Mathematics for Secondary Education Teachers

To know and to reflect on the most relevant content of the Mathematics, the subject of education at various levels of the secondary education.

Block 1 (1,5 credits)
1.1 The main concepts and results of the Analysis in secondary education: Historical review, interpretations and various resources.
• Real numbers.
• Sequences of real numbers.
• Real functions of real variable.
• Differential and Integral Calculus.

Block 2 (2 credits)
2.1 Applications of geometry
• Activity proposals. Plane projections. Maps. Terrestrial planispheres: main representation systems. (Subject 43. BOE Nº226 of September 21, 1993, page 27414).
• Analysis, discussion and resolution of geometric problems of high school oppositions.
2.2 Discussion activity on the teaching of geometry in secondary school.
• Critical analysis of exercises and illustrations from textbooks.
• Discussion on the teaching of geometry in secondary education: synthetic geometry in ESO versus analytical geometry in high school. Dynamic geometry, the panacea?. Objectives, contents and mathematical senses of measurement and space of the ESO Mathematics curriculum (https://www.xunta.gal/dog/Publicados/2022/20220926/AnuncioG0655-190922-…).


Block 3 (1,5 credits)
3.1 The problems in mathematics. Types of problems. Strategies and tactics for solving problems.
• Problems versus exercises. Types of problems: recreational, contest and open-ended problems.
• Strategies in the solution of problems: draw figures and diagrams, examine special cases, formulate equivalent problems, modify the problem, change the method of reasoning (contradiction).
• Tactics for solving problems: symmetry (geometrical symmetry, algebraic symmetry), the extreme principle, transformations and invariants.
• Arithmetic and combinatorial problems: modular arithmetic, divisibility, counting techniques, pigeonhole principle and graph theory.

BASIC BIBLIOGRAPHY
- Durán Guardeño, A. J. (1996). Historia, con personajes, de los conceptos del Cálculo. Madrid, España: Alianza Editorial, Colección Alianza Universidad.
- Pólya, G. (1989). Cómo plantear y resolver problemas (título original: How To Solve. It?). México: Trillas Editorial (decimoquinta reimpresión).
- Zeitz, P. (2007). The Art and Craft of Problem Solving. Hoboken, NJ: John Wiley & Sons, Inc.

COMPLEMENTARY BIBLIOGRAPHY
BOOKS
- Aleksandrov, A.D., Kolmogorov, A., Laurentiev, M.A. y otros (Edición 2014). La matemática: su contenido, métodos y significado. Madrid, España: Alianza Editorial, Colección Alianza Universidad.
- Edwards Jr., C. H. (1979). The Historical Development of the Calculus. Springer-Verlag.
- Tao, T (2006). Solving Mathematical Problems: A Personal Perspective. New York, EEUU: Oxford University Press.
- Zeitz, P. (2007). The Art and Craft of Problem Solving. Hoboken, NJ: John Wiley & Sons, Inc.

JOURNALS
- BOLETÍN DAS CIENCIAS. Asociación de Ensinantes de Ciencias de Galicia (ENCIGA).
- NÚMEROS: http://www.sinewton.org/numeros/
- SUMA: http://revistasuma.es/

RESOURCES
- Proyecto Descartes: http://recursostic.educacion.es/descartes/web/
- Archivo “MacTutor” (The MacTutor History of Mathematics archive):
http://www‐history.mcs.st‐andrews.ac.uk/index.html
- GeoGebra: http://www.geogebra.org/
- Lléname: http://geometriadinamica.es/paginas/llename/aplicacion/llename.html
- Wolfram Research: http://www.wolfram.com/ (contiene enlace a Wolfram|Alpha)
- Demonstrations Project: http://demonstrations.wolfram.com/
- Subproyecto ED@D da "Red Educativa Digital Descartes": http://proyectodescartes.org/EDAD/index.htm
- Arránz, J.M., Losada, R., Mora, J.A. y Sada, M.: http://www.geometriadinamica.es/
- Inter2geo. Interoperate Interactive Geometry for Europe. http://inter2geo.eu/
- Máxima: http://maxima.sourceforge.net/ e http://andrejv.github.com/wxmaxima/index.html
- ESO Mathematics Curriculum
(https://www.xunta.gal/dog/Publicados/2022/20220926/AnuncioG0655-190922-…)

GENERAL INFORMATION
- CiUG (Comisión interuniversitaria de Galicia): http://ciug.cesga.es/index.html
- Currículo educativo Galicia: http://www.edu.xunta.gal/portal/guiadalomce
- Materiales curriculares: http://centros.edu.xunta.es/iescarril/matematicas/materiais_matematicas…
- Unidades didácticas ESO: http://proyectodescartes.org/EDAD/

VIDEOS, WEBSITES, COMPLEMENTS AND ASSOCIATIONS
- Sáenz de Cabezón, E., Canal DERIVANDO:
https://www.youtube.com/channel/UCH-Z8ya93m7_RD02WsCSZYA
- Grime, J., Canal SINGINGBANANA
http://www.youtube.com/user/singingbanana?feature=results_main
- Grima, C., http://claragrima.com
- Macho, M., http://www.ehu.eus/~mtwmastm/
- Apuntes Marea Verde: http://www.apuntesmareaverde.org.es/
- Libros de divulgación de Adrián Paenza: http://cms.dm.uba.ar/material/paenza
- Asociación Galega de Profesorado de Educación Matemática: https://www.agapema.org
- Ensinantes de Ciencias de Galicia: http://www.enciga.org

General
(CG1) To know the curriculum contents of the relative subjects to the corresponding educational specialisation.
(CG4) To seek, to obtain, to process and to communicate information (oral, printed, audiovisual, digital or multimedia), to transform it in knowledge and to apply it to the processes of teaching and learning in the matters specific to the specialisation studied.
(CG16) To work as a team with other professionals of the education, enriching its training.
(CG17) To develop habits and attitudes to learn to learn throughout its later professional development.
(CG18) To apply the acquired knowledges and the capacity to problem-solving to educational environments new or unfamiliar.

Specific
(CE-E1) To know the training and cultural value of the matters corresponding to the specialisation.
(CE-E2) To know the contents that take in the respective teachings.
(CE-E3) To know the recent history and developments of the matters and its perspectives in order to transmit a dynamic vision of the same.
(CE-E4) To know contexts and situations in which use or apply the diverse curriculum contents
(CE-E8) To promote a climate that provides the learning and puts in value the contributions of the students.
(CE-E9) To integrate the training in audiovisual and multimedia communication in the process of teaching-learning.

Transversal
(CT-1) To use bibliography and search tools of general and specific library resources, including the access for Internet.
(CT-3) To promote the skills for the work in cooperative and multi-disciplinary environments.
Teaching methodology.

The work in the classroom in the expositive lessons consists, basically, in teaching provided by the teacher. Ordinarily, in the same session will be devoted a time to the exposure or illustration of some theoretical question, and another time to the resolution of problems or exercises. Sometimes, the model will be approached to the one of the masterly lesson, in other cases, it will seek the involvement of all students in the discussion of the matters raised.
In the teaching in the classroom more interactive we seek active participation of students. It will have different formats, sometimes matters prepared by students will be dealt, not explained previously; other times they will be dedicated to the monographic discussion of matters of difficult understanding. General, they will have a more practical orientation that the expository sessions.
Eventually, we will ask the delivery in writing of some activities, or of the summary or comment of some proposed reading.
Tutorships in very reduced groups will be dedicated, individually or in groups, to solve doubts and particular difficulties that they go emerging, and to the customised follow-up of each student.
All the tasks of the student (studies, works, use of platforms, projects, readings, exhibitions, exercises, practical,...) will be guided by the teacher in the classroom and in the tutoring sessions.

STUDENTS WITH DISPENSE OF ATTENDANCE
TEACHING METHODOLOGY
Partially virtual tutoring, according to student availability. For this, the means provided for synchronous and/or asynchronous telematic teaching and the virtual classroom of the course will be used.

There will be a double method of evaluation: the punctual evaluation and the continuous evaluation,

1. Attitude and work in the classroom. 10%
Attendance, participation and attitude in the lectures and seminar classes.

2. Participation in classroom activities (debates, analysis of teaching materials, search for resources...). 10%
Design and oral presentation of an activity to deal with analysis contents in a secondary school mathematics subject. 20%

3. Individual participation in the classroom in the debate on the teaching of geometry in secondary school and written delivery of the questions previously proposed. 10%
Preparation and oral defense of an individual project of complementary activity for an ESO mathematics subject. 20%
Consult the instructions in the work folder in the virtual course.
Consult the instructions in the work portfolio in the virtual classroom.

4. Individual written and oral presentation of the resolution of two problems raised regarding problem solving techniques from block 3. 30%
Instructions are posted in the virtual classroom.
(2, 3 and 4 EQUIVALENT TO THE EXAMINATION OF THE SUBJECT)

In the case of fraudulent realization of exercises or tests, the one contained in the Regulations for the evaluation of the academic performance of students and review of grades will apply:
Article 16. Fraudulent performance of exercises or tests: The fraudulent performance of any exercise or test required in the evaluation of a subject will imply a failure grade in the corresponding call, regardless of the disciplinary process that may be followed against the offending student. Considering fraudulent, among others, the realization of plagiarized works or obtained from sources accessible to the public without reworking or reinterpretation and without citations to the authors and sources.

STUDENTS WITH DISPENSE OF ATTENDANCE
EVALUATION SYSTEM

1. Attitude and personal work. 10%
The students will deliver in the virtual classroom the tasks proposed in the classroom.

2. Delivery through the virtual classroom of small questions on teaching materials, didactic resources...etc. (Specific instructions will be published in the virtual classroom). 10%.
Design and oral presentation of an activity to deal with analysis contents in a secondary school mathematics subject. 20%

3. Individual delivery in the virtual classroom of the answers to the questions posed in the debate on secondary education: synthetic geometry in ESO vs. analytical geometry in high school. 10%
Delivery in the virtual classroom and oral defense of an individual project of complementary activity for a mathematics subject of the ESO. 20%
Consult the instructions in the work portfolio in the virtual classroom.

4. Individual written and oral presentation of the resolution of two problems raised regarding problem solving techniques from block 3. 30%
Instructions are posted in the virtual classroom.

(2, 3 and 4 EQUIVALENT TO THE EXAMINATION OF THE SUBJECT)

Class meeting:

Expositive lessons in large groups: 15
Interactive lessons in small groups reduced in normal classroom (seminars, debates, exhibitions, ...): 20
Interactive lessons in small groups in specialised classroom (computer science, laboratory, practices of field, ...): --
Tutorships in small groups: --
Tutorships in very small group: 5
Other sessions with the teacher. To specify: --
Work hours total in person: 40


Independent work:


Individual self-study or in group: 50
Writing exercises, conclusions or other derivative works of the subject: 15
Programming / experimentation or other work by technical means (computer, laboratory, data collection, ...): --
Preparation of debates, oral presentations or similar: 5
Library activities, recommended readings, seek bibliographic: 15
Other tasks proposed by the teacher. Specify: --
Total hours working personal: 85

Class attendance and participation in the proposed activities.

There will be an e-learning platform to support the teaching of this subject at USC.

The works realized pole students owe to delivered, preferably, through the Virtual Campus,.

Environmental responsibility:
- Avoid plastic lids or other unnecessary external packaging.
- Whenever possible, use staples instead of binding.
- Print both sides in “ink saving” quality.
- Do not use blank pages as separators for chapters or parts.
- Avoid annexes that do not have direct reference to the topics developed.

Gender perspective:
- It is recommended to make use of non-sexist language, both in the daily work of the classroom and in the academic work entrusted to us.

- Mandatory use of the rai email account.
- Mandatory use of institutional technological tools: Virtual Campus, Microsoft Office 365, and other tools provided by the faculty and authorized as institutional tools by the university.
- It will not be possible to use the mobile phone, except when it is used as a work tool following the instructions given by the teacher, the students being responsible for the legal and academic consequences that may arise from a non-accommodated use of it.
- Take into account that teaching-learning (classes / tutorials) is a private process, it is understood as a private process of communication and exchange between the teacher and the student body enrolled in the subject.
- Compulsory compliance with data protection regulations https://www.usc.gal/es/politica-privacy-protección-datos