International Journal of Computers for Mathematical Learning (2006)
DOI 10.1007/s10758-006-9106-7
Ó Springer 2006
MICHAL MAYMON EREZ and MICHAL YERUSHALMY
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‘‘IF YOU CAN TURN A RECTANGLE INTO A SQUARE, YOU
CAN TURN A SQUARE INTO A RECTANGLE ...’’ YOUNG
STUDENTS EXPERIENCE THE DRAGGING TOOL
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ABSTRACT. This paper describes a study of the cognitive complexity of young students,
in the pre-formal stage, experiencing the dragging tool. Our goal was to study how
various conditions of geometric knowledge and various mental models of dragging
interact and influence the learning of central concepts of quadrilaterals. We present three
situations that reflect this interaction. Each situation is characterized by a specific interaction between the students’ knowledge of quadrilaterals and their understanding of the
dragging tool. The analyses of these cases offer a prism for viewing the challenge involved
in changing concept images of quadrilaterals while lacking understanding of the
geometrical logic that underlies dragging. Understanding dragging as a manipulation that
preserves the critical attributes of the shape is necessary for constructing the concept
images of the shapes.
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KEY WORDS: dynamic geometry environment, dragging, mental models, critical
attributes, hierarchical relations
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1. LEARNING GEOMETRY WITHIN A DYNAMIC
GEOMETRY ENVIRONMENTS
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Several studies discuss the complexities of understanding basic
geometry concepts by learners at various ages (e.g., Burger and
Shaughnessy, 1986; Hasegawa, 1997; Monaghan, 2000; Senk, 1989;
Usiskin, 1982). Research also suggests that learning with the support
of dynamic geometry environments (DGE) offers new ways
for students to construct conceptual meaning (Choi Koh, 1999;
Jones, 2000; Pratt and Ainley, 1997; Vincent and McCrae, 1999). The
goal of the present study is to learn more about the way in which
younger students (aged 11–12 years) learn basic geometry concepts
within a DGE. Specifically, we explore how various conditions of
geometric knowledge and various mental models of dragging interact
to promote learning of central concepts of quadrilaterals. We start
with an explanation of our views on these aspects of understanding
geometry.
MICHAL MAYMON EREZ AND MICHAL YERUSHALMY
Learning geometry within a DGE introduces three views of
geometry: the formal one, that of the learner, and that of the software
environment.
1.1. The Formal Geometry
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By formal we mean definitions and attributes of geometric shapes.
We focus in this study on students of 11–12 years and on quadrilaterals. Teaching students of this age focuses on critical attributes of
basic shapes (triangles, quadrilaterals) and on the hierarchical relations among them. According to Hershkowvitz (1990), the concept’s
critical attributes are those that can be found in any example of the
concept. They are essential and critical because if one of them is
missing it is no long an example of the concept. For example, ‘‘four
sides,’’ ‘‘two pairs of parallel sides,’’ or ‘‘two pairs of equal opposite
angles’’ are some of the critical attributes of a parallelogram but ‘‘two
long sides and two short sides’’ or ‘‘two acute angles and two obtuse
angles’’ are not. There are parallelograms with four equal sides
(called rhombi) or with four right angles (called rectangles). Learning
in this sense means learning to analyze the attributes of different
quads, to distinguish between critical and non-critical attributes of
different quads, and also learning the hierarchy among quads.
Markman (1989) typifies four criteria of understanding hierarchical
relations: (1) the ability to classify the concept into different categories and label it with different names, e.g., that the rhombus can also
be called polygon, quadrilateral, parallelogram, or kite; (2) understanding the transitive relations between the concepts, e.g., that if a
square is a rhombus and a rhombus is a parallelogram, then a square
is also a parallelogram; (3) understanding the asymmetry of relations
among quadrilaterals, e.g., that every rectangle is a parallelogram,
but not every parallelogram is a rectangle; (4) understanding the
opposite asymmetry and transitive relations of the critical attributes
of the concepts: the critical attributes of the rectangle are included in
the critical attributes of the square, but the critical attributes of the
square are not included in those of the rectangle. Apparently this
view of the formal geometry is not unique. While hierarchy is the
common approach to classification of shapes, other proposals exist.
One alternative being studied is a partition classification of quads (de
Villiers, 1994). Such classification assumes that various subsets
of concepts are disjoint from one another. For example, squares,
YOUNG STUDENTS EXPERIENCE THE DRAGGING TOOL
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rectangles, and rhombi are not viewed as parallelograms. According
to a partition definition a parallelogram can be defined as ‘‘a quadrilateral with two pairs of opposite parallel sides, but not all angles or
sides equal.’’ Alternatively, the description ‘‘a quadrilateral with two
pairs of opposite parallel sides’’ can typify a hierarchical definition of
a parallelogram. The current study was part of a curricular sequence
that assumes hierarchy and its derived definitions, and therefore the
critical attributes of quadrilaterals are derived from the hierarchical
view.
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1.2. The Geometry of the Learner
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Young as well as older students hold concept images of quadrilaterals
that are not necessarily consistent with the formal definitions.
According to Vinner (1983) concept image is a collection of mental
representations that might include pictures, graphs, diagrams, symbolic forms, and attributes of the concept; it is created from the
students’ experimentation with examples and non-examples of the
concept. Sometimes the students’ concept image is consistent with
one of the concept’s definitions. More often the concept image is
prototypic (Burger and Shaughnessy, 1986; Hershkowitz, 1990).
According to Hershkowitz (1990), each concept has one or more
prototypical examples, usually the subset of examples that have all
the critical attributes of the concept, as well as other specific, noncritical attributes possessing strong visual characteristics. For example one may consider only an upright position square to be an
example of a square.
The prototypical phenomenon holds for both young and older
students. It is demonstrated by learners using the prototypical
example as their framework for observing and analyzing a range of
examples of the concept rather than using its definition and critical
attributes. According to Van Hiele’s model (1959), at the first level of
development of geometrical thought figures are recognized by their
shape as a whole and not by their attributes. At the second level,
learners can analyze the attributes of the shapes. At the third level,
they should be able to distinguish between critical and non-critical
attributes of the shape and to order shapes logically. Often students
face problems understanding the hierarchical relations of quadrilaterals (Burger and Shaughnessy, 1986; Currie and Pegg, 1998; Geddes
et al., 1982; Usiskin, 1982). This difficulty is manifested by students
not being able to address a shape by multiple names (de Villiers,
MICHAL MAYMON EREZ AND MICHAL YERUSHALMY
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1994) and not being aware of class inclusion relationships (a square is
a parallelogram, a rhombus is a kite, and so on) (Currie and Pegg,
1998). Nor do they understand opposing direction inclusion relationships between sets of examples on the one hand, and their set of
critical attributes on the other (Hershkowitz, 1990). Sometimes, as
noted by de Villiers (1994), students have no logical or relational
problems understanding the hierarchical classification but they prefer
to view it in a different way. They prefer to classify the quads
according to a partition classification and therefore their concept
images of quads follow that view. For example, they may hold a
concept image of a rhombus as a parallelogram with equal sides and
oblique angles or of a rectangle as a parallelogram with equal angles
but without four equal sides.
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1.3. The Geometry of DGEs
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Geometry software environments (e.g., Cabri, The Geometer’s
Sketchpad, The Geometric Supposer) offer ways of learning geometry
by generating examples, observing, and experimenting with examples
as the basis for generalized conjectures. The environments offer
geometric objects, tools to manipulate them, as well as measurement
and computation tools. This allows users to construct planar shapes
compatible with the ones constructed by straight edge and compass,
to measure lengths, angles, and areas, and to manipulate the construction by repeating the procedure on new shapes or by dragging
and transforming the shape.
In his essay ‘‘The right size bite,’’ Judah Schwartz (1995) argues
(as do Healy and Hoyles, 2001; Jackiw and Finzer, 1993;
Yerushalmy, 1999) that the nature of the primitives and of the tools
that a software program provides to its users are essential components of the functionality of the software in a learning environment.
We also assume that the nature of the primitives matters, and we
believe that to better understand the contribution of the primitives we
must observe the activities and the resources that learners act on and
bring with them. Dragging is a major tool of any DGE and an
essential component in the current study. Dragging allows changing a
shape by direct translation of parts of its components on the screen.
Dragging produces what appears like a continuous change, and at
any moment the dragged shape preserves the geometric relations
according to which it was initially defined. Thus, the critical attributes
associated to this definition are preserved during dragging but the
YOUNG STUDENTS EXPERIENCE THE DRAGGING TOOL
non-critical attributes are changed. Learning by DGE is closely related to theories of constructivist learning, in which construction of
meaning occurs through the learner’s active participation. We suggest
two major categories for the application of dragging:
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1. When the learner constructs a shape, dragging will preserve the
attributes of the elements and the relations among the elements as
defined by the learner. Dragging will then mirror (to a certain
extent) the attributes and the construction procedure. For
example, if a user constructs a parallelogram using a procedure that assumes two pairs of parallel lines, dragging the
constructed parallelogram will create a variety of parallelograms,
and the parallelogram cannot be ‘‘messed up.’’ At the same time, if
a parallelogram is constructed by adjusting two pairs of lines to
look parallel, most likely it will not remain parallel upon dragging.
Understanding dragging here means understanding that the
dynamic manipulations preserve the relations defined for the
construction and the associated attributes.
2. When learners drag a pre-constructed shape (given by the
teacher as a file or chosen from a menu of shapes) they do not
necessarily know the attributes upon which the shape was
constructed – neither the critical attributes of the shape nor the
construction procedure. But learners must be aware that dragging
does not change the characteristics of the shape. According to the
literature dealing with the role of diagrams, the dynamic diagram
should be conceived as a figure: a representation of a geometrical
object that corresponds to a class of drawings with the same
geometrical attributes (Laborde and Laborde, 1995; Parzysz,
1988). Here understanding dragging means understanding that the
dynamic manipulation preserves the critical attributes of the preconstructed geometric shape.
The two categories differ in the purpose for which the dragging
tool is used in the learning process. When the learner uses pre-constructed shape, the learning task is often to conjecture about the
attributes and relations that characterize it. Here theoretical knowledge is an outcome of experimenting; as Arzarello et al. (2002) term it
an ascending process. Using dragging to explore pre-constructed
shapes is often part of young students’ learning of geometry, where
they are expected to draw conclusions about the critical attributes
and relations of the shapes (Battista, 2001).
MICHAL MAYMON EREZ AND MICHAL YERUSHALMY
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We aim to expose and elaborate some complexities of learning the
geometry of quadrilaterals, which are probably rooted in the inseparable links between understanding the dragging tool and understanding attributes of quadrilaterals, and for this purpose we use
dragging according to the second category mentioned above that is to
say starting from pre-constructed shapes. This elaboration requires
first the clarification of the theoretical orientation of our approach to
cognitive processes.
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1.4. Mental Models
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There is a great deal of evidence that human beings are capable of
mentally representing objects or events; other words, they hold
mental models (English, 1997; Fischbein, 2001; Johnson-Laird, 1983;
Norman, 1983). To explain how such cognitive representations
become part of the learning process, Norman (1983) isolated the
target system, the conceptual model of the target system, and
the user’s mental model of the target system. The target system is the
system that the person interacts with. In our case the dragging tool.
The relevant conceptual model of the dragging tool is the understanding that when you drag a pre-constructed shape, for example a
parallelogram, it changes continuously while the dynamic manipulation preserves its critical attributes. By interacting with the target
system a person continuously modifies the mental model: existing
mental models are changed and new ones evolve. If students can
coordinate the mental model with the development of mathematical
meaning they understand that dragging preserves the critical attributes of the pre-constructed shape and that when attributes are not
preserved under dragging they are not critical attributes of it. Thus
they perceive the diagram as a figure.
We conjecture that in order to follow the systematic change that
preserves critical attributes one must be aware of the existence of such
attributes. We sought to learn how young students, who are just
attempting to study the critical attributes, follow the systematic
change that preserves critical attributes. Our goal was to analyze the
mental models of the dragging tool while dragging pre-constructed
shapes. In this process we were also interested in analyzing the
changes in students’ knowledge of critical attributes of different
quads and of the hierarchical relations among them.
YOUNG STUDENTS EXPERIENCE THE DRAGGING TOOL
2. METHOD
2.1. Participants
2.2. Interview Design
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To examine how young students interpret the dynamic behavior of
pre-constructed shapes and glean the critical attributes of quadrilaterals and their hierarchical relations, we used the Supposer to interview ten 5th-grade students (aged 11–12 years), who were successful
math students in a regular class at a public school. In the interview
they were asked only about quadrilaterals. When constructing the
tasks we took into consideration what students were expected to have
learned in previous years according to the standard Israeli curriculum
(in the Israeli math curriculum students begin learning about quadrilaterals in the 1st grade). Students were acquainted with various
Supposer tools, having used the software for a few weeks as part of
their geometry class. Before the interviews the class had used the
Supposer to learn about triangles. Students investigated triangles
using the measure and dragging tools. Observation of their classroom
activities suggests that in most cases they explored examples of static
on-screen triangles and only rarely manipulated the triangles by
dragging.
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The semi-structured interview was based on new tasks for the
participants, giving us the opportunity to analyze the learning process
through interaction with the interviewer. The first author conducted
all the interviews, which took place in school, after school hours.
Each interview began with a 10–15 min preliminary intervention
during which students were given a definition of a ‘‘properly constructed shape’’ that preserves the shape’s critical attributes upon
dragging, and two preliminary tasks. The term ‘‘properly constructed
shape’’ was central to the interview tasks, and students had not been
introduced to it in class. The main goal of the first task was to
demonstrate our meaning of the term. Interviewees experienced
dragging of a properly constructed isosceles triangle1 and were asked
to explain why it was a ‘‘properly constructed shape.’’ Using visual
and measurement cues, the interviewer guided the interviewees to
recognize that in a proper construction of an isosceles triangle two
sides or two angles of the triangle remain equal upon dragging. The
next task, shown in Figure 1,was designed to allow us to follow the
MICHAL MAYMON EREZ AND MICHAL YERUSHALMY
Three students, Gadi, Yossi and Rina, constructed a parallelogram in the Geometric Supposer. I
would like you to check, using the dragging tool, whether they constructed a proper construction of
a parallelogram.
Let’s check whether Gadi built a proper parallelogram. Is it a parallelogram? Now drag it
and check whether it is a proper construction. Why is it? Why isn't it?
Let’s check whether Rina built a proper parallelogram. Is it a parallelogram? Now drag it
and check whether it is a proper construction. Why is it? Why isn't it?
Let‘s check whether Yossi built a proper parallelogram. Is it a parallelogram? Now drag it
The second preliminary task.
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Figure 1.
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and check whether it is a proper construction. Why is it? Why isn't it?
shape
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The given
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students’ use of the term ‘‘properly constructed shape’’ and to learn
about their concept image of a parallelogram.
Attached to the task were pre-constructed shapes each was
adjusted to appear as a parallelogram. But only Gadi’s was a correct
construction of a parallelogram. Yossi’s construction was a trapezoid
that looked like a parallelogram and Rina’s was a random
quadrilateral that looked like a parallelogram (see Figure 2).
All the interviewees correctly identified who constructed the
proper construction of the parallelogram. In their arguments, the
students used various familiar relevant or irrelevant terms about
visual cues, measurements, and the hierarchy of shapes. After this
An example
of the shape
after dragging
Gadi (proper construction of
Yossi (improper construction Rina (improper construction
a parallelogram)
of a parallelogram: a
of a parallelogram: a random
trapezoid)
quadrilateral)
Figure 2.
The pre-constructed shapes of the second preliminary task.
YOUNG STUDENTS EXPERIENCE THE DRAGGING TOOL
Here is a correct construction of X. Do you agree that it is an X? Why?
Do you think you can turn it into Y by dragging it? Why?
Figure 3.
A generic interview task.
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intervention we proceded with the central interview tasks. A generic
interview task is presented in figure 3.
The first part of the task was intended to examine the students’
concept image of quadrilaterals. Next, the students conjectured and
explained why it was or was not possible to drag shape X and turn it
into shape Y. Then they checked their conjectures using the dragging
tool. By analyzing their explanations we drew conclusions about their
knowledge of hierarchical relations and about their understanding of
dragging.
Theoretically, if the students conceive the pre-constructed shape as
a figure and hold a mental model that is identical with the conceptual
model of the dragging tool, they can provide one of three possible
answers:
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1. It is possible to drag shape X so that it becomes shape Y because Y
is a kind of X. For example, you can drag a parallelogram so that it
becomes a rectangle and a kite so that it becomes a rhombus, etc.
2. It is not possible to drag shape X so that it becomes shape Y
because X is a kind of Y. For example, you cannot drag a rectangle
so that it becomes every type of parallelogram. Of course, a rectangle is already a parallelogram, but you cannot drag a rectangle so
that it becomes a parallelogram with acute and obtuse angles.
3. It is not possible to drag shape X so that it becomes shape Y
because there is no direct hierarchical relation between them except that they are both quadrilaterals, polygons, etc. For example,
you cannot turn a rectangle into a kite that is not a square.
Other possible answers are ‘‘It is possible to drag any quadrilateral
so that it becomes quadrilateral Y because they are both quadrilaterals’’ or ‘‘It is not possible to drag shape X so that it becomes shape
Y’’ although Y is hierarchically under X (e.g., X is a rectangle and Y
is a square). Such answers are expected if the interviewees do not
conceive the diagram as a figure, if they hold a prototypic concept
image, or if their concept image evolves according to a partition
definition.
MICHAL MAYMON EREZ AND MICHAL YERUSHALMY
2.3. Choosing the Data for Presentation
3. FINDINGS
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We analyzed the interviews according to two criteria: (1) familiarity
with the critical attributes of quadrilaterals and with the hierarchical
relations among them; (2) understanding the geometric logic underlying dragging. We chose to present three leading situations by
analyzing one or two representative episodes of each. The situations
represent an interaction between the students’ understanding the
geometric logic underlying dragging (or lack of such understanding)
and their familiarity with critical attributes of quadrilaterals
according to the hierarchical definition (or lack thereof).
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Our findings are organized along three cases. Case 1. From the
student’s answers we can characterize a mental model of dragging
that is similar to the conceptual model of the target system. The
student understands the geometric logic underlying dragging, knows
the critical attributes of different kinds of quadrilaterals according to
a hierarchical definition, and grasps their hierarchical relations. Case
2. The student holds concept images of quadrilaterals according to a
partition definition, and is partially successful in identifying the
hierarchical relations. His mental model of dragging does not reflect
attention to preserving the critical attributes of the shapes. Case 3.
This student holds a concept image of some of the quadrilaterals
according to a partition definition, and she is partially successful in
identifying the hierarchical relations between them. She seems to
have developed a mental model of dragging based on preserving the
critical attributes of the shapes, but she is confused and not
consistent.
3.1. Case 1
‘‘You can turn a parallelogram into a rhombus because a rhombus is a kind of
parallelogram ...’’
The episode is taken from the interview with Shiri. She knows the
critical attributes of all the quadrilaterals and their hierarchical
relations, and grasps the dynamic behavior as a behavior that
preserves the critical attributes of the shapes. She demonstrates
YOUNG STUDENTS EXPERIENCE THE DRAGGING TOOL
geometric understanding typical of van Hiele’s level three, and she
has a mental model of dragging that allows her to predict correctly
the dynamic behavior of the quadrilaterals on the screen.
3.1.1. Episode
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Michal: Is this shape a parallelogram?
Shiri: Yes
Michal: How do you know?
Shiri: Because it has two pairs of opposite sides that are parallel.
Michal: Do you think you can drag the parallelogram and create
a rhombus?
Shiri: Yes.
Michal: Why?
Shiri: Because a rhombus is a parallelogram.
Michal: What is the difference between a rhombus and a parallelogram?
Shiri. In a rhombus all the sides are equal. (Shiri tries to create a
rhombus and creates a square)
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3.
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5.
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On the screen:
On the screen:
11. Shiri: But in fact a square is also a rhombus.
12. Michal: What did you do?
13. Shiri: I made all the sides equal.
MICHAL MAYMON EREZ AND MICHAL YERUSHALMY
Michal: So what shape is this?
Shiri: This is a square or a rhombus.
Michal: What is the difference between a square and a rhombus?
Shiri: A square has equal angles too.
Michal: Do you think that you can create a rectangle?
Shiri: I can try.
Michal: Just a moment, before you try. Why can you?
Shiri: Because there the opposite sides are parallel too.
Michal: What is the difference between a rectangle and a parallelogram?
23. Shiri: In the rectangle all the angles are equal. (Shiri drags the
shape to look as a rectangle)
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36.
37.
38.
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Michal: Do you think that you can create a trapezoid?
Shiri: No.
Michal: Why not?
Shiri: Actually yes; there are trapezoids that are parallelograms.
Michal: What kind of trapezoids can you create?
Shiri: This is a trapezoid. (Points to the rectangle)
Michal: What is impossible to create?
Shiri: A...Kite.
Michal: Is it impossible to create a kite out of the parallelogram?
Shiri: It is possible because a rhombus is a kite.
Michal: So it is possible to create a rhombus and then it will be a
kite?
Shiri: Yes.
Michal: Again about the trapezoid. Is it possible to create all
kinds of trapezoids?
Shiri: No.
Michal: What kinds are possible?
Shiri: Some kinds are possible...there are trapezoids with only
one pair of opposite sides which can’t be created.
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27.
28.
29.
30.
31.
32.
33.
34.
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On the screen:
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19.
20.
21.
22.
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40. Michal: I constructed a proper construction of a square. Do you
think that it is possible to turn it into a rectangle?
41. Shiri: No.
42. Michal: Why not?
43. Shiri: It is already a rectangle...because the sides of the rectangle
are not always equal in length.
44. Michal: And in a square?
45. Shiri: They are.
46. Michal: So it cannot be turned into a rectangle?
47. Shiri: No.
48. Michal: Could it be turned to another shape?
49. Shiri: No.
50. Michal: Why not?
51. Shiri: Because in a square everything has to be identical and in
the others not always.
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Throughout the episode (and throughout the interview) Shiri
linked her geometric knowledge and the dynamic behavior of the
shapes. It seems that she conceived the quadrilaterals on the screen as
figures and based her answers on geometrical arguments. She correctly completed all tasks and made correct conjectures regarding the
dynamic behavior of the quadrilaterals on the screen. She conjectured
that (1) you can turn a parallelogram into a rhombus, a rectangle,
and a square because ‘‘there the opposite sides are parallel too’’ and
they are kinds of parallelograms (1–23). (2) You cannot turn a square
into other quadrilaterals (40–51) ‘‘because in the square everything
has to be identical, and in the others not always’’. (3) You cannot
turn a parallelogram into any kite (30–31). She seemed to know that
there is no direct hierarchical relation between them, but you can
create a rhombus and then have a kite (33–35).
Based on Shiri’s answers we can determine that her mental model of
dragging is of an action that preserves the critical attributes of the
shape. In our view, the student succeeded in coordinating the mental
model with the understanding of mathematical meaning: she seems to
hold a mental model of dragging that is identical with the conceptual
model. As stated earlier, the conceptual model of the dragging tool is
based on the understanding that the dynamic behavior is continuous
and rests on the critical geometry attributes of the shape. Shiri’s mental
model makes it possible for her to anticipate the behavior of the tool and
to interpret its results correctly and productively. This is the state to
which educators aspire, but it is not the usual case. Next we present
MICHAL MAYMON EREZ AND MICHAL YERUSHALMY
other cases that suggest some complexities involved in using the dragging tool as a way to learn quadrilaterals and hierarchical relations.
3.2. Case 2
‘‘If you can turn a rectangle into a square, you can turn a square into a rectangle...’’
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The following episodes are taken from an interview with Ran,
whose responses and actions suggest that he did not perceive the
geometrical logic underlying dragging. His responses were based
mainly on visual arguments and less on geometrical or logical ones. He
appears to hold concept images of quadrilaterals based on a partition
definition. We focus on parts of the interview dealing with the relations
between a rhombus and a square, and a rectangle and a square.
3.2.1. First episode
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On the screen:
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1. Michal: I constructed a proper construction of a rectangle. How
do you see that this is a rectangle?
2. Ran: I see, because there should be two long sides and two short
sides... two length sides, vertical short, shorter than the horizontal sides and all 90 degrees.
3. Michal: And all the angles 90 degrees?
4. Ran: Yes.
5. Michal: Do you think that you could drag the rectangle and
create a square?
6. Ran: Yes.
7. Michal: Why is it possible?
8. Ran: Because I move only this side. (Points to one of the horizontal sides)
9. Michal: Okay, try.
10. Ran: In fact it is a duplicated square, which was added to... (He
tries and creates a square)
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On the screen:
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Michal: Is it still a rectangle?
Ran: No it is a square.
Michal: So, is this shape not a rectangle now?
Ran: No, it is a square.
Michal: And if I...(Turn the shape on the screen to a rectangle that
is not a square) and now?
Ran: It is a rectangle because this is longer than that. (Points to
the sides) I can see it also by the measures. (Means the measures
of the sides)
Michal: Do you remember that at the beginning I said that a
proper construction is the one that preserves the attributes of the
shape? and just now I said that I constructed a proper construction of a rectangle. Are you suggesting a way to turn this
shape to something else?
Ran: Yes. I made it a square. It will always be a rectangle except
when it is a square – when all is equal.
Michal: Couldn’t everything be equal in a rectangle?
Ran: It may not.
Michal: What must be in any rectangle?
Ran: It has two parallel sides, it is a parallelogram, and it also
must must must have 2 long sides and 2 short ones.
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11.
12.
13.
14.
15.
A similar exchange took place about a rhombus. Ran conjectured
that you can turn a rhombus into a square, but a square is not a
rhombus because a rhombus may not have four equal angels.
Ran perceives rectangles and rhombuses according to a partition
definition (2, 22). His concept image of a rectangle is of a parallelogram with four right angles, two long sides and two short ones (2, 22).
His concept image of a rhombus is of a parallelogram with equal
sides and a pair of equal acute angles and a pair of equal obtuse
angles. He regards the inequality of the length of adjacent sides of the
rectangle and the size of adjacent angles of the rhombus as critical
attributes. The square does not possess those critical attributes, so
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Ran doesn’t recognize the square as a rectangle (13–14, 18–19) or as a
rhombus. Throughout the episode Ran did not change his concept
images of rectangles, rhombuses, or squares, nor did he infer the
hierarchical relations between them.
At this stage Ran did not perceive dragging as a process that
creates different examples of the same geometrical object; and he did
not perceive the dynamic diagram as a figure. He thought that he
could turn the rectangle into a square, which he deliberately did not
consider to be a rectangle (6, 14, 18). He seems to have applied a
mental model of dragging that let him assume that he could turn
a rectangle or a rhombus into a shape that is neither a rectangle nor a
rhombus. It may appear that Ran perceives a square as an instance of
a rectangle, but the next episode lends further support to our argument about Ran’s perception of dragging.
In the next episode we observe Ran’s answers to the reverse
problem: Can you turn a square into a rectangle?
3.2.2. Second episode
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1. Michal: I constructed a proper construction of a square. How do
we know that it is a square?
2. Ran: All are 90 degrees and all are equal.
3. Michal: Do you think you could create a rectangle from the
square?
4. Ran: Yes of course.
5. Michal: Why?
6. Ran: Because I just reduced the square into half and I get a rectangle.
7. Michal: Okay, try it.
8. Ran: It only rotates. No, it only shrinks, grows and rotates. (Ran
tries to turn the square to a rectangle by pulling its sides but the
square only rotates, increases and decreases)
On the screen:
YOUNG STUDENTS EXPERIENCE THE DRAGGING TOOL
25.
26.
27.
28.
29.
30.
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24.
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15.
16.
17.
18.
19.
20.
21.
Michal: Why is it impossible?
Ran: Because it only rotates and all are equal.
Michal: Does it surprise you?
Ran: Yes, I thought that if I drew here... (He means one of the
square’s sides)
Michal: It will let you.
Michal: And why do you think the computer doesn’t let you?
Ran: I don’t know.
Michal: If it would have let you, would it still be a square?
Ran: Yes. No, because not all would be equal.
Michal: Correct, and what did I say, that I made –
Ran: You made a square.
Michal: I made a square. So is it possible to turn a square into a
rectangle?
Ran: Only if you made a square that can be pulled. This one can’t
be pulled. It only shrinks, grows and rotates.
Michal: Can I make a rhombus?
Ran: All the angles are equal so it’s impossible. It will always
remain a square. The shape you constructed will always remain a
square, no matter what you do.
Michal: And other shapes that I constructed?
Ran: You could pull them.
Michal: The rectangle, for instance or, the rhombus: doesn’t it
always remain a rhombus when I pull it?
Ran: No, you can turn it to a square.
Michal: But now it’s no longer a rhombus?
Ran: No it’s no longer a rhombus.
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10.
11.
12.
Ran did not perceive the square as a rectangle or as a rhombus (3–6,
23–24). He believed that one can turn the square into a rectangle (any
kind of rectangle) by dragging although he knew that the square would
not be a square anymore (17–18). When he was not able turn the square
into a rectangle he was surprised and blamed the specific square we had
constructed: ‘‘Only if you constructed a square that can be pulled. This
one can’t be pulled. It only shrinks, grows and rotates’’ (21–30).
Ran apparently did not conceive the dynamic diagram as a figure
and did not understand that dragging preserves the critical attributes
of the shapes; he was surprised to realize that he could not create the
rectangle from the square (11–12). Ran’s mental model of dragging
regards the shape on the screen as a ‘‘rubber band’’ that can be drawn
to create other different shapes. Based on this model, he argued
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(22, 25–30) that we constructed a square that could not be pulled,
contrary to the rectangle and the rhombus, which could be pulled. He
seems to have expected coherence in the dragging manipulation:
being able to pull and ‘‘destroy’’ the square and turn it to a rhombus
or a rectangle, just as he could pull and ‘‘destroy’’ the rectangle and
rhombus and turn them to a square (which in his opinion is no longer
a rectangle or a rhombus). Based on these episodes we infer that
Ran’s mental model of dragging does not take into account the
critical attributes of the shapes.
This case shows that although Ran encountered a conflict between
his conjectures and the results of dragging on the screen, dragging
didn’t help him to construct the expected geometrical knowledge nor
did it change his perception of dragging. Ran operated according to a
partition definition of rectangles and rhombuses and did not grasp
the hierarchical relations between these and squares.
This case emphasizes the interaction between understanding
dragging and the development of knowledge in geometry. It seems
difficult to change the concept images of quadrilaterals without
understanding that a geometrical logic underlies dragging.
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3.3. Case 3
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‘‘You can’t create a square from a rhombus because the angles of the rhombus are
not equal ...’’
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In the following two episodes, Gali, like Ran, holds concept
images of some quadrilaterals according to a partition definition. An
essential difference between Ran and Gali is that the latter holds a
mental model of dragging that considers the geometrical logic
underlying dragging. Note how her concept images of quadrilaterals
change in the course of the interview.
3.3.1. First episode
1. Michal: I constructed a proper construction of a rhombus. Do
you think that you could create a square from the rhombus?
2. Gali: I don’t think so.
3. Michal: Why not?
4. Gali: Because the angles of the rhombus are not equal. But I can
try. (She creates a square)
YOUNG STUDENTS EXPERIENCE THE DRAGGING TOOL
On the screen:
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11.
12.
13.
14.
Michal: What is this?
Gali: It’s a square.
Michal: Is it a square?
Gali: Yes it is 100 percent a square.
Michal: So what, in fact, does it tell us?
Gali: It means that a rhombus is a kind of a square. They are
almost the same but they differ in their angles.
Michal: So is this shape still a rhombus?
Gali: It was a rhombus but not any more now.
Michal: Is it not a rhombus now?
Gali: No, now it is a square. (The interviewer changes the shape
back to a non-right-angle rhombus)
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5.
6.
7.
8.
9.
10.
15.
16.
17.
18.
19.
20.
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On the screen:
Michal: If I change it like this, is it a rhombus now?
Gali: It is a rhombus because it has two pairs of equal angles.
Michal: And if I change it like this? (Changes back to a square)
Gali: It’s not a rhombus. It’s a square.
Michal: Aren’t there two pairs of equal angles here?
Gali: Here there are, so this is a rhombus. It’s a rhombus and a
square as well.
21. Michal: So is it a rhombus and a square as well?
22. Gali: Yes, it’s a rhombus and a square as well.
Gali’s interview shows that she holds a concept image of a
rhombus according to a partition definition: a parallelogram with
MICHAL MAYMON EREZ AND MICHAL YERUSHALMY
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equal sides and non-right angles. She also has a correct concept image
of a square. Like Ran, Gali considered the rhombus attribute of nonright angles to be a critical one. Therefore, at the beginning of the
episode she did not recognize the square as a rhombus. During the
episode a learning process occurred, marking the beginning of a
change in her concept image of a rhombus, when she said that it was
a rhombus ‘‘...because it has two pairs of equal angles’’ (16). This
change in the terms used in relation to the angles of the rhombus
indicates her first successful attempt to verbally express the attributes
of a rhombus in a way that enabled her to consider a square to be a
kind of rhombus. The interviewer noted the change in wording and
guided Gali to the realization that a square also has the critical
attribute of the rhombus (19–20). Learning by conjecturing and
experimenting with pre-constructed shapes, and the help she received
through the interviewer’s leading questions, made it possible for Gali
to further develop her geometrical knowledge, and challenge her
partition view.
Gali seems to hold a mental model of dragging based on its
geometrical logic, namely that dragging preserves the geometrical
attributes of the shape. She established her conjectures about the
dynamic behavior of the rhombus by considering its geometrical
attribute, i.e., that its angles are not and may not be equal (1–4). But
her model was not complete or coherent, because she thought that
the shape on the screen – the right-angle rhombus – was a square
and no longer a rhombus (12–14). She showed inconsistency in her
grasp of the preservation principle underlying dragging when she
argued dragging the rhombus created a square (which is not a
rhombus). We assume that the source of her misunderstanding lies in
her partial geometrical knowledge. Her assumption that the angles
of the rhombus should not be equal exceeded her understanding that
dragging a rhombus creates only instances of a rhombus. Thus,
when she encountered a conflict between her geometrical knowledge
and her understanding of the dragging tool she ignored the principle
underlying dragging.
In the second episode Gali answered another question: Can you
create a trapezoid from a parallelogram? Gali correctly anticipated
the dynamic behavior of the properly constructed parallelogram
according to her concept images of a parallelogram and a
trapezoid.
YOUNG STUDENTS EXPERIENCE THE DRAGGING TOOL
3.3.2. Second episode
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1. Michal: So you were able to make a rectangle from a parallelogram.
2. Gali: Yes a rectangle.
3. Michal: A rhombus and a square.
4. Gali: Because of that it is written here, in the quadrilateral, in the
parallelogram. (Points the interviewer to the shapes’ menu showing
that a rectangle, rhombus and a square can be found under
‘‘parallelogram’’)
5. Michal: Why it is written there?
6. Gali: Because it is what you can create from a parallelogram.
7. Michal: I understand. Can you create a trapezoid from a parallelogram?
8. Gali: No, you can’t
9. Michal: Why not? What do you need to get a trapezoid?
10. Gali: You need one pair of sides which are parallel and one pair
which aren’t parallel
11. Michal: So can you create it if it is a proper construction of a
parallelogram?
12. Gali: No, you can’t.
13. Michal: Why not?
14. Gali: Because it will always be the same.
15. Michal: What do you mean by ‘‘the same’’?
MICHAL MAYMON EREZ AND MICHAL YERUSHALMY
16. Gali: It’s impossible because everything is parallel. You can’t
destroy and make it not parallel. It will always be dragged as a
parallel. (She drags the parallelogram and demonstrates that two
pairs of sides always stay parallel.)
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This episode also shows that Gali understood the dragging tool
and was able to conjecture and explain the dynamic behavior of the
parallelogram. She knew that it was possible to make a rectangle, a
square, and a rhombus from a parallelogram. She also linked between
the software menu and the dragging tool. She showed the interviewer
that all the quadrilaterals (rectangle, rhombus, square) appear in the
Shapes menu under Parallelogram because all of them are types of
parallelogram and can be created by dragging the parallelogram
(1–6). She seemed to hold a concept image of a trapezoid according to
a partition definition: a quadrilateral with only one pair of opposite
sides parallel. Therefore she said that it was impossible to create a
trapezoid from a parallelogram and didn’t consider the parallelogram
to be an example of a trapezoid (7–16).
The two episodes demonstrate that Gali understood the geometrical logic underlying dragging, and she always explained her
conjecturers with regard to the attributes of the quadrilaterals. But
her mental model of dragging was not consistent and stable, and
when she encountered a conflict between her geometrical knowledge and her understanding of the dragging tool she became
confused.
Both Gali and Ran perceived at first the square as not being an
example of a rhombus. Gali’s conjectures, unlike Ran’s, suggest that
she connected between the geometrical logic of dragging, the dynamic
changes in the diagram, and the attributes of the shape. We conclude
that this crucial difference between her and Ran reflects Gali’s
learning with the aid of the dragging tool. This difference made it
possible for her to change her geometric knowledge and perceive the
square as an example of a rhombus.
This case emphasizes the interaction between understanding
dragging and geometrical knowledge about the critical attributes of
shapes. It is essential to understand that dragging a proper construction of shapes creates only different instances of the same
shape and preserves its critical attributes. With this understanding,
students have an opportunity to infer the critical attributes of
different shapes, and change and construct their knowledge of
geometry.
YOUNG STUDENTS EXPERIENCE THE DRAGGING TOOL
4. DISCUSSION
4.1. Three learning interactions with the dragging tool
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Each of the three cases reflects a different and unique interaction
between understanding the dragging tool and geometrical knowledge,
which is manifest throughout an entire interview. We consider
the cognitive aspects that these cases highlight to be important to the
study of the learning of basic concepts in geometry with the aid of the
dragging tool.
Shiri has a mental model of the dragging tool that is consistent
with its conceptual model. By explaining the dynamic behavior with
relation to the critical attributes of the shape she demonstrated an
understanding of the dynamic behavior of properly pre-constructed
shapes and knowledge of the hierarchical relations among quadrilaterals
Ran and Gali demonstrate the challenge of attempting to simultaneously infer the critical attributes of quadrilaterals in the hierarchical perspective expected here and understand that dragging
preserves them. It was difficult for Ran to change his concept image
of quadrilaterals by experimenting with the dragging tool because he
failed to understand that dragging manipulation preserves the critical
attributes of the shape.
When Gali faced a conflict between her geometrical knowledge
and her understanding of the dragging tool she acted inconsistently
and argued that dragging changes the nature of the shape. But because she understood that there was a relation between the dynamic
behavior of the shape and its attributes, it was possible for her to
glean meaningful knowledge from her experiment and to overcome
the conflict resulting from her partition perspective and her mental
model of dragging.
4.2. Reflections on young students’ difficulty to construct knowledge
while learning with the dragging tool
In both the second and third cases students clung to their geometric
knowledge when faced with a conflict, which demonstrate the challenge of constructing geometrical knowledge while learning with the
dragging tool. Ran, for example, was surprised when the square did
not turn into a rectangle. He thought that as he could turn the
rectangle into a square he could also turn the square to a rectangle.
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The fact that it wasn’t possible did not cause him to change his
geometrical assumptions and to construct knew knowledge. In Gali’s
case, a conflict occurred when she could turn the rhombus into
square, contrary to her expectations. At first she argued that the
shape on the screen, the square, was no longer a rhombus. But after
further experimentation with the dragging tool, rewording the
rhombus attributes, and with the guidance of the interviewer she
argued that a rhombus could also be a square. In both cases the
dynamic behavior in itself was not fully internalized and did not
convince the students, who adhered to their concept images of
quadrilaterals.
Another conclusion regards the difficulty to change the concept
images of quadrilaterals without understanding that a geometrical
logic underlies dragging. Understanding that dragging preserves the
critical attributes of the shape is necessary for constructing the concept images of the shapes expected here.
The discussion of the three cases indicates that it is difficult to
change the concept image of quadrilaterals and at the same time
understand the mathematical logic of the dragging manipulation.
One reason for this is the students’ tendency to abide by their existing
knowledge and their difficulty in changing it when they meet different
and unanticipated results on the screen. Another reason has to do
with the kind of thoughts and skills young students need in order to
understand the dragging tool. When dragging a shape, students need
to track the visual changes and infer what has been changed and what
has been preserved. This is a complex action, as the students must
simultaneously think about geometrical attributes and trace specific
change and invariance. But young students’ thinking is concrete at
this stage, and they are influenced mostly by visual changes, so it is
difficult for them to infer what is preserved during dragging. During
the dragging manipulation the size and position of the shape change
continuously, and only the critical attributes are preserved. If
students do not think about these attributes during the dragging of
the parallelogram, and unless these attributes visually dominate the
diagram, they do not notice the preservation of the parallel sides, the
equal opposite sides, etc. Students need to focus on preserved attributes while for them the visual changes on the screen are more
dominant. Understanding the properly pre-constructed shape as
constantly representing the same object requires formal thinking.
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4.3. Pedagogical considerations
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These findings about the different types of learning and thinking
taking place while learning with a technological tool are consistent
with the literature on mental models. Norman (1983) argues that
mental models are incomplete and unstable, and that users of a
technological tool continue to modify the mental model of the tool in
the course of their interaction with it.
Analysis of the mental models that students develop while
experimenting with the dragging tool can assist educators in formulating activities and guiding students who receive instruction with the
aid of a dragging tool. It is important to create a learning environment that encourages young students to use different software
options in various situations. An environment of this type incites
controversy and causes the student to develop and design mental
models of the software rich in mathematical meanings. The learning
environment should stimulate students to conjecture, ask questions,
and discuss their thoughts and disagreements with their classmates.
The teacher has an important role in the learning environment;
‘‘conducting’’ the learning process and guiding the students. Based on
the interviews, we conclude that without guidance it is difficult to
construct new geometrical knowledge based on ‘‘free’’ investigation
and experimentation with the dragging tool.
The mathematics education literature lends support to learning
geometry with pre-constructed shapes by young students. Research
about learning with the ‘‘Shape Maker,’’ a specially designed
microworld for students at the pre-proof stage of geometrical
thinking, is one such example (Battista, 2001). In this environment
each class of common triangles and quadrilaterals has a ‘‘shape
maker,’’ a Geometer’s Sketchpad construction that can be dynamically transformed in various ways to produce different shapes within
that class. Battista (ibid.) found that as students manipulated and
reflected on their manipulations with the ‘‘Shape Maker’’ they abstracted certain actions and integrated these abstractions into a
mental model of the ‘‘shape maker’’ that constituted their construction of meaning for the device. Battista found that as the students
worked with the ‘‘Shape Maker’’ they moved from visual to propertybased thinking, and hence to thinking that utilized inference to relate
and organize both attributes and classes of shapes.
We demonstrated the vague discrimination between relevant
and irrelevant features of the shape and suggested that dragging a
MICHAL MAYMON EREZ AND MICHAL YERUSHALMY
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pre-constructed shape is not explicit. The students had difficulty in
identifying what was preserved when the shape was dragged. To
overcome this problem it is necessary to teach within what Vygotsky
(1978) called the ‘‘zone of proximal development.’’ When this type of
teaching and learning is assisted by software, it is important to rethink the design features that encourage students to develop better
distinctions between geometric attributes and infer what has been
preserved during the dragging manipulation. The following three
recommendations would make the geometric attributes more explicit
when investigating examples of the same shape.
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(1) Adding a ‘‘discrete’’ dimension to the continuous change of the
dragged shape. When the shape is dragged, after every fixed
interval the shape leaves traces (like the traces of the parallelogram in Figure 4, left) that persist on the screen. Such traces may
enhance the construction of the relevant distinction and provide
an environment more conducive to comparing the examples and
identifying the elements that are preserved during dragging.
(2) Visually defining the trail of the dragging to help the student see
that there is a consistency in the dynamic continuous change of
the shape. The visible trail would demonstrate one of the shape’s
critical attributes, as the parallel lines that restrict dragging only
to changing the length of two sides (Figure 4, middle).
(3) A simultaneous examination of randomly pre-constructed static
examples of the same shape. This idea is currently being tested in
another study we are conducting (Maymon Erez, in preparation)
to determine whether the simultaneous examination of several
random and static pre-constructed examples is an effective
method of encouraging young students to start deducing the
critical attributes of shapes (Figure 4, right).
In sum, this paper presented the complexities faced by young
students learning with the dragging tool and suggested some of the
Figure 4. Three recommendations. left: discrete traces; middle: visible trail; right:
random examples of a ‘‘shape’s family.’’
YOUNG STUDENTS EXPERIENCE THE DRAGGING TOOL
cognitive difficulties inherent in this process. By analyzing the design
of the dragging tool and the students’ thoughts and difficulties while
using it, we hope to have contributed to clarifying some of the
complexities and providing some answers.
NOTE
1
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Yerushalmy Michal
E-mail: [email protected]
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Michal Maymon Erez
Department of Education
University of Haifa
Mount Carmel, 31905,
Haifa, Israel
E-mail: [email protected]