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

Introduction

Motivation to do research

Innovation

•Learning styles (activity styles)

•Social learning processes

•Going beyond numeracy

• understanding abstractions

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.

Classroom Plane place of social and cognitive interaction(collection of interpersonal planes)

Interpersonal plane

Intrapersonal plane

Interaction between inter and intrapersonalplane Internalisation

Interaction between interpersonal planes

Figure 1: interaction in the classroom

Reflexive relationship

Social plane

Social meaning negotiation

Individual plane

Construction of higher mental

concepts

Social meaning is

reconstructed Previously individual

experience is compared

current one

Meaning in the mathematics classroom is

mediated through the use of tools and symbols

Tools

Technical

tools

Psychological

tools

Symbols

Specific to mathematical

activity

Symbols are used to

construct meaning but the

very process fills symbols

with meaning (Sfard 2000)

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

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

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

COGNITIVE

FUNCTION FORMS

LELVELS

LEANINGS

MODES

ORGANISATION

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.

Leaning styles describe the attitude in terms of tolerance and intolerance

towards restructuring processes (high and low tolerance)

Mode styles describe mathematical 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).

SOCIAL ACTIVITY STYLES

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

SCOPE LEANINGS

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

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

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

MAIN FINDINGS

The analysis 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.

• 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.

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

Meaning making

Retrieval of

previous

experience

introduction

of new

meanings

Application

of Rule

or other

abstraction

Justification

of meaning

application

Weak retrieval

Procedural

operational / graphical

terminology

Strong retrieval

Procedural

operational / graphical

terminology

conceptual

Operational/ graphical /terminology

explanation

simplistic

simplistic

complex

reflective

Strong automatisation Strong or weak automatisation

Concentration on operational structure and organisation

Increasing flexibility and complexity of styles

Retrieval of previous experience

Reconstruction processes

intolerance

procedural

operational

simplistic

intolerance

conceptual

Numerical terminology

simplistic complex

reflective impulsive

Increasing complexity as well as sufficient and efficient restructuring processes

RETRIEVAL AND INTRODUCTION OF MATHEMATICAL TOOLS These provide two frequent types of profiles which are summarised as

Retrieval of operation connected to tool

Retrieval / discussion of meaning of tool

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

Lead

s to

Lead

s to

Leads to

procedural conceptual

operational

numerical

impulsive

simplistic

strong retrieval

simplistic

Operational numerical explanation

complex

reflective

impulsive

intolerance

Strong automaticity Weak automaticity

Small styles Wide styles

Application

APPLICATION OF MATHEMATICAL TOOL

Application of operational

structure using key words

Restructuring operational structure of mathematical tool according to context

Application / automaticity of operational structure of

tool

(“you would add them up and divide them … by 2…” (searching for mean)

Discussion of context / automaticity of restructuring

process

(“divide by nine … because there are nine athletes”

(discussing mean in context))

Increasingly complex profile of activity styles

The application of mathematical tools shows two dominant profiles:

resu

lts in

resu

lts in

: “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?”

Procedural , strong retrieval,operational, “you would addthem up and divide them”

Teacher

Divided by what?

Sam

Impulsive,proceduralsimplistic

“2”

Frank

CooperationConceptual or

proceduralsimplistic

numerical “9”

Sam

Reflective,conceptual“oh ja nine”

Teacher

why

Frank

Conceptual, explanationpossibly complex

“because the are nineathletes”

Sam

Tom

Individual motivation“I don’t understand”

Frank

Impulsive,procedural

“a hundred metres”

Michael

Conceptual or procedural, operational, simplistic

“you add up the times forthe hundred meters”

Michael

Conceptual or procedural,simplistic, operational “you

add the two numbers up forthe men and the women”

Teacher

And you would add that upand divide it by nine

Frank

Reflective, co-operation

yes that’s what I said

Michael and Sam

Co-operation

“yes”

Tom

Impulsive, conceptual,operational

“you times I"

Teacher

What would you times

Frank

Conceptual, orprocedural numerical

“The eleven”

Sam

Impulsive,procedural

“by two”

Teacher

By what

Tom

Co-operation,procedural“By two”

Tom

Reflectiveconceptual

“no”

Frank

intolerance“you are confusing

me”

Sam

Procedural,numerical “by

two”

Sam

Frank

Social motivation,Conceptual, simplistic

“the time”

Sam

conceptualexplanation, numerical

“of the five men”

Conceptual, complexoperational, reflective,

numerical“oi, times that by ehm byfive and the other by four”

Michael

Conceptual,operational

“times this by five”

all

Operational,conceptual“and this by

four”

Tom

intolerance“you have to describe the

word mean to me”

The process continuous with the teacher question: “Of what is 10.15 the

average?”

conceptualimpulsive complex

simplistic

explanation

graphical

terminology

wide

styles

small styles

strong automaticity weak automaticity

Justification

JUSTIFICATION OF MATHEMATICAL TOOLS The variation of one particular profile dominates this area.

Discussion of tool within its context

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))

Lead

s to

Leads to

Increasingly complex profile of activity styles

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 profile signal 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 cognitive activity 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.

CLASSROOM activity STYLE PROFILES

INDIVIDUAL activity STYLE PROFILE

Individual activity style profile preferences are apparent as the student first

engages in group discussions as well as shows difficulties in following these.

Initially individual profiles change 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 the task 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.

RELATIONSHIPS BETWEEN 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 more guidance, 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

dominant classroom styles into their individual preferred styles. This allows more

advanced reconstruction processes of 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

interaction influences group and individual activity style profiles and therefore

the type of negotiated .

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 styles and 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)

MATHEMATICAL DEVELOPMENT

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

Indicators for mathematical development are:

Complexity of styles

Activation of styles

Flexibility between style combinations

Compatibility of styles /

combinations

Move from concrete use of tool to

generalisation of tool

Structure and success of processes of restructuring mathematical tools depend on:

Quality of processes of restructuring mathematical tools

Social styles

Influences

Development from focus on operation of tool to concept in which it is set.

Figure 3: style combinations and processes of restructuring

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 making processes are predominantly the

product of classroom discussions. These facilitate mathematical development

for the individual as well as for the group. Although the motivation for social

interaction may differ from student to student, its influence on 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.

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

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.com54 Cardigan Road London E3 5HT02089808270