Research about technology in mathematics education: an evolution

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Research about technology in mathematics
education an evolution

• Jean-baptiste Lagrange
• Equipe de didactique des mathématiques,
  université Paris 7

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These themes are not independant. They differ by
a specific entry into our general questionning,
but they are convergent. We adopt different
theoretical orientations (TSD, TA, activity
theory ). Our specificity is to cross these
approaches and to analyse the implications of
theoretical choices.
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An example

• Vandebrouk (1999)Lutilisation du tableau noir
  par des enseignants de mathématiques
• Cazes Vandebrouk (2006)An Emergent Inquiring
  Field the Introduction in the Classroom of
  Online Exercises Set

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A lecture about research in technologies.For
what purpose ?

• The role of artefacts in human
• Knowledge
• Cognition
• Social activity
• An Artefact
• Is a product of art or industry,
• Expresses a fundamental property of a living
  being to have a project and to inscribe this
  project in a production. (Monod)

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Artefacts influence conceptualisation

• Multiplication is not an operation
• It does not commute

Artefacts
Task The table and the calculator are two
artefacts that can play specific roles in the
conceptualisation of multiplication. - as an
operation, as a commutative operation.
Please specify these roles.
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T1 the role of artefacts in conceptualisation
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Reification

•  Many theoretical and empirical arguments may be
  employed to show that in mathematics, operational
  conception precedes the structural. What is
  conceived as a process at one level becomes an
  object at a higher level. 
• Sfard and Linchevski
• The gains and the pitfalls of reification

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Programming

• Important feature of computer technology
• Specific languages proposed as means to
  manipulate mathematical entities.
• A central assumption programming
• Helps learners to reflect on actions
• Favours conceptualisation (reification).

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ISETL and APOS

• A programming language associated with a specific
  theory
• An individual's mathematical knowledge is her or
  his tendency to respond to mathematical problem
  situations by
• reflecting on them
• constructing or reconstructing mathematical
  actions processes and objects
• organizing these in schemas to use in dealing
  with the situations"

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APOS
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Procepts (Tall)

• An elementary procept is the amalgam of
• a process,
• a related concept produced by that process
• a symbol which represents both the process and
  the concept.
• A procept consists of a collection of elementary
  procepts which have the same object.

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A more flexible approach

• The process involved must not first be given and
  encapsulated before any understanding of the
  concept can be derived.
• In introducing the notion of solving a
  differential equation, I have designed software
  to show a small line whose gradient is defined by
  the equation, encouraging the learner
• to stick the pieces end to end
• to construct a visual solution through
  sensori-motor activity.

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• This builds an embodied notion of the existence
  of a unique solution through every point,
• It provides a skeletal cognitive schema for the
  solution process before it need be filled out
  with the specific methods of constructing
  solutions.
• It uses the available power of the brain to
  construct the whole theory at a schema level
  rather than follow through a rigid sequence of
  strictly mathematical action-process-object.

• http//www.Bibmath.Net/dico/index.Php3?Actionaffi
  chequoi./C/champ.Html

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Theories about visualization

• To take advantage of the multiple representations
  of mathematical entities allowed by computer

• To favor more flexible approaches to
  conceptualization

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The idea of micro-world

• A more or less virtual space for learners
• freely conceptualise by considering questions and
  constructing solutions.
• Powerful enough as to evolve
• from the first vision linked to turtle geometry
  (Papert 1980)
• to recent projects like Mathlab (Noss Hoyles
  2006), based on the idea of building new
  representations.

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Papert

• constructionism shares constructivism's
  connotation of learning as "building knowledge
  structures" (and)
• then adds the idea that this happens especially
  effectively when learners are engaged in
  construction for a public audience".

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Weblabs
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Weblabs
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Guess my robot

• Nasko posted his response. He had built a robot
  that produced Rita's five terms, So, he posed a
  two-part challenge back at Rita
• Could she use his robot to generate a new
  sequence of five terms?
• Could she use her robot to generate the same
  sequence?
• Rita was totally surprised Nasko and Ivan had
  solved her challenge, but their robots seemed
  completely different from hers.
• She worked out what inputs Nasko must have given
  his robot, and showed that her robot could in
  fact generate the same output as his.
• She has made a new robot that subtracted one
  stream of outputs from the other and had watched
  the robots create a stream of zeros. She had
  generated thousands of zeros in this way and was
  convinced that this was a 'proof' of her
  conjecture that the sequences were the same.

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Situated cognition

• Because computer objects and representations
  generally differ from usual mathematics,
• Math Educators
• became aware that conceptualisation always
  depends on situations
• questioned the notion of abstraction (Noss
  Hoyles 1996), introducing the idea of connection.
  

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Computer symbolic systems

• Raised a lot of attention,
• Assumption Quick and easy actions in problem
  should
• dramatically reduce the part of meaningless
  technical manipulation
• favor conceptualization

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The spreadsheet

• Specific notation to express relationship between
  entities
• Dynamic execution
• Great potential for
• Introducing younger students to algebra,
• Preparing them to notions like variables,
  equations and functions.

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Difficulties when implementing tools in the
classroom.

• To benefit of the tools potential, a learner
  needs knowledge intertwining
• mathematical understanding and
• awareness about the tools functioning.
• Acquiring this knowledge is a non-obvious and
  time-consuming process, instrumental genesis.

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An example framing the graphic window.
Consider the function Use the graphic calculator
to obtain an accurate representation, Make
conjectures on its properties, Test and prove
these conjectures
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Instrumentation

• Distinction between tool, artefact, instrument
• Instrumental genesis (Rabardel)
• Interwoven mathematical and instrumental genesis

A human being
An artifact
Her/his knowledge
Its constraints
Her/his work method
Its potentialities
Instrumentation
Instrumentalization
An instrument
Part of the artifact schemes
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The anthropological approach

• Concepts first, then skills ??
• If mathematics instruction were to concentrate
  on meaning and concepts first, that initial
  learning would be processed deeply and remembered
  well. A stable cognitive structure could be
  formed on which later skill development could
  build. (Heid 1988, p. 4).

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Techniques and concepts

• Not so simple relationship
• Suppressing paper-pencil techniques
• also suppresses the possibility of reflection on
  these, useful for conceptualisation,
• brings difficulties related to teachers systems
  of values.

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Ruthven (2002)

• In the experimental classes, constitution of a
  quite different system of techniques
• The shift to reasoning in non algebraic modes of
  representation which characterized concept
  development in the experimental classes (p. 10)
• created new types of task,
• encouraged systematic attention to corresponding
  techniques
• The experimental course
• exposed students to () wider techniques
• helped them to develop proficiency in what had
  become standard tasks,
• even if they were not officially recognized as
  such, and had not been framed so algorithmically,
  taught so directly, or rehearsed so explicitly as
  those deferred to the final skill phase.

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Techniques

• a manner of solving a type of task in an
  institution
• a complex assembly of reasoning and routine.
• a pragmatic value
• an epistemic value

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New challenges to mathematics education

• Today fast developing web based technologies
• Internet based communication and social
  interaction.
• Self learning
• Learning in different institutions
• The position of the teacher using technology

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Observations of Gaps

• Strong institutional demand/
• few actual uses
• Potentialities/
• actual uses by teachers
• Teacher expectations/
• actual carrying-out of the lesson in the
  classroom

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Hypotheses

• Discrepancy between
• potentialities underlined by researchers
• from a didactical analysis
• teachers expectations towards supposed effects
  of technology,
• marked by aspirations regarding students
  activity
• Episodes marked by improvisation and uncertainty
• Hidden constraints and obstacles

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Task 4. Hidden obstacles

• Context
• Upper secondary level
• Non scientific students
• Reformed curriculum (sequences)
• Spreadsheet compulsory