Inclusive and Supportive Education Congress
International
Special Education Conference
Inclusion: Celebrating Diversity?
1st - 4th August 2005. Glasgow, Scotland
about the conference
programme
registration
accommodation
contact
|
Inclusive and Supportive Education Congress 1st - 4th August 2005. Glasgow, Scotland |
home about the conference programme registration accommodation contact |
Nicole Schnappauf
London, UK
nicoles757@aol.com
This paper discusses the theory and a few preliminary findings of my doctoral thesis on how students with Specific Learning Difficulties (SpLD) negotiate meaning through the mediation with tools and symbols in the secondary classroom. The project itself employs a socio cultural approach to learning. It proposes a theory investigates particular hypothetical activity styles and characteristics of SpLD as predictors of mathematical development in form of increasingly skilful participation in discourse. The abstractions constructed form socially negotiate meaning and their representations in form of tools (or artefacts) during meaning making processes, as well as the application of the abstractions to advance these, form the centre of theory and investigation. It describes SpLD as becoming apparent when meaning is not reconstructed in socially and culturally expected ways. The initial appearance during social interaction, makes learning difficulties a predominantly social rather than individual construct. The study does not focus on learning difficulties perse, but proposes a theoretical framework of teaching which may make these difficulties observable but at the same time may provide possibilities to restrict their influence on learning process.
This theory is based on practitioner research. It is informed through teaching experience as well as theoretical models from educational and cognitive psychology.
This paper presents a small selection of theory and findings from my doctoral thesis on how students with Specific learning difficulties (SpLD), such as Dyslexia and Asperger Syndrome, construct meaning in a discursive mathematics classroom setting. It investigates the use of hypothetical activity styles, which are assumed observable during classroom discourse and reflected in meanings reconstructed during social interaction. Furthermore, it theorises on the influence of difficulties associated with working memory and communication, on this type of discourse and meaning.
The research data was collected in my own GCSE mathematics classroom over a period of three years. The thesis is based on a period of two years and concerns meaning making processes of six students. The data is of qualitative nature and predominantly comprises transcripts of sequences of classroom interaction. Additional material such as interviews and questionnaires with students and teachers, as well as CAT and WISC assessments are used as further information, regarding learning preferences and the nature of each student’s specific learning difficulty. Today, however I will predominantly focus on the theory of this project and some findings, rather than the data itself. Whilst writing this paper, I was in the middle of my data analysis. Towards the end of this presentation, I will discuss some preliminary findings. I developed the theoretical framework before and during the data collection process. The analysis of the data provided further refinements to this theoretical framework.
1.1 Research question
My overall research question states:
Can hypothetical activity styles and characteristics of specific processing difficulties, function as predictors for the level of skilful discourse participation in mathematics classrooms?
It concerns:
1.2 Current research and teaching within SpLD
Research within SpLD and Mathematics focuses predominantly on numeracy and difficulties observed within. This project however, assumes that these problems are firstly not restricted to numeracy only, but may affect other areas of learning mathematics. Secondly and more importantly, there is a rather scattered amount of what is generally known on how meaning making experiences are abstracted, reapplied and further reconstructed within secondary mathematics education. Further exploration in general and special education is needed. I am hoping to shed some more light on the construction and application of abstractions during social interaction in the mathematics classroom.
This study includes students with varying strength and difficulties in mathematics. This project does not focus on the difficulties, which may arise but rather tries to develop a theory to situate and possibly explain factors, which influence mathematical learning processes differently to social and cultural expectancy.
The approach taken to teaching in this project differs to most approaches within “remedial” numeracy and mathematics teaching. Whilst most teaching in SpLD focuses on numeracy and ways or methods, “which work for an individual”, the teaching in this project proposes a method, which requires students to construct meaning and explore it further, under the supervision of a teacher.
In other words, whilst traditional teaching focuses on a set of ways to learn, this approach investigates learning, whilst constructing meaning and encourages each student to be actively involved in the way meaning is explored and reconstructed in the classroom. Mathematical learning and development is therefore seen in two ways. It does not solely or even predominantly focus, on the mastery of a given task but on successful and task sufficient reconstruction processes and therefore an increasing skilful discourse participation (Sfard 2002) . The latter promotes increasing independence of the group of students from teacher guidance as well as for each individual student from the group.
There is some evidence that middle to high ability student do better in co-operative learning situation and less able students in rather instructive settings. In my experience however, less able students take longer to adjust to a change in teaching method and respond well to a mixture of discourse and summarising guidance.
This project is using a sociocultural approach to teaching and research. It is based on Vygotsky’s (1987) idea that learning takes place on two planes, firstly the social or interpersonal one and secondly the individual or intra personal plane. Meaning is at first constructed during social interaction and internalised through individual reconstruction of social meaning. The relationship between both processes is reflexive. During participation, social meaning is challenged, due to previous individual experiences, social interaction in turn challenges individual concepts. Vygotsky (1987) proposed further, that learning is mediated through the use of psychological and technical tools.
Tools, in particular psychological tools, are at the centre of this project. Within theories of sociocultural approach, tools are either described as somewhat auxiliary means to negotiate and construct meanings, which are external or internal. External tools construct meaning but become redundant once meaning is constructed. An internal interpretation of tools, however makes these “part and parcel” of the process (Sfard and McClain 2002) . This means meaning constructed is inseparable from the tools used and the tools become an integral part of the meaning; they cannot be made redundant. I am assuming the latter description of tools.
Within socio-cultural theory, social interaction in the mathematics classroom using tools to negotiate meaning and to fill mathematical symbols is often referred to as communicating, symbolising and or discourse.
Before I dig deeper into the theory, I would like to provide a short description of tools, objects and socio-norms in the classroom. There are central to social interaction and the meanings reconstructed during this process.
- gestures, such as the indication of a shape or symbol or part pf the shape Spoken language and gestures to communicate in the classroom provide in this project the main tools to negotiate meaning.
3.2 Symbolising
The use of language, as a tool, fills mathematical symbols with meaning and in return, the use of mathematical symbols constructs meaning, which is attached to the mathematical symbol. This creates a circular relationship between meaning constructed and the use of symbols (Sfard 2002) . Mathematical symbols become objects and tools of mathematical meaning making processes. Language, as a tool shows insides into both social meanings as well as individual concepts, and reconstructs these further.
Similarly, to Vygotsky (1987), I am assuming that it is not simply the use and application of language in general, which promotes development but the particular language, which is used to do so. This means the particular use of language to negotiate meaning may provide further insights into the processes of meaning construction and application, or in other words in learning processes, as well as the structure of meaning itself. I will refer to the underlying structure of language and of course gestures, as activity styles.
3.3 Activity styles
Literature often discusses learning styles, in particularly but not exclusively cognitive ones, as indicators of individual ways of learning. Within SpLD education, learning style assessments, mainly inform of static self-report questionnaires, play an increasingly important role for assessments and IEPs. Dynamic learning style assessments are applied as and when the students are engaging in learning processes such as classroom interaction. Their application is rare and centres on the learning processes or the interaction of the individual with the learning environment.
I am using the term activity styles to emphasis the importance of social interaction and to describe classroom activity as an inseparable interference between individual and social processes. In other words I am exploring how the individual engages into social interaction and how social interaction shapes individual engagement in meaning making processes.
These activity styles may indicate individual or social preferences, but are shaped through previous and continuous social interaction. Clusters of words, which carry reconstruction processes of cultural meaning as well as personal and social meaning reflect activity styles. This means the structure of tools, in particular that of language, may provide insights in the use of activity styles during social interaction. The types of activity styles employed may vary between individuals due to previous individual experiences as well as to social norms that have been established over time in the classroom to do so.
For the purpose of this paper, I am using the following differentiations of activity styles:
(My study uses a wider range but has identified these as one of the most influential ones.)
- application of a rule or fragment relating to previous experience,
Each particular structure of language suggests a particular combination of activity styles as well as representation of a meaning, which is interpretable by the researcher. The application of activity styles may therefore provide insights in the learning processes individual students as well as the groups, based on the reconstruction of social meaning making processes through the researcher.
3.4 Abstractions
Meaning negotiated during the comparison and reconstruction of a number of similar experiences leads to generalisations or abstractions of a number of actual experiences. In return, the use of tools, to negotiate meaning reflects so called representations of abstractions or in other words possible structures o mental abstractions. Abstractions are unlikely to become directly visible but the appearance of representations to reconstruct and apply meaning carry reflections of abstractions. I am hoping to gain greater insight into the structure and nature of abstractions, by investigating representations as reflections of abstractions through activity styles. Abstractions are internal tools and belong to the circular process of the construction of meaning of mathematical symbols.
3.4.1 Abstractions, representations and activity styles
I am assuming that initial abstractions are closely related to particular tasks. Over time these may become more generalised due to the influence of different experiences. However, this does not mean that the abstraction become decontextualised (Van Oers 2004) . In contrary I am presuming that these abstraction carry reconstruction processes as parcels with them. These parcels can be unpacked as required. Proposing that the use of activity styles provides structural insights into meaning making processes may imply further that activity styles influence nature and structure of abstractions as well as releasing insights of these. The type of abstractions constructed may depend on the activity styles applied during meaning making processes.
The application of these abstractions may carry the particular use of language or use of tools according to the representation itself.
Increasingly skilful (in terms of successful and sufficient) participation in learning processes may advance abstractions and their numbers as well as provide connections between these. Each abstraction consists of a number of representations, such as a visual or operational discussion of mathematical symbols.
For example, the median may be presented in different modes of activity styles. A numerical style would focus on finding the middle number of a given set of numbers, in a visual way such as in comparing box-and –whiskers diagram. The identification of the median in cumulative frequency graph may than require both the use of a numerical style in finding the middle of the cumulative frequency as well as drawing the line to identify the median line, indicating a visual style. Furthermore, a conceptual representation would link the activity of finding the median to the context of the task and its setting whilst a procedural approach would use a memorised rule, without contextual link or justification.
3.4.2 Abstraction spiral
I am assuming the formation of abstraction spirals, which take care of the increasing number of representations for each abstraction. A spiral grows depending on the structure and packaging of abstractions. The connections between abstractions on one or more spirals may differ, due to the type of representations and packages attached to each abstraction. This leads to the proposal of two broad types of abstraction spirals.
The first type of abstractions or generalisations to construct abstractions takes care of meaning negotiated during a number of similar experiences. These are consequently organised in a manner that they can be restructured for a wide range of applications.
For example, the abstractions for the areas for quadrilaterals can be organised in a manner that they relate to the area of a basic rectangle. This means that the area of every regular quadrilateral can be reconstructed using the initial rule, provided the students are given appropriate guidance, tools and mathematical symbols, such as drawings which provide appropriate connections between representations and advance these.
On the other hand, students may use an abstraction of area of a rectangle during the initial reconstruction process but may not make connections between representations of their abstraction explicit. Overall, their abstractions related to area will than form a number of these existing next to each other, rather than forming a bigger spiral.
The abstractions may than be organised as loosely connected or even a number of smaller spirals next to each other with little connection between.
3.4.3 Connections between representations and abstractions
So far, there is little research on the nature or structure of these abstractions. For the purpose of the theoretical framework, I am proposing three types of connections, which may be apparent at any time during social meaning making processes.
The representations may be connected through
Abstractions, which become applicable for a wide range of experiences, are within this theory thought to most likely to indicate a greater number of explicit connections between representations. Abstractions with less explicit but a greater number of coexisting connections between representations are assumed to be smaller and more of these.
Conflicting representations may lead to further reconstructions or indicate open conflict between students or and individual and social interaction.
Going back to the use of tools, this means that the quality of tools used may next to other factors such as previous learning experience and therefore established preferences, affect the type of connections between representations and therefore structure and type of abstractions.
Smaller connections between representations as well as abstractions may lead to a number of coexisting abstractions with restricted application. There is little doubt that students can successfully complete the mathematics school curriculum by learning each required rule. However, it restricts school mathematics to the classroom.
Abstractions, on the other hand, constructed with predominantly explicit connections between them may enable more skilful participation in discourse, as they are proposed to be applicable for a much wider range of mathematical problems. This may lead to a much greater application of school mathematics outside the classroom.
So now what role do characteristics associated with Specific Learning Difficulties play during meaning making processes and the abstractions which describe meanings?
4.1 Interpretation of learning difficulties within a sociocultural approach
Starting from a sociocultural approach means, that learning difficulties become observable, as learning processes differ to what is socially and culturally expected. These difficulties are social in their nature as they become at first apparent during social interaction and secondly as the individual tries to reconstruct social and individual meaning to construct individual concepts. This could be interpreted as if there are difficulties, which restrict the negotiation and reconstruction of meaning in socially and culturally established and expected ways the possibility of becoming a skilful participant in discourse may be reduced.
I am assuming further, that each culture has preferred ways to engage in, pursue and process social interaction and these are reflected in the way meaning is negotiated, reconstructed and further applied. Ways of processing social constructs require individual mental functions, which may be disrupted due to neurological differences.
These alternations to socially expected norms affect how meanings are processed, stored and the availability of mental space to reconstruct. In other words mental operation such as auditory and visual processing and working (or short)-term-memory functions show difficulties. Within communication difficulties, this means reconstructing the meaning, which is associated with a number of socially and culturally established meanings, is insufficient or at least provides difficulties for one individual within a particular social setting.
For the purpose of this paper, I will focus on working memory only.
Research on working memory focuses on psychological and neurological theories how the individual processes and stores information on the basis of socially and culturally expected norms. Research on individuals with ‘dyslexia’ has provided enormous amounts of insights on neurological and psychological processing and storage of information. Interpretations of findings propose growing numbers of factors, which prevent ‘normal functioning’ in form of socially and culturally expected one.
This is not the place to dig deeper into research, but will relate the idea of automaticity, which is one of the current theories to understand and explain processing and memory difficulties, associated with dyslexia in language processing and more recently with learning number facts (e.g. Dowker 2004) . However, this project does not focus on difficulties predominantly but rather how they may obstruct culturally and socially expected ways of communicating in the classroom.
4.2 Automaticity
Theories on automaticity assume that students can automatically recall learnt number facts, provided they had been exposed to sufficient learning processes to do so (Dowker 2004) . This means in case of number facts a student can recall a required fact such as 4 x 8 without further effort. The process to store what often called knowledge, in a manner that it can be retrieved automatically without further effort is referred to as automaticity (Grigorenko 2001) . Students with specific learning difficulties may show difficulties in automatic retrieval and storage. Most research focuses on basic number facts. I am suggesting that difficulties in the construction of basic mathematical meanings and their abstractions may influence the construction of more complex abstractions in similar if not in ways that are even more comprehensive too. These require for example the retrieval of previously constructed meanings of a mathematical symbol.
Research in this area within SpLD is very scarcely scattered, an often focuses on observations on teaching methods and the mastery of given tasks rather than how students engage in social learning processes as well as the concepts constructed during these.
Transferring the idea of automaticity into a sociocultural theory would imply that the process of reconstruction becomes almost effortless. Participants can apply abstractions and release important package to restructure a new task and solve it appropriately. In other words, students engage into a meaningful and skilful discourse, releasing previous experiences with little effort. A current experience may or may not need effort to be compared to a previous experience. A reconstruction process, which requires the actual reconstruction of previous experiences, such as the reconstruction of a of the rule of the trapezium from scratch repeatedly, may be described as requiring effort.
Translating these back into the automaticity theories, means that a person cannot automatically release the required rule. This may result in either an incomplete or an inappropriate solution, a reconstruction of the situation if possible or simply a giving up.
This theory of course needs further development and I am hoping that the findings of the project will help to construct further hypotheses investigating the nature and connection of abstractions further.
My early research findings indicate that students with a preference to learn rules by heart tend to give up, as they cannot retrieve a previous experience. Those students however, who previously engaged into a related meaning making process show tendencies to reconstruct these previous experience in interaction with other participants and more often than not reach sufficient solution, even if they prefer a rule to a rule in context.
4.3 Working memory
Reconstruction of previous experiences and or previous and current experiences requires working memory to do so. Limitations of the working memory may influence the type of abstractions constructed or in other words may be observable in the types of connection between representations of abstraction, as well as the number of representations involved. Similar may be true for retrieval which requires effort. Almost effortless retrieval may require different forms of memory such as long-term memory and whilst comparing a previous experience to a current experience working memory will be required.
The need of extensive working memory during initial periods of meaning negotiations particular those, which involve more complex processes, may indicate a breakdown in working memory for students with what is often referred to as restricted memory space. The number of simultaneous processes required at any one time may result in an overload. An initial overload may lead to a social as well as individual preference to make on fewer explicit connections between representations but a greater number of coexisting ones.
I am proposing that once basic abstractions are established and explicit connections between representations have been made less memory space to reconstruct new experiences with similar previous ones will be required. This means that if particular types of learning processes are accessible, which enable the student to make enough explicit connections between representations as well as abstractions for more complex mathematical meanings, students can become more skilful discourse participants. This again emphasises on suitability and the nature of tools applied to construct meaning in the classroom.
Primary findings of research
Primary findings of research indicate that:
Making students skilful discourse participants (Sfard 2002) rather than just emphasising on the sufficient mastery of predictable classroom tasks is at the heart of this research project and my teaching in general. It requires teaching methods and classroom interaction, which promotes for the group as well as each individual
It requires appropriate language use and socio norms to do so.
These findings are preliminary findings from a small selection of sequences, however they already indicate support and refinement of the theory discussed above. I am hoping to develop the theory on abstractions and their representations further as well as formulate hypotheses on the influence of working memory and communication difficulties on abstraction processes.
This theory may challenge current practice in SpLD education but at the same time provide solutions for inclusion to promote learning which is not restricted to the classroom or the remedial of difficulties but as a method to make students aware of their difficulties and identify ways together to help overcome and minimise these, without excluding students from important activities.
Finally, I have to thank Catherine Allen as the Head of the Home School for trusting my teaching and researching and providing all the freedom and support to letting me experiment. Furthermore, I am grateful to all the students who I taught over a period of eight years in Mathematics and Science. They all put up with “all the talking about maths”, uncountable why?s, questionnaires and interviews on what they think is learning or knowledge. Little comments such as “she (the teacher) doesn’t tell us anything, we have to b….. work it (mathematical meanings) all out ourselves!” and successful GCSE grades, made it all worth it. A final thank-you goes to all my fellow colleagues who engaged in endless discussions on learning and who on occasion called me mad to use discourse with students who have great difficulties in talking.
Cobb, P. (2000). From Representations to Symbolising: Introductiory Comments on Semiotics and Mathematical Learing. Symbolizing and Communicating in Mathematics Classrooms; Perspectives on Discourse, Tools and Instructional Design. P. Cobb, E. Yackel and M. Kay. Mahwah, New Jersey; London, Lawrence Erlbaum Associates.
Dowker, A. (2004). What works for Children with Mathematical Difficulties? DfES publications / University of Oxford: 1-55.
Grigorenko, E. L. (2001). "Developmental Dyslexia: An Update on Genes, Brains, and Environments." Journal of Child Psychology and Psychiatry42(1): 91-125.
Sfard, A. (2002). "The Interplay of Intimations and Implementations: Generating New Discourse With New Symbolic Tools." The Journal of the Learning Sciences 11(2&3): 319-357.
Sfard, A. and K. McClain (2002). "Guest Editor's Introduction: Analyzing Tools: Perspectives on the Role of Designed Artifacts in Mathematics Learning." Journal of the Learning Sciences 11 (2-3): 153-161.
Van Oers, B. (2004). The Contextualisation of Inscriptions: an Activity-Theoretical Approach to the Transferabilitiy of Abstractions. Rethinking Abstraction and Decontextualisation n the Relationship to the "Transfer Dilemma" AERA, San Diego.
Vygostky, L. S. (1987). Mind in Society: The Development of Higher Psychological Processes. Cambridge, Harvard University Press.
home . about the conference . programme . registration . accommodation . contact
 |
 |
 |
 |
 |
 |