Publicación Cuatrimestral. Vol. 2, Año 2017, N
o
3 (77-94).
HEURISTIC INSTRUCTION FOR WAVE EQUATION PROBLEM-
SOLVING USING VARIABLE SEPARATION METHOD
M.Sc. Manuel S. Alvarez-Alvarado
1*
; M.Sc. Fernando Vaca-Urbano
2
1
Facultad de Ingeniería en Electricidad y Computación (FIEC), Escuela Superior Politécnica del Litoral
(ESPOL), Campus Gustavo Galindo Km 30.5 Vía Perimetral, Apartado 09-01-5863, Guayaquil, Ecuador.
*Autor para la correspondencia. Email: manuel.alvarez.alvarado@ieee.org
Recibido: 21-10-2017 / Aceptado: 27-12-2017
ABSTRACT
This study focuses on heuristic instruction as a method of logical thought processes. The objective is to raise
and establish a framework for problem-solving in the application of separation of variables for the wave
equation. This leads to a simple and easy pathway to make the topic at a manageable level for the students.
This paper proposes a base problem, over which the heuristic instruction is applied. Eventually, the result is
the determination of a system that allows a simple approach to avoid difficulties in solving problems in which
the separation of variables in the equation of the wave is employed. In order to determine the impact of heuristic
in students learning gain, two groups of 20 students each (experimental and control) were subject to a pre-
and post-test. To the control group, the scheme proposed was not employed. In contrast to the experimental
the proposed approach was employed. The results reveal that the heuristic scheme for wave equation
problem-solving has a significant impact on students’ learning process.
Key words: Physics, Heuristic Instruction, Differential Equations, Wave Equation.
INSTRUCCIÓN HEURÍSTICA PARA RESOLVER PROBLEMAS
DE ECUACIÓN DE ONDA MEDIANTE EL MÉTODO
SEPARACIÓN DE VARIABLES
RESUMEN
Este estudio se centra en la instrucción heurística como un método de procesos de pensamiento lógico. El
objetivo es plantear y establecer un marco para la resolución de problemas en la aplicación de separación de
variables para la ecuación de onda. Esto conduce a un camino simple y fácil para que el tema sea manejable
para los estudiantes. Este documento propone un problema de base, sobre el cual se aplica la instrucción
heurística. Eventualmente, el resultado es la determinación de un esquema que evite dificultades para resolver
problemas en los que se emplea la separación de variables en la ecuación de la onda. Con el fin de determinar
el impacto de la heurística en la ganancia de aprendizaje de los estudiantes, dos grupos de 20 estudiantes
cada uno (experimental y de control) fueron sujetos a una prueba previa y posterior. Para el grupo de control,
el esquema propuesto no fue empleado. Por el contrario, para el grupo experimental, se empleó el enfoque
propuesto. Los resultados revelan que el esquema heurístico para la resolución de problemas de ecuaciones
de ondas tiene un impacto significativo en el proceso de aprendizaje de los estudiantes.
Palabras clave: Física, instrucción heurística, ecuaciones diferenciales, ecuación de onda.
Ciencias Matemáticas
Artículo de Investigación
MSc. Manuel S. Alvarez-Alvarado, M.Sc. Fernando Vaca-Urbano
78
INSTRUÇÃO HEURISTICA PARA RESOLVER PROBLEMAS DA
EQUAÇÃO DE ONDA MEDIANTE O MÉTODO SEPARAÇÃO DE
VARIAVEIS
RESUMO
Este estudo se baseia na instrução heurística como um método de processos de pensamento lógico. O
objetivo é estabelecer um marco para a resolução de problemas na aplicação de separação de variáveis para
a equação de onda. Conduzindo um caminho simples e fácil para que o tema seja de fácil manipulação para
os estudantes. Este documento propõe um problema de base sobre o qual se aplica a instrução heurística.
Eventualmente, o resultado é a determinação de um esquema que evita as dificuldades para resolver
problemas nos quais é empregada a separação de variáveis na equação de onda. Com a finalidade de
determinar o impacto da heurística no aprendizado dos estudantes, dois grupos de 20 estudantes cada um
(experimental e de controle) foram sujeitos a uma prova previa e posterior. Para o grupo de controle, o
esquema proposto. Os resultados revelam que o esquema heurístico para a resolução de problemas de
equações de ondas têm um impacto significativo no processo de aprendizado dos estudantes.
Palavra chaves: Física, instrução heurística, equações diferenciais, equação de onda.
1. INTRODUCTION
Teaching Thinking means to be on the student to establish links between certain objects
and tasks, and the corresponding response actions (Cabrera et al., 2006). Problem-solving
is a complex mental process (Gilar, 2003) since is a pathway to apply the knowledge in real
situations. When it comes to solving-problems of partial differential equations in order to
describe a physical phenomenon, it becomes difficult for the student because this involves
the application of many criteria such as the variables separation, Fourier series, Laplace
transform, etc.
Nowadays, students solve problems based on a previous example given during the lectures.
Nevertheless, they do not realize about what they do, that is if some condition change in the
problem they may not be able to solve it. On the other hand, instructors must conscious that
they are not programing machines, and presenting problem-solving as a recursive process
is not the solution. Therefore, the relevance of this study lies in the development of a scheme
or algorithm for problem-solving of wave equation using the variable separation method,
which can be created based on basic concepts and logic criterion. The heuristic appears as
a comprehensive rule of practical reasoning which can be employed for problem-solving.
For any topic, both students and instructors must be protagonists of the heuristic instruction
in order to teach and learn in a conscious and planned way.
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The heuristic comes from the Greek εὑρίσκειν, meaning to find, discover, invent, etc. This is
a systemic strategy for immediately positive innovations, is defined as the art and science
of discovery and invention, in addition to problem-solving through lateral thinking or
divergent thinking, which constitutes creativity.
A problem is formed within a psychological structure, as follows: 1) It starts from data, 2) the
analysis of data, 3) Establishing relationships between data, 4) Debugging information, 5)
development of a particular strategy suited to the problem. Under this framework, choice
and decision making are crucial, as they provide guidance to possible solutions of the
problem, which is closely related with heuristics (Rocio & Elizabeth, 2004).
The use of heuristics instruction in the teaching-learning process is critical because it helps
to achieve: 1) student's cognitive independence; 2) The integration of new knowledge; 3)
Training of mental abilities such as intuition, productivity, originality of the solutions,
creativity, and so on. This last point, is the most relevant since at this level the student has
developed an intellectual ability to the highest level. This way of problem-solving was not
well received in academic circles, apparently due to their limited logical and mathematical
rigor. However, thanks to its practical potential to solve real problems were slowly opening
the door to heuristic methods, especially from the 60s of XX century (Abrosio, 2011).
Currently, in Math continue the development of heuristic methods and are increasing the
range of their applications, and their variety of approaches such as Celia & Richard (2017).
The relevance of the heuristics lies in the visualization and solve problems that were
previously too complex and unthinkable in previous generations.
2. METHODOLOGY
2.1. Research Participants
This study incorporated the participation of 40 students (34 men and 6 women) from an
Ecuadorian University. The students were between 20 and 25 years old and they were
registered in a course of Calculus III. Two groups were formed, group one (CG), which is
the control group and group two (EG), called experimental group. The main objective for this
lecture is to learn wave equation problem-solving using variable separation method.
2.2. Research Instrument
A quasi-experimental type design was used with a pre-test-intervention-post-test. The test
employed is as given in Annex 1 and the time given to finish the test was 30 minutes. The
MSc. Manuel S. Alvarez-Alvarado, M.Sc. Fernando Vaca-Urbano
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lecture given for the control group was using traditional teaching method, while for the
experimental group was heuristic instruction following the designed scheme. The students
are evaluated following the rubric given in Annex 2.
2.3. Hypothesis
H1: In problem-solving for the wave equation, the proposed heuristic scheme produces a
better academic performance on students, in comparison with traditional methodology
teaching scheme.
H01: In problem-solving for the wave equation, the proposed heuristic scheme produces the
same academic performance on students, as the traditional methodology teaching scheme.
2.4. Heuristic Scheme Proposed
In Math, a partial differential equation (sometimes abbreviated as PDE) is a relationship
between a mathematical function of several independent variables x,y,z,t, and the partial
derivatives of u with respect to these variables. PDEs are employed to describe the behavior
of physical processes that are usually distributed in space-time. The variable separation
method is relatively simple and powerful enough to solve not homogeneous differential
equations. The method is based on the following statement: Let the function ,
then there must be a function
such that:
(Zill, 1997; Chanillo, Franchi, Lu, Perez & Sawyer, 2017). Based on this criterion the heuristic
starts to take place.
This study does not develop a general theorem to solve problems using variable separation
method for the wave equation, rather it has developed a problem (given in Annex 1) that is
used as a base to develop a heuristic scheme. Process analysis in solving problems by
using heuristic instruction have revealed the following key points:
A. Read and interpret the problem then determine the objective
Start solving a problem is the most difficult part for many students because they ignore the
reading and focus their attention on mathematical formulations. They do not read the
problem carefully, and so, they cannot understand the problem and they do not know what
to do.
The heuristic instruction indicates to start from data (Montero, 2011), but this is not enough,
a proper interpretation of the variables involved in the problem is also needed, and
obviously, it is relevant to keep in mind that data need to be processed and this eventually
will lead to a result.
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For the problem in Annex 1, the data to be considered and interpreted are:
(1)
The objective is to determine
B. Understand and apply the variable separation method, to obtain a ratio and find
a wave equation solution
Once the data and the objective of the problem are set, the next step is to develop a plan in
order to get a solution. PDE allows to reach the solution of the problem, in the heuristic, this
is known as search strategies (Francisco, 1999). Applying the variable separation method
to the problem given in Annex 1:
(2)
The problem states
.This mathematical formulation in combination with (2):
(3)
Consequently, it is possible to do the following relationship:
(4)
Hence:
(5)
(6)
C. Use the initial and boundary conditions
By imposing that
, a proper interpretation to use the initial and boundary
conditions is needed. In heuristics instruction, this is known as data analysis. Applying this
criterion to the given example:
1) From first the initial condition
, by inspection:
;
2) From the second initial condition
, by inspection:
;
3) From the boundary condition
, by inspection:
.
MSc. Manuel S. Alvarez-Alvarado, M.Sc. Fernando Vaca-Urbano
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D. Create a system of equations and solve them separately
Combining (5), (6) and the initial conditions:
(7)
(8)
At this point, note that the problem comes down to solving systems of equations separately
and then perform the operation
. Nevertheless solving the system is not easy as it
requires the development of high intellectual skills (Zbigniew, 2004).
E. Determine a model to solve the differential equation
From this point, the hardest part of problem-solving begins, since it requires previously
known expressions and association models within this heuristic instruction are termed as
intellectual skills based on known models. It is known that a differential equation of the type
, has a solution
, then:
(9)
(10)
F. Determine the wave equation and express it as a summation
Once
and are
known, it is possible to find
using (2), then:
(11)
Equation (11) can be expressed it in terms of summation:
(12)
G. Express the wave equation as Fourier series
Recalling the Fourier series:
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(13)
Where:
(14)
(15)
(16)
Then, (12) can be written as:
(17)
This within the heuristic is known as establishing relationships (Arslan, 2010).
H. Define the limits of the integral and solve it to find some missing coefficients
There are some coefficients that maybe missing in the wave equation and in order to
determine them physical parameter given in the problem are needed. The given problem
presents a rope with and . With these data, it is possible to solve the integral and
proceed to find the missing coefficients and determine the wave equation.
3. RESULTS Y DISCUSSION
3.1. Heuristic Scheme
As a result the following heuristic scheme for problem-solving the wave equation is obtained.
This is shown in Figure 1.
The grades obtained (over 10) in the pre- and post-test for the control and experimental
group are presented in Figure 2 and Figure 3, respectively. Then, a descriptive statistic of
the grades in the pre- and post-tests for the control and experimental group are obtained.
These are presented in Table 1 and Table 2, respectively.
MSc. Manuel S. Alvarez-Alvarado, M.Sc. Fernando Vaca-Urbano
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FIGURE 1. Heuristic scheme for wave equation problem-solving using variable separation method.
3.2. Pre- and Post-Tests Statistical Values
Figure 2. Control group grades obtained.
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
CG Pre-test CG Post-test
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Instituto de Ciencias Básicas. Universidad Técnica de Manabí. Portoviejo-Ecuador 85
Table 1. Descriptive Statistics of the Pre- and Post-Tests CG.
Statistical Values
Pre-test
Post-test
Participants
1.70
3.20
Mean
1.50
3.50
Median
1.00
5.00
Mode
1.26
1.54
Standard Deviation
1.59
2.38
Variance
4.00
5.00
Maximum
0.00
1.00
Minimum
4.00
4.00
Range
1.70
3.20
Figure 3. Experimental group grades obtained.
Table 2. Descriptive Statistics of the Pre- and Post-Tests EG.
Statistical Values
Pre-test
Post-test
Participants
1.30
7.10
Mean
1.00
7.00
Median
1.00
7.00
Mode
1.22
1.62
Standard Deviation
1.48
2.62
Variance
4.00
10.00
Maximum
0.00
4.00
Minimum
4.00
6.00
Range
1.30
7.10
3.3. “t” Test
A “t” test is employed to verify the hypothesis. The post-test data for the experimental and
control are contrasted using EXCEL 2013. The results are shown in Table 3 and it reveals
0
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
EG Pre-test EG Post-test
MSc. Manuel S. Alvarez-Alvarado, M.Sc. Fernando Vaca-Urbano
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a P value of 4.39699E-6 which is less than 0.05, therefore the null hypothesis is rejected in
favor of the alternative hypothesis.
Table 3. “t” test results
Teaching Scheme
Traditional
Heuristic
Mean
3.2
7.1
Variance
2.4
2.6
Observations
20
20
Pearson Correlation
-0.514287463
Hypothesized Mean Difference
0
df
19
t Stat
-6.339820507
t Critical one-tail
1.729132812
P(T<=t) two-tail
4.39699E-06
t Critical two-tail
2.093024054
3.4. Leaning Gain
In order to quantify the students’ learning gain, the Hake factor (g) was employed. The
grades obtained in the pre- and post-tests are related as follows (Dellwo, 2010 ):
(17)
where
is the grade obtained in the post-test and
is the grade obtained in the pre-
test.
The learning gain for both groups is presented in Figure 4. The results reveal that EG
presents more learning gain than CG.
Figure 4. Hake Factor.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
CG EG
0.47
0.82
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4. CONCLUSION
When it comes to solving problems, which aims to determine the wave equation, we need
certain skills related to differentiation and integration of real variable functions, and students
must recognize that there is no recursive procedure to solve the problem. Nevertheless, it
is possible to devise a strategy to facilitate to obtain the solution, and then develop a
methodology and summarize it in a comprehensive scheme. This scheme is nothing more
than a succession of heuristic indications with a view to have an orientation and to develop
the problem. It is relevant to emphasize that the scheme does not guarantee success in
solving the problem as largely, as indicates the heuristic is necessary for the individual to
develop intellectual operations such as: analyze, synthesize, compare and rank; also
requires forms of critical thinking and mathematical science: variation of conditions, search
for relations and dependencies, and considerations analogy. The scheme is very useful and
avoids certain difficulties in raising strategies in order to find its solution.
Concerning the learning gain obtained (Hake factor) for the CG and EG is 0.47 and 0.82,
respectively. Therefore, the proposed schematic heuristic instruction for wave equation
problem-solving using variable separation method brings a higher gain learning than
traditional methodology. Furthermore, in the “t” test, the p statistic value (4.39699E-06) is
less than 0.05, and following the theory related with this test, the null hypothesis is rejected.
Therefore, it is possible to state that in problem-solving for the wave equation, the proposed
heuristic scheme produces a better academic performance on students, in comparison with
traditional methodology teaching scheme.
5. REFERENCES
Arslan, S. (2010). Traditional instruction of differential equations and conceptual learning. Teaching
Mathematics and its Applications, 29, 94-107.
Abrosio, G. (2011). Heurística, racionalidade y verdad. CADERNOS UFS FILOSOFIA – Ano 7, Fasc. XIII, Vol.
9.
Cabrera, L., Marilú, J., Valdivia, M., Villegas, E., Mondéjar, J. & Miranda, L. (2006). La heurística en la
enseñanza de la matemática. Monografía de la Universidad de Matanzas “Camilo Cienfuegos”, pag. 4.
Celia, H. & Richard, N. (2017). Visions for Mathematical Learning: The Inspirational Legacy of Seymour Papert
(1928–2016). EMS Newsletter.
Chanillo, S., Franchi, B., Lu, G., Perez, C. & Sawyer, E. (2017). Harmonic Analysis, Partial Differential
Equations and Applications. Honor of Richard L. Wheeden.
Dellwo, D. (2010). Course Assessment Using Multi-Stage Pre/Post Testing and the Components of Normalized
Change. Journal of the Scholarship of Teaching and Learning, 10(1), 55-67.
MSc. Manuel S. Alvarez-Alvarado, M.Sc. Fernando Vaca-Urbano
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Francisco, C. (1999). La aplicación de procedimientos heurísticos y situaciones problémicas en la resolución
de problemas de Matemáticas I. Tesis de grado de maestría pag. 19.
Gilar,C. (2003). Teoría del aprendizaje. Tesis Doctoral Universidad de Alicante.
Montero, P. (2011). Aprendizaje metodológico de la solución de problemas a través de documentales
cinematográficos.
Rocio, A. & Elizabeth, V. (2004). Incidencia del razonamiento abstracto en el escogimiento de carreras
técnicas en los alumnos de los sextos cursos del instituto técnico superior “San Vicente Ferrer” de la
ciudad de Puyo provincia Pastaza, año lectivo 2002-2003.Tesis de grado pag. 22.
Zbigniew, M. & David, F. (2004). How to Solve It: Modern Heuristics. Springer, Second Edition, North Carolina,
pag. 1-20
Zill, D. (1997). Ecuaciones diferenciales con aplicaciones de modelado, 6ta Ed, México, pag. 70-77.
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Annex 1
Proposed problem with the corresponding solution
There is a string of length and is released, without impulse, from the position
We denote c, as the speed at which the waves travel up the rope. The equations that model this problem are:
Determine the solution describing the wave equation.
Solution:
Read and interpret the problem then determine the objective.
The data to be considered and interpreted is:
The objective is to determine the wave equation.
Understand and apply the variable separation method, to obtain a ratio and find a wave equation
solution.
The basic idea of the method is to seek solutions in the form of separate variables of the homogeneous part
of the problem to solve.
The mathematical formulation
, in combination with the variable separation method criterion:
,
Consequently, the following relationship can be done:
Thus:
MSc. Manuel S. Alvarez-Alvarado, M.Sc. Fernando Vaca-Urbano
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Use the initial conditions and boundary conditions.
By imposing that the function
and using this with the initial conditions:
The wave equation
, we get that
, then
The initial condition
, by inspection
.
The boundary condition
, by inspection
.
The boundary condition
, by inspection
.
Create a system of equations and solve them separately.
Therefore, it produces two separate problems:
,
Determine a model to solve the differential equation.
The solution, for the equation
is
and using the boundaries
conditions:
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Replacing values y
, on
:
The solution to the equation
is
and using the initial
condition:
Replacing the values of and
, on the equation
, we get:
Determine the wave equation and express it as a summation.
Now:
Express the wave equation as Fourier series.
This function is described in Eq. (3) is called the normal modes are the natural modes of vibration of the string.
The natural term means that due to the linearity of the homogeneous problem, the vibration of the string will
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always be a superposition (sum) of these infinite normal modes, making it possible to write the function as a
sum:
Recalling the Fourier series:
Where:
It is indicated as a condition of the problem:
So:
Define the limits of the integral and solve it to find some missing coefficients.
It can be associated with the Fourier series
and by inspection of the equation and the graph given in
the problem:, y
Then:
To solve the integral of Eq. (6) requires the integration by parts
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Then:
By replacing the limits of the integral Eq. (6) and solving:
Replacing the Eq. (5) and Eq. (7) into Eq. (4):
MSc. Manuel S. Alvarez-Alvarado, M.Sc. Fernando Vaca-Urbano
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Eq. (8) is the solution that describes the wave equation.
Annex 2
Rubric employed for the problem given in Annex 1
Task
Points
If the student get:
0-1
If the student use the initial and boundaries conditions as follows:
The initial condition
, by inspection
.
The boundary condition
, by inspection
.
The boundary condition
, by inspection
.
1-2.5
(+0.5 each)
If the student get:
or
or
or
2.5-6.5
(+1 each)
If the student express the wave equation as a Fourier series:
6.5-8.5
(depending on the process)
If the student get the wave equation solution:
8.0-10.0
(depending on the process)
