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Assessing students’ understanding of
parallel lines and related angle properties
in a dynamic geometry environment
CHENG, Lo Carol
True Light Middle School of Hong Kong
Introduction
Assessment is one of the important aspects in education, not just for
mathematical learning and teaching, but for learning and teaching in general.
Once we want to assess learners’ understanding and knowledge within or after
several learning processes, we need to do assessment.
Some mathematics educators (Ginsburg, Jacobs & Lopez, 1993) argue that
assessment in mathematics education should not just focus on rote
memorization of number facts or ability for computational procedures but
students’ thinking and learning potential in mathematics since they believe that
mathematics education should help students to think mathematically and train
up their mathematical thinking.
Based on the demand for assessment, more and more ways of assessment
are expected to be developed for serving different purpose. Besides, as
technology changes from time to time, it is quite predictable that mathematical
skills and knowledge required in curriculum will be shifted throughout the
world in the near future when new forms of technologies are introduced into
schools. For example, the introduction of calculator in our curricula allows
students to perform more tedious computation and they are expected to draw
conclusions from these numerical results rather than just know the techniques to
compute correctly.
With the insights into the development of dynamic geometry (abbreviated
as DG below), some researchers (e.g. Lee, Wong & Leung 2004, 2006) begin to
investigate the potential of DG not only as a supportive means to learning but
also as an assessment tool. Lee, Wong and Leung (2006) suggest that traditional
paper-and-pencil assessment focuses on assessing student’s ability to find an
unknown geometric measurement or to write a proof for an unfamiliar
geometric property. This assessment practice has influenced teachers’ practices
which seem to be more focusing on product-oriented curriculum rather than
process-oriented curriculum. It seems to have led to one of the reasons why
teachers may be hesitant to bring DG into classroom since DG is more likely to
be used to explore and form conjectures in learning geometry concepts. They
propose that using a new kind of DG manipulative tasks as an assessment may
bring more insights to both potential use of DG and assessment in geometry. A
dynamic geometry assessment platform has been established to serve as an
alternate means for teachers to do both assessments and teaching in geometry.
Some of the DG tasks which are related to parallel lines and some interesting
findings about students’ performance thereon will be discussed in this paper.
Before we proceed, we may need to have a glance at some important factors,
suggested by Duval, in learning and teaching of geometry.
Learning and Teaching of Geometry
Comparing to numerical operations or elementary algebra, learning and
teaching of geometry may be considered as one of the more complex and less
successful. Duval (1998) tries to identify reasons for why and how geometry
should be taught in school. He explains that geometry involves three cognitive
processes which fulfill specific epistemological functions:
- Visualization process refers to space representation for illustration,
exploration or verification of different geometric situations;
- Construction processes are related to actions for constructing a
configuration according to restricted tools and geometrical requirements;
- Reasoning is related to discursive processes for proof and explanation.
As he further mentioned that “these three processes are closely connected
and their synergy is cognitively necessary for proficiency in geometry” (Duval,
1998, p.38), he believes that these three kinds of processes must be developed
separately; various ways of seeing figures and reasoning should be included in
the curriculum.
Seeing, constructing and describing a geometrical figure and its properties
with mathematical sense is not an easy task for many students. Furthermore,
different figures for the same mathematical situation may induce variations in
performances. Duval (1995) uses four cognitive apprehensions: perceptual,
sequential, discursive and operative as an analytical framework to explain why
students’ performance varies with different geometrical figures. He believes that
there are several ways of looking at a drawing or a visual stimulus array and
points out that “mathematical perception is not simple and it overlaps the
different apprehensions” (Duval, 1995, p.155). The following is a summary of
his four apprehensions:
(i) Perceptual Apprehension
It is about physical recognition (shape, representation, size, brightness, etc.)
of a perceived figure. We should also discriminate and recognize
sub-figures within the perceived figures since a relevant discrimination or
recognition of these sub-figure units may help and give cues for problem
solving in geometrical situations.
(ii) Sequential Apprehension
It is about construction of a figure or description of its construction. Such
construction depends on technical constraints and also mathematical
properties since construction of a figure may merge different figural units. It
is believed that construction can help recognition of relationships between
mathematical properties and technical constraints.
(iii) Discursive Apprehension
Mathematical properties represented in a drawing can only be clearly
defined with speech determination. Making denomination and hypothesis is
very important for one to know and derive the mathematical properties
within the representation; otherwise, it is only an ambiguous representation.
We should be aware that there is a gap between what the figure shows and
what it represents and, as Duval (1995, p.146) mentioned, “what the
perceived figures represent is determined by speech acts”. Different
conclusions and deductions may be drawn from the same drawing if the
given denominations and hypotheses changed.
(iv) Operative Apprehension
It is about making modification of a given figure in various ways:
 the mereological way: dividing the whole given figure into parts of
various shapes and combine these parts in another figure or sub-figures;
 the optic way: varying the size of the figures;
 the place way: varying the position or its orientation.
These modifications can be performed mentally or physically. A relevant
way of figural changes can give insights to the solution of a problem.
However, operative apprehension seems to be the most difficult one among
the four since there are various possible figural modifications, finding an
appropriate and relevant one to bring solution to a given question becomes
a difficult task for many students.
Duval (1995) believes that these apprehensions are important for students
to become proficient in geometry and they are closely related to each other as he
has mentioned that “operative apprehension does not work independently of the
others, particularly of discursive apprehension. …Special and separate learning
of operative as well as discursive and sequential apprehension are required, and
a mathematical way of looking at figures only results from coordination
between separate processes of apprehension over a long time” (Duval, 1995,
p.155).
Visualization and Reasoning
Visualization is an important skill for us to analyse geometric problems
and situations; but seeing does not imply reasoning. Visualization may be an
intuitive aid for finding a proof or explanation to a problem but sometimes it
may also be misleading, especially for those who are inexperienced in dealing
with mathematical propositions and geometric relations. Duval (1998) begins to
investigate the relationships among different apprehensions, especially
perceptual with discursive and perceptual with operative. Besides, he also
claims that there are several triggering or inhibiting factors which may influence
the visibility of relevant re-configuration of a given figure.
Duval (1998, p.45) believes that there are three cognitive processes in
geometry involved in problem solving and in proof:
1. a purely configural process which means operative apprehension;
2. a natural discursive process which is performed in speech through
description, explanation, argumentation;
3. a theoretical discursive process which is performed through deduction.
It is quite obvious that there is always a gap between the natural discursive
process, which is closer to “everyday” language and sometimes can be designed
as “figural reasoning”, and the theoretical discursive process, which is
performed in a purely symbolical register or in the natural language register.
Even though the deductive process is a difficult task to many students, we still
think it is worthy to be taught since we believe that those who can discover such
process can experience the logical necessity of the conclusion and the power of
this way of reasoning.
Duval (1998) thinks that visualization and reasoning is a “double gap”
between naïve behaviour and mathematical behaviour. The main problem of
teaching and learning geometry is how to get pupils to step over this double
gap.
Tasks related to parallel lines in a DG platform
Based on Duval’s ideas of visualization and reasoning, I have designed
fourteen DG manipulative tasks which involve parallel lines and related angle
properties to assess students’ understanding and also their problem solving skills
in a DG platform. Altogether 88 students from three classes (namely, two from
S3 and one from S1) have participated in this DG test. It is reasonable to believe
that students may act differently in a DG test compared with traditional
paper-and-paper test. Besides, figures (including shapes and measurements) in
the DG tasks can be varied through dragging a point; such variations may help
to differentiate students’ abilities in dealing with geometry relations. Two tasks
concerning orientation preference will be discussed in the following sections.
Task Design and Preliminary Observations
I. Testing Orientation Preference
Task 2a (shown in Figure 1.1) involves the theorem about the
supplementary property of the pair of interior angles on the same side of a
transversal through a pair of parallel lines. It is designed to assess whether
students may have preference to orientation of the parallel lines (horizontal or
vertical in particular) in solving a problem. The coordinates of point P,
measurement of  and  are recorded to facilitate further investigation.


Figure 1.1
Task 2a (orientation preference)
Figure 1.2
Measurement of  (referring to a
horizontal pair of parallel lines)
Figure 1.3
Measurement of  (referring to a
vertical pair of parallel lines)
Figure 1.4
Figure 1.5
More students have chosen to have
a horizontal pair of parallel lines
A student making opposite angles equal
From Figure 1.4, we note that more students have chosen to have a
horizontal pair of parallel lines by making the angle  supplementary to the
fixed angle 75. There are 43 out of 88 chose to use the “upper-lower” pair
while only 9 chose the “left-right” pair. Also, there are 27 students chose both
orientations, i.e. making a parallelogram. It may suggest that students prefer to
choose horizontal orientation for making parallel lines. It is interesting to note
that 4 students of S3 have made the opposite angles equal rather than making a
pair of parallel lines.


Figure 2.1
Task 2b
Task 2b (as shown in Figure 2.1) is similar to Task 2a as it is also used to
test students’ orientation preference in dealing with geometric figures. In Task
2b, two theorems related to making parallel lines are also used for comparison.
We may assume that making corresponding angles equal should be a more
fundamental and essential concept than theorem of interior angles
supplementary since no calculation is needed for applying the first one. But,
would orientation preference have any influence on students’ choice? This
becomes an interesting question to us.
Figure 2.3: Measurement of 
Figure 2.2: Measurement of 
(corresponding angles with respect to a (interior angles with respect to a
horizontal pair of parallel lines)
vertical pair of parallel lines)
Figure 2.4
Coordinates of P in Task 2b
Among the 88 responses, there are 27 students who are looking at interior
angles for solving the task while there are 21 using corresponding angles.
Besides, there are 26 students making two pairs of parallel lines, i.e. both
horizontal and vertical pairs. In Figure 2.4, it is shown that more students will
choose vertical orientation because of the corresponding angles theorem but still
more students prefer to use the interior angles theorem because of the horizontal
orientation.
These DG tasks are differently designed from those in paper-and-pencil
form. Since there are more than one answers can be used, we might understand
students’ perception about parallel lines and its related angle properties by
observing students’ choice for answers. Besides, traditional types of questions
usually ask students to find unknowns or prove certain geometric relations by
deduction. In these DG tasks, students are encouraged to do more exploration
by dragging. One of the exploration tasks will be discussed in the following.
II. Exploring by dragging

Figure 3.1a
Figure 3.2
Figure 3.3
Figure 3.1b
Figure 3.1c
45 out of 88 students chose 116 for the value of 
Students’ response to the task: making  = 116
Students cannot figure out the answers from Figure 3.1a until they drag the
point P to find the value of angle A in Figure 3.1b first. This exploration process
is a crucial step to solve the task. After this, they need to figure out the angle
sum and make  become 113 shown as Figure 2.1c. Students’ performances in
this task are just as poor as expected. Dragging to obtain extra information from
existing figure has not been trained in our school, many students actually have
hesitation in dragging process since dragging actually change the shape of the
original figure which may create uncertainty to students who are not familiar
with DG and geometric relations. In fact, many students chose to look at the
pair of opposite angles as a pair of interior angles in this task. Figure 3.2 shows
that there are 45 students choosing 116 as the answer. From these responses,
we notice that students are familiar with the concept of making sum of two
angles 180 but the position of the interior angles pair seems to be a big
question for them (see Figure 3.3). Besides, many students may not be able to
figure out the measurement for angle A and they may just tackle this task by
visual judgment. We note that there are 15 students making the angle ranged
from 110 to 112 which can make the two lines look almost parallel.
Conclusion
These DG manipulative tasks are mostly considered to be easy for most
students; however, they do give us more thought about students’ perceptions of
parallel lines. Introduction of DG manipulative tasks and an assessment
platform using DG to our curriculum may not provide a better way of assessing
but it definitely helps us to review these fundamental concepts of parallel lines
in a different way. I believe that the use of these DG manipulative tasks can
bring more insights about our school geometry teaching and learning and it
leaves much room for us to explore.
References
Duval, R. (1995). Geometrical pictures: Kinds of representation and specific
processings. In R. Sutherland & J. Mason (Eds.), Exploiting Mental
Imagery with Computers in Mathematics Education (pp. 143–157),
Springer published in cooperation with NATO Scientific Affairs Division.
Duval, R. (1998). Geometry from a cognitive point of view. In M. Carmelo & V.
Vincio (Eds.), Perspective on Teaching of Geometry for the 21st Century:
An ICMI Study (pp. 37–52). Dordrecht: Kluwer Academic Publishers.
Ginsburg, H. P., Jacobs S. F. & Lopez L. S. (1993). Assessing mathematical
thinking and learning potential in primary grade children In M. Niss (Ed.),
Investigations into assessment in mathematics education: An ICMI study
(pp. 157–167). Dordrecht: Kluwer Academic Publishers.
Lee, A. M. S., Wong, K. L. & Tang, K. C. (2004). Exploring the use of dynamic
geometry manipulative tasks for assessment. In W. C. Yang, S. C. Chu, T.
de Alwis & K. C. Ang (Eds.), Proceedings of the 9th Asian Technology
Conference in Mathematics (pp.252–261). National Institute of Education,
Singapore, Dec 13–17.
Lee, A. M. S., Wong, K. L. & Leung, A. Y. L. (2006). Developing learning and
assessment tasks in a dynamic geometry environment. In C. Hoyles, J.-B.
Lagrange, L. H. Son & N. Sinclair (Eds.), Proceedings of the Seventeenth
International Commission on Mathematical Instruction Study Conference –
Digital Technologies and Mathematics Teaching and Learning: Rethinking
the Terrain (pp.334–341). Hanoi University of Technology, Hanoi, Vietnam,
Dec 3–8.
Author’s e-mail:
[email protected]