Inclusive and Supportive Education Congress International Special Education Conference Inclusion: Celebrating Diversity? 1st - 4th August 2005. Glasgow, Scotland

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An Examination of Quality of Inclusive Pedagogies in Primary
Mathematics Classes

David Evans    PhD
Monica Wong    MTch (Hons)
University of Sydney, Australia

Contact Details:

Associate Professor David Evans
Faculty of Education and Social Work    A35
University of Sydney     NSW     2006    Australia
d.evans@edfac.usyd.edu.au

Acquisition of mathematical knowledge during the school-age years is important for later functioning in society (Fuchs, Fuchs & Courey, 2005). For students with disabilities and difficulties learning, acquisition of proficient mathematical skills is no less important. As these students become active members of their communities they require mathematical knowledge and skills to carry and engage in activities.  

There are few studies that have systematically studied the teaching and learning practices of teachers in mathematics classrooms in regards to students with disabilities and learning difficulties. Kilpatrick, Swafford and Findell (2002) in an extensive report of quality mathematical programs provided an extensive overview of quality mathematics programs for students in general. Little information about programs for students with special education needs was cited.

Swanson (1998) provided an overview of instructional practices that are linked to effective outcomes for students with learning disabilities (i.e., segmenting of knowledge, management of processing demands, use of technology, careful sequencing of skills). This meta-analysis used only a small set of studies to support these conclusions from the area of mathematics, and concern was raised in general about the quality of research undertaken.

This paper will outline the results of a study that examined the pedagogical features from a set of mathematics classes for students in a range of classroom in primary schools (i.e., Kindergarten to Year 6) in Sydney, Australia. In examining the teaching within these classrooms, specific attention was initially given to providing an overview of the practices used in the classes observed. Following this examination, specific attention was given to those strategies that were used to cater for students with specific needs in learning mathematics. Finally, conclusions will be made in regards to the practices observed, what the literatures claims to be effective practices, and where future research could be focused to assist develop mathematics instruction that leads to mathematical proficiency for all students in the primary classroom.

Prior to discussing the methodology of this study, and the results from the ensuing analysis, the framework for mathematical proficiency will be discussed. As part of discussing this framework, a model of quality teaching will also be highlighted. This model of quality teaching is based on one developed by the New South Wales Department of Education and Training, and will be used to analysis classroom data collected.

Mathematical Proficiency

Over the past years, reading has been the focus of much research in the area of students experiencing difficulties (Jitendra, 2005). In the past five years, research into mathematics programs for students experiencing difficulties has become more common. An increasing number of publications is addressing instruction, programmatic and assessment issues relating to mathematics, and the application of mathematical knowledge – numeracy. In addition, government initiatives have listed of mathematics and/or numeracy as an area of focus (e.g., Adelaide Declaration, 1999).

The increased focus on developing mathematics knowledge for all students has required the area to be examined carefully. A central part of this examination has been the publication by the National Research Council of the report Adding it Up: Helping Children Learn Mathematics (Kilpatrick, Swafford & Findell, 2001). This report used the term mathematical proficiency to represent what all students need to learn if they are able to learn from using mathematical knowledge in their world. Mathematical proficiency is posed as five, intertwined, interdependent strands:

• Conceptual understanding: - comprehension of mathematical concepts, operations and relations
• Procedural fluency – skills in carrying out procedures flexibly, accurately, efficiently and appropriately
• Strategic competence – ability to formulate, represent and solve mathematical problems
• Adaptive reasoning – capacity for logical thought, reflection, explanation, and justification
• Productive disposition – habitual inclination to see mathematics as sensible, useful and worthwhile, coupled with a belief in the value of diligence and in one’s own efficacy. (Kilpatrick et al., 2001, p.116)

The development of mathematical proficiency in a classroom is therefore a complex process, and examining it in the classroom limited by the methodological issues. Mathematical proficiency could be examined in terms of each strand outlined by Kilpatrick et al. (2001). This approach to examining mathematical proficiency in the classroom does not permit the complexity of the learning mathematics to be fully examined or understood. A richer examination must permit the differing strands of mathematical proficiency to be examined, as well as their interaction to be recognised as part of the analysis of teaching practices being examined.

In achieving this balance in analysing classroom data recorded, and to retain validity with the teaching profession, a framework for evaluating classroom teaching was required that examined the multi-dimensionality of the teaching process. The NSW Department of Education and Training Quality Teaching Model was adopted for the initial analysis. This model examines eighteen elements of teaching, of which nine were chosen for this project. These nine elements, as shown in Table 1, were chosen because they permitted the issues of conceptual understanding and procedural proficiency to be examined through deep knowledge and deep understanding. The model also permitted the other strands to be examined: strategic competence (knowledge integration, connectedness), adaptive reasoning (self-regulation, engagement, knowledge integration) and productive reasoning (high expectations, explicit quality criteria, engagement).

Utilising the Quality Teaching model for structuring the analysis of classroom observations strengthened the external validity of the study. This is, in working with class teachers the researcher was able to pedagogical issues to current developments within their professional learning area. In terms of meeting the needs of students experiencing difficulty learning, a more specific model of analysis was required. This was achieved by analysing video data that examined intervention components of mathematics programs, instructional behaviour of teachers, and the use of environment to support learning identified in the literature as effective in catering for students with special education needs.

Components of effective instruction . Teaching behaviours that have been identified as being effective in meeting the needs of students experiencing difficulties learning have been reported across the years (e.g., Fuchs & Fuchs, 2003; Heward, 2003; Swanson, 2005; Swanson, Hoskyn, & Lee, 1999; Kaufmann, 2005). These behaviours cover issues relating to the way teachers prepare to teach a lesson, how they conduct themselves during a lesson, and what they bring to support learning from the community. A list of these features is shown in Table 2, along with a description of what the behaviour may look like in a classroom.

The components of effective interventions for students experiencing difficulties listed in Table 2 were taken from a review of the literatures conducted by Swanson et al (1999). These components of effective interventions were identified from an extensive meta-analysis of the literature, and currently represent a comprehensive review of what the literature is reporting as effective for students experiencing difficulties learning. Further, these components of effective interventions cover the areas of planning or curriculum design (e.g., segmenting, sequencing), instruction (e.g., strategy cues, directed questioning, drill and practice, controlling task demands, modelling) and use of school community resources to assist meet the needs of students (e.g., technology, supplements to the teacher).

Method

Classrooms . This project aimed to capture a series of video images from a range of classrooms in primary schools in Sydney, Australia. The collection of these data is ongoing, and for the purposes of this paper four classes will be included in discussions. These four classrooms were from the inner city area of Sydney, and represent a cross section of socio-economic areas in this part of a large metropolitan city. Two classes were included from Stage 1 (or Years 1 and 2), and two from Stage 3 (Years 5 and 6).

Teachers . The teachers taking the classes observed each had more than five years of teaching experience. In one case, the teacher (male) was the Assistant Principal in the school of approximately 350 students, while the other teacher (female) in this school was on a junior class full-time. The other teachers were from a similar sized public school, representing the same gender and classes.

Measures . The primary source of data in this study was digital video images taken in the target classes. Each teacher was videoed working in their classroom during a series of consecutive lessons. During each lesson, the teacher wore a wireless microphone that was connected to the video camera so that teacher language and commentary could be recorded and synchronised with the video images.

The initial analysis of data was conducted using the NSW Department of Education and Training Quality Teaching model. The lessons were specifically analysed along those elements outlined in Table 1 and notes made to support the judgements.

Following this initial analysis, data were analysed using a series of components taken from the work of Swanson et al (1999), who listed a series of features of effective instruction needed to cater for students with difficulties learning. These features are listed and described in Table 2.

Procedure .   A number of schools were approached to be part of this project. There were varied positive, responses to these requests, and there was general consensus that it was a worthwhile project to be undertaking. Obtaining final consent from teachers, however, was a time when a number of persons rethought the process and, as per the ethical conduct of the project, withdrew their original consent to be part of this project. While this outcome was something that the researcher was happy to take as part of the ethical conduct of the research process, it did pose a question as to what it was about being videoed conducting your everyday job. The one issue that did appear to come to the surface in reflecting on this phenomenon was that mathematics was a primary school subject that teachers tended to feel uncomfortable about, and videoing of lessons was something out of their comfort zone.

On obtaining written permission from teachers and students, a schedule of videoing was arranged with the class teacher. It was generally agreed that a series of lessons were to be videoed, permitting ongoing development of concepts and procedures to be recorded, and for students to become adjusted to presence of a video camera in the classroom. For each teacher, a series of at least five lessons were recorded with the most being eight. The videoing of a lesson was generally 30 to 45 minutes in length.

Following the recording of a lesson, the lesson was transferred in digital format to an external hard drive. This material was backed up onto a CD ROM to guard against possible loss of materials, and equipments failures. At all times materials were kept in safekeeping, ensuring anonymity of subjects. (During the project, the camera, microphones, and sundry materials were stolen, curtailing the videoing program for some weeks. No data were removed, and the ethical integrity of the project as maintained.)

Data analysis was conducted using editing software on an iMac computer. The research and an assistant worked through lessons for each teacher, coding teacher behaviour and the their behaviours. This was first conducted using the Quality Teaching model (NSW Department of Education and Training), and then the components of effective interventions identified for this project. These data were validated by both observers, and tallied.

Discussion of Results  

The early analyses of data indicate that the teachers who have participated in this study demonstrated a range of skills that are supported by the Quality Teaching model. Teachers across the year levels were shown to actively engage students in a range of activities and tasks, and language used during these interactions was often adjusted to the levels of students.

The concepts of deep understanding and deep knowledge were two important aspects of this project as they linked closely to the mathematical proficiency model posed by Kilpatrick et al. (2001). It was seen across year levels and teachers that these features were used in varying degrees. The teachers in lower grades appeared to reduce the number of concepts they focused on during a lesson, while the teachers in the older grades involved a larger number of concepts. While this appeared typical, the differentiation of this practice across students within a classroom was not apparent. Those students who had difficulty in learning concepts may have benefited through teachers controlling the demands of the task by breaking tasks into smaller units, adjusting language to be clear and consistent, and for a greater range of practice opportunities. Achieving this balance of intervention variables across a classroom is complex, and assisting teachers to develop this complexity of skills is a possible focus for future research projects.

Another observation from early analyses of data showed that teachers engaged with students in a range of ways. One teacher working with a class Year 5 and 6 students, for example, used his charm and humour to engage and motivate students in an activity, and then to move students into a zone just beyond their current level of performance. The activities set were often linked to events within the student’s community, therefore linking closely with productive disposition and adaptive reasoning. Using the components of effective intervention from Swanson et al. (1999), these activities showed evidence of segmenting tasks into smaller units and moving back to the whole again. This observation was, however, not common and did not appear to be differentiated in regard to student need.

The analysis of data is ongoing, and further results will be forthcoming. The emerging data appears to show that teachers in mainstream classroom are keen to manage the educational and social demands that emerge from the diverse student needs. Components of effective intervnetion that need to be further investigated from this study include how teachers promote strategic knowledge (therefore linking into the conceptual understanding strand of the Kilpatrick et al. (2001) model). Other components include controlling the processing demands, engagement, use of supplements to teaching and technology, the use of questioning across lessons and what purpose (e.g., acquisition of knowledge, practice and review).

These issues will be addressed in the emerging data collected from this data, and from schools that will be brought into the project. What will be a greater challenge once a picture of what is happening in classes for students experiencing difficulties learning is established, is how these results can be used to inform ongoing professional knowledge development for teachers. As has been shown in this project, that teachers are generally keen to learn, but are possibly unsure of how to go about.

References

Fuchs, L. & Fuchs, D. (2003). Enhancing mathematical problem-solving of students with mathematics disabilities. In H. Swanson, K. Harris, & S. Graham (Eds.), Handbook of Learning Disabilities. New York: Guilford.

Fuchs, L., Fuchs, D., & Courey, S. (2005). Curriculum-based measurement of mathematics competence: From computation to concepts and applications to real-life problem-solving. Assessment for Effective Intervention, 30, 33-46.

Heward, W. (2003). Ten faulty notions about teaching and learning than hinder the effectiveness of special education. Journal of Special Education, 36, 186-205.

Gersten, R. & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. Journal of Special Education, 30, 18-28.

Jitendra, A. (2005). Mathematics assessment: Introduction to the special issue. Assessment for Effective Intervention, 30, 1-2.

Kuaffman, J., Landrum, T., Mock, D., Sayeski, B., & Sayeski, K. (2005). Diverse knowledge and skills require a diversity of instruction groups: A position statement. Remedial and Special Education, 26, 2-6.

Kilpatrick, J., Swafford, J. & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington: National Research Council.

Swanson, H. (2005). Searching for the best model for instructing students with learning disabilities. In T. Skrtic, K. Harris & J. Shriner (eds.), Special education policy and practice: Accountability, instruction and social changes. Denver, CO: Love.

Swanson, H., Hoskyn, M., & Lee, C. (1999). Interventions for students with learning disabilities: A meta-analysis of treatment outcomes. New York: Guilford.

Specific data are not reported in this version of this paper due to requirements still to be met as part of the ethics applications made with local education authorities. Specific data will be reported at the conference, and an up-dated version of this paper available for those interested.

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