# Mathematics in the School Curriculum

The First Inter-American Conference on Mathematics Education (Bogota, December 4-9, 1961) was a milestone in the introduction of Modern Mathematics in Colombia. This movement declined in Colombia by the end of the 1970s, in part, for reasons associated with the educational model that it implied (e.g. behaviorism, management of the curriculum, production of educational materials, teacher preparation), and in part for difficulties with the abstract nature of the Mathematics itself as the basis for curricular reform (e.g. set theory, structure and modern algebra). With this decline, space was opened for a new reform that can be understood as the Colombian response to what is commonly referred to as the “Back to Basics Movement”. This reform was supported by arguments taken from Piagetian theory and arguments against set theory as the curricular referent for school Mathematics. Instead, arguments supporting system theory^{1} as the most appropriate curricular referent. Therefore, a solid psychological theory was used to explain the pedagogical processes that take place in the classroom. As a result of this reform, in the second half of the 1980s, the MEN promoted a new optional approach, that the Mathematics curriculum could be organized in relation to five mathematical systems (numerical systems, geometrical systems, measurement systems, data systems and logic systems), to which were added two topics (sets, and relations and functions). For each of those, details about contents, sequence, level of depth, interrelations and development of the focus were developed.

When the implementation of this approach had been in effect for only a few years, the country underwent a substantial change with the introduction of a new Constitution, which naturally affected the vision and implementation of education. In the development of this new political charter, laws^{[1]} ^{[2]} were established that defined the bases for educational transformation. One of the transformations implied the definition and adoption of Curricular Guidelines for Mathematics (Colombia 1998) that more than programs of study, constitute road maps that, respecting the cultural diversity consecrated in the Constitution, orient the efforts of educational institutions.

The Guidelines did incorporate some aspects of the previous reforms while proposing new theoretical and methodological elements in an attempt to update the curricular structure of school Mathematics. Among the elements are three that stand out. First, is the introduction of the different types of mathematical thinking^{[3]} (numerical, spatial, measurement, variational, and random). Second, is the contexts in which school Mathematics should be developed (mathematical, daily life and from other sciences). Finally, there is the insistence on the importance of the development of processes (solving and posing of problems, reasoning, communication, modeling, and the elaboration, comparison and practicing of procedures). Together these permit the learning of Mathematics in contexts significant to students, using problem situations as the central axis for said contextualization.

Among the theoretical elements and methodologies of the Guidelines is the call for interdisciplinarity. This is not only from the perspective of teacher preparation, but in classroom practice given that in this document elements of teacher professional knowledge and ways that teachers work in the classroom are discussed. At one point in the document the MEN points out that “... the future teacher should receive a preparation intrinsically interdisciplinary that is distinct from what has happened in the past [that is], a conglomeration of courses that students must add up at their own risk” (Colombia 1998, p. 124). And a Calculus course, for example, is added which should include its history, its epistemology, and its teaching from a modern sense of how it should be the result of inquiry in interdisciplinary and even inter-institutional work groups.

Parallel with what is reported above, in the last twenty years the Colombian education system has had an ongoing series of discussions, shaped by educational policies, on the development of basic competencies^{[4]} (focused primarily on competencies in Mathematics, Spanish language, and Natural and Social Sciences), (general and specific) workplace competencies,^{[5]} and citizenship competencies,^{[6]} These competencies seek to create an equilibrium between a solid academic preparation, and preparation for work and citizenship.

In this competencies development framework, specifically for the case of education in Mathematics, early in the new century a document was published with basic competency Standards for Mathematics (Colombia 2006). In that document the concept of competency was presented broadly “as a set of socio-affective and psychomotor understandings, abilities, attitudes, knowledge and cognitive dispositions appropriately related among themselves to facilitate a flexible, effective and sensible performance when faced with new and challenging activities”. In this sense, more than speaking of “mathematical competence”, the idea of “mathematically competent” was proposed. Those responsible for the education system were invited to see Mathematics as a human activity inserted into, and the result of, cultural processes characteristic of the time and place. They were also invited to view Mathematics as the result of successive processes of reorganization of the practices of people in relation to the quality of their lives.

The term competency promulgated in the Standards document highlighted other dimensions associated with school Mathematics. In that sense, Valero (2006) points out that:

The adoption in Colombia of the language of mathematical competence emphasizes dimensions of Mathematics Education that had not necessarily been so explicit in the past. As Vasco (2005) noted, matters of quality and equity, of the social and cultural value of Mathematics, and its contribution to the development of citizens and the consolidation of democracy in the country are dimensions now being highlighted (p. 1).

Thus, the Standards (Colombia 2006) call for mathematical development to not consume itself with disciplinary contents, but instead that the school should be rehabilitated to offer an ideal mathematical development in the development of the citizen: A citizen is formed when Mathematics is learned. The notion of a mathematically competent citizen works on the least pragmatic dimensions in relation to the notion of competency (knowing what to do in a given context). This is done in pursuit of a more holistic perspective, where the focus is the understanding of Mathematics on the part of the individual. Hence, there is the development of a set of abilities, capacities, conceptualizations, forms of action, etc., that permit in-formed (formed from within) decision making with Mathematics and from Mathematics.

- [1] To provide language to unify the different branches of Mathematics and other sciences, theconcept of “system” was proposed as the basis for organizing the curricular processes inElementary and Secondary Education, emphasizing that the approach to any mathematical systemshould include at least three components: the concrete, the symbolic and the abstract. For detailssee Vasco (1994).
- [2] Law of Higher Education or Law 30 in 1992, and the General Law of Education or Law 115 in1994.
- [3] For a detailed synthesis see Obando (2004).
- [4] Basic competencies seek to generate conceptual constructions and the capacity to utilize scientificand humanistic knowledge in processing, interpreting and solving problems related to the surroundings, school environment, and science and technology.
- [5] Workplace competencies are oriented to the development of a set of knowledge and techniquesthat prepare the individual to be a productive member of society. The general competencies arecross-cutting and transferable to any context in which they are present in any academic orworkplace activity. On the other hand, specific workplace competencies refer to particular contextsrelated to activities characteristic of a group of professions.
- [6] Citizenship competencies refer to the development in the individual of a set of values, actions andbehaviors needed by society, a critical and reflexive nature in facing situations that presentthemselves in the ongoing practice of citizenship, and an active participation in the life of thecommunity.