1. Introduction
In the late 1980s, Dynamic Geometry Systems first appeared, this software developed specifically
for the teaching and learning of plane geometry, including tools that allowed users to manipulate
figures, mathematical objects on the screen directly and automatic [7-9, 17]. The first dynamic
geometry software packages were The Geometer's Sketchpad (GSP) and Cabri Geometry. In recent
decades, there are students who are interacting directly with Dynamic Geometry Environment (DGE)
to gain knowledge is being studied by many mathematics educators around the world. With the
support of softwares such as GSP, Cabri, and Geogebra, the transition from the traditional paper-andpencil based graphical environment to the "virtual" graphical environment based on on-screen data,
through dragging, has the potential to deeply influence how students perceiving and reasoning in
geometry. From here, educators are beginning to pay attention to using computers to teach geometry
and discussing the effectiveness of using dynamic geometry softwares in education as well as exploiting
the strengths of software compared to teachers using traditional chalkboards [3, 6, 13, 14, 17].

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Nguyen Dang Minh Phuc, Nguyen Huu Hau and Huynh Thi Ai Hang
62
HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1075.2017-0176
Educational Sciences. 2017, Vol. 62, Iss. 12, pp. 62-73
This paper is available online at
GENARATING ABDUCTIVE CONJECTURES THROUGH CONDITIONAL LOCUS
MATHEMATICS PROBLEMS IN DYNAMIC MATHEMATICS REPRESENTATIONS
Nguyen Dang Minh Phuc
1
, Nguyen Huu Hau
2
and Huynh Thi Ai Hang
3
1
Department of Mathematics, Hue University of Education
2
Department of Training Management, Hong Duc University
3
Secondary school Nguyen An Ninh, Ho Chi Minh City
Abstract. Dynamic mathematics representations are effective in supporting students to conduct
investigations to generating abductive conjectures. This study aims to construct conditional locus
problems to support students in exploring mathematical representations to generate abductive
conjectures. The empirical results showed that these conjectures are proposed in the exploration
and cooperation process among students, which is optimized and strengthened by dragging
modalities.
Keywords: Abduction, conjecture, dynamic mathematics representations, locus mathematics.
1. Introduction
In the late 1980s, Dynamic Geometry Systems first appeared, this software developed specifically
for the teaching and learning of plane geometry, including tools that allowed users to manipulate
figures, mathematical objects on the screen directly and automatic [7-9, 17]. The first dynamic
geometry software packages were The Geometer's Sketchpad (GSP) and Cabri Geometry. In recent
decades, there are students who are interacting directly with Dynamic Geometry Environment (DGE)
to gain knowledge is being studied by many mathematics educators around the world. With the
support of softwares such as GSP, Cabri, and Geogebra, the transition from the traditional paper-and-
pencil based graphical environment to the "virtual" graphical environment based on on-screen data,
through dragging, has the potential to deeply influence how students perceiving and reasoning in
geometry. From here, educators are beginning to pay attention to using computers to teach geometry
and discussing the effectiveness of using dynamic geometry softwares in education as well as exploiting
the strengths of software compared to teachers using traditional chalkboards [3, 6, 13, 14, 17].
2. Content
2.1. The origin and theoretical base of conditional locus mathematics problems
From the benefits of DGE, many mathematics educators pay close attention to incorporating
localized problems into the DGE, thereby generating conditional locus mathematics problems - one
kind of problem special designed on DGE to help students exploring and solving problems in a more
intuitive and profound way. Conditional locus mathematics problems have been developed since the
Received: September 27, 2017. Revised: December 10, 2017. Accepted: December 16, 2017.
Contact: Nguyen Huu Hau, e-mail address: nguyenhuuhau@hdu.edu.vn
Genarating abductive conjectures through condictional locus mathematics representations
63
study of mathematics educator Arzarello et al. In the late '90s, Arzarello and his colleagues studied
and classified the different dragging modalities used by leaners during the process of solving
geometric problems in Cabri [1].
In recent years, mathematics educators have been encouraging the use of technology in classroom
to foster the math skills of each student [2, 4, 14, 15, 18]. Some studies in geometric teaching and
learning show that DGE can promote the way leaners thinking, how the learners’ constructions were
built to use dragging tools. A DGE can be powerful enough to examine an open-ended problematic
situation [1, 5].
Involving the designing of open-ended problems to create conditional locus mathematics
problems, we focus primarily on the contributions of mathematics educators such as Allen Leung,
Baccaglini - Frank and Mariotti. According to Leung [10, 11], the discovery through dragging
principle in DGE is process-oriented and user-centered. This process is open to teaching and learning
geometry ability for exploring and experiencing, supplementing for teaching and learning approaches
and inductive inferences.
2.2. Conditional locus mathematics problems in DGE
As with regular locus problems in the paper-pencil environment in which students have been
introduced in the junior high school level, the mathematics problem of conditional locus in DGE is
basically finding a set of points that met given conditions. In geometry, finding a set of points is
descripting in detail of that set. For example, the locus is a circle, a line, a segment
However, the difference between the locus problem on the paper-pencil environment and the
problem of conditional locus on DGE is that the locus problem in the paper-pencil environment is
usually expressed as: A configuration has a number of fixed elements, and one (or several) element
that changes to a certain requirement (point moves on a circle, a line turns around a point ...), this
changed factor will lead to the movement of a number of other factors, then seeking to find the loci of
the relevant factors; meanwhile, in DGE, when moving one element, the students not only have to
look at the behavior of other factors depend on it, but also requires finding locus of this mobility
element so that it still reserved some given properties.
Example 1. Construct three points A, B, and C on the GSP screen, construct the lines AB and AC.
Then, construct the line l parallel to AC through B, and a perpendicular line to l through C. Name the
intersection of the two lines is D, considering the quadrilateral ABDC. Guessing the special
quadrilaterals that ABDC can become to and finding the locus of point A so that ABDC becomes to the
special quadrilaterals.
Figure 1. Maintaining-Dragging point A so that ABDC is rectangular
Nguyen Dang Minh Phuc, Nguyen Huu Hau and Huynh Thi Ai Hang
64
From the statement of problem, the construction of A, B, C showed that these three points can
move freely, not necessarily fixed. Specifically, when moving point A, students discover for
themselves what factors will change and what remains unchanged, dragging A so that "ABDC keeps
still a rectangular", enable tracing, and predicting specific shape of traces of A. From this, the learner
not only seeing the visualization of the regular locus problems but also having chance to investigate a
new kind of locus problems, characteristic of the DGE, generating abductive conjectures.
2.3. Abductive reasoning, generating abductive conjectures
Abduction describes a special type of reasoning that leads to the realization of experience
phenomena through the formation of hypotheses/conjectures to explain them. In the description of
Magnani [12], abduction is the process of inferring facts/rules and assumptions to make a problem
reasonable, to discover and explain a phenomenon.
When the solvers discovered an open-ended problematic situation in DGE and was required to
construct an abductive conjecture for a given geometric object, they try to find the invariants, that is,
the properties of the figure are constant while dragging a given point, and try to link two (or more)
given geometric invariants (or easily derive directly from the assumptions) by a conditional
proposition [2-4, 12]. In fact, when invariants are discovered through maintaining-dragging, there is
often an order in detecting them: the initial invariant properties (level 1) are maintained through
dragging, the invariants need to detect (level 2) will be appeared when initial invariants are
maintained.
The process of generating abductive conjectures of learners as they investigating the conditional
locus mathematics problem on dynamic mathematical representations is a matter of considerable
research in two respects: (1) Designing dynamic mathematics representations in appropriate topics for
investigators to help them generating abductive conjectures, and (2) The process of generating
abductive conjectures of investigators. From this point of view, we focus on two research questions:
How to build conditional locus mathematics problems on dynamic mathematics
representations to support students in generating abductive conjectures?
How are students’ conjectures generated when they solve the conditional locus mathematics
problems on dynamic mathematics representations?
2.4. Dynamic mathematics representations
Mathematics representations on the GSP differ from what we draw on paper with universal tools
not just because of structural accuracy. GSP remembers the relationships among different objects in
that structure when dragging free objects. For example, in Example 1, it remembers the line l parallel
with AC through B, remembers that the BD is perpendicular to DC line. According to Finzer & Jackiw [7],
DGEs are characterized by three properties:
Manipulation is direct. In Example 1, you grabbed point A and dragged it. The cognitive
distance between what is on the screen and the mathematics behind it is minimal. You do not feel
inclined to say, “I’m moving the mouse, which drags this small circle on the screen, which changes the
coordinates of point A.” You say, “I’m dragging the point A.”
Motion is continuous. Change takes place during the drag. When point A moves, the objects
associated mathematically with A also follow, and you can see all the intermediate states.
Genarating abductive conjectures through condictional locus mathematics representations
65
The environment is immersive. Your experience is that you are involved with the objects you
are manipulating—surrounded by them, exploring them, playing with them. The interface is minimally
intrusive so that your focus is on how to accomplish your mathematical goals, not on how to drive the
technology.
Dynamic mathematics representations are mathematics representations designed on dynamic
geometry softwares. According to Baccaglini - Frank and Mariotti [2], we can generally look at two
different worlds: the mathematical world of Euclidean geometry, and the empirical world, the learner's
experience on DGE. A DGE can be a potential bridge between the two worlds, providing teachers with
new insights and tools to overcome their difficulties. This is because DGE not only motivates students
to develop their abilities but also gives teachers the opportunity to tailor their teaching to their special
needs. Students who are neglected with mathematics learning may be more focused on computer
mathematics problems. Students who have difficulty learning mathematics, they can derive the results
from their mistakes when formulating abductive conjectures through investigations on dynamic
mathematics representations.
2.5. Conjectures generating base on dynamic mathematics representations
2.5.1. Dragging on dynamic geometry environment
The use of dragging methods has been extensively studied in earlier papers by mathematics
education researchers through the observation of students’ natural behaviors (Arzarello et al., 2002)
and teaching experiments to introducing dragging modalities [2, 4, 6].
Dragging in DGE can be done by users, through the computer mouse; they can determine the
movement of different objects in two ways:
A direct motion of a fundamental element (eg. a point) represents the change of this factor in
the plane. Users can select the appropriate elements and drag it on the screen.
An indirect motion of an element that occurs when this element moves on the screen,
depending on the movement of a selected base point (or object) being dragged.
2.5.2. Dragging Modalities
With the study of the Maintaining Dragging Model, Baccaglini-Frank and Mariotti ([2]) have
proposed four modalities described below.
Wandering/random dragging (Italian: ‘‘trascinamento libero’’): randomly dragging a base
point on the screen, looking for interesting configurations or regularities of the Cabri-figure;
Maintaining dragging (Italian: ‘‘trascinamento di mantenimento’’): dragging a base point so
that the figure maintains a certain property;
Dragging with trace activated (Italian: ‘‘trascinamento con traccia’’): dragging a base point
with the trace activated;
Dragging test (Italian: ‘‘test di trascinamento’’): dragging base points to see whether the
constructed figure maintains the desired properties. In this mode it can be useful to make a new
construction or redefine a point on an object to test a formulated conjecture.
Example 2. Given 3 points A, M, K on the GSP screen. Construct the point B that is symmetrical
with A through M and C is symmetrical with A through K. Let D is the symmetry of B over K. Drag
the point M and predict quadrilateral ABCD may become, and find the locus of point M so that ABCD
is that particular quadrilateral.
Nguyen Dang Minh Phuc, Nguyen Huu Hau and Huynh Thi Ai Hang
66
Firstly, we wandering/random drag the point M to see whether the quadrilateral ABCD has any
particular shape. By the way, ABCD is a parallelogram. In addition, for a certain position, ABCD will
“look like” a rectangle. After fixing the special type of ABCD as a rectangle, we start maintaining
dragging the point M at other positions so that the ABCD is retained a rectangle. Next, we start
dragging with trace activated for M, where M is moving so that ABCD remains rectangle, traces on
the screen allow us to predict the traces of point M “looks like” a circle.
Figure 2. Traces of point M “look like” a circle
After dragging with trace activated, it is possible to predict that the trace of point M is the circle
of diameter AK. We start dragging test to see if the traced circle is maintained the desirable properties
or not. If so, we can propose a prediction: "If M is on the AK circle, and then ABCD will be a
rectangle”.
2.6. Invariants perception on dynamic mathematics representations
In DGE, when a figure is constructed, it can be altered (through drag) while preserving all
graphical properties along with all of its consequent properties under Euclidean geometry. Leung [10]
described mathematical experience as “the discernment of invariant pattern concerning numbers
and/or shapes and the re-production or re-presentation of that pattern”. An important feature of DGE is
the ability to visually represent all geometry invariants at the same time caused by dragging. Dragging
in DGE involves selecting an element of a dynamic figure (a shape built follow a set of properties in a
DGE) with a point and moving along with changing the position of the selected objects (and thus,
other possible objects) on the screen.
To illustrate this, we can imagine simply as follows. Let the triangle ABC with E and F
respectively are the midpoints of AB and AC. When dragging A, B, or C, the midpoints E and F are
still centered, meaning that "AE = BE" and "AF = CF" are preserved. However, the invariant "EF is
parallel to BC" is also preserved because EF is the median of the triangle. Perception through these
relationships can lead to the following abductive conjecture: "If a line is constructed with the
beginning and the end points being the midpoints of two sides of a triangle, then it is parallel with the
third edge of the triangle". Leung [10, 11] has given the following terms:
Level-1 invariants: Aspects of a dynamic figure, potentially corresponding to geometrical
properties, which are perceived as constant during variation of the figure through dragging. For
example, "AE = BE" and "AF = CF" level-1 invariants of the dynamic figure.
Level-2 invariants: Invariant relationships among level-1invariants. For example, “AE = EC, CF
= FB causes (or leads to) EF // AB”.
Genarating abductive conjectures through condictional locus mathematics representations
67
2.7. Research Design
This section describes the research process; students participated in the experiment, research tools.
2.7.1. Research process
Analyzing, classifying conditional locus mathematics problems, especially problems that
support the possibility of generating abductive conjectures.
Designing conditional locus mathematics problems, integrating into mathematical tasks to
become teaching activities, preparation of learning sheets (worksheets), questionnaires, and
experimental recording equipment.
Conduct data collection and data analysis to answer research questions.
2.7.2. Students participated in the experiment
The experiment was conducted in the second semester of the 2014-2015 school year for students
of grade 9 of Nguyen Van Linh junior high school, Hue city, Vietnam. In this study, we used a case
study. We selected 30 students to conduct experiments on the computer directly interacting with the
GSP to observe in detail the process of solving the conditional locus mathematics problems of each
student base on the problems designed in the following section. Students’ names are abbreviated when
presented in the data analysis results section.
2.7.3. Research tools
The research tools include dynamic mathematics representations designed on the GSP integrated into
the worksheets and student surveys. Within the scope of this paper, we present only a worksheet
containing a conditional locus mathematics problem with the tasks.
Worksheet content
a. Introduction
Construct three points A, B and C on the GSP screen, draw a line through A and B and a line
through A and C. Then line l is parallel to AC through B, and a line that perpendicular to l through C.
Name the intersection of these two lines as D, considering the quadrilateral ABDC.
Figure 3. The quadrilateral problem
Task 1: Drag point A to change its position. Take a survey to get a comment on the factors that do not
change their positions and the factors that change their positions when point A moved. Then complete
the following table.
While dragging point A
What points move?
What points do not
move?
What properties
reserved?
What properties did
not reserve?
Nguyen Dang Minh Phuc, Nguyen Huu Hau and Huynh Thi Ai Hang
68
Task 2: Generate conjectures about the quadrilaterals that ABDC can become.
Task 3: Drag point A so that ABDC becomes a special quadrilateral form that found in task 2, trace
and predict the path of A while keeping ABDC in that special quadrilateral.
Task 4: Find the locus of point A so that ABDC is the special quadrilateral and explain it in your own
way.
2.7.4. Data collection and data analysis process
The data collected included: students group worksheets, questionnaires, and students’ recording
data. For data analysis, we use a student-with-computer approach with dynamic mathematics
representations under the support of teachers. This approach is demonstrated by the following model:
Figure 4. Model for data analysis
2.8. Data analysis results
a. Task 1. Finding invariants
Finding invariants is an important task in determining the locus of the point of drag. With
wandering/random dragging, students focused on the change of the figure while moving the point A.
The result of student group A. - L. after investigation process is as follows:
For group Th. - Tr., corresponding results are "Points A and D moved," "Points B and C did not
move," the invariant are "AC // BD, AC is perpendicular to CD, BD is perpendicular to CD”. The non-
invariant is "the measure of angles CAB, DBA” and “AB is not parallel to CD".
Through random dragging, which is only available in DGE, students can easily see level-1
invariants such as dragging point A and points B and C remain unmoved. Moreover, according to the
construction, AC is parallel with BD and the mea