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Transcript of Mathematical Learning Processes of Secondary School Students with Dyslexia and Related Difficulties...
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
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 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:[email protected] Cardigan Road London E3 5HT02089808270