# EXPLORIA, a New Way to Teach Maths at University Level as Part of Everything

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## Abstract

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## 1. Introduction

#### 1.1. Affective Domain

#### 1.2. Rejection towards Mathematics

#### 1.3. Trends in Learning: STEM, STEAM, STREAM

#### 1.4. Active Methodologies

#### 1.5. Previous Works. STEAM Projects in Educational Systems

## 2. EXPLORIA Project. A New Way of Conceiving the University

#### Project EXPLORIA in the Degree of Engineering in Industrial Design and Product Development

## 3. Research Objectives

## 4. Materials and Methods

#### 4.1. Research Design and Data Analysis

#### 4.2. Participants

#### 4.3. Scope of Application

#### 4.4. Instrument

## 5. Design and Implementation of EXPLORIA Pilot Project

- Act I: Shape
- Act II: Volume
- Act III: Colour

- Act IV: Space
- Act V: Structure
- Act VI: Project

#### 5.1. Mathematics in EXPLORIA

#### 5.1.1. Description of Sessions and Timing

- In the theoretical session of mathematics the concepts of proportionality are explained. Golden ratio.
- In physics, a practical exercise to measure the proportions of the body is carried out.
- The authors who use the golden number in art are studied and analysed.
- In representation of shapes the concept of proportion is used to establish the proportionality of the shape.

- In mathematics we develop the concepts of vector and matrix as set of coordinates of an object. The isometries.
- In Basic Design, the matrices associated with the shape are applied and also to modular structures.

- In the maths session we study how to build tessellations using isometries. The concept of homothecy is explained.
- In the basic design subject, tessellations are developed. Exercises using homothecy: gradation and similarity are also performed.

- In the theoretical session of mathematics, the concept of function is developed, seen from two approaches: approximation and the matrix that represents a function.
- The concept of composition of functions is studied making reference to the composition of isometries.
- In physics the concept of function is used to approximate the CLO.
- In basic design, the concept of function and composition of functions is used to develop tessellations with a higher level of complexity than in session 3.

- In mathematics, there is a session in which derivatives are explained and optimization problems are carried out.
- The concepts are applied to basic design with problems related to space. In physics with problems related to electricity.

- In the theoretical part of mathematics, the concepts of integral are explained as a problem of approximation of areas.
- In basic design the integral is applied to calculate an approximation to the volume of an object using the serial planes (cross-sectional area).

- In mathematics, Pappus Guldin’s theorem is explained.
- In another basic design session, students analyse shapes of revolution.
- In the Physics session they calculate the volume and surface area of the shapes of revolution seen in basic design.

- Mathematics: The concepts related to Venn diagrams are explained. Logical operations. RGB diagrams are studied.
- With the logical operations they study the primary, secondary additive and subtractive colours used with Photosop in basic design.
- Representing shapes are used to make additive and subtractive mixtures.

- Mathematics: Basic trigonometry is explained. HSV/HSL models in which saturation and hue are explained as polar coordinates.
- In basic design they use different colour gradations and modify them using saturation and hue and making the changes indicated in the maths part.

- Mathematics: they study the colour wheel and establish its polar coordinates. Converting to Cartesian coordinates. The law of sines and cosines is used to establish the distance on the colour wheel.
- The distance on the colour wheel is used in basic design to find visual harmonies.

- Maths session: vector spaces are explained. RGB is used as a generator space to explain the different combinations of a given value and generate the colour range in RGB. The concept of base and the change of base are studied, in particular, the opponent colour space.
- In physics the concept of linear combination seen in mathematics is used to measure colours of objects by using an app. This session looks at how to describe them according to their RGB ratios.
- In representation of shapes, the opponent space is related to ‘discomfort’ and negative emotions.

#### 5.1.2. Assessment Methodology

#### 5.1.3. Some Milestone Project Examples

- VITALIA: This project is for the Milestone III. Well-being. The students developed a bottle of water that it is able to measure the quantity of water that you drink in a day and alerts you when you should drink because to be hydrated is important for the well-being, see Figure A1.
- Youmood: This project is for the Milestone III. Well-being. The students develop a product with a bottle of water where each box or bottle has a color and each color is related with art and emotions, see Figure 2.

## 6. Results

#### 6.1. Student Perception Survey

- “By coordinating and intertwining subjects, it is easier to relate concepts among them”
- “I think it improves learning, due to its method based mainly on experimentation.”
- “In my opinion this new method due to being more practical is more focused on the world of work and what we will do in the future.”
- “It is a new method that allows you to see the application of all the theory given in subjects such as mathematics and physics”
- “It is another way of acquiring knowledge in a more practical way in which it becomes more enjoyable and dynamic every day than doing it in an exam”
- “This new methodology in my opinion is much better, since in all subjects there are things that have to do with each other, this means that you can see how the knowledge that is given can be applied in different fields”
- “I think this new methodology makes us more involved in the learning process”
- “It is very effective, because you really learn the meaning and use of mathematics in the professional field”

#### 6.2. Comparison of Academic Results in the Last Three Years

## 7. Discussion

## 8. Conclusions and Further Developments

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**VITALIA. A bottle of water that it is able to measure the quantity of water that you drink in a day and alerts you when you should drink.

**Figure A2.**YOUMOOD. A product with a bottle of water where each box or bottle has a color related with art and emotions.

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**Figure 2.**“Sustainable City” STEAM project developed in [33].

Subject | ECTS |
---|---|

Maths | 6 |

Physics | 6 |

Art | 6 |

Basic design | 6 |

Art History | 6 |

Item | Content |
---|---|

1 | Vectors |

2 | Matrices and applications |

3 | Isometries. Compositions of isometries. Ratio |

4 | Functions |

5 | Differential calculus |

6 | Introduction to integral calculus |

7 | Basic and advanced trigonometry |

8 | Linear Algebra |

Cuatrimestral Term 1 | STEAM Classification | Cuatrimestral Term 2 | STEAM Classification |
---|---|---|---|

Physics | S,T,M | Physics Extension | S,T,M |

Maths | M | Maths Extension | M |

Art History | A | Anthropology | S |

Basic design | A,S,T | Design Extension | A,S,T |

Shape representation | A | Descriptive geometry | A,S,T |

Week | Content | Subjects |
---|---|---|

Act I.—Shape | ||

S1 | Proportion, Vectors | Physics, Art, Basic Design |

S2 | Flat isometries. Matrices connected | Basic Design |

to isometries. Inverse matrix | ||

S3 | Tessellations (DB) Homothecy (gradation) | Basic design |

S4 | Composition of isometries. Functions. | Basic Design, Physics |

S5 | Milestone I.—port | |

Act II.—Volume | ||

S6 | Differential Calculus | Physics, Basic Design |

S7 | Integral Calculus: Approximation of areas and volumes | Physics, Basic Design |

S8 | Shapes of revolution. Calculation of areas and volumes. | Physics, Basic Design |

Pappus-Gouldinus theorems. | ||

S9 | Milestone II.—Light | |

Act III.—Colour | ||

S10 | Venn Diagrams. Primary, secondary colours | Basic Design, Physics |

Additive and subtractive colours. RGB System | Shape Representation | |

S11 | Flat trigonometry. HSV/HSL Models | Physics, Basic Design |

Polar coordinates. Change from RGB to HSV/HSL | ||

S12 | Laws of sines and cosines. Polar distance. | Basic Design, Physics |

Chromatic harmonies. Colour Wheel | Shape Representation | |

S13 | Vector spaces, Generator system: RGB space. | Basic Design, Physics |

Combination of colours Bases of a vector space. | Shape Representation | |

RGB space. Change of coordinates. Opponent space. | ||

S14 | Milestone III—Well-being |

ID | Question |
---|---|

1 | In the previous year, did you study mathematics? |

2 | What was your perception of mathematics before the course? |

3 | Have you improved your perception of mathematics after the course? |

4 | It has been more motivating to solve math problems with the EXPLORIA |

methodology than with the traditional method. | |

5 | I feel more involved in this type of learning. |

6 | My perception of learning has improved. |

7 | Would you propose to keep on using the new learning model next year? |

8 | In the connection of mathematics with the rest of the subjects, what connection has |

helped you the most to understand it? | |

9 | What do you think of the application of this new teaching experience? |

Question | Yes | No |
---|---|---|

1 | 18 | 1 |

Question | Very bad | Bad | Neutral | Good | Very good |
---|---|---|---|---|---|

2 | 1 | 3 | 7 | 6 | 2 |

Question | Strongly Disagree | Disagree | Neutral | Agree | Strongly Agree |
---|---|---|---|---|---|

3 | 0 | 0 | 3 | 9 | 7 |

4 | 0 | 1 | 0 | 11 | 7 |

5 | 0 | 0 | 1 | 10 | 8 |

6 | 0 | 1 | 0 | 11 | 7 |

7 | 0 | 0 | 1 | 5 | 13 |

Question | Physics | Basic Design | Art | Shape Representation |
---|---|---|---|---|

8 | 11 | 7 | 1 | 0 |

Course 2020–2021 | Course 2019–2020 | Course 2018–2019 | |
---|---|---|---|

Students attending | 93.6% | 85.2% | 91.3% |

Pass | 94% | 70% | 74% |

Fail | 0% | 30% | 26% |

Average pass mark | 7.85 | 6.55 | 6.49 |

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**MDPI and ACS Style**

Romero, P.D.; Montes, N.; Barquero, S.; Aloy, P.; Ferrer, T.; Granell, M.; Millán, M. EXPLORIA, a New Way to Teach Maths at University Level as Part of Everything. *Mathematics* **2021**, *9*, 1082.
https://doi.org/10.3390/math9101082

**AMA Style**

Romero PD, Montes N, Barquero S, Aloy P, Ferrer T, Granell M, Millán M. EXPLORIA, a New Way to Teach Maths at University Level as Part of Everything. *Mathematics*. 2021; 9(10):1082.
https://doi.org/10.3390/math9101082

**Chicago/Turabian Style**

Romero, Pantaleón D., Nicolas Montes, Sara Barquero, Paula Aloy, Teresa Ferrer, Marusela Granell, and Manuel Millán. 2021. "EXPLORIA, a New Way to Teach Maths at University Level as Part of Everything" *Mathematics* 9, no. 10: 1082.
https://doi.org/10.3390/math9101082