Silvia Solano
email: sp.solano10@uniandes.edu.co

Pedro Gómez

María José González
email: mariaj.gonzalez@unican.es

Alignment of mathematics curriculum to standards at high schools in Colombia

Keywords: Alignment, Mathematics curriculum, Standards

. These official documents have been the baseline to decide how to conduct the math curriculum parameters in schools. Specially, the basic standards of competence document provides teachers with a set of guidelines for what students are expected to know and be able to do, defining the intended curriculum. However, there is a lack of coherence between these parameters and what the teachers design in their school curricula. We analyzed the alignment between the mathematics curriculum of a sample of Colombian schools and the document of basic standards of competences at high schools in Colombia. We found an inadequate alignment between the schools' mathematics curriculum and the official document of basic standards of competence.

Law 115 of 1994 and curricular autonomy

The curriculum in Colombia has suffered many different changes or transformations based on political parameters or requirements. In 1994, the Ministry of National Education (MEN) stated the Law 115. This law organized the educational services in Colombia, through general rules. In this law, the government established the curricular autonomy of schools. This means that schools have autonomy to make and adapt their curricula. Also, the Ministry of National Education has proposed some guidelines in terms of learning expectations called Basic Standards of Competence [START_REF] Men | Estándares básicos de competencias en lenguaje, matemáticas, ciencias y ciudadanas[END_REF].

Basic Standards of Competence (2006)

Mathematical competences are related to mathematical thinking and mathematical systems. The standards are based on these mathematical thinking and systems. These guidelines contemplate the competences that a student must achieve when a cycle of two school years ends.

Alignment to standards

In the past, the most common educational use of the concept of alignment referred to the match between an assessment instrument (or instruments) and a curriculum [START_REF] Webb | Criteria for Alignment of Expectations and Assessments in Mathematics and Science Education[END_REF]. Alignment does not only refer to a comparison between one assessments instruments with a curriculum, but extends to a set of assessment instruments or the assessment system. However, Webb has defined the alignment as "the degree to which expectations and assessments are in agreement and serve in conjunction with one another to guide the system toward students learning what they are expected to know and do" (1997, p. 12). An area plan is a curricular document that has the objectives, the methodology, the distribution of time, and the evaluation criteria for a subject, in our case, the area of mathematics. The area plan becomes the roadmap that guides the implementation of the mathematics curriculum within schools. The alignment with standards is a characteristic of Colombian area plans. We assume the alignment with the standards as the measure to which the mathematics area plans approach the contents proposed in the standards. With this attribute, we seek to determine how much is covered of the subjects in the area plans according to what it is stated in the official document of the standards. For a specific syllabi and specific topics, the alignment should indicate to what extent that area plan meets the standards proposed to these issues.

Methodology

We collected a sample of 212 syllabi throughout the country taking into account three main variables: geographical area (rural and urban), type of institution (public and private) and type of secondary education (academic and technical). In addition, we built code trees based on the didactic analysis carried out on three subjects of school mathematics. The topics chosen are the conic sections, the derivative and the descriptive statistics. The process of coding the 212 area plans consisted in identifying the text segments that allude to the proposed topics. Once these text segments were identified, they were assigned to label each one with a code of the code tree in N-Vivo program. Similarly, we coded the standards document as if this document were an area plan. After coding the documents, we proceeded to make a comparison between each area plan and the standards. Once the sources were compared, we proceeded to count the codes that each syllabi document shares with the standards document. We constructed a statistical variable with this measure. The value of this variable for an area plan is equal to the quotient between the number of codes that the area plan shares with the standards document and the total number of codes assigned to the standards document.

Results

We found that the sample mean of the variable is 29,19%, with a standard deviation of 13,36% which means that there is a low alignment and a great variability in the alignment percentage of the area plans and the standards document. At 95% confidence, we found a confidence interval of the population mean of Colombian area plans of (27.31%, 31.07%). The area plans should be aligned to the curricular parameters and standards; however, we found evidence that these plans do not fit what the Law established causing a gap on secondary education, mainly on tenth and eleventh grades.


Eszter Varga
email: evarga@bpg.hu

The potential of Problem Graphs as a representational tool with focus on the Hungarian Mathematics Education tradition

Keywords: Series of Problems, Hungarian mathematics education, teacher design capacity

Introduction

Capturing teachers' principles and working methods in instructional design can pose some challenges, as well as its representation and dissemination. In this work in progress, we suggest a new representation tool, the Problem Graph, that has the potential to reveal some of the Teacher Knowledge embedded in series of problems and task sequences. Problem Graphs are developed with dual purpose: as an analytical tool to explore the structure of existing resources through a priori analysis, and as a design tool that can help teachers plan their instructional sequences and long term learning trajectories. The potential of Problem Graphs in developing teachers' design capacity were also investigated in two pilot experiments with the objective of introducing them in teacher education and PD programs.

Initially, this work is strongly rooted in the Hungarian Mathematics Education tradition, but both the approach in question and the representation tool is relevant to the international community for its special handling of Mathematical Knowledge for Teaching [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF].

Background

My present and prospective work with Problem Graphs is a part of a large scale research project launched by the Hungarian Academy of Sciences. 1 The project focuses on the Hungarian "Guided Discovery" approach, that was experimentally implemented nationwide by a reform movement led by Tamás Varga in the 1960s and 70s (see Gosztonyi et al., available in pre-Proceedings of ICMI 24;[START_REF] Varga | Mathematics education in Hungary today[END_REF]. This approach is highly recognized among Hungarian experts. The research project is assigned to describe the approach and examine its background as reflected in the recent theoretical frameworks as well as revisiting the reform and communicating it to both national and international audiences.

In a preliminary study, Gosztonyi identified Series of Problems as a characteristic aspect of teachers' work in the Guided Discovery approach. The research group started to analyze Series of Problems as a special type of resource, using the frames of the Documentational approach [START_REF] Gosztonyi | Teachers documentary work with series of problems[END_REF]Gueudet, Pepin, & Trouche, 2012). In this course of work, a need emerged for an efficient visual representation of the inner structure of Series of Problems, to which the graph representation came up quite naturally as one of the possible answers. Problem Graphs were built to some of the examined Series of Problems and discussed in details with experts. In these structures, nodes represent the individual problems, while edges show the explored connections. The connections reflects Mathematical Knowledge for Teaching, one-way arrows show precedentconsequence relations. For the standards of further classification, the domains suggested by Ball et al. might be a good starting point, but this work is only at its first steps at the moment.

The poster will demonstrate that the name Series of Problems covers a notion that is more complex than a task sequence, and seems to be indigenous in the Hungarian tradition. Nevertheless, the graph representation can be an effective tool to support the analysis of other learning sequences independent from this cultural context.

Dealing with problem Graphs led to the idea that this type of work on set of problems could benefit teachers' professional development greatly, especially if conducted in groupwork. Planning on investigating this opportunity, we conducted two pilot experiments with 10 and 18 participating teachers. In both experiments, teachers worked in groups on the same subset of a Problem Field which was examined previously by the research group. During the sessions, the teachers built a Problem Graph from the selected problems, followed by a whole group discussion and a questionnaire in the second experiment.

Poster Content

The poster will summarize some major principles of the Guided Discovery approach as a context. It will also give a working definition for Series of Problems. Limitations in space will not allow to provide full examples for Series of Problems, but some key problems will be highlighted. In the main part, Problems Graphs will be displayed, with one extensive example representing a Series of Problems from the field of Elementary Geometry in the center, with some specification of the connections. Graphs built by teachers from the same problems will also be presented, accompanied with some teacher responses from the questionnaires. This will allow us to draw some conclusion from the pilot experiment regarding the potential usage of the Problem Graph.

1

  The poster is made with the financial support of the MTA-ELTE Complex Mathematics Education Research Group, working in the frame of the Content Pedagogy Research Program of the Hungarian Academy of Sciences (ID number: 471028).