of Number systems in general (Animasahun and Akinsola, 2007). The numbers in Box P are all odd numbers, which differ by 4. The first time a player gets in this box, if the opposite team player shouts “bhekari”, the player must be able to say ‘1”. The next time the player comes back to the same box, the number will be 5, after which it will be 9. These numbers must be remembered by both teams because each time the numbers are asked, both the running and ball throwing teams must be sure of the count. The set of numbers in Box Q are all even numbers. The numbers in Box R are not only odd numbers, but they are also prime numbers, except 15. Those in Box S are even numbers as well as multiples of 4. It can be observed that these numbers form a sequence as shown in the Table 1 below:

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]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: May 30, 2019

**O. O. Adetola**

Forestry Research Institute of Nigeria, P.M.B 5054 Jericho Hills

Ibadan, Nigeria

Journal Full Text PDF: Impacts and Mitigation of Acid Rain on Forest Trees (A Short Review in Nigeria).

**Abstract**

Acid rain has been described as one of the most serious environmental problems in nature. Acid rain is mainly a mixture of sulphuric and nitric acids depending upon the relative quantities of oxides of sulphur and nitrogen emissions. The interaction of these acids and other atmospheric components releases protons hence, increasing soil acidity. Lowering of soil pH mobilizes and leaches away nutrient cations and increases availability of toxic heavy metals. Such changes in the soil chemical characteristics reduce the soil fertility, which ultimately causes the negative impact on growth and productivity of forest trees and crop plants. Acidification of water bodies causes large scale negative impact on aquatic organisms including fishes. Acidification has some indirect effects on human health also. Acid rain affects each and every components of ecosystem. Acid rain also damages man-made materials and structures. By reducing the emission of the precursors of acid rain and to some extent by liming, the problem of acidification of terrestrial and aquatic ecosystem has been reduced during last two decades.

**Keywords:** Acid Rain, Acidification, Forest Trees & Photosynthesis.

**1. Introduction**

According to [16], acid rain has direct impacts on forest ecosystems and their inhabitants. The damage to the forest trees and plants is widespread. Acid rain damages leaves as it falls. Acid rain runoff from the trees and forest floor infiltrates the forest’s water supplies; runoff that doesn’t enter the water supply is absorbed by the soil. The consequence of this is just as it is for any soil or water source infected with acid rain. The plants and creatures die off, and the creatures that rely on those plants and smaller creatures lose their food source and die as well. Acid rain is an environmental pollutant in developed countries where burning of fossil fuel release sulphur and nitrogen oxides into the atmosphere [2]. Burning of fossil fuels in industries and transport sector, industrialization and urbanization have led to increase in concentrations of gaseous and particulate pollutants in the atmosphere leading to air pollution [5, 10]. The problem of acid rain is believed to result from the washout of oxides of Nitrogen, Sulphur and other constituents present in the atmosphere. Main sources of these oxides are coal fired power stations, smelters (producing SO2) and motor vehicle exhausts (producing NOx). These oxides may react with other chemicals and produce corrosive substances that are washed out either in wet or dry form by rain as acid deposition. Initially events of acidic rainfall were frequent only around industrial areas. But with the increased use of tall stacks for power plants and industries, atmospheric emissions are being transported regionally and even globally [6, 11].Acid rain affects the quality of human life, threatens the environmental stability and the sustainability of food and timber reserves, thus posing an economic crisis. Acid rain has broad economic, social and medical implications and has been called an unseen plague of the industrial age [1]. The first incidence of acid rain seems to have coincided with onset of the industrial revolution in the mid 19th century. Acid rain problem was observed in England then as a regional phenomenon in Scandinavia in the late 1960’s [7]. By 1965, the pH of rainwater in Sweden was about 4 or less and it was reported in 13th UN conference on the Human Environment held at Stockholm in 1972. This was the beginning of acid rain research. It was suggested that rain and snow in many industrial regions of the world are between five and thirty times as acidic as would be expected in an unpolluted atmosphere [8]. In 1974, over the northeast United States, the pH of rain and snow was found to be around 4.0 [9] .

**2. Effects of Acid Rain on Trees and Causes of Acidification**

Scientists and foresters over the years have discovered that forests grow more slowly without knowing why. The trees in these forests do not grow as quickly as usual. Leaves and needles turn brown and fall off when they should be green and healthy. Researchers suspected that acid rain may cause the slower growth of these forests in association with other conditions such as air pollutants, insects, diseases and drought. Acid rain usually kills trees by damaging their leaves, limiting the nutrients available to them, or poisoning them with toxic substances slowly released from the soil. Leaves turn the energy in sunlight into food for growth. This process is called photosynthesis. When leaves are damaged, they cannot produce enough food energy for the tree to remain healthy. Once trees are weak, they can be more easily attacked by diseases or insects that ultimately kill them. Weakened trees may also become injured more easily by cold weather.

Acid rain can harm other plants in the same way it harms trees. Food crops are not usually seriously affected, however, because farmers frequently add fertilizers to the soil to replace nutrients washed away. They may also add crushed limestone to the soil. Limestone is a basic material and increases the ability of the soil to act as a buffer against acidity. Acid rain affects the availability of nutrients in soils by increased leaching and removal of nutrients and active metabolites in leaves [12]. The destruction of forest by acid rain has a chain effect this include Soil erosion, sedimentation of waterways, water supply deterioration and dependent wildlife destruction.

Sulphur dioxide (SO2) and oxides of nitrogen and ozone to a large extent are the primary causes of acid rain. These pollutants originate from human activities such as combustion of burnable waste, fossil fuels in thermal power plants and automobiles. These constituents interact with reactants present in the atmosphere and result into acid deposition. The man-made sources of SO2 emissions are the burning of coal and petroleum and various industrial processes [4]. Other sources include the smelting of iron and other metallic (Zn and Cu) ores, manufacture of sulphuric acids, and the operation of acid concentrators in the petroleum industry. The levels of NOx are small in comparison to SO2, but its contribution in the production of acid rain is increasing. Main natural sources of NOx include lightening, volcanic eruptions and biological processes, other sources are power plant and vehicle emission.

The chemical reaction that results in the formation of acid rain involves the interaction of SO2, NOx and O3. When the pollutants are vented into the atmosphere by tall smoke stakes, molecules of SO2 and NOx are caught up in the prevailing winds, where they interact in the presence of sunlight with vapours to form sulphuric acid and nitric acid mists. These acids remain in vapour state under the prevalent high temperature conditions. When the temperature falls, condensation takes the form of aerosol droplets, which owing to the presence of unburnt carbon particles will be black, acidic and carbonaceous in nature. This matter is called “acid smut”. The presence of oxidizing agents and the characteristics of the reaction affects the rate of acid generation [3].

Fig 1: Formation of Acid Rain

Source: Climate Justice 7. Gas Flaring poisons communities. The above picture illustrates the cyclical process behind acid rain.

**3. Causes of Acid Rain in Nigeria**

The oil rich Niger Delta area of Nigeria has been associated with gas flaring which is a major producer of Sulphur Dioxide and its Nitrogenous counterpart involved in the acidification of rain water. Also the south western part of Nigeria is known for high no of vehicular movement and heavy industrial presence which also produces toxic gases and heavy metals. Soil temperature over 300c leads to decreased agricultural yields, with the major impact being desiccation and damage to the micro flora [13, 15]. Air quality was also affected, with damage to vegetation, the microclimate surface and groundwater, as a result of the high concentration of volatile oxides, carbon, nitrogen, sulphur oxide and particulates that exceeded the standard set by FEPA in 1991 [14] . Nitrogen dioxide reacts with water from rain to form nitric acid (HN0 3), falling as acid rain, harmful to human health and the environment as it causes acidification of drinking water in reservoirs, corrosion of metals, and damage to crops and the Niger Delta forest.

**4. Mitigation Strategy of Acid Rain on Forest Stand**

Emission control: The most important solution for acid rain problem is reduction of SO2 and NOx emissions. The use of fuel that is low in Sulphur is not really practical because the world supply of low Sulphur fuels is limited. Various techniques are available to reduce Sulphur emission from non-ferrous smelters. Oxides of nitrogen can also be reduced through reduction or better control of combustion temperature.

**5. Conclusion**

Acid rain has deleterious effect on ecosystem, which includes decline in growth of trees as well as other plants including crops, reduction in aquatic flora and fauna. Marble, limestone and sandstone can be easily destroyed by acid rain. Acid rain can also affect indirectly the human health. Soil fertility is negatively affected due to acid rain as a result of leaching of essential nutrient cations and increase of availability of toxic heavy metals. Acid rain problem has been tackled to some extent in the developed world by reducing the emission of the gases causing acid rain. Such efforts need to be done in Nigeria most especially in the Niger Delta oil rich region where gas flaring happens daily and the South western part due to heavy industrial site and high vehicular movement.

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