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Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, as well as the strategies used by students and their connection with generalization. This paper jointly describes the strategies and representations of generalization used by a group of 33 sixth-year elementary school students, with no former algebraic training, in two generalization tasks involving a functional relationship. The strategies applied by the students differed depending on whether they were working on specific or general cases. To answer questions on near specific cases they resorted to counting or additive operational strategies. As higher values or indeterminate quantities were considered, the strategies diversified. The correspondence strategy was the most used and the common approach when students generalized. Students were able to generalize verbally as well as symbolically and varied their strategies flexibly when changing from specific to general cases, showing a clear preference for a functional approach in the latter.

This article aims to delve into the meaning of school mathematical concepts by their semantic analysis. Concretely, this paper aimed primarily to describe and characterize the meaning of derivative of a function at a point expressed by mathematics pre-service teachers. In order to achieve this goal, we use a semantic framework, in which the meaning is composed of: (a) conceptual structure, (b) representation systems, and (c) sense and usage. This qualitative, descriptive study analysed the replies of 37 pre-service teachers to questions involving the definition, requirements, and structural concept of derivatives. It also explores the sense and usage attributed by participants to this mathematical concept. At least five profiles of meanings for derivatives could be constructed from the results. The most prominent findings included the prevalence of the geometric interpretation of the concept and participants' failure to mention elements and requisites essential to the definition of derivatives.

This paper describes the experience of applying a new teaching strategy to a basic mathematics course at the Open University of Catalonia. This strategy requires students to engage with practice and assessment tests with automatic feedback. The main aim of this research is to look into the extent to which the key elements of this teaching strategy influence the students' course completion and academic achievement. In order to achieve this purpose, qualitative and quantitative variables over one term have been measured weekly. Our findings suggest that this teaching strategy significantly reduced the number of students who dropped out and improved academic results. More specifically, regular practice activities at the beginning of the semester help to engage students in the completion of the course. (Contains 4 figures and 2 notes.)

First-year university students exhibit considerably different levels of proficiency, often lacking the necessary prerequisites to attend certain courses. This is especially true for mathematics courses. While motivation alone plays a role, by far the most significant challenge consists in harmonising the different (and often incompatible) backgrounds of the students. In response, universities have made various attempts to tackle this issue. Many institutions now offer intensive courses prior to academic entry or even complete introductory courses occupying the whole first term, whether in a classroom or (partially) online. Massive Open Online Courses (MOOCs), while relatively unknown until a few months ago, have brought about a real educational revolution. The MOOC is an educational model that brings online learning content to any person willing to participate. At Pompeu Fabra University (UPF) we were intrigued by this new educational paradigm and wondered whether it could be useful in filling the gap between high school and university. Therefore, we recently designed and implemented such a course: while remaining available to anyone, the course specifically targets our prospective students who in turn should tackle first-year linear algebra having acquired an adequate background. In this paper we describe the first MOOC launched by UPF, "Decoding Algebra". We provide an overview of its design, the technological means we used, our students' profile and the lessons from this experience. More importantly, by focusing on our incoming students, we examine whether MOOCs can be a good approach to level out the differences among first-year university students.

This paper analyses the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is defined as an indicator of the generic competence: Use of Technology. Additionally, we show that using CAS could help to enhance the following generic competences: Self Learning, Planning and Organization, Communication and Writing, Mathematical and Technical Writing, Information Management and Critical Thinking. A follow-up at several stages of the resolution of a problem-example shall be performed; analysing each contribution of Maxima to the competences' acquisition. The problem-example concerns an electronic device for a building surveillance system, with different possibilities according to the number of sides of the building and other parameters. (Contains 3 figures.)

Technology and, in particular, computer algebra systems, allows us to change both the way we teach mathematics and the mathematical curriculum. Curiously enough, unlike what happens with linear system solving, algebraic system solving is not widely known. The aim of this paper is to show that, although the theory lying behind the "exact solve" commands (Grobner bases) is far more complex than the Gauss method, understanding and interpreting the output of the corresponding commands is not much more difficult than in the linear case, and can be introduced to non-specialists even at high-school level, if the computer is used as a black-box. Let us underline the case of statistical packages, which allow non-specialists to perform complex statistical studies. Similarly, computer algebra systems open a whole new world of exciting possibilities, covering a variety of fields. (Contains 4 figures.)

Introduction: In the present-day didactic process of transition from natural numbers to integers, a certain kind of measure is involved which is neither easy nor appropriate to integrate into the familiar numerical systems. We refer to measures and comparisons for which we need a third numerical system--what we will call "relative natural" numbers; this numerical system presents five structural differences from the systems of natural numbers and of integers. In this paper we present part of a study whose purpose was to make evident that such differences are also cognitive. Method: Qualitative meta-analysis and didactic analysis are used for theoretical study and for analysis of responses, for the descriptive study and correspondence analysis of responses to structured questionnaires. Results: The algebraic structures and the epistemology and phenomenology of relative natural numbers and of integers are different. Subjects distinguish between the two types of numbers by responding differently to situations which pertain to one or the other type of numerical structure. Discussion: Results confirm our conjectures, and call for a new organization of the additive numerical field, in which additive relativity and relative natural numbers would be considered, as is the case with proportionality and fractions in the conceptual field of multiplication. (Contains 3 figures, 4 tables, and 6 footnotes.)

In this paper, we describe an educational experience in the context of the Master's degree that is compulsory in Spain to become a secondary education mathematics teacher. Master's students from two universities in Madrid (Spain) attended lectures that addressed—emphasizing the concourse of a dynamic geometry software package—some historical, didactic and mathematical issues related to linkage mechanisms, such as those arising in the 18th and 19th centuries during the development of the steam engine. Afterwards, participants were asked to provide three different kinds of feedback: (i) working on an assigned group task, (ii) individually answering a questionnaire, and (iii) proposing some classroom activity, imagining it would be addressed to their prospective pupils. All three issues focused on the specific topic of the attended lectures. In the framework of Mason's reflective discourse analysis, the information supplied by the participants has been analyzed. The objective was to explore what they have learned from the experience and what their perception is of the potential interest in linkages as a methodological instrument for their future professional activity as teachers. This analysis is then the basis upon which to reflect on the opportunities (and problems) that this particular bar-joint linkages methodological approach could bring towards providing future mathematics teachers with attractive tools that would contribute to enhancing a STEAM-oriented education. Finally, the students' answers allow us to conclude that the experience was beneficial for these pre-service teachers, both in improving their knowledge on linkages history, mathematics, industrial, technological and artistic applications, and in enhancing the use in the classroom of this very suitable STEAM context. [ABSTRACT FROM AUTHOR]

The progress of Mathematics during the nineteenth century was characterised both by an enormous acquisition of new knowledge and by the attempts to introduce rigour in reasoning patterns and mathematical writing. Cauchy’s presentation of Mathematical Analysis was not immediately accepted, and many writers, though aware of that new style, did not use it in their own mathematical production. This paper is devoted to an episode of this sort that took place in Spain during the first half of the century: It deals with the presentation of a method for numerically solving algebraic equations by José Mariano Vallejo, a late Spanish follower of the Enlightenment ideas, politician, writer, and mathematician who published it in the fourth (1840) edition of his book Compendio de Matemáticas Puras y Mistas, claiming to have discovered it on his own. Vallejo’s main achievement was to write down the whole procedure in a very careful way taking into account the different types of roots, although he paid little attention to questions such as convergence checks and the fulfilment of the hypotheses of Rolle’s Theorem. For sure this lack of mathematical care prevented Vallejo to occupy a place among the forerunners of Computational Algebra. [ABSTRACT FROM AUTHOR]

Abstract: This paper focuses on a number of sources that could have inspired a very interesting kind of diagrams of coefficients of algebraic expressions appearing in a Catalan manuscript of ca. 1520. These diagrams are used in some problems to represent and operate on the algebraic expressions involved in the solving process. In this research, it has been necessary to delve into different medieval traditions. [Copyright &y& Elsevier]

Abstract: Predicting both PV module and generator performances under natural sunlight is a key issue for designers and installers. Five simple algebraic methods addressed to predict this behaviour in Mediterranean climates have been empirically validated. Firstly, the calibration in STC of all significant electrical parameters of both a monocrystalline and a polycrystalline silicon PV modules was entrusted to an accredited independent laboratory. Then, a 12-month test and measurement campaign carried out on these modules in the city of Jaén (Spain, latitude 38°N, longitude 3°W) has provided the necessary experimental data. Results show that (a) crystalline silicon PV module outdoors performance may be described with sufficient accuracy – for PV engineering purposes – only taking into account incident global irradiance, cell temperature, and using any one of two simple algebraic methods tried in this paper and (b) regardless the used method, poor results may be achieved if the PV specimens under study are not electrically characterised in STC prior to analysing their outdoors performance. Even so, the methods recommended in (a) perform best. [Copyright &y& Elsevier]

The blind separation problem of linear time-dependent mixtures is addressed in this paper. We have developed a new algorithm based on the minimization of the mutual information as well as a penalized term which ensures an a priori normalization of the estimated sources (outputs) and so, avoids the scale indeterminacy. The criterion minimization is done using a well-known gradient approach. Finally, some numerical results are shown to illustrate the performance of the penalized algorithm compared to the Babaie-Zadeh approach presented in (Proceedings of IWANN, Granada, Spain, June 2001, pp. 834–842). [Copyright &y& Elsevier]

Background: Verification uses logical algorithms to detect potential errors before laboratory results are released to the clinician. Even though verification is one of the main processes in all laboratories, there is a lack of standardization mainly in the algorithms used and the criteria and verification limits applied. A survey in clinical laboratories in Spain was conducted in order to assess the verification process, particularly the use of autoverification. Methods: Questionnaires were sent to the laboratories involved in the External Quality Assurance Program organized by the Spanish Society of Clinical Biochemistry and Molecular Pathology. Seven common biochemical parameters were included (glucose, cholesterol, triglycerides, creatinine, potassium, calcium, and alanine aminotransferase). Results: Completed questionnaires were received from 85 laboratories. Nearly all the laboratories reported using the following seven verification criteria: internal quality control, instrument warnings, sample deterioration, reference limits, clinical data, concordance between parameters, and verification of results. The use of all verification criteria varied according to the type of verification (automatic, technical, or medical). Verification limits for these parameters are similar to biological reference ranges. Delta Check was used in 24% of laboratories. Most laboratories (64%) reported using autoverification systems. Autoverification use was related to laboratory size, ownership, and type of laboratory information system, but amount of use (percentage of test autoverified) was not related to laboratory size. Conclusions: A total of 36% of Spanish laboratories do not use autoverification, despite the general implementation of laboratory information systems, most of them, with autoverification ability. Criteria and rules for seven routine biochemical tests were obtained. [ABSTRACT FROM AUTHOR]

One of the main objectives in mathematics education is to motivate students due to the fact that their interest in this area is often very low. The use of different technologies, as well as gamification in the classroom, can help us to meet this goal. In this case, it is presented the use of two techniques, which are a digital escape room, using Genial.ly and a breakout, for learning algebra in the third course of secondary education. To carry out the experience, a comparison of the course 2018/2019 and the course 2019/2020 is made. Students of both courses completed an exam of algebraic fractions and then an equations exam; the exams were done in the course 2019/2020 taken after having used the techniques. The results show that there are low significant differences between both courses in the algebraic fractions exam qualifications, but those of the equations exam were significant, with a difference of almost two points, which led us to conclude that the use of this techniques improved the qualifications of students. Finally, the perception that students had of the experience was very positive, as shown by the data obtained and allowed them to improve their knowledge as well as their motivation and teamwork. [ABSTRACT FROM AUTHOR]