Conceptualization of derivative using GeoGebra

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Conceptualization of derivative using GeoGebra

• Enrique de la Torre Fernández
• (Universidade da Coruña, Spain)
• Adelina Silva Muslera
• (Universidad Autónoma de Querétaro, Mexico)

GeoGebra Conference 2009
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Introduction

• Our concern is about the use of technologies in
  the mathematics classroom.
• We questioned ourselves mainly as the students
  use technology to conceptualize the differential
  and integral calculus.
• We need a software that will adjust to that
  concern and a methodological frame that fit in
  our plans of research.
• This it is a first result of that concern.

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Instrumental approach

• The instrumental genesis is described like the
  process by means of which a device is transformed
  into an instrument by the subject.
• A device is a material or abstract object, given
  to the subject.
• An instrument is a psychological construction,
  building from the device it is any object that
  the subject associates with its action to make a
  task.
• The instrumental genesis has two components,
• the instrumentalization is relative to the
  discovery, personalization and transformation of
  device by the subject.
• and the instrumentation is relative to the
  emergency and evolution of the schemes of
  students for the accomplishment of a given task.

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Instrumental approach

• The instrumental approach has been developed and
  applied in the context of algebra software (CAS),
  where different types of restrictions have been
  identified, what often are learning opportunities
  and which provide data for the professor and for
  the researcher.
• This idea supposes that technology is a mean
  where abstract relations and structures are
  expressed.
• The dynamic geometry software (DGS) is also a
  mean where it is possible to express the
  geometric relations by the way of to drag with
  the mouse or the facility to redefine, as the
  algebra software does with the algebraic
  expressions .

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Context and methodology

• 28 first course students of Chemistry in the
  Faculty of Sciences (University of Corunna,
  Spain).
• Time destined for practices, one hour for week.
• 10 sessions
• Subjects were derivatives, applications of
  derivatives, integral and applications of
  integral.

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Context and methodology

• 16 computers in the classroom.
• Students had access to paper and pencil.
• Some of them made use of their personal
  calculators.
• Software
• GeoGebra.
• CamStudio (to record the work the students
  develop in computers).

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Context and methodology

• A interpretative qualitative research
  (ethnographic methods ).
• We use the participant observation like a
  strategy for collecting data.
• We want to understand the reality within a
  context given, from the meaning of people implied
  in that context.

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Example of derivatives

• Second day two students manipulate a computer
  and generate a video, also each one of them wrote
  their answers with paper and pencil.
• Problems proposed
• Is the function derivable at x2?
  Justifies your answer.
• To find the equation of a tangent to the function
  f(x) 3x314x23x8 what go through the
  coordinates origin.

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Example of derivatives

• The solution to the first exercise it could be
  obtained directly when they analyze the function
  and they see that function is not defined at x2.
  
• Nevertheless, as they have the computer on, the
  first thing they make is to design the function.
•  They use the commando Function
• Function x1/2x2-8
• The software indicates a mistake.

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Example of derivatives

•  For many minutes, the students rewrite the
  sentence, placing parenthesis in the numerator
  and denominator, and trying the menu for to
  create a slider.
• This is the so called fishing behavior they
  test without affecting its organization or her
  verification, hoping that in a reasonable time
  something interesting will be.

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Example of derivatives

• Techniques of paper and pencil for the first
  exercise

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Example of derivatives

• Techniques of paper and pencil for the first
  exercise

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Example of derivatives

• Techniques of paper and pencil for the second
  exercise

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Example of derivatives

• Techniques of paper and pencil for the second
  exercise

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Conclusions

• One of the potentialities of GeoGebra, the
  functions graphs, becomes a restriction for the
  students.
• When the students make derivatives, usually they
  do not see the graph of the derivative function,
  and when the derivative function is showed in the
  screen, it is something new for them.
• This can be denominated a obstacle in the sense
  that it is a lack of balance in the conceptual
  aspects .

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Thank you!