Analysis of Industrial Engineering Students’ Perception after a Multiple Integrals-Based Activity with a Fourth-Year Student
Abstract
:1. Introduction
2. Methodology
2.1. Context and Participants
- On a Likert scale [26] of 1–5, students answered the following question: “I think my level of digital competence is high”. The average response value was 3, and the standard deviation was 0.75.
- Students were asked if they had previous experience using the package Wolfram Mathematica [27], of which 94.1% answered no.
- Finally, students were asked if they were currently enrolled in any second, third, or fourth subjects, of which 88.2% answered no.
2.2. Analysis of the Methodology
2.2.1. Interdisciplinary Problems to Teach Calculus
2.2.2. Computer Tools
3. Development of Our Activities and Results
3.1. Real-Life Engineering Problems
- -
- Industrial robots and manipulators. One of the most important features of these automatons is their work range, i.e., the surface or volume represented by all the robot’s positions in space. To calculate this, a manipulator position study is carried out obtaining a cloud of points with its limit positions. These data are interpolated, and, with multivariable integration, the range surface or volume can be obtained.
- -
- Self-driving cars and LIDAR sensors. These active sensors scan their surroundings with an infrared laser, mapping thousands of points. The point cloud obtained is treated in a similar way to the previous problem.
- -
- Metrology. Laser interferometers can be used to measure roughness. These perform a micrometric scan of the part to be analyzed creating a point cloud that approximates its surface. To calculate the volume of material that needs to be removed in a grinding operation to improve its finish, it is necessary to apply multivariable integration.
- -
- Calculation of structures. It is increasingly common to see constructions with a quadric surface shape, since they have very interesting properties. For instance, the cooling towers of nuclear power plants are shaped like hyperboloids, to boost natural convection and expel hot gases outside.
3.2. Estimating the Volume of a Building Excavation, from a Point Cloud Collected by Topography Measurements
- (i)
- Finding an interpolating polynomial that fits the point cloud set up by the previous coordinates. For this, the least squares adjustment method is used. We calculate the coefficients of the polynomial (introduced in Appendix A) so that they minimize the quadratic error between the interpolating function and the data points. The approximation obtained is shown in Figure 2. This is a fourth-order polynomial in x and y, with 12 terms (12 unknown coefficients determined).
- (ii)
- Determining the equations of the different lines between the perimeter points, in order to establish the limits of integration with which the volume will be calculated later.
- (iii)
- Subdividing the prismoid formed by the coordinates of each plot point and the origin plane in different integrable prisms. Its edges are defined by the equations of the line obtained previously. The total volume of the excavation is calculated as the sum of the volumetric integrals of these bodies.
3.3. Questionnaire
4. Discussion on the Results
4.1. Questionnaire Given to First-Year Students
4.2. Participant Observation
“I consider that these teaching methodologies are very enriching for both students and teachers. The mathematics subjects taught in the first years of engineering are usually very theoretical and overwhelm many students. In my opinion, exploring real-world problems and challenges from early grades drives students to obtain a deeper knowledge of the subjects they are studying and encourages them to develop confidence and motivation. Personally, I would have loved to participate in this type of activity when I first entered college.
At the beginning of the activity, I noticed how the students seemed curious and were excited to do something new, outside the usual routine. As the session progressed, although some of them were rather clueless, most students remained attentive and really seemed to be interested in what we were doing. Overall, the atmosphere was quite welcoming. I would highly recommend other senior students to continue carrying out these experiences.”
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
Appendix A
- Estimating the volume of a building excavation
- The comments in the code used in the classroom are in Spanish for academic purposes. In this work, they have been translated into English to allow readers to understand them.
- (* From the Excel file, we import data that the students previously calculated in another third-year subject. *)
- data = Import[“C:\\data_palomares.xlsx”][[1]]
- {{−0.937085, −5.90457, 0.492498}, {13.5512, −7.28953, 0.326174}, {25.4282, −9.473, 0.168468}, {18.6554, −40.5825, 3.69048}, {−6.18117, −16.894, 1.41553}, {−13.2269, −33.8948, 3.37329}, {−7.95319, −66.4691, 6.04395}, {12.8675, −61.5239, 5.9983}, {2.85255, −84.6279, 7.25412}, {−0.12187, −93.8704, 7.74539}, {−26.3122, −86.2181, 7.02484}, {−19.8547, −56.455, 5.19855}, {1.05622, −39.5746, 3.50618}}
- data = {{−0.93708456`, −5.90457219`, 0.49249848`}, {13.551175`, −7.28952502`, 0.32617372`}, {25.4281851`, −9.47299653`, 0.16846814`}, {18.6554001`, −40.5824811`, 3.69047693`}, {−6.18117493`, −16.8939747`, 1.41553061`}, {−13.2268731`, −33.8947756`, 3.37329311`}, {−7.95318955`, −66.4691303`, 6.04395385`}, {12.8674808`, −61.5238784`, 5.99830393`}, {2.85254857`, −84.6279358`, 7.25411933`}, {−0.12186975`, −93.8704137`, 7.74539119`}, {−26.312194`, −86.2180903`, 7.02483645`}, {−19.8547003`, −56.4550142`, 5.1985514`}, {1.056217`, −39.574576`, 3.506184`}};
- (* We must choose how we want the polynomials to be. We want the polynomials to be like the one below (for example). *)
- pol[x_, y_] = a1 + a2*x + a3*y + a4*x^2 + a5*x*y + a6*y^2 + a7*x^3 + a8*x^2*y + a9*x*y^2 + a10*y^3 + a11*x*y^3 + a12*x^2*y^2;
- (* Thus, the system of vectors, the generators of our vector space, is formed by means of the below phi functions. *)
- phi[i_, x_, y_] =
- Which[i == 1, 1, i == 2, x, i == 3, y, i == 4, x^2, i == 5, x*y,
- i == 6, y^2, i == 7, x^3, i == 8, x^2*y, i == 9, x*y^2, i == 10,
- y^3, i == 11, x*y^3, i == 12, x^2*y^2];
- (* Now, we can calculate the coefficients of the polynomial that reaches (or is closest to all the points from our Excel file, depending on the dimensions of our problem). We evaluate our basis in all the points and solve the least squares problem. *)
- A = Table[ phi[j, data[[i, 1]], data[[i, 2]] ], {i, 1, Length[data]}, {j, 1, 12}];
- b = Table[ data[[i, 3]], {i, 1, Length[data]}];
- LeastSquares[A, b]
- {0.0565949, 0.0797015, −0.070381, −0.00583912, 0.00981652, 0.000672554, 0.000107678, −0.000217221, 0.00022677, 5.86501*10^−6, 1.46394*10^−6, −1.51735*10^−6}
- (* This problem can also be solved as an optimization problem, and Plot3D can be used to draw our polynomial and compare it with the real points as shown in Figure A1. *)
- NMinimize[ dist, {a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12} ]
- {0.00415069, {a1 -> 0.0565949, a2 -> 0.0797015, a3 -> −0.070381, a4 -> −0.00583912, a5 -> 0.00981652, a6 -> 0.000672554, a7 -> 0.000107678, a8 -> −0.000217221, a9 -> 0.00022677, a10 -> 5.86501*10^−6, a11 -> 1.46394*10^−6, a12 -> −1.51735*10^−6}}
- (* Finally, we are ready to approximate the volume. *)
- ord1 = Ordering[data]; (* Order of the coordinates *)
- data2 = {{−26.312194`, −86.2180903`, 7.02483645`},{−19.8547003`, −56.4550142`, 5.1985514`},{−13.2268731`, −33.8947756`, 3.37329311`},{−0.93708456`, −5.90457219`, 0.49249848`},{13.551175`, −7.28952502`, 0.32617372`},{25.4281851`, −9.47299653`, 0.16846814`},{18.6554001`, −40.5824811`, 3.69047693`}, {−6.18117493`, −16.8939747`, 1.41553061`},{12.8674808`, −61.5238784`, 5.99830393`}, {2.85254857`, −84.6279358`, 7.25411933`},{−0.12186975`, −93.8704137`, 7.74539119`},{−26.312194`, −86.2180903`, 7.02483645`}};
- (* We calculate the lines that will be used in the integration. *)
- For[i = 1,i < Length[data2],i++,
- m[i] = (data2[[i + 1,2]]-data2[[i,2]])/(data2[[i + 1,1]]-data2[[i,1]]);
- b[i] = data2[[i,2]]-m[i]*data2[[i,1]]
- ];
- Integrate[ Z[x, y],{x,data2[[1,1]],data2[[2,1]]},{y,(m [11]*x + b [11]),(m [1]*x + b [1])}] +
- Integrate[ Z[x, y],{x,data2[[2,1]],data2[[3,1]]},{y,(m [11]*x + b [11]),(m [2]*x + b [2])}] +
- Integrate[ Z[x, y],{x,data2[[3,1]],data2[[4,1]]},{y,(m [11]*x + b [11]),(m [3]*x + b [3])}] +
- Integrate[ Z[x, y],{x,data2[[4,1]],data2[[5,1]]},{y,(m [11]*x + b [11]),(m [4]*x + b [4])}] +
- Integrate[Z[x, y],{x,data2[[5,1]],data2[[11,1]]},{y,(m [11]*x + b [11]),(m [5]*x + b [5])}] +
- Integrate[Z[x, y],{x,data2[[11,1]],data2[[10,1]]},{y,(m [10]*x + b [10]),(m [5]*x + b [5])}] +
- Integrate[Z[x, y],{x,data2[[10,1]],data2[[9,1]]},{y,(m [9]*x + b [9]),(m [5]*x + b [5])}] +
- Integrate[Z[x, y],{x,data2[[9,1]],data2[[6,1]]},{y,(m [8]*x + b [8]),(m [5]*x + b [5])}] +
- Integrate[Z[x, y],{x,data2[[6,1]],data2[[8,1]]},{y,(m [8]*x + b [8]),(m [6]*x + b [6])}] +
- Integrate[Z[x, y],{x,data2[[8,1]],data2[[7,1]]},{y,(m [7]*x + b [7]),(m [6]*x + b [6])}]
- 10070.1
- (* The result is similar to others obtained using engineering software. *)
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Degree Program | ECTS * | Academic Year | Students | |
---|---|---|---|---|
Number | Percentage of the Total Study Group | |||
Bachelor in Mechanical Engineering | 240 | 4 | 9 | 52.9% |
Bachelor in Electrical Engineering | 240 | 4 | 1 | 5.9% |
Bachelor in Electronic and Automatic Engineering | 240 | 4 | 4 | 23.5% |
Double degree | 276 | 4 | 3 | 17.6% |
Code | Question | Category |
---|---|---|
Q1 | I think mathematics is a field closely related to engineering. | Motivation |
Q2 | I think that integral calculus is related to other subjects in my degree course. | Motivation |
Q3 | I think that integral calculus will be important in my professional career. | Motivation |
Q4 | The math classes are fun and catch my interest. | Motivation |
Q5 | I think a math class with practical applications, like this one, is more useful than a conventional class. | Usability |
Q6 | I think computers are needed when teaching math. | Usability |
Q7 | The activities of this subject serve to organize my learning and to be able to approach solving problems of different types in a structured way (and not just learn mathematical content). | Critical thinking |
Q8 | Knowing the formulas of the surface and the length of the line in different types of coordinates (cartesian, parametric, and polar) can help me solve real-life problems. | Critical thinking |
Q9 | Activities that mix engineering and mathematics should also be included when learning other subjects. | Usability |
Q10 | I think my computer skills are high. | Usability |
Q11 | I think it is a good idea for a senior classmate to teach a math class. | Usability |
Q12 | When I am a fourth-year student grade and see the usefulness of mathematics in other subjects, I would like the opportunity to give a class to first-year students. | Usability |
Question | Mean | Standard Deviation | Standard Error | |
---|---|---|---|---|
Q1 | Pre-test | 4.29 | 0.77 | 0.19 |
Post-test | 4.47 | 0.62 | 0.15 | |
Q2 | Pre-test | 3.77 | 1.09 | 0.26 |
Post-test | 4.35 | 0.49 | 0.12 | |
Q3 | Pre-test | 3.06 | 1.25 | 0.30 |
Post-test | 4.12 | 0.78 | 0.19 | |
Q4 | Pre-test | 2.24 | 1.15 | 0.28 |
Post-test | 3.18 | 1.13 | 0.27 | |
Q5 | Pre-test | 3.76 | 1.03 | 0.25 |
Post-test | 4.06 | 0.75 | 0.18 | |
Q6 | Pre-test | 3.41 | 0.87 | 0.21 |
Post-test | 3.65 | 0.93 | 0.22 | |
Q7 | Pre-test | 3.35 | 0.79 | 0.19 |
Post-test | 3.59 | 0.87 | 0.21 | |
Q8 | Pre-test | 2.71 | 0.99 | 0.24 |
Post-test | 3.82 | 1.01 | 0.25 | |
Q9 | Pre-test | 3.77 | 0.97 | 0.24 |
Post-test | 4.18 | 0.81 | 0.20 | |
Q10 | Pre-test | 3.00 | 1.00 | 0.24 |
Post-test | 2.88 | 1.11 | 0.27 | |
Q11 | Pre-test | 4.00 | 0.71 | 0.17 |
Post-test | 4.24 | 0.75 | 0.18 | |
Q12 | Pre-test | 2.65 | 1.37 | 0.33 |
Post-test | 2.71 | 1.31 | 0.32 |
Test | Q1 | Q2 | Q3 | Q4 | Q5 | Q6 | Q7 | Q8 | Q9 | Q10 | Q11 | Q12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Kolmogorov–Smirnov | 1.00 | 0.454 | 0.046 | 0.240 | 0.954 | 0.954 | 0.954 | 0.046 | 0.954 | 1.00 | 0.954 | 1.00 |
Median test | 1.00 | 0.707 | 0.396 | 0.120 | 1.00 | 0.493 | 0.493 | 0.016 | 0.463 | 0.463 | 1.000 | 1.000 |
Mann–Whitney U | 0.586 | 0.131 | 0.012 | 0.024 | 0.540 | 0.433 | 0.413 | 0.003 | 0.259 | 0.786 | 0.375 | 0.838 |
Kruskal–Wallis | 0.542 | 0.087 | 0.008 | 0.200 | 0.495 | 0.403 | 0.366 | 0.002 | 0.223 | 0.774 | 0.332 | 0.830 |
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Giménez, A.R.; Martín-Vaquero, J.; Rodríguez-Martín, M. Analysis of Industrial Engineering Students’ Perception after a Multiple Integrals-Based Activity with a Fourth-Year Student. Mathematics 2022, 10, 1764. https://doi.org/10.3390/math10101764
Giménez AR, Martín-Vaquero J, Rodríguez-Martín M. Analysis of Industrial Engineering Students’ Perception after a Multiple Integrals-Based Activity with a Fourth-Year Student. Mathematics. 2022; 10(10):1764. https://doi.org/10.3390/math10101764
Chicago/Turabian StyleGiménez, Anuar R., Jesús Martín-Vaquero, and Manuel Rodríguez-Martín. 2022. "Analysis of Industrial Engineering Students’ Perception after a Multiple Integrals-Based Activity with a Fourth-Year Student" Mathematics 10, no. 10: 1764. https://doi.org/10.3390/math10101764