The black doubt that lurks in every pedagogues heart is not so much whether she is teaching correctl

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Research and development of geometry in Gr K-9 on
three Continents

• The black doubt that lurks in every pedagogues
  heart is not so much whether s/he is teaching
  correctly as whether WHAT s/he is teaching is
  worth teaching at all. The real danger is not
  that we shall teach the right things
  inefficiently, but that we shall teach the wrong
  things more and more efficiently.

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Research Model Design Research

• Instructional Design
• (Guided by domain specific instructional theory)

• Classroom-based Analyses
• (Guided by interpretive framework)

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Instructional designSpatial Operational
Capacity Model (SOC)

• Representation format
• Transformation Type
• Dimension of stimulus/response
• Type of task
• Type of motion
• Habits of mind

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Classroom-based analyses

• Nature of the didactical contract

Peers
Learner
Tools
Teacher
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Timeline of the research
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Learning and teaching geometry in an integrated
static/dynamic environment

• The rest of the discussion will utilize
  reflection as a sample analysis to clarify
  aspects of the pre-mentioned methodology and
  theoretical framework that guided this research.

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Paper folding and pencil and ruler construction
activities
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Hands-on activities with plastic shapes
?
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Reflection activities with pencil, paper and
mirrors.

• Reflect the following shapes by using a mirror.
  Do it here in your book.

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Activities in the textbook and from there onto
sketchpad

• Look at the drawing of the quadrilateral (this
  one is called a parallelogram) below.
• Draw in your book, not on the computer yet. What
  would it look like if that drawing is reflected
  in the direction of the arrow, over the positive
  y-axis? Draw it on the picture below.
• Start with reflection to the right (first quad),
  then down (Fourth quad) then to the left (third
  quad). Draw what you think here.
• You have to think very carefully before you draw
  it in your book!! THIS IS CALLED THINKING AHEAD,
  OR MAKING A CONJECTURE

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Activities via the textbook and working on
Sketchpad
Hold the Shift key in and Click on B , and click
on point I at the intersection of the mirror line
and the line linking B and B?. Go to the Measure
menu and click on Distance It is written on the
screen. Write it down here as well. CI
.. Hold the Shift key in and Click on
B? , and click on point I at the intersection of
the mirror line and the line linking B and B?. Go
to the Measure menu and click on Distance It is
written on the screen. Write it down here as
well. IC'.. An ant stood at the
intersection of the mirror line and the line
AA?. The ant looked to the A? and wanted to turn
to the top of the mirrorline at F. What would the
size of the turn be that it had to
make? .
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Application activities on Sketchpad

• Task 3 Make a tile based on rotations only.
• 1. You can start with a triangle. Rectangle,
  hexagon or square. This is one way of doing it by
  starting with a rectangle.
• 2. The result could look like the bottom picture

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Findings

• Subject specific

• Didactical specific

• Curriculum specific

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Conclusion

• The ability to penetrate below the surface of a
  subject is to uncover the deeper reasons behind a
  phenomenon. Deeper understanding of a problem not
  only leads to the answer more quickly, but also
  explains WHY the answer takes the particular form
  that it does.