This chapter contains some programs and specific software that could help lecturers to solve some of the proposed problems. Algorithms and mathematical contents are included in corresponding chapters in the book.
A multiple integral is often defined as an operator in which the integrand involves a function of more than one variable and which requires for evaluation repetition of the integration process. A so-called simple integral finds the area under a curve in one dimension. A double integral can find the area under a two-dimensional curve (a surface). Similarly a triple integral can find the sum of some value in a solid (if you are given a density function, it can find the total mass of a solid). The basic applications of double integral is finding volumes. The basic application of triple integral is finding mass of a solid. Generally, a solid has some mass but it depends on its density as the density is not constant but varying. In this chapter applications of multiple integrals to mechanical engineering will be presented and discussed.
In 2017, we proposed a European project: Rules_Math, in order to find new rules to assess mathematical competencies. This proposal was a consequence of what we were facing separately in our daily classes. We teach mathematics in several engineering schools, and we want to change the way of teaching and learning for engineering students. Some university teachers from different departments usually teach mathematics as we have learned mathematics. They only give master classes but students usually want to write numbers and formulas to learn and practice mathematical reasoning and to distinguish mathematical symbols. To communicate in, with, and about Mathematics, they need mathematical thinking, and to use aids and tools for mathematical activity. We have included in this papers our proposal to make this possible, and to assess mathematical competencies.
This article presents a Breakout played with students of the Master in Teaching of Compulsory Secondary Education and Higher Secondary School, at the University of Salamanca, as part of their training as future teachers. Both the difficulties and the advantages that this type of educational activities bring out are analyzed. Moreover, some possibilities of its possible utilization in the classroom, at different educational levels, are also discussed and presented.
En los últimos años la enseñanza y el aprendizaje de las matemáticas en ingeniería, así como la evaluación de las competencias de matemáticas que un ingeniero necesitará en su labor profesional, ha cobrado gran importancia. Las destrezas y herramientas matemáticas que utilizará un ingeniero son diferentes a las de un matemático, puesto que la forma en la que cada uno desarrollará su carrera será también diferente, por eso debe ser distinta la forma de aprenderlas. En este estudio detallamos las actuaciones que estamos llevando a cabo para llegar a un consenso en la forma de evaluar las competencias de matemáticas en educación superior para ingeniería.
Have you ever thought what would happen if a ransomware infected your laptop? This type of virus kidnaps files and encrypts them, and the only way to recover the data is by paying in bitcoin or some other cryptocurrency. This situation is undoubtedly terrible. All of your work, projects, and personal files will not be available (unless you pay the ransom). The first time students watched a video that contextualizes this stressful situation, they thought that they had been attacked by a computer virus. Fortunately, the panic only lasted a few seconds. This is the way to start a game called breakout or escape room. The difference between these two words is that, in the first case, the goal is to open a padlocked box, while the objective of an escape room is to find the key that allows to get out of a room. Both games are similar, containing riddles, puzzles, problems and some clues that would help the players to find the solution. This study analyses the use of a breakout game for educational purposes, more specifically in a university context. The experiment conducted mixes game–based learning methodologies with engineering students learning Linear Algebra, Calculus or Cryptography, which has allowed us to obtain promising results about the usage of this methodology.
The concepts taught during a Statistical Methods course make use of different mathematical skills and competencies. The idea of presenting a real problem to students and expect them to solve it from beginning to end is, for them, a harder task then just obtain the value of a probability given a known distribution. Much has been said about teaching mathematics related to daily life problems. In fact, we all seem to agree that this is the way for students to get acquainted of the importance of the contents that are taught and how they may be applied in the real world. The definition of mathematical competence as was given by Niss (Niss 2003) means the ability to understand, judge, do, and use mathematics in a variety of intra– and extra – mathematical contexts and situations in which mathematics plays or could 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. In the OEDC PISA document (OECD, 2009), it can be found other possibility of understanding competency which is: reproduction, i.e, the ability to reproduce activities that were trained before; connections, i.e, to combine known knowledge from different contexts and apply them to different situations; and reflection, i.e, to be able to look at a problem in all sorts of fields and relate it to known theories that will help to solve it. The competencies that were identified in the KOM project (Niss 2003; Niss and Højgaard 2011) together with the three “clusters” described in the OECD document referred above were considered and adopted with slightly modifications by the SEFI MWG (European Society for Engineering Education), in the Report of the Mathematics Working Group (Alpers 2013). At Statistical Methods courses often students say that assessment questions or exercises performed during classes have a major difficulty that is to understand what is asked, i.e, the ability to read and understand the problem and to translate it into mathematical language and to model it. The study presented in this paper reflects an experience performed with second year students of Mechanical Engineering graduation of Coimbra Institute of Engineering, where the authors assessed Statistical Methods contents taught during the first semester of 2017/2018 and 2018/2019 academic years. The questions in the assessment tests were separated into two types: ones that referred only to problem comprehension and its translation into what needed to be modelled and calculated and others where students needed only to apply mathematical techniques or deductions in order to obtain the required results. The research questions that authors want to answer are:
There are eight Mathematical competencies proposed by B. Alpers et al. in the Framework for Mathematical Curricula in Engineering Education, previously proposed in the Danish KOM. However, some of the competencies identified are difficult to evaluate in higher education since there is a large number of students in the classrooms and therefore the personal contact between the teacher and the students is limited. When we want to evaluate the competency “communicating in, with, and about mathematics” or “make use of aids and tools for mathematical activity”, we need other moments of evaluation besides written exams. With this intention we developed a practical project during the semester for all the students who wanted a continuous evaluation. The project consisted in the development of a video lesson about one of the contents studied in the curricular unit. The students had to create an initial script and then produce the video where we were able to evaluate the referred mathematical competencies. At the end of the semester, we identified if the content developed in the video lesson by the student was acquired and is a part of their knowledge. For this, in the written evaluations made by each student the exercises that contained the video lesson content were analysed and verified if they were correctly solved. Students also presented their videos to their peers and to the teachers and their presentation was also evaluated.
With the general objective of highlighting the changes produced in the teaching of mathematics in European engineering studies, this paper presents examples of the University of Salamanca (Spain) and Gazi University (Turkey). The use of technological tools for the teaching and learning of Calculus or Algebra has allowed a change in the objectives and form of content-based teaching such as integral calculus, differential equations or linear algebra, where we have move from training in calculation strategies to reasoning processes for the application of those contents to engineering situations. However, these changes are not always visible in the assessment instruments that are used, therefore, it is considered necessary to go deeper into diverse assessment ways and instruments, that are valid in the different European countries, as proposed in the RULES_MATH project.
We have been working, for some years, in the application of mathematics to different engineering problems with the main goal of improving our students’ motivation and make possible for them to connect and apply what they learn in Calculus classes to other engineering subjects. We describe a case study with students of Electrotechnical Engineering from the Coimbra Institute of Engineering (ISEC) and from Biologist and Geological Engineering degrees students from the University of Salamanca (USAL) in a Calculus course. The study involves the application of mathematical software in the calculus of a real situation: the estimation of the surface and length of an irregular figure: the Iberian Peninsula. Furthermore, we include the results of a pre‐ and post‐test control group study, before and after using that application in the classroom. The results indicate that those experiences were very enriching for the students in different degrees and in both countries. We tested at different centers from two different countries with similar results.