# Differential and Integral Calculus in First-Year Engineering Students: A Diagnosis to Understand the Failure

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## Abstract

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## 1. Introduction

- The use of learner-centred methods.
- Contextualization of mathematics using real-world examples.
- Development of strategies to overcome gaps in basic and elementary knowledge.
- Applying face-to face classes and student conversation.
- Improvement of students’ motivation, involvement and self-efficiency.
- Different learning styles approaches.

“An individual’s mathematical knowledge is her or his tendency to respond to perceived mathematical problem situations by reflecting on problems and their solutions in a social context and by constructing or reconstructing mathematical actions, processes, and objects by organizing these in schemas to use in dealing with situations”.[45] (p. 5)

“Mathematical competence means the ability to understand, judge, make and use mathematics in a variety of intra and extra mathematical contexts in situations in which mathematics plays or can play a role. Necessary, but certainly not sufficient, prerequisites for mathematical competence are lots of factual knowledge and technical skills in the same way as vocabulary, orthography, and grammar are necessary but not sufficient prerequisites for literacy”.[48] (pp. 6–7)

- Thinking mathematically (C1), reasoning mathematically (C2), posing and solving mathematical problems (C3), and modelling mathematically (C4), which make up the ability to ask and answer questions in and with mathematics.
- Representing mathematical entities (C5), handling mathematical symbols and formalism (C6), communicating in, with, and about mathematics (C7), and making use of aids and tools (C8) are concerned with “the ability to deal with and manage mathematical language and tools” [48].

## 2. Portuguese Context in the Access to Engineering Studies

## 3. Differential and Integral Calculus in ISEC

## 4. The Study

#### 4.1. Methodology

#### 4.2. Instruments

- Contents taught in the CU-DIC of each degree of engineering at ISEC.
- Assessment methodologies.
- Students’ competencies that will be acquired.
- Type of classes integrated by each CU (theoretical, theoretical-practical, or tutorial orientation).
- The number of hours of on-site classes with students and unaccompanied work.
- Attendance classes’ rate, considering only the highest lowest rate found with an indication of the week in which each occurrence was confirmed.
- Attendance rate in examination moments are given by the ratio between the number of students participating in the assessment (P) and the number of enrolled students in the CU (I), i.e., P/I;
- Dropout rate in the examination moments given by the ratio between the number of evaluated students (A) and the number of students participating in the assessment (P), i.e., A/P.
- Pass rate in examination moments are given by the ratio between the number of students approved (Ap) and the number of evaluated students (A), i.e., Ap/A.

#### 4.3. Sample

## 5. Results

#### 5.1. General Scope

- Complementary course related to essential knowledge in mathematics (CU pre-calculus frequency).
- Attendance of CU-DIC in sliding regime.
- Integration in OCP, taking into account compulsory frequency in pre-calculus unit.

#### 5.2. Private Sector—Biomedical and Informatics

## 6. Conclusions and Future Work

- The time for on-side classes of the different CU-DIC taught in engineering degrees at the ISEC is distributed by theoretical, practical, and theoretical-practical classes, except EC and GSC that only consider theoretical-practical classes. EM and EI consider practical classes and EE includes tutorial guidance classes.
- The common contents to all CU-DICs under study are trigonometric functions, primitivation, definite integral, and improper integrals, which constitute the core of the mathematical knowledge that teachers understand as essential.
- We can also infer that the competencies defined in the FCU do not seem to influence the success of mathematics in the engineering courses since the pass rates do not have significant deviations between degrees.
- We can also conclude that the CU-DIC in the various degrees in engineering of the ISEC register very variable assessment attendance rates. Although there is no relationship between the attendance rates at the examinations and the pass rates, it should be noted that fewer and fewer students attend classes and exams—a fact that will influence the final pass rate, which is found to be low.

- Student’s profile attending and participating in different examination models proposed by teachers.
- Relationship between the attendance rates (at the classes and examinations) and economic, social, cultural, technology development conditions.
- Teaching/learning strategies to be applied, aimed at reaching students who do not carry out the examination and understand the consequent reasons that led them to dropout.
- The set of basic and elementary level knowledge that students need to master upon entering higher education.
- Mistakes made in basic and elementary knowledge that allow the definition of a structured intervention in overcoming gaps.
- Environments that lead to meaningful learning and involve all actors (teachers and students) in the educational process.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Pass rate distribution for the 1st semester of the academic years 2011/2012 to 2014/2015 and all undergraduate degrees in ISEC.

**Figure 6.**Pass rate distribution for the 1st semester of the academic years 2015/2016 to 2016/2017 and all undergraduate degrees in ISEC.

Aspect | Results |
---|---|

General scope | The hours of on-site classes are distributed by theoretical, practical and theoretical-practical classes, the latter model being the most used in all the degrees studied. Contents common to all CU-DICs under study are Basics Concepts of Differential and Integral Calculus, curricular plans integrate other more specific topics according to the degree. Only Reasoning Mathematically competence is common to all degrees. The attendance rate decreases notably over the two semesters, reaching a dropout rate of over 90% in some cases. The overall average of the pass rate in both semesters is low, closely 58%. |

Private Sector—Biomedical and Informatics | EBiom results show a high attendance rate for exams. EI results shows a low attendance rate for exams. Distributed assessment is the preferred modality of students with a significant pass rate, in both degrees. |

Ex | AD | |||
---|---|---|---|---|

Mean | Std. Deviation | Mean | Std. Deviation | |

Biomedical | 45.03% | 12.13% | 93.84% | 5.2814% |

Informatics (1stS) | 18.48% | 9.15% | 71.66% | 6.30% |

Informatics (2ndS) | 24.29% | 3.52% | 64.37% | 11.46% |

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**MDPI and ACS Style**

Bigotte de Almeida, M.E.; Queiruga-Dios, A.; Cáceres, M.J.
Differential and Integral Calculus in First-Year Engineering Students: A Diagnosis to Understand the Failure. *Mathematics* **2021**, *9*, 61.
https://doi.org/10.3390/math9010061

**AMA Style**

Bigotte de Almeida ME, Queiruga-Dios A, Cáceres MJ.
Differential and Integral Calculus in First-Year Engineering Students: A Diagnosis to Understand the Failure. *Mathematics*. 2021; 9(1):61.
https://doi.org/10.3390/math9010061

**Chicago/Turabian Style**

Bigotte de Almeida, Maria Emília, Araceli Queiruga-Dios, and María José Cáceres.
2021. "Differential and Integral Calculus in First-Year Engineering Students: A Diagnosis to Understand the Failure" *Mathematics* 9, no. 1: 61.
https://doi.org/10.3390/math9010061