NUMERACY IN SPAIN A GENERAL OVERVIEW

NUMERACY IN SPAIN A GENERAL OVERVIEW CREA Centre of Research in Theories and Practices that Overcome Inequalities Universitat de Barcelona - SPAIN crea@ub.edu Contents Introduction General framework of the adult educational system in Spain Level 1: State Level 2: Regions (Comunidades Autónomas) - Specific case: Catalonia Numeracy in Spain Numeracy and the Adult Education System in Catalonia 1. Introduction The adult educational system in Spain is decentralized. It means that central Government establishes the general criteria by law, but each region develops their own adult educational system. So, actually there are several systems at the same time. In this report we are going to give a general overview of this system. Firstly some issues related to the general educational system in Spain will be presented. Secondly the main general programs will be highlighted. Thirdly, the Catalonian educational system will be focused on and an explanation will be provided on how the adult educational system is organized in Catalonia. Finally numeracy will be discussed, and the main elements in the adult education curriculum will be expounded, with regards to Mathematics. 2. General framework of the adult educational system in Spain Level 1: State Adult education in Spain offers different kinds of activities. There are both regular and non-regular courses. Also there are courses oriented towards training adults in order to pass specific exams, in order to obtain professional or academic qualifications. Adult education is geared towards people over 16 years old. The Organic Quality of Education Law (Ley Orgánica de Calidad de la Educación) regulates adult education in Spain. This law is based on a lifelong learning concept. Article III is entirely dedicated to adult education. The main objective is to provide adults with basic training in order to open up new possibilities such as promoting adults access to other educational levels, such as access to University courses, 1

achieving qualifications and developing their capacity to get involved in social, cultural and political events. The specific objectives of adult education in Spain are the following: - To update the educational background and to facilitate access to different educational levels to each person who may require it. - To develop programs and courses related to specific educational needs from excluded groups. - To improve professional qualifications and/or to acquire training in order to develop other professions. - To develop people s capacity to participate in social, cultural and political events. These are the main objectives for adult education in Spain. People can find both standard and on-line courses. So, there is standard education and distance learning also. Article 52. Objective. Permanent education offers lifelong learning to all members of the public in order to acquire, update, complete and extend their abilities and skills for their professional and personal development. To do this, the administration of education will collaborate with the civil service departments with competencies in terms of adult education, and especially in employment management. (...) Public administrations prioritises all these people who have not completed their basic education, for various reasons. Also people over the age of 16 would be able to participate in adult education if they cannot attend a standard educational centre because of their job or other circumstances. (...) The administration of education will promote specific programs for the Spanish language, culture and other co-official languages in order to integrate the immigrant population. (Source: LOGSE General Arrangement of the Educational System Law). The provision of Adult Education in Spain includes the following options: - Basic Adult Education. - Professional training. - Mentored lessons. - Courses to prepare for exams in order to access other levels of education. - Spanish classes for the immigrant population. - Non-formal educational courses. 2

Table 1. The Adult Education system in Spain: a brief overview. ACCESS V.I.A. People over 18 years old. LEVEL 1 LEVEL 2 LITERACY CONSOLIDATION OF KNOWLEDGE AND ACQUISITION OF INSTRUMENTAL TECHNIQUES LITERACY AND NEO-READERS GLOBAL MODULE I GLOBAL MODULE II GLOBAL MODULE III Topics LANGUAGE AND LITERATURE MATHEMATICS SOCIO-NATURAL SCIENCES There is no temporary Deadline: 1 scholar year (It is possible to do it over 2 years with limit to do it justification) Regular attendance Regular attendance or distance courses Duration: 12 weeks Duration: 16 weeks Tutoring: once a week Teachers Students file (contents): V.I.A., assessment report, grades Level 2: Regions (Comunidades autónomas) specific case: Catalonia In Catalonia adult education is defined as lifelong learning. Adult learning refers to all kinds of training activities which allow people to develop their abilities and skills. These courses also allow adults to obtain the Secondary Education Graduate certificate (Graduado en Educación Secundaria GES ). Academic centres (schools for adults but also local centres, etc.) have open and flexible courses on offer. Therefore it is possible to create tailored courses in accordance with individual needs. It is also possible to obtain certificates through distance learning, with a program called Graduï s. Ara en secundària! (You have graduated! Now for the next stage!). Everyone over the age of 18 is able to participate in the adult education system, except people who have already completed standard education courses. It is also possible to enrol people over the age of 16, in order for them to prepare for standard exams for entry into vocational courses. An interdepartmental program in Catalonia manages Adult Education. This means that different departments from the Catalonian government are involved in Adult Education. There are over 5,000 actions taking place, divided into the following fields: - Initial education (ensenyaments inicials) - Basic Adult Education - Vocational training - Education for leisure and culture - Training courses 3

- Courses for people with special needs - Courses for in-service adult teachers In Catalonia there are 5 main domains in adult education: introductory courses, an instrumental training level, secondary education, and teaching geared towards preparing for exams in order to access other educational levels ( such as university, for example) and courses on information society competences. The introductory course includes: the Catalan language, the Spanish language, IT and foreign languages. The Instrumental training level includes reading skills, writing skills, mathematics and general culture. The Secondary education level also includes all these topics, but they are dealt with more in-depth. The third option, preparing for exams in order to access other educational levels, includes: a) access to secondary level vocational training levels (proves d accès a cicles formatius de grau mitjà); b) access to higher vocational training levels (proves d accès a cicles formatius de grau superior); and c) access to Universities. Lastly, courses to learn about information society competences mostly include IT skills and foreign languages. Table 2: Adult Education in Catalonia Introductory courses Instrumental training level Secondary Education Training to prepare exams in order to access other educational levels Courses on Information society competences 3. Numeracy in Spain in relation to the education system Numeracy is an important part of the Spanish curricula. Mathematics is one of the few topics which is present at almost every educational level. The main mathematical objectives within the Spanish curricula are the following (Alsina, 2001): - To incorporate the language and arguments typical of mathematical reasoning (numerical, graphical, geometrical, logical, algebraic, probabilistic) in order to be able to communicate in a precise and rigorous way. - To use logical ways of thinking, to formulate and verify conjectures, to make inferences and deductions, and to 4

organize and link information related to daily life and problem solving. - To quantify real situations by virtue of data, measures and numbers while making the appropriate computations. - To elaborate personal strategies for analyzing concrete situations and for problem solving using various resources and instruments, analyzing the final results obtained. - To use elementary techniques for collecting data to obtain information about phenomena and situations and to represent this information by means of numerical or graphical forms in order to evaluate it. - To recognize the diversity of different situations and to see how different points of view can be used to explain it (deterministic / random, finite / infinite, exact / approximate). - To identify shapes and spatial relations present in reality, analyzing geometrical properties and relationships. - To identify the mathematical elements (statistical data, graphics, maps, computations, etc.), which are present in news, pools, commercials, etc., critically analyzing their functions and possibilities for a better understanding of messages. - To approach day-to-day situations and problem solving in a mathematical way (exploring alternatives, using a precise language, etc.). - To know and value personal mathematical skills in order to deal with situations that require their use or in which one can enjoy the creative, manipulative, aesthetic or useful aspects of mathematics (Alsina, 2001: 3-4). As Alsina (2001) claims, the main mathematical concepts in Spanish curricula are: numbers and operations, measures, estimation, calculus of magnitudes, representation and organization of space, interpretation, representation, analysis of information and treatment of the element of chance. Students should be aware of all these ideas, because they are present in the compulsory education system. In Spain children learn to solve equations (at compulsory levels just up until quadratic equations, then teachers introduce linear algebra systems and calculus), problems based on the Pythagorean theorem, situations about similarity (through proportions and applications of Tales theorem), different kinds of word problems, percentages, and so forth. In the last few years the curricula has been more focused on problem solving and a conception of contextualized mathematics, in terms of realistic mathematics and everyday mathematics. 4. Numeracy and the adult education system in Catalonia In Catalonia adult numeracy is focused on the practical application of Mathematics. Mathematical concepts, relations and basic operations are used as tools, in order to interpret and to solve problems in everyday life. So the kinds of activities mainly include practical situations and problems within a context, in the sense of realistic mathematics. Mathematics also has a social component. Because 5

of its nature, Mathematics forces people to go backwards and forwards between concrete to abstract ideas. This dynamic encourages the development of logical reasoning and cognitive schemata. So, Mathematics in Catalonia is focused on situations analysis, problem solving, discussions about different resolutions, etc. The starting point is always everyday situations, so that adults people have the opportunity to make connections with their own life experience. There are several adult education centres where the curricula is based on dialogical mathematics or multicultural considerations. Here is an overview of both adult basic education and secondary education Mathematics curricula.. Adult basic education (ABE) curricula in Mathematics The general objectives of the ABE curricula in Mathematics in Catalonia are the following: - To value and use Mathematics as a tool to interpret information from the real world in a critical way. - To search for and to select information in order to share it, to discuss different interpretations of it and to build different strategies to solve problems. - To value Mathematics as a way to collaborate with partners, and to encourage self-confidence. - To carry out the whole process involved in solve a problem, from understanding the problem to checking the answer. - To identify, interpret and use different languages and mathematical codes. - To use mental calculus as well as computer calculus (with a calculator or with a computer). - To interpret numbers as quantitative characters, by using the correct scale from an estimation of the magnitude order. - To identify, represent and classify geometrical shapes, by using t knowledge about the properties and relationships of these shapes, in order to improve spatial understanding. - To identify graphs and tables by using basic techniques to collect data, and by using them to represent and analyse real phenomena. The contents of the ABE curricula in Mathematics in Catalonia are the following: -Procedures- 1. Mathematical procedures: a. To observe, to experiment and to collect data. b. To classify, to compare, to arrange and to measure data. c. To estimate and to check data. 2. Mathematical languages: a. To understand mathematical vocabulary and to use the most regular codes (numbers, units of measurement, etc.). 3. Techniques for calculations: a. Exact calculus and approximations. 6

b. The use of calculus tools. c. Introduction to new technologies as tools for solving problems. 4. Problem solving: a. To build strategies for problem solving. b. To use arithmetic, geometry and measurements for problem solving. -Concepts, ideas- 1. Natural numbers and operations. a. Cardinals & ordinals. b. Position value. c. Addition, subtraction, multiplication and division. 2. Fractions and decimal numbers. a. Fractions as part of a unit or a quantity, as an expression of measurement and as a division. b. Decimal numbers as fractions. c. Percentages. 3. Magnitudes and measurements. a. Magnitudes: length, surface, volume, time, temperature and money value. b. Measurement units. 4. Geometrical shapes: a. Shapes on the plane and in space. Elements of shapes. b. Geometrical relations. c. Location and direction. 5. Introduction to statistics. a. Statistical data. b. Elementary parameters: mean, mode, median. c. Graphics: data representation. -Values, norms, attitudes- 1. To value individual and collective efforts. 2. To value work in small groups as a way to enrichment. 3. Scientific attitude. 4. To consider Mathematics as a tool to represent the real world. 5. To value Mathematics as a tool to plan and to organize concrete situations in the real world. 6. To use estimation, rounding-up, and mental calculus. 7. The value of error and its usefulness within the learning process. 8. To consider the usefulness of technological tools as calculators, computers, etc., as tools to facilitate mathematical routines. Secondary curricula in Mathematics The curricular design at this level begins with the knowledge acquired in the previous steps (adult basic education). The main goal is to promote the development of a scientific attitude. The curriculum combines both theoretical and 7

practical components. It is focused on values such as the promotion of the pleasure of observing, understanding, and discovering. This approach promotes people s real world understanding. The curriculum expects to offer a scientific-technologic view of society and also to promote a scientific way of reasoning and understanding. Mathematics is presented along with science and technology. The general objectives at this level are the following: - To understand Mathematics as an open and dynamic science which addresses problem solving, with its own history. - To value the instrumental character of Mathematics for other fields of knowledge. More specifically, to underline the capacity of mathematical models to describe, to interpret and to understand the physical and natural world and also society. - To acquire the skills needed to compare, to use, and to evaluate different mathematical models. To apply mathematical methods in a creative way when facing new situations. - To recognize the role of the current technological media in Mathematics, such as tools to carry out mechanical procedures. - To understand and to value one s own mathematical skills and to use them in a flexible way. To have enough resources to use to solve problematic situations (either in everyday situations or mathematical situations). - To observe both quantitative and geometric aspects from the real world and also from society in a systematic way. To organize data in a clear way that makes their analysis and understanding easier. - To express and to interpret the data collected correctly, as well as graphs, tables, and so forth. - To use different mathematical languages (algebra, statistics, geometry, etc.) when it is useful in order to represent and describe questions and their answers. - To understand and to use terminology, symbolism, and scientific tools correctly in order to understand scientific compositions. - To observe, describe and classify objects, organisms, phenomena and processes related to the physical environment. - To value scientific attitudes like curiosity, objectivity, observation and the different processes of scientific investigation. The main contents of secondary curricula in Mathematics in Catalonia are the following: -Procedures- 1. Languages and processes a. The use of different mathematical languages: numerical expressions, algebraic expressions, functional language, graphic language, and geometrical language. b. Translation. c. Classification. Arrangement. d. Application of deductive and/or inductive methods. 8

e. Problem solving. 2. Techniques to take measurements and to calculate them a. Techniques for symbolic and graphical number representation. b. Techniques to take direct measurements (with the use of tools) and indirect measurements (through algorithms, and scalar representations). c. Exact calculus and approximations. The use of a scientific calculator. d. Elementary ways to solve equations. e. Permutations. f. Techniques to calculate statistical parameters. 3. The use of geometrical models. a. The application of geometrical models to interpret real situations. b. Scalar representation in the plane of geometrical shapes. c. Break plane shapes into more elemental shapes. d. Plane representation of spatial shapes. 4. Collecting data. a. Techniques to collect data. b. Direct observation of objects, phenomena and procedures. c. Observation of figures, models and scale models. d. The use of technological media. 5. Data processing. a. Building tables (of items and frequencies). b. Graphic representation of phenomena in the X-Y axis. c. Specific techniques for graphic representation of statistical data. d. Graphic and symbolic representation of technical data. e. Creation and use of formula to make links between variables. f. Analysis of functional dependences. g. Analysis of statistical variables. h. Interpretation of schema, symbols and symbol sequences. i. Random phenomena processing. Calculus of probabilities. j. Interpretation of data collected. -Concepts, ideas- 1. Numbers. a. Natural numbers. Divisibility. b. Whole numbers. c. Rational numbers and decimal numbers. d. Numbers and the use of the scientific calculator. e. Magnitudes. f. Equations. 2. Plane and space. a. Elements and organization in the plain. b. Elements and organization in the space. c. Metric relations and trigonometric. 3. Dependence between variables. a. X-Y axis. General characteristics of graphs. b. Functions: dependence and associated ideas. c. Lineal function. 9

d. Quadratic function. e. Increase and decrease of different kind of functions. 4. Elemental statistics and chance. a. Basic concepts in statistics. b. Parameters of centrality / dispersion c. Random phenomena. d. Probability: concept and main principia. -Values, norms, attitudes- 1. To make questions and research problems receptive to mathematical understanding. 2. Systematization of work with mathematics. 3. To value mathematical tools. 4. To use technological media with wisdom. 5. Confidence in one s own ability to solve real problems in a mathematical way. 6. Interest in self-study and lifelong learning. At the end of the secondary level, adults should be able to understand different types of numbers and their properties, as well as different ways to represent numbers (whole numbers, fractions, percentages, scientific notations, etc.). Adults should be able to use algorithms, solve equations, recognize different kind of problems, make connections between Mathematics and the real world and use Mathematics to represent real situations and try to find answers to real problems. Therefore, and to conclude, in Catalonia numeracy means to have a mathematical understanding of the real world, as well as having mathematical skills to solve problems in everyday life. References Alsina, C. (2001). Reference levels in School Mathematics Education in Europe. National presentation: Spain. EMS. European Mathematical Society. Committee on Mathematics Education. http://www.emis.de BOP. (2002) Decret d 1 d agost pel qual s estableix l ordenació curricular de la formació bàsica de les persones adultes. http://www.gencat.net. Ministerio de Educación. (2006). http://www.mec.es 10