DEVELOPMENT OF DIDACTICAL MATERIAL FOR CALCULUS TEACHING AND ITS APPLICATION ON ENGINEERING
Nicolás Góngora Salazar, Dionisio Humberto Malagón Romero
Facultad de Ingeniería Mecánica, Universidad Santo Tomás
Corresponding author’s information: dionisiomalagon@usantotomas.edu.co
ABSTRACT
The lack of competences development related to engineering problems solving is one of the most frequent criticisms made to the education of new engineers in Universities in Colombia. In this study, 64 mechanical engineering students from Universidad Santo Tomás (Bogotá, Colombia) were allocated to an experimental group. They were divided into six small groups and participated in three practical sessions. These sessions were designed based upon the guidelines of curricular integration and active learning proposed by CDIO and IT-based education posed by the Colombian Government. In every practical session, students faced a mechanical engineering-related problem, supported by calculus and a specialized software, in order to solve the problem under the construction of an environment of sciences integration, emphasizing math. The participants took a couple of similar diagnostic tests before and after each session. These tests were intended to quantify their performance in regards to the integration of four competences from Bloom’s taxonomy, namely: interpret, explain, execute and implement, included within the levels understand and apply. 56% of the students showed they have enough skills in problem-solving in accord with the final test results, in comparison with the 38% evidenced in the diagnostic test, resulting in a satisfying improvement of 18%. This is an advance in the formation of new engineer in the Colombian context.
KEYWORDS
Problem solving skills, computer base math, engineering learning. Standards: 3, 7, 8.
INTRODUCTION
The skill in solving engineering-related problems is the fourth one with greatest deficit in recent engineering graduates (Saavedra Guevara & Vega Hernández, 2014). In Colombia, the improvement of math learning has been stressed since 1961 (García, 1996). In 1994, the Colombian Ministry of Education restructured the model, proposing a set of curricular guidelines based on basic standards of competences and performance assessment (Murcia & Henao, 2015). In spite of such restructuring, Colombia achieved the lowest score in problem solving out of the 44 countries participating in PISA (National Center of Education Statistics, 2014), (OECD, 2012). Nowadays, the Government seeks for the internet massification and the digital ecosystem development (Gobierno de Colombia - MinTIC, 2014a) as tools for the sake of teaching personalization and the opening of teaching positions based on information access (Gobierno de Colombia - MinTIC, 2014b)
This problem had begun during elementary school and continuing during university step. At tertiary level, math learning is one of the most relevant issues not only in Colombia but generally in Latin America (García, 1996). The disconnection between reality and the career environment is the main cause (García, 1996), which generates high failure indicators in university courses (Caramena Gallardo, 2010). The implementation of curricular integration, as proposed by CDIO, which connect math and engineering is a strategy that might contribute to the resolution of this problem.
At a global level, different institutions have designed and applied multiple strategies aimed at strengthening math and problem-solving skills, using concepts integration, which has led to satisfactory outcomes. In this way, the program in industrial engineering from Sao Paulo University suggested a model of learning through engineering implementation projects for first-year students. It emphasized the role of differential calculus, which resulted in a positive experience (Pereira, Barreto, & Pazeti, 2017). Since 2013, when PBL (Project Based Learning) was implemented, there has been tested multiple strategies like activities for improving communication skills and the building and presentation of a real prototype as a solution for the courses semesterly works based on real engineering problems. The evidenced evolution was so significantly positive than PBL was extrapolated to other three engineering programs inside Universidad de Sao Paulo (Pereira et al., 2017).
Other alternative for math teaching is Computer Based Math (CBM), which represents a different way for math teaching using machine-computer maths into a crucial part of all curriculums in order to present students tools for applying math for high level problem solving without thinking about how hard to hand-solve it is (Computer Based Math Org., 2014). This a long-term project boosted by Wolfram Research, developers of Mathematica and Wolfram Alpha, for transforming worldwide teaching in 25 years.
Multiple studies which involved math and engineering teaching methodologies based on computer has resulted in success. For example, Chemistry Engineering Department of Universidad de Huelva, Spain, proposed and implemented a novel teaching methodology for improving the calculation of pumping power by using Mathcad software (Cuadri, Marín-Alfonso, & Urbano, 2018) with second-year chemical engineering students. Surveys applied to students indicated that the computational tool and the training courses related with, highly improved their ability to solve calculation of pumping power (engineering problem) as the mathematical difficulties were handled by software focusing them on the understanding of the theoretical basis of the phenomenon (Cuadri et al., 2018).
Another success case comes from Universiti Putra Malaysia where a similar methodology were implemented with Linear Algebra students and Maple software, which produced good results on the understanding of usually-hard concepts as eigenvalues and eigenvectors. As the software did the algebraic manipulation, the students could focus on the important properties and principles of the concepts improving the understanding (Kilicman, Hassan, & Husain, 2010).
This approaches leads to nowadays real-world math’s which requires complex models that should not be approached by hand. However, what is found in today’s math education is that it spends 80% of the curriculum time to develop hand-calculation skills focused on algebraic manipulation even having computer technology for math software (Titterton & Computer Based Method Org., 2014). Most of the times, this translates in high expertise for mechanical algebraical processes but not into the understanding of basic or complex topics.
The aim of this article is evaluating a strategy which conjugates curricular integration and CBM for enhancing the learning of math in the Mechanical Engineering at Universidad Santo Tomás in Bogotá Colombia based on previous works but focusing on engineering problem solving.
METHODOLOGY
Participants
64 mechanical engineering students (15% out of the total amount) from Universidad Santo Tomás (Bogotá, Colombia), studying between third and eighth semesters, took part of this research project. The sample size was determined using Cochran’s Formula with margin of error of 10%, confidence level of 90%, response distribution of 50% population size of 430 students, suggesting 59 students for sample.
Procedure
The students took a diagnostic test in which they solved four mechanical engineering-related problems. The length of the test was twenty minutes and the use of computational tools like Mathematica 11.0 and Geogebra online was allowed.
problem. The result of the practical sessions was a Mathematica file with an algorithm designed by each student which solved the problem via Calculus and software.
Finally, they took an ending test to determine the effect of the study cases on the development of mathematical competences. The questions and conditions of the ending test were similar to the first one’s and allowed investigators to quantify the math skills development levels using two Bloom’s Taxonomy levels: understand and apply.
The diagnostic and final tests
The diagnostic and final tests were made to determine the level of mathematical competences development based on two levels from Bloom’s taxonomy (BT): understand and apply (Radmehr & Drake, 2018). Each diagnostic test contained four questions linked (one by one) with the subcategories interpretation and explanation, execution and implementation belonging to the levels understand and apply. In Table 1, the structure of the diagnostic and final tests are displayed.
Table 1. Structure of diagnostic and final tests
Category in accord with Bloom’s taxonomy
Subcategory Related question Related problem
Understand
Interpret
Can students use a graphic representation to obtain optimum
performance variables?
Establishment of optimum performance conditions for a hydraulic device from a graph
Explain
Can the students use proposed basic models to solve problems?
Establishment of
quantities in an entry and exit model
Apply
Execute
Can the students build and develop basic
models for solving known problems?
Establishment of optimum performance conditions for a thermal machine from a
mathematical function
Implement
Can the students build and develop basic models for arguing and solving unknown problems?
Establishment of variables of production for maximizing
usefulness from models with mathematical inconsistencies
Study cases
All study cases were solved by the students by means of the Computer Based Math model (CBM), which proposes a step-by-step process of four key stages: problem definition, problem transfer to math, calculation of responses throughout a computer and results interpretation (Titterton & Computer Based Method Org., 2014), (Kadry & Shalkamy, 2012).
The three main study cases were the result from the evolve of problems faced by students in three previous courses of the program of Mechanical Engineering into more complex problems and the creation of alternative solving methods. The transformation was made based on CDIO standards (CDIO, 2016) 3: integrated curriculum; 5: design-implement experiences and 7: integrated learning experience, so the new problems allowed students to interact with facts proposed by CDIO in these ways:
The problems had been solved by the students in their previous courses using traditional calculus methods characterized for being iterative and manuals. So, it was built a new methodology based not only on the previous course concepts but in the conjunction of math knowledge for quickly obtaining of good answers. Also, the study cases were complexed to stop being common functional validations and start being design problems. As the problems became more complex and the traditional methodologies turned useless, the students focused on building and applying new alternative methodologies which involved not only the concepts learnt on a specific engineering curse but integrated knowledge of math and optimization models. Those alternatives were transformed into algorithms with math models implementations that allowed the obtention of fast and optimal answers.
Further than the understanding of the engineering concepts required for solving the problems, the selection and adaptation of the study cases did not represent a big challenge. The original cases were presented and analyzed in each previous course because of its correspondence with the objectives of the course itself. The main purpose of the study cases was to establish a comparison between the traditional methodologies and the new ones supported by integrated concepts, emphasizing math, and software.
Introductory case: animation of an airplane take-off
The introductory case is an original idea for the investigation and its purpose was to motivate the students showing them how a simple mathematical function and the use of key concepts as derivate are useful to achieve interesting results.
The students received a set of dots whose union produces a fighter aircraft perimeter (Figure 1). Overall, the take-off is similar to the arctangent function close to the origin. The students used this abstraction to construct an animation in Mathematica 11.0. There, the graphic object of the airplane followed the coordinates (x(t), arctan(x(t)) and inclined according to the curve tangent, obtained from the derivative, achieving an optimal representation of the rise.
Figure 1. Airplane take-off animation frame worked in Case I
Case I: Diameter establishment of a recessed beam of circular profile submitted to dynamic load
In the second case study, the students determined the minimum diameter of a solid, ground and hot rolling circular profile of Steel 1005, guaranteeing the proposed configuration resistance to the dynamic loads to which it would be submitted, with a safety factor equal to 1.50 ± 0.05. This tolerance was defined to highlight the flairs of calculus and the software given that traditional methodologies lead to solutions with sporadic safety factors, hindering the attainment of hundredths tolerance (Budynas & Nisbett, 2013).
One of the two possible methods to analyze the system, in accordance with materials resistance theories by means of singularity functions, leads to a two-variable equation (see equation 1), which is translated in the effort the profile must resist (σ) (Budynas & Nisbett, 2013).
σ(𝑥, a) = F*𝑏^2*𝑙^3*((2𝑎 + 𝑙) – 𝑎*𝑙) – [{F(𝑥 − 𝑎) if 𝑥 ≥ 𝑎] (1)
Where F was the applied force on the profile at a distance units from left support and b distance units measured by right support, l was the length of the profile and x the analysis point.
Figure 1. Maximum and minimum stress points of σ(x, a) (1) obtained by one of the participating students using Mathematica 11.0 for study case 1
quantities to determine the resistance of the material submitted to loads, and built a routine of programming in Mathematica 11.0 for iterating the diameter quickly up to achieve the desired safety factor, without exceeding the proposed tolerance.
The case study objective was showing the students the flairs given by calculus and the software to solve tough problems, by comparing the problem solving through the optimization with traditional teaching methodologies. The building an algorithm based on math can simplify any iterative process as computer iterate itself in accord to inputs.
The students who wrote and used an algorithm (See Figure 2) just needed to execute the program to get the estimated safety factor. If the obtained safety factor did not meet the requirements, the inputs could be changed quickly and the algorithm executed again in order to get closer to the answer.
Figure 2. Part of the code wrote by one of the participating students in Mathematica 11.0 for the determination of safety factor for study case 1
Case II: Hydraulic pumps system setup establishment according to experimental data
In this case study, the students established the setup of a hydraulic pump system to transport a specific fluid flow, considering as a basis the experimental data collected by a group of fluid mechanics students for a three-phase feed pumps available at the Hydraulics laboratory from Universidad Santo Tomás.
In the experimental practice, the upstream and downstream pump pressure, the intensity, the capability demanded by the three-phase engine and the pump shaft angular speed were measured for different flows. The working fluid was water at standard conditions.
V1^2*rho/2 + P1 + rho*g*z1 + PB = v2^2*rho/2 + P2 + rho*g*z2 + PL (2)
Where P was the pressure measured at each analysis point (1 or 2), PB the power delivered to the fluid because of the pump in terms of pressure head, PL the power losses because of roughness in terms of pressure heads, V the speed of the fluid at each analysis point (1 or 2), Z the height measured from a reference system at each analysis point (1 or 2), g the gravity acceleration and rho the density of the fluid.
Figure 3. Efficiency of the pump as a function of work discharge built by one the participating students using Excel 2016 for determination of process variables in study case 2
This case of study aimed the students to construct a model to solve an engineering-related problem based on their results and group discussion.
Case III: Establishment of the diameter of a conductor cable insulation and comparison with real cable
The students tackled a heat transfer problem associated with the establishment of the diameter of an optimum thermal insulation layer for a rubber electric current conductor cable or reference 3X14 600V, manufactured by Cablecol Colombia, submitted to specific conditions, comparing its outcomes with the diameter measurement of a real cable. The students developed a model departing from the Fourier law in cylindrical coordinates (Cengel Yunus, 2002) (see equation 3) to establish the diameter of the thermal insulation layer that in accord with the theory guaranteed the best condition for heat transfer. Later, they measured the diameter of the real cable using a slide gauge by themselves, showing a match between the results reached via the mathematical model and the real cable dimensions.
dQ/dt = -k * ∫[r (∂T/∂r) dA] (3)
This case of study intended the students to arrive at the same solution of a real problem provided by the industry through modelling mathematically from differential geometry and differential equations.
R² = 0.999
-20% 0% 20% 40% 60% 80% 100%
0.00 20.00 40.00 60.00 80.00 100.00 120.00
Pump
e
ff
ic
ie
n
cy
(
%
)
Figure 4. Heat transfer curve for a conduct cable as a function of thermal insulating layer radius obtained by one of the participating students using Mathematica 11.0 for study case 3
RESULTS AND ANALYSIS
Diagnostic input test
The diagnostic input test evidenced that 72% of the students used a graphic representation to obtain optimum performance variables. Out of the 64 participants, 46 identified the point in a curve, graphically represented in the plane similar to the result achieved via the optimum performance system. Nonetheless, only 15% of the students (7) who identified the optimal point in the graph, facing a similar problem without graphic representation, constructed and developed a mathematical model that allowed them to obtain optimum performance variables.
Plot 4. Interpret and explain indicators quantification
On the other hand, 55% of the students (35) utilized simple mathematical models posed by the test to solve a problem. Yet, only 13% of the participants (8) constructed and developed a mathematical model that permitted them to attain the best response, even surpassing mathematical inconsistencies pertaining to the unknown environment they were facing:
7 7 32
0 10 20 30 40 50
Identified the optimal point on a curve and:
N
u
mb
e
r o
f stud
e
n
ts
who
:
Interpret and explain
Had troubles with the math model to get the optimal point
Plot 5. Execute and implement indicators quantification
Difficulties found
The diagnostic input test and subcategories of the second level of Bloom’s Taxonomy allows to identify the main difficulties presented by students when solving engineering problems.
As 72% of students properly obtained the optimum performance variables using a graphic representations, it is possible to say that most of the students has a suitable level of the interpret skill. This allows them to transform the way they represent the information as they took graphical quantities to convert them into numerical values. So far, only 15% of the students who has an appropriate interpret level (11% of the total amount of students) could be told to have an adequate level of the execute skill. Even having properly interpret skills, most of the students could not build and solve a mathematical model in order to get the same answers as the ones obtained via graphics. When the students did not have a graphical support and were challenged to prepare their own model for getting optimal solutions for a known problem, weakness appeared.
When the students had an already proposed model, the results were much better than when they had to build it and solve it. 55% of the students correctly managed the proposed models for solving the problem. Thus, it is possible to assert than the half of the students have a properly level of the explain skill, as they used already proposed math models for solving problems. Nonetheless, only the 13% of the students could build and calculate their own math model for unknown problems. So, only the 13% of the students evidenced to have an accurate level of the implement skill. If comparing this result with the one related with execute skill, there appears a question: why solving unknown problems looks easier than already known ones?
Summarizing, most of the students of the Mechanical Engineering Program have good interpreting skills; half of the students evidences good explaining skills, only 13% showed good implement skills and single 11% displayed good execute skills.
Evaluation output test
After the three practical sessions, the final test evidenced a satisfactory increase in percentage amount of students who used graphic representation to obtain optimum performance variables
27 8
0 5 10 15 20 25 30 35 40
Used appropriately proposed math models and
N
u
mb
e
r o
f stud
e
n
ts
who
:
Execute and implement
(93%). Furthermore, 62% of the students identified the point properly and constructed and developed a mathematical model that allowed them to obtain optimum performance variables in a second problem.
Similarly, despite the decrease in percentage amount of students that used a simple mathematical model proposed (44%), the percentage amount of participants satisfactorily facing an unknown problem related to mathematical inconsistencies, by constructing and developing models, rose up to 26%.
The following plot will compare the outcomes of both the diagnostic input and output tests, according to the related subcategories in accord with Bloom’s Taxonomy:
Plot 6. Comparison between indicators before and after practical sessions
Analysis
Significantly, most of the participants (93%) evidenced to have skills in using graphic representations for obtaining optimum solutions in comparison with the initial 72%. This indicator illustrates an increase in the students’ interpretative capacity since they have transformed the way they represent information to fit better within problem solving activities. Likewise, the meaningful rise of participants who utilized models proposed for problem solving (47%) shows improvement in students’ competence to explain owing to the use of concepts integration and computational tools. In turn, the fourth indicator shows that students enhanced their skill in solving unknown problems departing from posing and developing a mathematical
72%
15%
55%
13% 93%
62%
44%
26%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Interpret Explain Execute Implement
Stu
d
e
n
ts
q
u
an
ti
ty
(
%
)
Associated subcategory in accord with BT
Results comparison
Before practial sessions
model (implement). However, the number of handicaps to solve known problems from the proposal and development of a mathematical model (execute) rose.
The enhancement of interpretation and explanation skills is translated, in accord with BT, in the increase of students’ capacity to understand concepts and therefore, problems. The three practical sessions made it possible to enhance this skill in the group of students. Also, the fact that the students showed greater skill in facing new problems rather than known problems previously worked, evidences how they welcomed the proposed methodology in the practical sessions while the traditional one was relegated to the background.
This proves that teaching methodologies which includes multidisciplinary concepts integration and software use in order to lay aside algebraical manipulation, can result into good levels of development of the student’s skills of interpret, explain and implement; key abilities for facing and solving unknown engineering problems. The methodology proposed by this investigation represents the conjunction of the good results obtained by the enforcement of CDIO standards: 3 ;5 and 7 in real cases as PBL applied in Sao Paulo University, and the advantages of CBM learning in order to get successful understanding as evidenced on Universidad de Huelva, but empowers it as is not limited to a single math course but to the developing of problem solving skills applicable to any engineering challenge.
CONCLUSIONS
Participating students evidenced the empowerment of their skills for solving unknown engineering problems when they were introduced into new strategies related with the integration of different concepts from Engineering Sciences and Basic Sciences, emphasizing math, supported by specialized computational technologies.
The 56% of the students showed they have enough skills in problem-solving in accord with the final test results, in comparison with the 38% evidenced in the diagnostic test, resulting in a satisfying improvement of 18%. Then, the use of curricular integration, design-implement practices and integrated learning experiences as strategies for engineering teaching combined with interactive use of specialized math software for problem solving, can result in their improvement of the interpret, explain and implement skills for problem solving.
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BIOGRAPHICAL INFORMATION
Nicolás Góngora Salazar is a last-year student of the Mechanical Engineering program of Universidad Santo Tomás, Bogotá, Colombia. His current focus is on the use of technology in education and the teaching of math application on engineering.
Corresponding author
Ph.D. Dionisio Humberto Malagón Romero Universidad Santo Tomás
Programa de Ingeniería Mecánica
Carrera 9 N 51 – 11. Bogotá, COLOMBIA +57 304-364-75-85
dionisiomalagon@usantotomas.edu.co
