Introduction

1. Mathematical Learning Processes of Secondary School Students with Dyslexia and Related Difficulties Nicole Schnappauf Maths, Physics and SEN teacher and consultant EdD student Kings’ College London nicschnap@aol.com

2. Introduction Motivation to do research • Innovation • Learning styles (activity styles) • Social learning processes • Going beyond numeracy • understanding abstractions

3. RESEARCH QUESTION What is the relationship between group processes and mathematical activity styles when students with specific learning difficulties engage in mathematical discussion? The first aim was to explore possible dynamic group activity style profiles during class discussions. The second aim was to explore individual dynamic activity style profiles apparent in during social interaction. The third aim was to explore the relationship between activity style profiles,mathematical activity and development of the group as well as the individual.

4. Classroom Plane place of social and cognitive interaction (collection of interpersonal planes) Interaction between inter and intrapersonal plane Internalisation Interpersonal plane Interaction between interpersonal planes Intrapersonal plane Figure 1: interaction in the classroom

5. Reflexive relationship Social plane Social meaning negotiation Social meaning is reconstructed Previously individual experience is compared current one Individual plane Construction of higher mental concepts

6. Meaning in the mathematics classroom is mediated through the use of tools and symbols Symbols Tools Specific to mathematical activity Symbols are used to construct meaning but the very process fills symbols with meaning (Sfard 2000) Technical tools Psychological tools

7. Psychological tools “language, various systems for counting, mnemonic techniques; algebraic symbol systems; works of art, … all sorts of conventional signs” (Vygotsky, 1981, pp136-7).   May at first be external auxiliary means but become part and parcel of meaning negotiated and cannot be separated from meaning Language, gestures and social norms play the most important role in my project Language is differentiated into General mathematical classroom language

8. Technical tools e.g. protractors, calculators are external auxiliary means but may carry psychological tools with them such as concept on angles Mathematical symbols a dot or line on a piece of paper, an algebraic expression, a graph, a drawing, etc. Mathematical symbols signify mathematical concepts and at the same time act as tools to negotiate meaning

9. activity styles • Based on Sternberg et al (1997) ‘self government of mind’ • Dynamic activity styles • Social and neurological origin but altered through social interaction • Profiles of styles are not hierarchical but task appropriate or individual preference • Assessment during participation in meaning making processes • Describe the social interaction between participants and the individual within the social to negotiate meaning

10. FUNCTION FORMS LEANINGS ORGANISATION COGNITIVE LELVELS MODES

11. COGNITIVE ACTIVITY STYLES Function styles describe the interaction with a mathematical tool regarding the context and relation to previous experience they are set (conceptual) in as well as their operational structure (procedural). Form styles describe the complexity (complex) and simplicity (simplistic) of restructuring process as well as the impulsivity and reflectivity of the participants during the learning process. Level styles describe the preference to exploring the context of a task or tool is set in or simply the scanning for key words.

12. Leaning styles describe the attitude in terms of tolerance and intolerance towards restructuring processes (high and low tolerance) Mode styles describemathematical modalities,interpretations and organisations of mathematical tools and their applications (operational, explanation, terminology, numerical, visual). Organisation of meaning describes the nature of retrieval and generalisation process, either regarding the context and meaning of a tool (weak retrieval or weak automaticity) or regarding the structure of the tool (strong retrieval or strong automaticity).

13. SOCIAL ACTIVITY STYLES SCOPE LEANINGS Scope of styles describe social interaction between all participants in terms of cooperation and competition (cooperation / competition) Leanings of styles describe the level of tolerance towards social interaction during restructuring process

14. How do Specific Learning Difficulties fit into all of this? • reconstruction of mathematical meaning or mathematizing is a cultural activity = each culture has preferred activity styles to do so • Hence, difficulties in engaging into culturally and historically established activities means that the individual doesn’t function in a culturally expected ways • I am assuming these differences to be caused by mental functioning, which is different to culturally expected. • Specific Learning Difficulties = difficulties in engaging into culturally established and expected ways of meaning making

15. My current study proposes these to be mental functions, which each individual uses the engage, process and store current and past experiences from social and individual activities. • Working memory Baddeleys and Loggies (1998)definition as “the moment-to-moment monitoring, processing and maintenance of information” or in this case negotiation of meaning • Automaticity – as the effortless retrieval and reconstruction of previous experiences, and their application on current experiences such as the area of a rectangle to calculate the area of a triangle • Communication – as the understanding of meanings of words, changes of meanings in different contexts as well as the reading of social interaction

16. METHOD Teacher research: • researching own GCSE classroom • Open ended instructional approach Lesson content: • GCSE mathematics / statistics for higher and intermediate tier Data collection: • Transcripts of classroom interaction over 36 mathematics lessons • Student Questionnaires on learning and understanding mathematics • Staff Questionnaires on each students learning and understanding • Students individual work during class discussions Analysis: • Qualitative data analysis using narrative analysis Setting: • Independent secondary school for students with SpLD in London

17. MAIN FINDINGS Theanalysis suggests • that hypothetical activity styles are important but not sole facilitators of processes negotiating meaning through tools and symbols and bringing meaning to symbols. • That their increasingly complex organisation into task appropriate profiles makes them indicators of skilful participation in mathematics classrooms and therefore indicators of mathematical development.

18. Existence of individual dynamic activity style profile within group processes as well as a dynamic group activity style profile • Availability of particular styles on either plane influences or even predicts the processes of restructuring mathematical tools. • Choice and combination of activity styles influence the sufficiency and effectiveness of restructuring processes of mathematical tools. • Changes in style profiles suggest mathematical development.

19. Meaning Making I divided for the purpose of this study meaning negotiation processes with tools and symbols and the filling of symbols with meaning into three areas of mathematical activity. These are • Retrieval of previous experiences and the introduction of new symbols and meanings • The application of rules and abstractions made during previous experiences • The justification and explanation of meaning negotiated as well as the application of rules and abstractions

20. Meaning making Retrieval of previous experience introduction of new meanings Application of Rule or other abstraction Justification of meaning application

21. Retrieval of previous experience

22. Reconstruction processes

23. RETRIEVAL AND INTRODUCTION OF MATHEMATICAL TOOLS Retrieval of operation connected to tool Retrieval / discussion of meaning of tool These provide two frequent types of profiles which are summarised as Leads to Leads to Leads to Automatised knowledge (“you add them all up and divide them by how many there is” (discussing mean) Taken-as-shared knowledge beyond further justification Restructuring of mathematical tool using key words (“it is the middle number” (retrieval of median)) Restructuring of mathematical tool in context (“ ..it will be steeper … because there will be more less tall ones” (discussing cumulative frequency)) Increasingly complex activity style profile

24. strong retrieval impulsive simplistic impulsive procedural simplistic conceptual complex reflective Operational numerical operational numerical explanation intolerance Weak automaticity Strong automaticity Small styles Wide styles Application

25. APPLICATION OF MATHEMATICAL TOOL The application of mathematical tools shows two dominant profiles: Application of operational structure using key words Restructuring operational structure of mathematical toolaccording to context results in results in Discussion of context / automaticity of restructuringprocess (“divide by nine … because there are nine athletes” (discussing mean in context)) Application / automaticity of operational structure of tool (“you would add them up and divide them … by 2…” (searching for mean) Increasingly complex profile of activity styles

26. : “5 male athletes have an average time of 10.15 seconds for running 100 metres. Four women have an average time of 11.25 seconds for running 100 metres. What is the average time taken by the nine athletes for running 100 metres?”

30. The process continuous with the teacher question: “Of what is 10.15 the average?”

31. impulsive conceptual complex simplistic wide styles explanation graphical terminology small styles strong automaticity weak automaticity Justification

32. JUSTIFICATION OF MATHEMATICAL TOOLS The variation of one particular profile dominates this area. Discussion of tool within its context Leads to Leads to Tool in particular context (“because it tells me the most often one ... the train comes” (justification of use of mean)) Generalisation of tool (“ weight is more spaced out …. That means you get more different ones..” (discussion of cumulative frequency graph)) Increasingly complex profile of activity styles

33. CLASSROOM activity STYLE PROFILES The classroom analysis indicates a rich variety of activity style profiles, which at times seems to be an entity in itself. These enable students with a range of individual style preferences to engage in restructuring processes at different times and at varying levels. Changes in at least parts of the profilesignal changes in the organisation of discussion and meaning making processes of mathematical tools. Hence this indicates hierarchical or at least more appropriate activity styles profiles for different and increasingly demanding restructuring processes. Some social activity styles may alter and expand cognitiveactivity style profiles of the group and as a result advance the reconstruction of mathematical knowledge of all participants. Other social activity styles are indicators of the incompatibility of individual and group profiles.

34. INDIVIDUAL activity STYLE PROFILE Individual activity style profile preferences are apparent as the student firstengagesin group discussions as well as shows difficulties in following these. Initially individual profileschange during interaction processes and appropriate classroom styles. Over time initial classroom preferences can become individual preferences. Individual style profiles may consist of either individual style preferences or a selection of styles, which are appropriate to context of thetask in discussion. The styles used are refined during discussion. Individual motivation to engage in classroom processes differs. Classroom processes provide a platform to engage with meaning making processes as well as to confirm individual concepts. The latter often dominates the structure of the classroom profile.

35. RELATIONSHIPSBETWEEN PROFILES The analysis shows individual differences within the classroom plane. Greater and more complex style combinations facilitate the classroom plane as a forum of discussion, while restricted style combinations require moreguidance, instruction as well as discussion. Students who combine styles available on the classroom plane into complex style combinations indicate successful restructuring processes and may even influence those of others. Over time students may internalise dominantclassroom styles into their individual preferred styles. This allows more advanced reconstruction processesof meaning for the group and the individual. Changes in profiles, which advance meaning making processes and the content of mathematical tools, describe mathematical development. The nature of social interactioninfluences group and individual activity style profiles and therefore the type of negotiated .

36. Meaning making processes and their success within all three areas depend on Combination of styles (some combinations are more successful than others) Importance or appropriateness of styles (some styles are more adequate than others in particular areas of mathematical activity) Compatibility between stylesand their combinations (on the classroom plane as well as during the internalisation process of the individual) Flexibility between styles combinations (for some more advanced tasks it is preferable to change between styles and profiles during the task)

37. Individual Increasing independence from the group processes Internalising of style combinations of group processes to individual processes Increasing success in restructuring and generalising group Increasing independence from teacher intervention Increasing flexibility of movement between styles Increasing compatibility between individual students’ initial style combinations and those of their peers MATHEMATICAL DEVELOPMENT Indicators for mathematical development are:

38. Structure and success of processes of restructuring mathematical tools depend on: Social styles Compatibility of styles / combinations Complexity of styles Activation of styles Flexibility between style combinations Influences Quality of processes of restructuring mathematical tools Development from focus on operation of tool to concept in which it is set. Move from concrete use of tool to generalisation of tool Figure 3: style combinations and processes of restructuring

39. IMPLICATIONS FOR TEACHING The findings suggest for teaching: The group discussions provide a vital place for the reconstruction of mathematical tools for the group as well as the individual. The identification of activity style profiles provides a dynamic tool to investigate the understanding, development and achievement of a student or a group of students. Social interaction provides a platform for a student to discuss his or her understanding as well as engage in discussions, which he or she could not have done by him or herself. Furthermore, conceptual meaning makingprocesses are predominantly the product of classroom discussions. These facilitatemathematical development for the individual as well as for the group. Although the motivation for social interaction may differ from student to student, its influenceon individual and group style profiles are significant. However the analysis implies that not all group processes are accessible for all students at all times; Some students need support to follow class discussions.

40. PRESENT RESEARCH At present I am exploring: • a possible relationship between different types of processing difficulties and dynamic hypothetical activity styles. • a possible relationship between activity styles, processing difficulty and mathematical development • The of abstractions students made and how these are applied • The data collection took place in my own classroom with year 11 • It focuses on Space Measure and Shape

41. REFERENCES Vygotsky L.S. (1987). The collected works of L.S. Vygotsky. Vol.1: Problems of general psychology. Including the volume Thinking and speech. (R.W. Rieber & A.S. Carton, Eds., N. Minick, Trans.). NY: Plenum Press. Sternberg, R.J. (1997) Thinking Styles, Cambridge: Cambridge University Press RESEACHER DETAILS Nicole Schnappauf Head of Mathematics and Science Home School of Stoke Newington Educational Doctorate student at King’s College London (final year) Contact details: nicschnap@aol.com 54 Cardigan Road London E3 5HT 02089808270