Download The Effect of Dynamic Geometry Software_Redacted A+ 81

The effect of dynamic geometry software
on education in early secondary school
Mathematics
2
Table of Contents
Introduction .................................................................................................................................. 4
Issues in geometry education ................................................................................................. 5
The power of dynamic geometry software and the dangers that this brings ........ 7
Remarks .......................................................................................................................................... 8
Teaching Sequence...................................................................................................................... 9
Methodology and Ethics ..........................................................................................................11
Discussion of results .................................................................................................................12
Conclusion ....................................................................................................................................25
References ....................................................................................................................................27
3
The effect of dynamic geometry software on education in
early secondary school
Introduction
As the longest standing area of study in modern mathematics, the longevity of
geometry is unquestionable (Royal Society, 2001). In comparison, algebra and
analysis are mere toddlers, or at the very least adolescents. Although the
popularity of Euclidean Geometry as a curriculum element of secondary
education may have had its ebbs and flows during the second half of the 20th
century (Whiteley, 1999) before this time it was considered the dominant force
in mathematics education, with Euclid’s Elements forming the basis for the
majority of the curriculum. It is perhaps unsurprising that this should be the
case, as we inherently exist in a spatial realm and therefore might expect that our
education might be influenced by this fact.
More so than this, however, it should be noted that historically geometry was the
first branch of mathematics formalised into an axiomatic structure. Within the
domain of education, it is also the earliest example of an axiomatic structure that
allows learners to engage with the concept of deductive proof and the decline in
its presence in the classroom brought with it a decline in the presence of the
concept of proof. Since this notable decline in the emphasis on geometry in the
secondary curriculum, there has been resurgence in its popularity; which has
taken place alongside the dramatic rise in the use of computers both personally
and in education. This has offered the chance for a revolution in the style of
teaching available to mathematics educators as the availability of dynamic
geometry software (DGS) has become more widespread.
The most popular of the options available at the moment, as evidenced by it’s
repeated use in recent literature (Koyuncu et al., 2014) is Geogebra. This is in no
small part due to the fact that it is a free and open source piece of software that is
very easily deployable in any situation, even with the most limited funding
4
available. The importance of this to geometry education cannot be overstated:
where once it was possible to demonstrate three or four examples to encourage
the investigation of a conjecture, a teacher can now generate infinitely many in
much less time. Or possibly even more powerfully, where the resources are
available, the learners can do this for themselves. Unfortunately though the
uptake of this power has not been as widespread as might be liked, as (Ruthven
et al., 2008) demonstrate in educators differing attitudes to the use of DGS both
as a demonstrative and exploratory tool in the classroom. It is therefore the aim
of this study to investigate the effect that DGS can have in the classroom
alongside traditional methods of instruction and work in order to encourage the
use of these powerful techniques.
Issues in geometry education
Geometry forms a major part of the National Curriculum in the England; of the
five subject areas directly referenced, geometry is second only to number in
terms of breadth of content (National Curriculum, 2013). As with every other
branch of mathematics, geometry education is not without its difficulties. Of all
of the branches of mathematics that are presented to learners in compulsory
education, it is of interest that geometry is faced with so many of the same
obstacles. Unlike most of the other areas of study, the element of spatial
reasoning that is present in geometry is of course familiar to learners at a very
early stage. It is argued by Piaget (1953) that the earliest geometric
understanding is topological in nature, and although these arguments have been
disputed more recently it is important that educators often fail to take into
account the wide array of spatial awareness available to learners that is gleaned
from early life independently of education (Bryant, 2009).
One of the fundamental issues with mathematics education is the inherently
abstract nature of mathematical objects. Any academic pursuit, and vocational
pursuits for that matter, will involve the inclusion of some extent of semiotic
representations. The difference when it comes to mathematics is of course the
5
fact that the only way to access any mathematical object is through the use of
these representations (Duval, 2006). Duval argues that mathematics is unique in
its development and engagement in that it has only been able to progress along
with the complexity and suitability of the semiotic representations available.
Although this is present in all branches of mathematics, including even the most
basic numeracy, it is most obvious in the presentation of geometry. This is
because, one of the strangest contradictions of the presentation of geometry is
that we use concrete signifiers to represent abstract mathematical objects that
can never have a physical realisation, for example that of a table as a rectangle.
Duval highlights the “cognitive IMPOSSIBILITY OF DISSOCIATING any semiotic
representation content and its first represented object when there is no other
possible access to mathematical object than semiotic” (2006, p.124) (emphasis in
original). A conflict arises here, when necessarily attempting to convert between
representations in order to work with mathematical objects effectively.
The issues present in the difficulty of dissociation of semiotic representations of
mathematical objects are often not helped by classroom teachers. It is the
prevalence of the presentation of mathematical objects in particular fashions
that often can cause some of the most salient and enduring misconceptions. It is
of course to the issues of presenting geometric figures in a constant horizontal
and perpendicular fashion that is referred to here: isosceles triangles must have
their unequal side as their base, a square is represented so often with its sides
horizontal and vertical that in any other configuration it could only be a
rhombus. The representation of geometric objects form an important part of
presentation of geometric ideas in the classroom (David and Tomaz, 2011). As
David and Tomaz argue: “drawings of geometrical figures have a powerful role in
structuring and modifying the mathematical activity in the classroom” (2011,
p.413). We notice that they not only argue that representation forms an
important pedagogical role in formation of ideas, but can also aid in the practical
side of guiding and modifying the activities that take place. An important
consideration that is made by David and Tomaz is the assumption that the rules
for presentation and manipulation of geometric objects will be implicitly learnt
as part of the presentation of curricular material. They advocate instead for the
6
idea that “one should intentionally and explicitly take the norms and rules for the
use of drawings of geometrical figures as an object of teaching in the
mathematical activity in the classroom” (2011, p.430). In the following section
we will discuss the power of DGS to address these issues.
The power of dynamic geometry software and the dangers that this brings
It is through the use of DGS that we hope to overcome some of the prevalent
issues in geometry teaching, as well as to generally broaden the enjoyment and
engagement of learners in the classroom. A key feature and important aspect of
geometry education highlighted by McDowell and Kennedy 1997 is the
importance of accurate demonstration of figural representations. Their answer
to this is to engage learners with a geoboard, a solution that remains relevant
today. Geoboards offer the opportunity for representations to achieve accuracy
by providing a grid of pins laid out in parallel lines on which rubber bands may
be manipulated. This allows the construction of various geometric shapes
without requiring the learner to perform the inherently inaccurate task of
drawing. However, currently we also have access to the much more powerful
way to create accurate representation of geometric figures through the use of
DGS.
One further power of DGS in the classroom lies in its ability to change the way in
which learners engage with problems and how they seek to solve them (Koyuncu
et al., 2014), however this research only looked at the engagement of those with
already advanced skills in geometric reasoning. It is unclear what effect DGS may
have on learners at an earlier stage, or on those learners with low attainment. A
further issue highlighted by Koyuncu et al. (2014) is the tendency for students to
rely on the measurement tool of DGS in order to attempt to prove results instead
of as a method of generating conjectures that could be proven with deductive
methods. It is therefore of utmost importance that use of the measurement tool
in this way be actively discouraged.
7
A key argument in the use of DGS is whether or not the power available should
be used and carefully controlled by the instructor or whether it is advantageous
to encourage a more exploratory use by the students (Ruthven et al., 2008).
Ruthven et al. demonstrate the unwillingness of many teachers to allow students
to engage directly with the software and instead prefer to control the
environment in order to avoid the appearance of “mathematical anomalies”
present in the software. Some educators on the other hand actively encourage
the demonstration of these anomalies as a way to prompt further mathematical
discussion. Interestingly, regardless of the decisions made on this front, it is
apparent that the use of DGS “took classroom activity beyond established official
norms” (2008, p.297). It is noted here that students’ engagement with the
software directly forced students to engage analytically with the rules required
to construct geometric figures, thereby directly engaging with the issues
presented by David and Tomaz.
Ruthven et al. also note the parallels present in the implementation of DGS in the
classroom with that of curriculum material in general. Although the designers of
the curriculum may have specific aims in mind, it is up to individual practitioners
how the material will be presented to learners. Similarly, although the designers
and proponents of DGS place emphasis on the dynamic power of the drag tool, it
is rarely implemented in this fashion in the classroom. For the most part,
practitioners focus on the drag tool as a method of generating a multitude of
static examples in what Ruthven et al. term the arithmetical register rather than
using the drag tool to construct more dynamic arguments lying in the geometric
register. Even further to this, it is noticed that the use of DGS can undermine the
traditional progression of the curriculum. As the software opens up more
possibilities in solving a problem, it can become unclear how this should be
engaged with, that is whether or not the full capability of the software should be
harnessed and a new progression for the material should be considered.
Remarks
8
This review has considered some of the key issues that are reflected in geometry
education and the role that DGS can play in alleviating these issues while also
considering the possible ramifications. In trying to consider the ways in which
DGS can affect classroom practice it is important to consider the pedagogic value
of the tasks presented, however it is clear that great potential to change and
improve geometry education is made available in this domain (Leung, 2011). The
importance of geometry education within the current curriculum has also been
considered and although it has been argued that pre-university education in
geometry is out-dated with regards to current geometric progress (Hestenes,
2010), it is clear that the fundamental techniques that can be presented within
geometry, i.e. the concept of deductive proof and the axiomatic structure of
mathematics are relevant throughout mathematics. However it should be noted
that the Royal Society has argued that an axiomatic approach to geometry
education may be inappropriate at a pre-university level, encouraging instead
for a basis on obvious facts which in itself could be considered an exploratory
axiomatic reasoning structure (Royal Society, 2001).
Teaching Sequence
The effect of DGS on geometry education has been explored through the teaching
of a sequence of four lessons on the concepts of Shape and Angle to a lower
secondary school class (11-12 years old) with unexceptional attainment levels.
The core aim of the sequence was to challenge and expand the use of semiotic
representations, with which learners are familiar, through the use of DGS.
Lesson 1
The aim of lesson 1 was to offer insight into the concept of an angle as a measure
of a turn and not as the space between two lines as well as to improve the
powers of estimation and measuring of angles.
Lesson 2
9
Lesson 2 aimed to introduce parallel angle rules as well as the equality of
vertically opposite angles while engaging with a weak form of proof involving
obvious facts as recommended by the Royal Society (2001). A demonstration of
the equality of corresponding and alternate angles is presented through the
dynamic manipulation of parallel lines through the use of Geogebra.
Lesson 3
Lesson three offers the opportunity for the learners to interact with DGS
independently through an activity that allows the manipulation of the corners of
a trapezium to try to generate other quadrilaterals while the two parallel sides
were fixed to be parallel. This engages with the arguments of Duval (2006) by
offering learners the chance to directly interact with representations of
geometric objects dynamically in order to appreciate the cascading structure of
the classification of quadrilaterals as presented in figure 1.
Trapezium
Kite
Parallelogram
Rhombus
Rectangle
Square
Figure 1: Cascading structure of classification of quadrilaterals. Arrows
represent inclusion.
Lesson 4
Lesson 4 attempted to use DGS to introduce formal proof while connecting the
concepts of shape and angle by presenting proofs of the sums of angles in a
triangle and quadrilateral.
10
Methodology and Ethics
Data was collected through a combination of techniques. A test was given to the
class at the beginning of the first lesson and end of the last lesson of the sequence
in order to gauge the knowledge and skills of the students before and after the
sequence. Each test has the same structure with only the specific examples used
varying in order to offer the most reliable comparison possible:
1. Estimation and classification of an acute angle
2. Estimation and classification of an obtuse angle
3. Finding a missing angle on a straight line when given one angle
4. Finding a missing angle in a triangle
5. Finding a missing angle in a quadrilateral
6. Giving all possible classifications of a square presented with opposite vertices
lying on vertical and horizontal lines, rather than having the sides horizontal and
vertical
The first test was used as a means of adapting the presentation of material in the
sequence to best benefit learners and feedback was given after the second test in
order to allow learners to reflect on their progress and areas that need
improvement. A homework task was used to assess the functional skills
developed by the learners and feedback was given to allow them to develop.
Observations of students throughout the lessons both during activities and in
their responses to questions posed by the teacher will also be used as part of the
discussion of the results of the sequence of lessons presented. A focus group of
three students was held after the third lesson and a questionnaire was given out
to the entire class as the end of the sequence in order to attempt to ascertain the
effectiveness of the sequence and the individual activities presented in engaging
learners with the desired concepts.
All learners were given the opportunity to withdraw their participation from the
research project and all names that follow have been anonymised. It should also
11
be noted that the research project was used to improve the education of learners
at all points and did not compromise on the quality of their learning. The
sequence aimed to deliver the content required by the Department for
Education, in particular:
“Pupils should be taught to:
•
•
•
•
derive and illustrate properties of triangles, quadrilaterals, circles, and other
plane figures [for example, equal lengths and angles] using appropriate
language and technologies
apply the properties of angles at a point, angles at a point on a straight line,
vertically opposite angles
understand and use the relationship between parallel lines and alternate
and corresponding angles
derive and use the sum of angles in a triangle and use it to deduce the angle
sum in any polygon, and to derive properties of regular polygons”
(National Curriculum, 2013, p.8)
Discussion of results
The space between two lines
In the first lesson, the question “what does an angle measure?” was posed to the
class. Of the eleven responses heard, ten referred to the “space”, “area” or
“distance” between two lines and only one response gave any mention of the
concept of direction. No response indicated that an angle may measure a turn
and in counting the responses, Quinn’s comment that “it’s that cornery shapey
thingie” has been ignored. While the definition of an angle involving the concept
of space may not be entirely incorrect, it relies on the abstract concept of two
rays meeting and cannot be directly applied to line segments, which forms the
majority of the representation of angle in secondary education. It also does not
offer much insight into the measure of angle (Clements and Burns, 2000). It was
therefore decided that it would be of benefit to students understanding of shape
and angle to shift their view of angle to the more dynamic concept of a turn.
The first dynamic demonstration offered was designed to combat this view by
presenting two line segments that meet at a point and can be extended or
contracted by dragging. This demonstration allows the dynamic change of the
space or distance between the two lines whilst fixing the angle between them
12
thereby prompting a discussion in the class about whether or not an angle
measures these properties. In an exit ticket question during lesson three, the
same question was posed and all students responded by writing an answer down
before handing it in. In this collection of responses, fourteen students responded
that an angle could measure a turn while eight retained the concept of the space
between two lines. A further four responses gave either numeric answers or
other off topic responses. It is worth noting that although three of the
respondents who originally gave an answer that an angle measures space stuck
to this definition; the remaining seven amended their responses on the exit
ticket referring instead to angles measuring a turn.
During the focus group Bogart notes, “When the lines change length the angle
doesn’t change… so it can’t be about space or area”. This indicates a shift in
perception that was facilitated by the dynamic nature of the presentation.
However, during the demonstration of the equality of corresponding angles an
issue arose that caused considerable confusion in the classroom. The circular arc
labels for the angles in question were colour coded in order to ease reference to
each, but were of slightly different sizes so that when overlaid both were still
visible (see figure 2).
Figure 2: Overlapping angle labels
13
During the focus group, both Billy and Bogart argued that the green angle is
larger than the red angle because the edge of the red label was visible. Bogart
notes that both the red and green angles “had the same turns… I think the red
one’s just a little bit smaller”. There appears to be a disconnect between the
concept of an angles as a turn and the representation of the angle with an arc in
which the radius of arc can vary. Indeed, even though Thomas argued rather
vehemently that the angles are equal, he did so under the pretence that Bogart
was wrong to say “three teachers who got degrees in mathematics… are wrong”.
It would appear that Thomas submits to the authority of knowledge held by his
teachers rather than to connect with the demonstration itself, then going on to
attempt to question the interviewer for an opinion. This example does not
represent a unique situation, and it is one that requires some consideration as
“The difference between knowing you are right because you have checked it by
another method, and being dependent on a teacher or textbook to find out if you
are right, may be of central importance to adolescents” (Watson, 2006, p.60).
However, Thomas was also the only participant to refer back to the previous
demonstration that the angle is irrespective of the length of the line segments
that meet and recognise it’s relevance in this situation. This discussion agrees
with the issues discussed by Duval (2006) showing the importance that the
semiotic representation of objects has on the perception of learners. Here, the
labelling of an angle with a circular arc has the danger of drawing attention to
the radius of the arc instead of the angle through which the arc turns.
“Does that make it a parallelogram?”
During the course of the sequence, learners were presented with the task of
classifying and estimating the size of various angles presented with their line
segments oblique to the page. When presented with this situation, many learners
resorted to rotating the page in order to have one line segment horizontal before
attempting to make an estimate. In particular when estimating an angle on
slightly smaller than a right angle, Lewis expressed that he “wasn’t sure if it was
a right angle, but then I flipped it and found out it wasn’t”. Other learners
attempted to use a book as a reference for a right angle when attempting to
14
tackle the same problem. In contrast to this, when presented with the activity of
lesson 3 (henceforth referred to as the trapezium activity) in which the
trapezium was manipulated Lewis ended up with the example of an oblique
rectangle in figure 3 (note that this is a recreation of the situation, as the original
could not be saved).
Figure 3: An oblique rectangle found during an investigation (recreation)
Presented with this situation, the following interaction was observed between
Lewis and a teacher:
“Lewis: It’s sort of a rectangle, but like that. Does that make it into a parallelogram?
Teacher: What makes a rectangle a rectangle?
Lewis: Right angles!
Teacher: So what does that mean?
Lewis: It’s still a rectangle.”
It can be seen here that there is an agreement with the work of Koyuncu et al.
(2014) in the effect that DGS can have on the methods used as compared with
traditional pen and paper techniques. When presented with a pen and paper
task, learners were able to use physical reference points as well as to manipulate
the representation in order to make it more familiar. When the similar situation
arose with DGS (that is an oblique figure) this no longer became possible, or at
least it was non trivial. In this situation the learner was offered a more direct
opportunity to engage with the abstract definition of a rectangle in order to
15
understand the nature of the problem. The potential that this has to have an
effect on the way in which learners engage with mathematics is of great interest
and it would be valuable for further investigation to be carried out on the ways
in which DGS can alter perceptions of learners early in their mathematical
career. Similar situations arose with several other learners in which oblique
squares were thought to be rhombuses, or rhombuses with oblique lines of
symmetry were thought to be kites. During the course of the lesson, no learner
was able to recognise that since one pair of sides was forced to be parallel, the
only kite that could be constructed was also a rhombus, even under considerable
prompting from a teacher.
The aim of the lesson was made clear during a plenary session in which the
attention of all learners was drawn to the cascading nature of the classification of
quadrilaterals. The teacher used the activity to highlight the transition that takes
place when the figures are manipulated; and many learners were able to
appreciate the relationships that were shown. In one particular case a learner
shouted out excitedly “a square is a special kind of rectangle!” However there
were two cases that the learners were very reluctant to accept: that a
parallelogram is a special type of trapezium and that a rhombus is a special type
of kite. In the first case, it was apparent that the reason for this reluctance
stemmed from an interpretation of “one pair of parallel sides” being an exclusive
definition, in that having two pairs of parallel sides implied that there couldn’t be
a single pair. The issue with the comparison between a rhombus and a kite on
the other hand appeared to be far more visual in nature; they seemed
uncomfortable with the fact that the two had a different appearance when
traditionally represented figurally. The contrast in these two situations is of
great interest, that in the first case learners engage directly with the abstract
definitions, albeit with a small misinterpretation caused by ambiguous language,
while in the second case they almost disregard the definition and instead focus
directly on the visual representation. It is interesting that during the focus group,
none of the participants was able to give an adequate description of a kite while
they were able to do the same for other quadrilaterals. It became clear during the
discussion of the definition of a kite that was presented to them that they could
16
not fully appreciate the connection between the language of the definition and
the figural representation of a kite. We can see that although the use of DGS has
offered a chance to engage learners in the abstract language of mathematics, it is
unable to do so exclusively and the importance of semiotic representations still
remains a key constraint, as does the clarity of the definition of the object.
“Is this still a quadrilateral?”
Through the use of the trapezium activity a situation arose in which a
mathematical anomaly became apparent in the DGS. Because the shape was
constructed to be free flowing and allow the user to manipulate it into other
figures, the only constraint on the points used was the fact they should lie on two
parallel lines. This resulted in a situation as presented in figure 4.
Figure 4: "Is this still a quadrilateral?"
17
Although the activity had been constructed so that there was very little
opportunity to move beyond the bounds of expected behaviour, learners were
still able to manipulate the figure into a plane figure that no longer qualified as a
quadrilateral. This kind of situation vindicates the concerns presented by
teachers as reported by Ruthven et al. (2008) in allowing learners direct access
to the manipulation of software. In this case the teacher in question used the
opportunity to discuss with the whole class the necessary features of a
quadrilateral and whether or not this figure fit into these. After this discussion
the class was closely observed and no further instance of this situation was
apparent, the learners were able to focus their efforts on searching for
quadrilaterals directly without being distracted by cases that were not part of
the current investigation. Considering the inherent complexity in the software, it
is likely that, regardless of measures taken, learners may be able to manipulate
the figures in a way that was not originally intended. Considering this, it becomes
clear that if a teacher intends to use DGS to allow learners to directly manipulate
figures, it is necessary to be prepared for these potential situations. In this
situation, the teacher used this anomaly as a learning opportunity as many
teachers noted they would in Ruthven et al., and it becomes clear that in order to
allow learners direct access to the power that interacting with DGS has, a teacher
must be prepared to do so in order to be successful as it is unlikely that it will be
possible to prevent learners from finding anomalies completely.
Static examples and dynamic proofs
So far most of the examples considered in the use of DGS have been dynamic in
nature. Attention will now be turned to more static examples and their
effectiveness in convincing learners of mathematical facts. The two static
examples demonstrated the summative property of angles at a point and the sum
of angles in a quadrilateral (see figures 5 and 6). These two examples were used
to promote discussion amongst the class rather than to offer a deductive proof of
results. However, in both cases, the demonstration was seen to be convincing to
the majority of the class. In the second case, the class were given the opportunity
to work on examples of finding missing angles in quadrilaterals before a
deductive proof was shown and all learners were will to take as fact that the sum
18
of angles in a quadrilaterals is 360°. This agrees with the statements of Koyuncu
et al. (2014) that learners may often rely on the measurement tool of DGS in
order to form general conclusions. This calls into question the practice of the use
of the measurement tool to inspire discussion and further investigation, as even
in this case there is a danger that learners will cling onto the exemplifying cases
more than the deductive proof. On the other hand, there is still power offered in
using DGS in this way. During the demonstration of the invariance of the sum of
angles in a quadrilateral several learners gave examples in which they thought
this property might fail to hold. The advantage in using the measurement tool
was that these examples could be immediately constructed and demonstrated to
still have the property in question. This offers the chance for learners to
investigate the possibility that a property may not be universal quickly and
effectively.
Figure 5: the summative property of angles on a straight line
19
Figure 6: The sum of angles in a quadrilateral is 360°
In addition to making use of static examples to form hypotheses, DGS was also
used to generate analytic proofs of the sum of angles in a triangle and in a
quadrilateral. In these cases, a pre-generated workbook was not used and
instead the proof was generated during the class discussion by making use of the
construction tools as the proof developed. This engages with the arguments of
David and Tomaz (2011) that learners should be presented directly with the
norms for drawing and working with geometrical figures by making use of the
dynamic geometry software to construct the proofs in a procedural fashion.
However the value of these activities in this context must be questioned. As the
Royal Society (2001) argues, it may be inappropriate to draw attention to this
form of axiomatic proof early on in education. A compromise must be made
between encouraging deductive reasoning and not allowing incorrect
conclusions to be drawn from faulty reasoning whilst ensuring that the
deductions made are appropriate for the setting. It is the attempt here to argue
that these proofs are appropriate for the situation in question as they rely
exclusively on obvious facts, or definitions that are familiar to learners from an
early age. That is, the only reliance on prior knowledge in this sequence is that
angles on a straight line sum to 180°.
“Half-parallel”
20
The importance of semiotic representations in geometry has already been
discussed in some detail. During the progress of this study, this was evident in
the way in which learners engaged with tasks and demonstrations presented to
them. In almost every case, learners engaged solely with the semiotic
representation that was used by the teacher. However, during the focus group, it
became clear that Thomas did not approach geometric discussions in the same
way. Rather than focusing solely on that which was in front of him, he instead
discussed geometric concepts in relation to physical representations such as
describing the idea that “Ethan and me are half-parallel right now” and trying to
demonstrate the invariance of an angle with respect to the length of the line
segments using his thumb and forefinger. It was unclear exactly what was meant
by half parallel, but it appeared to be related to an idea that one could look at
rays that never met because they terminated before their point of intersection as
being half parallel. Although this concept is does not represent a correct
mathematical definition, it does at least represent the flexibility of his perception
of semiotic representations. At one point he discussed the representation of a
parallelogram by noting “unless you draw it really sloppily, it’ll have two parallel
lines” (it was later clarified that he did indeed mean two pairs of parallel lines). It
seems that Thomas had been able to separate the semiotic representation of a
geometric object from the object itself quite successfully. As a result, he was the
most successful of those interviewed in comprehending the implications of the
various discussions held.
It is interesting, however, to note that this success in comprehension of abstract
representation did not translate well into functional skills. Thomas had little
success in applying the knowledge and comprehension he displayed on the two
tests and homework that were used to assess the progress of learners. One must
therefore be critical of Duval (2006) in the importance that is placed on semiotic
representation in geometry education. If the aim is to produce functionally
literate learners who are able to perform well on standardised tests, then the
relevance of appreciation of the flexibility of semiotic representation may not be
the most important point to consider, and instead focus may be better spent
21
elsewhere. On the other hand, it is probable that the aim of a teacher will often
be a compromise between developing functional literacy and mathematically
deductive reasoning skills. In this situation then, it is necessary to consider the
balance that must be struck between these two aims.
Test and homework results
The first test was completed by 27 learners, while the second was completed by
23. In both the first and second test, the most successfully completed questions
were the first two in which an angle had to be classified and estimated. In the
first test, all but one student was able to classify and give an appropriate
estimate of an acute angle and all but two could do the same for an obtuse angle
and in the second test every student successfully completed the same questions.
The success in the questions involving finding missing angles was much more
varied. In the first test, ten students were unable to correctly identify a missing
angle on a straight line, twelve were unable to correctly identify a missing angle
in a triangle, and only four could correctly identify a missing angle in a
quadrilateral. Of these issues, the majority appeared to be unrelated to numeracy
issues with only a handful of examples attempting to perform the correct
procedure but failing due to a calculation error. In the second test every student
was able to calculate the missing angle on a straight line, all but one was able to
calculate the missing angle in a triangle (the incorrect response was due to a
calculation error evidenced by the working present) and fourteen learners were
able to correctly calculate the missing angle in a quadrilateral.
Some of the most interesting insights can be gleaned from the final question held
on both tests that attempted to question the relationship between semiotic
representation and interpretation of the nature of the figure. Learners were
presented with a square with opposite vertices lying on horizontal and vertical
lines; that is in the traditional orientation that would be used to represent a kite
or a rhombus. Of the 27 responses in the first test, 16 responded that the object
represented was a kite, a rhombus or both; with 10 of these also responding that
the figure could be described as a square. Four responses chose only to describe
the shape as a square and seven described the shape as either a parallelogram or
22
rectangle, or both. No response indicated that the figure could be described as a
trapezium, nor did any response include a combination that included a response
of either kite or rhombus and rectangle or parallelogram, nor did any response
indicate that the figure could be described both as a square and as a rectangle.
The results of the second test were largely similar with 12 respondents
indicating that the shape was either a kite or rhombus or both. This time
however there was considerably more overlap between the different
perceptions of what the figure could represent. Six of these responses that
included kite or rhombus also included a parallelogram and two responses
indicated that the figure fit the characteristics of a trapezium. Unlike the first test
where the most categories indicated by a response were three, the second test
included a response that included five of the six descriptions available, only
neglecting to include a trapezium.
The results of the homework task set to the students also lends some insight into
the approach that learners took when attempting to solve questions involving
finding missing angles. Of the 23 pieces of work observed, eight of these involved
an attempt to measure the missing angle with a protractor in at least one of the
questions involved, contradicting established facts and the rest of the
information presented in the question. This is despite the fact that no part of the
teaching sequence involved physical measurement with a protractor and that
this was actively discouraged in each lesson. We can see from this the prevalence
of a reliance on measurement by learners as described by Koyuncu et al. (2014),
this time however it relates to an attempt to solve problems not through the use
of DGS, but instead through traditional pen and paper methods. Here we see that
although there is a difference in many ways with the approach that learners take
in solving problems between pen and paper methods and in DGS (Duval, 2006),
this is a concern that is highlighted in both situations.
A further concern noted from the homework questions was the ability of
learners to solve parallel line questions that were presented as they had been
during lessons, but that this did not extend to questions such as that in figure 7.
In this instance, learners were expected to recognise the application of parallel
23
line rules in the context of missing angles in a quadrilateral. It is important to
note that this is not the only non standard feature of the presentation, it is also
the case that the line segments do not extend beyond their points of intersection
which was not the case in most of the examples they had seen. This question
presented the learners with the most challenge with only two correct responses,
both of which featured no working to be verified. It is therefore difficult to
ascertain how the learners came to their conclusion or if they did so on their own
at all.
Figure 7: A question that proved very challenging
As a part of the questionnaire offered to learners, they were asked what was the
most difficult part of the sequence presented to them. Of the 12 respondents, 6
indicated that the conceptual understanding of kites proved most difficult. On
the other side of this coin, 8 respondents claimed that the easiest part of the
sequence for them to understand was the work on the definition of angles. This
is surprising, considering the fact that a much smaller proportion of learners
revised their concept of angles to represent a turn rather than a space. In one
case it would appear that a lack of familiarity caused the learners to feel that they
lacked understanding, whilst in the other they appear to have ignored a
presentation of material that contradicted an established mode of thought.
Success of the teaching sequence
In terms of functional skill development, the test results and homework
questions offer evidence that the sequence was able make progress with the
education of the learners. The majority of learners were able to respond to
question involving missing angles at a point, in triangles and quadrilaterals as
24
well as in parallel line problems. However one of the key aims of the sequence
was also to develop the learners understanding of the semiotic presentation of
quadrilaterals and how this relates to their definition and label. In this instance
the sequence would appear to have not been successful as there was little change
in the perception of learners evidences by the closing test of the sequence. The
change that was present did not seem to be consistently developing towards an
understanding of the cascading nature of the classification of quadrilaterals. It is
thought that this is in no small part due to the long established norms of
presentation and language that have been seen by the learners in earlier
education and that such a short sequence was unable to overcome these issues.
Conclusion
This study has engaged with the issues in geometry education mainly through
the use of dynamic geometry software while also attempting to investigate the
effect that this has on the perception and progress of learners. Of key interest in
the results of the study is the difficulty learners face in the flexibility of their
understanding of trapezia and kites. In the instance of trapezia there was a
strong reliance on a narrow interpretation of the definition that did not appear
to allow for much reinterpretation. Whilst in the instance of a kite, the lack on
familiarity with the definition created a dependence on the visual representation
of the geometric object that did not allow for a theoretical discussion on the
properties the shape should hold. We see a strong resonance with this concept of
the cognitive dissonance presented by Duval (2006). It was the hope that
engagement with dynamic geometry software would allow learners to resolve
this dissonance in a constructive manner, and in the case where the relationship
between the abstract definition and semiotic representation was clear to
learners there was some limited success, even if later testing did not give much
representation of this fact. However, in those cases where learners faced more
difficult with the resolution between the abstract and the visual, there was
almost no progress whatsoever in this realm.
It is apparent from this study that the effects observed by Koyuncu et al. (2014)
in mathematically and geometrically literate participants also appears to apply
25
to those earlier on in their educational progress. The learners observed in this
study engaged far more with the abstract definitions of geometric object when
working with the dynamic geometry software, as they were given no opportunity
to use any physical form of measurement or reference point. However, when
working on pen and paper and in particular when working on problems
unsupervised by a teacher, learners relied very heavily on measurement tools
over deductive reasoning despite active discouragement. It becomes apparent
that although the learners can be prevented from using the measurement tool by
limiting the way they can engage with dynamic geometry software, this is not
possible in pen and paper problems. This fact could prove useful for teachers
hoping to find a way to move learners away from their dependence on
measurement.
It also became clear from this sequence that dynamic geometry software offers a
strong potential for dynamic investigations that can offer a foundation of
understanding for learners. However it is important to note that there was still
an issue present with the fact that dynamic geometry software still forms a
semiotic representation of an abstract geometric concept. The issues observed in
learners attempting to discern whether or not two angles are equal when the
radii of the arcs used to notate them differed evidence this fact. We see that this
again offers strong agreement with the arguments of Duval (2006). We see that
although dynamic geometry software offers a great deal of powerful tools, these
may not prove entirely useful in attempting to resolve this fundamental cognitive
dissonance.
It is hoped that this study has made clear the great effect that dynamic geometry
software can off in changing the level and nature of engagement of learners,
while also noting the areas that must be carefully considered when doing so.
26
References
Bryant, P. (2009) 'Paper 5: Understanding space and its representations in mathematics.', in Terezinha
Nunes et al. (eds.) Key Understandings in Mathematics Learning: Introduction and Summary of
Findings. Nuffield Foundation. p. 11. [online]. Available from:
https://books.google.com/books?id=fwpBMwEACAAJ&pgis=1 (Accessed 21 December 2014).
Clements, D. H. & Burns, B. A. (2000) Students’ Development of Strategies for Turn and Angle
Measure. Educational Studies in Mathematics. [Online] 41 (1), 31–45. [online]. Available from:
http://link.springer.com/article/10.1023/A%3A1003938415559 (Accessed 8 February 2015).
David, M. M. & Tomaz, V. S. (2011) The role of visual representations for structuring classroom
mathematical activity. Educational Studies in Mathematics. [Online] 80 (3), 413–431. [online].
Available from: http://link.springer.com/10.1007/s10649-011-9358-6 (Accessed 11 December
2014).
Duval, R. (2006) A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics.
Educational Studies in Mathematics. [Online] 61 (1-2), 103–131. [online]. Available from:
http://link.springer.com/10.1007/s10649-006-0400-z (Accessed 13 December 2014).
Hestenes, D. (2010) 'Modeling Theory for Math and Science Education', in Richard Lesh et al. (eds.)
Modeling Students’ Mathematical Modeling Competencies SE - 3. [Online]. Springer US. pp.
13–41. [online]. Available from: http://dx.doi.org/10.1007/978-1-4419-0561-1_3.
Koyuncu, I. et al. (2014) INVESTIGATING PLANE GEOMETRY PROBLEM-SOLVING
STRATEGIES OF PROSPECTIVE MATHEMATICS TEACHERS IN TECHNOLOGY AND
PAPER-AND-PENCIL ENVIRONMENTS. International Journal of Science and Mathematics
Education. [Online] [online]. Available from: http://link.springer.com/10.1007/s10763-0149510-8 (Accessed 19 December 2014).
Leung, A. (2011) An epistemic model of task design in dynamic geometry environment. ZDM.
[Online] 43 (3), 325–336. [online]. Available from: http://link.springer.com/10.1007/s11858011-0329-2 (Accessed 21 December 2014).
McDowell, E. & Kennedy, J. (1997) Geoboard Quadrilaterals. Mathematics Teacher. 91 (4), 288–290.
[online]. Available from: http://eric.ed.gov/?id=EJ564635 (Accessed 21 December 2014).
National Curriculum (2013) National curriculum in England: mathematics programmes of study: key
stage 3. [online]. Available from: https://www.gov.uk/government/publications/nationalcurriculum-in-england-mathematics-programmes-of-study (Accessed 13 February 2015).
[online]. Available from: https://www.gov.uk/government/publications/national-curriculum-inengland-mathematics-programmes-of-study (Accessed 13 February 2015).
Piaget, J. (1953) How Children Form Mathematical Concepts. Freeman. [online]. Available from:
http://books.google.co.uk/books/about/How_Children_Form_Mathematical_Concepts.html?id=A
p2nHAAACAAJ&pgis=1 (Accessed 21 December 2014).
Royal Society (2001) Teaching and learning geometry 11-19. [online]. Available from:
https://royalsociety.org/policy/publications/2001/teaching-geometry/ (Accessed 21 December
2014). [online]. Available from: https://royalsociety.org/policy/publications/2001/teachinggeometry/ (Accessed 21 December 2014).
Ruthven, K. et al. (2008) Constructions of dynamic geometry: A study of the interpretative flexibility
of educational software in classroom practice. Computers & Education. [Online] 51 (1), 297–
317. [online]. Available from: http://dl.acm.org/citation.cfm?id=1371264.1371426 (Accessed 19
December 2014).
27
Watson, A. (2006) Raising Achievement In Secondary Mathematics. McGraw-Hill Education (UK).
[online]. Available from: http://books.google.com/books?hl=en&lr=&id=IOyXLN4OzkC&pgis=1 (Accessed 14 February 2015).
Whiteley, W. (1999) The decline and rise of geometry in 20th century North America. Proceedings of
the 1999 Conference of the Mathematics Education Study Group of Canada. [online]. Available
from: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.8162 (Accessed 8 February
2015).
28