Mathematics Learning Opportunities For Students With Intellectual Disabilities In Chilean Special Schools

Abstract

In Chile, students with learning disabilities (LD) go to special schools or to mainstream schools with Integration Programmes. Even though there are data regarding the access of students to the educational system, there is not much knowledge related to the learning opportunities (LO) that these students are given. In this context, a multiple case study was developed, aiming to explore the learning opportunities in mathematics that are created within special education centers. This work presents the findings of the analysis of interviews to teachers and principals, as well as the findings of the analysis of the classroom materials of three cases, in terms of the type of cognitive demand and the curricular coverage of the mathematical tasks performed in notebooks and textbooks in the span of a semester. As shown in the results, teaching LD students must be focused on the development of the concept of number, as well as basic operations, aiming to use money in the community. Activities developed in special schools feature a low level of cognitive demand and the repetitive use of the same type of tasks

Keywords: Learning opportunitiesintellectual disabilitycognitive demand

Introduction

One of the main problems for the theory and practice of mathematics education is equity (Forgasz & Rivera, 2012), especially if we come to the understanding that equity cannot happen without access to this education (Bishop & Forgasz, 2007). This is why in the research on equity in mathematics education, finding how students access to the learning goals of the curriculum gains more importance, and particularly, those students that are at risk of being excluded whether due to personal of contextual conditions, which is the case of people with learning disabilities (LD) (Ainscow, Booth & Dyson, 2006).

In Chile, LD students go to special schools or to mainstream schools with Integration Programmes. Even though there are data referring to the access of students to the educational system, there is no much knowledge regarding the Learning Opportunities that these students are given. In this context, this work presents the results of a bigger research (Howard, San Martin, Salas, Blanco and Díaz, in press), whose main goal is to know and describe the LOs in math produced in special schools for LD students. The research shows the results derived from the analysis of classroom material (notebooks and textbooks) and from semi-structured interviews to the teaching and management staff of Chilean special schools.

Mathematics provides students with the language that allows them to interpret, describe, analyze, predict and solve everyday problems. Even though the learning objectives of the mathematics curriculum are established for the whole universe of students, international studies have shown that LD students do not have access to high educational quality mathematics programs (Gervasoni & Lindenskov, 2011). The research has clearly stablished that a high amount of these students are deprived of the chance of learning mathematics, because it is considered an inadequate field of study for them (Faragher, Brady, Clarke & Gervasoni, 2008).

The abstract and conceptual nature of mathematics raises particular challenges to LD students, especially in problem solving. Therefore, these students require a large variety of actions that help increasing their learning opportunities (Sarama & Clements, 2009).

One way to study the access of students to the learning that they receive is through the concept of Learning Opportunities (LO). The LOs were originally defined as a measure towards the possibility of studying a particular subject, or learning how to solve a determined type of problem given in an evaluation (McDonnell, 1995). Later on, the LOs were broadened towards the supply of proper opportunities for every group of students, including the analysis of resources, school conditions, the curriculum and the teaching experiences (Banicky, 2000). Schmidt and McKnight (1995) have indicated that the LOs analysis may inform the distance between the designed curriculum and the one that is actually implemented, and that the LOs depend, among other factors, on variables that are associated to teachers, the characteristics of students and the school (Howard et al., in press)

The international studies that have sought to explore de LOs have been performed through different techniques, such as questionnaires aimed to teachers (Cogan & Schmidt, 2015; Schmidt & McKnight, 1995) and analyses of the notebooks of students (Ruiz-Primo, Li & Shavelson, 2001). These analyses have allowed the development of a fuller approach to the LOs, since it provides information regarding the processes through which teachers develop a given objective, besides including contents and skills. Thus, the study of the notebooks of the students allow the identification the type of mathematical task, the skills involved in the proposed activities, and it also allows to analyze the cognitive demand (Stein, Grover & Henningsen, 1996) of the type of task, and therefore, the LOs that teachers give to their students. The level of cognitive demand of a given activity refers to the type of cognitive processes the student requires to fulfill it. According to the Stein et al. (2000) taxonomy this level may vary within a continuum of four increasingly complex levels: memorization, offline procedures, on line procedures and “do math”. These studies have remarked the importance of the activity of teachers in the generation of opportunities of learning through the selection of tasks and mathematical activities (Sullivan, Clarke, Clarke & O’Shea, 2010), due to the influence of their perceptions and the objectives that underlie their actions and decisions (Carrillo, Contreras, Zakaryan, 2013) in the LOs of students (Sullivan et al., 2010). In this regard, the analyses of the characteristics of teachers, particularly their beliefs, have garnered increasing interest, given their influence in the pedagogical practice (San Martín, 2012; Thompson, 1992).

Regarding the theoretical development of the beliefs linked to mathematics, three dimensions have been identified (Schmidt & McKnight, 1995): beliefs on mathematics as object of study, beliefs on the nature of teaching mathematics, and beliefs regarding the learning of mathematics.

In this scenario, the present work shows the partial results of a bigger research, which seeks to answer the question, what opportunities are provided in special schools for students with intellectual disabilities to develop the learnings defined in the national mathematics curriculum?

Problem Statement

There is a lack of systematic studies regarding the learning opportunities provided to students with intellectual disabilities attending special education schools

Research Questions

What opportunities are provided in special schools for students with intellectual disabilities to develop the defined learning in the national math curriculum?

Purpose of the Study

Know and describe learning opportunities in mathematics that are given in special schools for students with intellectual disabilities

Research Methods

This research used a multiple case study design (Yin, 1994) through two elements: a record of the classroom material and semi-structured interviews.

Three cases were studied. Each of them correspond to a first basic cycle mathematics course from different special schools in the Chilean Metropolitan Region.

In each case, the work material (notebooks and textbooks) of the first semester was collected from four students from three first basic cycle courses through images In addition, the teacher of each of these courses, the technical pedagogical coordinator and the principal of each school were interviewed. Nine interviews were conducted in total. An analysis of the content of every activity/exercise registered in notebooks and textbooks was performed, which covered two aspects: i) the identification of the learning objectives (current national curriculum) and ii) the identification of the level of cognitive demand. The codification was developed by two researchers through independent sessions and three sessions devoted to discussion and recoding. The level shown in the analysis was satisfactory, according to a statistical Cohen Kappa (k), that fluctuated between .62 and .97, being k = 85 the mean value.

The interviews delved into the description of the design, development and evaluation processes of the mathematics classes and the learning of students, as well as the meanings given by the interviewees to mathematics, its teaching and learning, specifically with LD students. Once the transcription of every interview was made, an analysis of the content was performed, through the systematic implementation of the constant comparative method (Glaser & Strauss, 2009).

Findings

The presentation of the results is structured in two sections. While the first one is focused on the analysed classroom material (notebooks and textbooks), the second one presents to the results of the analysis of the interviews.

Conclusion

The current work explored and analysed the learning opportunities from both the available classroom material and the declared beliefs and practices of classroom teachers and the management staff of the Chilean special schools, regarding mathematics and its teaching and learning process. To do so, a descriptive qualitative method which sought to understand from singular experiences the core aspects of the study problem was used.

Results show that: i) in these educational contexts, mathematics teaching comes up as a process that instead of adapting to the curricular demands, it needs to adjust to the characteristics and needs of the students; ii) the teaching is focused on the learning of numbers and basic operations, since they are both perceived as key factors for the use of money within the community; iii) the activities developed tend to be repetitive and well-trodden, with a low level of cognitive demand and that promote the development of the representing skill, and in a lesser degree, the modelling skill; iv) teaching planning is fuelled by diagnostic evaluations that help in the designing of annual course planning and its subsequent adaptation to specific student cases. However, the lack of no instruction time constitutes an obstacle these design processes must face; v) the learning evaluation is based on an individual look of the process, which is carried out through qualitative and quantitative instruments to measure half-yearly learning results. Results also show, even though not so highlighted, that the teaching methodologies are flexible and cover strategies that can be both ludic and more traditional and individual, such as working in notebooks, textbooks and worksheets; results also indicate that within the mathematics class, technological resources and particular materials are used, aiming to help in the manipulation and counting.

Results show educational beliefs and practices underpinned by the development of learning processes, which in turn are based on a restricted and less rigorous curriculum (Nolet & McLaughlin, 2000), that showcase an intensive use of particular materials (Sarama & Clements, 2009). These beliefs and practices show the need of performing individual curricular adaptations. Even though there are times when individual adaptations are needed, this stand of analysing each case on an individual basis may stem from a work logic that is very common in the cognitivist special teaching, which focuses on the particular skills without meddling with other approaches that perceive teaching and learning as a co-construction performed by each of its participants.

Results expose the limitation of the LOs given to LD students in mathematics, which in turn limits the educational equity, restricting mathematics specifically to the concept of number associated to the use of money.

Finally, it is of importance to pin down the need to develop future research to complement the results shown in this work, in order to generate guidelines for the educational practice in mathematics for LD students. It is to be expected that the sum of these elements will contribute in moving towards a true equal access to the LOs in mathematics, based on the right to quality education for all students.

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