ORDER OF CONSTRUCTION in DYNAMIC GEOMETRY ENVIRONMENT

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ORDER OF CONSTRUCTION in DYNAMIC GEOMETRY
ENVIRONMENT

• Varda Talmon
• The University of Haifa, Israel
• CET- The Center for Educational Technology,
  Israel

Under supervision of Prof. Michal Yerushalmy
ATCM 2004, Singapore
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Introduction

• This presentation is a part of a larger study (My
  PhD dissertation) that examines various
  instruments (Artigue, 2002 Mariotti, 2002
  Rabardel Bourmaud, 2003) developed by users
  while using dragging in DGEs1. It addresses one
  aspect of dragging the connection between
  dynamic behavior and the sequential order of
  construction.

1 This presentation deals with two DGEs The
Geometers Sketchpad (Jackiw, 1995), and The
Geometric Supposer (Schwartz, Yerushalmy,
Shternberg (1998).
Talmon V. ATCM 2004, Singapore
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Dynamic Behavior

• Dragging produces a Dynamic Behavior (DB) for
  each element in the construction.
• DB refers to the degree of freedom of the dragged
  element (is it possible to drag the element, and
  if so, along what path?) and to the response of
  related elements (changes and invariance during
  the dragging).

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background
Three sets of decisions determine the Dynamic
Behavior of a figure within DGE

• Decisions based on the axiomatic system and
  concept of Euclidean Geometry
• Design decisions
• User's decisions

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Decisions based on the axiomatic system and
concept of Euclidean Geometry
Example The DB of a figure reflects the
definition on which its construction procedure is
based.
Two pairs of parallel sides
For Example, these two parallelograms
Two diagonals that intersect one another in the
middle
The construction of the first parallelogram is
based on the proposition that if the two
diagonals of a quadrangle intersect one another
in the middle, than the quadrangle is a
parallelogram. The second construction is based
on the proposition that if a quadrangle has two
pairs of parallel sides, than the quadrangle is a
parallelogram.
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Design Decisions
For Example The Supposer main menu includes a
Shapes Menu that provides pre-constructed well
known geometric objects. This allows students to
inquire a figure even if they do not know how to
construct it.
Let us assume that a student is asked to inquire
the properties of the rectangle. Using the Shapes
Menu, the student browses through the
parallelograms family and gets a pre-constructed
rectangle. The student can also use the Quad Menu
and construct his own quadrangle by determining
the dimensions of its diagonals.
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Design Decisions
The DB The first rectangle is dynamic. All its
vertices and sides are draggable. The second
rectangle is fixed and obviously cannot be
changed by dragging.
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Design Decisions
On the other hand, the option Commands in the
Construct menu of the Sketchpad is enabled only
when the appropriate prerequisites have been
selected (the objects used to define the
construction).
This requires pre-knowledge User must know a
procedure to construct the figure based on the
geometric properties of the object.
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Design Decisions
As we just have seen, the DB depends on the
designers decisions about the modes of the
construction.
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Users Decisions
To construct a figure within DGE, the student has
to create a geometrical procedure of the
construction. One of his main decisions concerns
the sequential organization of the procedure
(order of construction).
The connection between Dynamic Behavior and the
Order of Construction will be our focus in the
time left for this presentation.
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hierarchy of dependencies Dynamic Behavior
The sequential organization of a procedure in DGE
produces a hierarchy of dependencies, as each
part of the construction depends on something
created earlier (Jones, 2000). This hierarchy of
dependencies is one of the main factors that
determine DB within DGE (Jackiw Finzer, 1993
Laborde, 1993). This order is often explicitly
organized by terms such as parent and child.
An element that is related to a previous one
The previous element
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Order on paper Order in DGE
On paper the order of construction is central in
complex or advanced constructions, but it is not
critical in many basic constructions. For
example, a point on a line or a line through a
point are geometrically identical in a paper
sketch, but these two constructions have
different DBs.
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The study Users expectations

• Goldenberg et al. (In preparation) raise the
  following question
• what do users expect would be the effect of
  dragging point B on point E placed arbitrarily on
  a line perpendicular to AB.

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Users expectations Order of construction

• We took this question further and conducted a
  study on the issue of understanding the DB of
  geometrical objects in a Dynamic Environment. One
  of the main questions of the study was

what are users expectations of DB and
interpretations of the connection between the
order of construction and the DB of its elements
in terms of parent-child relations?
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The Study

• The main tool of the study was an interview
  consisting of three parts
• To perform a given procedure within one of the
  two following environments the Sketchpad or the
  Supposer.
• To predict the dynamic behavior of the
  constructed geometric object. We used
  transparencies to demonstrate the predictions.
• To drag several points of the object and explain
  the specific behavior they saw on the screen.
• 15 ninth-grade pupils and 10 M.A. Math-Education
  students participated in the study.

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Shoval A 9th grade student
After following a given construction procedure
using the Supposer, Shoval predicts the DB of
point C.
Varda Let us look at point C. Can we drag point
C? Shoval Yes, we can. But the whole shape will
be preserved. Varda First of all, tell me if we
can drag point C. Shoval Where to? Varda
Wherever you like. Shoval We can drag it
point C but the construction will move too.
To demonstrate his predictions, I placed a
transparency on the computer screen, and Shoval
copied the figure he constructed and (while the
transparency was still on the screen) made a
sketch representing his conjecture about the
shape of the figure after dragging point C.
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Shovals transparency page
Here is his drawing

• Shoval predicts that point C (a point on a line)
  can be dragged freely.
• He predicts that dragging point C would affect
  other elements of the construction including
  point Cs parents (it relocates B)

• Shoval copies a triangle (an object he didnt
  construct)
• He predicts that dragging point C would translate
  the whole triangle

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Avis transparency page
Avi A 9th grade student

• Avi predicts that dragging point B would not
  relocate Its child (E) but would affect its
  parents (Break AB).

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We found 7 Categories of Reverse-Order
predictions of DB
Prediction 1 A free point (without any
constraints) cannot be dragged
Prediction 2 A constrained point (a point on a
figure, e.g. on a segment or a circle) can not be
dragged.
Prediction 3 A constrained point can be
dragged without any constraints.
Prediction 4 Dragging a child changes the
location of its parent.
Prediction 5 The existence of a child affects
the Dynamic Behavior of its parent.
Prediction 6 Dragging a parent does not change
the location of the child.
Prediction 7 Dragging a point changes the
location of a free point.
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Examples of the 7 Categories of Reverse-Order
predictions of DB
Prediction 2 C cant be dragged
Prediction 3 C can be dragged Freely
Prediction 1 A (B) cant be dragged

• Procedure A
• AB (j) is an arbitrary segment
• C is a point on AB (j)
• (k) is a perpendicular to AB (j) through C
• E is a point on (k)
• EB (l) is a segment between B and E

Prediction 4 Dragging E Relocates C
Prediction 7 Dragging C relocates A (B)
Prediction 5 C cant be dragged because of k
Prediction 6 Dragging A doesnt relocate C
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Explanations
Two possible reasons
1. The extent to which the figure-image held by
the interviewees is reflected in their reverse
order predictions.
We found that interviewees tend to relate to
figures that can be seen on the screen but are
not logically recognized by the software. They
use these figures to explain their predictions.
For example, three segments are treated as a
triangle.
2. The extent to which the dragging image held
by the interviewees is reflected in users'
predictions as a mathematical transformation.
The action of dragging was sometimes interpreted
by users as a mathematical transformation such as
Translation, rotation or homothetic. This was
reflected in their predictions as they seemed to
deduce properties of dragging from the properties
of these three transformations.
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Why does this matter?

• Dynamic Geometry Environments (DGEs) have become
  standard tools for students, teachers, and
  mathematicians.
• Moreover, using DGE established new norms of
  learning geometry and deepen our understanding of
  tool-situated learning.
• Dragging geometric constructions is one of the
  fundamentals of these environments and
  unquestionably enhances the esthetics and power
  of DGEs.
• At the same time it changes the nature of the
  dependencies declared in the construction. Thus
  it introduces new complexity to learning using
  DGE.
• It seems that Euclidean axioms and theorems only
  partially influence the DB of constructions.

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Why does this matter?
Therefore
To understand more fully the stages of
instrumental genesis of DGE and software design
decisions, questions such as, What do users
observe while dragging one element of the
construction?, How do users interpret DB, and
what are their perceptions of it?, should be
further investigated.
The study defines categories and terms that could
hopefully contribute to the crystallization and
depth of further research questions as well as to
curriculum and software design considerations.
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Acknowledgment
Special thanks to Michal Yerushalmy, for being a
teacher and a friend, and to Beba Shternberg,
Nicholas Jackiw and Paul Goldenberg for
discussing with me the ideas that appear in this
presentation.
Talmon V. ATCM 2004, Singapore