Cognitive strategies and CAS

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Cognitive strategies and CAS
Computer Algebra and Dynamic Geometry Systems in
Mathematics Education RISC, Castle of Hagenberg,
Austria. July 11-13, 2009.

Csaba Sárvári, Mihály Klincsik, Zsolt
Lavicza
University of Pécs Pollack Mihály Faculty of
Engineering
University of Cambridge Faculty of Education
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Cognitive strategies

• help learners make and strengthen associations
  between new and already known information
• make possible adaptation of attained knowledge in
  new situation
• facilitate the mental restructuring of
  information.

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Cognitive strategies

• guessing
• analysing, synthesising
• reasoning inductively and deductively
• taking systematic notes
• reorganising information
• hypothesis testing
• searching for clues in surrounding material and
  ones own background knowledge
• searching for new rules
• trying
• making new structure
• completion of knowledge
• elaboration
• activating of information
• recalling of the previous knowledge
• experimenting

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Metacognition

• Metacognition refers to ones knowledge
  concerning ones own cognitive processes or
  anything related to them, e.g., the
  learning-relevant properties of information or
  data.
• J. H. Flavell (1976)
• thinking about thinking
• Most fundamental metacognitive strategies
• connect the new information with previous
  knowledge
• conscious selection of thinking operations
• planning, monitoring, controlling and evaluation
  of thinking processes.

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Self-regulated learning

• The term self-regulated can be used to describe
  learning that
• is guided
• by metacognition
• strategic action (planning,monitoring, and
  evaluating personal progress against a standard),
  and
• motivation to learn.

Self-regulated learners

• are cognizant of their academic strengths and
  weaknesses, and
• they have a repertoire of strategies they
  appropriately apply to tackle the day-to-day
  challenges of academic tasks.

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Cognitive schemes and knowledge-representation
network

• Cognitive schemes (epistemological-, thought-
  schemes)
• are such building blocks of our thinking that are
  meaningful by themselves and having independent
  meanings.
• are actively direct ones cognition and thinking
  while they are constantly changing in relation
  to the acquired knowledge.
• are not independent components of ones
  consciousness, but they establish an ever
  changing relation-system calledknowledge-represen
  tation network.

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Knowledge representation network

• Efficiency of the mathematical knowledge can be
  approached
• by evaluating the organization of knowledge
  elements.
• A concept is comprehended if the concept is well
  represented and bounded with other knowledge
  elements. Consequently,
• the thickening of the knowledge-representation
  web is the result of the development and
  modification of interrelated cognitive schemes.

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Enlarging of the knowledge representation network

• The inner representation network

Before the learning event
During the learning event
After the learning event
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Cognitive strategies and CAS

• The aim of our presentation to show how we can
• give - using CAS additional tools for students
  
• to be able to apply efficient (additional)
  cognitive strategies
• to gain a metacognitive knowledge about self,
  the task and strategies and how to use these
  strategies
• to develop their ability of self-regulated
  learning.

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Examples

• Our examples are the following
• using Student Calculus1 Package by integration
  to demonstrate how the students can efficient
  learn to integrate applying with different
  strategies the tools of the package
• minmax approximation to show how can be used CAS
  to combine seemingly distinct areas of
  mathematics, to help the experimental work
• investigating solutions of differential
  equationsto present how we can use CAS to gain
  additional strategies to investigate the
  different solutions of differential equations.

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Flow chart of the problem solving process
Links between elements of the mathematical theory
Problems
Give the domain of existence of the solutions on
x-axis
Show, there are two solutions go through to the
origin (00).
From local investigation to global picture
Determine the limits of the solutions as x tends
to 8!
Determine the asymptotes of the solutions as x
tends to 8!
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Aspects of the guided learning tour
Experimentations with plotting different solutions
Links to the previous knowledge-element (black
box, building the problem space)

• Domain of existence of functions

• Solutions of differential equation

• Initial values

• Limiting value of functions

• Maple commands dsolve, solve, subs, plot,
  odetest, limit, DEplot