Abstract:

The first year of a university degree, which usually represents a considerable change for students who enter university from other educational levels, is crucial because this is precisely when students must acquire and consolidate the competencies and skills necessary to progress satisfactorily in their university studies. However, on some degree courses such as mathematics, the results obtained in the first year are much worse than those obtained in the final years of secondary education or similar, and some specific first year subjects have a high failure rate.

Students in their first year of an undergraduate degree in mathematics (or physics and similar) are, to a large extent, confronted with a sudden change in the way in which concepts and notions are presented and the manner in which they are expected to learn and assimilate these. In earlier educational stages, mathematical knowledge is often learnt through repetition of very similar problems, where stress is placed on obtaining an immediate result, often without investigating the reasons underlying the procedure. In addition, a relatively high percentage of first year students only conduct a superficial revision of the information contained in their notes, texts and other sources of information, or even of the evaluation tests set throughout the year. Furthermore, many students cram the subject syllabus, i.e. they usually study by heart, and thus most of the main concepts are forgotten over time. Therefore, many students clearly require a complement to teaching that helps to mitigate or solve the deficiencies, questions and shortcomings that come to light during their first year of university.

In this context, this study is focused on the detection of the main conceptual and operational errors made by students when solving standard problems or exercises in theoretical and practical tests given during an initial mathematical analysis subject forming part of a mathematics degree.

In fact, the main goals pursued with this initiative are the following:
- to train all first year students in competences where their understanding is, in general, insufficient to achieve their goals, trying to enhance their grasp of the main concepts and techniques of the subject in question, especially in the initial part of the course;
- to help students identify the main areas of difficulty and misunderstanding in the subject matter. This should enable students to revise and analyse their notes or bibliographical resources from a different perspective to that used during their first reading;
- to revise and broaden students’ knowledge about the subject matter in question;
- to encourage self-assessment of the theoretical and practical tests sat throughout the year;
- to increase students’ capacity to assess their own knowledge, including that acquired in previous stages;
- to detect the specific difficulties students encounter with the problems set (from the point of view of the teaching staff);
- to detect those concepts that students have difficulty understanding (from the point of view of the teaching staff) in order to spend more time on them in subsequent theoretical explanations.

Keywords:

Teaching in mathematics, Teaching in mathematical analysis, Analysis of a real variable, Higher education, Learning and teaching Methodologies, Common errors.